The History of Trigonometry
Trigonometry was not invented.
It was converted, three times over, by people who each
wanted something different from it.
Start hereWelcome, and how to read this book
This is the story of where trigonometry came from, written to be read alongside your Trigonometry course.
Most textbooks give the history a sentence. The Greeks did something with triangles, the Arabs saved it, then it turns up in your calculator. All three parts are true and all three are badly misleading, in a way that makes the mathematics harder to understand rather than easier.
Here is the version this book argues for. Trigonometry was not invented. It was converted, three times over, by people who each wanted something different from it. It starts as a table of chords, straight lines across a circle, built by Greek astronomers who wanted to predict where a planet would be. It becomes a table of half-chords in India, which is the sine, and that change is not cosmetic: a half-chord attaches to a right triangle and a chord does not. It becomes six functions and a subject of its own in the Islamic world, taken out of astronomy and given its own theorems. And then, much later and much more slowly than the story usually allows, it stops being a table at all and becomes a function: something you can graph, differentiate, and add to other functions to build a wave.
Every one of those conversions is still visible in your course. The reason a sine is a ratio and not a length is the second one. The reason there are six functions and not two is the third. The reason your calculator has a radian mode is the fourth, and the radian is younger than the telephone.
This book is a companion, not a replacement. Read a chapter before the unit it belongs to, or arrive from a link in your course book and read one box. Both ways work, and nothing here assumes you have already read the rest.
Three parts, and they are used differently:
Part I, the story. Eight chapters, in order, roughly 13 hours of reading end to end and about 100 minutes a chapter. They are meant to be taken one at a time, beside the unit they belong to.
Part II, the master timeline. 260 dated events across ten eras. Nobody reads this front to back. You arrive at it to place somebody.
Part III, reference. Twelve appendices, A to L, the last of them the glossary: 185 people, 94 terms and symbols, 84 logged disagreements between sources, 332 sources, and every calculation re-derived in code so you can check it yourself.
Every factual sentence in this book carries a pointer to the source it came from, like this: (S001). Click one and it takes you to the full reference in Appendix J. Where two sources disagree, the book says so and shows both, rather than picking a favorite and sounding confident. 73 of the 84 logged disagreements are still open. That is not a defect. An open disagreement recorded is scholarship; an open disagreement forgotten becomes a fact.
Where something could not be checked, the book says that too, in plain words, rather than quietly leaving it out or quietly asserting it.
A date is a claim somebody made in a document, not a fact from nowhere. Two careful historians reading the same tablet can date it differently and both be reasonable. When you see the Disputed label in this book, that is what it means: not "nobody knows anything", but "the evidence genuinely underdetermines the answer, and here is the shape of the argument."
Chapter 1 makes this concrete straight away. The object that chapter is built around carries no date on it at all, and everything anyone says about when it was written is inference from its handwriting and its vocabulary.
A few reading tools, available everywhere: every section has a Listen button to hear it read aloud, and every underlined term shows its meaning on hover.
It uses your device's own voice, not a recording. That has three consequences worth knowing before you rely on it for a long drive.
It remembers where you stopped. Start a section, stop part way, come back later, and a Resume listening button appears on that section and picks up from the paragraph you were on.
It keeps going with the screen off, and your lock screen shows which section is playing, with pause and play controls.
There is no audio file to download, and this book is not going to pretend otherwise. Reading all eight chapters aloud is about nineteen hours of speech, which is roughly two hundred megabytes: more than twenty times the size of this entire book. Because the voice is generated on your device, the Listen button works with no signal at all, but only while the page is open. If you need audio in a car with no reception, open the book once while you still have signal and leave the tab open.
The pronunciation buttons in Appendix C are different: those are 209 real recordings, made when the book was built and embedded in it, so they play offline with no device voice involved.
The top bar holds the rest: Aa sets text size, line spacing, and the reading voice, Night flips to a dark theme, and ↓ Print / PDF prints the whole book or one section. To search, use the Search box in the sidebar, or your browser's own find-on-page (Ctrl+F, or Cmd+F on a Mac).
Everything in this book works offline: open it once with internet and it keeps working without one. The links out to Trigonometry are the one exception, since they go to another book.
Quick questions, quick answers
Do I have to read it in order?
No. Part I rewards reading in order, because the three conversions happen in sequence and each one only makes sense after the last. The other two parts are built to be arrived at from a link. Every chapter stands on its own.
Where does this fit with my course?
Appendix I maps every story in this book to the exact section of your Trigonometry course it belongs with, 50 placements in all, each one a link you can click.
Is any of this on the test?
Ask your teacher. This book exists to make the mathematics make sense, not to add memorization. Knowing that the word sine is a Latin translator's mistake for an Arabic word meaning a fold will not be graded, but it may make the subject feel less like it arrived from space.
Why do some claims have a Disputed label?
Because the sources disagree and the disagreement is not settled. Appendix F lists all 84 of them with both readings side by side.
Can I check the math for myself?
Yes, and you should. Appendix E re-derives every calculation in this book, step by step, and each one was verified in code before it shipped.
I think I found a mistake.
Tell your teacher. A sharp catch helps every reader after you, and this book is more checkable than most, on purpose.
Jump straight to Chapter 1, which starts with a clay tablet that sat in a New York library for fifty years under the heading "Commercial account". The style key just below is here for whenever you want to know what a mark means.
Front matterThe style key: what every mark means
This book uses color and shape on purpose, not for decoration. Here is the full key, so nothing is a mystery.
The goals box. One opens each of the three parts with a short list of what you are about to be able to do. Navy bar.
The Welcome's jump-straight-in pointer, a filled soft sky-blue panel. It appears once, and it takes you to Chapter 1.
The precise meaning of a term, at the moment it first matters. The word being defined is shown in bold navy. Teal bar.
A claim you can rely on, with the evidence behind it named. Steel-blue bar on a pale panel.
A question placed before the passage that answers it, wherever possible using a wrong answer a competent person once published as the distractor. Guess first, then open the reveal. Eight of these, one per chapter. Guessing wrong and finding out is how the answer sticks.
The chapter's last word: one open recall question, with the answer. Eight of these too, one per chapter.
A story you have probably heard that is wrong, or a claim that gets repeated everywhere and should not be. Amber bar, the book's one caution color.
The link back to your own textbook: one per chapter, naming the exact section of Trigonometry where this story does the most good. Appendix I holds all 50 placements.
The three or four things to carry forward, closing every chapter and ending with one sentence pointing at the next.
The marks inside a sentence
| Mark | What it means |
|---|---|
| S001 | A source pointer. Every factual sentence carries one. Click it and you land on the full reference in Appendix J, where 332 sources are listed. It is a box, not a superscript, so it is big enough to tap on a phone. |
| Disputed: c. 490 BCEDisputed | Sources disagree and it is not settled. The word "Disputed" is written out rather than left to italics, because a screen reader does not announce italics and the large-print setting removes them. |
| Bold navy | The point of the sentence, or a new technical term at the moment it is defined. |
| Italic | A title, a word in another language, or ordinary emphasis. Italic does not mean disputed. It used to in an earlier draft; the visible label carries that job now. |
| jayb | A word in its original language, tagged so a screen reader pronounces it in that language rather than in English. |
| Math set in a serif | All symbols and equations are typeset, for example . |
Dates, and why some of them look strange
England kept the Julian calendar until 1752, while most of Europe had moved to the Gregorian one in 1582. For most of the seventeenth century that puts English dates ten days behind continental ones, and the English year began on 25 March rather than 1 January. This matters for Newton. A letter dated 1 February 1665 in London and one dated 11 February 1665 in Paris can be the same day, and a London document dated February 1665 may belong to what we would call 1666.
Where a date is given in this book in the form 1665/6, that is the convention for a date between 1 January and 24 March, when the English and modern year numbers disagree. Where it matters to an argument, the chapter says which calendar it is using.
Abbreviations, spelled out
| Short | Full form |
|---|---|
| BCE / CE | Before the Common Era / Common Era. The same years as BC and AD, named without the theology. |
| c. | From circa, "around". An approximate date. |
| fl. | From floruit, "flourished". Used when birth and death years are unknown but the person is documented working in a period. |
| O.S. / N.S. | Old Style / New Style: the Julian and Gregorian calendars, as above. |
| IPA | International Phonetic Alphabet, the pronunciation notation in Appendix C. |
| MS / MSS | Manuscript / manuscripts. A handwritten document, as opposed to a printed one. |
A note on dashes: this book uses commas, colons, and parentheses instead of long dashes on purpose, so a dash is never confused with a minus sign.
A note on spelling: US spelling throughout, except inside a direct quotation, inside an archive's own wording of its rights terms, and in a proper name. Those three keep whatever the original used.
Front matter
Table of Contents
One place to jump anywhere. Open a part to see what is in it; every line is a link. Part I is the story and rewards reading in order. Parts II and III are reference, built to be arrived at from a link rather than read front to back.
Front Matterhow to read this book▸
Part I: The Story8 chapters, about 13 hours end to end▸
- 0Opener and Objectives
- Chapter 1. Before Trigonometry (Mesopotamia and Egypt, c. 3000 to 300 BCE)
- Chapter 2. The Chord Table (Greek Astronomy, c. 300 BCE to 400 CE)
- Chapter 3. The Half-Chord Becomes the Sine (India, c. 400 to 1600 CE)
- Chapter 4. Six Functions and a Discipline (The Islamic World, c. 750 to 1500)
- Chapter 5. The Road Not Taken (China and Japan, c. 100 BCE to 1750)
- Chapter 6. The Word, the Names, and the Tables (Europe, c. 1100 to 1620)
- Chapter 7. From Table to Function to Wave (1660 to today)
- Chapter 8. Where Every Word and Every Theorem Came From
Part II: The Master Timeline260 dated events, ten eras▸
- 0Opener and Objectives
- How to read the timeline
- 3000 BCEBefore the function: Mesopotamia and Egypt
- 300 BCEGreece builds the chord table
- 400India turns the chord into the sine
- 750The Islamic world completes the six functions
- 1200Other roads, and the long transmission
- 1500Europe names the functions and prints the tables
- 1620From table to function
- 1750Analysis, the wave, and the radian
- 1850Measurement, notation, and the modern classroom
- 1950The machine age, and the tablets read again
Part III: Reference185 people, 94 terms, 332 sources▸
- 0Opener and Objectives
- Appendix A. People register
- Appendix B. Symbols, words, and eponyms
- Appendix C. Pronunciation quick reference
- Appendix D. Whose name is on it, and who did the work
- Appendix E. Every number in this book, re-derived and checked in code
- Appendix F. Disagreements between sources
- Appendix G. Stories that check out, and stories that do not
- Appendix H. Image catalog and rights
- Appendix I. Where each story goes in your course
- Appendix J. Bibliography
- Appendix K. What could not be read, and what to request
- Appendix L. Glossary
Back Matterhow this book was built▸
The story: five thousand years, and one idea converted three times
Trigonometry was not invented. It was CONVERTED, three times over, by people who each wanted something different from it: a chord table for Greek astronomers who wanted to predict a planet, a half-chord in India that could attach to a right triangle, six functions and a subject of its own in the Islamic world, and finally a function you can graph, differentiate and add to other functions to build a wave. Every one of those conversions is still visible in your course.
- Say what a chord table is, why the Greeks built one, and what it could not do.
- Explain why halving the chord was not a cosmetic change, and what it made possible.
- Name the six functions, and say who first brought all six into one system.
- Trace the word sine from Sanskrit through Arabic to a Latin mistranslation.
- Say when trigonometry stopped being a table and became a function, and what changed.
- Name at least four places outside Europe where this mathematics was worked out first.
- Connect a historical episode to the section of your own course where it belongs.
Chapter 1
Before Trigonometry (Mesopotamia and Egypt, c. 3000 to 300 BCE)
The people in this chapter
Faces where a face survives. Every name links to its full entry in Appendix A.
A tablet filed under "Commercial account"
In 1943 a scholar named Isaac Mendelsohn (MEN-dul-sun) cataloged the cuneiform tablets in the libraries of Columbia University. Item 322 got three lines:
"322. Clay tablet, left-hand edge broken away, bottom of right-hand corner, and a piece of columns 3 and 4 chipped off; fairly well preserved, dark-brown. 8.8 x 12.5 cm.; on obverse 4 columns with 16 lines, reverse blank. Content: Commercial account. No date." (S002, p. 172, quoting Mendelsohn 1943, p. 71)
Two years later Otto Neugebauer (NOY-guh-bow-er, 1899 to 1990) and Abraham J. Sachs (SAKS, 1915 to 1983, dates approximate) published that same tablet as Text A in Mathematical Cuneiform Texts. That was the first full scholarly edition, what specialists call the editio princeps, "first edition". The tablet went on to become one of the most argued-over objects in the history of mathematics (S002, S031). It is now called Plimpton 322 (PLIMP-tun three twenty-two). It holds fifteen rows of Pythagorean triples, written roughly 1,200 years before Pythagoras was born.
Mendelsohn was not careless. That is the point. The Assyriologist whose work reshaped how everyone reads this object, Eleanor Robson (EL-uh-nor ROB-sun, born 1969, approximate), explains why: "Given the striking similarity in format and lexis between Plimpton 322 and the other early tables from Larsa mentioned above, it is hardly surprising that its true character passed unnoticed by dealer, owner, and cataloguer alike" (S002, p. 172). Lexis means vocabulary. The tablet looks like a grain account because a clerk trained to write grain accounts wrote it. He used the layout grain accounts used, and the words a clerk used for a delivery of barley.
This chapter is about the toolkit people had before anyone had a trigonometric function. No sine, no cosine, no chord, no table of angles. Instead: ratios, tables, gradients, a base-60 number system, a 360-part day, and superb arithmetic. That toolkit built pyramids and predicted planets. It has also been oversold for a century. One of those oversells collapsed in print, in the lifetime of people now teaching. Watching that collapse is the most useful thing a student can take from these two thousand years.
In 2017 two mathematicians made front pages worldwide with the claim that this chapter's tablet is "the world's only completely accurate trigonometric table" (S003). The tablet contains no angles. Before you meet the five readings of it: what did the people who wrote Plimpton 322 most likely think they were writing?
I have a guess▸
A scribal-school working document. Robson's case, which most specialists accept, reads the tablet in the format and vocabulary of the other teachers' tables from the same city: a list built from reciprocal pairs, laid out the way Larsa's school documents are laid out (S001, S002). What it is FOR is still argued; what it is not is a table of angles, because nothing in the whole Mesopotamian corpus measures an angle.
If you guessed a trigonometric table, you guessed the 2017 headline, and you are in good company: that reading was published by professionals (S003). The chapter states it precisely, then walks through what it has to assume.
What the mathematics was for
Nobody in Larsa or Thebes woke up wanting a sine. They wanted four things, and every object in this chapter serves one of them.
Land had to be divided and defended. Fields were sold, inherited, split between heirs, and argued over in court. A surveyor who could not lay out a right angle in a narrow strip of ground was useless. Daniel F. Mansfield (MANZ-feeld, living) quotes an Old Babylonian poem in which one scribe humiliates another over this exact incompetence: "Go to divide a plot, and you are not able to divide the plot; go to apportion a field, and you cannot even hold the tape and rod properly. The field pegs you are unable to place; you cannot figure out its shape, so that when wronged men have a quarrel you are not able to bring peace ... Among the scribes, you (alone) are unfit for the clay" (S004, p. 999, quoting Vanstiphout 2003, p. 589). That is the job description. Geometry is a peacekeeping tool.
Stone had to be cut to a stated slope. A pyramid face is thousands of blocks. Every one must be dressed to the same angle by masons who cannot see the finished shape and have no protractor. Egypt solved this with a ratio in mixed units, the seked (SEK-ed).
Time had to be told and the calendar kept. Night hours came from watching named stars cross the meridian. Day hours came from shadows. Months and years came from the moon and had to be reconciled with the sun.
The sky had to be predicted. Babylonian astronomers wanted to know where Jupiter would be, when the moon would first appear, and how long daylight would last. They got extraordinarily good at this using arithmetic alone.
Not one of those four problems requires a trigonometric function. That is worth sitting with, because the usual story of mathematics runs "people needed X, so they invented X." Here people needed to measure the world. For three thousand years they did it without the tool we now think of as the tool for measuring the world.
How a looted tablet reached New York
Plimpton 322 has no archaeological context, because it was never archaeologically excavated. It came through the antiquities trade. The dealer was Edgar J. Banks (BANKS), a trained Assyriologist turned antiquities trader. He said it came from "Senkereh," the modern name for the ruins of the ancient city of Larsa (LAR-suh; Senkereh is SEN-keh-reh) in southern Iraq. Banks sold it to the New York publisher George Arthur Plimpton (PLIMP-tun, 1855 to 1936) for about ten dollars around 1922 (S001, S003, p. 397). The Institute for the Study of the Ancient World gives the acquisition as "1922/1923" rather than a single year (S013). Plimpton left his collection, including tablet 322, to Columbia University in 1936, shortly before his death (S001, S012, S013). Mendelsohn cataloged it in 1943 as a commercial account (S002). Neugebauer and Sachs published it in 1945 (S001, S002).
Because there is no findspot record, Robson had to date the tablet from the object itself: handwriting, spelling conventions, and above all document format. Her method is worth spelling out, because it is the same method a historian uses on an undated letter. Old Babylonian administrative tables from the Larsa region share a set of habits. They are landscape (wider than tall), and they put a heading over each column. They sort entries into descending order, and they run their calculations left to right. The last column is headed MU.BI.IM, Sumerian for "its name," and holds the line count. Her key comparison object is YBC 4721, a grain account written at Ur in 1822 BCE and now at Yale. It does all five (S001, pp. 110 to 111). Plimpton 322 does all five too.
That places the tablet in the roughly sixty years before Hammurabi (ham-uh-RAH-bee, reigned to 1750 BCE) besieged and captured Larsa in 1762 BCE. That capture is the latest date the city could have produced it (S001, S002). The exact window shifts a little between publications, and the shifts are worth recording rather than smoothing. Robson 2002 gives the format range for the comparable Larsa tables as 1822 to 1784 BCE. Robson 2001 gives the earliest attested Larsa administrative tables as 1837 to 1784 BCE. Mansfield and Wildberger cite Robson 2001 as dating Plimpton 322 "between 1822 and 1762 B.C.E." (S001, S002). Robson herself says she can "confidently date" it within that method.
Cross-curricular hook (history, civics and ethics). Have students look up what ten dollars bought in the United States in 1922. Then ask what the tablet lost when looters dug it out instead of archaeologists. The answer is everything except the object: which room it sat in, what was beside it, and what building it belonged to. Was that building a school, an office, or a house? That missing context is what the whole scholarly argument in this chapter is fighting over. The class can then argue the harder question, which has no settled answer. Who owns an object like this now? And what should a museum do with a purchase made a century ago from a dealer who never said where the thing came from?
The object itself
Plimpton 322 as preserved is a clay tablet measuring about 12.7 by 8.8 cm. Robson: "It is a clay tablet, measuring some 12.7 x 8.8 cm as it is preserved, ruled into four columns" (S002, p. 171). Mendelsohn's 1943 measurement, quoted above, was 8.8 by 12.5 cm. Same object, measured the other way round, and a couple of millimeters short.
The front (the obverse) carries a heading line and then fifteen numbered rows. Neat vertical lines rule them into four columns, and neat horizontal lines separate the rows. Mansfield and Wildberger describe it: "The main body of the obverse is ruled by neat horizontal lines into fifteen equally spaced rows containing sexagesimal numbers ... The vertical lines continue on the bottom and reverse, which are otherwise empty" (S003, p. 397). The reverse is blank apart from those continuing rulings. The Cuneiform Digital Library Initiative has published a high-resolution composite photograph of the obverse, all four edges, and the reverse. On the reverse you can read a pencilled "322" and the ink stamp "PLIMPTON LIBRARY" (S033).
The left edge is snapped clean off along one of the vertical rulings. So what survives is the right-hand portion of a wider document. Everything anyone says about the missing part is reconstruction.
The glue, and a disagreement about it
There is modern glue in that break, and the two camps read it in opposite directions.
Robson: "There is a clean break here, along one of the vertical rulings which divide the surface of the tablet into columns. Traces of glue remain in this break, and it has been implied that the other fraction of the tablet must therefore have been lost in modern times, deliberately or otherwise" (S002, p. 172). She does not accept that implication. Her reading: "it was not unusual for unscrupulous early 20th-century antiquities dealers to manufacture 'whole' tablets out of disparate fragments in order to attract a higher price for them ... In the case of Plimpton 322 it is likely that Banks, who had been a professional Assyriologist, was scrupulous enough to remove the extraneous matter before putting the tablet up for sale" (S002, p. 172). On that account, the glue held some unrelated scrap that a dealer had stuck on to fake a complete tablet. Banks, who knew better, took it off.
Mansfield and Wildberger read the same smear the other way: "the left-hand edge showing clear evidence of being broken, and indeed remnants of modern glue suggests that the break occurred in recent times" (S003, p. 397). On that account the tablet's own missing left portion may still have existed in the twentieth century.
One physical fact, two incompatible stories, and no way to decide between them from the object alone. Keep this in view. It is a miniature of the whole Plimpton 322 argument, where everyone agrees on the evidence and disagrees on what it means.
How much is missing
Robson estimated the lost width from the curvature of the surviving clay. A tablet is not flat. It is a slightly domed pillow, and the curve of the surviving part limits how wide the whole could have been. Her conclusion: "To judge from the curvature of the extant part of Plimpton 322 there is probably room for two columns of roughly the width of Columns II and III in the missing portion, thereby adding no more than about 5 cm to the width of the tablet" (S002, p. 189).
That measurement does real work. It kills Schmidt's proposal of an eight- or nine-column table, "which would mean restoring to Plimpton 322 a total width of 25 to 30 cm and a sharply asymmetric curvature" (S002, p. 189). Whatever was in the lost portion, there was room for about two columns, not eight.
What is in the four surviving columns
Read right to left, the four columns are:
- Column IV: a line count, 1 to 15. Each entry is written ki n, Sumerian ki meaning "its place," so "its place, 1," "its place, 2," and so on. The column heading is MU.BI.IM, Sumerian for "its name," the standard heading for the line-numbering column in Larsa administrative tables from 1822 BCE onwards (S003, S001, p. 400).
- Column III: headed íb-si8 ṣiliptim (IB.SI8 tsih-LIP-tum). íb-si8 is Sumerian for "square-side". Robson translates the heading as "square-side of the diagonal" (S001, p. 114). Mansfield is more cautious: "the word ib-si refers to the result of some operation" (S003).
- Column II: headed íb-si8 of the width, "square-side of the width" (S001, p. 114).
- Column I: a long number in base 60, decreasing steadily down the table. Its two-line heading is broken at the start of both lines, and it has been retranslated three times.
Columns II and III give two of the three numbers of a Pythagorean triple. The third, the long side, is never written down at all. Row 1 has 1,59 and 2,49 in base 60, which is and . The missing long side is 120, and .
The Akkadian word behind Column III, ṣiliptum, comes from the verb ṣalāpum, "to strike through." It means the diagonal of a rectangle. By a habit of thought that runs through all Mesopotamian geometry, it also means the rectangle itself. A figure and the line that defines it share a name (S002, S003). That is not sloppiness. It is a different theory of what a shape is, and it will matter enormously in a few pages.
Row 1, worked all the way through
Column I row 1 reads 1,59,00,15. Sexagesimal notation writes no zeros of position and no decimal point, so a modern reader has to decide where the point goes. Here it goes after the leading 1. That leading 1 is about to become the most contested wedge in the history of mathematics. Written with a semicolon for the sexagesimal point and commas between places:
Now the triple from Columns II and III. Short side , long side (not written on the tablet), diagonal :
The same number to every place. Column I row 1 is (S003, p. 398).
Now put the long side along the base and call the base angle . Then , and
Check the second identity against the tablet:
The identity is sitting in a column of a clay tablet from about 1800 BCE. That single fact is why this object has caused a century of trouble, and why the trouble is so easy to misread. The relationship is there in the numbers. Whether anyone in Larsa thought of it as a relationship between angles is a separate question. The rest of this chapter is mostly about that gap.
The broken edge decides which function it is
Everything above depends on there having been a leading "1" in each Column I entry, in the zone that is now broken away. If the 1s were never there, Column I is : the square of the short side over the square of the long side. In modern terms that is . If they were there, Column I is , which is .
Robson set the problem out plainly in 2002: "Part of the tablet has broken away at the beginning of the first column but, depending on whether you believe the column has fully survived or not, it holds the square of either the hypotenuse or the shortest side of the triangle divided by the square of the longer side l. Whether it lists d2/l2 or s2/l2, this column is in descending numerical order" (S001, pp. 106 to 107).
The answer is now settled. Robson reached it in 2001, and John P. Britton (BRIT-un, 1939 to 2010), Christine Proust (krees-TEEN PROOST, living), and Steve Shnider (SHNY-der, living) confirmed it in their 2011 review. That review is the paper that fixed the current consensus reconstruction of the whole tablet (S003) [event E-2011-britton-proust-shnider-review]. Mansfield and Wildberger summarize: "Some scholars have argued that the leading 1's which are obscured by the break are not actually there, and that the column instead contains β2. Closer inspection has shown, in agreement with the column heading, that the 1's were originally there and likely contributed to the break (Robson, 2001, 191; Britton et al., 2011, 524)" (S003, p. 398).
Two things in that sentence deserve attention. First, "likely contributed to the break": a column of numbers each beginning with a wedge sitting right on a ruled line is a weak edge. It snapped there. Second, "in agreement with the column heading." The heading itself tells you which function it is, once you can read the heading.
So Column I is of the base angle, not . Every popular account that says Plimpton 322 is "a table of tangents" is wrong twice over: wrong about which ratio, and wrong about the word "table."
The four columns, row by row, and an honest limit
I can give you row 1 in full because I have checked it against the published transliterations. For rows 2 to 15 the research behind this book preserves the structure of the table, the location of every scribal error, and the restored base angles as Robson printed them. They do not preserve the full sexagesimal numerals of Columns I, II, and III row by row. I am not going to invent them. What follows is what I have verified.
The errors first. Mansfield and Wildberger tabulate them: "There are several errors in the tablet, due either to calculation or copying mistakes." Their Table 1 prints the erroneous entries in round brackets under the corrections, at row 2 (Columns I and III), row 8 (Column I), row 9 (Column II), row 13 (Column III), and row 15 (Column II) (S003, p. 398). Robson diagnoses the row 13 error as an ordinary copying slip from rough work to fair copy: the scribe "misreads the final three wedges of 3 13 and transfers 3 12 01 onto the good copy" (S002, p. 190). Anyone who has ever miscopied a line of their own working will recognize it.
Now the angles. Robson's Figure 4 sets out all three rival restorations of the tablet side by side. One of them, the trigonometric restoration (not her own preferred reading), carries a column headed "a" giving the base angle in degrees for each row. Those printed values are:
| Row | Base angle (degrees) | Step down from previous row | Scribal error noted in this row |
|---|---|---|---|
| 1 | 44.76 | not established | not established |
| 2 | 44.25 | 0.51 | Columns I and III |
| 3 | 43.79 | 0.46 | not established |
| 4 | 43.27 | 0.52 | not established |
| 5 | 42.08 | 1.19 | not established |
| 6 | 41.54 | 0.54 | not established |
| 7 | 40.32 | 1.22 | not established |
| 8 | 39.77 | 0.55 | Column I |
| 9 | 38.72 | 1.05 | Column II |
| 10 | 37.44 | 1.28 | not established |
| 11 | 36.87 | 0.57 | not established |
| 12 | 34.98 | 1.89 | not established |
| 13 | 33.86 | 1.12 | Column III |
| 14 | 33.26 | 0.60 | not established |
| 15 | 31.89 | 1.37 | Column II |
(S001, Figure 4, column "a"; S003, p. 398; differences computed here by subtraction from the printed column)
Read the third column. The steps run from 0.46 degrees to 1.89 degrees, with the biggest jump between rows 11 and 12. The famous claim that the rows step down by one degree each is a loose approximation, not a fact about the tablet. The claim traces back to a passing remark by Neugebauer and Sachs: "we start out with almost half a square (because the value of b : l which corresponds to the first line is 0;59,30) and gradually diminish the angle between l and d step by step, the lowest value being almost exactly 31 degrees" (S031, 1945, p. 39, quoted at S002, p. 179). Mansfield and Wildberger give the range as "from 45 degrees to 59 degrees, with each row separated by about 1 degree" (S003, p. 400). Those two ranges are not in conflict. They measure the two different acute angles of the same triangles. Robson's column gives the base angle running 44.76 down to 31.89 degrees. The complements of those run 45.24 up to 58.11 degrees, which is Mansfield and Wildberger's range. Both sides are describing the same fifteen triangles from opposite corners.
The heading over Column I, and what it says
The heading is broken at the start of both lines, and its translation has been revised three times in eighty years. The core sense has never changed.
Robson gives the Akkadian as takilti ṣiliptim ša ištēn innassaḫuma pātum illû and translates: "The takiltum-square of the diagonal from which 1 is torn out, so that the short side comes up" (S001, pp. 114 to 115). She adds a striking admission about the state of the field: "Surprisingly, no one has been able to improve convincingly on the translation made by Neugebauer and Sachs when they first published Plimpton 322 [20, p. 40]. They were uncertain about the first word and the last word, as well as what was missing at the beginning of the second line" (S001, pp. 114 to 115).
The current preferred version is Britton, Proust and Shnider's, quoted by Mansfield and Wildberger: "The original translation of the damaged column I heading has been successively improved upon by Robson (2001, 192) and most recently Britton et al. (2011, 526), who translate it as: The takiltum of the diagonal (from) which 1 is subtracted and (that of) the width comes up" (S003, p. 398).
Line the heading up against the arithmetic and the two match exactly. Take the quantity built on the diagonal, subtract 1, and the corresponding quantity for the width comes up. That is:
The heading is the proof that the leading 1s were there. If Column I held , subtracting 1 from it would give a negative number. There is no negative number anywhere in Old Babylonian mathematics.
The disputed word is takiltum (tah-KIL-tum). Robson derives it from the verb kullum, calling it "a technical derivative of the verb kullum ('to multiply lengths together into areas')," and renders the whole thing as the "holding-square." François Thureau-Dangin (frahn-SWAH tuh-ROH dahn-ZHAN, 1872 to 1944) held that it "may just mean square" (S001, S002, S003). The word is still open.
Five readings of fifteen rows
Robson lays out three interpretations of Plimpton 322 that were live when she wrote. Mansfield and Wildberger added a fourth in 2017. Mansfield modified his own in 2021. Each is worth stating in the form its advocate gave it, because the popular versions have drifted.
1. A trigonometric table (advocates: general histories, not the first editors). Robson's summary of the position: "if Columns II and III contain the short sides and diagonals of right-angled triangles, then the values in the first column are tan2 or 1/cos2 and the table is arranged so that the acute angles of the triangles decrease by approximately 1 degree from line to line" (S001, pp. 107 to 108). She is careful about attribution, and this is one of the most-corrected points in the whole literature. She traces the popular version to D. E. Joyce's 1995 web page and to general histories, not to Neugebauer and Sachs. Of the 1945 editors she writes: "Nowhere, however, in their concluding discussion of 'historical consequences' did they mention 'trigonometry' or 'angle', but kept their comments to the (equally dubious) 'purely number theoretical character' of Plimpton 322" (S002, p. 180). Neugebauer is regularly blamed for a claim he did not make. What he and Sachs did claim is that the tablet is a piece of number theory. Robson thinks that is also wrong, but it is a different wrongness.
2. Generating functions (advocates: Neugebauer and Aaboe; developed by Price). Choose , coprime, not both odd, and form
which generates every reduced Pythagorean triple exactly once (S001, pp. 107 to 108). Robson's comment on the research program this launched is dry: "The quest has then been to find how p and q were chosen." Derrick de Solla Price (duh SOL-uh PRICE, 1922 to 1983, dates approximate) proposed in 1964 a specific generating scheme whose parameters yield 38 rows. The surviving tablet preserves the first 15 of them. That reconstruction has been influential well beyond Price's own reading. Mansfield and Wildberger adopt the row count while rejecting the interpretation. They "agree with the many authors (de Solla Price, 1964; Conway and Guy, 1996; Britton et al., 2011) who have argued that the top, bottom and reverse of the tablet were intended to be filled with an additional 23 rows" (S003, p. 407).
3. Reciprocal pairs (advocates: Bruins, then Robson). Evert Marie Bruins (BROWNS, 1909 to 1990, dates approximate) proposed in 1949 and again in 1955 that the tablet derives from pairs and that both terminate in base 60. Buck, Friberg, and Schmidt each rediscovered the idea independently around 1980. The pairs run in descending order from 2;24 and 0;25 down to 1;48 and 0;33,20, and from each pair you build
(S001, pp. 107 to 108). Robson rebuilt this reading in 2001 with a cultural and linguistic argument rather than a numerical one. It is now the mainstream position.
4. Normalized right triangles (advocate: Jöran Friberg). Jöran Friberg (YUR-an FREE-berg, living), of Chalmers in Gothenburg, argued that Plimpton 322 is a systematic classification of normalized right triangles, equivalently a table of square-root parameters. He was also the first to propose that the two missing columns held the ratios and . Those are the short side and the diagonal of a triangle whose long side is set to 1 (S002, S003). That proposal has outlived the debate it was born in. Mansfield and Wildberger build directly on it: "we are in agreement that the missing columns I and II probably contained β and δ" (S003, p. 407). Friberg is one of the people this chapter is meant to rescue from the footnotes. He is usually mentioned, if at all, as one of three names who rediscovered Bruins's reciprocal pairs around 1980. His own positive contribution was the idea that the lost left-hand columns held normalized side ratios. It is the single reconstruction detail that both sides of the 2017 argument accept.
5. Exact ratio-based trigonometry (advocates: Mansfield and Wildberger, 2017). Their claim, in their own words: "We propose that P322 is a different kind of trigonometric table which lists right triangles with long side 1, exact short side β and exact diagonal δ, in place of the approximations sin θ and cos θ" (S003, p. 396). The crucial move is that this reading abandons angle entirely and keeps only ratios of sides. Whether that is still "trigonometry" is exactly what the fight is about, and their own answer changed.
How Robson killed reading 2
Not with mathematics. With office practice.
"the fact that Plimpton 322 follows the same formatting rules as all other tables from ancient Larsa leads us to dismiss Neugebauer's theory of generating functions. If the missing columns at the left of the tablet had listed p and q, they would not have been in descending numerical order and would thus have violated those formatting rules" (S001, p. 111).
Every Larsa table sorts its leading column into descending order. Columns of and would not be sorted that way. So whatever stood in the missing space, it was not and .
This is an unusual argument in the history of mathematics, and a good one. The strongest evidence about a document is often not its content but its genre. Genre means what kind of document it is, and what rules documents of that kind follow. A modern parallel: you can often tell a receipt from a poem without reading a single word. If someone tells you a particular receipt is secretly a poem, the burden is on them.
Robson's case against the trigonometric reading
Robson's argument is not that the numbers fail. The numbers work. is sitting there in Column I. Her argument is that the concept the trigonometric reading requires did not exist in the Old Babylonian period. She makes it on three fronts.
Front 1: the circle was built from the outside in
Old Babylonian geometry approached a circle through its circumference, not through a rotating radius. The evidence is a group of school tablets on which the area of a circle is computed from the circumference. With taken as 3, the rule is
That coefficient 0;05 is exactly what the scribes used. Robson reads this off YBC 7302 and YBC 11120, where the working is present and no radius is drawn or named. She reads it off Haddad 104 as well, and there the area is computed from the circumference even when the diameter is already known (S002, pp. 182 to 183). That last case is the telling one. Knowing the diameter and still going through the circumference is a habit of thought, not an accident.
The vocabulary agrees. The diameter, tallum (TAL-lum), "regularly crops up in OB problems about circles," but the radius, pirkum (PEER-kum), "is never mentioned" (S002). There is exactly one exception, the coefficient list TMS 3 from Susa (SOO-suh). There the radius appears last, after area and diameter, treated as the short transversal perpendicular to the diameter, "never conceptualised as a rotatable line" (S002).
The word for a circle carries the whole idea. kippatum (kip-PAH-tum), from kapāpum, "to curve," means both the two-dimensional disc and the one-dimensional circumference that defines it. ṣiliptum does the same doubling, meaning both a diagonal and the rectangle it cuts (S001, S002). In this system a figure is its defining line. There is no center in the definition, so there is nothing for an angle to open from.
Her conclusion: "In short, to treat Plimpton 322 as a trigonometric table of any kind does extreme violence to Criteria 1 to 2. The Old Babylonian circle was a figure, like all OB geometrical figures, conceptualised from the outside in. In such a situation, there could be no notion of measurable angle in the Old Babylonian period. Without a well-defined center or radius there could be no mechanism for conceptualising or measuring angles, and therefore the popular interpretation of Plimpton 322 as some sort of trigonometric table becomes meaningless" (S002, pp. 182 to 183).
Front 2: scribal practice and what scribes wrote
Robson's second front is the corpus. If Old Babylonian scribes had a concept of measured angle, it would leave traces across thousands of tablets. You would expect angle units, angle problems, and angle coefficients in the coefficient lists. The corpus has none. What the corpus does contain is gradients, which are ratios of lengths, and a rough practical sense of squareness. She is scrupulous about not overstating this, and popular writing garbles the nuance badly. Here it is verbatim:
"neither do I mean that there was no concept of angle at all in ancient Mesopotamia. Gradients were used to measure the external slope of walls and ramps in formulations like 'for every 1 cubit depth the slope (of the canal) is 1/2 cubit' (YBC 4666, rev. 25). There was also a rough distinction made between right angles and what we might call 'wrong angles', namely, those configurations for which the Pythagorean rule held true or not, with probably a 10 to 15 degree leeway" (S001, p. 113).
She also insists that the radius was perfectly well known as a physical thing: BM 15285 "depicts several circles whose deeply impressed centres reveal that they were drawn by means of rotating compasses" (S001). Scribes drew circles with compasses. They did not build their geometry on the line the compass swings.
So the precise claim is narrow and strong. Scribes had slope. They had right angles to a tolerance of maybe 10 to 15 degrees. They had compasses. What they did not have was angle as a measured quantity you could tabulate and look up. A trigonometric table is nothing but a tabulation of measured angles.
Her footnote quotes Jens Høyrup (YENS HOY-rup, born 1943, approximate) putting it more bluntly: "the trigonometric claim is as meaningless as a claim that the sequence 1 to 2 to 3 to 4 in itself, and with no corresponding angles listed, constitutes a table of tangents" (S002, p. 183 n. 20). The point is exact. A table of tangents needs two columns, angles and ratios. Plimpton 322 has no angle column, and no tablet anywhere supplies one.
Front 3: the tablet is shaped like an administrative document
The third front is the one already sketched: format. Landscape orientation, headings over columns, descending sort, calculation running left to right, MU.BI.IM at the end. Robson's comparison piece, YBC 4721, is a grain account from Ur written in 1822 BCE. It obeys every one of those conventions, and so does Plimpton 322 (S001, pp. 110 to 111). The tablet is not laid out like an astronomical table or a geometrical treatise, because those genres did not exist. It is laid out like a spreadsheet from the accounts office, because that is the only tabular format the scribe knew.
This is also why Mendelsohn's 1943 catalog entry, which reads as a comic failure, is nothing of the kind. He filed it as a commercial account because it looks exactly like one.
And a long-range warning about angles
Robson adds a point that reaches far past Mesopotamia and is worth pinning to the wall: "Nearly two millennia after Plimpton 322 was written, Ptolemy conceptualised the circle as a diameter rotating about its centre in order to simplify his calculations of chords of arc, but those chords were functions of arc, not of angle (Toomer [31, p. 47])" (S001, p. 112). Even Ptolemy (TOL-uh-mee), the person most often credited with founding trigonometry, tabulated chords against arcs, not against angles. The idea that the fundamental input is an abstract angle is later than almost everyone assumes.
She also disposes of a persistent confusion: "we can dismiss immediately any suspicion that Plimpton 322 might be connected with observational astronomy ... the accurate and detailed programme of astronomical observations for which Mesopotamia is rightly famous began a thousand years later, at the court of the Assyrian kings in the eighth century BCE" (S001, p. 107 n. 1). Plimpton 322 has nothing to do with the sky. It predates Babylonian systematic astronomy by about a thousand years.
Robson's own reading, and what she admits she cannot show
Her positive proposal has two parts: how the numbers were generated, and what the object was for.
Generation. "We have found, then, the most historically, culturally, and linguistically convincing of our three interpretations of Plimpton 322: a list of regular reciprocal pairs, each four places long or shorter, was drawn up in the usual decreasing numerical order on the missing part of the tablet. They were used to find the short sides s and diagonals d of triangles with long sides of length l = 1 by the method of completing the square. One of the intermediate results was recorded in the first extant column. Then common factors were eliminated from the triples produced to give the coprime short sides and diagonals listed in Columns II and III" (S001, p. 116).
Her anchor for this is a different tablet entirely: YBC 6967, an igi-igibi problem (EE-gee ee-GEE-bee, the Sumerian words for a reciprocal and its partner), which opens "[A reciprocal] exceeds its reciprocal by 7." The procedure on YBC 6967 uses the word takiltum for exactly the construction the Plimpton 322 heading names. That is the linguistic evidence: the same technical term, in the same operation, on a tablet where the whole procedure survives.
Function. "On balance, then, Plimpton 322 was probably (but not certainly!) a good copy of a teachers' list, with two or three columns, now missing, containing starting parameters for a set of problems, one or two columns with intermediate results (Column I and perhaps a missing column to its left), and two columns with final results (II to III)" (S002, pp. 201 to 202).
Her model for what a teacher's problem list looks like is BM 80209 from Sippar (SIP-ar): "It repeats a few school mathematics problems over and over, each time giving a different set of numerical data that will yield a tidy integer answer" (S001, p. 117). Anyone who has written a worksheet recognizes the genre instantly.
And then she stops, on purpose: "That is perhaps as far as we can go on present evidence: without closer parallels we run the risk of crossing the fuzzy boundary from history to speculation. The Mystery of the Cuneiform Tablet has not yet been fully solved" (S002, pp. 201 to 202).
That sentence is the model for how to write history honestly. It is also the reason Robson belongs in this chapter as a person and not just as a citation. Her paper's title, "Neither Sherlock Holmes nor Babylon," is a warning against exactly the kind of triumphant detective story that Plimpton 322 keeps attracting, and she ended the paper by refusing to declare victory.
She is also careful about the human being who wrote the tablet, in ways most accounts skip: "it is virtually certain that our author was male: all the known female scribes from ancient Mesopotamia lived and worked much further north, in central and northern Iraq" (S001, p. 116). And she rejects both labels usually pinned on him. He was not a professional mathematician, because "the professionalisation of academic disciplines is a phenomenon of the very recent past," in her words. Nor was he an amateur mathematician of the sort found in Classical Antiquity and the Middle Ages, because "There is not one example of this type of individual in the whole of Mesopotamia's three-thousand year history" (S001, p. 116). He was a scribe. Mathematics was part of a job.
Jens Høyrup, and why cut-and-paste geometry matters
Høyrup is the third name to feature here at length, and his contribution is the one that makes Robson's reading of the heading possible.
For most of the twentieth century, Old Babylonian problem texts were translated into modern algebra. A tablet says "I added the area and the side of my square and got 0;45," and the standard translation ran . Then came the observation that the scribe's procedure matches the quadratic formula. This made the Babylonians look like algebraists who happened to lack symbols.
Høyrup showed that this is a mistranslation of the thinking. He read the technical vocabulary closely and found that the words are not algebraic at all. They are the vocabulary of physically cutting and moving pieces of a rectangle. You take a strip off one side, swing it round, paste it onto another, and complete a square. The operations are geometrical actions on a real figure. The "algebra" is a modern overlay (S001, S002).
Why this matters for Plimpton 322 in particular: takiltum only makes sense inside that world. As a piece of symbol-pushing it is an unmotivated intermediate quantity. As a piece of cut-and-paste geometry it is the square you are holding while you complete the figure. That is why Robson renders it "holding-square", and why the heading can say in one breath that you subtract 1 from the diagonal's quantity and the width's quantity "comes up" (S001, pp. 114 to 115). The number comes up because the shape comes up.
Høyrup's reading also supplies a general lesson worth stating to students directly: translating an ancient text into modern notation is not neutral. It can preserve every number and destroy the idea. His terminology shows up elsewhere in this material too. Mansfield uses Høyrup's collection of Akkadian technical terms. One of them is mutarrittum (moo-tar-REET-tum), literally "direction of the plumb line," used as a metaphor for the perpendicular side of a shape (S004, via Høyrup 2002). A culture that names the perpendicular after a hanging weight is telling you where its geometry came from: building sites.
The 2017 claim, stated precisely
On 24 August 2017, Mansfield and Norman John Wildberger (WILD-ber-ger, living), both at UNSW Sydney, published "Plimpton 322 is Babylonian exact sexagesimal trigonometry" in Historia Mathematica (S003). The press coverage was enormous and mostly reported the headline. Their actual claims are worth reading.
From the abstract: "We trace the origins of trigonometry to the Old Babylonian era, between the 19th and 16th centuries B.C.E. This is well over a millennium before Hipparchus is said to have fathered the subject with his 'table of chords' ... we demonstrate that Plimpton 322 is a powerful, exact ratio-based trigonometric table" (S003, p. 395).
The headline consequence they draw: "If this interpretation is correct, then P322 replaces Hipparchus' 'table of chords' as the world's oldest trigonometric table, but it is additionally unique because of its exact nature, which would make it the world's only completely accurate trigonometric table" (S003, p. 396).
Their reconstruction: six columns holding , , , , and the row number, over 38 rows, of which 15 survive (S003, p. 407). The first two columns are Friberg's proposal. The 38 rows are Price's.
Their description of the shapes: "the right triangles corresponding to the Pythagorean triples turn out to have a steadily decreasing reciprocal slope, or ukullû βn = bn / ln from 59.30 to 37.20. In terms of angles, which would be foreign to OB thinking, the range of inclination goes from 45 degrees to 59 degrees, with each row separated by about 1 degree" (S003, p. 400). Note "which would be foreign to OB thinking." They concede Robson's point about angle and then argue that a ratio table without angles is still a trigonometric table.
The Akkadian term they lean on is ukullû (oo-KOOL-loo), identified by Thureau-Dangin in 1938 from about half a dozen instances. It is the reciprocal slope, run over rise, "usually in terms of a certain length per cubit," and its literal sense is reported as "fruit" (S003, and see the etymology note: the gloss is at second hand from Thureau-Dangin). A related term, indanum (IN-dah-num), gives the change in width per unit height of a trapezoid, that is . It is a property of the whole shape rather than of any one side (S003, from MS 3052 no. 1 via Friberg 2007). These are real Mesopotamian slope concepts, and Mansfield and Wildberger are right that they exist. The question is whether slope-per-cubit adds up to trigonometry.
And their own explicit concession, which almost no press report carried: "There is no known historical evidence that confirms how P322(CR) was actually used. This is a question that must be answered by archeology, or by further discoveries on existing tablets. But we can open the possibility for a trigonometric interpretation" (S003, p. 410).
They also quote Robson's central objection in full before disputing it: "As Robson (2002, 112) points out: '... there was no conceptual framework for measured angle or trigonometry [in the OB era]. In short, Plimpton 322 could not have been a trigonometric table.' But the possibility that P322 is an exact sexagesimal trigonometric table, without the assumption of a circle-based measurement system based on angles, has not been considered until now" (S003, p. 410).
Their objections to Robson, which are not silly
The 2017 paper is not junk, and its criticisms of the teacher's-list reading are technical and pointed. Four of them (S003, pp. 409 to 410):
- "The key operation upon which all quadratic problems hinge is the extraction of a square root. But ... the numbers on P322 are just too big to allow students to reasonably obtain the square roots of the quantities required." If these are exercise parameters, the exercises are unreasonably hard.
- "if P322 is a table of parameters for quadratic problems, then this does not explain the comprehensive and systematic ordering by ukullû. In addition, why would such a table be ordered by the answers and not the questions?" A worksheet generator sorted by its own outputs is odd.
- "This hypothesis also fails to explain the purpose of columns II and III."
- Against Friberg's factor-reduced-core hypothesis specifically: "entries b5, b11, d11 and b15 contain regular factors and hence are not factor reduced cores."
A student should see that last list, because the honest shape of this dispute is not "sound scholar versus crank." It is a real technical disagreement in which the minority position has arguments that the majority position has not fully answered.
They also make a comparison that got quoted everywhere. Plimpton 322, they say, is a more powerful practical table than the sine table of Madhava of Sangamagrama (MAH-duh-vuh, c. 1340 to c. 1425), because its entries are exact ratios rather than rounded values (S003). Whether "more powerful" survives contact with the fact that the tablet lists only 15 shapes is a fair thing to argue in class.
The rebuttals, and the retraction
The most-cited public criticism appeared within days, from Evelyn Lamb (EV-uh-lin LAM, living), a mathematician and science writer. She published it in her Roots of Unity blog at Scientific American, under the title "Don't fall for Babylonian trigonometry hype" (S014). Her objections come on four distinct grounds, and they are worth separating:
- Factual. The tablet contains well-known errors, which sits badly with the claim that it is "the only completely accurate trigonometry table." A corrected version, she notes, "would not be a revolutionary replacement for modern trig tables" anyway.
- Methodological. Robson's principle applies: ancient mathematical texts and artifacts "must be viewed in the light of their mathematico-historical context."
- Logical. Mansfield's argument that base 10 has only two exact fractions, and , ignores that terminates exactly. He also applies a looser standard to base 60 than to base 10. Lamb suggests this may not be "an honest mistake."
- Motivational. The interpretation is driven by Wildberger's "rational trigonometry" framework, which "has almost no traction in the mathematical community."
On the factual point, note that the authors themselves supply the ammunition. Their own Table 1 lists errors at rows 2, 8, 9, 13, and 15 (S003, p. 398). A table with five bad entries out of fifteen rows is not a completely accurate anything.
Here is the honest state of the literature, and it surprises people. I searched for a formal, peer-reviewed published rebuttal of the 2017 paper and did not find one. What exists, in the sources I read, is Lamb's blog post and Mansfield's own later paper. If a formal published response exists somewhere, I did not locate it. Nobody should claim "scholars published rebuttals" in the plural without checking further. That is an uncomfortable fact about how a discipline responds to a claim that is popular in the press and unpopular among specialists. The correction happened in a blog and in the original author's own follow-up, not in a journal reply.
Mansfield retracts, in print, in his own words
In 2021 Mansfield published "Plimpton 322: A study of rectangles" in Foundations of Science (accepted 25 June 2021, online 3 August 2021) (S004). In section 3.5.3, headed "No Trig as We Know It," he wrote:
"The word trigonometry is universally used to mean the study of the ratios of the sides of right triangles as functions of angle (sin, cos, tan and the reciprocal ratios csc, sec, cot), and it is quite clear that these functions are not part of Mesopotamian mathematics. Any interpretation of Plimpton 322 as a table of trigonometric functions is rightly dismissed by Robson as anachronistic, who is far from alone on this point. There is vast consensus that Plimpton 322 is not about trigonometry as we know it (Van Brummelen, 2009, p. 14)." (S004, pp. 997 to 998)
He keeps one disagreement, and it is a fair one:
"However, it is misleading to say that '... there could be no notion of measurable angle in the Old Babylonian period ...'. Scribes measured and understood just one angle: the right angle." (S004, p. 997)
And he keeps one residual hope for the 2017 reading, carefully downgraded:
"While Plimpton 322 cannot be a trigonometric table in the usual sense, it may still relate to practical mensuration as proposed by Mansfield & Wildberger (2017)." (S004, p. 998)
His 2021 hypothesis is a different animal from his 2017 one: "Plimpton 322 was a theoretical investigation into a certain problem in contemporary land measurement." He even flags his own scare quotes around the keyword "proto-Trigonometry" (S004).
Put the two papers side by side. Same author, same tablet, four years apart, headline claim withdrawn by name in a peer-reviewed journal. That is not a scandal. It is the system working: a strong claim, specialists pushing back with evidence about how the culture operated, and a revision published under the claimant's own signature. Students are told constantly that scholarship self-corrects. Here they can hold both documents and read the correction line by line.
How to present the dispute fairly. The shared ground is far larger than the disagreement. Everyone agrees on the numbers in the four columns. Everyone agrees on where the errors are. Everyone agrees the entries were generated from reciprocal pairs or an equivalent procedure. Everyone agrees the tablet has nothing to do with observational astronomy. The live disagreement is over one word, "trigonometry," and over what the object was for. That is a much more interesting argument than "was it a trig table, yes or no."
Si.427: a surveyor using Pythagorean triples
Mansfield's evidence for his one remaining disagreement with Robson, that scribes did understand and measure the right angle, is a different tablet.
Jean-Vincent Scheil (vann-SAHN SHAYL, 1858 to 1940, dates approximate), a Dominican Assyriologist, discovered and cataloged Si.427 ("see four twenty-seven"; Si is the museum's code for Sippar tablets in Istanbul) on the 1894 French archaeological expedition to Sippar. He published a partial edition in 1895 and cataloged it in 1902. The complete edition waited 125 years, until Mansfield's 2020 paper in the Journal of Cuneiform Studies (S004, p. 1001). The tablet is now on display in the İstanbul Arkeoloji Müzeleri (is-tan-BOOL ar-keh-oh-LOH-yee myoo-zeh-leh-REE).
It is a round Old Babylonian field plan, from the same broad period as Plimpton 322, c. 1900 to 1600 BCE (S004, S025, S011). The published photograph shows a roughly circular tablet about 11 cm across, judged against the museum's own millimeter scale bar. It carries a large crack running diagonally, ruled rectangular subdivisions across its face, and cuneiform annotations. Someone has inked "Si.427" on the tablet itself (S034).
What it shows, in Mansfield's words: "Si.427 is one of the most complete examples of applied geometry from the ancient world and can be found on display at the İstanbul Arkeoloji Müzeleri. Like earlier field plans, Si.427 retains the subdivision into rectangles, right trapezoids, and right triangles. But unlike earlier field plans it concerns the sale of private land and the measurements have been made with unusually high precision. The rectangles themselves are most remarkable because they actually have opposite sides of equal length, which is unique and suggests that OB surveyors had devised a way to create perpendicular lines more accurately than before" (S004, p. 1001). To UNSW's press office he described it as "the only known example of a cadastral document from the OB period, which is a plan used by surveyors to define land boundaries" (S025).
The method is the interesting part. The surveyor laid out his boundaries using two scaled Pythagorean triples: "The surveyor who wrote Si.427 used the (7:30, 18, 19:30) = 1:30 x (5, 12, 13) and (4, 7:30, 8:30) = 30 x (8, 15, 17) diagonal triples. Why were (5, 12, 13) and (8, 15, 17) chosen instead of the simple (3, 4, 5) triple? ... in Si.427 the regions are too narrow to accommodate the width of the (3, 4, 5) triple, which is probably why other triples were chosen" (S004, p. 1001).
Read that as a practical constraint and it comes alive. A 3-4-5 triangle big enough to be accurate is a wide triangle, and the strips of land here are narrow. A 5-12-13 is long and thin, so it fits. The surveyor picked his triple to suit the shape of the field.
Mansfield draws a sharp contrast with Roman practice: "Instead of creating a small auxiliary shape, OB surveyors would create a whole region with the dimensions of a diagonal triple," where the Roman surveyor Balbus (BAL-bus) used a 3-4-5 rectangle as a disposable scaffold and then threw it away (S004, p. 1001).
Framing this correctly for students. Si.427 shows the applied use of Pythagorean triples in surveying, a thousand years before Pythagoras. It does not show trigonometry. Mansfield's careful statement is that "OB surveyors created accurate perpendicular lines from a variety of diagonal triples," and he phrases his link to Plimpton 322 as a conjecture: "Was Plimpton 322 inspired by this cadastral interest in rectangles with regular sides? Perhaps, but we cannot hope for definitive answers to such questions" (S004, p. 1001). Headlines that called Si.427 "the oldest applied geometry" or tied it to trigonometry outran the paper they were reporting.
One name, and the press got it backwards. Coverage of Si.427 in 2021 widely reported a personal name, Sin-bel-apli, as the surveyor who wrote the tablet. The full 2020 edition has since been read for this book, and it says something different. The name is there, on the reverse, bottom, lines 8 to 10: "total (surface area) of the field, together with various marshes / of Sin-bel-apli / apart from 1 ese 1 iku field acquired through purchase" (S483).
Sin-bel-apli owned the field. He did not survey it. The surveyor is never named anywhere on the tablet. Mansfield notes that Sin-bel-apli is "a common Old Babylonian name well-attested at Sippar," which is part of why the tablet is taken to be Old Babylonian at all (S483). Two smaller things worth knowing if you go to the paper: Mansfield spells the name two ways on the same pages, and the transliteration of those lines is not his own reading but one "taken from a note attributed to Marten Stol" (S483).
And the tablet carries no date. Mansfield states it plainly: "According to Robson's criteria, Si. 427 is a real field plan, even if it is undated" (S483). Its Old Babylonian placing comes from its format, its language and that well-attested name, not from anything written on it. So when this chapter says c. 1900 to 1600 BCE, that is an inference from the object's type, not a reading off its face.
Where base 60 came from
The honest answer is that nobody chose 60.
Robson traces the ancestry to the proto-cuneiform accounting tablets from Uruk (OO-rook), written just before 3000 BCE, which run several number systems at once: "There were four sets of units for counting different sorts of discrete objects, another set for area measures, and another for counting days, months and years. There were also four capacity measure systems for particular types of grain ... Each counting or measuring system was context-dependent: different number bases were used in different situations, although the identical number signs could be used in different relations within those contexts. One of the discrete-object systems was later developed into the sexagesimal place value system, while some of the other bases were retained in the relationships between various metrological units" (S020, pp. 150 to 151).
Stop on that. The same sign meant different amounts depending on which system you were writing in: sheep counted one way, grain another, fields another. A scribe carried several number systems in his head at once. He switched between them by context, the way a cook switches between grams and teaspoons. Base 60 was not designed. It is the descendant of one of those systems, and the others survive as the odd ratios between units.
Consolidation came from the state. Metrology, in the quotation that follows, means the whole set of units a society measures with. In the late 24th century BCE, under the dynasty of Akkad, "the traditional metrological systems were overhauled and linked together, with new units based on divisions of sixty. Brick sizes and weights were standardised too" (S020, p. 152). That is an imperial bureaucracy imposing a standard, in the same spirit as metrication. Standardized brick sizes are the giveaway: this is a construction and taxation reform that happens to be a mathematical one.
The date of place-value notation has moved. This is one of the places where a textbook written in 1970 is now wrong: "it was long thought that the sexagesimal place system ... was an innovation of the following Old Babylonian period ... However, we now know that it was already in use by around 2050 BCE, and that the conceptual framework for it had been under construction for several hundred years" (S020, p. 152). That is the Ur III period, evidenced at Ur and Girsu (GEER-soo), about three centuries before Plimpton 322. Robson names the older view explicitly as superseded.
Neugebauer's methodological warning is the one popular accounts skip, and it reframes the whole question:
"But it is not enough to realize that the 60-division is only one of several contemporary norms between higher and lower units. The essential point lies in the use of the place value notation, regardless of the value of the ratio between consecutive units. No historical theory of the origin of the sexagesimal system is acceptable if it does not account also for this extraordinary feature, namely, the use of the same small number of symbols for different values, depending on the arrangement. A variety of 'bases' is well known from number words and number writing all over the world. The place value notation, however, is the most striking feature of the Babylonian system. A problem of this kind cannot be solved by speculation, but only by a systematic analysis of the written documents." (S006, ch. I, p. 18)
In other words: asking "why 60?" is asking the second-most interesting question. Bases of 12, 20, and 60 turn up all over the world. Place value, where the same wedge means 1 or 60 or 3600 depending on where it sits, is the rare invention, and it appears here first.
Neugebauer's own suggestion for the 60 is modest and metrological: "A very important feature of cuneiform numerical notation is the existence of special signs for 1/2, 1/3, 2/3 and 5/6 which are in very common use also in later periods, even occasionally in mathematical texts. These 'natural fractions' undoubtedly play an important role in the arrangement of metrological units. Obviously one will group higher units in such a form that they admit directly the forming of these most common parts. This leads naturally to a grouping in 12 or 30 or 60. All these ratios do occur in one or another of the parallel systems of units in Mesopotamian metrology" (S006, notes to ch. I, p. 26). If your everyday fractions are halves, thirds, two-thirds, and five-sixths, you want a unit that splits into all of them, and 60 does.
Other theories are in circulation. A survey of them exists in a student paper, which I read as a summary of positions rather than as an authority (S026). The competing accounts run like this. One says 60 was chosen for its divisibility by 2, 3, and 5, an explanation traceable to Theon of Alexandria (THEE-on) in the fourth century CE. It is a compromise between a quinary (base 5) and a decimal system, argued by Thureau-Dangin. It has a linguistic origin, argued by Powell (POW-ul). It has an astronomical origin. Or it has an origin in the units of measure, which is Neugebauer's position. The survey's author argues the astronomical theory fails on chronology, because "sexagesimal mathematics had predated its application in measuring the sky by more than a millennium." That chronological point is independently supported by both Robson and Neugebauer (S020, S006), so it is safe even though the survey is not.
Cross-curricular hook (computing, design and everyday measurement). Sixty has twelve divisors: 1, 2, 3, 4, 5, 6, 10, 12, 15, 20, 30, 60. Ten has four: 1, 2, 5, 10. Have a class compute in both bases. In base 60 it is 0;20, exact and finished. In base 10 it is 0.3333..., forever. Then ask why a culture that did every division by looking up a reciprocal in a table would care enormously about which fractions terminate. Then take the survivals one at a time. Clocks: 60 seconds, 60 minutes, and hours in twelves. Navigation: latitude and longitude in degrees, minutes and seconds, and one nautical mile defined as one minute of arc along a meridian. Angle measure: 360 degrees, each of 60 minutes, each of 60 seconds. Everything else in science went metric two centuries ago. These did not, because they are embedded in instruments, charts, training, and law. That is path dependence, the same reason a keyboard designed in the 1870s still says QWERTY.
Cross-curricular hook (language, technology and information). Clay is an information technology, and a good one. It is free, it is local, and it takes a sharp impression from a cut reed. When a city burns, the archive bakes instead of vanishing. So we have hundreds of thousands of Mesopotamian documents and almost no Egyptian ones from the same centuries. Compare the storage media in the room: paper, hard drive, cloud account. Ask which of them will be readable in four thousand years, and what would have to be true for the answer to be "any of them." Then look at Plimpton 322 as a document design problem rather than a mathematical one. Headings across the top, one row per record, sorted by a key column, and a line-number column at the end so nothing gets lost. That is a spreadsheet, invented before glass, and Mendelsohn filed it as an account for exactly that reason.
The 360-part circle: what really happened
The familiar story is that the Babylonians thought the year had 360 days, so they cut the circle into 360 parts. It is tidy, memorable, and not what the sources show. The real sequence has five steps and takes about 1,500 years.
Step 1: the day, not the circle. MUL.APIN (mool-AH-pin, named from its opening words, the Plow star) is the great Mesopotamian astronomical compendium, composed before about 750 BCE. That date is a terminus ante quem, a latest-possible date. The tablets from Huzirina that carry it were written towards the end of the eighth century, so "MUL.APIN could not have been composed any later than about 750 bc" (S007). Hermann Hunger (HAIR-mun HOONG-er, living) and John M. Steele (STEEL, living) produced the 2019 edition, which is the current standard. They note that star-list analyzes have suggested composition dates "between 1300 and 1000 bc" and "about 1300 bc with an uncertainty," but they present these as arguments about when the observations were made, not as settled dates for the compendium (S007). Hunger made the 1989 edition with David Pingree (PING-ree, 1933 to 2005, dates approximate).
MUL.APIN measures short time intervals in three units, and Hunger and Steele give the chain: "Time intervals of less than a day are measured in bēru (DANNA), UŠ and NINDA, where there are 12 bēru in a day, 30 UŠ in a bēru, and 60 NINDA in an UŠ: Day, 12, bēru, 30, UŠ, 60, NINDA" (S007, Introduction, p. 9).
Multiply it out:
There is the 360. A bēru (BAY-roo, written DANNA, DAN-nuh) is a "double hour," one twelfth of a day, so two of our hours. An UŠ (oosh) is one thirtieth of that, which is four minutes. A NINDA (NIN-dah) is one sixtieth of an UŠ, which is four seconds. Every one of those is a unit of time. The 360 is a division of the day.
Step 2: the schematic year. MUL.APIN also computes on an idealized 360-day year. The British Museum holds BM 86378, its copy of tablet 1 of MUL.APIN (registration 1899,0610.108; 86 by 60 by 16 mm; dated 1000 to 500 BCE; excavated in southern Iraq and acquired in 1899 from Messrs Mann & Bishop). The museum's own record describes it as "a list of the three divisions of the heavens, the dates (in the ideal 360-day year) of the rising of principal stars" (S010). Hunger and Steele refer throughout to "the 360-day schematic calendar" and "the 360-day schematic year" (S007, pp. 14 to 15).
Note the word "ideal." Nobody in Babylon thought the year was 360 days long. They watched the sky for centuries. They ran a lunar calendar with intercalary months, extra months slotted in, to keep it in step with the sun. The 360-day year is a calculating device: twelve months of thirty days, chosen because it makes the arithmetic clean. It is the same move as a physics problem that says "ignore air resistance."
Step 3: what MUL.APIN computes with. Its mathematical toolkit has four parts. First, the zigzag function, described by Hunger and Steele as "linear increases and decreases between maximum and minimum values in uniform steps" (S007, p. 9). Second, a scheme for the length of the shadow cast by a gnomon (NOH-mon, a vertical rod) on the solstices and equinoxes, at tablet II ii 21 to 42. Third, schemes for lunar visibility. Fourth, the 3:2 ratio of longest to shortest daylight. There is nothing resembling a chord and nothing resembling a trigonometric function (S007).
Step 4: UŠ becomes a unit of arc, around 400 BCE. This is the step the popular story deletes, and it is the interesting one. Steele dates the uniform zodiac, twelve equal signs of thirty UŠ each, to "sometime in Babylonia during the late fifth century BC," and records three things. First, "Beginning in around 400 BC, zodiacal signs are almost always given in reports of observations of the first or last appearance" of planets. Second, "the zodiac is divided into 12 signs each containing 30 UŠ". Third, "the position of the moon or a planet is specified by the number of degrees (UŠ) within a sign of the zodiac" (S016). He locates its origin precisely where you would now expect: "The concept of the zodiac, therefore, comes out of the schematic calendar."
So the same word, UŠ, took a second job. It had been one 360th of a day. It became one 360th of the way round the zodiac. The bridge is the sky itself. The heavens turn at a steady rate, so marking off a fraction of a day is marking off a fraction of a circle. This is why astronomers to this day give positions in both hours and degrees. One hour of right ascension equals exactly 15 degrees of arc, since .
Before that shift, and alongside it, Babylonian observers recorded angular separations between celestial bodies in cubits and fingers, a body-part measure, not in degrees. Steele records the conversion from a cuneiform source: "BM 41004 Obv. 15 (Neugebauer and Sachs 1967, Text E) which contains a parallel statement but using cubits rather than degrees (1 cubit = 2 degrees)" (S015, p. 206 n. 10). One cubit equals two degrees. That is the sky measured by holding up your arm, and it is what the Astronomical Diaries are full of.
Step 5: Neugebauer's verdict. Flatly, in his own notes: "The division of the circumference of the circle into 360 parts originated in Babylonian astronomy of the last centuries B.C. The sexagesimal number system as such is many centuries older and has nothing to do with astronomical concepts" (S006, notes to ch. I, p. 25).
The earliest surviving Greek witness
Hypsicles of Alexandria (HIP-sih-kleez) flourished c. 175 BCE (disputed: the Dictionary of Scientific Biography places him in "the first half of the second century B.C., about 175 B.C."; Neugebauer says "about 150 B.C."). He wrote a short work called the Anaphorikos (an-af-or-ih-KOSS, On Ascensions), a treatise on how long each part of the zodiac takes to rise.
The DSB calls it "noteworthy in being the first work in which the ecliptic is divided into 360 parts or degrees," and quotes Hypsicles directly: "The circle of the zodiac having been divided into 360 equal arcs, let each of the arcs be called a spatial degree," with temporal degrees defined the same way (S019).
Read that quotation slowly, because the phrase "spatial degree" is doing something specific. Hypsicles has to specify that he means a degree of space, because his readers would otherwise assume he meant a degree of time. The two kinds of degree, temporal and spatial, are both live in his text, and he keeps them apart. That distinction is the fossil of the whole Babylonian history above, preserved in a Greek sentence.
The DSB adds that the 360 division "was almost certainly borrowed from Babylonia" and is "testimony to the existence of links between Greek and Babylonian astronomy in the second century B.C." (S019). That is not a guess. Neugebauer, working independently and on other evidence, shows Hypsicles running Babylonian arithmetic: "In the second century B.C. we find Hypsicles using the System A of rising times for the computation of the length of daylight in Alexandria, the only modification being that M:m is now given the value 7:5" (S006, ch. VI, p. 183). Hypsicles is demonstrably using a Babylonian scheme, with one parameter changed to suit his latitude. That makes it very likely he took the 360 division from the same place.
The word itself then traveled: Babylonian UŠ, to Greek moira, to Latin gradus, to English degree (S019, S006).
The accurate sentence
The 360 division is Mesopotamian in origin. But it began as a division of the day into 360 time-degrees of four minutes each, recorded in MUL.APIN before about 750 BCE. It ran alongside a 360-day schematic year used as a computing convenience. It became a division of arc only with the uniform zodiac around 400 BCE. And it reached Greek astronomy with Hypsicles in the second century BCE.
The popular story is not so much wrong as collapsed. It deletes 1,500 years and the one genuinely interesting step, where a unit of time picked up a second career as a unit of angle.
Three distinctions students and teachers routinely get wrong:
- A bēru is a double hour, one twelfth of a day. In MUL.APIN it is a time unit. It is also used elsewhere as a distance unit. It is never a degree.
- An UŠ is one 360th of a day in MUL.APIN, that is four minutes. Only later, in mathematical astronomy, does the same word serve as the unit of ecliptic arc we translate "degree."
- The Babylonians of the Old Babylonian period did not divide "the circle," as an abstract geometrical object, into anything. They divided the day, and much later the zodiacal band. Robson's argument earlier in this chapter is exactly that: nobody thought of the abstract circle in a way that would make a 360 division of it meaningful.
Babylonian astronomical tables: no chord, no angle
This deserves its own flat statement, because it is the load-bearing negative fact of the chapter.
Babylonian astronomy was extraordinary. Systematic observation began at the Assyrian court in the eighth century BCE. The Astronomical Diaries are the longest continuous scientific record ever compiled. They begin with the diary for 652 BCE, which is where Sachs and Hunger's volume I starts (S007). By the last centuries BCE, Babylonian astronomers could predict lunar and planetary phenomena using "purely numerical methods, without direct empirical input" (S007, p. 14).
And in all of it there is no chord function, no half-chord, no sine, and no function of an angle. The toolkit is arithmetical: zigzag functions, step functions, ratios, period relations, and schematic calendars. MUL.APIN's core tool is the zigzag function (S007, p. 9). The later systems are numerical schemes. Even Mathieu Ossendrijver (ma-TYUR OSS-en-dry-ver, living), whose result below qualifies the picture, frames his own work as a challenge to the view that Babylonian astronomical methods were "of a purely arithmetic nature," and that tells you what the standing consensus is (S018).
A note on the limits of that claim. This is an absence-of-evidence statement bounded by what I read: the introduction to Hunger and Steele's MUL.APIN, Steele on Geminos, Steele on the zodiac, and Ossendrijver's two papers. I did not read the Astronomical Diaries themselves. My statement that the series begins at 652 BCE rests on the title of Sachs and Hunger's volume I rather than on the volume. Nothing in what I read reports a chord table, a table of half-chords, or any tabulated function of an angle in cuneiform. Take that as a well-supported claim about a corpus I did not personally survey, which is how such claims should always be labeled.
Jupiter, trapezoids, and a result filed in the wrong drawer
In 2016 Ossendrijver published a startling result in Science, and it gets misfiled constantly.
Working with Late Babylonian procedure tablets in the British Museum, he identified five texts carrying a common procedure. Text A is BM 40054. Text B is BM 36801, BM 41043 and BM 34757. Text C is BM 34081 plus 34622 plus 34846 plus 42816 plus 45851 plus 46135. Text D is BM 35915. Text E is BM 82824 plus 99697 plus 99742. "The tablets date from 350 to 50 BCE" (S017). His 2018 follow-up gives the wider range c. 400 to 50 BCE for the astronomical procedure texts generally (S018).
What the procedure does: "Jupiter's displacement along the ecliptic is computed as the area of a trapezoidal figure obtained by drawing its daily displacement against time" (S017).
The numbers. Jupiter's daily velocity along the ecliptic falls from degrees per day at first appearance to degrees per day sixty days later. The distance covered over those sixty days is the area of the trapezoid.
A second trapezoid covers the following sixty days and gives , so Jupiter moves in the first 120 days after first appearance (S017).
The 2018 paper goes further: "the time when Jupiter reaches half the total distance is computed by bisecting the trapezoid into two smaller ones of equal area" (S018). Bisecting a trapezoid like that is a subtle operation. Ossendrijver traced its ancestry back to Old Babylonian and Kassite school mathematics: the trapezoid-bisection problems on UET 5, 858 from Ur, YBC 4675, AO 17264 (Kassite, 1400 to 1200 BCE), and TMS 26 sections 3 to 4 from Susa (S018). So a technique taught to schoolchildren around 1800 BCE resurfaces 1,500 years later as a tool for tracking a planet.
Ossendrijver's own comparative claim: "These computations predate the use of similar techniques by medieval European scholars by at least 14 centuries. The 'Oxford calculators' of the 14th century CE ... are credited with formulating the 'Mertonian mean speed theorem'." He notes that Nicole Oresme (or-EM) later "devised graphical methods" using trapezoids in the same way (S017).
File this correctly. The area under a velocity-time graph equals the distance traveled. That is an integration idea. It belongs beside the Oxford Calculators' mean speed theorem and Oresme's diagrams, and it belongs in a calculus classroom. It has nothing whatever to do with ratios in a right triangle. Any teaching text that lists Ossendrijver's result under "Babylonian trigonometry" has put it in the wrong drawer. The mistake matters, because it turns a real and remarkable result into support for a claim it does not support.
Egypt: measuring slope without measuring angle
Egypt attacked the slope problem from a different direction and solved it cleanly.
The papyrus, and the man who signed it
The main source is the Rhind Mathematical Papyrus, named for Alexander Henry Rhind (RIND, rhymes with "find," 1833 to 1863), a Scottish lawyer and antiquarian who acquired it at Thebes around 1858. The British Museum bought it from David Bremner (BREM-ner) in 1865 (S008, S009). It is now in two sections. EA10057 is 33 cm high by 296 cm long, in a frame 45.5 by 331 by 5.5 cm. EA10058 is 32 cm high by 198.5 cm long, in a frame 43 by 216 by 4.5 cm. The museum dates both to 1550 BCE. Neither is on display. The first was last shown in Room 62 until 1997 (S008, S009). Thomas Eric Peet (PEET, 1882 to 1934) reckoned the original roll, before it was cut in two, ran about 543 cm, with 14 sheets on the recto each 383 to 400 mm wide (S005, p. 3).
The scribe signed his work. His name was Ahmose (AH-mohs, also written Ahmes or AH-mess; flourished c. 1550 BCE). He dated his copy to year 33 of the Hyksos king Apophis (uh-POH-fis), whose full name was Aauserre Apophis. Peet dates him only broadly: he "must have ruled at some time between 1788 and 1580 b.c." The British Museum says 1550 BCE. So the two authorities are 230 years apart at the outside, and the Egyptian absolute chronology of this period is unsettled (S005, S008).
Ahmose also says that he was copying an older document, from the reign of Nimaatre, that is Amenemhat III (ah-men-EM-hat), whom Peet dates to "about 1849 to 1801 b.c." (S005). Peet believes him. So the mathematics in the papyrus may be three centuries older than the copy that carries it.
Name Ahmose in class. He is one of a tiny handful of named individuals in ancient mathematics whose name survives on his own work, and he is usually flattened into "an Egyptian scribe." He is not the author of the mathematics. He says so himself. He is a copyist, and in a manuscript culture that is a technical profession. He had to read a document three hundred years old, in an older spelling system, and reproduce it accurately. That included the red-ink conventions that mark where one problem ends and the next begins. When he slips, as he does at problem 59, the slip is a copying slip, and we can see it. The entire Egyptian mathematical record we possess rests on the competence of two or three men like him.
What a seked is
The Egyptian slope measure is the seked (SEK-ed, written skd, also transliterated seqed). It is the horizontal run per one cubit of vertical rise, expressed in palms and fingers.
Peet gives the units: "The royal cubit is in this papyrus divided into 7 palms (Nos. 56 ff.), a palm being the breadth of the four fingers, and the palm again into 4 fingers (Nos. 58 and 59)" (S005, p. 25). So:
Run over rise is the cotangent of the angle the face makes with the horizontal. So a seked is a cotangent expressed in mixed units, with the rise fixed at one cubit and the run measured in sevenths of a cubit.
Why build it that way? Peet answers with the mason standing at the block:
"The object is to provide the mason with a simple practical rule for dressing at the required angle the stones of the outer facing of the pyramid. All he has to do in this case, when confronted with a solid block of stone, parallelopipedal in shape, is to measure one cubit upwards on the outer edge, and then 5 1/25 palms inwards at right-angles. The line joining the point thus reached to the point from which he started gives the correct angle of dressing." (S005, p. 98)
He is emphatic about the direction of the ratio, because getting it backwards inverts everything. In his note to problem 60: "the skd is the reverse of this measurement, namely the divergence from the vertical in a vertical height of one cubit" (S005, p. 102). Peet is also arguing against an earlier reading by Eisenlohr (EYE-zen-lore). Eisenlohr took the seked to be a cosine of the edge rather than a cotangent of the face. Peet's reconstruction wins because it matches both the arithmetic and the mason's procedure. Gillings (GIL-ingz), quoted by Mansfield and Wildberger, states the modern consensus: "The seked of a right pyramid is the inclination of any one of the four triangular faces to the horizontal plane of its base, and is measured as so many horizontal units per one vertical unit rise. It is thus a measure equivalent to our modern cotangent of the angle of slope. In general, the seked of a pyramid is a kind of fraction, given as so many palms horizontally for each cubit vertically, where 7 palms equals one cubit" (S003, p. 400).
The words themselves, with Peet's hedges intact. The Egyptian technical vocabulary here is not securely understood, and Peet says so repeatedly. On seked: "if the s is causative it may be formed from ḳd 'to build' or 'form' ... skd would then mean 'that which forms', i.e. the measurement which builds up the pyramid." He immediately undercuts his own suggestion: "The word used for batter is equally obscure." On ukha-thebet (OO-kha THEB-et, written wḥꜣ-tbt), which he takes to be the side of the square base: "a compound formed of the verb wḥꜣ 'to seek' and the noun tbt 'a sandal'." And on the whole enterprise of arguing from these names: "Unfortunately no corroboration can be obtained from the names of the measurements themselves" (S005). The third term in the family is peremus (PER-eh-moos, written pr-m-ws), the vertical height. Peet's summary of how the three fit together: "We can make the skd correspond with this ratio if we take the pr-m-ws to be the vertical height DG, and the wḥꜣ-tbt to be a side of the base, for the angle of slope (DEG) of a side QDR is one whose cotangent is half a side of the base divided by the vertical height" (S005, p. 98).
A word meaning "seek a sandal" for the base of a pyramid is either a lovely metaphor for the footprint of a building or a false etymology. The honest position is that we do not know which. Peet, who did know Egyptian, would not commit. Nobody writing a textbook should commit harder than Peet did.
Cross-curricular hook (design, technology and the workplace). The seked is a complete manufacturing instruction. No protractor, no angle, no irrational number, no square roots. A worker with a cubit rod and a set square executes it exactly. Have a class cut blocks to a stated seked and stack them into a face. Mixed units are not a failure to invent decimals. They are a way of stating a ratio in numbers a person can measure with the tools that are on site. Then push it further into a history lesson: a pyramid is a labor-management problem before it is a geometry problem. Thousands of workers, a fixed slope that must not drift, blocks cut by different gangs at different times that have to meet at an edge. The specification is a ratio in palms rather than an angle in degrees for one reason. The ratio is checkable by any worker at any block, with no instrument that has to be calibrated. That is a quality-control decision, and it is the same reason a modern roof pitch is given as "4 in 12" rather than as 18.43 degrees.
Worked example: RMP problem 56
Peet's translation: "Example of reckoning out a pyramid 360 in length of side and 250 in its vertical height. Let me know its batter. You are to take half of 360: it becomes 180. You are to reckon with 250 to find 180. Result 1/2 + 1/5 + 1/50 of a cubit. A cubit being 7 palms, you are to multiply by 7 ... Its batter is 5 1/25 palms" (S005, p. 97). "Batter" is Peet's English word for the seked. It is a builder's term for the inward lean of a wall.
The units of 360 and 250 are not stated in the text, and Peet notes this. (He also has a slip of his own here, writing "the dimensions 300 and 250 are given in no particular unit" and then correctly saying two lines later "The 360 is halved, result 180." Even the standard critical edition contains typos, which is a useful thing for students to see.)
The method, step by step:
Check the Egyptian fraction form, since it is the part students find alien: , , , and . Egyptian arithmetic wrote every fraction as a sum of distinct unit fractions, meaning fractions whose top number is one. So 0.72 has to be assembled from halves, fifths, and fiftieths. Two modern treatments confirm the answer of palms independently (S027, S028).
The modern check. The face rises 7 palms for every 5.04 palms it runs in.
You get the same angle straight from the original numbers, without converting to palms at all, since too. Converting to palms changes the units, not the information. That is the whole point of the seked: it is the same ratio, re-expressed in a form a mason can lay off with a rod.
(That angle is my own arithmetic from Peet's numbers, not a claim made by any ancient or modern source. Ahmose never computed an angle, because he had no such quantity.)
Worked example: RMP 57 and 58, which are a 3-4-5 triangle
Problem 57 runs the calculation backwards. Peet: "A pyramid 140 in length of side, and 5 palms and a finger in its batter. What is the vertical height thereof? You are to divide one cubit by the batter doubled, which amounts to 10 1/2. You are to reckon with 10 1/2 to find 7, for this is one cubit ... Make two-thirds of 140, namely 93 1/3. This is the vertical height thereof" (S005, p. 99).
Five palms one finger is palms, since a palm is four fingers. Ahmose's route is odd, and Peet flags it. Instead of halving the base and then applying the seked, he doubles the seked and applies it to the whole base. Peet's comment: "instead of finding EG from the datum EF by halving, and then determining DG by means of the batter, the batter is doubled (twice 5 1/4 palms = 10 1/2) and the proportion used is DG : EF :: 7 : 10 1/2" (S005, p. 99).
Doubling the seked and using the full base is algebraically identical to halving the base and using the seked, since both amount to dividing by two once. It saves a step when the base is an awkward number and the seked is not.
Problem 58 is the exact inverse: given height and base 140, find the seked. The papyrus ends "Total 5 palms 1 finger. This is the batter" (S005, p. 99).
Peet notes the numbers were chosen kindly: "The numbers are suitably chosen, for 70 is precisely 3/4 (3/4 + 1/4 in Egyptian fraction form 1/2 + 1/4) of 93 1/3" (S005, p. 99). That is a textbook writer picking figures that come out clean, and it is evidence that these are exercises rather than site records.
Now look at what those figures are. Half-base 70 and height stand in the ratio 3 to 4. The slant height of the face completes the triangle:
And is , scaled by . So
Problems 57 and 58 are the 3-4-5 triangle wearing cubits and palms.
And now the discipline. Nothing in the papyrus says "3-4-5." Nothing names that angle. Nothing indicates that Ahmose or his source noticed the triple at all. He is doing proportion with a fixed conversion factor of 7. For these particular numbers the proportion lands on the most famous right triangle there is. That may be design, since a textbook writer picking clean numbers might well reach for the 3-4-5 without saying so. Or it may be luck. The evidence does not decide, and this is exactly the point at which popular accounts stop being careful and start writing "the Egyptians knew the 3-4-5 triangle."
RMP 59, 59b and 60
Problem 59 is where Ahmose slips as a copyist. He ran two problems together by failing to switch to red ink for the second opening. What the manuscript shows as one problem is really two. Problem 59 as written gives "the vertical height whereof is 12 and the side 8," which Peet says is a transposition of the data: "In order to give a batter of 5 palms 1 finger it is the height which must be 8 and the side 12." Problem 59b is the reverse operation: given base 12 and a seked of 5 palms 1 finger, find the height, answer 8 (S005, pp. 99 to 100). Both are the same 3-4-5 slope as 57 and 58, on smaller numbers.
Problem 60 is the odd one, and it matters more than its content suggests because of what it has been used to claim. Peet's translation: "A cone (?) of 15 cubits in its base and 30 in its height. Let me know its batter ... The result is 4. This is the batter thereof" (S005, p. 102).
Peet's diagnosis is that the text is corrupt: "We ought now to divide this by the height, 30, from which we should get the answer 1/4 cubit or 1 3/4 palms. But instead of this the scribe divides the 30 by the 7 1/2 and declares the resulting 4 to be the batter. There is clearly something wrong here."
Work it both ways and the problem is obvious:
The scribe inverted the ratio. The seked is run over rise, and this "4" is rise over run. So someone was bound to notice that the number 4 looks like a tangent rather than a cotangent, and to argue that Egyptians sometimes measured slope the other way up. Peet shuts that down as hard as he can: "The problem affords no case for the belief that the batter was in some cases measured by the tangent instead of the cotangent of the base angle" (S005, p. 102). One corrupt line in one problem is a copying error, not a second Egyptian slope convention.
Peet also argues that the object in problem 60 is not a pyramid at all. The word is iwn, written with a house determinative, a sign that tells you what class of thing the word names. Peet leans towards a cone (S005, p. 102). So the one problem in the group that appears to support a tangent reading is textually broken and probably about a different solid.
RMP 56 is not the Great Pyramid
This confusion is everywhere, so here is the arithmetic. Take the Great Pyramid's figures as Mansfield and Wildberger give them: "The side of the Great Pyramid at Giza had an original height of 280 cubits and a width of base of 440 cubits ... we can verify that the seqed or ukullû of the side of the pyramid would have been 220 : 280, which gives indeed the famous value of 5 1/2 : 7, or 5 palms and 2 fingers per cubit" (S003, p. 414).
The 51.84 degree figure matches Petrie's (PEE-tree) survey of the monument (S027).
Set the two side by side:
| Object | Seked | Face angle from horizontal | Source |
|---|---|---|---|
| RMP problem 56 | palms (5.04) | about 54.25 degrees | (S005, S027, S028, p. 97) |
| Great Pyramid at Giza | palms (5.5) | about 51.84 degrees | (S003, S027, p. 414) |
Different seked, different pyramid, about two and a half degrees apart. RMP 56 is a textbook exercise with round numbers, not a description of Khufu's monument, and it never claims to be.
The Moscow papyrus and Berlin Papyrus 6619
Two other Egyptian mathematical documents get dragged into trigonometry's prehistory. Neither belongs there, and it is worth saying what they do contain so the claim can be checked.
The Moscow Mathematical Papyrus, in the Museum of Fine Arts in Moscow, dates from the 12th Dynasty. Peet wrote in 1923 with access only to photographs and preliminary reports, since Struve's edition came in 1930. He describes it as containing "19 problems, some of which give us new types of calculation unknown till now ... Four of these problems are geometrical ones. The first shows how to define the length of the sides of a quadrilateral, when the relation of the sides and the area of the quadrilateral are known. The two next give a method of calculating the area of a triangle: a method already known to us" (S005, p. 3).
His own verdict is deflationary: "though the papyrus is of the highest interest owing to its early date and admirable state of preservation (in part at least) it contains nothing, with the exception of the problem of the truncated pyramid, which will greatly modify the conception of Egyptian mathematics given to us by the already published papyri and fragments" (S005, p. 3).
That one exception is impressive. The truncated pyramid problem takes a frustum, a pyramid with its point sliced off, whose top surface is 2 cubits square, whose bottom is 4 cubits square, and whose height is 6 cubits. It computes the volume by
which is correct if is the vertical height. Peet: "If the height is the vertical height of the frustum the solution is correct, and in this case the Egyptian has here made a very notable achievement." Work it through:
and the papyrus closes with the line "See, there you have it, 56" (S005). A correct formula for the volume of a frustum, in about 1850 BCE, with no derivation preserved.
Peet is also emphatic that the Moscow papyrus shows the right triangle was thoroughly understood: "in the Moscow papyrus the right-angled triangle is clearly perfectly well understood and treated as half a rectangle: the two sides enclosing the right-angle are actually called the 'length' and 'breadth' respectively, terms manifestly taken from the terminology of a rectangle" (S005, p. 92). Note how closely that matches the Mesopotamian habit: a right triangle is half a rectangle, and its legs borrow the rectangle's vocabulary. Two independent traditions, the same conceptual move.
What the Moscow papyrus does not contain: any seked problem, any slope ratio, any angle, any table.
Berlin Papyrus 6619 is a set of four fragments, published by Schack-Schackenburg (SHAK SHAK-en-boork). Its provenance, meaning where it was found, is not stated. Peet lists it and describes its content as second-degree equations: "Equations of the second degree where there is virtually only one unknown were also understood. In the Berlin Papyrus 6619 we have to divide 100 square [cubits into two squares]" (S005, pp. 4 and 21). He also cites it, with the Kahun papyrus and the Moscow papyrus, as evidence that square roots were known in Egypt: "No example of square root occurs in Rhind, but Pap. Berlin 6619, Pap. Kahun Pl. VIII, l. 40, and Pap. Moscow show that the idea of square root existed" (S005).
Splitting 100 into two squares is a nice problem, and its solution turns on a Pythagorean triple, since . It is algebra, not trigonometry. There is a more recent study of this papyrus (Miatello 2012, mee-uh-TEL-oh). I could only reach its abstract, so nothing in this chapter rests on it.
Peet was writing before Struve's edition of the Moscow papyrus and before modern re-readings of Berlin 6619, so his account is early. On the one point that matters here, that neither document contains slope or angle material, it has not been superseded.
The instruments
Egypt has surviving, cataloged, dated hardware for sighting stars and telling time by shadow. None of it reads out an angle. That last sentence is the whole reason this section exists.
| Object | Object number and date | What the catalog says | Link |
|---|---|---|---|
| Egyptian merkhet, bronze with hieroglyphic text inlaid with electrum | 1929-585, dated 600 BCE, made in Egypt | Inscribed for "Bes, son of Khonsirtis, an astronomer priest of the god, Horus of Edfu in Upper Egypt"; used "as a timekeeping tool to determine hours via the sun's position during day and stellar observations at night, and as a surveying instrument for establishing building axes"; 25 x 93 x 22 mm, 0.098 kg; credited to Dr Howard Carter; "fitted with a replica plumb bob, probably at some point during the early twentieth century" (S021) | https://collection.sciencemuseumgroup.org.uk/objects/co500/egyptian-merkhet |
| Copy of a merkhet and bay | 1913-573 | "A merkhet is a bar with a plumb line used for measuring star transits"; a bay is "a forked stick used for alignment"; together used for "astronomical timekeeping"; original in Berlin's Royal Museum; credited to Capt. H. G. Lyons (S022) | https://collection.sciencemuseumgroup.org.uk/objects/co1219/copy-of-a-merkhet-and-bay-instruments |
| Ancient Egyptian altitude sundial, or shadow clock, in pine | 1926-992, 1000 to 800 BCE, made in Qus, Egypt | "The horizontal bar on top of the two uprights is turned to face the sun so that it casts a shadow on the base which is marked to indicate equal units of time"; 125 x 1190 x 120 mm (S023) | https://collection.sciencemuseumgroup.org.uk/objects/co473/ancient-egyptian-altitude-sundial-or-shadow-clock-in-pine-wood |
The merkhet (MER-khet) and the bay (BAY) work as a pair. The bay is a forked stick: you sight a star through the notch. The merkhet is a bar carrying a plumb line, which gives you a true vertical to sight against. Two observers with these instruments can fix the moment a named star crosses the meridian, the imaginary north to south line overhead. That gives you the hour at night. They can also lay out a building axis on a fixed direction (S021, S022). What the pair delivers is an alignment and a moment. It does not deliver a number of degrees, because there is no scale on it to read.
The shadow clock is the same story in the daytime. The bar casts a shadow along a base "marked to indicate equal units of time" (S023). The graduations count hours, not angles.
Mesopotamia used the gnomon, a vertical rod casting a shadow. MUL.APIN contains "a mathematical scheme for the length of a shadow cast by a gnomon on the dates of the solstices and equinoxes" at tablet II ii 21 to 42, and later Babylonian texts include "further expansions of the scheme for the length of shadow cast by a gnomon" (S007, pp. 12 and 15). The dedicated study is Steele's 2013 article on shadow-length schemes in SCIAMVS 14. I identified it from the MUL.APIN bibliography but could not read it.
The crucial framing point, and the best single illustration in this chapter. Read that description again: it is a scheme for the shadow's length, tabulated against dates. Not the sun's angle. To record where the sun is, a Babylonian astronomer wrote down how long the shadow was, in cubits. Turning a shadow length into a solar altitude requires an arctangent. Nobody had one, or wanted one. The gnomon and the merkhet are the concrete proof that "measuring the sun's height" and "measuring an angle" were not the same operation in the ancient world. Give a class a meter stick in a schoolyard and have them record shadow lengths hourly for a day. They will produce a Babylonian shadow table. Then ask them to convert it to angles and watch what tool they reach for.
Egyptian sundials have a documented history, and two moments in it are worth having. The earliest written evidence for a sundial in Egypt comes from the reign of Thutmose III (thoot-MOH-suh; Vodolazhskaya, voh-doh-lazh-SKY-uh, gives his reign as 1521 to 1473 BCE, which is not the most common chronology), in a campaign narrative describing an army setting out at noon "when the shadow of the sun turns" (S024). A soldier's account of a march is our first sundial record. Second, a sundial appears in the tomb of Seti I (SET-ee) in the Valley of the Kings, dated to about 1300 BCE (S024). Beyond those, the Egyptian Museum in Berlin holds a green slate sundial from Thutmose III's reign and a Faiyum sundial dated 1000 to 600 BCE. And in 2013 a University of Basel expedition led by S. Bickel (BIK-ul) and E. Paulin-Grothe (poh-LAN GROH-tuh) found a semicircular limestone tile with a hole and fan-shaped lines in the Valley of the Kings (S024). That source is published in a low-visibility journal and its Egyptian chronology is not mainstream, so treat the dates as indicative.
People who usually get left out
Six of the people in this chapter deserve more than a citation.
Ahmose (flourished c. 1550 BCE) is the only ancient person in this chapter whose handwriting we can read and whose name we know from his own hand. He copied the Rhind papyrus in year 33 of Apophis, and said he was copying a document from Amenemhat III's reign about three centuries earlier. He used red ink to mark the start of each problem, a formatting convention with a job to do. His slip at problem 59, where he forgot the red ink and merged two problems into one, is visible to us 3,500 years later. That makes him the earliest identifiable person in this book to make a mistake we can still diagnose. Everything English speakers know about Egyptian mathematics runs through his copying (S005, S008).
Eleanor Robson (born 1969, approximate) changed how one object is read and, more usefully, how a whole class of object should be read. She wrote two papers: "Neither Sherlock Holmes nor Babylon" in Historia Mathematica in 2001, and "Words and Pictures: New Light on Plimpton 322" in the American Mathematical Monthly in 2002. Both argue that you cannot interpret a mathematical artifact without knowing what kind of document it is, what its words meant to the person who wrote them, and what the writing conventions of its time and place were. The 2002 paper began life as an invited address at the Joint Mathematics Meetings in New Orleans on 10 January 2001. She made this argument to a room full of working mathematicians, not only to historians. Her method is transferable to any object in any chapter of any history of mathematics, and her willingness to end with "The Mystery of the Cuneiform Tablet has not yet been fully solved" is the part students should copy (S002, pp. 201 to 202).
Jens Høyrup (born 1943, approximate), at Roskilde, showed that translating Old Babylonian problem texts into modern algebra preserves the numbers and destroys the thought. His cut-and-paste reading treats the operations as physical rearrangements of a rectangle. Reading them that way is what makes the Plimpton 322 heading intelligible, and what makes takiltum a meaningful word rather than an unmotivated intermediate. He matters here for a reason beyond Mesopotamia: he is the standing warning that modernizing notation is an interpretive act, not a neutral one (S001, S002).
Christine Proust (living), at the CNRS in Paris, co-authored the 2011 review with Britton and Shnider. It established the current consensus reconstruction of Plimpton 322 and the current translation of its Column I heading (S003) [event E-2011-britton-proust-shnider-review]. Notice where her work sits in this chapter's argument. Her generation procedure is the one Mansfield and Wildberger extended in 2017 to build their reconstruction. Her paper's reading of the leading 1s is the one that establishes Column I as , which is the technical fact the trigonometric reading needs. The person whose work both sides depend on is not the person whose name went in the headlines (S003, S030).
Thomas Eric Peet (1882 to 1934) produced in 1923 the critical edition and translation of the Rhind papyrus that this chapter, and most English-language work on Egyptian mathematics, still runs on. What makes him worth naming is his temperament. He hedges his etymologies in public. He flags his own uncertainty about seked and ukha-thebet. He rules out a tempting interpretation of problem 60 that would have made his material more interesting, on the grounds that the text is corrupt. He notices that a scribe merged two problems by forgetting red ink and says so. A century of Egyptian mathematics scholarship is built on his restraint (S005).
Jöran Friberg (living), at Chalmers in Gothenburg, is the quiet load-bearer of the Plimpton 322 debate. He independently rediscovered the reciprocal-pair generation around 1980, and he read the tablet as a systematic classification of normalized right triangles. He was also the first to propose that the two lost columns held and , the short side and diagonal of a triangle with long side 1 (S002, S003). Both Robson's reading and Mansfield and Wildberger's build on parts of his work. He also supplied, through his 2007 publication of MS 3052, the trapezoid term indanum that turns up in the slope discussion (S003).
Three more who deserve a line each. Isaac Mendelsohn, whose 1943 catalog entry is quoted at the top of this chapter, was right about the genre and wrong about the content. That is a more interesting failure than it looks. Kenneth Voils (VOYLZ) worked out a reciprocal-pair analysis that Buck signaled in 1980 would appear in Historia Mathematica, and it never did: "Voils' work, although signaled by Buck [1980, 344] to appear in Historia Mathematica, never actually made it into print, so it is not always possible to disentangle his theory from Buck's (mis)interpretation of it" (S002, p. 188). Unpublished work still shapes a field, badly, through other people's summaries of it. And Donald Knuth (kuh-NOOTH, born 1938), better known for computer science, argued in a 1972 paper in a computing journal that Old Babylonian scribes used linear interpolation. His evidence was AO 6770 in the Louvre. That tablet poses an interest problem: how long does it take 1 gur of grain to double at 20 percent per annum? The scribe solves it by interpolating linearly between the three-year and four-year totals, then converting the remainder to months. Mansfield and Wildberger reproduce Neugebauer's transcription and note that Knuth's paper, "in a computing machinery journal has perhaps been missed by some" (S003, pp. 417 to 418). Where you publish determines who reads you.
Words worth knowing, and how to say them
Read these aloud in class. Half of the confusion in this material is people not being sure how to pronounce the evidence.
Places. Larsa (LAR-suh), the southern Iraqi city where Plimpton 322 was probably written, known in modern times as Senkereh (SEN-keh-reh). Sippar (SIP-ar), the central Iraqi city where Si.427 and BM 80209 come from. Uruk (OO-rook), the southern city whose proto-cuneiform accounts, from just before 3000 BCE, are the ancestors of base 60. Girsu (GEER-soo) and Ur, where the Ur III sexagesimal place-value evidence comes from. Nippur (nih-POOR), Eshnunna (esh-NOO-nuh) and Susa (SOO-suh), the other main sources of Old Babylonian mathematical tablets.
Languages and systems. Sumerian (soo-MEER-ee-un), the older language, written in cuneiform, whose signs are used as shorthand in later Akkadian (uh-KAY-dee-un) texts. Sexagesimal (sek-suh-JESS-ih-mul), meaning base 60.
Units and instruments. bēru (BAY-roo), the double hour, one twelfth of a day, written DANNA (DAN-nuh). UŠ (oosh), one thirtieth of a bēru, so four minutes of time, and later one degree of arc. NINDA (NIN-dah), one sixtieth of an UŠ. ziqpu (ZIK-poo), from a word meaning "culminating," the stars named for crossing the meridian, listed in MUL.APIN I iii 49 to I iv 9. seked (SEK-ed), the Egyptian run-per-cubit-of-rise. merkhet (MER-khet), the plumb-line sighting bar, used with the bay (BAY), the forked sighting stick. gnomon (NOH-mon), a shadow-casting rod.
Technical words on the tablets. Read down this list and notice how many of them mean two things at once. A Babylonian scribe did not distinguish a square from its side, or a disc from its edge, because for his purposes the word named the whole situation.
- takiltum (tah-KIL-tum), the disputed word in the Plimpton 322 Column I heading, from kullum, "to multiply lengths together into areas." Robson renders it "holding-square."
- ṣiliptum (tsih-LIP-tum), from ṣalāpum, "to strike through": the diagonal of a rectangle, and also the rectangle.
- mitḫartum (mith-HAR-tum), the reflexive of maḫārum, "to be equal and opposite": it means both "square" and "the side of a square."
- kippatum (kip-PAH-tum), from kapāpum, "to curve": both the disc and the circumference.
- ukullû (oo-KOOL-loo), literally reported as "fruit," used for reciprocal slope, run over rise, per cubit.
- tallum (TAL-lum), the diameter, common in circle problems.
- pirkum (PEER-kum), the radius, almost never mentioned.
- igi (EE-gee) and igibi (ee-GEE-bee), a reciprocal and its partner.
- MU.BI.IM, Sumerian "its name," the heading over the line-count column.
- ki (KI), "its place," written before each row number.
- íb-si8 (IB.SI8), "square-side," in the headings of Columns II and III.
Works and titles. MUL.APIN (mool-AH-pin), the Babylonian astronomical compendium. Enūma Anu Enlil (eh-NOO-muh AH-noo EN-lil), the great omen series. Anaphorikos (an-af-or-ih-KOSS), Hypsicles' treatise on rising times.
What the evidence does not support
Claim: "Plimpton 322 is the world's first trigonometric table." Historia Mathematica published the strongest version of this claim in 2017. Its own lead author withdrew it in Foundations of Science in 2021, in print, by name: "it is quite clear that these functions are not part of Mesopotamian mathematics. Any interpretation of Plimpton 322 as a table of trigonometric functions is rightly dismissed by Robson as anachronistic" (S004, pp. 997 to 998). The problem is not the numbers. Column I really is of the base angle, and the heading really does encode . The problem is that Old Babylonian geometry defined figures from the outside in, had no functional radius, had no unit of angle, and had no tabulated angles anywhere in its corpus (S002, pp. 182 to 183). A table of ratios with no angles attached is not a trigonometric table any more than the list 1, 2, 3, 4 is (S002, p. 183 n. 20). What survives of the 2017 claim is the weaker and defensible idea that the tablet relates to practical measurement of rectangles (S004, p. 998).
Claim: "The Babylonians divided the circle into 360 parts because the year has 360 days." Nothing I read supports that as stated. The sources show a division of the day into 360 UŠ, by way of 12 bēru of 30 UŠ each, in MUL.APIN before about 750 BCE (S007, p. 9). They show a separate 360-day schematic calendar used as a computing convenience, which nobody mistook for the real year (S010). They show UŠ becoming a unit of arc only with the uniform zodiac around 400 BCE (S016). And they show Neugebauer stating flatly that the 360-part circle "originated in Babylonian astronomy of the last centuries B.C. The sexagesimal number system as such is many centuries older and has nothing to do with astronomical concepts" (S006, notes to ch. I, p. 25). The schematic year is real and it is connected, since Steele traces the zodiac's twelvefold structure back to it. But "360 days therefore 360 degrees" compresses 1,500 years of development into a slogan, and it deletes the step where a time unit became an arc unit.
Claim: "The seked proves the Egyptians had trigonometry." A seked is a cotangent, so the ratio is one of the six trigonometric ratios. But there is no table, no angle, no function, no interpolation, and no general method beyond a fixed conversion of 7 palms to the cubit. RMP 56 to 59 convert between a slope ratio and a set of dimensions (S005, pp. 97 to 101). That is proportional reasoning with a unit conversion, which is what a builder needs and all a builder needs. And problem 60, the one place a tangent seems to appear, is textually corrupt. The scribe divided 30 by instead of the reverse, and Peet is explicit that "The problem affords no case for the belief that the batter was in some cases measured by the tangent instead of the cotangent of the base angle" (S005, p. 102). Peet also thinks the object in that problem is a cone rather than a pyramid.
Claim: "RMP 56 describes the Great Pyramid." It does not. RMP 56 gives a seked of palms, a face angle of about 54.25 degrees. The Great Pyramid's seked is palms, about 51.84 degrees (S003, S027, p. 414). Two and a half degrees apart, and a monument you can measure yourself settles it.
Claim: the named surveyor of Si.427. RESOLVED, and against the press. Press coverage in 2021 supplied the name Sin-bel-apli for the surveyor who wrote Si.427, and it spread widely. The full 2020 edition has now been read for this book. The name is on the tablet, but it belongs to the owner of the field, and the surveyor is not named at all (S483). The same edition states that the tablet is undated. An earlier version of this book withheld the name because no source in hand contained it; that was the right call on the evidence then available, and the evidence has since arrived.
Claim: "the rows of Plimpton 322 step down by one degree each." Approximately, and unevenly. The printed angle column of the trigonometric restoration steps down by amounts ranging from 0.46 to 1.89 degrees, with the biggest jump between rows 11 and 12 (S001, Figure 4, differences computed here). The observation goes back to a passing remark by Neugebauer and Sachs about the first and last rows (S002, p. 179). In retelling it has hardened into a regularity the tablet does not have.
Claim: "Neugebauer said Plimpton 322 was trigonometry." He did not. Robson: "Nowhere, however, in their concluding discussion of 'historical consequences' did they mention 'trigonometry' or 'angle', but kept their comments to the (equally dubious) 'purely number theoretical character' of Plimpton 322" (S002, p. 180). She thinks their actual claim is also wrong, but it is a different claim. Attributing the trigonometric reading to the first editors is a citation error that has been copied for decades.
Claim: "Babylonian astronomers used trigonometry to track Jupiter." No. They computed the area under a velocity-time graph, which is an integration idea. They also bisected trapezoids to find the halfway time, which is a subtler one (S017, S018). Both are remarkable and neither is trigonometry.
What I could not read, and what that costs
Three sources central to this chapter were closed to me, and every claim that depends on them is second-hand. Saying so is not a formality. It tells you exactly which sentences to check first if you are building on this.
Britton, Proust and Shnider 2011, "Plimpton 322: A review and a different perspective," Archive for History of Exact Sciences 65, 519 to 566 (S030). Paywalled. This is the paper that fixed the current consensus reconstruction of the tablet and the current translation of the Column I heading. It is load-bearing for the middle third of this chapter. Everything I have reported from it, including the translation "The takiltum of the diagonal (from) which 1 is subtracted and (that of) the width comes up" and the finding that the leading 1s were present, is quoted from inside Mansfield and Wildberger 2017 or Mansfield 2021, with their page citations passed through (S003, p. 398, citing Britton et al. 2011, pp. 524 and 526). I have not checked those quotations against the original.
Neugebauer and Sachs 1945, Mathematical Cuneiform Texts, American Oriental Series 29 (S031). No open copy located. This is the first edition of Plimpton 322 and the origin of the whole modern discussion. Every quotation of it in this chapter, including "we start out with almost half a square ... and gradually diminish the angle between l and d step by step, the lowest value being almost exactly 31 degrees," reaches me through Robson or Mansfield (S002, p. 179, quoting Neugebauer and Sachs 1945, p. 39).
Mansfield 2020, "Perpendicular lines and diagonal triples in Old Babylonian surveying," Journal of Cuneiform Studies 72, 87 to 99 (S483). Now read. This is the full edition of Si.427, and it closes three of the four things an earlier version of this chapter had to leave open: the reading of the text, the identity of the name on it (the field's owner, not the surveyor), and the surveyor's name (there is none). It also answers the date question in the negative: the tablet is undated, and its Old Babylonian placing is an inference from format, language and onomastics (S483). What remains unverified is its exact physical dimensions, which this chapter still estimates from the published photograph against the museum's scale bar (S034).
Four smaller gaps, for completeness. I read only the introduction and bibliography of Hunger and Steele's MUL.APIN, so my statements about that work's contents are their summary statements, not my reading of the tablets (S007). I did not read the Astronomical Diaries themselves, so the 652 BCE start date rests on the title of Sachs and Hunger's volume I. I read Robson 2001 from a mirror at uruk-warka.dk rather than from the publisher, because ScienceDirect and the Oxford repository were both blocked. The mirror reproduces the journal typesetting with pages 167 to 206 and the DOI, but anyone repeating this work should verify against the publisher copy. And I read Steele 2018 on the zodiac as a fetched summary rather than line by line, which is why the "around 400 BCE" date for the uniform zodiac carries slightly lower confidence than the rest of the 360-degree story (S016).
One more, of a different kind. I searched for a formal peer-reviewed rebuttal of Mansfield and Wildberger 2017 and did not find one. The published criticism I could read consists of Evelyn Lamb's Scientific American blog post and Mansfield's own 2021 revision (S014, S004). If a journal reply exists, I did not locate it.
Primary sources a class can look at directly
Every object in this chapter is photographed and online. Students should look at the actual thing before reading anybody's interpretation of it.
| Object | Holding institution and shelfmark | URL | Notes |
|---|---|---|---|
| Plimpton 322, composite photograph of obverse, four edges and reverse | Rare Book and Manuscript Library, Columbia University; photographed for CDLI; CDLI P254790; Columbia call number "Plimpton Cuneiform 322" | https://cdli.mpiwg-berlin.mpg.de/dl/photo/P254790.jpg | 2159 x 2886 px JPEG. The reverse carries the ink stamp "PLIMPTON LIBRARY" and a pencilled "322". CDLI states no license for its images, so rights are unknown; do not assume reuse is permitted (S033) |
| Plimpton 322, obverse | Rare Book & Manuscript Library, Columbia University Libraries; gift of George Arthur Plimpton, 1936 | https://exhibitions.library.columbia.edu/files/original/05b5e75547e2e2fa241ed1d31ff410cd.jpg | 2000 x 1780 px JPEG (S012) |
| Plimpton 322, obverse and reverse | Lent to ISAW, NYU by Columbia University | https://isaw.nyu.edu/exhibitions/before-pythagoras/items/plimpton-322/ | From the 2010 exhibition Before Pythagoras: The Culture of Old Babylonian Mathematics (S013) |
| Si.427, obverse, with museum scale bar | İstanbul Arkeoloji Müzeleri; Si.427 | https://media.springernature.com/lw1200/springer-static/image/art%3A10.1007%2Fs10699-021-09806-0/MediaObjects/10699_2021_9806_Fig2_HTML.png | Figure 2 of Mansfield 2021, CC BY 4.0; photograph courtesy of the museum (S034) |
| Rhind Mathematical Papyrus, first section | British Museum EA10057 | https://www.britishmuseum.org/collection/object/Y_EA10057 | 33 x 296 cm; dated 1550 BCE; purchased from David Bremner 1865 (S008) |
| Rhind Mathematical Papyrus, second section | British Museum EA10058 | https://www.britishmuseum.org/collection/object/Y_EA10058 | 32 x 198.5 cm; same acquisition history (S009) |
| MUL.APIN tablet 1 | British Museum 86378, registration 1899,0610.108 | https://www.britishmuseum.org/collection/object/W_1899-0610-108 | 86 x 60 x 16 mm; 1000 to 500 BCE; excavated in southern Iraq, acquired 1899 (S010) |
| Egyptian merkhet, bronze | Science Museum Group 1929-585 | https://coimages.sciencemuseumgroup.org.uk/349/937/large_smg00082665.jpg | 600 BCE; CC BY-NC-SA 4.0 (S021) |
| Egyptian altitude sundial, pine | Science Museum Group 1926-992 | https://coimages.sciencemuseumgroup.org.uk/272/914/large_b003209.jpg | 1000 to 800 BCE, made in Qus; CC BY-NC-SA 4.0 (S023) |
- Mesopotamia and Egypt had superb base-60 arithmetic, exact reciprocal pairs, and slope reckoned as a ratio (the Egyptian seked): run per unit of rise, never an angle.
- Plimpton 322 is real, exact, and argued over: five readings, from teacher's aid to the 2017 trigonometric claim, are laid out with what each must assume.
- Nothing in this world measures an angle, and no function of one exists anywhere in it. What it bequeathed instead: 60, 360, and the habit of tabulating.
Next: Greek astronomers, who did measure angles, build the first table of a function of one, and it is not the sine.
The 360 on your protractor and the 60s inside every minute and second of arc are this chapter's number system, still running. Your course opens with triangles measured by ratios, and the oldest mathematics behind it measured nothing else. It starts in Trigonometry 1.0
Egypt specified the slope of a pyramid face without any angle. What did the seked measure instead?
Show the answer▸
The horizontal run for one cubit of rise, counted in palms and fingers. A slope was a ratio in mixed units: a cotangent-shaped quantity centuries before anything like a cotangent, and the reason a "gentler" pyramid has a larger seked, not a smaller one.
Chapter 2
The Chord Table (Greek Astronomy, c. 300 BCE to 400 CE)
The people in this chapter
Faces where a face survives. Every name links to its full entry in Appendix A.
A column of numbers that still checks out
Open Ptolemy's Almagest (Mathematical Systematic Treatise) at Book I, chapter 11, and you see a column of numbers. The third row reads 1;34,15. Three rows on: 3;8,28. Keep going for eight pages and past 360 rows, and the last entry is 120;0,0.
That column is the oldest surviving trigonometric table on Earth, finished in Alexandria around 150 CE (S061). Scribes copied it by hand for seven hundred years. Printers set it in Venice in 1515, then in Greek at Basel in 1538. The numbers still check out. In sexagesimal, which counts in sixtieths rather than tenths, 3;8,28 means . The modern value of the same quantity is . Ptolemy is off by about one part in 25,000, which is smaller than the last digit he bothered to write down.
Here is the thing that trips up most students, and it needs saying before anything else. There is not a single sine in that table. Ptolemy (KLAW-dee-us TOL-uh-mee, c. 100 to c. 175 CE) had no sine, no cosine and no tangent, and neither did anyone else in the Greek world. Toomer, who produced the standard English translation, states it flatly in his introduction: "The sole trigonometrical function used by Ptolemy is the chord" (S061, Introduction p. 6). What Ptolemy had was the chord: the straight line joining the two ends of an arc, a stretch of the circle's rim. One function, tabulated once, used for everything.
Get that one idea and the whole era opens up. Miss it, and every page of Greek astronomy looks like a sine table with the labels scrambled.
Ptolemy's chord table was finished around 150 CE and then copied by hand, over and over, for more than a thousand years before printing. Open a printed edition today and recompute a row from scratch. How well does the second-century arithmetic hold up?
I have a guess▸
It still checks out. The table is the oldest surviving trigonometric table on Earth, and its entries reproduce from the mathematics at the precision Ptolemy printed (S061). This book re-derives rows of it in Appendix E and in the tables of this chapter; the misprints of individual copies are the exception, not the rule.
If you guessed that hand copying must have corrupted the numbers beyond use, that is the reasonable guess, and the survival is the story: the table format, a value plus its sixtieths, is unusually good at exposing a copyist's slip to anyone who checks a difference column.
The problem: predicting an eclipse is a geometry problem on a sphere
Nobody built a chord table because triangles are interesting. They built it because they had a sky to predict.
An eclipse is a scheduling problem in three dimensions. The Moon has to be in the right place at the right time, seen from one city on a spinning globe. And "the right place" is a point on a sphere. So is every related question a working astronomer faced. Where on the horizon does this star rise? How many hours of daylight does this city get on this date? Where will the Moon be nine days from now? When does the Sun cross the equator?
Greek astronomers had geometry for spheres, and had had it for centuries. They lacked one thing. They had no way to turn an angle into a length without drawing the figure and measuring it with a ruler. Drawing works once. It does not work when you need the answer for 360 different angles, to four significant figures, tonight.
A chord table is that converter. Ptolemy says as much in the opening line of I.10: he is setting the table out "for the user's convenience" (S061, I.10). Notice who the user is. Not a geometer proving things. An astronomer with a calculation to finish.
Calendars pushed in the same direction. To say when the spring equinox falls you need the Sun's position to better than a day, which means numbers, not diagrams. Hypsicles' little book on how long each sign of the zodiac takes to rise, which we will come to, is a calendar problem wearing a geometry costume. So is Eratosthenes measuring the Earth. The Greeks did not invent trigonometry and then look for uses. They had the uses first, for two hundred years, and built the tool late.
What a chord is, and why the halving is there
Draw a circle. Pick an arc of degrees on it. Join the two endpoints of that arc with a straight line. That straight line is the chord of , written .
The length depends on the size of the circle, so you have to fix a radius. Ptolemy fixes the diameter at 120 parts, which makes the radius 60. In modern terms:
The halving is where students get lost, so here is where it comes from. A chord of 60 degrees corresponds to a sine of 30 degrees, because the chord is keyed to twice the angle you care about. Drop a perpendicular from the center of the circle to the chord. It cuts the chord in half and it cuts the arc in half. You now have a right triangle: hypotenuse , angle at the center, and opposite side equal to half the chord. So half the chord is , and the whole chord is twice that. That is the entire content of the formula.
Ptolemy built a notation around the same doubling. He measures angles in units "of which 2 right angles are 360". Toomer's translation calls these demi degrees and marks them with a double circle, keeping the ordinary degree sign for the units in which a right angle is 90. An angle of ordinary degrees is demi degrees. Toomer explains why Ptolemy bothers: "This enables him to switch smoothly from the triangle to the circle (and hence to the chord table)" (S061, Introduction p. 8). An inscribed angle is half the central angle standing on the same arc. So an angle in demi degrees is the number of ordinary degrees in the arc it subtends, that is, the arc it stretches across. Ptolemy never has to write "now multiply by two". The units do it for him.
Two warnings, and both are load-bearing.
First: is a modern restatement, not a Greek formula. No Greek text states it. No Greek text can state it, because no Greek text has a sine. Duke uses this modern shorthand himself when he explains Neugebauer's reasoning, writing it as "" (S063). And that is what it is: a translation device, so we can check ancient arithmetic on a calculator. Toomer makes the same point from the other side. Ptolemy's choice of means, he says, that the table "in some ways resembles a sine table with " (S061, note 61 to I.10). Resembles. It is not one. Convert a Ptolemaic chord to a modern sine and you are doing something Ptolemy would have recognized as a ratio, but not as an operation. The object you are converting to did not exist for him.
Second: Ptolemy has no cosine either, and it shows in how he works. Faced with a triangle that is not right-angled, a modern student reaches for the law of cosines. Toomer: "In modern trigonometry we would use the cosine formula. Ptolemy has no equivalent, so he drops the perpendicular HK, thus transforming the problem into one of solving only right triangles, which is his standard procedure" (S061, Introduction p. 8). And with no tangent, "he has to use 'Pythagoras' theorem' to find the hypotenuse of the right triangle in question" (S061, Introduction p. 9). Every problem in the Almagest gets chopped into right triangles. Each one sits in a circle whose diameter is that triangle's hypotenuse, taken as 120 parts. Then the table does the rest.
There is one interesting exception. Toomer notes that Ptolemy "knows the equivalent of the sine formula, namely that in the general triangle the sides are proportional to the chords of the doubles of the opposite angles, but uses it surprisingly infrequently. An example is IX 10" (S061, note 16). The law of sines, in chord clothing, was available to him. He mostly did not want it.
The words the Greeks used
Ptolemy never uses the Greek word khordē (χορδή). This is not a quibble about vocabulary. It tells you what he thought he was doing.
Look up χορδή in the standard Greek lexicon, Liddell, Scott and Jones. Here is the entry, in order: I, "guts, tripe"; II, "that which is made from guts", subdivided into 1, "string of gut", especially "string of a lyre or harp", with sense b "musical note", and 2, "sausage or black-pudding". That is the whole entry. LSJ records no geometrical sense at all (S089). The French lexicon of Bailly agrees, giving only "corde à boyau, corde d'un instrument de musique" and "andouille, boudin" (S089). The word's family tree runs back through Proto-Indo-European to Sanskrit hirā "vein", Latin haruspex (the priest who read entrails), and Old Norse gǫrn "entrails" (S089). Earliest citation for the "string" sense is Odyssey 21.407, Odysseus stringing his bow.
So the English word "chord" descends from Greek χορδή by way of Latin chorda, meaning a gut string on an instrument. The geometrical application is not Greek usage. Saying "the Greek word for chord was khordē" is a sentence about the history of English, dressed up as a sentence about Greek mathematics.
What Ptolemy wrote was eutheia en kuklō, "the straight line in the circle". His two chapter headings in Book I, as they stand in Heiberg's Greek critical edition, are:
- ι΄ Περὶ τῆς πηλικότητος τῶν ἐν τῷ κύκλῳ εὐθειῶν, "On the size of the straight lines in the circle" (chapter 10)
- ια΄ Κανόνιον τῶν ἐν κύκλῳ εὐθειῶν, "Table of the straight lines in the circle" (chapter 11) (S062)
Toomer flags the same thing in a translator's note: "'chords': literally 'straight lines in a circle'" (S061, note 51 to I.9). Every time you read the word "chord" in an English Almagest, a translator put it there.
The verb matters too. In the body of I.10 the lines subtend the arcs: τὰς ὑπὸ τὰς καθ' ἡμιμοίριον παραυξήσεις τῶν περιφερειῶν ὑποτεινομένας εὐθείας, "the straight lines subtending the arcs at half-degree increments" (S062, H31). The participle is from hypoteinousa (hy-po-TAY-noo-sa), built from ὑπό "under" plus τείνω "stretch": literally "stretching under" (S062). It is the same word that gives us "hypotenuse", the side stretching under the right angle. The Greek picture is physical. A line is pulled taut underneath a curve.
I checked this the crude way. A string search for the letters "χορδ" across the full digitized text of Heiberg's Syntaxis mathematica Pars I, covering Almagest Books I to VI, returns zero occurrences (S062). That search runs on scanned Greek type, and optical character recognition of Greek is imperfect. So this is strong evidence rather than proof, and I have flagged it as such below. But it matches the lexicon, and it covers the six books in which the chord table is built and used hardest.
One more piece of vocabulary, for later. When Menelaus of Alexandria (men-uh-LAY-us, first century CE, disputed, see below) defined the spherical triangle, he did not call it a triangle. Pappus of Alexandria (PAP-uss, active c. 320 CE) reports that Menelaus in the Sphaerica calls the figure a τρίπλευρον, tripleuron (TRIP-lyoo-ron), a "three-side", from τρι- "three" and πλευρά "side" (S068, p. 263). He kept the ordinary word τρίγωνον, trigōnon, "three-angle", for the flat kind. A mathematician who invents a new word for an object is telling you the object is new.
Before the table: the toolkit that already existed
The chord table arrives late. By the time anyone tabulates anything, Greek mathematicians have been solving angle problems for two centuries. Their tools are not trigonometry and do not want to be. Six of those people belong in this chapter, and most of them get one line in a textbook.
Euclid, and the law of cosines without a cosine
Euclid (YOO-klid, active c. 300 BCE) has no angle measure in the Elements and no trigonometric function of any kind. He still states the law of cosines. Twice.
Elements II.12, verbatim: "In obtuse-angled triangles the square on the side opposite the obtuse angle is greater than the sum of the squares on the sides containing the obtuse angle by twice the rectangle contained by one of the sides about the obtuse angle, namely that on which the perpendicular falls, and the straight line cut off outside by the perpendicular towards the obtuse angle" (S076).
Elements II.13, verbatim: "In acute-angled triangles the square on the side opposite the acute angle is less than the sum of the squares on the sides containing the acute angle by twice the rectangle contained by one of the sides about the acute angle, namely that on which the perpendicular falls, and the straight line cut off within by the perpendicular towards the acute angle" (S077).
Read them side by side. The two propositions are identical except for three swaps: obtuse for acute, greater for less, outside for within. In modern symbols II.12 says , where is the bit of the extended base cut off by the perpendicular. David Joyce's commentary makes the connection explicit. The proposition "parallels the law of cosines", because " equals , the cosine of an obtuse angle being negative", and "this proposition and the next may be considered geometric versions of the law of cosines" (S076). Joyce is careful to add that negative numbers and trigonometry both arrived long after Euclid. So this is a modern reading of an ancient text, not a claim about what Euclid thought he was writing. (Joyce's page for II.13 carries no separate commentary of its own; the law-of-cosines remark is on the II.12 page and covers both propositions.)
Two propositions, one obtuse, one acute, and the only difference between them is a sign. That sign is the cosine changing sign at 90 degrees. Euclid could not write it, so he wrote the proposition twice.
Aristarchus, bounding the Sun
Aristarchus of Samos (a-ris-TAR-kuss, c. 310 to c. 230 BCE) wrote On the Sizes and Distances of the Sun and Moon. It contains, in Heath's judgment, the earliest surviving reasoning that does trigonometry's job without trigonometry's machinery.
The treatise opens with six hypotheses. Two of them drive everything:
- Hypothesis 4: "That, when the moon appears to us halved, its distance from the sun is then less than a quadrant by one-thirtieth of a quadrant." Heath glosses this: less than 90 degrees by of 90 degrees, that is by 3 degrees, so the angle is 87 degrees (S069).
- Hypothesis 6: "That the moon subtends one fifteenth part of a sign of the zodiac." A sign is 30 degrees, so this makes the Moon 2 degrees across (S069).
From Hypothesis 4, Proposition 1: "The distance of the sun from the earth is greater than eighteen times, but less than twenty times, the distance of the moon (from the earth); this follows from the hypothesis about the halved moon" (S069).
The geometry is clean. At half moon, the Earth, Moon and Sun form a right angle at the Moon. Say the angle at the Earth between Moon and Sun is 87 degrees. Then the ratio of distances is , which a calculator gives as 19.1, sitting between 18 and 20. Aristarchus has no cosine. He gets the bounds by inscribing and circumscribing polygons and comparing ratios of lines with ratios of angles.
The observation is badly wrong, and the honest thing is to say so. The true elongation of the half moon is about 89.85 degrees. That puts the Sun roughly 390 times as far away as the Moon, not 19. The angle is nearly impossible to observe, because judging the instant of half phase by eye is hopeless. Hypothesis 6 is wrong too, and Archimedes says so. Heath: "Another passage of the Sand-reckoner of Archimedes states that 'Aristarchus discovered that the sun's apparent size is about one 720th part of the zodiac circle.' This, again, is a valuable contribution to our knowledge of Aristarchus, for in the treatise On the sizes and distances of the sun and moon he makes the apparent diameter not 1/720th of the zodiac circle, or 1/2°, but one-fifteenth part of a sign, that is to say 2°, which is a gross over-estimate" (S069, pp. 311 to 312). So on Archimedes' testimony, Aristarchus had the right figure of half a degree somewhere else. In his own book he used a value four times too big. Nobody has explained that satisfactorily.
The method survives the bad inputs, and the method is what matters here. Heath calls the key section "Trigonometrical equivalents". He then pins down what Aristarchus is helping himself to: "there lie at the root of Aristarchus's reasoning certain propositions assumed without proof, presumably because they were generally known to mathematicians of the day. The most general of these propositions are the equivalent of the statements that: If is what we call the circular measure of an angle, and is less than , then (1) The ratio decreases as increases from 0 to , but (2) the ratio increases as increases from 0 to " (S069, p. 333). Heath credits Paul Tannery with working out the trigonometrical equivalents of the individual propositions.
Look at what that means. Aristarchus assumes, as common knowledge among working mathematicians around 260 BCE, the monotonicity of . That is the inequality Ptolemy will need four hundred years later to nail down the chord of one degree. Ptolemy will prove it rather than assume it. Inside Proposition 7 the machinery is bare: "Now, since GE has to EH a ratio greater than that which the angle GBE has to the angle DBE", with the angles given as 15 sixtieths and 2 sixtieths of a right angle (S069).
Duke converts the results into modern language: "By considering circumscribed and inscribed triangles and assuming a bound on Aristarchus effectively establishes bounds on as , and, although he does not mention it, this also establishes bounds on as ." And: "Later, in Propositions 11 and 12 Aristarchus proves using similar methods that and , always understanding, of course, that what we write as sine and cosine was to Aristarchus a ratio of sides in a right triangle" (S064).
Check those against a calculator. , and the bounds are and . , and the bounds are and . , and . Every bound holds. This is trigonometry a century and a half before anyone tabulated anything, and it is the honest place to start the story.
Archimedes, and the 96-gon
Archimedes of Syracuse (ar-kih-MEE-deez, c. 287 to 212 BCE) is in this chapter for two reasons: a number and a technique. Heath fixes his dates from the tradition that he was 75 when Roman soldiers killed him at the fall of Syracuse in 212 BCE. That tradition, he says, "enables us to fix his date at about 287-212 B.C." (S068).
The number is . In Measurement of a Circle, Proposition 3, Heath's translation: "The ratio of the circumference of any circle to its diameter is less than but greater than " (S070). In decimals, , and the true value 3.1415927 sits between them.
The technique is repeated bisection, and it is the technique a chord table needs. Archimedes starts from a regular hexagon and halves the angle four times. Heath's text: "the angle BAG which is the result of the fourth bisection of the angle BAC, or of one-third of a right angle, is equal to one-fortyeighth of a right angle... Therefore BG is a side of a regular inscribed polygon of 96 sides" (S070). Six sides, halved four times, gives .
Now the point that gets missed. The side of an inscribed regular -gon is a chord, of the arc . Archimedes' hexagon-to-96-gon chain computes, in order, the chords of 60, 30, 15, 7.5 and 3.75 degrees, in a circle whose radius he keeps track of throughout. Duke spells the consequence out: "Archimedes begins with a hexagon and an estimate for and by successive halving implicitly computes about two-thirds of the entries needed to populate a chord table with spacing. It would be straightforward to generate the remaining entries by starting with a square and an estimate of " (S063, n. 3). And elsewhere: "using Archimedes' method, and in many cases the very numbers that appear in his text, anyone could have assembled the table in increments of that was used in India and might have been used by Hipparchus. The two steps needed to go beyond Archimedes are (a) a normalization convention, and (b) an interpolation scheme" (S064).
A normalization convention means picking a radius and sticking to it. An interpolation scheme means filling in the gaps between computed entries. Both are bookkeeping. The mathematics was finished by 240 BCE.
Heath adds a warning about the text that is easy to skip and should not be. The Measurement of a Circle we have is not the whole book. Heath reports Tannery's suggestion that Archimedes and Apollonius may already have compiled a table of chords. In doing so he refers to "the work of which we possess only a fragment in the Measurement of a Circle" (S068, p. 253). Whatever else Archimedes did with this machinery, we do not have it.
That Tannery suggestion is worth quoting in full, because it shows how old this argument is. Heath, writing in 1921: "Tannery indeed suggested that not only Apollonius but Archimedes before him may have compiled a 'table of chords', or at least shown the way to such a compilation, Archimedes in the work of which we possess only a fragment in the Measurement of a Circle, and Apollonius in the ὠκυτόκιον, where he gave an approximation to the value of closer than that obtained by Archimedes; Tannery compares the Indian Table of Sines in the Sūrya-Siddhānta, where the angles go by 24ths of a right angle (1/24th = 3° 45', 2/24ths = 7° 30', &c.), as possibly showing Greek influence" (S068, p. 253). Those numbers, 3 degrees 45 minutes and 7 degrees 30 minutes, are the step sizes Neugebauer, Toomer and Duke will build the R = 3438 argument on, fifty and eighty years later. The observation is a century old.
Eratosthenes, and the angle in the shadow
Eratosthenes of Cyrene (eh-ruh-TOS-thuh-neez, c. 276 to c. 194 BCE) measured the Earth with a stick. His measurement is a central-angle argument with no trigonometry in it, which is why it belongs here.
Heath's account: he observed "(1) that at Syene, at noon, at the summer solstice, the sun cast no shadow from an upright gnomon... while (2) at the same moment the gnomon fixed upright at Alexandria (taken to be on the same meridian with Syene) cast a shadow corresponding to an angle between the gnomon and the sun's rays of 1/50th of a complete circle or four right angles... the distance from S to A was known by measurement to be 5,000 stades; it followed that the circumference of the earth was 250,000 stades" (S068, pp. 106 to 107). Heron's Dioptra is Heath's authority that the measurement appeared in a separate lost work, On the Measurement of the Earth.
The arithmetic is one line:
No table, no function, no chord. Only one fact: arcs on a circle are proportional to the angles they subtend at the center. That is Euclid VI.33, and every one of these people had it.
The ancient sources already disagreed about the answer. Heath: "This is the figure given by Cleomedes, but Theon of Smyrna and Strabo both give it as 252,000 stades. The reason of the discrepancy is not known" (S068, p. 107). Heath's own guess is that somebody adjusted 250,000 upward to 252,000. That is because 252,000 is divisible by 60 and gives a round 700 stades per degree, which is handy if you are building tables. MacTutor prints 250,000 (S084). Two round numbers, two ancient authorities each, and the standard modern history of Greek mathematics says openly that nobody knows why they differ.
Hypsicles, and the day the circle got 360 parts
Hypsicles of Alexandria (hip-SIK-leez, c. 190 to c. 120 BCE, weakly grounded) wrote the Anaphorikos (On Ascensions), on how long each sign of the zodiac takes to rise above the horizon. It is the earliest extant Greek work, Heath says, "in which we find the division of the ecliptic circle into 360 'parts' or degrees" (S068, p. 214).
His dates are shaky and should be presented that way. MacTutor gives c. 190 to c. 120 BCE; Heath declines to commit and only places him after Apollonius (S068, S084).
Here is the passage, in Heath's translation, and it is worth reading slowly:
"The circle of the zodiac having been divided into 360 equal circumferences (arcs), let each of the latter be called a degree in space (μοῖρα τοπική, 'local' or 'spatial part'). And similarly, supposing that the time in which the zodiac circle returns to any position it has left is divided into 360 equal times, let each of these be called a degree in time (μοῖρα χρονική)" (S068, p. 214).
Moira (MOY-ra) means a portion or a share. A "spatial moira" is our degree of arc. A "time moira" is one 360th of a day, which is four minutes. Note that Hypsicles needs both, because his problem is a conversion problem: how much sky rises in how much time.
Heath catches something in the grammar. "From the word καλείσθω ('let it be called') we may perhaps infer that the terms were new in Greece" (S068, p. 214). A writer who stops to name a thing is naming it for the first time. That single imperative verb is most of the evidence for when Greek astronomy adopted the 360-part circle.
Where did 360 come from? Heath rehearses four competing explanations. First, that it comes from the 365 days of the year, rounded. Second, that it comes from the inscribed regular hexagon (six arcs), with each arc divided sexagesimally, that is by 60. Third, that it is a purely arithmetical Babylonian construction. Fourth, a variant of that involving sundials (S068). He endorses Tannery's answer: "It is certain that both the division of the ecliptic into 360 degrees and that of the nychthemeron into 360 time-degrees were adopted by the Greeks from Babylon" (S068). "Nychthemeron" is the night-and-day cycle taken as one unit. Heath says "certain". The question is in fact still open, and a student should be told that a 1921 "certain" is a strong scholarly opinion, not a proof.
Theodosius, and a sphere book with no trigonometry in it
Theodosius of Bithynia (thee-oh-DOH-see-us, dates disputed) wrote a Sphaerica in three books. It is the best example in this chapter of what spherical geometry looks like before spherical trigonometry exists.
Heath puts a section heading on it: "No actual trigonometry in Theodosius" (S068, p. 250). His text is careful about the boundary: "It is perhaps hardly correct to say that spherical triangles are nowhere referred to in Theodosius, for in III. 3 the congruence-theorem for spherical triangles corresponding to Eucl. I. 4 is practically proved; but there is nothing in the book that can be called trigonometrical. The nearest approach is in III. 11, 12, where ratios between certain straight lines are compared with ratios between arcs" (S068, p. 250).
That last clause is the diagnostic. Comparing a ratio of lines with a ratio of arcs is what Aristarchus did. It is one step short of a function. You can bound a quantity that way. You cannot tabulate it.
On his date, Heath: "our Theodosius was earlier than Menelaus (fl. about A.D. 100), who quotes him by name... Vitruvius mentions a Theodosius who invented a sundial 'for any climate'; and Strabo, in speaking of certain Bithynians distinguished in their particular sciences, refers to 'Hipparchus, Theodosius and his sons, mathematicians'. We conclude that our Theodosius was of Bithynia and not later in date than Vitruvius (say 20 B.C.)", adding that "the character of his Sphaerica suggests a date even earlier rather than later" (S068, pp. 245 to 246). He is often called "Theodosius of Tripolis", which Heath traces to a confusion in the Byzantine encyclopedia Suidas. Bithynia is right; Tripolis is a copying error that stuck for a thousand years.
The Strabo passage that anchors him is itself textually unstable, and this is a good thing for students to see. The Greek of Strabo 12.4.9 in Meineke's Teubner edition, as served by Perseus, reads: ἄνδρες δ' ἀξιόλογοι κατὰ παιδείαν γεγόνασιν ἐν τῇ Βιθυνίᾳ Ξενοκράτης τε ὁ φιλόσοφος καὶ Διονύσιος ὁ διαλεκτικὸς καὶ Ἵππαρχος καὶ Θεοδόσιος καὶ οἱ παῖδες αὐτοῦ μαθηματικοί, that is, "Hipparchus". The Loeb English translation by H. L. Jones at the same passage reads: "Xenocrates the philosopher, Dionysius the dialectician, Hippocrates, Theodosius and his sons the mathematicians, and also Cleochares the rhetorician of Myrleia, and Asclepiades the physician of Prusa" (S079). Hipparchus in the Greek, Hippocrates in the English. Heath cites it as "Hipparchus" (S068). I have not resolved which reading modern editors prefer, and I am not going to pretend otherwise. If you want to use Strabo to prove that Hipparchus came from Bithynia, you need to settle that question first.
Hipparchus: what is attested, and what is reconstructed
Hipparchus of Nicaea (hip-PAR-kuss, c. 190 to c. 120 BCE) is routinely called the founder of trigonometry. Before you accept that, look at what survives.
The observations. Ptolemy quotes dated observations by Hipparchus throughout the Almagest, and Toomer's index lets you read off the range. They start in astronomical year -146, that is 147 BCE, with an observation of -146 September 26/27 at midnight. They run through to -126 July 7. Toomer's index also lists a Mercury observation of -127 August 5 and lunar observations of -126 May 2 and -126 July 7 (S061). Twenty years of dated work, all of it known only because Ptolemy copied it out. That is the firm anchor for his life, and it is the only one.
The dates. No source I read gives independent evidence for "born about 190 BCE" or "died about 120 BCE". MacTutor prints "born 190 BC, died 120 BC" without an argument (S084). Treat 190 and 120 as conventional round figures placed either side of the attested observations.
The places. Nicaea in Bithynia, modern İznik in Türkiye, on the Strabo passage discussed above, with its textual problem. He worked mostly on Rhodes and probably visited Alexandria (S061, S068, S084).
The surviving work. One. MacTutor: "Only one work by Hipparchus has survived, namely Commentary on Aratus and Eudoxus" (S084). It is a three-book critique of the astronomy in a popular poem. Karl Manitius edited it in Greek with a facing German translation for Teubner in 1894, and the scan is free (S078). It is not a trigonometry text. It contains no table.
The chord table. Gone. Every copy.
So what is the evidence that he made one? A single sentence, at second hand, written five hundred years after his death. Theon of Alexandria (THEE-on, active c. 370 CE), commenting on Ptolemy's Syntaxis, mentions it in passing. Heath quotes him: "(1) Theon of Alexandria says on the Syntaxis of Ptolemy, à propos of Ptolemy's Table of Chords in a circle (equivalent to sines), that Hipparchus, too, wrote a treatise in twelve books on straight lines (i.e. chords) in a circle, while another in six books was written by Menelaus" (S068, p. 257).
That is it. That is the entire direct evidence that Hipparchus wrote on chords: twelve books, subject stated, contents lost. It also happens to be the entire direct evidence that Menelaus wrote six books on the same subject.
Heath, who had read everything, is guarded about what follows from this. His sentence: "Even if he did not invent it, Hipparchus is the first person of whose systematic use of trigonometry we have documentary evidence" (S068, p. 257). And on the shape of the lost table: "We have no details of Hipparchus's Table of Chords sufficient to enable us to compare it with Ptolemy's, which goes by half-degrees, beginning with angles of 1/2°, 1°, 1 1/2°, and so on" (S068, p. 259).
Compare the reference-work version. MacTutor: "it seems highly probable that Hipparchus was the first to construct a table of chords and thus provide a general solution for trigonometrical problems" (S084). Both sentences are defensible. Only one is a report of evidence. The other is an inference with a probability word in front of it. By the time it reaches a textbook, the probability word has usually been deleted.
There is one further trace of pre-Ptolemaic chord work, and it is concrete enough to be worth teaching. Heath: "Heron in his Metrica says that 'it is proved in the books about chords in a circle' that, if and are the sides of a regular enneagon (9-sided figure) and hendecagon (11-sided figure) inscribed in a circle of diameter , then (1) , (2) very nearly, which means that was taken as equal to 0.3333... and was made equal to 0.28" (S068, p. 259, citing Heron Metrica I 22, 24). Heron is quoting a treatise, by title, that is not Ptolemy's, on chords in a circle, and giving two of its results. The true values are against Heron's 0.3333, and against 0.28. Both are a little low, both are usable. Somebody before Ptolemy had a book of these.
The R = 3438 argument: four positions, two people, fifty years
This is the best place in the chapter to watch professional historians disagree in print, and one of them disagree with himself.
Ptolemy uses a circle of radius 60. Hipparchus, on the standard reconstruction, used 3438. That number does not come from any table. It is reverse-engineered from two odd ratios.
In Almagest IV.11 Ptolemy reports the results Hipparchus obtained for the Moon's motion from two separate sets of three eclipses, called eclipse trios. The ratios are from one trio and from the other (S063). Those denominators are not round, and they are nowhere near 60 or 120.
Position one: Toomer, 1973 (or 1974). G. J. Toomer, later the translator of the standard English Almagest, proposed that Hipparchus worked in a circle whose circumference was 21,600 units. That is one unit per arcminute of the whole circle, since . Duke states the origin: "In 1973 Toomer suggested that the basic scale of ancient (ca. 500 AD) Indian sine tables, a circle of radius , was a remnant of even earlier Greek trigonometry, and that evidence for Greek use of that scale might be found in the otherwise curious ratios, and , for the lunar anomaly that Ptolemy attributes to Hipparchus (Toomer 1973). Toomer was building upon an earlier suggestion by Neugebauer that ancient Indian sine tables compiled at angular intervals of might have been derived from early Greek chord tables compiled at angular intervals of (Neugebauer 1972), using the fundamental relation " (S063).
Why 3438? Divide 21,600 by :
Duke gives the rationale in those terms: "the rationale for the value is that it is the radius of a circle whose circumference is " (S064). There is a quirk worth showing students, because it is the kind of thing that makes people suspect a mistake when there is none. Duke, in a footnote: "More precisely, chord tables are in fact based on a circle of a given diameter, and for a circle of circumference 21,600 the diameter is, to the nearest integer, 6875, while the radius is, again to the nearest integer, 3438. The apparent discrepancy, that , is simply an accident of rounding" (S063, n. 2). Twice 3438 is 6876, not 6875. Both are correct roundings of and . Rounding does not commute with doubling.
As a teaching aside, not as an ancient rationale: 3438 is also the number of arcminutes in one radian, rounded to the nearest integer. One radian is arcminutes. That identity is a modern way of putting it. The ancient rationale, as reported, is the circumference convention.
Position two: Toomer, 1984, against himself. In a footnote to his own translation of Almagest IV.11, Toomer walked the hypothesis back. He had followed a previous editor's emendation of a corrupted time interval. That corruption fed straight into his 1973 recomputation. His own words:
"I carelessly followed his interpretation and emendation in Toomer[2], in which I used Hipparchus' intervals to recompute the ratios for the eccentric and epicyclic models... Now, however, using the correct time interval of hours for II-III, I find much better agreement with the above ratio, as I shall show in detail elsewhere... These calculations not only vindicate Hipparchus' computational abilities, but cast doubt on my claim that he was operating with a chord table with base R = 3438" (S061, note to IV 11 at H347).
Read that again. The man who proposed R = 3438 printed a retraction of it in a footnote, inside the standard scholarly edition of the primary source. Every specialist would see it there. Almost no textbook writer would.
Position three: Duke, 2005, rescuing it from the other trio. Dennis Duke of Florida State University went back to the same material for Centaurus in 2005. His summary of what Toomer's correction did: "Using the longer (correct) interval, Toomer found that the ratio attributed to Hipparchus by Ptolemy was essentially correct, and so Hipparchus in fact did the geometry correctly, but the resulting numbers in the ratio no longer supported the hypothesis that Hipparchus was using a chord table based on a circle of radius 3438" (S063).
Toomer had worked the second trio. Duke worked the first, which he calls Trio A: "Toomer analyzed this trio more or less correctly, and found , clearly suggesting that (a) his hypothesis that Hipparchus had used a chord table of radius was correct". Duke's own verdict: "The conclusion appears unavoidable and firm: the numbers 3144 and 3438 are unambiguously linked" (S063).
Then he says something unusually honest. The thread is thin: "it is entirely fortuitous that the evidence revealing the 3438 base of his chord table was revealed in the reported number 3144 of the Trio A ratio" (S063). The whole reconstruction survives because one ancient number happened not to get simplified before Ptolemy copied it.
Position four: Duke, elsewhere, undercutting the framing entirely. In The Very Early History of Trigonometry, the same author raises the possibility that Hipparchus was not working with chords at all. "We cannot, however, be sure whether Hipparchus used the same chord construct as Ptolemy, or perhaps just gave the ratio of side lengths corresponding to a set of angles." And his closing sentence goes further: "Coupled with the fact that the sin and not the chord is used also in the Indian texts, this suggests that the chord was introduced later rather than sooner, and certainly offers no encouragement to anyone claiming that Hipparchus used chords or that the sine was invented in India as an 'improvement' over the chord" (S064).
Four positions, from two scholars, one of whom argued both sides. That is what an open question looks like, and it teaches more than any settled result in this chapter.
A bibliographic curiosity worth flagging. Toomer's paper is dated inconsistently in the literature. His own bibliography in Ptolemy's Almagest prints: "Toomer [2]: G. J. Toomer, 'The Chord Table of Hipparchus and the Early History of Greek Trigonometry'. Centaurus 18, 1973, 6-28" (S061). Duke cites it as 1973 in both papers (S063, S064). The publisher's own Crossref record gives an issued date of March 1974, volume 18, pages 6 to 28, DOI 10.1111/j.1600-0498.1974.tb00205.x (S088). Cite it as 1974 if you follow the publisher, but expect 1973 throughout the older literature. Do not assume two different papers when you meet both dates. I was not able to read the paper itself. The publisher's site returned HTTP 403. So everything reported here about Toomer's 1973 argument comes at second hand, from his own later footnotes and from Duke (S087).
The 7.5 degree step is inference too, and from a different direction. Neugebauer noticed that Indian sine tables step by 3.75 degrees, which in chord terms is 7.5 degrees, and proposed a Greek ancestor (S063). Duke prints a reconstructed 25-row table on that basis, and labels it carefully: "a possible replica of Hipparchus' table", not a reconstruction of an attested one (S064). Duke's own description of the underlying claim is equally hedged. He writes: "Hipparchus' working set of tools included tables with 23 (non-trivial) entries of side ratios in angular increments of , corresponding to chords in increments of " (S064). Note "side ratios", not chords, even here.
No ancient source states a step size for Hipparchus's table. None. The number 7.5 comes from India, six hundred years later, via a nineteenth and twentieth century argument about influence.
Here is Duke's reconstruction in full, so you can see what is being claimed and check its internal consistency. The one number that anchors it: at 60 degrees the chord of a circle equals its radius. So a table with must read 3438 at 60 degrees, and this one does.
Try the second-easiest check and it does not come out. The chord of a semicircle is the diameter, so the last row should read , and it reads 6,875. Every other row in the table is within one unit of , and so is this one, but it is the row where the answer is exact and no rounding is involved. Duke's paper could not be obtained for this book, so whether the 6,875 is his or a transcription slip on the way here is not established. Treat it as a reminder that a reconstruction is an argument, not a document.
| Arc | Chord |
|---|---|
| 0 | 0 |
| 450 | |
| 15 | 897 |
| 1,341 | |
| 30 | 1,780 |
| 2,210 | |
| 45 | 2,631 |
| 3,041 | |
| 60 | 3,438 |
| 3,820 | |
| 75 | 4,186 |
| 4,533 | |
| 90 | 4,862 |
| 5,169 | |
| 105 | 5,455 |
| 5,717 | |
| 120 | 5,954 |
| 6,166 | |
| 135 | 6,352 |
| 6,511 | |
| 150 | 6,641 |
| 6,743 | |
| 165 | 6,817 |
| 6,861 | |
| 180 | 6,875 |
Erased, overwritten, and read again in 2022
In 2022 a team announced that they had found part of Hipparchus's lost star catalog under a layer of scraped-off ink.
The object is the Codex Climaci Rescriptus, a palimpsest: parchment scraped clean and written over, because parchment was expensive and old text was not. Its history is a chapter in itself. The Museum of the Bible's record for accession MS.000149.1-.86 lists 137 folios of ink on vellum, with three datable layers. There is a Western Palestinian Aramaic layer from around the 500s, a Greek layer from around the 700s, and the Syriac layer from around the 800s to 900s. Scribes made the book at St Catherine's Monastery in Sinai. The Scottish twin sisters Agnes Smith Lewis and Margaret Dunlop Gibson acquired it in Egypt between 1895 and 1906, and left it to Westminster College, Cambridge in 1926. The Green Collection bought it in 2010 and gave it to the Museum of the Bible in 2012 (S083). The 2022 paper reports that the 146 known folios sit in three places. The Museum of the Bible holds 137, St Catherine's Monastery holds 8 (Syriac NF 38), and the Mingana Collection at Birmingham holds 1 (MSyr637) (S065).
Underneath the Syriac, on folios 47 to 54 and 64, is something else. This undertext, the erased writing that the later scribe wrote over, is a codex holding Aratus's Phaenomena plus star coordinates. Paleographers date it to the fifth or sixth century CE. It was scraped away "by the 9th or 10th c., when it was re-used to write Syriac translations" (S065).
Victor Gysembergh, Peter Williams and Emanuel Zingg imaged it. They used "a MegaVision spectral system that featured an E7 50MP digital back". It took 42 shots per page, across wavelengths from 365 nm in the ultraviolet to 940 nm in the near infrared, plus fluorescence imaging (S065). Ink that has been scraped away still leaves chemical traces in the parchment. Those traces absorb and fluoresce differently from the surrounding skin, at wavelengths the eye cannot see. Stack the exposures, run the right image processing, and text invisible in ordinary light comes back.
On folios 48r and 53v they read coordinates for the constellation Corona Borealis, the Northern Crown. The recovered text gives it in length as spanning "9°1/4 from the first degree of Scorpius to 10°1/4 in the same zodiacal sign", and in breadth "6°3/4 from 49° from the North Pole to 55°3/4". Four individual stars appear. β Coronae Borealis sits at right ascension 210°30′ and ι Coronae Borealis at right ascension 220°15′. π Coronae Borealis is at codeclination 49°, δ Coronae Borealis at codeclination 55°45′ (S065).
Their claim is that these are Hipparchus's, and they give four grounds (S065):
- The vocabulary. The text uses mēkos (MAY-kos, "length") and platos (PLA-tos, "breadth") for the east-west and north-south extension of a constellation, which is how Hipparchus's surviving Commentary on the Phaenomena talks.
- The coordinate system. The coordinates are equatorial (right ascension and codeclination), not ecliptic, which is the older convention and the one Hipparchus is thought to have used.
- The epoch. "These coordinates are accurate to within 1° for the epoch of Hipparchus' star catalogue (ca. 129 BCE)."
- Precedent in the Aratus Latinus, a later Latin descendant of the same tradition. Of seven coordinates appearing in both Hipparchus's Commentary and the CCR or Aratus Latinus texts, "perfect agreement is observed in four cases".
Where does "around 129 BCE" come from? Not from the palimpsest, which carries no date. It is inferred from Ptolemy's statement that about 265 years elapsed between Hipparchus's measurement and Ptolemy's own of 137 CE. That puts the epoch near 128 BCE. Both the 2022 paper and its 2024 critics work from that Ptolemaic figure (S065, S066). The dating is a chain: Ptolemy's round number, minus 265 years, compared against precession.
The strong version of the 2022 claim is that this catalog "would make Hipparchus' Catalogue significantly more accurate than his successor Claudius Ptolemy's" (S065).
Then, in 2024, the rebuttal, in the same journal. Gerd Grasshoff and Susanne M. Hoffmann published an astronomical re-analysis and did not hedge. Their abstract: "(a) we disagree with their astronomical dating and find inconsistencies by using the given numbers, and (b) the terminology and the data format used in the palimpsest do not match Hipparchus or anybody else. Therefore, the palimpsest does not prove anything about Hipparchus's star catalogue nor did Hipparchus use rectangular constellation borders. Specifically, the constellation of Corona Borealis, typically depicted as a circle since Babylonian times, is not considered a rectangle by Hipparchus" (S066).
That last objection is pleasingly concrete and easy to explain to a class. The recovered text describes Corona Borealis with a length range and a breadth range, which is a bounding box. Corona Borealis is a crown. It has been drawn as an arc or a circle since Babylonian star lists. Why would the astronomer who wrote the Commentary box it?
Their conclusion, at p. 345: "Only one of four numbers matches Hipparchus's time (50%-75% don't) and to interpret it this way, we need to suppose a mistake by the scribe. This suggests that the CCR fragment is corrupted and cannot be interpreted in connection to any specific historical author". And at p. 346: "It is not to be considered as the 'lost star catalogue of Hipparchus' but as an erased text that was corrupted in the moment of its writing" (S066).
Gysembergh, Williams and Zingg published a reply in 2025, "A note on the new evidence for Hipparchus' star catalogue", in the same journal at 56(3), 287 to 290. I could not read it past the landing page, and the abstract was truncated. So I am not going to tell you what is in it (S067). The dispute is live and unresolved as far as I could verify.
What is not in dispute is that the images exist and are free. The Museum of the Bible has released color, multispectral and "yellow tracing" images of folios 47r through 64v under a stated license: "These works are licensed under a Creative Commons Attribution-ShareAlike 4.0 International License" (S083). A class can look at the actual evidence in this argument. You cannot usually say that about a dispute in the history of astronomy.
Ptolemy: the dates, the anchors, and the book
Claudius Ptolemy worked in Alexandria in Roman Egypt. Toomer's estimate is the standard one: "he lived from approximately A.D. 100 to approximately A.D. 175" (S061). Both ends are soft, and it is worth knowing what holds them up.
The firm anchors are three. First, Ptolemy's own observations recorded in the Almagest run from 127 to 141 CE (S061). That is a working career of at least fourteen years, so he was an adult by 127. Second, the Canobic Inscription is a stone Ptolemy erected at Canopus in Egypt. Its own text dates it to the tenth year of the emperor Antoninus, that is 146/7 CE (S061). Third, Toomer, following N. T. Hamilton, argues that the Canobic Inscription shows a stage of Ptolemy's theory earlier than the Almagest. That makes it a lower bound on the date of the book (S061).
That priority argument is the interesting one. The inscription and the Almagest give different parameters for some of the same quantities. If the inscription is earlier, the Almagest was written after 146/7 CE. Toomer draws the conclusion: "the Almagest can hardly have been published earlier than the year 150" (S061). Without Hamilton's argument you would have only a bare terminus post quem, the earliest date the book could have been written, taken from the latest observation Ptolemy uses, 141 February 2. So "around 150 CE" for the Almagest is not a guess. It is a stone, plus a comparison of parameter values, plus an inference about which set came first.
The book's real title is the Mathēmatikē Syntaxis, the Mathematical Systematic Treatise. "Almagest" is what came back from Arabic, where the Greek superlative megistē, "greatest", picked up the Arabic definite article al- and never let go. The book Europe read for four hundred years wore an Arabic name for a Greek work. That tells you the route it took.
The table itself
Ptolemy states the design in the first paragraph of I.10, and it is worth reading as a specification document:
"For the user's convenience, then, we shall subsequently set out a table of their amounts, dividing the circumference into 360 parts, and tabulating the chords subtended by the arcs at intervals of half a degree, expressing each as a number of parts in a system where the diameter is divided into 120 parts. [We adopt this norm] because of its arithmetical convenience" (S061, I.10 at H31).
Four decisions in one sentence:
360 parts to the circumference. Inherited, on Heath's account, from Babylon by way of Hypsicles.
Half-degree steps. From to , which is 360 rows. He stops at 180 because past a semicircle the chord starts shrinking again and repeats values he already has.
Diameter 120 parts, so radius 60. Toomer's note explains the "arithmetical convenience": "The principal convenience is that the radius is 60 parts, or 1,0 in the sexagesimal system. Hence in some ways this resembles a sine table with R = 1" (S061, note 61). In base 60, sixty is written as one-zero. A radius of 60 is a radius of 1, shifted one place. Multiplying and dividing by the radius becomes a shift, not a calculation. This is the same reason computers use powers of two.
Three sexagesimal places. Each chord is written as whole parts, sixtieths, and thirty-six hundredths: 1;2,50 means . Toomer's note 68 records a modern recomputation. Glowatzki and Göttsche ran the whole table by machine and found "that Ptolemy must have carried out his calculations to five sexagesimal places to achieve the [accuracy he did]" (S061). He worked to five places and published three, which is what a careful numerical analyst does today.
There is a third column, and students should notice it. Toomer's translation of I.10 at H47: "They will be arranged in sections of 45 lines to achieve a symmetrical appearance. The first column [in each section] will contain the arcs tabulated at intervals of 1/2°, the second the corresponding chords in units of which the diameter contains 120, and the third the thirtieth part of the increment in the chord for each interval. [This last] is so that we may have the average increment corresponding to one minute [of arc], which will not be sensibly different from the true increment [for each minute]" (S061).
Half a degree is thirty minutes. Divide the row-to-row jump by thirty and you get a per-minute rate of change. That is a linear interpolation column, printed alongside the data, roughly eighteen hundred years before anyone called it that. It also means the table is effectively good to the arcminute, not the half degree: 21,600 usable values from 360 printed rows.
Here are the first ten rows, in Toomer's rendering, cross-checked against Heiberg's Greek at H48 (S061, S062). The Greek gives the same values in Greek numerals. The half-degree row reads ∠′ | ο λα κε | ο α β ν. Then α | α β ν, then α∠′ | α λδ ιε, then β | β ε μ, and so on.
| Arc | Chord | Sixtieths column |
|---|---|---|
| ° | 0;31,25 | 0;1,2,50 |
| 1° | 1;2,50 | 0;1,2,50 |
| ° | 1;34,15 | 0;1,2,50 |
| 2° | 2;5,40 | 0;1,2,50 |
| ° | 2;37,4 | 0;1,2,48 |
| 3° | 3;8,28 | 0;1,2,48 |
| ° | 3;39,52 | 0;1,2,48 |
| 4° | 4;11,16 | 0;1,2,47 |
| ° | 4;42,40 | 0;1,2,47 |
| 5° | 5;14,4 | 0;1,2,46 |
Watch the third column. It starts at 0;1,2,50 and decreases, slowly. That decrease is the curvature of the sine function showing up in a difference table. Chords grow almost linearly for small arcs, then start falling behind. Ptolemy's interpolation column records the falling behind to the fourth sexagesimal place.
Now convert two rows and check them. Take the half-degree row first:
An error of fourteen millionths of a part, on a quantity of about half a part. Now the 3 degree row, where the numbers are bigger and the check is easier to feel:
Convert the true value back into sexagesimal and you get 3;8,28.44, which rounds to what Ptolemy printed. He is correct to the last digit he wrote. Relative error: about four parts in a hundred thousand.
The landmark rows further down the table are where the geometry shows through, because each one is the side of a regular polygon inscribed in the circle.
| Arc | Chord (sexagesimal) | Chord (decimal) | 120 sin(arc/2) | Figure |
|---|---|---|---|---|
| 36° | 37;4,55 | 37.081944 | 37.082039 | side of the regular decagon |
| 60° | 60;0,0 | 60.000000 | 60.000000 | side of the hexagon, equal to the radius |
| 72° | 70;32,3 | 70.534167 | 70.534230 | side of the regular pentagon |
| 90° | 84;51,10 | 84.852778 | 84.852814 | side of the square |
| 120° | 103;55,23 | 103.923056 | 103.923048 | side of the equilateral triangle |
| 144° | 114;7,37 | 114.126944 | 114.126782 | supplement of 36° |
| 180° | 120;0,0 | 120.000000 | 120.000000 | the diameter itself |
The 60 degree row is the table's built-in sanity check. An inscribed regular hexagon has sides equal to the radius. So must be exactly , and the table says 60;0,0 (S061). The 180 degree row must be the diameter, and it says 120;0,0. If a copyist has mangled your manuscript, those two rows will tell you.
Ptolemy expects you to check him, and says how. From I.10 at H47: "It is easy to see that, if we suspect some scribal corruption in one of the values for the chord in the table, the same theorems which we have already set out will enable us to test and correct it easily, either by taking the chord of double the arc [of that] of the chord in question, or from the difference with some other given chord, or from the chord of the supplement" (S061). Three independent recomputation routes, offered in advance, for a document he knows will be copied by hand for centuries. That is error-correction design.
It did not entirely work. Toomer records at least one internal inconsistency. The chord of 72 degrees appears in the text of I.10 as 70;32,3, and the table agrees at 70;32,3. But Heiberg changed the table value to 70;32,4 on other manuscript evidence. Toomer wonders in a note whether this points to an earlier version of the chord table (S061).
One practical caution for anyone re-keying this table. The freely available scan of Toomer's translation has unreliable optical character recognition in the table pages. It renders the degree row as "10 38 49" where the value should be 10;58,49. It renders the degree row as "7 30 54" where it should be 7;50,54. I checked every value printed above against the Greek or against the internal arithmetic. A student edition should be re-keyed from a clean copy (S061, S062).
Ptolemy's theorem, and the four rules that fill 360 rows
Ptolemy fills the table with five geometrical results, applied in a fixed order. Only one of them is famous.
Rule zero: the polygons. Euclid gives the sides of the inscribed decagon, pentagon, hexagon, square and equilateral triangle in terms of the radius. Ptolemy quotes the resulting chords: "the side of the decagon, which subtends 36°, has 37;4,55 where the diameter has 120"; "the side of the pentagon, which subtends 72°, contains 70;32,3". Then the hexagon, "equal to the radius, contains 60", and "Crd 90° ≈ 84;51,10 and Crd 120° ≈ 103;55,23" (S061, I.10). Five rows, free, from a book three hundred years old.
Rule one: the supplement. "If any chord be given, the chord of the supplementary arc is given in a simple fashion, since the sum of their squares equals the square on the diameter" (S061, I.10). This is Thales plus Pythagoras. An angle in a semicircle is right, so a chord and its supplement are the two legs of a right triangle whose hypotenuse is the diameter. In symbols:
Ptolemy's worked case is 36 to 144 degrees. Follow the arithmetic:
which is what he prints (S061, I.10). Modern check: . One rule, and the whole table folds in half: every entry below 90 degrees gives you one above it.
Rule two: Ptolemy's theorem. For any quadrilateral inscribed in a circle, with diagonals and :
In words: the product of the diagonals equals the sum of the products of the two pairs of opposite sides (S061, I.10).
Toomer is careful about where it comes from: "This proposition, commonly known as 'Ptolemy's Theorem', is not in fact attested before him" (S061, note 59). Not proven to be his; not found earlier.
The proof needs one auxiliary line. It is worth doing in class, because it uses nothing but similar triangles and the inscribed-angle theorem. On the diagonal , mark the point such that . Now:
- Triangles and have by construction, and because both stand on arc . So they are similar, and .
- Triangles and have (add to each of the equal angles above), and because both stand on arc . So they are similar, and .
Add the two results:
Rule three: the chord of a difference. Now put the theorem to work. Set and at the two ends of a diameter, so . Let arc and arc , with . Then , , and because is a diameter, and . The unknown side subtends arc . Ptolemy's own account: "AB.GD + AD.BG = AG.BD... AD.BG is given by subtraction. And AD is a diameter. Therefore chord BG is given" (S061, I.10). Rearranged:
Substitute the modern identities and , write and , and watch it collapse:
Ptolemy's theorem is the sine subtraction formula, in a circle of diameter 120 instead of a unit circle.
Ptolemy's own first use of it: the chord of 12 degrees, "since we have those of 60° and 72°" (S061, I.10). Work it with his own tabulated values:
His table gives , and the modern value is (S061, I.11). Agreement to six figures, from a theorem about quadrilaterals.
Rule four: the half-arc. Given a chord, find the chord of half the arc. Ptolemy's construction gives two relations: "" and "", where subtends half the arc of the given chord (S061, I.10). In modern form, with the given arc:
Check that it is the half-angle formula. Write :
Run it on a real case. Take the 36 degree entry, whose supplement 144 degrees we computed above, and halve it to get the chord of 18 degrees. (This is my calculation using Ptolemy's rule, not a value quoted from his table.)
Modern value: , which in sexagesimal is 18;46,20. Ptolemy's chain is one unit low in the last place, which is what you expect after squaring, subtracting and taking a root on three-place data. Halve again for 9 degrees, again for 4.5, again for 2.25, and again. The halving never stops, which is how he gets down to fine arcs.
Toomer flags something curious about this rule, and it matters for Hipparchus. Ptolemy could have derived the half-arc formula from his own theorem in a couple of lines. He does not. He imports a separate theorem instead. Toomer identifies its ancestry: "Although Ptolemy's formula for the chord of the half-angle can easily be derived from his general theorem, he introduces instead another theorem, which goes back to Archimedes. It is a plausible inference that this is because the latter theorem was the sole basis of earlier chord tables, notably Hipparchus', as I have argued" (S061, note 60). And in note 59: "It remains uncertain whether any of the earlier chord tables (e.g. Menelaus') used any geometrical basis beyond the half-angle theorem" (S061).
If Toomer is right, then the structure of Almagest I.10 is a fossil. Ptolemy kept the older tool in the toolbox next to his new one. The shape of his chapter preserves the shape of books that no longer exist.
Rule five: the chord of a sum. Proved from the cyclic quadrilateral with a diameter (S061, I.10). The result is the mirror of rule three, with a plus sign:
Five rules. Add, subtract, halve, take supplements, start from the polygons. With those, and enough patience, you can compute the chord of any arc that is a multiple of one and a half degrees. Start from 72 and 60, difference to get 12, half to get 6, half to get 3, half to get 1.5.
And then Ptolemy hits a wall.
The worked example: squeezing the chord of one degree
He needs half-degree steps. His machinery gets him to 1.5 degrees and stops. Getting from 1.5 to 0.5 means dividing an arc by three, and he says plainly that this cannot be done:
"If a chord, e.g. the chord of 1 1/2°, is given, the chord corresponding to an arc which is one-third of the previous one cannot be found by geometrical methods" (S061, I.10).
Toomer's note gives the modern reason, and students should hear it. Ptolemy was right, for a reason nobody could state for another seventeen hundred years: "the problem of finding Crd from given Crd can be reduced to a cubic equation of the kind which cannot (except for a few particular values of ) be solved by Euclidean geometry (using straight line and circle)" (S061, note 63). Trisecting an arc is trisecting an angle. Wantzel proved it impossible with straightedge and compass in 1837. Ptolemy could not prove it, but he could see it was not working, and he did not fake it.
So he does what a numerical analyst does. He stops trying to be exact and proves an inequality instead.
The lemma. "If two unequal chords be given, the ratio of the greater to the lesser is less than the ratio of the arc on the greater to the arc on the lesser" (S061, I.10). He proves it as . In symbols, for arcs :
Chords grow more slowly than their arcs. This is the monotonicity of that Aristarchus had helped himself to without proof four hundred years earlier. Ptolemy proves it.
Now the squeeze. He has two chords already: , got by halving 1.5 degrees, and , got from the main chain. One degree sits between them, and he traps it from both sides.
Upper bound. Take and , so . The lemma gives :
Ptolemy's own words: "GA < 1;2,50 (for 1;2,50 = 4/3 × 0;47,8)" (S061, I.10).
Lower bound. Take and , so . The lemma gives , so :
Ptolemy's words: "AB > 1;2,50 (for 1;34,15 = 1 1/2 × 1;2,50)" (S061, I.10).
The conclusion. "Since the chord of 1° was shown to be both greater and less than the same amount, we can establish it as approximately 1;2,50 where the diameter is 120. By the preceding propositions we can also establish the chord of 1/2°, which we find to be approximately 0;31,25" (S061, I.10).
That last step is the half-arc rule applied once more, and with in hand the whole 360-row table can be generated.
The modern check.
High by thirty-eight millionths, which is about four parts in a hundred thousand.
Why this is the best thing in Book I. Look at how narrow the window is. His two bounds are 1.047222 and 1.047407, a gap of 0.000185. One unit in the last sexagesimal place he prints is . He bracketed the answer inside a window narrower than the precision he was reporting. Once he had the two bounds, the rounded three-place value was forced. There was only one number he could write.
One honest wrinkle, and it is a good discussion point rather than an embarrassment. Ptolemy's stated lower bound, 1.047222, sits slightly above the true value 1.047184. The lemma is correct and the logic is correct; the slippage comes from his input data. His tabulated is itself high by about 0.00008 against the true 1.570752, and two thirds of that error carries straight through. Run the same argument on exact values and you get , and the inequality holds as it should. This is what happens when you push rounded data through an exact method. It is the same lesson as any error-analysis exercise in a modern numerical methods course.
Ptolemy's pi, in the wrong book
Ptolemy's value for is 3;8,30, and it is not in Book I. It is in Book VI, chapter 7, among the eclipse calculations, because that is where he needed it. Do not attribute it to the chord table chapters.
His words: "we assumed that the ratio of the circumference to the diameter is 3;8,30 : 1, since this ratio is about half-way between 3 1/7 : 1 and 3 10/71 : 1, which Archimedes used as rough [bounds]" (S061, VI 7 at H513). He then uses it immediately: "the circumferences of the disks are, according to the ratio 1 : 3;8,30" (S061).
High by about 24 parts in a million. Good to four decimal places.
He did not average Archimedes' bounds. Check: and , whose midpoint is 3.141851. Ptolemy's 3.141667 is not that. He picked 3;8,30 because it is a tidy sexagesimal number sitting between the bounds. And he wrote "about half-way", which is honest.
The connection back to the chord table is the method, not the number. Archimedes got his bounds by taking a hexagon and bisecting the angle four times to reach a 96-gon (S070). Ptolemy fills his table by taking the polygons and halving arcs repeatedly. Same operation, different bookkeeping.
Menelaus, and the theorem that runs spherical astronomy
Everything so far has been flat. A chord table on its own solves plane triangles. The sky is not flat, and the tool that made Greek spherical astronomy work is a theorem about six arcs.
Almagest I.13 proves two compound-ratio results about a figure formed by four great circles crossing on a sphere. Neugebauer named them Menelaus Theorem I and Theorem II. Toomer's translation gives them as statements [13.5] and [13.6] (S061):
Three things to notice.
First, every arc is doubled. is the chord of twice the arc, which in modern terms is . The doubling is the demi-degree convention again, now permanently baked into the formulas of Greek spherical astronomy. Translate every into and the factors of cancel out of the ratios, leaving a clean statement about sines of arcs.
Second, the two theorems are the same configuration read two different ways. Toomer's note 84: "The theorem connecting six great circle arcs on the surface of a sphere in a Menelaus Configuration... is due to Menelaus, whom Ptolemy mentions in the Almagest only as an observer. It appears (in both forms) as Prop. III 1 of his Sphaerica (ed. Krause pp. 194-7). These two forms have been labeled by Neugebauer (HAMA 28) as Theorem I (= 13.6), where four inner parts of the Menelaus Configuration are related to two outer parts, and Theorem II (= 13.5), where four outer parts are related to two inner parts" (S061).
Third, and this is the practical point, Ptolemy applies it immediately. In I.14 he uses Theorem I to compute declinations, that is, how far north or south of the celestial equator a point on the ecliptic sits (S061). From there it runs the rest of the book: rising times, day lengths, the geometry of eclipses. Sidoli calls it, without qualification, "the fundamental theorem of ancient spherical trigonometry". He describes it in one sentence: "It asserts a compound proportion that holds for combinations of the chords of six arcs of great circles forming a concave quadrilateral on the surface of a sphere" (S072, p. 44).
Who proved it is a live question. Nathan Sidoli argues that the tidy attribution does not survive contact with the sources. "The Sector Theorem is now generally known as the Menelaus Theorem... it has now become clear that we do not possess any version of it that can simply be taken as that which Menelaus wrote" (S072, p. 43). And: "An examination of Menelaus' version of the sector theorem and its function in his Spherics, however, shows that it is unlikely that he intended the sector theorem to be read as his original contribution. An examination of the astronomical evidence in the writings of Hipparchus and others gives support to the conclusion that the sector theorem was known to, and used by, Hipparchus" (S072, p. 44).
The historiography swung. Sidoli records it: "Before Neugebauer published A History of Ancient Mathematical Astronomy, it was common to hear doubts expressed as to Menelaus' authorship of the theorem. For example, Bulmer-Thomas [1974, 299] argues that it was known before Menelaus in the Dictionary of Scientific Biography. After 1975, however, it became more common for scholars to simply accept that Menelaus had written the theorem. Toomer [1984, 69, n. 84], for example, states this as a fact" (S072). One influential book in 1975 turned a contested attribution into a textbook fact, and a 2006 paper turned it back into a question.
On the name: the standard Arabic term is al-shakl al-qaṭṭāʿ, "the sector figure" (S072, n. 3). English textbooks often call it "the rule of six quantities". I could not source that phrase to any of the scholarship I read, and I say so again below.
Menelaus himself, and a book that survives in the wrong language
Menelaus of Alexandria is the most underrated person in this chapter, and the state of the evidence about him is a lesson in itself.
His dates are a mess, and the disagreement should be shown, not smoothed. MacTutor gives c. 70 to c. 130 CE. Rashed and Papadopoulos place him "in the first century of our era". Heath will not commit beyond his observations and writes "fl. about A.D. 100" (disputed) (S068, S074, S084). The range c. 70 to c. 140 CE that circulates widely is a modern convention I could not trace to any source I read.
The one hard fact is two nights of observing at Rome. In the first year of Trajan, 98 CE, Menelaus watched the Moon pass in front of the star Spica on the night of Mechir 15/16 in the Egyptian calendar. Days later he recorded a second occultation, one body hiding another, this time of a star "in the forehead of Scorpius" on Mechir 18/19. Ptolemy reports both in Almagest VII 3, and they are the only firmly dated events in his life (S061, S068, S074). Toomer's index gives the two nights as -98 January 11 and 98 January 13/14 in his notation. Rashed and Papadopoulos give "approximately January 14, 98 A.D." for the first (S061, S074).
Sit with what that means. A man who wrote the founding treatise of spherical geometry is dated by two nights when the Moon passed in front of a star. Somebody else wrote them down, fifty years later.
Book I of the Sphaerica defines the spherical triangle, for the first time anywhere. Heath: "In this Book for the first time we have the conception and definition of a spherical triangle. Menelaus does not trouble to give the usual definitions of points and circles related to the sphere, e.g. pole, great circle, small circle, but begins with that of a spherical triangle as 'the area included by arcs of great circles on the surface of a sphere', subject to the restriction (Def. 2) that each of the sides or legs of the triangle is an arc less than a semicircle" (S068, pp. 262 to 263). He skips the elementary definitions because his readers have Theodosius. He does not skip this one, because nobody had it.
And, as noted above, he does not call it a triangle. Pappus reports that Menelaus uses tripleuron, "three-side", keeping trigōnon for the plane figure (S068, p. 263).
Ptolemy never credits him for the theorem. Toomer's note is dry about it: Menelaus is a man "whom Ptolemy mentions in the Almagest only as an observer" (S061, note 84). The most important theorem in Ptolemy's Book I is named after a man Ptolemy cites only for two nights of star-watching.
The Greek text of the Sphaerica is gone. Not damaged. Gone. The Qatar Digital Library's essay puts it flatly: "no Greek manuscript of the text is known" (S082).
What survives is a translation chain, and it is the best language-and-history story in this chapter. Rashed and Papadopoulos: "Before the Greek original was lost, this work, which is at the foundations of spherical geometry... was translated several times into Arabic, between the 9th and the 10th century" (S074). The named links:
- Manuscript colophons, the notes a scribe leaves at the end of a copy, attribute the Arabic original either to Ḥunayn ibn Isḥāq or to his son Isḥāq ibn Ḥunayn, the great translators of ninth-century Baghdad (S074).
- al-Māhānī (active 853 to 866) rectified the second Arabic translation (S074).
- al-Harawī, in the tenth century, revised al-Māhānī's edition. The Sector Figure is Proposition 66 in his version (S074).
- al-Dimashqī produced a separate translation in the late ninth century (S074).
- Ibn ʿIrāq (died c. 1036) produced a rectification of al-Dimashqī's translation (S074).
- Jacob ben Makhir ibn Tibbon, around 1273, made a Hebrew translation from the Arabic (S074).
- Maurolycus printed a Latin version at Messina in 1558, and Edmond Halley published an edition at Oxford in 1758, working mainly from Jacob ben Makhir's Hebrew (S068).
- Rashed and Papadopoulos, in 2017, published the modern critical edition. It prints a fragment of an early Arabic translation and the al-Māhānī/al-Harawī version, with English translations, as volume 21 of the De Gruyter series Scientia Graeco-Arabica (S074, S075).
Greek to Arabic to Hebrew to Latin to English, eleven hundred years, at least four languages and three religions, and the mathematics still works.
A disagreement worth recording, about how lost "lost" is. Rashed and Papadopoulos say the Greek "was lost". Bernard, reviewing the edition for Bryn Mawr Classical Review, says "only fragments of the Greek text survive, preserved in later authors" (S075). The Qatar Digital Library says no Greek manuscript is known (S082). These are compatible if you distinguish a manuscript of the work from quotations in other authors. But the phrasings differ, and a student text should say so rather than pick one. Bernard also notes what is still missing: "we still await critical editions of the Latin and Hebrew versions before we can hope to fully understand the medieval transmission of the text" (S075).
One link in the chain is online, in high resolution, for free. British Library MS Or. 13127 is Kitāb Mānālāwus fī al-ashkāl al-kurrīyah, al-Harawī's recension of the Spherics. A scribe named Ismāʿīl copied it at Damascus, and the colophon dates it: 4 Rabīʿ II 548 in the Islamic calendar, which is 29 June 1153 CE. It runs to 55 folios plus 4 endleaves and carries 118 spherical figures. The Qatar Digital Library has digitized it as 126 page images, with IIIF support. The rights statement is "Public Domain" under Creative Commons (S082).
A class can open a Damascus manuscript from 1153 and look at a diagram of a spherical triangle drawn by hand. The Greek behind it is gone. This is how the theorem reached us.
Ptolemy's other projection books, both lost in Greek
The Almagest is not the only Ptolemaic text that matters here, and the other two make the same point about survival.
The Planisphaerium is the first known treatise that "develops a plane diagram of the celestial sphere using methods mathematically related to stereographic projection". Sidoli and Berggren add that "although Ptolemy wrote the text, the methods contained in it probably go back at least as far as Hipparchus" (S071, p. 38). This is the mathematics behind the astrolabe: flatten a sphere onto a disk in a way that keeps circles as circles. The tenth-century Byzantine encyclopedia Suidas gives the Greek title as Ἅπλωσις ἐπιφανείας σφαίρας, Simplification of the Sphere (S071, p. 37 n. 1).
And the Greek is gone: "There are currently no known manuscripts containing the Greek text of Ptolemy's Planisphere" (S071, p. 37). The oldest witness is an anonymous Arabic translation, probably from the ninth-century Baghdad translation movement. Hermann of Carinthia made a loose Latin version in the twelfth century from a different Arabic version, dated 1143, and Isaac Hebreus made another in 1518. A medieval Hebrew translation appears to have been made from the Latin. One of Sidoli and Berggren's manuscript sigla is "I: Istanbul, Aya Sofya 2671" (S071, S080).
The Analemma (Περὶ ἀναλήμματος) is about plotting the celestial coordinates of the Sun or another body for any latitude at any time. That is what you need to build a sundial. The Ptolemaeus Arabus et Latinus entry: "The original text is lost in Greek, except for a number of fragments, and survives in full in Latin only", the Latin being William of Moerbeke's thirteenth-century translation. Heiberg edited the surviving Greek fragments in 1895 and again in volume II of the collected Opera in 1907 (S080).
Two of Ptolemy's three geometrical astronomy books survive only in translation. The one that survived in Greek is the one that became the school textbook.
Pappus, Theon, Hypatia: the chain that got the book to us
Books survive because institutions teach them. That is the whole story of how we have the Almagest, and it is also, in a grim way, the story of why we do not have Hipparchus.
Toomer states the mechanism: the Almagest's "success contributed to the loss of most of the work of Ptolemy's scientific predecessors, notably Hipparchus, by the end of antiquity, because, being obsolete, they ceased to be copied. Whereas Hipparchus' works are still used by Ptolemy's younger contemporaries, Galen and Vettius Valens, by the early fourth century (and probably much earlier), when Pappus wrote his commentary on it, the Almagest had become the standard textbook on astronomy" (S061).
Nothing burned. Nobody banned anything. A better book superseded Hipparchus's twelve books on straight lines in a circle, so scribes stopped copying them. After a few generations of not being copied, a papyrus text is gone. A better textbook killed the books it replaced.
The teaching tradition itself is documented. Toomer: the Almagest's position "as the standard textbook in astronomy for 'advanced students' in the schools at Alexandria... is amply demonstrated by the partially extant commentaries on it by Pappus (c. 320) and by Theon of Alexandria (c. 370)" (S061).
Pappus of Alexandria (active c. 320 CE) wrote a commentary on the Almagest, partially extant, and he is a key witness for lost material elsewhere. He is the reason we know that Menelaus called the spherical triangle a tripleuron (S068). Commentators are not decorative. When the primary text is gone, the commentator is the primary text.
Theon of Alexandria (active c. 370 CE) is the reason this chapter can be written at all. He taught the Almagest and wrote a commentary on it, running to eleven or thirteen books depending on how you count. The transmission line that reaches us runs through the school he represents (S061). And the single sentence in that commentary about Hipparchus's twelve books and Menelaus's six is the entire direct evidence for pre-Ptolemaic chord treatises (S068, p. 257). One aside, by a teacher, five hundred years after the fact.
Hypatia (hy-PAY-shuh, c. 370 to March 415 CE) was Theon's daughter, a working mathematician and astronomer in Alexandria, and she was murdered there in March 415 (S084). She is also the subject of more confident overstatement than anyone else in this chapter, so it is worth setting out what the evidence is.
The entire evidentiary basis is a subtitle in one manuscript. Alan Cameron's 1990 study locates it: the formula appears "(in the oldest MS., Laur. 28.18 = L) to the heading to Book III of Theon's commentary on Ptolemy's Almagest" and reads:
Θέωνος Ἀλεξανδρέως εἰς τὸ τρίτον τῆς μαθηματικῆς Πτολεμαίου Συντάξεως ὑπόμνημα ἐκδόσεως παραναγνωσθείσης τῇ φιλοσόφῳ θυγατρί μου Ὑπατίᾳ (S073, p. 106).
The manuscript is Laurentianus 28.18, in Florence. Books I and II of the same commentary carry headings that do not mention Hypatia. Cameron adds, at p. 115: "According to Rome (the only person who has looked at the MSS. for the rest of the commentary) there are no more such subheadings to Books IV-XIII" (S073).
What the heading says, on Cameron's reading, is not what most books say it says. The Greek has a genitive absolute, ἐκδόσεως παραναγνωσθείσης, "the edition having been revised", set against ὑπόμνημα, "commentary". Cameron, p. 115: "her father's commentary that she edited, but the text of the Almagest. If it had been the commentary, why introduce the misleading ἐκδόσεως? What we should then have expected is the neuter παραναγνωσθέν agreeing with ὑπόμνημα: 'the commentary of Theon revised by... Hypatia.' What we in fact get is a genitive absolute and an antithesis between commentary and edition: 'the commentary of Theon, with the edition revised by... Hypatia'" (S073).
So: the commentary is Theon's; the edition, meaning the working text of Ptolemy that the commentary sits alongside, was revised by Hypatia. Cameron is careful about what that revision was: "It will not (of course) have been any more of a critical edition in the modern sense than Theon's of Euclid. It was just a simplified and 'corrected' text for the use of students" (S073).
The heading also rules two things out. Cameron, p. 110: "the precise form of the heading to Book III seems to exclude the two obvious forms this assistance might have taken: namely that they wrote the commentary jointly between them... or that Hypatia completed or revised Theon's work after his death... Whatever Hypatia did, she evidently did while Theon was still at work, for it is Theon who mentions her role... And even in the case of Book III, Theon still claims the authorship of his commentary (Θέωνος ὑπόμνημα)" (S073). So it is not co-authorship and it is not posthumous editing.
And the internal evidence is thinner still. Cameron, p. 111: "Clearly there is no support here for the hypothesis of a substantial revision of Book III by Hypatia. Indeed, the text of Book III as extant gives little enough support to the idea that it was revised by anybody, even Theon... It is very singular that the only book expressly described as having been revised should show the least signs of having been revised at all" (S073).
Other scholars read it differently, and that should be said. Cameron sets out the positions he is arguing against. A. Rome held that Hypatia revised or edited Theon's work. Wilbur Knorr held that she inserted new material here and there rather than revising the whole; he assigned Books III and IV to her on stylistic grounds, and concluded that Theon delegated the publication of those books to her. A. Tihon came close to Knorr's position (S073). MacTutor still states plainly that Hypatia "assisted her father Theon of Alexandria in writing his eleven part commentary on Ptolemy's Almagest" (S084).
What is not in dispute is her competence, and Cameron affirms it himself: "Though most famous as a philosopher (she is so described by Theon), Hypatia was no mean mathematician and astronomer: she wrote commentaries on Diophantus, Apollonius of Perga, and Ptolemy's Handy Tables" (S073).
Three commentaries, on three hard technical works, one of them Ptolemy's own working tables. That is a serious mathematical career, and it does not need inflating.
From Toledo to Venice to Basel
The route the Almagest took to modern readers is worth tracing, because each step is datable and each step changed the text.
Ninth century, Constantinople and Baghdad. The two oldest surviving Greek manuscripts are both ninth century. Toomer lists the ones he collated: "A Parisinus graecus 2389. Mainly uncial, ninth century. B Vaticanus graecus 1594. Minuscule, ninth century. D Vaticanus graecus 180. Several hands..." (S061). Toomer also states an editorial disagreement with Heiberg worth recording: "The basis of my translation is the Greek text established by Heiberg. I have, however, found it necessary to make several hundred corrections to that text... I am convinced on grounds of internal consistency that [manuscript D] represents a sounder tradition than that of the mss. ABC, generally preferred by Heiberg" (S061). Several hundred corrections, and a different view of which manuscript to trust: the standard English Almagest is not Heiberg in English.
1175, Toledo. Gerard of Cremona completes the Latin translation of the Almagest, made from Arabic, at Toledo (S061). Toledo in the twelfth century is where Arabic learning crossed into Latin Europe, with Christian, Muslim and Jewish scholars working in the same city. Gerard's Latin, not the Greek, is what medieval Europe read.
10 January 1515, Venice. Petrus Liechtenstein prints Gerard's Latin translation as the Almagestum Cl. Ptolemei Pheludiensis Alexandrini. Toomer: "It was also the version in which the Almagest was first printed (Venice, 1515)" (S061, S086). So the first printed Almagest is a Latin translation of an Arabic translation of a Greek original. It appeared three hundred and forty years after Gerard finished his work, and thirteen hundred and sixty-five years after Ptolemy wrote his. A complete scan of the Deutsches Museum copy sits on the Internet Archive under a Public Domain Mark. The Ptolemaeus Arabus et Latinus project has a folio-by-folio viewer (S085, S086).
1538, Basel. Hervagius prints the Greek text for the first time. Toomer: "The sixteenth century saw the wide dissemination of the Greek text (printed at Basel by Hervagius, 1538)" (S061). Twenty-three years after the Latin. I could not locate a digitized copy of this edition, and I say so again below.
1898, Leipzig. J. L. Heiberg publishes the Greek critical edition, Claudii Ptolemaei opera quae exstant omnia, Vol. I.1: Syntaxis mathematica, for Teubner. Freely scanned (S062).
1984, Princeton. G. J. Toomer's English translation, 693 pages, the version anyone reading in English now uses (S061).
The manuscript and print record
Almost everything in this chapter that can be looked at, can be looked at online, and most of it is free. That is new, and it changes what a classroom can do. Vat.gr.1594, the ninth-century Vatican Ptolemy, is served with a IIIF manifest at https://digi.vatlib.it/iiif/MSS_Vat.gr.1594/manifest.json, so any IIIF-capable viewer can open it. Any student can zoom to the parchment (S081).
| Item | Institution and shelfmark | URL | Rights as stated |
|---|---|---|---|
| Ptolemy, Opera, saec. IX, 582 imaged openings, oldest fully digitised Greek Almagest witness | Biblioteca Apostolica Vaticana, Vat.gr.1594 (Polonsky Foundation Digitization Project; IIIF manifest at digi.vatlib.it/iiif/MSS_Vat.gr.1594/manifest.json) | https://digi.vatlib.it/view/MSS_Vat.gr.1594 | "Free use of this image is only for personal use or study purposes"; publication rights on request |
| Menelaus, Spherics, al-Harawī recension, Damascus 1153 CE, 55 fols., 126 images, 118 figures | British Library, Or. 13127, via Qatar Digital Library | https://www.qdl.qa/en/archive/81055/vdc_100023511683.0x000065 | Public Domain (Creative Commons) |
| Codex Climaci Rescriptus, color and multispectral images of fols. 47r to 64v | Museum of the Bible, MS.000149.1-.86 | https://www.museumofthebible.org/ccr-creative-commons-licensed-images | "These works are licensed under a Creative Commons Attribution-ShareAlike 4.0 International License" |
| Codex Climaci Rescriptus, object record and provenance | Museum of the Bible, MS.000149.1-.86 | https://collections.museumofthebible.org/artifacts/32858-codex-climaci-rescriptus-uncial-0250 | See institution's Rights and Reproductions page |
| Almagestum (Gerard of Cremona's Latin), Venice: Petrus Liechtenstein, 1515, folio images | Ptolemaeus Arabus et Latinus, print 1 | https://ptolemaeus.badw.de/print/1/70/1r | Not stated on the folio viewer |
| Almagestum, Venice 1515, complete scan of the Deutsches Museum copy | Internet Archive | https://archive.org/details/almagestumcl.ptolemeideutschesmuseum | Public Domain Mark 1.0 |
| Heiberg's Greek critical edition, Syntaxis mathematica Pars I, 1898 | Internet Archive (Google-digitised) | https://archive.org/details/claudiiptolemae00ptolgoog | Public domain |
| Manitius, Hipparchus, In Arati et Eudoxi Phaenomena, Greek and German, Teubner 1894 | Internet Archive (University of Illinois copy) | https://archive.org/details/ipparchoutonarat00hipp | Public domain |
| Heath, Aristarchus of Samos, 1913, with Greek text and translation of On Sizes and Distances | Internet Archive | https://archive.org/details/aristarchusofsam00heat | Public domain |
| Heath, The Works of Archimedes, 1897 | Internet Archive | https://archive.org/details/worksofarchimede00arch | Public domain |
| Heath, A History of Greek Mathematics vol. II, 1921 | Internet Archive | https://archive.org/details/historyofgreekma029268mbp | Public domain |
A note on the Manitius volume, for anyone tempted to use it. The page scans of Hipparchus's Commentary are readable, but the optical character recognition of the Greek is unusable (S078). You can look at it. You cannot search it.
What the evidence does not support
"Hipparchus used a chord table with and a step of 7.5 degrees." This is the single most repeated claim about Greek trigonometry, and it is a reconstruction wearing a fact's clothing. No ancient source states either number. is reverse-engineered from two ratios that Ptolemy attributes to Hipparchus in Almagest IV.11, and from Indian sine tables written around 500 CE. The 7.5 degree step is inferred from Indian tables stepping by 3.75 degrees in the sine. Duke labels his own reconstruction "a possible replica of Hipparchus' table" (S064). Toomer, who proposed , later wrote that his corrected recalculation "cast doubt on my claim that he was operating with a chord table with base R = 3438" (S061, note at H347). Duke rescued the hypothesis from a different eclipse trio in 2005 and called the link between 3144 and 3438 "unavoidable and firm" (S063). Then, in another paper, he argued that the evidence "certainly offers no encouragement to anyone claiming that Hipparchus used chords" at all (S064). Write "on the standard reconstruction", or "Toomer proposed, and Duke defends". Do not write "Hipparchus used".
"The 2022 palimpsest proves Hipparchus's star catalogue was better than Ptolemy's." The 2022 paper claims the recovered coordinates are Hipparchus's and accurate to within 1 degree for around 129 BCE (S065). In 2024, in the same journal, Grasshoff and Hoffmann disagreed. They argued that "the terminology and the data format used in the palimpsest do not match Hipparchus or anybody else", and that "only one of four numbers matches Hipparchus's time". On their reading the fragment "cannot be interpreted in connection to any specific historical author" (S066). A reply appeared in 2025 which I could not read (S067). Teach the discovery, which is real and remarkable. Teach the attribution as a finding under argument, with both papers named.
"Hypatia wrote a trigonometry text." There is nothing behind this at all. No source I read attributes any trigonometrical work to her. Cameron, who is generous about her abilities, lists what she wrote: "commentaries on Diophantus, Apollonius of Perga, and Ptolemy's Handy Tables" (S073). Number theory, conic sections, and a set of astronomical tables. No trigonometry text.
"Hypatia co-wrote Theon's Almagest commentary." The evidence is one subtitle in one manuscript, Laurentianus 28.18, at the head of Book III. Cameron argues on grammatical grounds that it credits her with revising the edition of Ptolemy's text, not the commentary. Its form, he argues, excludes joint authorship and posthumous revision (S073, pp. 110 and 115). Rome, Knorr and Tihon read it more expansively, and MacTutor still prints the loose version (S073, S084). The honest sentence is this. In one manuscript, the heading to Book III describes the commentary as Theon's, with the edition "revised by my philosopher-daughter Hypatia". Everything beyond that is inference that the leading modern treatment considers too strong.
"The Greek word for chord was khordē." The Greeks did not use χορδή geometrically. LSJ records guts, gut string, lyre string, musical note and sausage, and no geometrical sense whatever (S089). Ptolemy's phrase is eutheia en kuklō, "the straight line in the circle", and his chapter headings say so (S062). A search for the letters "χορδ" across Heiberg's Greek text of Almagest Books I to VI returns zero hits (S062). Our word "chord" comes from the Greek for a gut string by way of Latin chorda, but the geometrical meaning was attached later and elsewhere.
"Ptolemy's value of is in Book I, with the chord table." It is in Book VI chapter 7, among the eclipse calculations (S061, H513). This is a small point that is wrong in a lot of places.
"Menelaus proved the Menelaus theorem, and that settles it." Sidoli: "it has now become clear that we do not possess any version of it that can simply be taken as that which Menelaus wrote", and the astronomical evidence "gives support to the conclusion that the sector theorem was known to, and used by, Hipparchus" (S072, pp. 43 to 44). Before Neugebauer's 1975 book, doubting the attribution was the normal scholarly position (S072).
"Eratosthenes got 250,000 stades." He got 250,000 according to Cleomedes, and 252,000 according to Theon of Smyrna and Strabo. Heath's verdict on the difference: "The reason of the discrepancy is not known" (S068, p. 107). Give both numbers.
What I could not verify
Everything above is anchored to something I read. These are the things I could not pin down, and they are listed so that nobody later mistakes silence for confirmation.
Toomer's Centaurus paper itself. The publisher's page for Centaurus 18 (1974), 6 to 28, returned HTTP 403. Everything reported here about Toomer's 1973/1974 argument comes at second hand, from his own footnotes in the 1984 Almagest and from Duke's two papers. I have not cited the paper as evidence for anything (S087, S088).
The phrase "rule of six quantities". English-language accounts often give this as the traditional name for the Menelaus sector theorem. I searched Sidoli 2006, the Rashed and Papadopoulos preview, the Bryn Mawr Classical Review review, and the Qatar Digital Library essay. The Arabic name I found attested is al-shakl al-qaṭṭāʿ, "the sector figure" (S072, n. 3). I could not source "rule of six quantities" to any of them, and I could not find an Arabic phrase behind it. Treat it as a modern English label until somebody sources it.
The "broken chord" theorem attributed to Archimedes through al-Bīrūnī. This story appears in many popular accounts: that al-Bīrūnī preserves a theorem about a broken chord and credits it to Archimedes. I found a paper title on academia.edu and ResearchGate, neither of which returned readable full text through this connection. I also found several university lecture pages that are not citable scholarship. I could not read al-Bīrūnī's Kitāb Maqālīd ʿilm al-hayʾa or his Book on the Derivation of Chords in a Circle in any edition or translation. The attribution is unverified here, and I have not repeated it as fact.
A digitized copy of the Greek editio princeps, Basel 1538. The editio princeps is the first printed edition of a text. Toomer states that Hervagius printed the Greek Almagest at Basel in 1538, and I have recorded that on his authority (S061). Searches of e-rara (which returned HTTP 500), the Munich Digitization Center and the Internet Archive did not turn up a confirmed scan. The edition is attested; the scan is not.
Parisinus graecus 2389, Heiberg's manuscript A, ninth century, mainly uncial, and one of the two oldest witnesses to the Greek Almagest. I could not confirm a digitized copy on Gallica or in the BnF archives-et-manuscrits catalog. The ark I tried returned HTTP 403. Vat.gr.1594, the other ninth-century witness, is fully digitized, so the manuscript record in this chapter is half online and half not.
The 2025 reply by Gysembergh, Williams and Zingg. Landing page only, abstract truncated. Recorded to show the literature continues, used as evidence for nothing (S067).
The LSJ entry, in print. perseus.tufts.edu blocks automated fetching. I read the entry for χορδή instead at lsj.gr, which reproduces LSJ alongside Pape, Bailly, Dvoretsky, Liddell-Scott, Autenrieth, Middle Liddell and Frisk (S089). A print LSJ should be checked before this goes to press.
The first geometrical use of "chord". I could not establish when Latin chorda, or a vernacular descendant, first meant "line joining two points on a circle". Answering it needs the medieval Latin Almagest translations, a medieval Latin dictionary such as the DMLBS, or the OED, none of which I could read here. So the chapter says confidently that the Greeks did not use χορδή geometrically, and says nothing confident about when somebody started to.
Whether χορδή occurs anywhere in Ptolemy at all. My zero-occurrence search covers only Heiberg's Syntaxis Pars I, Books I to VI, and it runs on optical character recognition of Greek type. That is strong evidence, not proof (S062).
Hipparchus's dates and Menelaus's dates. No source I read gives independent evidence for Hipparchus born c. 190 BCE or died c. 120 BCE. Toomer's index establishes only that his observations run from 147 to 127 BCE. For Menelaus, the only hard datum is the pair of Rome observations of January 98 CE. The widely repeated range c. 70 to c. 140 CE could not be traced to a source I read (S061, S074, S084).
For the classroom
Astronomy and the calendar, together. Give students the actual problem before the actual tool. When does the equinox fall this year, to within an hour? How long is today, at this latitude? Every chord table in this chapter exists because somebody needed an answer better than "about". Hypsicles' Anaphorikos is a calendar problem in geometric clothing: how much of the zodiac rises in how much time. That is why he had to define both a degree of space and a degree of time in the same breath (S068, p. 214). Eratosthenes' measurement works because he picked a day when the answer is clean, the summer solstice, and a place where the Sun is overhead, Syene (S068, pp. 106 to 107). Calendar reform and astronomy are one subject, and were until quite recently.
Multispectral imaging: materials science, optics and computing in one object. Ask why erased iron-gall ink shows up at 940 nm when it is invisible at 550 nm. You are teaching absorption spectra with a manuscript instead of a cuvette. Then ask the computing question: the team took 42 exposures per page across the ultraviolet, visible and near infrared, plus fluorescence (S065). What do you do with 42 registered images of the same page? You are now teaching image registration, band selection and principal component analysis, on a problem where the answer matters and can be checked. The images are released under Creative Commons Attribution-ShareAlike 4.0, so a class can download the actual data (S083). And the historical question sits right there next to the technical one: two teams looked at the same recovered numbers and drew opposite conclusions (S065, S066). Recovering data is not the same as interpreting it.
Alexandria as a funded research institution. Count the people in this chapter attached to one city: Ptolemy, Menelaus, Hypsicles, Eratosthenes, Euclid, Pappus, Theon, Hypatia, and Aristarchus for part of his life. That is not a coincidence, and it is not about individual genius. Alexandria had a library, a salaried staff, a syllabus, advanced students and continuity across six hundred years. Then ask the harder question: why did most of the mathematics still get lost? Toomer's answer is uncomfortable and precise. The Almagest's "success contributed to the loss of most of the work of Ptolemy's scientific predecessors, notably Hipparchus, by the end of antiquity, because, being obsolete, they ceased to be copied" (S061). Nothing was destroyed. A better textbook came along and the older books stopped being assigned. After a few generations of not being copied, they were gone. That is a live question about digital preservation too.
One text across four languages, and what survives the trip. Trace Menelaus's Sphaerica from lost Greek, through at least two ninth and tenth century Arabic translations. Then through al-Māhānī's rectification, al-Harawī's revision, and Ibn ʿIrāq's correction of al-Dimashqī. Then into Jacob ben Makhir ibn Tibbon's Hebrew around 1273, into Maurolycus's Latin at Messina in 1558, and into Halley's Oxford edition of 1758, made mainly from the Hebrew. Finally into Rashed and Papadopoulos's English of 2017 (S068, S074, S075). Then open British Library Or. 13127, copied at Damascus and dated 4 Rabīʿ II 548 AH, that is 29 June 1153 CE, and look at the diagrams (S082). Ask what survived that journey and what did not. The theorems survived. The diagrams survived, redrawn every time. The Greek prose style, the word choices, the author's voice: gone. This is a translation studies lesson, a history lesson and a mathematics lesson using one object.
The geometry you are teaching anyway. Put Euclid II.12 and II.13 side by side on the board, in Heath's wording, and next to them write (S076, S077). Ask students to find the pieces. The term in II.12 is . The flip from "greater than" to "less than" between the two propositions is the cosine changing sign at 90 degrees. Euclid needs two propositions where we need one, and that is the price of not having signed numbers. Then do the same with Ptolemy's theorem and the sine subtraction formula, and derive one from the other in five lines, as above.
A numerical methods lesson hiding in Book I. Ptolemy could not compute exactly, so he trapped it between and . The gap between his two bounds was 0.000185, smaller than the 0.000278 that one unit in his last printed digit is worth. So he read off the answer (S061, I.10). That is bounding an intractable quantity to within reporting precision, which is what numerical analysis is. Have students redo it. Then have them redo it with exact values of the two input chords, and watch the stated lower bound come out slightly too high, because the input data were rounded. Error propagation, on a two-thousand-year-old worked example, with the original text in front of them.
- A chord joins the endpoints of an arc; fix the radius (Ptolemy: R = 60) and the chord of every arc becomes a function you can tabulate, the first one.
- The halving that puzzles students, , is the exact bridge between the Greek table and your sine.
- Hipparchus's table is attested but lost, and reconstruction is argued; Ptolemy's is extant, and his theorem plus four rules fills all 360 rows.
Next: India cuts the chord in half, and the half-chord, unlike the chord, attaches to a right triangle.
Every identity on your properties page descends from the four chord rules that generated Ptolemy's 360 rows. The supplementary-angle and half-angle patterns you use were table-building machinery first. They are collected in Trigonometry B.5
A chord belongs to a whole arc; your sine belongs to half of one. State the one-line identity that connects them.
Show the answer▸
, which is at Ptolemy's R = 60. Read right to left, it says a sine is half the chord of twice the arc, which is literally what the Sanskrit and Arabic words for sine meant.
Chapter 3
The Half-Chord Becomes the Sine (India, c. 400 to 1600 CE)
The people in this chapter
Faces where a face survives. Every name links to its full entry in Appendix A.
A radius that is not a round number
Open an Indian astronomy manuscript from the sixth century and you keep meeting the number 3438. It is the radius of every circle in the book. Nobody in the text explains it. It is not round. It is not a power of anything. It is not a count of days, or a number of gods.
It is 3437.7468, rounded up.
That number is this chapter in miniature. An astronomer working near modern Patna chose 3438 as the radius of the sky. Once you see why, you see what Indian mathematicians did to Greek trigonometry. You also learn something most students never get told. The sine you use in class has a unit system baked into it. That unit system, written down, is 3438.
The chapter runs from about 400 CE to about 1600. The early date is the earliest plausible one for the surviving Surya Siddhanta, or Treatise of the Sun. That book is an anonymous Sanskrit astronomy handbook (S121). The late date is when the last mathematicians of the Kerala school were still expanding sines as infinite series (S127). Across those twelve hundred years three things happen. The Greek chord becomes the half-chord, which is the sine. The half-chord gets a table, then an interpolation rule, then a closed-form approximation, then a power series. And the Sanskrit word for a bowstring travels through Arabic and Latin and comes out the other end as the English word "sine."
Indian sine tables fix the radius at 3438. Not 1000, not 3600, not any round number a committee would pick. Where does 3438 come from?
I have a guess▸
It is the radius measured in arcminutes of its own circle. A full circle is arcminutes, so , which rounds to 3438 (S121, S122). Sizing the radius in units of arc makes arc and sine nearly equal for small angles, which is exactly the property the radian buys you, twelve centuries early.
If you guessed an arbitrary convention, that is how most summaries wave it through. The chapter derives it, and the derivation is the reason this book calls R = 3438 the radian in disguise.
The problem: predicting the moon for a calendar that could not be wrong
Nobody invented the sine because it was pretty. Indian astronomers had a calendar to run. The Hindu ritual calendar is not a fixed grid of months. The sun and the moon drive it together. Three things have to be right. The moment of new moon. The moment the sun crosses from one zodiacal sign into the next. The timing and extent of eclipses. Get those wrong and rituals happen on the wrong day.
So the working astronomer had to predict where the moon and the planets would be, months in advance, by hand. The models were geometrical. A body moves on a circle whose center moves on another circle. That second circle is the epicycle. The ecliptic is the sun's yearly path across the sky. Turning that picture into a position on the ecliptic means finding the perpendicular from a point on a circle to a line. That perpendicular is a half-chord. Every epicycle correction in the whole tradition comes out as a half-chord. That is why the Surya Siddhanta stops in the middle of a chapter on mean motions. It hands you twenty-four numbers and a rule for filling in between them (S121).
Two more constraints shaped the mathematics. First, every calculation ran in sexagesimal arithmetic, meaning base 60. The units were degrees, arcminutes, and arcseconds, inherited from Babylonian astronomy by way of Greek astronomy. Second, a technical text in Sanskrit was a memorized object. That matters more than it sounds. Authors composed in metrical verse. Students learned it by heart and recited it. A scribe might write down a commentary, but the root text lived in people's heads. That is why Aryabhata packs twenty-four numbers into one line of verse. It is also why an eleventh century astronomer will happily accept a sine rule that is 3 percent wrong. The test he cares about is whether he can work it out while walking.
Cutting the chord in half
Hipparchus (hip-PAR-kuss, c. 190 to c. 120 BCE) and Ptolemy (TOL-uh-mee, c. 100 to c. 170 CE) tabulated the chord: the straight line joining the two ends of an arc. Indian astronomers tabulated half of that chord, for half of that arc. Halve the chord and you have the sine. That is the move the whole chapter turns on. The Sanskrit word for the chord is jya, "bowstring," and the technical form for the half-chord is ardha-jya, "half-chord" (S128).
That change is not cosmetic. Take a circle of radius and an arc of at the center. Join the ends of the arc: that is the chord, . Now drop a perpendicular from the center to the chord. It bisects the chord and it bisects the arc. So it cuts the whole figure into two congruent right triangles. Each triangle has hypotenuse , an angle at the center, and an opposite side equal to half the chord:
Read that from right to left, because that is the direction the history runs. The half-chord is . A chord is a right triangle with a mirror image stuck onto it. A half-chord is the right triangle by itself.
Once the object of study is a right triangle instead of a chord, the identities get shorter. Ptolemy's chord version of the difference formula is a nest of square roots. Chords of supplementary arcs are linked through the diameter. So every step drags a radical along. The half-chord version is the formula you memorized. Bhaskara II stated it in India in 1150, and Madhava in Kerala around 1400:
The stray in the denominator is there because these tables do not use a radius of 1. Divide through by and it disappears. That is exactly what European mathematicians did, six centuries later.
The other two functions arrive in the same package, named from the same picture. Koti-jya is the cosine. Koti means "the curved end of a bow," then "an extremity" in general, then, in trigonometry, "the complement of an arc to 90 degrees," which makes koti-jya the jya of the complementary arc (S128). Utkrama-jya, "reversed sine," is the versed sine or versine, . Picture the little bite of radius left between the center of the chord and the arc. Indian writers also called it the "arrow," isu or bana. If the arc is the bow and the chord is the bowstring, the versine is the arrow lying across it, pointing outward (S128, S130).
That picture is worth drawing on a board once. It explains the whole vocabulary at a stroke: bow, bowstring, half-bowstring, arrow.
R = 3438: the radian in disguise
Here is where the number in the hook comes from.
A full circle is 360 degrees, and each degree is 60 arcminutes. So a full circle is arcminutes of arc. Now measure the radius in those same arcminutes. You are sizing it in units of arc length, not in some separate unit of distance. The circumference is , so:
Round to the nearest whole arcminute and you get 3438. Walter Eugene Clark translated the Aryabhatiya in 1930. He states the convention flatly, without fuss: "The sines are given in minutes (of which the radius contains 3,438) at intervals of 225 minutes" (S122).
You might object that this uses our value of , not theirs. It does not matter. Aryabhata I (ARE-yuh-buh-tuh, 476 to 550 CE) gives his own value in Ganitapada II.10 of the Aryabhatiya (The Book of Aryabhata, 499 CE). His birth year follows from the text. His death year is conventional, and no source read for this chapter attests it. He says a circle of diameter 20000 has circumference 62832, which makes (S122). Run the same calculation with his number:
Also 3438. The choice fits his own geometry, not ours alone. The rounding hides no disagreement about .
Now the payoff. For a student about to meet radians, it is the most useful thing in this chapter.
Choosing arcminutes means one thing. You are measuring the sine in the same units as the arc. Both are lengths on the same circle, and both are counted in arcminutes. For a small arc, the sine and the arc are nearly the same length. In these units, then, they are nearly the same number. Watch it happen. The step between entries in every Indian sine table is 3 degrees 45 minutes. That is exactly 225 arcminutes. The first entry of every one of those tables is 225. The arc is 225 and the sine is 225. Over that first step the sine has not yet bent away from the arc. It is not far enough off to change the leading digits. The true value is , and the tables say 225 (S121).
That is the radian, twelve hundred years before anybody named it. A radian is the angle whose arc equals the radius. Aryabhata's convention says the same thing sideways. Pick the radius so that one unit of arc length is one arcminute of angle. Now angle and length are the same kind of number. The conversion factor between them is . And arcminutes is one radian written in arcminutes. Your calculator runs the same conversion in the opposite direction. Type sin(30) in degree mode and the machine turns 30 degrees into 0.5236 radians before it evaluates anything. The power series it uses works only in radians. Aryabhata built the conversion into the radius. We build it into the function.
Physics hook. This is the small-angle approximation . Students meet it again in the simple pendulum, in paraxial optics, and in every thin lens derivation. In those settings it is an approximation you apply and then apologize for. Here it is the design principle of the table. Have students compute and compare it with the tabulated 225. Then ask at which entry of the table the approximation has visibly failed. (Entry 2 is 449 against a true 448.749. By entry 6 the tabular 1315 is below the true 1315.666. The sine has fallen well behind the arc.) The tie to physics is direct. The "small angle" in a physics problem is small compared to one radian, and one radian is 3438 of the units these astronomers were counting in.
Not everyone used 3438. Varahamihira used 120, copying Greek practice. Brahmagupta used 150 in one work. Manjula used 488 on purpose, to make his table fit in a person's head. Each of those choices, covered below, tells you what its author was optimizing for.
The Surya Siddhanta: the oldest surviving half-chord table
The Surya Siddhanta holds the earliest surviving Indian sine table in half-chord form. It gives 24 entries at intervals of 3 degrees 45 minutes, with arcminutes. It adds 24 versed sines (S121, S128). Datta and Singh call it "the earliest Hindu treatise in which the above trigonometrical functions are now found recorded" (S128, p. 40).
The dating fight, in full
The date is c. 400 CE (disputed). The dispute deserves to be laid out rather than smoothed over. It shows how a text with no author and no colophon gets dated at all. A colophon is the note at the end of a manuscript. It gives the scribe, the place, and the date.
P. C. Sengupta edited the 1935 Calcutta reprint of Burgess's translation and wrote a long introduction to it. He concludes that "the earliest date of the Surya Siddhanta cannot be pushed up much higher than 400 A.D." (S121). That is a ceiling, not a birth certificate. He is saying that the surviving recension cannot be much older. A recension is the version of a text that has come down to us. He is not naming a year.
Against that, Sengupta reports a competing tradition. Sudhakara Dvivedi cited Nityananda, the author of the Siddhantaraja, for a date of Kali 3000 elapsed. That equals Shaka 421, which equals 499 CE, exactly Aryabhata's year. Sengupta adds the crucial caveat himself: Nityananda's reasons "are not stated" (S121). So the 499 date is a bare assertion in a later text, repeated by a nineteenth century scholar. No argument is attached that anyone can inspect.
A third position appears in Ebenezer Burgess's original 1860 apparatus, which the reprint preserves. It is John Bentley's attempt to date the text from the errors in its planetary numbers. The idea runs like this. A set of planetary parameters is most accurate near the epoch it was fitted to. So you can work backwards from the size of the errors to the date of composition. Burgess reproduces Bentley's argument and its much later date without endorsing it (S121). That method has a long history of producing dates other methods refuse to confirm. Burgess treats it as something the reader should know exists. He is not prepared to defend it.
Here is the honest summary for a classroom. The surviving Surya Siddhanta is a text of roughly the fifth century. Sengupta argues a floor for it. A competing precise date of 499 CE rests on an unexplained statement by Nityananda. Bentley's error-fitting method gives an outlier that its own translator declines to back.
The verses
The table is given in words, not numerals, because the text is verse. Burgess translates Surya Siddhanta II.17 to II.22 as a running list (S121):
"Two hundred and twenty-five; four hundred and forty nine; six hundred and seventy-one; eight hundred and ninety, eleven hundred and five; thirteen hundred and fifteen; ... Fifteen hundred and twenty; seventeen hundred and nineteen; nineteen hundred and ten; two thousand and ninety-three; Two thousand two hundred and sixty-seven; two thousand four hundred and thirty-one; two thousand five hundred and eighty-five; two thousand seven hundred and twenty-eight; Two thousand eight hundred and fifty-nine; two thousand nine hundred and seventy-eight; three thousand and eighty-four; three thousand one hundred and seventy-seven; Three thousand two hundred and fifty-six; three thousand three hundred and twenty-one; three thousand three hundred and seventy-two; three thousand four hundred and nine; Three thousand four hundred and thirty-one; three thousand four hundred and thirty-eight."
Then comes the labor-saving instruction that gives you the versed sines for free: "Subtracting these, in reversed order, from the half-diameter, gives the tabular versed-sines (utkramajyardhapindaka)" (S121, II.22).
That rule is exact, and it is worth seeing why. The versed sine of an arc is , and . The table is symmetric about the quadrant. So the sine of the complementary arc is another entry of the same list, read from the far end. The th versine is therefore minus the th sine. No new computation is needed. I checked all 24 entries against the printed versines. The relation holds exactly, with no exceptions (S121).
The full table
| n | arc | Hindu sine | difference | versed sine | modern R sin | error |
|---|---|---|---|---|---|---|
| 1 | 3 deg 45 min | 225 | 225 | 7 | 224.856 | +0.144 |
| 2 | 7 deg 30 min | 449 | 224 | 29 | 448.749 | +0.251 |
| 3 | 11 deg 15 min | 671 | 222 | 66 | 670.721 | +0.279 |
| 4 | 15 deg | 890 | 219 | 117 | 889.820 | +0.180 |
| 5 | 18 deg 45 min | 1,105 | 215 | 182 | 1,105.109 | -0.109 |
| 6 | 22 deg 30 min | 1,315 | 210 | 261 | 1,315.666 | -0.666 |
| 7 | 26 deg 15 min | 1,520 | 205 | 354 | 1,520.589 | -0.589 |
| 8 | 30 deg | 1,719 | 199 | 460 | 1,719.000 | +0.000 |
| 9 | 33 deg 45 min | 1,910 | 191 | 579 | 1,910.050 | -0.050 |
| 10 | 37 deg 30 min | 2,093 | 183 | 710 | 2,092.922 | +0.078 |
| 11 | 41 deg 15 min | 2,267 | 174 | 853 | 2,266.831 | +0.169 |
| 12 | 45 deg | 2,431 | 164 | 1,007 | 2,431.033 | -0.033 |
| 13 | 48 deg 45 min | 2,585 | 154 | 1,171 | 2,584.825 | +0.175 |
| 14 | 52 deg 30 min | 2,728 | 143 | 1,345 | 2,727.549 | +0.451 |
| 15 | 56 deg 15 min | 2,859 | 131 | 1,528 | 2,858.593 | +0.407 |
| 16 | 60 deg | 2,978 | 119 | 1,719 | 2,977.395 | +0.605 |
| 17 | 63 deg 45 min | 3,084 | 106 | 1,918 | 3,083.448 | +0.552 |
| 18 | 67 deg 30 min | 3,177 | 93 | 2,123 | 3,176.298 | +0.702 |
| 19 | 71 deg 15 min | 3,256 | 79 | 2,333 | 3,255.546 | +0.454 |
| 20 | 75 deg | 3,321 | 65 | 2,548 | 3,320.853 | +0.147 |
| 21 | 78 deg 45 min | 3,372 | 51 | 2,767 | 3,371.940 | +0.060 |
| 22 | 82 deg 30 min | 3,409 | 37 | 2,989 | 3,408.587 | +0.413 |
| 23 | 86 deg 15 min | 3,431 | 22 | 3,213 | 3,430.639 | +0.361 |
| 24 | 90 deg | 3,438 | 7 | 3,438 | 3,438.000 | +0.000 |
Four things in that table are worth pointing at.
The worst entry is number 18, at 67 degrees 30 minutes. The text says 3177 and the true value is 3176.298. The error is 0.702 arcminutes. Against the radius that is . Read as a pure sine, then, the table is good to about three and a half decimal places. You will often see the claim that these tables are "accurate to four decimal places," and it is a rounding of a rounding. Report the number, not the slogan.
Entry 8, at 30 degrees, is exact. Here on the nose, and 3438 happens to be even. So the sine is exactly half of 3438, which is 1719. Entry 24 is exact by definition. Everything else rounds an irrational number to a whole arcminute. Errors of a few tenths are the most you can hope to avoid.
The versines, those little bites of radius between chord and arc, run from 7 to 3438. The last one equals the radius, as it must: the versine of 90 degrees is .
And the differences, read down that fourth column, are 225, 224, 222, 219, 215, 210, 205, 199, 191, 183, 174, 164, 154, 143, 131, 119, 106, 93, 79, 65, 51, 37, 22, 7. Hold that list. It is the whole of Aryabhata's sine verse, and the next section is about it.
How the table was probably really built
The Surya Siddhanta offers a recursive rule of its own at II.15 to 16, and Burgess flatly refuses to believe it was the method of construction: "It is not to be supposed, however, that the Hindu sines were originally obtained by the process described in the text. That process was, in all probability, suggested by observing the successive differences in the values of the sines as already determined by other methods" (S121).
His reconstruction of the real method is a chain of halvings, and it is easy enough for a class to reproduce: "The sine of 90 degrees was obviously equal to radius, and the sine of 30 degrees to half radius: from the first could be found the sines of 45 degrees, 22 degrees 30 minutes, and 11 degrees 15 minutes; from the latter, those of 15 degrees, 7 degrees 30 minutes, and 3 degrees 45 minutes" (S121). Two known values, a half-angle formula, and repeated bisection build a scaffold. You get exact arcs at 3 degree 45 minute spacing. The rest of the quadrant follows from the complement rule and the addition theorem.
This is an 1860 opinion, and it should be labeled as one. But it is the kind of opinion that comes from someone who has just spent months inside the arithmetic.
Aryabhata I: twenty-four differences in one line of verse
Dating him from inside his own book
Aryabhata dates himself, which is rare and convenient. Kalakriyapada III.10 says that when three yugapadas and "sixty times sixty" years, that is 3600 years, of the yuga had elapsed, 23 years of the author's life had passed (S122). A yuga is one of the vast world ages of Indian cosmology. A yugapada is a quarter of one. Clark works the conversion and lands on 499 CE: "we arrive at the date 499 a.d. It is natural to take this as the date of composition of the treatise" (S122). If he was 23 in 499, he was born in 476.
Clark then complicates his own conclusion, and this is the part that usually gets dropped. He notes that Parameshvara, the Kerala commentator, quotes the Prakasikakara on the point. On that reading, 499 is not the date of composition at all. It is the epoch of the calculations, the zero point the planetary tables are reckoned from (S122). Those are different claims. Here is a modern parallel. An ephemeris is a table of predicted planet positions. One published in 2026 might use J2000.0 as its epoch. A reader a thousand years from now who dated the book to the year 2000 would be off by a generation.
So 499 CE for the Aryabhatiya is firm as an internal date. What that date marks is contested. The objection comes from inside the Indian commentarial tradition itself.
His location is also contested. He worked at Kusumapura, usually identified with Pataliputra, near modern Patna in Bihar. Bhaskara I calls him asmakiya, "of the Asmaka country," which lay between the Narmada and the Godavari rivers. So his origin and his workplace may not be the same. K. Chandra Hari has argued for a Kerala origin, and other scholars dispute it (S136). The prudent phrasing runs like this. He worked at Kusumapura, and he came from somewhere else we cannot pin down.
Numbers hidden inside syllables
The Aryabhatiya is 121 verses long. Into that space it packs a planetary theory and a sine table. It also packs algorithms for square roots and cube roots, an area rule, and a way to solve linear indeterminate equations. That is an entire astronomy course you could memorize in an afternoon and carry for life. Aryabhata needed large numbers in metrical verse. So he invented an alphabetic numeral system, a code that turns digits into syllables.
It is not the katapayadi system. That is a different, later scheme. Each consonant stands for a single digit. A whole phrase reads off as a decimal number. The Kerala mathematicians used katapayadi a thousand years later. It shows up further down this chapter, in the chronogram, a number hidden inside a line of verse, that encodes Madhava's radius. Confusing the two is the commonest error in popular accounts of Indian mathematics (S122).
Aryabhata's own scheme works by place classes. Clark translates the defining stanza, the paribhasa: "Beginning with ka the varga letters (are to be used) in the varga places, and the avarga letters (are to be used) in the avarga places. Ya is equal to the sum of na and ma. The nine vowels (are to be used) in two nines of places varga and avarga" (S122, stanza B). Unpacked, with Clark's exposition:
The varga ("square") places are the 1st, 3rd, 5th and so on, counting from the right. The avarga, or non-square, places are the 2nd, 4th, 6th and so on. The 25 consonants from k to m take the values 1 to 25. The semivowels and sibilants from y to h take 30, 40, 50, 60, 70, 80, 90, 100. A vowel attached to a consonant sets the place. It multiplies a varga letter by 1, 100, 10000 and so on, and an avarga letter by 10, 1000, 100000 and so on. Long and short vowels count the same. That leaves the poet free to fix the meter (S122).
Clark's own worked example is the number of revolutions of the moon in a yuga, from Dasagitika I.1. It is given as the single word cayagiyinusuchlr:
Clark adds two notes that a student should hear. First, on the ordering: "It happens here that the digits are given in order from right to left, but they may be given in reverse order or in any order which will make the syllables fit into the meter" (S122). The encoding is order-free because the place value travels with the vowel, not with the position in the word. Second, on scope: "The alphabetical notation is employed only in the Dasagitika" (S122). It is a compression format for one section, not a general-purpose numeral system. Aryabhata writes ordinary number words elsewhere.
Language and computing hook. This is a data compression scheme with a hard design brief. The output must be pronounceable. It must scan in a fixed meter. It must decode without ambiguity. And it must survive a thousand years of copying and recitation by people who may not understand it. Three modern schemes belong beside it. Base64 turns arbitrary bytes into printable characters. The NATO phonetic alphabet trades length for error resistance over a noisy radio link. The ISBN check digit catches a single transposed digit. Have a class encode their birth year in Aryabhata's scheme and say it aloud. Then ask what happens if a copyist drops a vowel. (Answer: a digit changes place, and the number can change by orders of magnitude. Verse meter is the error-detecting code. A dropped syllable breaks the meter, and the reciter hears it.)
The verse itself, and a numbering trap
Shukla and Sarma number the sine verse Gitikapada 12 in the 1976 Indian National Science Academy edition (S123). Clark numbers it I.10 in 1930 (S122). It is the same verse. Clark's numbering runs lower because he leaves three things uncounted. Those are the technical paribhasa stanza on the alphabetic notation, the invocation, and the colophon. Find "Gitikapada 12" in one edition and "I.10" in another, and do not conclude that Aryabhata stated the table twice. Cite the edition, not just the number.
Clark's translation of the verse is: "The (twenty-four) sines reckoned in minutes of arc are 225, 224, 222, 219, 215, 210, 205, 199, 191, 183, 174, 164, 154, 143, 131, 119, 106, 93, 79, 65, 51, 37, 22, 7" (S122).
And then Clark corrects his own translation in a note: "The numbers given here are in reality not the values of the sines themselves but the differences between the sines" (S122).
Shukla and Sarma print the same 24 numbers with the Sanskrit, which their scan renders as makhi bhakhi phakhi dhakhi nakhi nakhi nakhi hasjha skaki kisga sghaki kighva | ghlaki kigra hakya dhaki kica sga jhasa nva kla pta pha cha kaladhajyah || 12 ||, with the editors' apparatus recording manuscript variants at kisva, at hakya dhaki kica, and at kigra (S123). The numbers are certain. The exact syllables are less so, for two reasons. The Devanagari OCR of that scan is imperfect. And the manuscripts themselves disagree in three places. That is normal for a text copied by hand for fourteen centuries. It is also a useful thing for students to see. The mathematics is stable; the spelling is not.
Why differences, and what they sum to
Tabulating differences instead of values is the elegant move. Brevity is part of the reason. The largest difference is 225 and the smallest is 7. So every number in the verse is at most three digits, while the sines themselves run to four.
Brevity is not the whole reason, though. The differences are what you need anyway. To interpolate is to fill in a value the table does not list. For that you need the local difference. To build the table, you add the differences up. To compute an instantaneous rate of motion, as the astronomers did, you use the difference itself as a discrete derivative. Aryabhata tabulates the derivative and lets you integrate it yourself.
They sum to exactly 3438. Here are the running totals:
The full sequence of running sums is 225, 449, 671, 890, 1105, 1315, 1520, 1719, 1910, 2093, 2267, 2431, 2585, 2728, 2859, 2978, 3084, 3177, 3256, 3321, 3372, 3409, 3431, 3438. That is the Surya Siddhanta sine column, entry for entry, with nothing left over (S121, S122). Clark makes the cross-reference himself: "Compare Suryasiddhanta (II, 15-27)" (S122). The last difference is 7 rather than 0. That is the sine flattening out at the top of the quadrant without quite stopping.
The second differences are the differences between the differences. A second-order interpolation rule is built on them. They run 1, 2, 3, 4, 5, 5, 6, 8, 8, 9, 10, 10, 11, 12, 12, 13, 13, 14, 14, 14, 14, 15, 15. They increase steadily, and that is the discrete signature of . The amount by which the sine's growth slows is itself proportional to how large the sine has become.
| n | difference | running sum | second difference |
|---|---|---|---|
| 1 | 225 | 225 | 1 |
| 2 | 224 | 449 | 2 |
| 3 | 222 | 671 | 3 |
| 4 | 219 | 890 | 4 |
| 5 | 215 | 1,105 | 5 |
| 6 | 210 | 1,315 | 5 |
| 7 | 205 | 1,520 | 6 |
| 8 | 199 | 1,719 | 8 |
| 9 | 191 | 1,910 | 8 |
| 10 | 183 | 2,093 | 9 |
| 11 | 174 | 2,267 | 10 |
| 12 | 164 | 2,431 | 10 |
| 13 | 154 | 2,585 | 11 |
| 14 | 143 | 2,728 | 12 |
| 15 | 131 | 2,859 | 12 |
| 16 | 119 | 2,978 | 13 |
| 17 | 106 | 3,084 | 13 |
| 18 | 93 | 3,177 | 14 |
| 19 | 79 | 3,256 | 14 |
| 20 | 65 | 3,321 | 14 |
| 21 | 51 | 3,372 | 14 |
| 22 | 37 | 3,409 | 15 |
| 23 | 22 | 3,431 | 15 |
| 24 | 7 | 3,438 | n/a |
The recursion that does not reproduce his own table
Aryabhata also gives a rule for generating the differences, at Ganitapada II.12. Clark's translation: "By what number the second sine is less than the first sine, and by the quotient obtained by dividing the sum of the preceding sines by the first sine, by the sum of these two quantities the following sines are less than the first sine" (S122).
Clark's own worked start makes the machinery clear (S122): "Subtract 225 from 225 and the remainder is 0. Divide 225 by 225 and the quotient is 1. The sum of 0 and 1 is subtracted from 225 to obtain the second sine 224. Subtract 224 from 225 and the remainder is 1. Divide 225 plus 224 by 225 and the nearest quotient is 2. Add 2 and 1 and subtract from 225. The third sine will be 222."
In symbols, writing for the th difference and rounding each quotient to the nearest whole number:
This is a discrete version of , written in 499 CE. As mathematics it is impressive. The amount the difference shrinks at each step is proportional to the sine accumulated so far.
It also does not reproduce his own table.
I ran the rule exactly as Clark states it, rounding each quotient to the nearest integer, and it gives:
225, 224, 222, 219, 215, 210, 204, 197, 189, 181, 172, 162, 151, 140, 128, 115, 102, 88, 74, 60, 45, 30, 15, 0
Six entries agree with the verse. From entry 7 onward it drifts. The verse says 205 and the rule says 204, and the gap widens. By entry 12 the verse says 164 and the rule says 162. The last entry comes out 0 instead of 7, and the differences sum to
which is 70 short of the radius. A table built this way would put at 3368 instead of 3438. That is an error of 2 percent, far worse than any single entry of the real table.
Taking the floor of the quotient instead of rounding is worse still. That variant gives 225, 224, 223, 221, 218, 214, 209, 203, and so on. It diverges from the third entry onward.
Clark saw the problem and said so: "If this method is followed strictly there results several slight divergences from the values given in I, 10. It is possible to reconcile most of these by assuming, as Krishnaswami Ayyangar does, that from time to time the neglected fractions were distributed among the sines. But of this there is no indication in the rule as given" (S122). In other words, you can rescue the rule. Let the person doing the calculation carry the discarded fractional parts along and spend them later. The verse does not tell you to do that.
The conclusion is not that Aryabhata was careless. It is that the verse is a mnemonic reconstruction of a table built by other means. Burgess argued exactly the same for the Surya Siddhanta. Aryabhata has one other construction rule, a geometric one at Ganitapada II.11 ("One should divide a quarter of the circumference of a circle... From the triangles and quadrilaterals which are formed one will have on the radius as many sines of equal arcs as are desired"). On that rule Clark gives up: "The exact method of working out the table is not known" (S122).
Clark points to two older studies of exactly this problem. Both are worth naming for anyone who wants to go further. One is A. A. Krishnaswami Ayyangar in the Journal of the Indian Mathematical Society XV (1923 to 1924), pages 121 to 126. The other is S. N. Naraharayya, "Note on the Hindu table of sines," in the "Notes and Questions" section of the same volume, pages 105 to 113 (S122). I read neither for this chapter. They are leads, not evidence.
Varahamihira: a Greek radius in Ujjain
Varahamihira (vuh-RAH-huh-MIH-hih-ruh, c. 505 to 587 CE, disputed) worked at Ujjain, in the region of Avanti. He calls himself Avantyaka, "the man from Avanti." Scholars have identified his probable birthplace, Kapitthaka, with Kayatha and with Sankissa. The question is not settled (S138).
What the year 505 dates
The Pancasiddhantika (Treatise on the Five Astronomical Canons) refers to Shaka 427, which corresponds to 20 to 21 March 505 CE. That correspondence is firm. What it dates is not (S138).
Some scholars read Shaka 427 as Varahamihira's birth year, others as the date the work was composed (S138). Those readings put his career in different halves of the sixth century. Two further data points constrain it. The commentator Amaraja reports that Varahamihira died in Shaka 509, which is 587 CE. And al-Biruni, writing from Central Asia in the eleventh century, places him at 505 (S138). A birth in 505 with a death in 587 gives a life of 82 years and makes the Pancasiddhantika a mature work. A composition date of 505, with a death in 587, makes it an early one.
The book itself is a digest of five earlier astronomical systems. That is why it is called what it is. It is also our main window onto Indian astronomy just before and beside Aryabhata.
R = 120, and a Greek shortcut
Varahamihira's sine table does not use 3438. It uses 120, and his translator George Thibaut thought he knew why.
Thibaut's introduction is explicit about the unusual choice: "The most interesting feature of the table is that it bases on a subdivision of the Radius into 120 parts and of each of those 120 parts into 60; instead of subdividing the Radius, in the ordinary Indian fashion, into 3438'. It thus closely follows the Greek fashion of expressing the values of sines, only preferring to divide the Radius into 120 parts instead of sixty. In the majority of cases the agreement of the stated values of the sines with those given by Ptolemy is as close as possible" (S124, Introduction pp. xxxvi to xxxvii).
Then he offers a mechanism, and this is his own inference rather than a statement of the text: "in case of the table of sines basing on a Greek prototype, the plan of subdividing the Radius, and not the diameter, into 120 parts would have enabled the borrower to take over, without any change, the amounts assigned in the Greek table to the chords of the angles, and to insert them in his own table as the values of the sines of half those angles" (S124).
Work through why that is true. It is the cleanest illustration in this chapter of the chord-to-sine relation. Ptolemy divides the diameter into 120 parts, so his radius is 60. He tabulates . Varahamihira divides the radius into 120 parts and tabulates . Set and the two expressions are the same number. A copyist who understood nothing at all could take Ptolemy's chord for 30 degrees and write it down as the sine of 15 degrees. He would be exactly right.
Thibaut also warns the reader about his own editorial hand, which is the sort of admission that should always be quoted when it exists: "The Sanskrit text had in a few places to be emended to a considerable extent" (S124).
The table
| n | arc | PSi sine | modern 120 sin | error in sixtieths |
|---|---|---|---|---|
| 1 | 3 deg 45 min | 7' 51" | 7.848 | +0.10 |
| 2 | 7 deg 30 min | 15' 40" | 15.663 | +0.21 |
| 3 | 11 deg 15 min | 23' 25" | 23.411 | +0.35 |
| 4 | 15 deg | 31' 4" | 31.058 | +0.50 |
| 5 | 18 deg 45 min | 38' 34" | 38.573 | -0.36 |
| 6 | 22 deg 30 min | 45' 56" | 45.922 | +0.68 |
| 7 | 26 deg 15 min | 53' 5" | 53.075 | +0.52 |
| 8 | 30 deg | 60' 0" | 60.000 | 0.00 |
| 9 | 33 deg 45 min | 66' 40" | 66.668 | -0.11 |
| 10 | 37 deg 30 min | 73' 3" | 73.051 | -0.08 |
| 11 | 41 deg 15 min | 79' 7" | 79.122 | -0.29 |
| 12 | 45 deg | 84' 51" | 84.853 | -0.17 |
| 13 | 48 deg 45 min | 90' 13" | 90.221 | -0.25 |
| 14 | 52 deg 30 min | 95' 13" | 95.202 | +0.86 |
| 15 | 56 deg 15 min | 99' 46" | 99.776 | -0.58 |
| 16 | 60 deg | 103' 56" | 103.923 | +0.62 |
| 17 | 63 deg 45 min | 107' 38" | 107.625 | +0.52 |
| 18 | 67 deg 30 min | 110' 53" | 110.866 | +1.07 |
| 19 | 71 deg 15 min | 113' 38" | 113.632 | +0.10 |
| 20 | 75 deg | 115' 56" | 115.911 | +1.33 |
| 21 | 78 deg 45 min | 117' 43" | 117.694 | +1.35 |
| 22 | 82 deg 30 min | 119' [seconds lost in the scan] | 118.973 | not computed |
| 23 | 86 deg 15 min | 119' 45" | 119.743 | +0.42 |
| 24 | 90 deg | 120' 1" | 120.000 | +1.00 |
Three notes on that table.
Entry 22 is incomplete, and the gap is mine, not Varahamihira's. The seconds figure fell across a page break in the OCR of the scan I worked from. So the cell reads 119 parts and an unknown number of sixtieths (S124). The modern value is 118.973 parts. The neighboring entries run about half a sixtieth to one and a third sixtieths high. So the true reading is probably close to 119 parts 0 sixtieths. That is a guess, flagged as a guess. Anyone preparing this for print should recover the figure from the page image.
Entry 24 reads 120 parts 1 sixtieth, which is larger than the radius itself. A sine cannot exceed the radius, so this is an accumulated rounding artifact. Thibaut explains how it arises. In the text, "the amount of each sine is not stated directly, but has to be arrived at by the summation of the amounts of all the preceding sines" (S124). Varahamihira, like Aryabhata, tabulates differences. Twenty-four roundings, each up to half a sixtieth, pile up into a final value one sixtieth too big. That is a fine classroom example of error accumulation in a recursive computation. It also shows why a modern numerical analyst worries about summing a long list.
Finally, compare the precision. With the smallest unit in the table is one sixtieth of a part, that is of the radius. With the smallest unit is one arcminute, that is of the radius. So the R = 120 table is finer per printed digit but coarser in effect. Varahamihira carries only two sexagesimal (base 60) places. The Indian tables carry the equivalent of three and a half decimal digits. The errors in the table above run up to about 1.35 sixtieths. That is of the radius. That is roughly the same relative accuracy as the Surya Siddhanta.
The three identities, and how much of the popular claim survives
Varahamihira is routinely credited with three results: , , and the half-angle formula. The credit is partly deserved and partly not, and the parts need separating.
The complement rule is his, unambiguously. Datta and Singh quote him directly: "The Rsine of 90 degrees minus latitude is the Rcosine of the latitude," citing Pancasiddhantika IV.28 (S128, p. 46). That is . He states it as a working rule for one quantity, the observer's latitude. That is how such rules were always stated.
The half-angle formula is his. Thibaut translates Pancasiddhantika IV.5: "Lessen the Radius by the sine of three signs (i.e. Radius) from which (three signs) double the required arc has been previously deducted, and multiply the remainder by sixty; the result is the square (of the desired sine). By deducting that square from the square of the Radius you obtain the square of the cosine" (S124). Datta and Singh gloss the same verse and explain the odd-looking 60: "The factor 4R on the right-hand side has been stated by Varahamihira as 60 since he has taken the value of the radius to be equal to 120. In modern notations, the above formula becomes sin^2 theta = (1/2)(1 - cos 2 theta)" (S128). The multiplier is only because . Change the radius and the constant in the verse changes with it. These rules are stated in units, not in the abstract.
There is a second, geometric form of the same idea, at Pancasiddhantika IV.3 to IV.4 in Thibaut's numbering: "take the double of the arc whose sine you wish to find, deduct it from the quarter of the circle, diminish the Radius by the sine of the remainder, and add to the square of half of that the square of half the sine of double the original arc. The square root of that sum is the desired sine" (S124). Datta and Singh render this as
which is the Pythagorean theorem applied to the little right triangle whose legs are half the chord and half the versine (S128).
The Pythagorean identity is where the popular claim overreaches. Thibaut's translation of Pancasiddhantika IV.4 does contain it, in the form of a subtraction rule: "The square root of that sum is the desired sine. The 'constant' square lessened by that sum (is the square) of the remaining quantity (i.e. of the cosine of the given arc)" (S124). The "constant," dhruva, is defined at IV.2 as the square of the radius. So the sentence says . The identity is in the text. What is not in the text is any sign that Varahamihira treated it as a theorem worth stating for its own sake. It appears as a step in a computing recipe.
Two independent cautions sit on top of that. First, look at their section on relations between the functions. There Datta and Singh credit the explicit statement of this relation not to Varahamihira but to Lalla (Sisyadhivrddhida II.30) and to Brahmagupta (Brahmasphutasiddhanta XXI.20f.). They quote Varahamihira only for the complement rule (S128). Second, David Eugene Smith is the usual ultimate source for the popular claim, and he hedges it carefully: "It is further probable, from the efforts made to develop simple tables, that the Hindus were acquainted with the principles which we represent by the formulas sin squared phi + cos squared phi = 1, [half-angle formula], and [a third relation], the last two of these appearing in the Panca Siddhantika of Varahamihira (c. 505)" (S421, vol. 2, p. 615). Read that sentence slowly. Smith puts the Pythagorean identity in the "probable" group, a general inference from the existence of good tables. He says that only the last two relations appear in the Pancasiddhantika. The confident textbook sentence "Varahamihira stated " is a hardening of a hedge.
A numbering conflict, again. Thibaut and Dvivedi (1889) place the half-angle rule at IV.5 and the geometric construction at IV.3 to IV.4. Datta and Singh cite IV.5 for the half-angle rule but IV.26 for the geometric construction and IV.28 for the complement rule (S124, S128). The editions divide the chapter differently. Cite the edition.
Bhaskara I: a sine you can compute without a table
Tables are heavy. Somebody has to copy them correctly. One miscopied digit spreads through every calculation that uses it. So every mathematical culture wants a formula that gets you close without a lookup.
Bhaskara I (BAHS-kuh-ruh the First, c. 600 to c. 680 CE) supplied one, and it is startlingly good. Datta and Singh date his statement of it to 629 CE (S128). The MAA Convergence project gives his dates as c. 600 to c. 680 and does not date the formula (S144). He was a commentator in the Aryabhata school. No source read for this chapter establishes where he lived.
The rule is at Mahabhaskariya (Greater Book of Bhaskara) VII.17ff., and Datta and Singh translate it: "Subtract the arc in degrees from the degrees of the semi-circumference and multiplying the arc by the remainder, put down (the result) at two places, (at one place) subtract (the quantity) from 40500; by one-fourth of the remainder divide the quantity (at the second place) multiplied by the maximum value of the function; thus the value of the direct or reversed Rsine of an arc and its complement is obtained wholly" (S128).
Follow the instructions literally. Let the arc be degrees. Compute and write it down twice. Subtract one copy from 40500. Take a quarter of what is left. Divide the other copy, multiplied by , by that quarter. Datta and Singh write the result as
and note the origin of the constant: (S128). With , in degrees:
An algebraically identical form, got by multiplying numerator and denominator by 4, is
and the MAA Convergence project gives the radian version, , and I checked that it is the same function (S144). Keep both degree forms in view. The "what the evidence does not support" section below is about what happens when someone splices them together.
Two worked values
Take degrees:
The modern value is . Bhaskara I is low by 0.0012244, which is 0.17 percent. That is four multiplications and one division, with no table in the room.
Take degrees:
The modern value is 0.2588190. This time the formula is high, by 0.0015360.
And gives 0.8648649 against a true 0.8660254, low by 0.0011605.
How good is it, exactly
I evaluated the formula at every thousandth of a degree from 0 to 180 and compared it with the modern sine:
- The maximum absolute error is 0.0016318, attained at degrees. Both the formula and the sine are symmetric about 90 degrees, so it is attained at as well.
- The formula is exact at , 30, 90, 150 and 180 degrees. It is exact at 30 and at 150, which is why it feels so accurate in the middle of the range.
- In units, the worst error is 5.61 arcminutes, about eight times worse than the Surya Siddhanta table's worst entry of 0.702 arcminutes. That is the trade being made: portability bought with precision.
- Near zero the relative error settles at about 1.859 percent. The formula's slope at the origin is per degree. The true slope is per degree. So for small arcs, where astronomers often worked, the approximation runs about 2 percent high every time. A careful user would know not to use it there.
Who else used it
Datta and Singh trace the formula forward and backward (S128).
Brahmagupta restates it with the constant divided by 4, at Brahmasphutasiddhanta XIV.23: "Subtract the degrees of an arc or its complement from the semicircle (i.e. 180) and multiply (the remainder) by that; subtract one-fourth the product from 10125; divide the product by the remainder and multiply by the semi-diameter; (the result) is the Rsine of that (arc or its complement)," which is
Sripati says almost the same thing in the Siddhanta-sekhara III.17 (1039 CE). Bhaskara II gives a chord version: "From five times the fourth part of the square of the circumference subtract the 'first', and by the remainder divide the 'first' multiplied by four times the diameter; the quotient will be the chord of the arc," which Datta and Singh reduce to the elder Bhaskara's formula. Ganesa was still using it in the Grahalaghava of 1545. That is nine centuries after it was written down.
Bhaskara I himself credits the formula to Aryabhata. Datta and Singh: "From a statement of Bhaskara I it appears that this formula was known to Aryabhata I," citing his commentary on Aryabhatiya I.11 (S128). That is a report of a claim, not a separate check. Present it that way. No verse of the Aryabhatiya contains the formula.
Bhaskara II's verdict on it is the fairest one. He calls the result sthula, "rough," and then says "it simplifies operations" (S128). That is what a working computer says about a good approximation.
Brahmagupta: the second difference, and where the rule lives
Brahmagupta (BRUH-muh-GOOP-tuh, c. 598 to c. 668 CE) was born at Bhillamala in Gurjaradesa, modern Bhinmal in Rajasthan, and later worked at Ujjain (S137). His dates rest on his two dated books. They also rest on the statements that he was 30 when he wrote the first and 67 when he wrote the second. The books are the Brahmasphutasiddhanta of 628 and the Khandakhadyaka (Edible Bite) of 665 (S125, S137). Clark measures the earlier book's distance from Aryabhata for the reader: it comes "129 years after the Aryabhatiya" (S122).
The problem with linear interpolation
A table gives you 24 values. You will want thousands. So you interpolate. The obvious way is linear. If you are 40 percent of the way from one entry to the next, add 40 percent of the difference.
That works well where the sine is nearly straight, near zero. It works badly where the sine curves, near 90 degrees. The reason is visible in the second-difference column printed above. The differences themselves are changing, by as much as 15 out of 225 from one step to the next. Linear interpolation pretends they are not.
Brahmagupta's fix has two steps. Average the difference just passed with the difference about to be traversed. Then correct that average by an amount proportional to how far into the interval you are.
Where the rule is, and where it is not
The rule is routinely dated to 628 and attached to the Brahmasphutasiddhanta. Datta and Singh say that is wrong, in as many words: "The earliest Hindu writer to do so was Brahmagupta. It is perhaps noteworthy that this more correct method of interpolation does not occur in his bigger work, Brahma-sphuta-siddhanta, which was composed in 628 A.D. but in his earlier monograph Dhyanagrahopadesa as well as in his later work Khanda-Khadyaka written in 665 A.D." (S128). Their citation is "KK, Part 2, i.4; DhGr, 17," and they give the Sanskrit incipit, the opening words by which a verse is known, as Gatabhogyakhandakantaradalavikalavadhacchatairnavabhiraptya / tadyuti yutonam bhogyadinadhikam bhogyam (S128).
So the Dhyanagrahopadesa, an earlier and much shorter work, has it at stanza 17. The supplement to the Khandakhadyaka has it at Part 2, chapter I, stanza 4, numbered I.8 in Sengupta's translation of the Uttara-Khandakhadyaka. The famous big book of 628 does not have it at all.
The rule itself
Sengupta translates the Uttara-Khandakhadyaka passage: "Multiply the residual arc left after division by 900 (i.e., by 15 degrees), by half the difference of the tabular difference passed over and that to be passed over and divide by 900 (i.e., 15 degrees): by the result increase or decrease, as the case may be, half the sum of the same two tabular differences; the result which, whether less or greater than the tabular difference to be passed, is the true tabular difference to be passed over" (S125).
Write for the tabular step, for the leftover arc, for the difference just passed and for the one about to be used. The "true difference to be passed over" is
and the interpolated function value is
Datta and Singh give the equivalent rearrangement
and say it "agrees with the formula method of interpolation, correct up to the second degree" (S128). In modern language this is a Newton-Stirling interpolation formula carried to second order, written in 665 CE.
The sign matters, and the tradition knew it. For Rsines the differences are decreasing, so you subtract the correction. For versed Rsines the differences are increasing, so you add it. Datta and Singh credit the explicit statement of that sign rule to Bhaskara II (S128).
The worked example, from the source
Brahmagupta's table in the Khandakhadyaka is coarse on purpose. It has six entries at 15 degree steps, with .
| n | arc | tabular sine | difference | modern 150 sin |
|---|---|---|---|---|
| 1 | 15 deg | 39 | 39 | 38.823 |
| 2 | 30 deg | 75 | 36 | 75.000 |
| 3 | 45 deg | 106 | 31 | 106.066 |
| 4 | 60 deg | 130 | 24 | 129.904 |
| 5 | 75 deg | 145 | 15 | 144.889 |
| 6 | 90 deg | 150 | 5 | 150.000 |
(The scan of Sengupta's translation prints the difference list as "39, 86, 31, 24, 15, 5". The 86 is an OCR error for 36, and the sines confirm it, since .) Sengupta's own comparison is "the calculated 'sines' are 38.82, 75, 106.06, 129.94, 144.89, 150. Hence Brahmagupta's 'sines' are all accurate to the nearest integer" (S125). My computation of gives 38.823, 75.000, 106.066, 129.904, 144.889, 150.000, which confirms it. Note in passing that Sengupta's fourth value, 129.94, is a typographical slip for 129.90.
Now Sengupta's worked example: find the Rsine of 57 degrees from that six-entry table (S125).
Fifty-seven degrees is 3420 arcminutes. Divide by the step of 900 arcminutes: . So three whole intervals have passed, taking us to 45 degrees. There the tabular sine is 106, the difference just passed is , and the difference about to be traversed is . The leftover is arcminutes out of .
Sengupta checks it against logarithms: "As worked out from the logarithm tables the same = 125.80" (S125). My computation gives .
Now do it the naive way, to see what the second-order term is buying. Linear interpolation uses alone:
So linear interpolation is off by 0.60 and Brahmagupta's rule is off by 0.04. That is an improvement of about a factor of 15, in exchange for one extra multiplication and one extra subtraction. On a six-entry table. That is the whole argument for second-order interpolation in one example.
A priority claim, and whose it is
Sengupta thinks this is a first, and says so twice. In his comment on the passage: Brahmagupta "thus takes a decidedly improved step here and is undoubtedly the first man in [the history of mathematics to teach interpolation using the second difference]." In his preface: Brahmagupta "in the Uttara Khandakhadyaka teaches, for the first time in the history of mathematics, the improved rules for interpolation by using the second difference," citing his own paper in the Bulletin of the Calcutta Mathematical Society XXIII, No. 3 (1931) (S125).
That is Sengupta's claim, made in 1934. Attribute it to him rather than state it as established fact. What is not in doubt is the text and the date.
The successors nobody quotes
Two later Indian astronomers gave second-order interpolation rules of their own, different from Brahmagupta's. They are almost never mentioned (S128).
Govindasvami (go-VIN-duh-SWAH-mee, 8th century CE) gave one in his commentary on Mahabhaskariya IV.22. Parameshvara reproduces it in his commentary on the Laghubhaskariya, which is how it survived. Datta and Singh record a restriction that makes it more interesting rather than less: Govindasvami "prescribed it for the second sign only," that is, for arcs between 30 and 60 degrees (S128). That is a mathematician who has worked out where his formula earns its keep. He says so instead of overselling it. The sources read here establish nothing about his life.
Vatesvara (vuh-TAYSH-vuh-ruh, fl. 904 CE) gave another, at Vatesvara-siddhanta, chapter 2, section 1, verses 65 to 66 (S128). Again, the formula survives and the man does not.
The afterlife of Brahmagupta's rule includes two failures worth recording. Munisvara (1646) tried to improve it by iteration, and Datta and Singh say plainly that "his process of iteration is incorrect." Kamalakara (kuh-muh-LAH-kuh-ruh, fl. 1658) "severely criticised" the rule, and Datta and Singh add four words: "But he is wrong" (S128). Both are useful reminders. A tradition that lasts a thousand years contains arguments, including arguments that one side loses.
Brahmagupta's quadrilateral, and Bhaskara II's insult
Brahmagupta's other appearance in a trigonometry chapter is for a result that is not about sines at all. It sits next to Ptolemy's theorem, and it produced the sharpest exchange in Indian mathematics.
Colebrooke translates Ganita chapter XII, stanza 21: "The product of half the sides and countersides is the gross area of a triangle and tetragon. Half the sum of the sides set down four times, and severally lessened by the sides, being multiplied together, the square-root of the product is the exact area" (S126).
In modern notation, with the semiperimeter, the "exact area" is
which is Heron's formula for a triangle with a fourth factor bolted on. Colebrooke's worked examples include a tetragon, meaning a four-sided figure, with sides 39, 25, 25, 25 giving "exact area 768," and a trapezium with sides 60, 52, 39, 25 giving "exact area 1764" (S126).
The formula is correct, but only for a cyclic quadrilateral. That is one whose four vertices lie on a circle. Brahmagupta does not say so. The modern summary is careful about this: "Although Brahmagupta does not explicitly state that these quadrilaterals are cyclic, it is apparent from his rules" (S137).
Five centuries later Bhaskara II noticed the omission and was not gentle. Colebrooke's translation of Lilavati stanzas 169 to 172 (S126):
"Since the diagonals of the quadrilateral are indeterminate, how should the area be in this case determinate? The diagonals, found as assumed by the ancients, do not answer in another case. With the same sides, there are other diagonals; and the area of the figure is accordingly manifold. For, in a quadrilateral, opposite angles, being made to approach, contract their diagonal as they advance inwards: while the other angles, receding outwards, lengthen their diagonal. Therefore it is said, 'with the same sides, there are other diagonals.'"
That is a correct and strikingly modern objection. Four side lengths do not determine a quadrilateral. The figure can flex, and as it flexes the area changes. A quadrilateral with given sides reaches its maximum area exactly when it is cyclic. That is the fact Brahmagupta's rule silently assumes.
Then stanza 172: "Such a questioner is a blundering devil. Still more so is he, who answers the question. For he considers not the indefinite nature of the lines in a quadrilateral figure" (S126). Colebrooke's footnote glosses the word rendered "devil": "Pisacha (a demon or vampire). So termed, because he blunders."
There is a real link here to Ptolemy's theorem. The Kerala school worked it out explicitly. For a cyclic quadrilateral, the Ganita-Yukti-Bhasa derives that "the sum of the products of the two pairs of sides associated with a diagonal is equal to the product of the diagonal with the third diagonal," which is Ptolemy's theorem in Kerala dress. It also derives that "the area of the cyclic quadrilateral is equal to the product of the three diagonals divided by twice the diameter" (S127, ch. 7.10.2 to 7.10.3).
One honest caution. Popular accounts often say or imply that Brahmagupta's quadrilateral work is connected to chord tables and to Ptolemy. I found no passage in which Brahmagupta himself makes that connection, either in Colebrooke's translation of chapter XII or in any source read for this chapter. The Ptolemy connection is a modern observation about the mathematics, not a documented historical link.
Bhaskara II and the Jyotpatti
Bhaskara II (BAHS-kuh-ruh the Second, also Bhaskaracarya, "Bhaskara the teacher," 1114 to 1185 CE) is the last of the great siddhanta astronomers before the Kerala school. A siddhanta is a complete astronomical treatise, the standard book form of the tradition. His birth year is firm because he dates his own masterwork and gives his age. He completed the Siddhanta Siromani (Crown of Treatises) in 1150 at the age of 36 (S128, S139). The death year of 1185 is conventional.
Where he lived is contested. He names his home as Vijjadavida or Vijjalavida. Different scholars have placed that near Patan in the Chalisgaon taluka of Jalgaon district in Maharashtra, at modern Beed, in Karnataka, or in Telangana. MacTutor says he was born at Vijayapura and died at Ujjain (S139, S143). Nobody has settled it.
The addition and subtraction theorems
The Siddhanta Siromani contains a section called the Jyotpatti, "the generation of sines," inside the Goladhyaya, the book on the sphere. Datta and Singh translate its statement of the addition and subtraction theorems, citing Siddhanta-siromani, Gola, XIV.21f. (S128):
"The Rsines of any two arcs of a circle are reciprocally multiplied by their Rcosines; the products are then divided by the radius; the sum of the quotients is equal to the Rsine of the sum of the two arcs; and their difference is the Rsine of the difference of the arcs."
That is a complete and unambiguous statement of
in one sentence, with both signs, in 1150. Read it beside Ptolemy's chord-based version and the advantage of the half-chord jumps out. No square roots. No supplementary arcs. No case analysis.
The cosine theorem, attested by testimony rather than by text
You would expect the matching cosine formula. Datta and Singh could not find it in any printed edition of Bhaskara II, and they say so, then argue that he had it anyway: "Though we do not find this Rcosine theorem in the printed editions of the works of Bhaskara II, we are quite sure that it was known to him. For it has been attributed to him by his most relentless critic Kamalakara as well as by his commentator Munisvara" (S128).
The evidence is Kamalakara's own verse, which Datta and Singh quote and translate: Evamanayanam cakre purvam sviyasiromanau, Bhavanabhyamatispastam samyagaryo'pi Bhaskarah, that is, "This theorem, which is evident from the two Bhavanas, was stated before also by the highly respected Bhaskara in his (Siddhanta-)siromani" (S128).
Kamalakara goes further, claiming that "Many correct proofs of this theorem were given before by the learned authors of the [siddhantas]," and Datta and Singh add the sentence that every historian eventually has to write: "Unfortunately we have not been able to trace them as yet" (S128).
So a hostile witness writing five centuries later is what attests the cosine addition theorem in India. That is real evidence. It is also weaker than a verse, and the difference is worth teaching.
His own interpolation rule, with the reasoning attached
Bhaskara II also gives a second-order interpolation rule for a table at 10 degree intervals. Unusually, he shows why it works instead of just asserting it. Datta and Singh translate his gloss (S128):
"Half the sum of the (tabular) difference passed over and that to be passed will be the difference at the middle of those differences. But the difference to be passed is at the end of that interval to be passed. Hence proportion (should be taken) with their difference: If for an interval of 10 degrees, we obtain half the difference of them, then what will be obtained for (an interval of) the remaining degrees? Thus by the rule of three, 20 will be the divisor of the product of the remaining degrees and the difference of the (tabular) difference passed over and that to be passed. ... for in the calculations of Rsines the differences are in the decreasing order. But in the calculations of versed Rsines they are in the increasing order and hence the plus in this case. Thus (the rule) is proved."
Two things there. The divisor 20 is with degrees. It appears because half a difference is spread over a whole interval. And the closing clause, "Thus (the rule) is proved," is a mathematician insisting that a computational recipe is entitled to a justification. The Kerala school made that insistence into a genre.
Instantaneous motion, and an honest gap
Bhaskara II is often credited with the concept of tatkalika-gati, "instantaneous motion," and with an early recognition that the cosine measures the rate of change of the sine. I could not verify a primary quotation of him on this. No source read for this chapter turned up a translated verse stating the rule, either from the Siddhanta Siromani or from its own commentary, the Vasanabhasya. The one article that names the Vasanabhasya in this connection reached me only as an abstract. It supports nothing here.
What can be documented is this. The rule itself is older than Bhaskara II. It runs through the whole tradition in a form that is unmistakably a difference quotient of the sine.
Sengupta sets it out from the modern Surya Siddhanta II.47 to 49 (S121). The instantaneous daily motion of the moon is
where and are the mean daily motions of the moon and of its apogee, the point of its orbit farthest from the earth, and is the periphery of the epicycle. Sengupta derives it as a difference quotient outright: "the instantaneous daily motion = (l' - l)/r = n - (P/(360 r))[R sin{m - a + (n - n')r} - R sin (m - a)]" (S121). Divide the change in the sine by the change in the arc. You have the derivative in everything but name. The 225 in the denominator is the tabular step. Dividing by it turns a tabular difference into a rate.
The same rule appears in Lalla (LUL-luh, 8th century CE), Sisyadhivrddhida II.15, and Thibaut's translation of Pancasiddhantika IX has it too: "Multiply (the motion of the anomaly) by the difference of the sines of (at?) the anomaly, and divide by 22.5; reduce the result (to terms of the epicycle)" (S121). Thibaut's question mark is his own.
And Lalla, at Sisyadhivrddhida III.43, objects to the way Aryabhata's followers were using it: "the rule of dividing the daily motion in anomaly of the moon by 225 by which the pupils of Aryabhata have obtained the correction to her mean daily motion, gives the apparent daily motion of the moon for the day elapsed and it cannot be used for the current day" (S121).
Read that objection carefully, because it is better mathematics than it looks. A difference quotient computed across a finite interval gives you the average rate over that interval. That average belongs to the interval just passed. Using it as the rate right now, Lalla says, is a mistake. That is exactly the gap between and . He raised it as a practical objection in the eighth century, roughly a thousand years before anybody wrote .
The people who usually get cut
Textbook accounts of Indian trigonometry run Aryabhata, Brahmagupta, Bhaskara II, Madhava, and stop. Here are the ones who fall off the list, with what each of them did. In several cases the evidence about the person is thin to nonexistent. You get a name, a floruit (the span when someone is known to have been active), and a formula. That is worth saying out loud rather than padding.
Manjula, and the art of being deliberately inaccurate
Manjula (MUN-joo-luh, also Munjala, fl. 932 CE) deserves a classroom hour. Everyone else in this chapter pushed accuracy up. Manjula pushed it down on purpose. That is a better lesson about engineering than any of the precision records.
In the Laghumanasa (Brief Treatise for the Mind), II.2, he sets arcminutes, which is 8 degrees 8 minutes. On that radius he builds a five-entry table (S128). His rule is one line: "the sum of the signs successively multiplied by 4, 3 and 1 will give the degrees in the Rsines and Rcosines; such are the minutes" (S128).
Decode it. A "sign" is 30 degrees, so the quadrant is three signs. Walking up the quadrant you add 4 for the first sign, 3 for the second, 1 for the third. The running totals are 4, 7, 8. Then "such are the minutes" means you write the same number again as arcminutes. So the sine of one sign is 4 degrees 4 minutes, of two signs is 7 degrees 7 minutes, of three signs is 8 degrees 8 minutes.
Those look like angles. They are lengths, in the odd unit system Manjula has chosen. Multiply by 61 to convert them to arcminutes:
Check them. exactly, so the first entry is perfect. , so the third is perfect. The middle entry is where he pays. , and Manjula uses 427, which is 4.4 arcminutes high, about 1 percent. Datta and Singh notice the rounding and explain it as a design decision: Manjula "takes the value to be 7 deg 7 min obviously with the purpose of simplifying his rule" (S128). (The sentence in which they compute the accurate value is damaged in the scan I worked from. It reads "Accurately speaking jya 90 deg = 488 x 4/3 [sic]". The arithmetic above is mine.)
Their worked example is the Rsine of 76 degrees 30 minutes. That is two whole signs plus 16 degrees 30 minutes, so the third multiplier contributes of its value of 1:
which is what Datta and Singh print (S128). In arcminutes that is 460.55, against a true . So the answer is about 14 arcminutes low, roughly 3 percent. Almost all of that error comes from interpolating linearly across a 30 degree step, near the top of the quadrant where the sine bends hardest.
Three percent is terrible by the standards of the Surya Siddhanta. It is also beside the point of what Manjula was doing. His whole scheme is four numbers (4, 3, 1, and "same again as minutes"). It needs no table, no lookup, no long multiplication, and no division. You can run it in your head while walking. For a rough almanac calculation that is the right trade. Put Manjula's rule beside Madhava's eight-decimal table and ask a class which one is better. The answer is: for what?
Lalla
Lalla (8th century CE) appears above twice: for his rule on the moon's instantaneous daily motion, and for his objection to the way Aryabhata's school used it. Datta and Singh credit two writers with the explicit statement of . He is one of them, at Sisyadhivrddhida II.30 (S128). His book's title translates roughly as "Treatise for Increasing the Intelligence of Students," which is a good title. No source read for this chapter pins his dates closer than the eighth century, or says where he worked.
Govindasvami and Vatesvara
Covered above with Brahmagupta's rule. Both gave second-order interpolation formulas that are not Brahmagupta's. Both survive only inside other people's commentaries. Neither has a biography in the sources read here (S128).
Suryadeva Yajva and Sripati
Suryadeva Yajva (born 1191) and Sripati (fl. 1039) both state, in so many words, the two-way rule that makes the tables work: sum the differences forward and you get the sines, sum them backward and you get the versed sines. Suryadeva: "In order to get the direct sines, these tabular differences of sines should be added regularly from the beginning; and in order to determine the reversed sines, they should be added in the reversed order from the end" (S128). That is the Surya Siddhanta's "in reversed order" instruction, stated as a general principle six or seven centuries later.
Udayadivakara
Udayadivakara appears in Datta and Singh's comparative table of Indian sine tables with a radius given in thirds: 12375859''', which is 3437' 44'' 19''' (S128). A "third" is a sixtieth of an arcsecond. So this is a radius quoted to about one part in a hundred million. It is written as a single enormous integer. It is a rare unit choice. It also shows how sexagesimal notation lets you slide the unit anywhere you like.
Kamalakara
Kamalakara (fl. 1658) is a late figure who matters as a transmitter and a critic. He preserves the Rcosine addition theorem and credits it to Bhaskara II, as quoted above. He also gives two geometric proofs of the addition theorems (S128). He also attacks Brahmagupta's interpolation rule. Datta and Singh judge that attack mistaken (S128). He is quoted again in the etymology section below, on why mathematicians shortened ardha-jya to jya. Here is a working seventeenth century Sanskrit astronomer still arguing with a seventh century one, in the same technical vocabulary.
Kerala: a school, a lineage, and the series
Between roughly 1350 and 1600, in a small area of Kerala along the Nila river, a lineage of astronomers did three remarkable things. They derived the power series for the arctangent, the sine, and the cosine. They invented convergence acceleration, a way of squeezing a good answer out of a series that converges slowly. And they computed a sine table accurate to about eight decimal places. All of that happened roughly 250 to 300 years before Gregory, Newton, and Leibniz.
Madhava, whose books are gone
Madhava of Sangamagrama (MAH-duh-vuh, c. 1340 to c. 1425, disputed) is the founder. Three sources give three briefs: c. 1340 to c. 1425 (S140), c. 1350 to 1425 (S142), and simply "c. 14th cent." (S127). The article that gives the first of those says in the same breath that there is "no definite evidence to pinpoint the period during which Madhava flourished" (S140). There are two anchors. One is a reference to 1400 CE in his Venvaroha. The other is the Drgganita of his pupil Parameshvara, completed in 1430 (S140).
And his mathematical works are lost. MacTutor states it without softening: "His original mathematical writings are lost. Knowledge comes from later Keralese mathematicians, particularly Yukti-Bhasa by Jyesthadeva ... and Mahajyanayana prakara" (S142). The companion article adds the consequence: "most of Madhava's original works have been lost" and "which results are precisely Madhava's and which are those of his successors is difficult to determine" (S140).
So every series in this section reaches us at second hand, through people who name him. That attribution chain is documented. The Ganita-Yukti-Bhasa editors state that "The formulae vidvan etc. and stena etc., are attributed to Madhava by Nilakantha in his Aryabhatiya-bhasya, Ganitapada, 17 (see also, Yuktidipika, II. 437-438)," and that "Madhava has also given the tabulated sine values (for arcs in multiples of 225') ... in the rule sresthham nama varisthanam ... (cited by Nilakantha in his Aryabhatiyabhasya, Ganitapada, 12)" (S127). On a separate track, Datta and Singh's comparative table of Indian sine tables lists "Madhava, 3437' 44'' 48''', interval 225', to 3 [sexagesimal places]," sourced to "Nilakantha's com. on A, ii. 12" (S128). Two separate scholarly traditions point at the same commentary.
Even his home town is disputed. Sangamagrama means "confluence village." Many scholars identify it with Irinjalakuda, and the source that reports that view goes on to question it. The alternative is Kudallur, near Tirunavaya at the confluence of the Nila and Kunti rivers, whose name means exactly "confluence village" (S140).
The lineage, one name at a time
Parameshvara of Vatasseri (puh-ruh-MAYSH-vuh-ruh, 1360 to 1460, disputed) was a Namputiri Brahmin of the Vatasseri family. He lived on the north bank of the Nila river. He wrote a great deal. There is the Bhatadipika on the Aryabhatiya, commentaries on the Laghubhaskariya and the Laghumanasa, and a sine table carried to seconds (S127, S128). His dates are disputed inside a single book. The Ganita-Yukti-Bhasa prints "1360-1460" in its bibliography and "(A.D. 1360-1455)" in its lineage paragraph (S127). He also matters here as a witness. He is the commentator who quotes the Prakasikakara on what the year 499 dates in Aryabhata. He is quoted below on why ardha-jya got shortened to jya.
Damodara (DAH-mo-duh-ruh, c. 1410 to 1520) is the hinge of the whole story. He is almost never named. He was Parameshvara's son. He taught both of the men who wrote down what Madhava had done: Nilakantha Somayaji and Jyesthadeva (S127). Without Damodara there is no Kerala school as we know it. There is only a set of results with no route out of one family. Everything the sources read here say about him fits in that sentence. Rough dates from one source, a place (Kerala), a father, and two pupils. If you want a case study in how much a history can depend on a person about whom almost nothing is recorded, this is it.
Nilakantha Somayaji of the Kelallur family (NEE-luh-KUN-tuh so-muh-YAH-jee, 14 June 1444 to 1544, disputed) lived at Kundagrama, now Trikkandiyur in Tirur, Kerala. His birth date comes from a Kali-day figure of 1,660,181 (S141). The death year is unstable across sources, running from 1544 to 1560. Every candidate makes him a centenarian (S127, S141). He wrote the Tantrasangraha and the Aryabhatiyabhasya. He is our chief witness for Madhava's series and sine table.
The Tantrasangraha carries its own date as a Kali-day chronogram, a number encoded in words. The figure is 16,80,553, that is, 1,680,553 days from the epoch of the Kali era, which the Ganita-Yukti-Bhasa converts to 1500 (S127). Another reference work says the work was "completed 1501" (S141). So the date is 1500 (disputed; 1501 is the competing value). The disagreement is not about the chronogram, which everyone agrees on. It is about the conversion from a Kali day number to a Gregorian year. Chronograms are exact; their conversions are not.
Jyesthadeva (JYESH-tuh-DAY-vuh, 1500 to 1610, disputed) was of the Parangottu family, Sanskritised as Parakroda, a Namputiri family in the vicinity of Alattur and Trikkandiyur (S127). He wrote the Yuktibhasa, also called the Ganita-Yukti-Bhasa (Rationales in Mathematical Astronomy), in Malayalam. His dating is a small detective story and gets its own subsection below.
Sankara Variyar (SHUN-kuh-ruh VAH-ree-yar, 1500 to 1560) of the Trkkutaveli family is the reason we have the Sanskrit verses at all. He wrote the Yuktidipika (Lamp of Rationales), a commentary on the Tantrasangraha often dated around 1530. He also wrote the Kriyakramakari, a commentary on Bhaskara II's Lilavati (S127). Nearly every verse of Madhava quoted in the modern edition of the Yuktibhasa carries the tag "cited also in Yuktidipika" (S127). In that tradition the root text is the thing that gets copied. So a commentator who quotes his sources in full is doing archival work, whether he knows it or not.
Acyuta Pisarati (AH-chyoo-tuh pih-SHAH-ruh-tee, 1550 to 1621, disputed; the same book's bibliography prints "1500-1621") of Trikkandiyur was Jyesthadeva's pupil. He is also the direct documentary evidence for that relationship. His Uparaga-kriyakrama of 1592 names his "aged benign teacher Jyesthadeva" (S127). The date 1592 is firm. It is a chronogram in the text, explained by the text's own Malayalam commentary (S127). A single dedication in a single book, written by a student, fixes a whole generation of this chronology.
Dating the Yuktibhasa, and a rejected attribution
The modern editors argue the date of the Yuktibhasa at length. The argument is worth walking through. It shows what dating an Indian text involves (S127).
The evidence for placing Jyesthadeva at 1500 to 1610:
- A palm-leaf granthavari (a family or temple record) at Baroda.
- Acyuta Pisarati calling him pravayas, "very old," in the Uparaga-kriyakrama of 1592. If he was very old in 1592 and still alive, a birth around 1500 follows.
- A Kali chronogram in Jyesthadeva's Drkkarana corresponding to 1608, implying he was still working then.
Put together, those give a working life from about 1530 to 1608. That is a lifespan of about 110 years. That is why the shorter range given elsewhere, c. 1500 to c. 1575, is more comfortable and less well evidenced.
Against that stands an older printed edition which assigned the Yuktibhasa to somebody called Brahmadatta, in 1750. The evidence for that attribution was the verb alekhi in a closing verse. The modern editors reject it, reading alekhi in its natural sense, "written" in the sense of "copied" (S127). In other words, the closing verse records who made that manuscript copy, in 1750, not who composed the work. Every reader of old manuscripts meets this problem. A colophon tells you about the copy in your hand. Mistake it for a claim about the composition and you can move a text by two and a half centuries. You can also hand it to the wrong person.
Malayalam prose against Sanskrit verse
Every text so far in this chapter is Sanskrit verse: compressed, metrical, and memorizable. It is also closed to anyone without a Sanskrit education. In practice that meant closed to almost everyone.
The Yuktibhasa is in Malayalam, the spoken language of Kerala, and it is in prose. Its modern editors describe it as "written in the local language Malayalam, besides, it is in the form of an expository text which includes detailed explanations and proofs of various results" (S127). The title says so too: yukti means "rationale" or "reasoning," and bhasa means "language" in the sense of the vernacular. "Reasoning, in the local language."
The form follows from the purpose. Sanskrit verse is built for storage and transmission. It has fixed meter, the fewest syllables it can manage, and error detection built into the rhythm. Malayalam prose optimizes for teaching. You cannot fit "here is why this correction term is self-consistent, and here is a test you can run on a candidate correction to see whether it is accurate" into a metrical couplet. Jyesthadeva was not writing a reference card. He was writing a course, and the proofs are the point.
Language and history hook. Ask a class who gets to read a book written in Sanskrit verse in sixteenth century Kerala. Then ask who gets to read one written in Malayalam prose. Choosing a language is choosing an audience. European scientists faced the same choice when they began publishing in Italian, French, and English instead of Latin. Galileo published the Dialogo in Italian in 1632. That decision is inseparable from what happened next. Then ask the harder question. What does a field lose when its results are stored in a form only specialists can decode? And what does it gain?
The arctangent series
The Yuktibhasa states the arctangent series, known by its incipit istajyatrijyayorghatat, and also cited at Yuktidipika II.206 (S127):
Jya divided by koti is sine over cosine, which is the tangent. So with this is
the series often named after James Gregory, who published it in 1671. Put , so that degrees and . Out comes the series often named after Leibniz.
The derivation in the text is geometric and constructive. It divides the side of the circumscribing square into equal parts and sums the pieces. Then it lets grow (S127). The circumference series below use the same device. That device does the work a modern proof does with term-by-term integration of .
Four series for the circumference, and one more
Madhava did not stop at one series. This is the part that surprises people who know only the "Madhava-Leibniz" name. The Yuktibhasa quotes several different series for the circumference, each as a Sanskrit verse. Each is traceable also to Sankara Variyar's Yuktidipika (S127). I evaluated all of them.
One. vyasad varidhinihatat prthagaptam tryadyayugvimulaghanaih / trighnavyase svamrnam kramasah krtva paridhiraneyah (Yuktidipika II.290): "The diameter is multiplied by 4 and is divided, successively, by the cubes of the odd numbers beginning from 3, which are diminished by these numbers themselves. The diameter is now multiplied by three, and the quotients obtained above, are added to or subtracted from, alternatively" (S127). That is
Summing the terms with odd numbers up to 199 gives 3.1415929036, an error of .
Two. dvyadiyujam va krtayoh vyeka harad dvinighnaviskambhe / dhanam rnamante 'ntyordhvagataujakrtirdvisahita harasyardham (Yuktidipika II.292): "The squares of even numbers commencing from 2, diminished by one, are the divisors for four times the diameter. ... The quotients got by the first (division) are alternately added to or subtracted from twice the diameter" (S127). That is
Summing to the even number 398 gives 3.1416052162, an error of . Slower than the first.
Three. dvyadescaturadervakaturadhikanam nirekavargascet harah / kunjaragunito viskambhah svamatikalpito bhajyah / phalayutirekatra vrttirbhajyadalam phalahinamanyatra (Yuktidipika II.293 to 294): "Eight times the diameter is divided separately by these and the results are added together. This will give the circumference. The same sum subtracted from four times the diameter will also give the circumference" (S127). Two series in one verse:
Both have all terms of one sign, which is unusual and useful. With 200000 terms they give 3.1415901536 and 3.1415951536, errors of and . Notice that they bracket from below and above by the same amount. So their average is far better than either.
Four. samapancahatayo ya rupadyayujam catuhghnamulayutah tabhih / sodasagunitat vyasad prthagahatesu visamayuteh / samaphalayutimapahaya syadistavyasasambhavah paridhih (Yuktidipika II.287 to 288): "The fifth powers of the odd numbers (1, 3, 5 etc.) are increased by 4 times themselves. The diameter is multiplied by 16 and it is successively divided by the (series of) numbers obtained (as above)" (S127). That is
This one is dramatically the best of the four. Working the first two terms by hand shows why:
That is two terms, from a series whose ancestor needs thousands of terms to come as close. Summed to 200000 terms it gives 3.1415926535897913 against the modern . The error is . That is the limit of double-precision arithmetic, not the limit of the series.
And one more. The cakriya form quoted at Yuktidipika II.212 to 214 is
which is the arctangent series evaluated at , that is, at 30 degrees. Sixty terms give 3.141592653589794, an error of (S127). Abraham Sharp used this series in 1699 to get 71 digits of . It was in Kerala first.
The three correction terms
The Kerala mathematicians knew perfectly well that the plain alternating series is useless for computation. needs about terms for ten digits. So Madhava gave antya-samskara, "final corrections": a closing term you add after truncating, chosen to cancel most of the error you just committed.
Three of them are recorded, each better than the last. Truncate after the term , with even, and then add, with the sign of the first omitted term, one of:
The third is stated in the verse ante samasamkhyadalavargah saiko gunah sa eva punah / yugagunito rupayuto samasamkhyadalahato bhaved harah (Yuktidipika II.295 to 296): "At the end ... (apply another correction with) the multiplier being the square of half of the next even number plus 1, and the divisor being four times the previous multiplier with 1 added and multiplied by half the even number" (S127). Read literally, the multiplier is and the divisor is , which is the third expression above. That last step is worth checking rather than taking: .
Here is what each correction is worth, as an error in :
| terms used | no correction | first correction | second correction | third correction |
|---|---|---|---|---|
Take . That is fifty terms of a series whose raw value is not correct to two decimal places. Add one correction term and you have to about twelve decimal places.
Work the case in full, because the arithmetic is small enough to see:
The modern value is . Ten terms plus one correction give seven correct decimal places. Ten terms without the correction give . That is not correct to even one decimal place. The correction is doing nearly all of the work.
The Ganita-Yukti-Bhasa records the accuracy of the continued-fraction form of the correction: "this correction term leads to a value of pi = C/d, which is accurate up to 11 decimal places, when we merely evaluate terms up to n = 50 in the series," and adds that "the value of pi = C/d, given in the rule vibudhanetra..., attributed to Madhava in Kriyakramakari (p. 377), is also accurate up to 11 decimal places" (S127). The value most often quoted for that katapayadi chronogram is . I computed it as 3.1415926535922 against , an error of . That is correct to 11 decimal places, exactly as stated. One caveat: I read those digits in secondary literature, not in the Ganita-Yukti-Bhasa itself. Treat the exact numerator as unconfirmed, and the accuracy claim as confirmed.
The correction terms are argued for, not asserted
Popular accounts usually skip the mathematically most interesting passage in the whole Kerala corpus. It is about how you would know whether a correction term is any good.
The Yuktibhasa (S127, ch. 6.8): "First, it has to be verified whether this stated correction itself is accurate or not. Such verification (might be made as follows): Obtain the result after division by a certain odd number, keep it in two places, and apply the correction at one place. To the result, at the other place, apply (add or subtract) first the result obtained by dividing by the next odd number and to that apply the correction corresponding to the next even number. If the circumferences obtained in both cases are equal, then the correction can be taken as accurate."
That is a self-consistency test. Truncate at one point and correct. Then truncate one term later and correct again. If the correction is right, both routes give the same answer. The text then reasons about the sthaulya, the leftover inaccuracy, of a candidate correction. It uses that leftover to derive the next and better correction.
This is recognizably the logic of error analysis. Any student who has met the alternating series estimate in a calculus course has met a weaker version of it.
The sine and cosine series, as an algorithm
The Yuktibhasa gives the sine and cosine series not as a formula but as a procedure. It reads the way you would describe an operation on a counting board (S127, ch. 7.5.5):
"The required arc is the first result. When this is squared, halved and divided by the radius, the second result is got. ... In this manner, derive successive results by multiplying the previous result by the arc, and dividing by corresponding successive numbers 1, 2, 3 etc. and by radius. Now, place below the first result the odd results, viz., the third, the fifth etc., and place below the second result the even results, viz., the fourth, sixth etc. Then subtract successively the bottom result from the one above it, the remainder from the one still above it. ... These will be the required Rsine and Rversine."
Two columns, subtracted alternately from the bottom up. One column is the sine series, the other the versine series. The explanatory notes give the versine column in modern form as (S127).
For the sine on its own the text gives the recursion by its incipit nihatya capavargena ("having multiplied by the square of the arc"): "Multiply the required arc by the square of the required arc and divide by the square of radius. Then, divide also by 6, which is the product 2 and 3. The result is Rsine-arc difference. The further results are derived similarly, the multiplier is the square of the arc, and the divisor is the square of radius. Still another divisor is the product of the (corresponding) even number and the next odd number" (S127).
Unwind that. Start with the arc . Each new term multiplies by and divides by (even number) times (next odd number): , then , then . Those products are exactly what turns into into . The rule is the recursion for the Taylor coefficients of the sine, stated as an instruction instead of a formula. There was no formula notation to state it in.
The vidvan coefficients
For real computation Madhava supplied a nested polynomial with five constants, known by the mnemonic word vidvan. Nilakantha credits it to him in the Aryabhatiyabhasya (S127). Take the quadrant as arcminutes. Then
and the source says of it: "The above expression gives the values of Rsine of any arc accurately up to the thirds" (S127).
The five constants are the Taylor coefficients , with , rounded to thirds of arc (a third is a sixtieth of an arcsecond):
| coefficient | Madhava's value | exact value | difference |
|---|---|---|---|
| 2220' 39" 40''' | 2220' 39" 39.56''' | 0.44 thirds | |
| 273' 57" 47''' | 273' 57" 47.05''' | 0.05 thirds | |
| 16' 05" 41''' | 16' 05" 40.87''' | 0.13 thirds | |
| 0' 33" 06''' | 0' 33" 05.60''' | 0.40 thirds | |
| 0' 00" 44''' | 0' 00" 44.54''' | 0.54 thirds |
Every one of the five is the correctly rounded value. I evaluated the nested expression at every whole arcminute from 0 to 5400. The largest error is 0.000104 arcminutes, that is 0.375 thirds, and it falls at the end of the quadrant. As a pure sine that is about , or roughly seven to eight decimal places. It comes from five constants and four multiplications.
The nesting is Horner's scheme. A numerical analyst would choose the same arrangement today, and for the same reason. It minimises the number of multiplications and keeps the intermediate quantities small.
The radius, and the sine table
Madhava's radius is not 3438. It is
that is, 3437 arcminutes plus 44 sixtieths plus 48 three-thousand-six-hundredths, and so on to seven sexagesimal places. The Ganita-Yukti-Bhasa records that the verse srirudrah sridharah srestho devo visvasthali bhrguh encodes this value in katapayadi notation, the scheme in which each consonant stands for a digit: "If the diameter is 21600', then a fairly accurate value of radius is 3437' 44'' 48''' 22iv 29v 22vi 22vii given in the katapayadi notation by the verse" (S127).
I expanded that sexagesimal number. It equals 3437.7467707849364, against the true . The error is about 0.15 units in the seventh and last sexagesimal place. Aryabhata rounded this number to 3438 and worked in whole arcminutes. Madhava kept eleven significant figures and packed them into a line of verse a student could memorize.
His 24-entry sine table is carried to thirds. The intervals are the same 3 degrees 45 minutes as every table in this chapter.
| n | arc | Madhava Rsine | error (thirds) |
|---|---|---|---|
| 1 | 3 deg 45 min | 224' 50" 22''' | +0.17 |
| 2 | 7 deg 30 min | 448' 42" 58''' | +0.42 |
| 3 | 11 deg 15 min | 670' 40" 16''' | -0.05 |
| 4 | 15 deg | 889' 45" 15''' | -0.61 |
| 5 | 18 deg 45 min | 1105' 01" 39''' | +0.06 |
| 6 | 22 deg 30 min | 1315' 34" 07''' | -0.44 |
| 7 | 26 deg 15 min | 1520' 28" 35''' | -0.46 |
| 8 | 30 deg | 1718' 52" 24''' | -0.19 |
| 9 | 33 deg 45 min | 1909' 54" 35''' | -0.19 |
| 10 | 37 deg 30 min | 2092' 46" 03''' | -0.49 |
| 11 | 41 deg 15 min | 2266' 39" 50''' | -0.21 |
| 12 | 45 deg | 2430' 51" 15''' | +0.41 |
| 13 | 48 deg 45 min | 2584' 38" 06''' | +0.47 |
| 14 | 52 deg 30 min | 2727' 20" 52''' | -0.38 |
| 15 | 56 deg 15 min | 2858' 22" 55''' | -0.11 |
| 16 | 60 deg | 2977' 10" 34''' | +0.27 |
| 17 | 63 deg 45 min | 3083' 13" 17''' | +0.06 |
| 18 | 67 deg 30 min | 3176' 03" 50''' | +0.03 |
| 19 | 71 deg 15 min | 3255' 18" 22''' | +0.42 |
| 20 | 75 deg | 3320' 36" 30''' | -0.20 |
| 21 | 78 deg 45 min | 3371' 41" 29''' | -0.15 |
| 22 | 82 deg 30 min | 3408' 20" 11''' | +0.07 |
| 23 | 86 deg 15 min | 3430' 23" 11''' | +0.35 |
| 24 | 90 deg | 3437' 44" 48''' | -0.37 |
The maximum error is 0.61 thirds, at entry 4, which is arcminutes. Relative to the radius that is . Read as a pure sine, the table is good to about eight significant decimal places. Set that beside the Surya Siddhanta's . The gain is a factor of about 4000 over roughly a thousand years.
Two entries worked out in full, so the units are not mysterious.
Entry 8, the sine of 30 degrees, which we can check exactly because :
He is short by two tenths of a third of arc. A third of arc is of a right angle. Put another way, if the quadrant were a wall 100 meters long, the error would be about a sixtieth of a millimeter.
Entry 24, the sine of 90 degrees, which is the radius itself:
which is the truncation of his own recorded radius to three sexagesimal places.
The addition theorem, credited to Madhava
The jive-paraspara-nyaya, "the rule of the mutual sines," is the Rsine addition and subtraction theorem, and in Kerala it is credited to Madhava. The Ganita-Yukti-Bhasa notes: "The jive-paraspara-nyaya is the famous rule of Madhava (cited for instance, in Tantrasangraha 2.12), which corresponds to the following result. If s1 and s2 are two arcs then R sin(s1 +/- s2) = (1/r)[R sin(s1) R cos(s2) +/- R cos(s1) R sin(s2)]" (S127).
That is the same theorem Bhaskara II stated in 1150. Three centuries later, in another part of India, it is restated, renamed, and re-proved. There it serves as a lemma in the derivation of the series. Results in this tradition get rediscovered and reattributed constantly. That is what happens when transmission runs teacher to pupil instead of through print.
Charles Whish, John Warren, and who gets noticed
In 1834 Charles M. Whish (WISH, fl. 1830s), a civil servant of the East India Company in Malabar, published "On the Hindu quadrature of the circle" in the Transactions of the Royal Asiatic Society, volume 3, pages 509 to 523. The paper announced to European scholarship that Kerala mathematicians had infinite series for centuries before Europe (S127).
He promised a follow-up paper on the proofs in the Yuktibhasa. He never published it.
Then nothing happened for a hundred years. The modern editors of the Yuktibhasa put it bluntly: the Kerala work "was completely ignored by modern scholarship for over a century till it was resurrected by the pioneering work of C. T. Rajagopal and his collaborators in the 1940's" (S127).
Whish was not even the first to report it. John Warren's Kala Sankalita (Madras, 1825), at pages 92 to 93 and 330 to 331, already recorded Indian knowledge of infinite series, nine years earlier (S127). The same source generalizes: "it appears that from the early decades of the 19th century many British observers had noticed and reported on the Indian mathematicians' knowledge of several infinite series" (S127). In a footnote, the Ganita-Yukti-Bhasa also flags a caution about the year the Whish paper was published. It leaves that caution unresolved (S127).
The recovery came through C. T. Rajagopal (rah-juh-GO-pahl) and his collaborators from the 1940s onward. Two papers by Rajagopal and M. S. Rangachari are often cited as the point at which the Kerala material entered the international record. Both are in the Archive for History of Exact Sciences: volume 18 (1978), pages 89 to 101, and volume 35 (1986), pages 91 to 99 (S127). One small bibliographic wrinkle will trip up anyone chasing the citation. The Ganita-Yukti-Bhasa lists the 1986 paper twice, once as "35, 91-99" and once as "35(2), 91-99," so the issue number is uncertain even inside a single book. I obtained neither paper in full text for this chapter. Nothing here rests on their contents.
History hook. This is a good place to ask a class what makes a published result get taken up. Whish's paper was in a mainstream London journal. It was in English, by a named author, with the mathematics in it. Almost nobody who could use it read it, for a century. Warren's earlier report fared no better. The material was not secret. It was not lost. It was not in a language European scholars could not read. It did not interest the people who received it. It did not fit what they expected the history of mathematics to look like. Then compare. What would it take today for a result published in a real journal to sit unused for a hundred years? Does that still happen?
Did Kerala mathematics reach Europe?
Popular accounts run two claims together. Keep them apart, because one is documented and the other is not.
Claim one: priority. Madhava and his successors obtained the arctangent, sine, and cosine series, together with convergence-acceleration techniques. They did it roughly 250 to 300 years before Gregory, Newton, and Leibniz. This is not in dispute. The sections above document it from primary texts. Anyone with a calculator can check every number in those sections.
Claim two: influence. Those results reached Europe and influenced the European inventors of calculus. This hypothesis has a plausible opportunity structure. As of the sources read for this chapter, it has no documentary evidence.
Here is the case for it, stated as strongly as its own advocates state it. Dennis Almeida and George Gheverghese Joseph point to a Jesuit presence in Kerala from 1540 to 1670. Francis Xavier arrived at Goa in 1540 and Matteo Ricci in 1578. Johann Schreck, who had studied with Viete, and Antonio Rubino were also in India. Almeida and Joseph point to the Jesuit historical archive in Rome (ARSI) and to the collections "Goa 38, 46 and 58" held there. And they argue that "At some point in their stay in India these Jesuits went to the Malabar region including the city of Cochin, the epicentre of developments in the infinitesimal calculus" (S135). A companion survey adds two arguments. The first is methodological: "A key development of pre-calculus Europe, that of generalisation on the basis of induction, has deep methodological similarities with the corresponding Kerala development (200 years before)," in their words. The second is about motive: Jesuits were encouraged to acquire mathematical knowledge in Kerala (S131).
Now the evidence. No manuscript or documentary chain has been produced. The survey that makes the methodological argument is sympathetic to the hypothesis, and it says so itself in one sentence: "There is no evidence of direct transmission by way of relevant manuscripts" (S131). Almeida and Joseph concede the point in their own paper: "Even if there is documentary evidence in 16th century European manuals used for navigation, map-making and calendar construction of the use of approximate series derived from the discoveries of the Kerala School, it would hardly have been directly communicated to European mathematicians" (S135). That is the strong claim's own authors conceding that the strong claim is not established.
Now what most historians conclude. David Bressoud's position is quoted as "there is no evidence that the Indian work of series was known beyond India, or even outside of Kerala, until the nineteenth century" (S134). Victor J. Katz treats the question as open. He says that "the question of whether European mathematicians were influenced by the Kerala School remains an ongoing topic of scholarly debate" while allowing that communication routes make transmission a possibility. The same survey summarizes the state of play as "While communication routes and chronologies make this transmission a possibility, there is currently no direct manuscript evidence demonstrating that such a transfer took place" (S134). Katz has separately suggested influence running the other way, from Ibn al-Haytham into Kerala. He notes that the careers of Newton and Leibniz are well documented, with "no indication that their work was not their own" (S134).
I could not obtain Bressoud's 2002 paper in the College Mathematics Journal in full text, so that quotation reaches you at one remove. I could not obtain Kim Plofker's Mathematics in India either, nor the Rajagopal and Rangachari papers, nor George Gheverghese Joseph's The Crest of the Peacock. Nothing in this chapter rests on any of them.
The practical upshot for a classroom: the names "Madhava-Leibniz series," "Madhava-Newton series" and "Madhava-Gregory series" record priority. They do not assert transmission, and they should not be taught as though they did. Priority and influence are different historical claims, and they need different evidence. That difference is one of the more useful things a history of mathematics can teach.
From bowstring to bosom: how jya became sine
The English word "sine" arrived by a chain of translations and, probably, one misreading. Every step is documented except the last, and the last is still disputed.
The Sanskrit words
Jya means "a bowstring," and hence "the chord of an arc," because the arc itself is called dhanu or capa, "a bow" (S128). Jiva is a synonym. The picture is a strung bow lying on the page. The curved arc is the bow; the straight chord is the string.
The half-chord is ardha-jya, "half-chord," or jyardha, "chord-half." Both forms are used, and Aryabhata uses ardha-jya (S136).
But the full forms are long, and Sanskrit verse charges by the syllable. So the tradition shortened ardha-jya to plain jya. For a while one word did duty for two different things, the chord and the half-chord. Then it stopped meaning the chord at all. Two mathematicians, five centuries apart, remark on this in as many words. Bhaskara II (1150): "It should be known that ardha-jya is here called jya." Kamalakara (1658): "Having seen the brevity, the half-chords are called jya by mathematicians in this branch of mathematics" (S128, p. 40). Parameshvara (c. 1430) makes the same point in his definition of ardha-jya (S128). Notice what those three quotations are. They are mathematicians explaining a piece of their own notation to students who might otherwise be tripped up by an older meaning. Every technical vocabulary picks up this kind of scar.
The full-chord sine also has a name of its own, krama-jya, "direct sine," as opposed to utkrama-jya, "reversed sine." Utkrama means "reversed," "going out," "exceeding," and the versed sine carries that name for a strictly practical reason. You get its tabular values by subtracting the tabular sines from the radius in reversed order. That is the instruction in Surya Siddhanta II.22 quoted earlier (S121, S128). Its synonyms are vyasta-jya, from vyasta, "reversed," and viloma-jya, and, most vividly, isu and bana, both meaning "arrow."
Bhaskara II explains the arrow: "What is really the arrow between the bow and the bowstring is known amongst the scholars here as the versed sine" (S128). Draw it once and the whole vocabulary locks into place. Arc equals bow. Chord equals bowstring. Half-chord equals sine. One little segment runs from the middle of the string to the middle of the bow, along the line of fire. That equals the arrow, and the arrow equals the versine. That is not a metaphor invented by a historian. It is the metaphor the practitioners used. They used it because they had all handled a bow.
The cosine is koti-jya, from koti, "the curved end of a bow," extended to mean "the extremity" and then "the complement of an arc to 90 degrees" (S128, S130). In commentaries it is often shortened to kojya.
And the name of the whole subject, in India, is not any relative of "trigonometry." It is jyotpatti-ganita: jya ("sine") plus utpatti ("construction, generating") plus ganita ("the science of calculation"), that is, "the science of calculation for the construction of sines," sometimes shortened to jya-ganita. Datta and Singh find it at Brahmasphutasiddhanta XII.66, in 628 CE (S128, p. 39). That name tells you what the subject was for. It was not for solving triangles, which is what "trigonometry" says. It was for building the table. Bhaskara II's chapter title Jyotpatti is the same word.
The Sanskrit word for the modern subject is trikonamiti, from trikona ("triangle") and miti ("measure"). It is a calque on the Greek, a word-for-word translation, and it is recent. Datta and Singh, writing in the 1930s, describe it as new: "In very recent years there has appeared the name Trikonamiti" (S128). So Sanskrit has two names for two different subjects. The old indigenous one covers the science of building sine tables. The modern imported one covers the school subject of triangle measurement. They are not synonyms.
Into Arabic, into Latin, into English
Arabic mathematicians borrowed jiva phonetically as jiba. Written Arabic does not normally write short vowels. So jiba appears on the page as three consonants, j-y-b. Now a Latin translator meets those three consonants. There is a perfectly ordinary Arabic word spelled the same way, jaib, meaning "bosom," "the fold of a garment," or "a bay." He reads that, and renders it into Latin as sinus. The Latin word means the same cluster of things. A fold in a garment, a bend, a curve, a bay, the fold of a toga about the breast (S128, S130, S133).
So the sine function is named after a fold in a garment. Blame a vowel that nobody wrote down.
English took sinus and made "sine." The first English attestation is from the 1590s, in Thomas Fale's Horologiographia, the Art of Dialing (S133). The Oxford English Dictionary citation, reached here through Miller's survey of earliest uses, gives the year as 1593 and the sentence as "This Table of Sines may seem obscure..." (S427).
Who made the substitution is not known
Three or four candidates are in play. No source read for this chapter produced a document that settles it.
- Plato of Tivoli, in his 1116 translation of al-Battani. This is the attribution you will most often find in older textbooks. It is also the one with the strongest evidence against it. Braunmühl points out, following Kästner, that the word appears exactly once in that whole translation, in the compound sinus versus. The running text says chorda and chorda versa throughout (S425).
- Robert of Chester, in his revision of al-Khwarizmi's astronomical tables, often dated 1145. Smith's own footnote in volume 1 of his History of Mathematics prefers this: "The term was probably first used in Robert of Chester's revision of the tables of al-Khowarizmi" (S422, vol. 1, p. 202, n. 4). Carl Boyer agrees, and adds the mechanism explicitly: "When Robert of Chester came to translate the technical word jiba, he seems to have confused this with the word jaib (perhaps because vowels were omitted); hence he used the word sinus" (S427).
- Gherardo of Cremona, the most prolific twelfth century translator, working at Toledo. Braunmühl leans this way but hedges the verb: Gherardo "was probably the one who, through his renderings of various astronomical works of the Western Arabs, was responsible for the introduction of the word sinus in the West" (S425). Smith's volume 2 says the same: "When Gherardo of Cremona (c. 1150) made his translations from the Arabic he used sinus for jaib, each word meaning a fold, and this usage, possibly begun even earlier, was followed by other European scholars" (S421, vol. 2, p. 616). Howard Eves also credits Gherardo (S427).
- Nobody in particular: some one of the many scholars who moved a mass of Arabic writing into Latin in the twelfth century. On this account the usage spread before anyone thought to record who started it.
Note that Smith contradicts himself between his two volumes. The sources this book cites for the Indian material name three candidates and do not adjudicate (S130, S132). The honest answer to "Gerard or Robert?" has three parts. Neither has been proved. The leading nineteenth and twentieth century authorities split. And the confidence found in most textbooks is not warranted by the documents. One useful qualification: the Arabic jaib was firmly in place well before any Latin translator touched it. Al-Khwarizmi's own small Sindhind already had the sine. Its Latin translator called the functions elgeib elmustewi (the straight sine) and elgeib elmacus (the versed sine) (S425).
The cosine, and a link that may not exist
The obvious story for "cosine" is that Sanskrit kojya passed into Latin as co-sinus. Datta and Singh argue for exactly that: "in Hindu works, particularly in the commentaries kotijya is often abbreviated into kojya. When jya became sinus, kojya naturally became ko-sinus or co-sinus" (S128).
The counter-evidence is that early medieval Latin texts call the cosine complementi sinus, "sine of the complement," and on that showing the resemblance to kojya is a coincidence, with the "co-" a clipping of complementum (S130). The later history supports the Latin route. Edmund Gunter suggested co.sinus in 1620. John Newton shortened it to cosinus in 1658 (S421). Two plausible stories, one piece of awkward evidence, and no decisive document.
Kardaja: a word that changed what it meant
One more Sanskrit word made the journey. It is the strangest case in the chapter. The word arrived in Latin meaning something different from what it meant when it left Sanskrit.
Krama-jya, "direct sine," was corrupted in Arabic to karaja or kardaja, and appears in Latin translations as kardaga, karkaya, gardaga, and cardaga. Along the way it stopped being the name of the function. It became the name of the tabular interval. That interval is 3 degrees 45 minutes, the step size of every table in this chapter, and sometimes 15 degrees (S128).
That drift makes sense once you picture how a reader met the word. Meeting a table headed "kardaja" over and over, once per row, you might reasonably conclude that a kardaja is the thing each row covers. The unit takes the name of the function.
Datta and Singh trace the earliest use through the Fihrist, the tenth century Arabic bibliographical catalog. A work of Ya'qub ibn Tariq (c. 770 CE) is titled "On the table of kardaja," and the table in it "was copied from the Brahma-sphuta-siddhanta of Brahmagupta" (S128). Al-Khwarizmi (825) uses the variant karaja (S128). That attribution reaches you at second hand. It is Datta and Singh's reading of the Fihrist. I did not read the Fihrist, or any edition of it. Treat the date and the title as medium confidence.
If it holds up, it is a specific, datable act of transmission. It runs the other way from the one everybody argues about. A Sanskrit table crosses into Arabic within about 140 years of Brahmagupta's book. A catalog records the crossing.
What the evidence does not support
Five claims that circulate widely and do not survive contact with the sources.
The 16x version of Bhaskara I's formula
You will find this online and in classroom handouts:
Test it at , where the answer must be 1:
A sine of 16. The error is a splice. Someone took the numerator of one algebraically equivalent form and the denominator constant of another. The two correct forms are
and both give exactly 1 at : the second gives (S128, S144). This is a good five-minute exercise for a class. The test that catches the error is one a student can invent unaided. Put in a value where you already know the answer.
That Brahmagupta's second-order interpolation is in the Brahmasphutasiddhanta of 628
It is not. Datta and Singh state that the rule "does not occur in his bigger work, Brahma-sphuta-siddhanta, which was composed in 628 A.D." and that it appears in the earlier Dhyanagrahopadesa and the later Khandakhadyaka of 665 (S128). The achievement is real and the priority claim (which belongs to Sengupta) may well be right. The date and the book, as usually given, are wrong by 37 years and one title.
That Kerala mathematics demonstrably reached Europe
Covered at length above. The priority is documented; the transmission is not. The strongest statement the evidence supports is that the opportunity existed. The advocates of the hypothesis concede that no manuscript chain has been produced (S131, S135).
That Varahamihira stated the Pythagorean identity
The relation is present in Thibaut's translation of Pancasiddhantika IV.4, as a computing step (S124). But the usual source for the confident textbook claim is Smith. Smith does not say that. He puts the Pythagorean identity in a group of principles the Hindus were "probable" to have known, inferred from the quality of their tables. He says that only the half-angle relation and one other appear in the Pancasiddhantika itself (S421, vol. 2, p. 615). Datta and Singh, meanwhile, credit the explicit statement of the relation to Lalla and Brahmagupta, not to Varahamihira (S128). What is secure for Varahamihira is the half-angle relation and the complement rule.
Smith's dates for the Sulba Sutras
The Sulba Sutras are the ritual geometry manuals that contain Pythagorean triples and altar constructions. Popular accounts often date them by way of Smith, who is also cited constantly for the Indian side of the Pythagorean theorem's prehistory. Read what Smith wrote. In volume 2 he says of the triples that "Although the date of these writings is uncertain, it is evident that the relations were known rather early in India," and his footnote on the date offers only a guess: "Perhaps the 4th or 5th century b.c." In volume 1 he says outright that "The dates of the Sulvasutra period are unknown," and that the texts "were changed more or less by such commentators as Apastamba, Baudhayana, and Katyayana" (S422). So the number that appears in secondary literature as "Smith dates the Sulba Sutras to the fifth century BCE" is a footnote guess. Its author says in his own main text that the dates are unknown. He also says later hands revised the texts. Smith also dates two Chinese classics in the same neighborhood, far earlier than modern scholarly consensus. That is a further reason not to lean on his chronology here.
What I could not verify
The style of this book is to say so in the text, not to leave things out quietly. Here is what remained open after I read the sources.
A primary quotation of Bhaskara II on instantaneous motion (tatkalika-gati). The tradition is documented, through the modern Surya Siddhanta II.47 to 49, through Lalla's Sisyadhivrddhida II.15 and his criticism at III.43, and through Pancasiddhantika IX (S121). What is missing is a translated verse from the Siddhanta Siromani or its Vasanabhasya stating the rule in Bhaskara II's own words. The one article that names the Vasanabhasya in this connection was available only as an abstract. It is therefore not used here. Anyone finishing this section should get the Wilkinson and Bapu Deva Sastri translation of the Goladhyaya, or Arkasomayaji's translation, first.
That Bhaskara II identified the cosine as the derivative of the sine, or stated an early form of Rolle's theorem. Both claims appear in a widely used reference work. But the fetched text rendered the relevant formula as an empty placeholder and gave no verse citation. The MacTutor biography of Bhaskara II mentions neither (S139, S143). Treat both as unverified.
Devanagari and Malayalam forms of most of the names in this chapter. This is a real gap. It should stay visible rather than get papered over. Of everyone named here, only Aryabhata has a Devanagari form from a fetched source, आर्यभट (S136). Only Madhava has a native-script form, माधवन्. The source labels it Malayalam, although the glyphs shown are Devanagari (S140). Fifteen others are missing entirely: no source read for this chapter displayed their names in their own script. They are Varahamihira, Brahmagupta, Bhaskara I, Bhaskara II, Parameshvara, Damodara, Nilakantha, Jyesthadeva, Sankara Variyar, Acyuta Pisarati, Lalla, Govindasvami, Vatesvara, Manjula and Kamalakara. I have deliberately not supplied one from memory. One source printed Bhaskara II's name as भाष्कर, which appears to be a typographical error for भास्कर. That too is left unresolved (S139). The pronunciations given throughout this chapter are plain-English guesses from standard transliterations. They do not come from a pronunciation authority. Check them with a Sanskrit or Malayalam speaker before print.
The seconds figure of entry 22 of Varahamihira's sine table, at 82 degrees 30 minutes. Lost across a page break in the scan; the minutes figure is 119; the modern value is 118.973 parts. Recover it from the page image before print (S124).
The exact numerator of Madhava's vibudhanetra value of . The source states three things: the rule exists, the Kriyakramakari credits it to Madhava at page 377, and it is accurate to 11 decimal places (S127). It does not print the number in the passage read. The commonly quoted is consistent with that accuracy, and I checked it. But I did not read those digits in a source.
The Fihrist attribution of a "table of kardaja" to Ya'qub ibn Tariq (c. 770). Reported by Datta and Singh; I did not read the Fihrist or any edition of it (S128).
Whether Brahmagupta anywhere connects his quadrilateral area rule to Ptolemy or to a chord table. No such passage appeared in Colebrooke's translation of Brahmasphutasiddhanta chapter XII. Nor could I obtain any scholarly statement to that effect. The Ptolemy connection is a modern observation about the mathematics (S126, S127).
A firm death date for Aryabhata I. The birth year of 476 follows from Aryabhatiya III.10. The conventional 550 is not attested in any source read for this chapter.
Manuscript images of any of these works. I located none. Every scan used for this chapter is of a printed edition or translation. There are eight of them:
- Burgess's Surya Siddhanta, in the 1935 Calcutta reprint with Sengupta's introduction
- Clark's 1930 Aryabhatiya
- the Shukla and Sarma edition of 1976
- Thibaut and Dvivedi's 1889 Pancasiddhantika
- Sengupta's 1934 Khandakhadyaka
- Colebrooke's 1817 Algebra, with Arithmetic and Mensuration
- the 1983 Indian Journal of History of Science printing of Datta and Singh's "Hindu Trigonometry"
- the 2008 Ganita-Yukti-Bhasa I did not see the palm-leaf originals behind them.
One further caution about the Datta and Singh text. The open PDF of their 1983 article is a page-image scan with no text layer. I read it through local optical character recognition. Every quotation from it in this chapter should be spot-checked against the page image before print. Two OCR casualties are already noted above: the difference "86" for 36 in Brahmagupta's table, and the garbled sentence about Manjula's rounding.
Full text of several works cited only as bibliography. Bressoud's 2002 paper in the College Mathematics Journal, Kim Plofker's Mathematics in India, the Rajagopal and Rangachari papers of 1978 and 1986, Ranjan Roy's 1990 Mathematics Magazine paper on the series formula for , and Joseph's The Crest of the Peacock were all sought and not obtained. No claim in this chapter rests on any of them.
- India cut the chord in half, and the half-chord (jya) attaches to a right triangle, which the chord never did. That is the sine, as a length.
- R = 3438 is the radius in arcminutes: the radian's central idea, chosen for computation a millennium before the radian's name.
- Aryabhata compressed the table into 24 differences in one line of verse; Brahmagupta interpolated with second differences; Bhaskara I gave a formula good to a third of a percent with no table at all.
Next: the Islamic world takes the sine out of astronomy, adds its five siblings, and makes the subject a discipline.
Your unit circle is this chapter's picture: the sine as the upright half-chord of a doubled arc is the y-coordinate you read off in Trigonometry 2.0
Aryabhata fit a 24-entry sine table into one line of verse. What did he list instead of the sines?
Show the answer▸
Chapter 4
Six Functions and a Discipline (The Islamic World, c. 750 to 1500)
The people in this chapter
Faces where a face survives. Every name links to its full entry in Appendix A.
A slab of marble on a minaret
In 1371 or 1372 a man climbed the main minaret of the Umayyad Mosque in Damascus. He was carrying a slab of marble about two meters long and one meter wide. He set it on a platform on the southern side. It stayed there, telling the city what time it was, for five hundred years (S182).
The slab is a sundial, and it is not a decoration. Cut into its surface are curves for the five daily prayers. They mark the start of each one, for every day of the year, at the latitude of Damascus. It carries three dials, not one. First the large main dial. Then a small northern dial for the seasonal hours and the afternoon prayer, and a small southern dial for equal hours (S182). The man who made it, Ibn al-Shatir (IB-n ash-SHAA-tir, c. 1304 or c. 1305 to c. 1375), held a salaried post at the mosque. His job title was muwaqqit, mosque timekeeper. Fragments of the original now sit in the garden of the National Museum in Damascus. On the minaret stands an exact replica, made in 1876 by a later muwaqqit named al-Tantawi (S182, S200).
To cut those curves you need three things, for each of 365 days. Where will the sun sit? How high will it climb? How long a shadow will it throw? All three at one instant: the moment a vertical stick's shadow has grown by exactly the stick's own length. That is a problem in spherical trigonometry, the geometry of triangles drawn on the surface of a globe rather than on a flat page. Solved 365 times, before the slab is even quarried.
This chapter is about the people who solved problems like that, and what happened to trigonometry while they did it. Two things happened. First, the function list filled up. Look at the unit circle chart in your textbook: sine, cosine, tangent, cotangent, secant, cosecant. Every one of them exists in recognizable form by the end of the tenth century. Each one arrived because somebody needed it to answer a practical question. Second, and this is the larger change, trigonometry stopped being a chapter inside astronomy and became a subject with its own book.
Two of the six functions on your calculator were born from an everyday object, centuries before the names they now carry. Which object gave trigonometry the tangent and cotangent?
I have a guess▸
A stick standing in the sun. The gnomon. The shadow of a standard twelve-finger gnomon is for sun altitude , and medieval astronomers tabulated exactly that: shadow tables, the cotangent and tangent as functions of shadows (S181). The names came much later, from a different picture: Fincke's 1583 tangens is the touching line.
If you guessed the tangent LINE, you guessed the source of the NAME, and the chapter's point is that name and function have separate biographies: the function is a shadow, credited to nobody in your textbook.
The five problems that paid for all of this
Ritual practice in Islam sets three computational problems, and they are hard ones.
The five prayers are defined by the sun. David A. King, the historian who has done most of the archival work on this material, puts it flatly: "The expression ilm al-miqat refers to the science of astronomical timekeeping by the sun and stars in general, and the determination of the times (mawaqit) of the five prayers in particular. Since the limits of permitted intervals for the prayer are defined in terms of the apparent position of the sun in the sky relative to the local horizon, their times vary throughout the year and are dependent upon the terrestrial latitude" (S182).
Read that last clause again, because it is the whole engine. Vary throughout the year means you need the sun's declination, how far north or south of the equator it stands, which changes daily. Dependent upon the terrestrial latitude means a table computed for Cairo is wrong in Damascus. So every city needs its own tables, computed from scratch.
The definitions themselves are shadow definitions. That is why the shadow functions in this chapter matter so much (S182):
- The Islamic day, and the maghrib interval, begin at sunset.
- Isha begins at nightfall and fajr at daybreak, both defined by twilight.
- Zuhr, the midday prayer, usually begins when the shadow of any object is seen to increase after the sun crosses the meridian. In Andalusian and Maghribi practice the rule is sharper: the shadow must increase over its midday minimum by one quarter of the object's length.
- Asr, the afternoon prayer, begins when the increase in the shadow equals the length of the gnomon, the upright stick that casts it. It ends when the increase is twice the gnomon length, or at sunset.
- A sixth prayer, the duha, was observed at mid-morning in some communities.
"When the increase in the shadow equals the length of the gnomon" is a cotangent condition dressed as a rule about sticks. If the sun's altitude is and the gnomon has length , the shadow is . The asr begins at the altitude for which
that is, when , where is the sun's meridian altitude that day. That altitude changes every day of the year. So the asr altitude changes every day of the year too, and computing it is a job.
The lunar calendar depends on seeing a thin crescent. King notes the reason the Islamic day begins at sunset: "the calendar is lunar and the months begin with the sighting of the crescent shortly after sunset" (S182). Will a crescent be visible tonight? Answering that means computing the moon's position in three dimensions, relative to both the sun and the horizon, near the moment of sunset. It is a harder problem than prayer times, and it comes up twelve times a year, forever.
The qibla is a direction on a sphere. The qibla is the direction of prayer: Muslims pray facing the Kaba in Mecca. In a flat world that would be an easy question. It is not a flat world.
What the qibla asks you to compute
Put a globe on the table. Stick a pin in Mecca and a pin in your own town. Now stretch a rubber band between the two pins so it lies flat against the surface. It will settle along one specific curve, the shortest path over the sphere. That curve is a great circle: the circle you get by slicing the globe through both pins and through the center of the Earth.
The qibla is the compass bearing of that rubber band as it leaves your town. Two things about it surprise people.
First, the bearing is not what you would read off a flat map. On the common rectangular map, a straight line from Chicago to Mecca points roughly east-southeast. The great circle leaves Chicago pointing northeast, over Greenland. Both statements are correct about their own surface. Only one of them is about the Earth.
Second, the bearing changes as you go. Set off from your town on a fixed compass heading and you will spiral, not arrive. The great circle's bearing at Mecca is different from its bearing at home. The qibla is a local quantity: it is defined at your feet.
In modern notation, the qibla angle measured from the meridian satisfies
where is your latitude, Mecca's, and the difference in longitude. Nobody in the ninth century wrote it that way, because nobody had that notation. What they had was a spherical triangle, a triangle whose three sides are arcs drawn on a globe. One vertex sits at the north pole, one at Mecca, one at home. Two sides are known from the two latitudes, and the angle between them is known from the longitude difference. One angle is wanted. Solving that triangle is the qibla problem. Everything else in this chapter is, in one way or another, a tool for solving triangles like it.
Two rival traditions answered it, and King insists they be kept apart (S231). Folk-astronomical methods use horizon phenomena. Face the point where a named star rises, or where the sun sets at the equinox. The rule comes from a "sacred geography" that divides the world into sectors arranged around the Kaba. Mathematical methods use coordinates and trigonometry. The two "inevitably" disagree, which is why the orientation of a medieval mosque so often puzzles a modern visitor with a phone compass. King is careful not to sneer at the first tradition, calling the sacred-geography scheme "an ingenious response" rather than a naive one (S231).
Geography and map-projection hook. This is the best classroom entry point into projections that exists, because it has a right answer that people cared about. Give students a globe, a length of string and a world map. Have them find the great-circle bearing from their own town to Mecca with the string. Then draw the straight line on the flat map, and measure both angles with a protractor. The gap between the two numbers is the whole content of "a map is a lie you choose." Then ask which map projection would make the string line straight. The answer is a gnomonic projection, on which every great circle is a straight line, at the cost of distorting everything else.
Shams al-Din al-Khalili, and trigonometry as civic infrastructure
The best single answer to "what did all this look like as work?" is the output of one man in one mosque.
Shams al-Din al-Khalili (SHAMS ad-DEEN al-kha-LEE-lee, c. 1320 to c. 1380) was a muwaqqit at the Umayyad Mosque in Damascus, the same institution as Ibn al-Shatir. He came up through the muezzin ranks. Several of the celebrated muwaqqits, al-Khalili among them, had previously been muezzins, the men who call the prayer (S216). Here is what he produced (S182).
Prayer tables for Damascus. Twelve separate functions, tabulated for each degree of solar longitude. The twelve are:
- The solar meridian altitude.
- Half the length of daylight, called the half diurnal arc.
- The number of hours of daylight.
- The solar altitude at the start of asr.
- The hour-angle at the start of asr.
- The time from the start of asr to sunset.
- The time from midday to the end of asr.
- The duration of night.
- The duration of evening twilight.
- The duration of darkness.
- The duration of morning twilight.
- The time remaining until midday from the moment the sun stands in the direction of Mecca.
Total: about 2,160 entries. (Twelve functions times 180 degrees of solar longitude is exactly 2,160, which suggests he tabulated half the year and used symmetry for the rest. That inference is mine, not King's.)
An hour-angle table. The hour-angle as a function of solar altitude and solar longitude , for the latitude of Damascus. About 10,000 entries (S182). The hour-angle is how far the sun is from the meridian, measured as an angle; convert it and you have the time. This table answers the question "I have measured the sun's altitude with an astrolabe, what time is it?" for every altitude on every day.
Universal auxiliary trigonometric tables. These are the remarkable ones, because they are not for Damascus. They work at any latitude. King writes them as functions of two arguments: roughly , plus a companion function . There is also an inverse-cosine function . All are to base . Over 13,000 entries, given to two sexagesimal digits. King's judgment on them: they are "invariably accurately computed" (S182). (The scan of King's chapter I worked from has imperfect character recognition in the displayed formulae. So the exact argument lists in , and should be checked against a clean copy before anyone prints them for students. The entry counts and the accuracy judgment are unambiguous in the text.)
What the auxiliary tables do is worth spelling out. Combine them in a fixed recipe: for the hour-angle, and a similar recipe for the azimuth. Together they "serve to solve numerically any problem which can, in modern terms, be solved by means of the spherical cosine formula" (S182). That is a general-purpose spherical trigonometry engine, made of paper, in the fourteenth century. You do not need to understand the spherical cosine rule to use it. You need only look up three numbers, subtract, and look up a fourth.
A qibla table. The direction of Mecca as a function of terrestrial longitude and latitude, apparently compiled out of the universal auxiliary tables (S182). Not a qibla for one city. A qibla for anywhere.
Add these up. One man, in one mosque, in one lifetime, produced something like 25,000 computed table entries. Each one is a spherical trigonometry problem solved by hand in sexagesimal arithmetic. That is the scale these people worked at, and it is why the accuracy of a sine table was not an academic question.
The job
The office al-Khalili and Ibn al-Shatir held is younger than you would guess. King's account of its origins is worth quoting at length, because it documents something rare: an entire salaried profession whose daily work was trigonometry (S182).
"In practice, at least before the thirteenth century, the regulation of the prayer-times was the duty of the muezzin ... They were appointed for the excellence of their voices and their character, and they needed to be proficient in the rudiments of folk astronomy. They needed to know the shadows at the zuhr and the asr for each month, and which lunar mansion was rising at daybreak and setting at nightfall, information which was conveniently expressed in the form of mnemonics; they did not need astronomical tables or instruments. ... In the thirteenth century there occurred a new development, the origins of which are obscure. In Egypt at that time we find the first mention of the muwaqqit, a professional astronomer associated with a religious institution, whose primary responsibility was the regulation of the times of prayer. Simultaneously, there appeared astronomers with the epithet miqati who specialized in spherical astronomy and astronomical timekeeping, but who were not necessarily associated with any religious institution."
So there were two flavours of the job: the muwaqqit on a mosque payroll, and the freelance miqati. "A long line of muwaqqits worked in the Mosque from the fourteenth to the nineteenth century," King writes of Damascus (S182). This is not a story about court patronage or aristocratic hobbyists. It is a story about municipal employees.
The first muwaqqit known by name appears to be Abu al-Hasan Ali ibn Abd al-Malik ibn Simun (IB-n see-MOON, died c. 1286), who served about thirty years at the Mosque of Amr in Fustat, old Cairo. That attribution needs a hedge. It comes to me from a reference article citing King's 1996 work at pages 298 to 299. I did not read those pages of King directly. King's own encyclopedia chapter, which I did read, says only that the muwaqqit first appears in Egypt in the thirteenth century. It names no first holder (S216, S182). Treat "first muwaqqit known by name" as well sourced at one remove, not as something I verified in the original.
Two more names belong here. Abu Ali al-Marrakushi (a-BOO a-LEE al-mar-raa-KOO-shee, active late thirteenth century) worked in Cairo as a miqati, not attached to any mosque. He compiled Jami al-mabadi wa-l-ghayat fi ilm al-miqat, which King glosses as "An A to Z of Astronomical Timekeeping" (S182). It is the defining compendium of the whole discipline: instruments, tables, sundial theory and spherical astronomy, gathered in one place. It set the pattern for centuries of practitioners after him. It was also the first of this literature that Europeans read, by way of the Sedillots in the nineteenth century (S182). His contemporary Shihab al-Din al-Maqsi (shi-HAAB ad-DEEN al-MAK-see) compiled the Cairo timekeeping tables and an extensive treatise on sundial theory (S182).
Careers hook. Ask a class to write the job advertisement. Required: spherical trigonometry, sexagesimal arithmetic, instrument making, sundial design, table construction, and the ability to be right in public every single day. Offered: a salary, a room at the mosque, and a professional title. The muwaqqit is a working mathematician with a payslip, four hundred years before anybody in Europe had that job description. The tables he left behind are what he was paid for.
A warning before we go further
It would be neat to say that Islamic trigonometry exists because of the qibla. The specialist who wrote the standard survey of the field says do not do that. Marie-Therese Debarnot, whose chapter on trigonometry in the Encyclopedia of the History of Arabic Science is the backbone of this chapter, writes: "When the historians of science mention its development in the Arabic period, they readily quote the qibla: this accounts poorly for the complexity of the calculation of the zij, owing to its heterogeneous composition and the prodigious expansion of astronomy in the ninth century" (S181).
A zij is an astronomical handbook: tables, plus the instructions for using them. It covers planetary positions, eclipses, calendar conversion, star coordinates and much else. The qibla is one problem in a zij. The zij as a whole, and the enormous ninth-century expansion of astronomy that produced dozens of them, is the real driver. Keep both facts. The religious sciences created a standing, paid demand for spherical trigonometry that no other culture in this book had. And the mathematics grew for astronomical reasons far broader than that demand.
What arrived in Baghdad, and what it was called
Two traditions arrived, from opposite directions.
From Greek astronomy came Ptolemy's Almagest. It brought a table of chords, the straight lines joining the ends of arcs, and Menelaus's theorem for spherical triangles. From Indian astronomy came the siddhantas, with the half-chord, and the half-chord won.
The Sanskrit for the half-chord is ardha-jya or jya-ardha, shortened to jya, "bowstring", with the synonym jiva. Arabic mathematicians took it over as a sound, not a translation. They wrote jiba, a word with no meaning in Arabic, spelled with the three consonants j-y-b (S207, S209, S228). Arabic script does not write short vowels. So j-y-b sits on the page, and a later reader supplies whatever vowels make a word. There is a perfectly good Arabic word spelled j-y-b. It is jaib, meaning a fold in a garment, a bosom, or a bay or cove in a coastline. When Latin translators came to render it, they translated the word they saw. And the Latin for a fold, a bosom or a bay is sinus. English clipped that to "sine" (S207, S209, S228).
The sine function is named after a fold in a piece of cloth, because of vowels nobody wrote down.
Who first made the substitution is not settled. Two rival attributions are reported. Howard Eves credits Gherardo of Cremona around 1150. Carl Boyer places the first appearance in Robert of Chester's Latin translation of 1145 (S228). I did not read either Eves or Boyer, and neither Britannica nor MacTutor names a translator at all, so this stays open.
The versed sine came across in the same package. Indian writers called the "arrow", and Arabic authors kept the image: sahm, arrow (S181). The picture is a bow. The arc is the bow, the chord is the string, and the versed sine is the arrow lying between them, along the line of fire. Arabic astronomers used it constantly, right through to the fifteenth century, as the practical substitute for a signed cosine (S181). Why did a function as odd as the versine get tabulated for a thousand years? Because nobody had negative numbers in their tables. And stays positive for every from 0 to 180 degrees, whereas changes sign at 90.
Al-Khwarizmi, and a book that only survives in translation
Al-Khwarizmi (al-khwaa-RIZ-mee, محمد بن موسى الخوارزميّ, c. 780 to c. 850) worked in Baghdad and compiled the Zij al-Sindhind, the earliest Arabic astronomical handbook whose sine table and gnomon shadow table we can trace (S193, S206, S181). The name of the book announces its ancestry: Sindhind is the Arabic form of siddhanta, the Indian genre.
Here is the honest position on it, and students should have it. The Arabic original of the Zij al-Sindhind does not survive. What we have are later Latin and Andalusian recensions. Other people made those revised, adapted, translated versions, in other places, sometimes centuries later (S181, S193, S206). When you read "al-Khwarizmi's sine table had such-and-such a form", the honest sentence is "the table in the surviving recensions of al-Khwarizmi's zij has such-and-such a form, and how much of it is his is a scholarly question." That is not a reason to ignore the work. It is a reason to be careful about what we attribute to the man.
One more caution, since al-Khwarizmi's tables are usually described as "his sine and cosine tables". Sources confirm sine tables and a gnomon shadow table. I found no source confirming a separately tabulated cosine under that name. Medieval Arabic practice used the "sine of the complement" rather than a distinct cosine function. So the phrase "his cosine table" is probably an anachronism imported by modern retellers (S181).
His name did survive, in a place nobody expected. Al-Khwarizmi is a nisba, a byname meaning "the man from Khwarazm", the region south of the Aral Sea. Latinised it became algorismus, then Old French algorisme in the thirteenth century, then French algorithme, then English algorithm. The English word is attested from the 1690s, meaning "the Arabic system of computation", and only later widened to mean any computational procedure at all (S227). Say "the algorithm" about a social media feed and you are saying the name of a ninth-century Baghdad astronomer. Three sound changes and a thousand years downstream.
The gnomon, the shadow, and the function nobody credits properly
Stand a stick vertically in the sun. The stick is a gnomon, the Greek word for a shadow-casting pointer. Its shadow is short at noon and long in the evening. The relation between the shadow length , the gnomon length and the sun's altitude above the horizon is
Now notice the problem hiding in that formula. The number you get out depends on how long your stick is. Medieval astronomers standardized the stick: twelve fingers, because twelve divides nicely. So the tables they built are tables of : not a function, a stick.
Here is what such a table looks like, in modern arithmetic, so you can see the shape of the thing:
| solar altitude | shadow in fingers, 12 cot h | R tan h with R = 60 | same in sexagesimal |
|---|---|---|---|
| 10 degrees | 68.055 | 10.580 | 10;34,47 |
| 20 degrees | 32.970 | 21.838 | 21;50,18 |
| 30 degrees | 20.785 | 34.641 | 34;38,28 |
| 45 degrees | 12.000 | 60.000 | 60;0,0 |
| 60 degrees | 6.928 | 103.923 | 103;55,23 |
| 80 degrees | 2.116 | 340.277 | 340;16,37 |
(Values computed by me for comparison, not transcribed from a manuscript.)
Both columns carry the same information. Only the second one is a function of the angle alone. The first depends on the angle and on a stick you happened to choose. Debarnot puts her finger on why that matters: "The gnomon, a magnitude as arbitrary as the radius of the sphere, with its own units, is clearly an obstacle to any idea of generalization and the introduction of a useful function, tan or R tan" (S181).
That sentence resolves a priority question that school textbooks get wrong.
The al-Khwarizmi tangent-table claim does not survive contact with the specialists
You will read, in many places, that al-Khwarizmi produced the first table of tangents. Here is Debarnot on what those tables are:
"A table of the shadow as a function of altitude, in two places, by degrees, for a gnomon of 12 fingers, thus corresponding to the function theta to 12 cot theta, can be found in the zij of al-Khwarizmi (the author of the famous algebra, Baghdad, beginning of the ninth century) and of al-Battani (Raqqa, end of the ninth century). In the two treatises it is only applied to reciprocal calculations of the altitude and the shadow" (S181).
Take that apart. The tables in al-Khwarizmi's zij and in al-Battani's zij are shadow tables for a twelve-finger gnomon, that is . They are given to two sexagesimal places, by whole degrees. And they are used for one thing only: converting back and forth between the sun's altitude and the length of a shadow. There is no general function here. No application to anything but shadows, and no sign that anyone thought of the quantity as an object in its own right.
Calling that "the first tangent table" is like calling a table of the weight of a cubic meter of water "the first density table". The number is in there. The concept is not.
Habash al-Hasib, who did it
Habash al-Hasib al-Marwazi (HA-bash al-HAA-sib al-mar-WA-zee, حبش الحاسب) is the most under-credited person in this chapter, and possibly in this book. His name is a description of him: al-Hasib means "the Calculator". Al-Marwazi says he came from Merv, in Khurasan. He worked in Baghdad and Samarra.
His dates are a mess, and the mess should be stated rather than hidden. The Springer Biographical Encyclopedia of Astronomers gives roughly c. 796 to c. 894 (disputed), which is a life of nearly a century (S187). Other reference sources give a death date of c. 869 (disputed) (S219). Debarnot, working from the text itself, dates his surviving zij to "at least after 869", which sits badly with a death in 869 (S181). These cannot be reconciled from anything I read. What is not in doubt is the mathematics.
In his Mumtahan zij, Debarnot writes:
"In the same period, Habash al-Hasib does not have a table in his zij for the shadow of the sun. He calculates the altitude by the traditional 'diameter of the shadow', for a gnomon of 12 fingers. It is in this treatise, however ... that the general notion of the tangent of an arc appears, with a definition, a table and several applications. The way in which Habash introduces it makes us think that he did not take it from a predecessor" (S181).
Read the first sentence twice, because it is the opposite of what you expect. Habash does not have a solar shadow table. He has something better and more abstract: the shadow of an arc, any arc, detached from the sun and from any particular stick. He defines it, he tabulates it, and he uses it in several different calculations. Debarnot's specification of the table: "The 'table of the shadow' (function R tan) of the zij of Habash is for three places, by half-degrees from 0;30 degrees to 89 degrees" (S181).
Three sexagesimal places. Half-degree steps. From 0 degrees 30 minutes up to 89 degrees. That is 178 entries of , and it is the first tangent table in any surviving tradition. Debarnot's judgment that "he did not take it from a predecessor" is the strongest statement a careful historian makes about independence. She makes it here.
Habash also compiled the Jadwal al-taqwim, a set of auxiliary tables. His sine table already had entries at every fifteen minutes of arc, with a fourth column made up of 0 and 30. That fourth column is his way of packing the half-step values into the same grid: "Habash ... transposes the table of chords just as it is; for the sine, it has entries every fifteen minutes and a fourth column formed from 0 and 30" (S181). Remember that detail. It matters later, when Abu al-Wafa gets credited with the fifteen-minute interval.
There is a third claimant to the tangent, and it should be recorded. The Springer encyclopedia entry on Abu al-Wafa says he "introduced the tangent function for the first time" (S189), and the claim also appears in Sesiano, with Habash flagged as the rival (S214). Debarnot places the introduction squarely with Habash in the ninth century. Abu al-Wafa's role, a century later, was to make the tangent enter "definitively" into astronomical calculation through the shadow figure (S181). Britannica states the priority flatly in Habash's favor and dates it to about 860: "The first table of tangents and cotangents was constructed around 860 by Habash al-Hasib ('the Calculator'), who wrote on astronomy and astronomical instruments" (S209). Note that Britannica's 860 sits earlier than Debarnot's dating of the surviving zij to after 869. So even the date of the achievement is unsettled.
Three claimants, then. Al-Khwarizmi: shadow table only, so probably not a tangent table at all. Habash: general shadow of an arc, defined, tabulated, applied. Abu al-Wafa: the shadow figure and its systematic exploitation, a century later. No source I read resolves the priority against Habash. The best-supported reading is Habash for the function, Abu al-Wafa for its systematic use.
The Arabic word throughout is zill, "shadow" (S181, S207). Latin translators split it in two, by where the shadow fell. Umbra recta, the upright shadow, falls on the ground from an upright gnomon. Umbra versa, the turned shadow, falls on a wall from a horizontal one (S207). Which of the two is the cotangent and which the tangent depends on the geometry of the dial. Both names are still stamped on the backs of old astrolabes and quadrants, in a little grid called the shadow square. Abu al-Wafa's own pair of names is different again: he calls them the "first" or "versed" shadow and the "second" or "extended" shadow (S181). The word "tangent" itself is much later, and not Arabic at all. It comes from Latin tangere, "to touch", because the line touches the circle at one point. Thomas Fincke introduced it in 1583 (S181, S207, S228). Viete objected that the term was ambiguous (S181). "Cotangent", short for complementi tangens, the tangent of the complementary arc, is credited to Edmund Gunter in 1620 (S207, S228).
Al-Battani at al-Raqqa
Al-Battani (al-bat-TAA-nee, محمد بن جابر بن سنان البتاني, before 858 to 929), known in Latin Europe as Albategnius, ran the longest and most careful observing program of the ninth century. He was born at Harran and worked at al-Raqqa on the Euphrates, in Syria. He observed there from 877 to 918 (S183, S188, S202). Forty-one years. His birth date is approximate and the sources disagree: one gives "before 858", another c. 858, another c. 850. The death year, 929, is firm (S188, S202, S213). He died at Qasr al-Jiss near Samarra.
Forty-one years of observation is the point. Ptolemy's parameters had been in use for seven hundred years. Al-Battani's program let him correct them from data. The resulting handbook, the Sabi Zij, became one of the most heavily used astronomical books ever written in Arabic. (The epithet al-Sabi refers to his family's Sabian background at Harran.)
Its trigonometry is what concerns us. The Sabi Zij contains a table of shadows for a twelve-finger gnomon, by degrees, to two sexagesimal places. It is the same object as in al-Khwarizmi, used for the same altitude-and-shadow conversions (S181). Britannica states the underlying rule as al-Battani wrote it and then translates it:
"Al-Battani's rule, s = h sin (90 degrees minus theta)/sin theta, is equivalent to the formula s = h cot theta ... he constructed a 'table of shadows', essentially a table of cotangents, for each degree from 1 degree to 90 degrees" (S209).
That first expression is worth pausing on, because it is the quotient relation in ninth-century clothing. Written in modern symbols it says
Al-Battani did not have a cotangent as a separate thing to be divided by. He had sines, and he had the sine of the complement, and he formed the ratio when he needed it. The tertiary literature also credits him with the relations and in explicit form (S213, S226).
Two claims about al-Battani need flags.
The "table of cosecants" claim is wrong. English Wikipedia says al-Battani "produced the first table of cosecants for each degree from 1 degree to 90 degrees, which he referred to as a 'table of shadows'", citing Eli Maor's Trigonometric Delights (S213). This contradicts Britannica and contradicts Debarnot, both of whom read the same table as a cotangent table. Worse, it contradicts itself: the formula quoted a few lines away in the same text, , is a cotangent, not a cosecant. A cosecant is . I record the conflict, and I judge the "cosecant" reading to be an error that has spread through tertiary sources (S181, S209, S213, S226).
The spherical law of cosines attribution is weakly sourced. Al-Battani is often credited with knowing , the spherical cosine rule. The support I could find is tertiary: Wikipedia and a Wikipedia-derived mirror, citing Ben-Menahem (2009), whose pages I did not read (S213, S226). Debarnot's specialist chapter, which is the standard survey of exactly this subject, does not credit al-Battani with the spherical cosine rule. I did not read Nallino's edition of the Arabic text. So: possibly true, not verified here.
The qibla formula attributed to him has the same status. Tertiary sources give it as
where and are the differences in longitude and latitude from Mecca. Both sources note that the rule is an approximation, and that it ignores the sphericity of the Earth. Both cite Glen Van Brummelen's 2013 article "Seeking the Divine on Earth: The Direction of Prayer in Islam" in Math Horizons (S213, S226). That article is paywalled and I did not read it. You can see for yourself why the formula is an approximation. It treats the patch of Earth between you and Mecca as flat. The qibla then becomes the direction of a right triangle with legs and . That is fine for Damascus, which is close to Mecca. It falls apart for Cordoba, and it is nonsense for Beijing.
Al-Battani's afterlife in Europe was large. In 1116 Plato of Tivoli translated the zij into Latin as De motu stellarum (On the Motion of the Stars). Under that title it became one of the main channels through which Arabic astronomy reached Latin readers (S202). The dating comes from MacTutor without a manuscript citation, so treat it as medium confidence. The modern scholarly edition is Carlo Alfonso Nallino's Al-Battani sive Albatenii Opus astronomicum, published in Milan in three parts between 1899 and 1907. Part 3, the Arabic text, appeared first, in 1899. Part 1, the translated chapters with commentary, came in 1903. Part 2, the translation of all the tables, came in 1907 (S220). If you want to check anything about al-Battani against the source, Nallino is where you go. It is on the Internet Archive.
The tool everybody wanted rid of: Menelaus's sector figure
Before the reforms of the tenth century, every spherical astronomy calculation in the Ptolemaic tradition went through one theorem. That theorem is the sector figure, or transversal figure, of Menelaus. It relates six arcs on a configuration of four great circles, in a statement about compound ratios of chords. It works. It is also awkward. You have to build the right configuration of four circles for your problem. Then identify which six arcs to use. Then manage a relation among six quantities to extract one.
Thabit ibn Qurra (THAA-bit ib-n KUR-ra, أبو الحسن ثابت بن قرة بن زهرون الحراني الصابئ, 826 or 836 to 18 or 19 February 901) attacked it head on in On the Sector Figure (al-shakl al-qatta). His birth year is disputed, 826 or 836 depending on source, with the Springer encyclopedia hedging at c. 830 (S198, S230). He worked at Harran and Baghdad. His treatment was exhaustive. He worked Menelaus's theorem through eighteen cases, and he reduced the spherical version to an identity about projection (S181, S198). Eighteen cases sounds like drudgery, and to a modern eye it is. That is exactly the symptom the next generation was reacting against. When your fundamental tool needs eighteen separate case analyzes, the tool is wrong.
Al-Sijzi (as-SIJ-zee, c. 945 to c. 1020) came from Sijistan in eastern Iran and worked at Shiraz and in Khurasan. He took the same figure and gave, in the words of his encyclopedia entry, "a systematic mathematical approach to establishing the 12 relations that emerge from the transversal figure in spherical trigonometry" (S197). Twelve relations, systematically derived rather than found one at a time. His collected rules are what prompted Abu Nasr ibn Iraq to write his Book of Azimuths (S197). Al-Sijzi is invisible in school histories and central in the actual sequence of events. He made the mess legible enough that somebody could see how to replace it.
Two more names from the same decades, on al-Biruni's own testimony. Al-Nayrizi (an-nay-REE-zee, died c. 922) and Abu Jafar al-Khazin (a-BOO JAA-far al-KHAA-zin, died c. 961 to c. 971) had already recovered the rules of the Almagest "in a much easier way". They used methods based on right-angled figures, and they did it before the famous dispute of the 990s (S181). Al-Khazin also left behind a faulty construction. Abu Nasr later corrected it, and in correcting it produced the first known use of the polar triangle (S181). Progress here is not a relay race. It is several people circling the same idea for fifty years.
Abu al-Wafa and the shadow figure
Abu al-Wafa al-Buzjani (a-BOO al-wa-FAA al-booz-JAA-nee, 10 June 940 to 15 July 998, with the death year given as 997 or 998 depending on source) was born at Buzjan in Khurasan. He moved to Baghdad around 959 and spent the rest of his life there (S189, S210, S214). (I did not find the Arabic script of his name confirmed in any source I read, so it is not given here.)
His central contribution is a single figure that does two jobs. It is worth building carefully, because it is the moment the tangent becomes a real function.
Take a spherical triangle with a right angle, and a second right triangle associated with it by the configuration Abu al-Wafa constructs. Write , , and with capitals for the scaled versions used in the tables, so that . His double theorem then gives, for corresponding parts of the first triangle and of the second (S181):
One figure, two theorems. The first is the rule of four quantities. It is the workhorse relation that gets you from any three parts of a right spherical triangle to a fourth, without touching Menelaus. The second is the tangent rule. This is the first place in this history where a tangent appears in a general theorem rather than in a shadow calculation. Abu al-Wafa called it al-shakl al-zilli, "the shadow figure" (S181). The general spherical sine theorem then falls out of these as a corollary (S181).
This is why Debarnot says the tangent enters astronomical calculation "definitively" with Abu al-Wafa even though Habash defined and tabulated it a century earlier (S181). Habash gave the world a function. Abu al-Wafa gave it a theorem it was needed in.
Chapters I.5 and I.6
Abu al-Wafa wrote his own Almagest, of which the first seven treatises survive. Its opening treatise is where the six lines get organized (S181). Debarnot describes chapters I.5 and I.6:
"Abu al-Wafa geometrically defines the 'shadow' of an arc, which he calls his 'first shadow' or his 'versed shadow' ... The same figure is used to introduce the 'second shadow', or 'extended shadow', of the arc considered ... and to establish all the elementary relations between tangent, cotangent, sine and sine of the complement, some of them expressed with the help of two 'shadow diameters' ... our old secant and cosecant. Abu al-Wafa also notes that taking the norm as unity, the 'shadow' will be equal to the ratio of the sine to the sine of the complement" (S181).
Unpack that and you have the modern chart. From one geometric figure he defines the first shadow (tangent) and the second shadow (cotangent). He relates both to the sine and the sine of the complement (cosine). And where the shadow line meets the circle he has two shadow diameters. Those are the hypotenuses of the two shadow triangles, and they are exactly our secant and cosecant. The names are worth keeping, because they are more honest than ours. The secant of an angle is the diameter of the shadow triangle. Calling it "the cutting line" (Latin secans, from secare, to cut) tells you less.
The last clause of the quotation is the quotient identity, arrived at by choosing the radius to be 1. Here it is: "taking the norm as unity, the 'shadow' will be equal to the ratio of the sine to the sine of the complement". That is . Setting the radius to 1 is a bigger deal than it looks. As long as or , every identity carries stray factors of that you have to track and divide out. Set and the identities become the ones in your textbook.
And this is where the usual story is eight centuries out. You will read, in good places, that Euler fixed the radius at 1 in 1748. The specialist history puts it in the tenth century: "This possibility and its advantages had been known and implemented already in Abu'l-Wafa's work as well as in the Table of Minutes of al-Biruni's teacher Abu Nasr Mansur, but it was seldom used afterward by Islamic astronomers" (S481, p. 145). Al-Biruni's own sine table in the Qanun al-Mas'udi uses a unit radius and is, in Van Brummelen's words, "hence equivalent to a modern one" (S481, p. 145).
Now the honest part, because this book would rather you could check it than be impressed by it. The Abu al-Wafa half is solid: Debarnot reaches it independently from the Arabic (S181). The Abu Nasr Mansur half rests on one 1972 article that nobody working on this book has read, which Van Brummelen cites and nothing else. And the chain behind the whole claim is longer than it looks: Van Brummelen is reading modern scholarship, whose footnote goes to von Braunmühl in 1900 (S425), who says in his own footnote that he worked from an 1892 French extract of a Paris manuscript. That is four links from the evidence. One more limit worth stating: Abu al-Wafa's sine and tangent tables are missing from the only surviving manuscript, so what is attested is a statement in a text, not a table you can go and look at.
So what did Euler do, if not this? Something different, and bigger. The tenth century used as a scaling choice inside a table, and then mostly dropped it: Van Brummelen deflates its own advantage as "equivalent to saving the occasional shift of a decimal point" (S481, p. 145). Euler stopped treating sine as a length in a circle at all and made it a number. That is what turns the unit circle from a convenience into a definition, and it is the change your own textbook inherits.
His addition and subtraction formula, from Almagest I.6, is stated in the -scaled form (S181):
Divide through by and you have , unchanged.
One correction to the record on the secant and cosecant. It is often said Abu al-Wafa "introduced" them (S214). Debarnot notes that among the ninth-century auxiliary functions already being tabulated were "purely trigonometric" ones "like the inverse of the sine, formerly called the cosecant" (S181). So a cosecant-shaped function was on paper before him. What is his is the systematic treatment. He defines all six lines from one figure, and he establishes the relations among them rather than collecting them.
The sine of half a degree, and the number everybody misquotes
Building a sine table means starting from a few angles you can get exactly. Then you halve and add your way down to a small step. The bottleneck is always the same. You can bisect from and as far as you like, but you cannot reach 1 degree or half a degree exactly. Trisecting an angle is not a compass-and-straightedge operation. Everybody in this chapter runs into that wall. What separates them is what they do at the wall.
Abu al-Wafa's answer is a bracket. He squeezes the value between an upper and a lower bound and takes the middle. Debarnot: "Abu al-Wafa thus obtains: 0;31,24,55,52,2 < Sin 1/2 degree < 0;31,24,55,57,47, and then, by half sum, Sin 1/2 degree = 0;31,24,55,54,55. The calculation is not perfectly exact, but the method gives a bracketing that is almost six times finer than the procedure in the Almagest applied to the same values" (S181).
Let me show what that is worth, because this single number is the source of a claim that gets badly garbled. Read the sexagesimal as , and divide by to get the sine:
The two agree through and part company at the ninth decimal digit. So the value is correct to eight decimal places, and the error is about . In sexagesimal terms, the true value is , so his answer is right in four sexagesimal fractional places and wrong in the fifth. (Arithmetic mine, run at 40 digits of working precision.)
Both of those statements are worth having, because the "eight decimal places" figure is real but attaches to this one number, not to his table.
The sine table, and the correction
Debarnot on the table: "The sine table in the Almagest of Abu al-Wafa was forecast to four places with entries every fifteen minutes" (S181).
Four sexagesimal places is a resolution of , about , so roughly six to seven decimal digits. Entries every fifteen minutes of arc means 360 entries across the quadrant.
The claim you will meet online is "a sine table at 15-arcminute intervals accurate to 8 decimal places". That splices two different facts together. The interval is right. The precision is not: the table is to four sexagesimal places, roughly six to seven decimal digits. The eight-decimal figure belongs to the single value , computed above. And the fifteen-minute interval was not his innovation either. Habash's sine table already had entries every fifteen minutes a century earlier (S181).
An eclipse, watched from two cities
In 997 Abu al-Wafa in Baghdad and al-Biruni in Kath, in Khwarazm, arranged to observe the same lunar eclipse (S181, S189). This is one of the most elegant experiments in the book, and the logic is simple enough to teach in ten minutes.
A lunar eclipse happens when the Earth's shadow falls on the moon. That is a single physical event, occurring at one instant. Every place on the night side of the Earth sees it at that same instant. Local time is not the same everywhere. The sun sets it, and the sun stands at a different height in Baghdad than in Kath. So let two observers each record the local time at which the eclipse begins, then compare notes. The difference between their two local times is exactly the difference in their longitudes, converted at fifteen degrees per hour.
That is how you measure the distance between two cities without walking between them. And you need it: every one of al-Khalili's tables, every qibla calculation, every prayer time, depends on knowing where places are. Longitude was the hardest coordinate in geography for the next seven hundred years.
Physics and computing hook. This is the same idea as GPS, run backwards. A GPS receiver knows the time a signal was sent and the time it arrived, and turns a time difference into a distance. Abu al-Wafa and al-Biruni knew a distance was unknown and an event was simultaneous, and turned a time difference into an angle. Both are the same trick: a shared event plus two clocks equals a position.
The sine law argument, and why it is really three arguments
Somewhere around 990, in the small scientific community at Rayy in northern Iran, an argument broke out about who had discovered a theorem. It has been running ever since, mostly because the people arguing about it now are not always arguing about the same theorem.
Al-Biruni recorded the original quarrel in the Maqalid ilm al-hay'a (Keys of Astronomy), which he compiled between 994 and 1004 (S181). Debarnot summarizes: "in his Keys he echoes discussions which were taking place amongst the small scientific community of Rayy about a theorem over which al-Khujandi disputed priority with Abu al-Wafa and which he named 'the canon of astronomy'. It concerns the formula that we know as the 'rule of the four quantities'" (S181).
Note what that last sentence rules out. The historical dispute, the one that happened, between people who were alive at the time, was about the rule of four quantities. Not the general spherical sine law. When later writers say "there was a priority dispute over the spherical law of sines", they have swapped in a different theorem.
Here are the three things that get conflated.
One: the general spherical sine theorem. Abu Nasr Mansur ibn Ali ibn Iraq (a-BOO NASR man-SOOR ib-n ee-RAAK, ابونصر منصور بن علی بن عراق جیلانی, c. 960 to c. 1036, with the birth year variously given as c. 950, c. 960 or 970) states it outright (S195, S205). He came from Gilan, worked in Khwarazm and later Ghazna, and was al-Biruni's teacher and then his collaborator. His Risala on spherical arcs, which he himself called On spherical triangles, is dedicated to al-Biruni. Debarnot says the whole work is "arranged on the basis of the general theorem of sines". Abu Nasr states that theorem as: "In every spherical triangle formed by arcs of great circles, the sines of the sides are proportional to the sines of the arcs measuring the angles opposite them" (S181). From it he derives four relations for the right triangle. In modern notation the theorem is
with , , the sides measured as arcs and , , the angles. There is no ambiguity here and no dispute about the wording. Abu Nasr states the general theorem, in a work whose whole structure rests on it.
Two: the rule of four quantities. This is al-Khujandi's claim, and it is the only one that was ever contested between living people. Kushyar ibn Labban (KOOSH-yar ib-n LAB-baan, active around 1000 at Rayy) reworked al-Khujandi's version of the theorem. He attached to it the name that stuck: al-shakl al-mughni, "the figure that dispenses [with the quadrilateral]" (S181). That name is a small masterpiece of technical labelling. The quadrilateral it dispenses with is Menelaus's sector figure. The theorem's selling point, encoded in its name, is that you no longer have to build a four-circle configuration to solve a triangle. Kushyar is another of this chapter's invisible people. He named the central object of tenth-century spherical trigonometry, and almost nobody outside the specialist literature has heard of him.
Three: the double shadow figure. This is Abu al-Wafa's, and it is a different theorem from either of the others. The section above sets it out: two relations from one configuration, one of them involving tangents, with the general sine theorem as a corollary (S181).
Once you separate the three, most of the confusion evaporates. Abu Nasr states the general sine theorem. Al-Khujandi and Abu al-Wafa argued about the rule of four quantities. Abu al-Wafa's own distinctive contribution is the shadow figure, which nobody else claimed.
The common-source hypothesis, and why Debarnot rejects it
Some historians have proposed an older source that we have lost. On that account, the three men, in Khwarazm, Baghdad and Rayy, all took the material from it. Debarnot rejects it, and she does so on al-Biruni's own testimony:
"The possibility of a common source has been put forward to explain the coincidence in the contributions of the three astronomers from Khwarizm, from Baghdad and from Rayy. This hypothesis is contrary to al-Biruni's statement in his Keys of Astronomy ... In reality, the similarity of the statements has no other origin than the contents of astronomical texts" (S181).
That last sentence is the historically interesting one, and it is a good lesson about how mathematics moves. Three people converged on similar theorems, and not because they copied a lost book. They were all reading the same astronomical texts and hitting the same obstacle in them. The problem shapes the solution.
Debarnot is also blunt about the participants. On the general sine theorem she writes of "what is conventionally known, more or less correctly, as the discovery of the general theorem of sines", and on al-Khujandi: "Abu Mahmud al-Khujandi is not a mathematician of the first level. The necessary reform will be the combined work of Abu Nasr ibn Iraq and Abu al-Wafa al-Buzjani" (S181).
Other positions exist and should be on the record. MacTutor writes: "Abu'l-Wafa may have discovered this law first and Abu Nasr Mansur may have learnt it from him ... which of the two has priority is hard to determine and will almost certainly never be known with certainty", and discounts al-Khujandi as "essentially a practical astronomer, unconcerned with theoretical problems" (S205). The Springer encyclopedia entry on Abu Nasr credits him with "discovering the Law of Sines (for both plane and spherical triangles) and the polar triangle concept" and says he "disputed priority with his teacher over the sine law discovery" (S195). Wikipedia adds a fourth claimant, al-Tusi, who lived two hundred years later (S230). No winner is picked here.
Abu Nasr also states the plane sine law, because al-Biruni asked him
Here is a detail that deserves to be famous, and is not. The plane law of sines, the one in your textbook, appears explicitly in Abu Nasr's work. It is there because al-Biruni wrote to him and asked whether the spherical result held for flat triangles too. Debarnot quotes the reply:
"'when you knew' writes Abu Nasr 'that in the triangles formed by arcs of great circles of a sphere, the ratio of the sine of one side to the sine of the other side was equal to the ratio of the sine of the angle opposite to the first side to the sine of the angle opposite to the second, you asked if the rule was general for all triangles, I mean that they were formed by arcs or straight lines. Our answer is yes...'" (S181)
He then proves it by dropping an altitude, obtaining the relation in the form (S181).
That is the historical order, and it is the reverse of the teaching order. In school you learn the plane sine rule first and never meet the spherical one. Historically the spherical version came first, because it was the one astronomy needed. The plane version turned up as an easier special case, in reply to a letter.
Two people who thought the tangent was a bad idea
The tangent did not arrive to applause. "We also see, in this period, al-Khujandi and Kushyar, the two astronomers who met in Rayy, rejecting the tangent theorem of Abu al-Wafa, objecting that the use of the table of shadows is incorrect because of the rapid increase in their difference, a variation concretized by the lengthening of the shadows of the gnomon" (S181).
Their objection is not stupidity. It is numerical analysis, and it is correct. When you use a table you do not have the exact value you want. You have two neighboring entries, and you interpolate: you estimate the value in between, usually along a straight line. Straight-line interpolation works when the function is nearly straight over one step. The sine is nearly straight over a small step everywhere. The tangent is not. It runs away to infinity at 90 degrees. By 80 degrees, consecutive entries in the table above are already jumping by huge amounts. Look at the 12-finger shadow column again. Between 10 and 20 degrees the shadow shrinks by 35 fingers; between 80 and 90 degrees it shrinks by 2. Interpolating linearly in a table like that gives you garbage in one region and nonsense in the other.
So Kushyar and al-Khujandi were saying: your new theorem forces us to read a table that cannot be read accurately. They lost the argument, and they were right about the problem. It got solved later, by better interpolation. That is why al-Biruni's second-order interpolation rule matters as much as any theorem in this chapter.
Al-Khujandi's sextant, and a very large hole in the ground
Abu Mahmud al-Khujandi (al-khoo-JAN-dee, ابومحمود خجندی, c. 940 or c. 945 to c. 1000) came from Khujand, in what is now Tajikistan, and worked at an observatory near Rayy (S194). Whatever Debarnot thinks of him as a mathematician, as an instrument builder he was extraordinary.
He built the al-suds al-Fakhri, the Fakhri sextant, named for his patron. It is a 60-degree arc, one sixth of a circle, hence "sextant". Its diameter is about 43 meters, and it is set in the meridian plane (S194). Think about what that means physically. To read an angle finely you need a big arc. On a big arc one degree is a long way, and the marks are far apart. On a 43-meter-diameter arc, one degree is about 37 centimeters of physical scale. So a single arcminute is over 6 millimeters, a distance a human eye can split. The instrument is a machine for converting money and masonry into precision.
In 994 he used it to measure the obliquity of the ecliptic, the tilt of the Earth's axis relative to its orbit. He got 23;32,19 degrees: 23 degrees 32 minutes 19 seconds, or 23.5386 degrees in decimal (S194).
History hook. The Fakhri sextant, the Maragha observatory, and the Samarkand observatory are all the same kind of object. Each is a large, expensive, permanent research facility, funded by a ruler and staffed by salaried specialists. Each produces results no individual could produce alone. Ask a class what a government gets out of paying for one. The honest answers include calendar control, astrology, prestige, and navigation. A modern science-funding agency would give the same answers in different words.
Al-Biruni measures the Earth from a hilltop
Al-Biruni (al-bee-ROO-nee, ابوریحان محمد بن احمد البیرونی, 4 September 973 to c. 1048 or c. 1050) is the outstanding figure of this chapter. The calculation below is the outstanding worked example in this book. His death date is disputed: the Springer encyclopedia gives c. 1050, while the paper I used for the Earth measurement gives 13 December 1048 (S185, S190). He was born at Kath in Khwarazm, now Beruniy in Karakalpakstan, and spent his later life at Ghazna.
First, the old method, and why he abandoned it
The classical way to measure the Earth is to measure the length of one degree of a meridian. Walk north until the pole star has risen by one degree, and measure how far you walked. Under the caliph al-Mamun, between 813 and 833, a team did exactly this in the desert of Sinjar, between the places named Wamia and Tadmor. The team included Khalid ibn Abd al-Malik al-Marwarrudhi (KHAA-lid ib-n abd al-MAA-lik al-mar-wa-ROO-thee) and Ali ibn Isa al-Asturlabi (a-LEE ib-n EE-saa al-as-tur-LAA-bee) (S185).
The transmitted results are not one number but three. Sources quoting the expedition, including al-Biruni himself and Ibn Yunus, give the length of one degree three ways: 56 Arabic miles, 56 and one quarter miles, and 57 miles. One mile is 4,000 black cubits (S185). That spread is the honest state of the record. Nothing in the sources says which figure the expedition reported.
The method's weakness is obvious. It needs a long, flat, surveyable baseline and a large team. And any error in the distance measurement runs straight into the answer. Al-Biruni wanted something better.
The Nandana observation, in his own words
Around 1018, at the fort of Nandana in the Salt Range in what is now Pakistan, he found the geography he needed. Here is his description, in the Jamil Ali translation:
"I changed to another way owing to having found in a region in India a mountain peak facing toward a wide flat plain whose flatness served as the smooth surface of the sea. Then on its peak I gauged the intersection of heaven and Earth [the horizon] in the prospect, and I found it by an instrument to incline from the East-West line [southern astronomical horizon] a little less than 1/3 1/4 of a degree, and I took it as 34'. I derived the height of the mountain taking the summit in two places, and I found it to be 652 1/20 cubits" (S185).
He made two measurements. First, the height of the hill, found by sighting the summit from two places on the plain: a standard surveying operation. Second, the dip of the horizon, meaning how far below true horizontal you have to look, from the summit, to see the edge of the plain.
The dip is the clever part. From sea level, the horizon sits at eye level. Climb, and the horizon sinks below eye level, because you are now looking over the curve. How far it sinks depends on how high you are and on how sharply the Earth curves. Measure the sink and the height, and the curvature drops out.
His numbers: mountain height cubits (652 cubits, 3 minutes and 18 seconds, which is 652.055 cubits in decimal), and horizon dip arcminutes (S185).
For scale in modern units. One black cubit is 493 millimeters, made of 24 digits. One mile is 1,972 meters, or 4,000 cubits. One farsang is 5,916 meters, or three miles (S185). So the hill is meters, which is a believable hill.
The derivation
Draw three points: the center of the Earth , the mountaintop , and a point on the horizon. The point is where the line of sight grazes the surface. The line is tangent to the Earth at , so the angle is a right angle. The distance is the Earth's radius . The distance is . And the angle at , between and , equals the dip angle . Both are the angle between the vertical at and the line , measured in the same plane.
That gives a right triangle with hypotenuse and adjacent side :
Al-Biruni gets there by applying the sine law twice inside that triangle, not by writing a cosine formula. The content is the same (S185).
Look at the denominator, because that is where the whole problem lives. is 34 arcminutes, a tiny angle. So sits just under 1, and sits just above 0. You are dividing by a number near zero, and every error in gets multiplied enormously. The measurement is easy. The arithmetic is unforgiving.
Running his numbers
Put cubits and arcminutes into the formula, using a modern cosine:
(Computed by me at 40 digits of working precision; the paper that reprints al-Biruni's data recomputes it as 13,331,728.352 cubits, which agrees (S185).)
At 493 millimeters per cubit that is 6,572.5 km, about 3.2 percent too large against a true mean Earth radius of 6,371 km.
But that is not the number al-Biruni states. He states 12,851,369.845 cubits (S185). At 493 millimeters per cubit that is 6,335.73 km. That is about 0.55 percent low against the mean radius of 6,371 km, and about 0.66 percent low against the equatorial radius of 6,378.1 km.
So his stated answer is far better than his own data warrant. Something between the data and the answer is not what the modern retelling assumes.
Reverse-engineering the discrepancy
The formula has only three ingredients: , , and the cosine of . We have and from his own text. So solve for the cosine he must have used. From :
Now feed that value back through his formula and check it reproduces his answer:
Exactly his stated figure, to the third decimal place. So the tables he used gave the cosine of 34 arcminutes, equivalently the sine of 89 degrees 26 minutes, as . The correct value is , that is . His table is wrong in the fourth sexagesimal fractional place: 2 where it should be 26.
Four sexagesimal places into a value close to 1, in a formula that then divides by . That is the entire story. A defect in the fourth place of a table would be invisible in almost any other calculation. Here it becomes a 3.7 percent shift in the Earth's radius, and by accident it shifts it toward the right answer.
The independent check
There is a way to confirm that 12,851,369.845 is the number he worked with, rather than a figure introduced by a modern editor. He also gives the Earth's girth, that is its circumference, as 80,780,039 cubits 1 minute 33 seconds. The paper notes this is "clearly, the result of rounding pi up to 22/7" (S185). So test it:
His girth figure, to the cubit. Two independent quantities in his own text, radius and circumference, are consistent with each other through his own value of . That is as close to internal proof as this kind of forensic reading gets.
A typographical error in a modern paper, stated plainly
The paper I used for all of this gives the same explanation of the discrepancy, and identifies the erroneous sine value in al-Biruni's table. But it prints that value as "0 degrees 59' 59'' 9''' 2'''' 28'''''", that is (S185).
That digit string does not work. Run it through:
That is smaller than al-Biruni's stated radius by a factor of about 4.65. It is not close to any number in the story. The digit string that does reproduce his answer, exactly, is .
I judge the "9" in the published paper to be a typographical error or a scanning slip for "49". I am saying that plainly because the alternative is to print a number that a student could check in ten minutes and find wrong. This is a correction to a modern source, not a claim about al-Biruni.
How much of the accuracy was luck
The paper's own verdict is worth passing on, because it is unusually honest for a piece of history-of-science writing (S185).
First, the dip. Correct for atmospheric refraction, which bends the line of sight, and the apparent dip from the candidate peak is about 32 arcminutes rather than 34. Second, look again at al-Biruni's own words: he found the dip to be "a little less than 1/3 1/4 of a degree". One third plus one quarter of a degree is 35 arcminutes. So he is reporting something a little under 35, and he rounded down to 34. Third, and decisively, a single arcminute of dip changes the computed radius by hundreds of kilometers. The paper's conclusion: "it is only by chance ... that the compensating features of the lucky scenario chosen led him to a figure so close to the truth".
That is the right way to teach this. Al-Biruni's method is excellent: one man, one hill, two measurements, no expedition, no baseline survey. His result is excellent. And the path from his data to his result runs through a defective table entry and a rounding decision that happened to cancel. All three of those statements are true at once, and the third does not diminish the first.
One more conflict, on the kilometer figure
MacTutor states that al-Biruni "determined Earth's radius as 6,339.6 km" (S204). The paper gives 6,335.725 km (S185). The two differ because they assume different cubits. Dividing 6,339.6 km by 12,851,369.845 cubits implies a cubit of 493.30 millimeters rather than 493.0. Neither source explains its choice. And a third figure, 12,803,337 cubits, circulates widely in popular retellings. I could not verify it against any source I read, and it does not reproduce the 80,780,039-cubit girth. Recorded, not resolved.
Al-Biruni's other trigonometry
The Earth measurement is one calculation in a career that produced roughly 150 works. The trigonometric core is in three books.
Al-Qanun al-Masudi (The Masudic Canon), composed at Ghazna between 1030 and 1040 and dedicated to Sultan Masud, is his major astronomical work. Its third treatise, in ten chapters, is on plane and spherical trigonometry (S181). Ten chapters of trigonometry inside an astronomy book is a halfway house: not yet an independent treatise, no longer a chapter of preliminaries. In it he keeps Abu al-Wafa's simplification (S181).
Tahdid nihayat al-amakin (The Determination of the Coordinates of Positions), of 1018, is the geographical work in which the Nandana measurement appears. King calls it "the most sophisticated book ever compiled by a Muslim on mathematical geography" (S231).
Kitab fi ikhraj ma fi quwwat al-asturlab ila al-fil is his book on the astrolabe, and it contains his stereographic qibla construction (S183).
Two ways to find Mecca without spherical trigonometry
Al-Biruni's qibla constructions show off a technique that has almost disappeared from teaching. Take a three-dimensional problem, project it flat, then do plane geometry.
By orthogonal projection, in the Qanun. He locates the zenith of Mecca on the celestial sphere, the point directly overhead at Mecca. Then he constructs its orthogonal projection, its straight-down shadow, onto the plane of the local horizon. The line from that projected point to the center of the horizon circle is the qibla. To get there he marks off the town's latitude along the meridian to find the pole. Then Mecca's co-latitude, to reach a point on the diurnal circle of Mecca's zenith. Then he locates the zenith itself on that circle. Finally he rotates every circle involved into a single plane, so the whole construction becomes planar (S183).
By stereographic projection, in the astrolabe book. Stereographic projection maps the sphere onto a plane from a single point. Its magic property is that it is conformal: it preserves angles, everywhere. So if the quantity you want is an angle, you can project the problem flat and read the angle off the flat picture without distortion. Al-Biruni projects the spherical triangle formed by Mecca, the pole and the town from the south pole onto the plane tangent at the north pole. The two sides running through the pole become straight segments, and the third side becomes an arc. The qibla azimuth is the angle between that arc and the segment from the projected town to the projected pole (S183).
Related work. Habash al-Hasib and al-Biruni both knew the Analemma methods of Diodorus and used them for the qibla. Al-Biruni set out Habash's method in a writing addressed to al-Sijzi. Ibn al-Haytham (IB-n al-HAY-tham) gave a similar solution in his Qawl fi istikhraj samt al-qibla. And al-Kharaqi (al-kha-RAA-kee, died 1158) improved al-Biruni's method, so that azimuth lines no longer had to be engraved on the astrolabe plate (S183).
Instruments hook: the astrolabe as an analogue computer. That last detail is the giveaway. An astrolabe is a stereographic projection of the sky, engraved in brass, with a rotating star map on top. Because the projection is conformal and maps circles to circles, the horizon, the altitude circles and the azimuth lines all come out as circles. An instrument maker can scribe every one of them with a compass. Rotate the top plate to today's date and time, and the instrument shows you the sky. You read off altitudes, times, and directions by inspection. No arithmetic. The person who designed the plate did the mathematics once, and the metal stores it. Al-Kharaqi's improvement, removing the need to engrave azimuth lines, is a hardware optimization: fewer engraved curves, same answers. That is exactly the trade an engineer makes when moving work from run time into design time.
Ibn al-Nadim's Fihrist carries a list of astrolabe makers, which we come to at the end of this chapter. Those are the people who built these machines.
Interpolation, and a puzzle
Al-Biruni's Qanun replaces linear interpolation between table entries with a rule equivalent to a second-order, quadratic scheme. He describes it for the sine and for the tangent, then generalizes it to any table. He also indicates that the procedure can be iterated to higher orders (S181).
Recall Kushyar and al-Khujandi's objection to the tangent: you cannot interpolate linearly in a table that curves sharply. Second-order interpolation is the answer to exactly that objection. It fits a parabola through three neighboring entries instead of a line through two, so it tracks curvature. It is the reason the tangent table becomes usable.
Debarnot flags a puzzle that historians have not settled. A correct quadratic interpolation formula, equivalent to Newton's second-order formula, was already in Brahmagupta's Khandakhadyaka, a work al-Biruni knew well and quoted often (S181). So why does he present it as something to be worked out? No answer is on offer. Record the oddity.
The enneagon, and a cubic
Treatise III of the Qanun contains a chapter on constructing a regular enneagon, a nine-sided polygon. That is a trisection problem in disguise: the central angle is 40 degrees, and 40 is one third of 120, an angle you can construct. Al-Biruni gives two geometric constructions, and both reduce to cubic equations: one satisfied by , and , satisfied by (S181). In the language of the day, that reduction is bi-l-jabr wa-l-muqabala: by restoration and balancing, that is, by algebra. Both cubics are forms of the trisection equation. The following chapter gives four different calculations of the chord of 1 degree, two of them by way of the enneagon. It solves them by iteration (S181).
Keep that in mind for four hundred years, because al-Kashi will set up the same cubic and solve it far more powerfully.
One thing I could not verify. Al-Biruni is often credited with a treatment of Archimedes' broken-chord theorem, sometimes with a claim that he collected many proofs of it. The closest confirmation I obtained is Debarnot describing his Book on Chords as "the very geometrical treatise of al-Biruni on Chords, dedicated to some theorems related to a broken line inscribed in a circle". MacTutor lists the theorem among his achievements without elaborating (S181, S204). I obtained no account of the proofs themselves. Recorded as unverified.
Ibn Yunus in Cairo
Ibn Yunus (IB-n YOO-nus, أبو الحسن علي ... بن يونس ... الصدفي المصري, c. 950 to 6 June 1009) worked in Fustat, old Cairo, under the Fatimid caliphs, and compiled the Hakimi Zij, named for the caliph al-Hakim (S196, S230).
His sine table is sexagesimal to four places with entries every sixth of a degree, that is every ten arcminutes (S181). Debarnot's verdict on its quality is unflattering: "careless calculations make the table inexact, the error sometimes exceeding unity in the number in the fourth place" (S181).
His sin 1 degree is an original piece of work, and it shows how these people thought. He starts from and , both constructible exactly, and bisects four times from each, producing a bracket (S181):
Then he interpolates linearly at two thirds of the way across the bracket, giving . Then he applies a correction, assuming the error affects and equally and in opposite directions. That gives his final value, (S181).
Check it. The true value is with , which Debarnot rounds correctly to at six places (S181). So:
- His bracket, to , contains the true value. Good.
- His interpolated value, , is too high by 17 units in the fourth sexagesimal fractional place.
- His corrected final value, , is too low by 7 units in that same place.
His correction pushed him past the answer and out the other side. He would have done better to stop at the interpolation. (Comparison mine; the sexagesimal values are Debarnot's.)
His interpolation rule. He also introduced a quadratic interpolation rule, using the half-degree entries in his tables (S181). It belongs to the same general movement away from linear interpolation as al-Biruni's rule.
The Cairo tables. The corpus of timekeeping tables associated with Cairo is one of the largest computational artifacts of the medieval world. King describes "approximately 200 pages of highly accurate tables with over 10,000 entries" for Cairo. In his encyclopedia chapter he describes a fourteenth-century corpus, built on an earlier and perhaps less extensive set "set by the tenth-century astronomer Ibn Yunus". That corpus grew to "some 200 manuscript folios and containing over 30,000 entries", was used for several centuries, and no two surviving copies of it are identical (S182, S196). That last clause is a whole seminar in itself. Every user copied this working reference book by hand, corrected it, extended it and abridged it. The "text" is really a family of texts.
King also notes that Ibn Yunus's real sophistication in spherical astronomy came through "several hundred formulas likely derived by means of orthogonal projections and analemma constructions rather than formal trigonometric rules" (S196). And a striking negative fact. Ibn Yunus "paid no attention to the function tabulated by Habash". He went back to the sun's-shadow table, , in the Hakimi Zij (S181). The tangent was available. He did not want it.
The prosthaphaeresis claim, which is a misreading
Ibn Yunus is widely credited with inventing prosthaphaeresis, the technique of turning multiplication into addition using the product-to-sum identity
The technique is real and important. Before logarithms, it was the only way to replace a laborious multiplication of many-digit numbers with a look-up, an addition and a halving. If Ibn Yunus had it in 1000 CE, that would be a major fact.
The attribution traces to Anton von Braunmühl, who wrote that "Already via Ibn Yunus (950 to 1009) ... we have found an application of the second of these formulae" (S212). Kuehn and McCarthy, surveying the origins of prosthaphaeresis, reject it and quote David King directly: "This assertion is incorrect and it can be traced to Delambre's misunderstanding of the material in Chapter 15 of the Hakim Zij" (S212). Their conclusion is that Johannes Werner (1468 to 1522) "must be regarded as an independent inventor of the Prosthaphaeretic Methods" (S212). Wikipedia records the same origin, in Delambre's 1819 reading of the text (S230).
Debarnot's position is close to King's but not identical, and both belong on the record. She writes that "it will be improper to confuse for instance the simplification occasionally applied by Ibn Yunus or al-Kashi when in some rules of astronomy they replace some products of sines and cosines by sums, using the process of calculation called 'prostapheretic', known in Europe in the sixteenth century", and gives the formula in her notes (S181). So Debarnot accepts that such substitutions occur in Ibn Yunus, while warning against reading them as prosthaphaeresis in the European sense. King says the attribution is incorrect.
For a student text, here is the safe statement. The identity is not securely attested as a computational method in Ibn Yunus. The claim that it is traces to a misreading by Jean-Baptiste Delambre in 1819.
Ibn Mu'adh al-Jayyani, in Andalusia
Everything so far has happened between Baghdad and Central Asia. Ibn Mu'adh al-Jayyani (IB-n moo-AATH al-jay-YAA-nee, 989 to after 1079) worked at the other end of the Islamic world, in Jaen in Andalusia. He spent a period in Cairo around 1012 to 1016, and he was a judge by profession (S181).
His Book on the Unknown Arcs of the Sphere is an independent treatise on spherical trigonometry, and its independence is the interesting part. It establishes all six relations for the right spherical triangle without going through Menelaus. It uses the polar triangle to handle the case where all three angles are given. And it makes two choices of notation that no eastern author made (S181):
- He tabulates , not . That is a modern tangent, normalized, with no radius hanging off it.
- He refuses the word "shadow" entirely. No zill, no first and second shadows. The function is a function, not a gnomon.
Together, those two decisions detach the tangent from its physical origin more completely than anyone in the east had managed. And they do it in a treatise that owes nothing structural to Menelaus.
Debarnot's comment on the discovery of the treatise is candid: "The recent discovery of this small original treatise brings us more questions than answers about the difficult question of transmission to the West" (S181). The obvious question is whether Ibn Mu'adh, writing in Spain in the eleventh century, is a link in the chain that carried this material into Latin Europe. The honest answer is that nobody knows yet.
A useful counterweight sits nearby. Jabir ibn Aflah al-Ishbili (JAA-bir ib-n AF-lah al-ish-BEE-lee), a twelfth-century Sevillian, wrote an al-Majisti which "does not even mention the tangent". Yet Regiomontanus used it as a source, and it gave Europe "Geber's theorem", the relation (S181). So one Andalusian author normalizes the tangent two centuries early, and another ignores it completely. The one who ignores it is the one who reaches Europe. Transmission is not a filter that keeps the best ideas.
Maragha, and the first trigonometry textbook
Nasir al-Din al-Tusi (NAA-sir ad-DEEN at-TOO-see, نصیر الدین طوسی in Persian, نصیر الدین الطوسی in Arabic, 1201 to 1274) was born at Tus in Khurasan and spent years at the Ismaili fortress of Alamut. After the Mongol conquest he became scientific adviser to the Ilkhanid ruler Hulagu. The day-level dates conflict: MacTutor gives 18 February 1201 to 26 June 1274, the Springer encyclopedia 17 February 1201 to 25 June 1274 (S199, S203).
He talked Hulagu into building an observatory. Construction at Maragha began in 1259 and the facility was operational in 1262, with al-Tusi as its first director (S199, S203). It had instruments, a library, a staff, and Chinese astronomers among the personnel, which MacTutor notes explicitly (S203). It is the direct institutional ancestor of Ulugh Beg's observatory at Samarkand.
The Treatise on the Quadrilateral
His Kitab al-Shakl al-Qatta (Treatise on the Quadrilateral, sometimes Treatise on the Complete Quadrilateral) is the book that matters here. Its date is 1260 (disputed): Debarnot, the peer-reviewed specialist source, gives 1260; ProofWiki gives c. 1250 (S181). I prefer Debarnot's 1260, for two reasons: it is the peer-reviewed source, and a manuscript copy dated to the corresponding Hijri year, 658, is recorded in the trade. But the conflict is real and unresolved.
Note the title. The book is named after Menelaus's quadrilateral, the sector figure, and yet it is the book in which spherical trigonometry finally stops needing it. Debarnot lays out the structure (S181):
"The Treatise on the Quadrilateral, in five books ... Books I, II and IV concern compound ratios and Menelaus's theorem, both plane and spherical ... Book III, about the lemmas necessary for spherical calculation, briefly evokes the resolution of plane triangles using only the sine theorem. It is book V in particular which constitutes the proper trigonometric part."
So three of the five books are still Menelaus. Book V is where the new subject lives.
The six cases, in book V chapters 5 to 7
This is the passage that gets al-Tusi described as the man who completed right-spherical-triangle trigonometry. Take a spherical triangle with the right angle at , and sides , , opposite the corresponding vertices. Use capitalized , , , for the -scaled functions with . Debarnot gives the six relations with Braunmühl's numbering (S181).
Chapter 5 derives three of them from al-shakl al-mughni, the figure that dispenses with the quadrilateral:
Chapter 6 derives the other three from al-shakl al-zilli, Abu al-Wafa's shadow figure:
Chapter 7 then applies all six to the resolution of right spherical triangles, "carried out again either by the formulae in Chapter 5 or by those in Chapter 6". It finishes with the resolution of any spherical triangle, reduced to right triangles, using the polar triangle (S181).
A caution about the typography. The scan of Debarnot's chapter I worked from has imperfect character recognition in places. So re-check the exact placement of in relations (III) and (VI) before this is printed for students. Use a clean copy, or Caratheodory's French translation of al-Tusi. What is not in doubt is the structure. Six relations: three derived from one figure in chapter 5, three from the other in chapter 6, and all six applied in chapter 7.
The intellectual shape of that structure is the real achievement. Two figures generate six relations; six relations solve every right spherical triangle; every spherical triangle reduces to right ones; therefore the subject is finished. That is not an astronomy result. That is a mathematics result about a class of objects.
The polar triangle, incidentally, is not his. "The use of the polar triangle, applied to the resolution of an ordinary triangle with given angles, was first noted in the Treatise on the Quadrilateral (1260). It was the first known use of the duality principle developed in Europe in the time of Viete (1593) ... We know now that the method exposed by Nasir al-Din is due to Abu Nasr and that it goes back to a construction by al-Khazin" (S181).
The "first independent treatise" claim, weighed
MacTutor states it strongly: this work "is really the first in history on trigonometry as an independent branch of pure mathematics" and "the first in which all six cases for a right-angled spherical triangle are set forth" (S203).
That is a judgment, not a documented fact, and the specialist framing is more careful. Debarnot writes that the end of the tenth century "truly marks a turning point", after which trigonometry "will become the object of independent treatises", and she treats the Treatise on the Quadrilateral as the mature representative of "a new type of work", not as the first of its kind (S181). She names three earlier works in the same line: an anonymous compilation from the late eleventh century, Ibn Mu'adh's Book on the Unknown of the Arcs of the Sphere, and a text standing next to al-Biruni's Keys (S181).
So here is the fair statement. Al-Tusi's is the best-organized, most complete and most influential of the independent trigonometry treatises. It is not demonstrably the first.
There is a competing claim from a different tradition, and it is fair to name it here without settling it. Historians routinely award the same honor, "the first work to treat trigonometry as a subject independent of astronomy", to Regiomontanus's De triangulis omnimodis. That book was written in the 1460s, in Latin. Al-Tusi died two hundred years before Regiomontanus began writing. The two claims are not straightforwardly comparable. They concern different languages, different audiences, and different transmission histories. And the Latin work has the extra distinction of being the first such systematization to be printed. Whether Regiomontanus knew al-Tusi's book is not settled by anything I read. Nothing in the Arabic-side sources used for this chapter can adjudicate it, and this chapter does not pretend to.
One more conflict to record. MacTutor gives the plane law of sines, , as appearing in al-Tusi, and Wikipedia states that "In his On the Sector Figure, appears the famous Sine Law for plane triangles" (S203, S230). Debarnot's chapter is emphatic in the other direction: "A common point in all these treatises is the almost complete absence of plane trigonometry" (S181). She credits the explicit plane sine law to Abu Nasr ibn Iraq, in the reply to al-Biruni quoted above, and treats al-Tusi's book III as merely "briefly evoking" plane triangles by way of the sine theorem (S181). Both positions recorded; no winner picked.
Samarkand
Ulugh Beg (oo-LOOG BEG, 22 March 1394 to 27 October 1449) was born at Sultaniyya, in what is now Zanjan province in Iran. He was the grandson of Timur. His dates are firm, which is a relief after Habash. He governed Samarkand. And he did something no other ruler in this book did: he built a research institute and then worked in it himself (S192, S208).
The observatory. Its construction date is disputed. The Springer encyclopedia says 1420, and the Dictionary of Scientific Biography says 1424. The Stanford project page says it was built four years after the madrasa of 1417, which gives 1421 (S192, S208, S224). The building was circular, roughly 46 meters across (S192).
The great sextant. Its centerpiece was a Fakhri sextant in the meridian plane, set in a trench about two meters wide cut into a hillside. Mounted that way, the instrument could stay permanently in place and never flex (S208, S224). The Springer encyclopedia gives its radius as about 40 meters. The Dictionary of Scientific Biography gives 40.04 meters, and calls it "the largest astronomical instrument in the world of that type". Wikipedia says about 36 meters (disputed) (S192, S208, S230). The scale that a 40-meter radius buys you is worth stating in physical units: 70.2 centimeters per degree, 11.7 millimeters per arcminute, and 1 millimeter per five arcseconds (S208). The solar position was readable to 5 arcseconds (S192). The site was lost for centuries. V. L. Vyatkin rediscovered it in 1908 (S208).
The results. Ulugh Beg's team measured the obliquity of the ecliptic as 23 degrees 30 minutes 17 seconds. The true value at the time was 23 degrees 30 minutes 48 seconds (S192). The star catalog contains 1,018 stars, according to the Dictionary of Scientific Biography and the Stanford project. Wikipedia describes it as 992 stars redetermined plus 27 taken from al-Sufi (S208, S218, S224). The Springer encyclopedia calls it "the only large-scale observations of star coordinates made in the Islamic realm in the medieval period" (S192). His collaborators included Qadi Zada al-Rumi (KAA-dee ZAA-da ar-ROO-mee), al-Kashi, and Ali Qushji (a-LEE KOOSH-jee) (S208, S218, S224).
The tables. The Zij-i Sultani was completed in 1437 (disputed). The Stanford project page and Wikipedia's Ulugh Beg article give 1437. Wikipedia's article on the Zij-i Sultani gives 1438 to 1439 (S218, S224, S230). Its trigonometric tables give sine and tangent at one-arcminute steps up to 45 degrees, and at five-minute steps from 45 to 90 degrees. Deviations from the true values stay "within seconds" (S208).
A precision correction is needed here, because this one is widely garbled. The Springer encyclopedia says the sine is given "to five sexagesimal places" for every arcminute; the Stanford project page says "the trigonometric tables were calculated to five places for both the sine and tan functions and the spherical trigonometric functions were computed to three places"; Wikipedia says "correct to at least eight decimal places" (S192, S224, S230).
Those are the same statement in two number systems, not two competing claims. Five sexagesimal fractional places is a resolution of
which sits between eight and nine decimal digits. So: five sexagesimal places, roughly eight decimal places. The claim you sometimes meet, that Ulugh Beg's tables run to eight sexagesimal places, is a slip. Eight sexagesimal places would be a resolution of about , which nobody in the fifteenth century tabulated for a million entries.
An arcminute step from 0 to 45 degrees is 2,700 entries per function, and each entry is right to about a part in a billion. That is the state of the art in numerical tables until European table-makers of the sixteenth century, working with radii of 10,000,000 and larger, catch up.
Al-Kashi computes the sine of one degree
Jamshid al-Kashi (al-KAA-shee, in full Ghiyath al-Din Jamshid ibn Masud al-Kashi, c. 1380 to 22 June 1429) came from Kashan in Iran and joined Ulugh Beg at Samarkand around 1420 (S191, S201). His death date is given flatly as 22 June 1429 by MacTutor and hedged with "possibly" by the Springer encyclopedia (S191, S201). He is the best computational mathematician in this book, and possibly anywhere before the seventeenth century.
July 1424: al-Risala al-muhitiyya (Treatise on the Circumference), in which he computes to nine sexagesimal fractional places and also converts it to sixteen decimal places (S181, S201). Debarnot describes his method. It is not Archimedes' approach. Instead he expresses as the limit of a nested-radical expression: divided into a repeated square root , with "an error calculation" bounding the result (S181). A course-reading source describes the same computation from the other side, as a polygon of sides (S232). That source is an unattributed PDF and I flag it as low authority. But the arithmetic is at least adequate to the claim, since
which agrees with to 18 decimals (my computation).
1427: Miftah al-hisab (Key of Arithmetic), his major arithmetic textbook, which contains his triangle-solving rules (S217, S232).
c. 1427 to 1429: Risala al-watar wa'l-jayb (Treatise on the Chord and Sine), dated to about 1427 by Britannica. MacTutor describes it as his final work, which "may have remained unfinished at his death and was possibly completed by Qadi Zada" (S201, S211).
The problem
You cannot construct from constructible angles. Getting to 1 degree from 3 degrees means trisecting, and trisection is not a ruler-and-compass operation. Everyone in this chapter hits that wall. Ptolemy bracketed. Abu al-Wafa bracketed better. Ibn Yunus bracketed and then over-corrected. Al-Biruni set up the cubic and iterated on it.
Al-Kashi turned the wall into a machine.
The identity
Start from the triple-angle formula for the sine:
Set . Now is known: you can reach 3 degrees exactly by bisecting and subtracting from constructible angles. So the equation above is a cubic in the unknown , with a known right-hand side.
In table units with , everything picks up factors of , and Aaboe gives al-Kashi's form as (S184):
The coefficient is , exactly the factor you get when converting into units (check: , and , the same number).
The rearrangement that makes it work
Here is the move. Do not solve the cubic. Rearrange it so the unknown appears alone on the left, with the cube buried on the right where it barely matters.
In sexagesimal, with meaning and meaning 900, that is (S184).
Now look at the right-hand side. The term is around 2,826. The term is around 1.15, since is around 1.047. The cube contributes less than one twentieth of one percent of the numerator. So a rough guess at on the right produces a much better on the left, and you can go round again. This is what a numerical analyst calls a fixed-point iteration: feed an answer back into the formula until it stops changing. Its convergence depends on how flat the right-hand side is as a function of . Here it is about as flat as it gets, and each pass gains you roughly one more sexagesimal digit.
Al-Kashi's value of is (S184). (Miriam Chelebi's copy, of which more below, has , an error in the fifth place.) Multiplying by 900 gives the constant, which Aaboe writes as , that is and then the fractional part (S184). So the scheme is
Two iterations, worked
Start from nothing at all, , and drop the cube:
The integer part is settled: the first digit is . Nothing after it is trustworthy yet, so keep only what is settled and go again with :
The first fractional digit is now settled: . Keep and go again:
and the second fractional digit is settled: .
Aaboe traces exactly this sequence in al-Kashi's own presentation: , , , matching the rule as Chelebi states it. Chelebi reports (S184). (I reran the scheme above at 50 digits of working precision and reproduce those three digits.)
Aaboe also gives an informal convergence argument. The error introduced by truncating at each stage is less than 6 units in the next digit. And the derivative of the right-hand side is far smaller than 1 near the fixed point, so the errors shrink (S184). That is a correct modern convergence criterion for a fixed-point iteration, stated in 1429 in the language of digits.
Computing hook. This is Newton's method's less famous cousin. A modern square-root routine still uses it in spirit: rewrite the equation as , guess, substitute, repeat, and stop when the digits stop moving. Have students implement the loop in three lines of Python and watch the digits lock in one at a time. Then have them try the naive rearrangement instead, and watch it diverge. That is the lesson: not every rearrangement of the same equation converges, and choosing the right one is the skill.
The answer, and a problem with the digits
Run it far enough and you get, in the form usually quoted:
That last digit is wrong, and this book can show you why. Expand far enough and the true string is
The tenth digit is 28, so truncating at nine places and rounding at nine places both give 26,18. The widely circulated 26,17 is off by one unit in the ninth sexagesimal place: it lands below the true value where 26,18 lands below it, three times closer. Van Brummelen prints 26,18 (S481). Every calculation in this paragraph is redone in Appendix E, at sixty digits, so you can check it rather than take it.
Keep the wrong digit in view rather than quietly fixing it. A number this good, copied this often, drifting by one unit in the ninth place is exactly how a small error survives for a century: nobody re-derives a value that is obviously excellent.
With , that is , against the true value . Line them up:
The first 17 decimal places agree, , and the 18th differs (2 against 0). Counted in significant figures, 16 significant digits are correct and the 17th is off by two units. The absolute error is about . (My arithmetic, at 50 to 60 digits of working precision.)
Put that in perspective. It is better than the double-precision floating-point arithmetic in the computer you are reading this on, which carries about 15 to 17 significant decimal digits. Al-Kashi computed a sine to the precision of a modern double, by hand, in sexagesimal, in the 1420s.
Now the flag, and it is a serious one. The only source I read that prints the digits is Aaboe (1951), and what Aaboe prints is different. He quotes a marginal gloss in the India Office copy of al-Kashi's Zij-i Khaqani, folio 32 recto, in a translation he credits to Dr Ahmad Fakhri (S184):
"The ancients found no method for obtaining accurately the sine of the third of an angle whose sine is known. We have developed a method and have written an essay explaining it to its limit. We have extracted the sine of one degree by that method; it is 1,2,49,43,11,14,44,16,19,16 ninths. This is an annotation of the author; may God forgive him."
The digits there end , not . And the Aaboe version is numerically wrong from the eighth sexagesimal place onward. Its decimal value is , which misses the truth by about : roughly three hundred times worse than the commonly quoted form.
Now the part that took a further source to settle. The commonly quoted is not the numerically correct string either. It is very close, and for a century that was close enough that nobody rechecked it. The true expansion runs , so nine places truncate and round alike to . Van Brummelen prints (S481). Line the three up against the truth:
| Reading | Where it comes from | Error against the true value |
|---|---|---|
| Van Brummelen 2009 | ||
| quoted everywhere | ||
| Aaboe, as read here |
Two explanations are available and this book could not distinguish them. Either the India Office gloss is itself a scribal corruption, faithfully transcribed by Aaboe. Or the scanned copy of Aaboe read here has corrupted digits. What separates them is the manuscript, and neither this book nor, as far as it could tell, any source it read has gone back to it and printed the digits from the page.
So two different questions have two different answers, and it is worth holding them apart. What is the number? Settled: it ends , and you can check that yourself in Appendix E. What did al-Kashi write? Not settled, and this book does not pretend otherwise.
One further thing my own arithmetic turned up, offered as an observation rather than a claim about any source. Aaboe prints to seven sexagesimal places, . That is the true value correctly rounded at the seventh place. But if you run the iteration with exactly that rounded input, it converges to , not to . Run it with an unrounded and it converges to , matching the quoted value to the last digit. The conclusion is unavoidable, and flattering to al-Kashi. Whatever he used, he carried to more places than the seven that reach us through Aaboe. A seven-place input cannot support a nine-place output. He understood that his answer could be no better than his input.
How precise, according to whom
The sources disagree on how to state the precision, and the disagreement should be visible.
- Aaboe, on al-Kashi's , describes "correct sexagesimal, or 17 correct decimal, digits" and notes "he gives both representations, the consistent use of decimal fractions being an invention of his" (S184).
- Britannica says al-Kashi "calculates the sine of 1 degree correct to 10 sexagesimal places" (S211).
- Debarnot says "Al-Kashi would have determined the first ten places of Sin 1 degree", and adds that the commentator "confines himself to showing the calculation of the first five places from an exact value of Sin 3 degrees to eight places" (S181).
- The Springer encyclopedia says 18 decimal places (S191).
- MacTutor says he computed "to the same accuracy as he had computed pi", that is sixteen decimals (S201).
Ten sexagesimal places, meaning one integer digit plus nine fractional ones, is the figure that Aaboe, Debarnot and Britannica agree on. It is the one to use. In decimal terms my own arithmetic on the value gives 17 correct decimal places and 16 correct significant figures.
The value can be checked the same way. The commonly quoted sexagesimal string is . I did not find a source that prints it, so the digits themselves are unverified. But they evaluate to against a true of , an error of about . Halving gives against . So the first 16 decimal places of agree and the 17th differs. That matches MacTutor's "sixteen decimal places" exactly, which corroborates the digit string indirectly without confirming it.
Two notes on scale. His own published sine table, in the Zij-i Khaqani, is exact to four sexagesimal places with entries every minute of arc (S181). So the eighteen-digit sine of one degree was never printed in a table. He computed it to give the tables a foundation that would not drift. And in the main text of the Zij-i Khaqani, using the older bracketing method, he arrives at , which is wrong from the fifth sexagesimal place (S184). The new method is not a marginal improvement on the old one. It is a different class of tool.
The law of cosines, and a name that is younger than your teacher
Miftah al-hisab of 1427 contains al-Kashi's triangle-solving rules, and the relevant one is what we call the law of cosines:
Wikipedia's history section reports that al-Kashi "repeated essentially al-Tusi's method, now consolidated into one formula". It gives explicit detail for finding an unknown side from two sides and the included angle, citing Azarian (2000) and Aydin, Hammoudi and Bakbouk (2020) (S217). I did not read either of those, so attribute the priority framing to its source rather than stating it flatly.
What is well attested is the name. In France the plane law of cosines is called le théorème d'Al-Kashi. And that naming is recent. The French Wikipedia article on the loi des cosinus states that the Frenchified name of the Persian mathematician appeared in French school textbooks "dans les années 1990". Until then the names used were théorème de Pythagore généralisé (generalized Pythagorean theorem) and loi des cosinus (S217, S440).
So the honorific is about thirty-five years old. It is a school-textbook coinage of the 1990s, not a historical tribute paid by al-Kashi's contemporaries or by nineteenth-century historians. That is worth knowing. The phrase "the French call it al-Kashi's theorem" is often used as evidence of how his contribution was recognized. What it shows is a curriculum decision made in living memory. Both facts are interesting. They are different facts.
Miriam Chelebi, the reason we have the algorithm at all
Debarnot makes a sobering point about the iteration you have just been shown: "This algorithm is only known through a commentary of the astronomical tables of Ulugh Beg. It is therefore difficult to know to what extent al-Kashi is taking from a predecessor" (S181).
That commentary is the work of Miriam Chelebi (MEER-yam che-le-BEE, also written Mirim Celebi, died 1524 or 1525). He was a grandson of Ulugh Beg's collaborator Qadi Zada al-Rumi, and he worked in the Ottoman lands about a century after al-Kashi's death (S184). (The name looks feminine to an English eye; the source describes him as a grandson, and I follow it.) His commentary on Ulugh Beg's tables is the channel through which the whole method reaches us. It is where Aaboe's reconstruction begins.
Aaboe reaches the opposite conclusion to Debarnot on the attribution. After showing that Sedillot's puzzling "A tab-Eddin-Djemshid" is a misreading of "Ghiath ud-Din Jamshid" caused by missing diacritical dots in a manuscript, he writes that "That this elegant method is due to al-Kashi seems quite certain" (S184). Debarnot leaves open whether al-Kashi took it from a predecessor. Both positions on the record; no winner picked.
There is a human detail on the same India Office folio worth carrying. After al-Kashi's own annotation comes a gloss by a later scribe. He reports seeing al-Kashi's own book "in which he had commanded the extraction ... of this sine of one degree by the method of algebra", and he copies a note from its margin: "This is the sine of one degree; it was extracted by inspired strength from the Eternal Presence." A further gloss on the same page refers to "the martyrdom of the martyred Sultan Ulugh Beg" (S184). Somebody, reading a page of sexagesimal arithmetic, thought it looked like revelation. Ulugh Beg was assassinated in 1449.
Al-Ijliyyah, and what to do with one sentence of evidence
Al-Ijliyyah (al-IJ-lee-ya) was an astrolabe maker in the tenth century. Everything known about her fits in a single clause of a single book.
The book is the Fihrist (Catalogue) of Ibn al-Nadim (IB-n an-na-DEEM), completed in 987. It surveys everything its compiler could establish about the books and learned people of his world. It contains a list of astrolabe makers. Here is the passage, in Bayard Dodge's translation, page 671 (S186):
"... Shuja' ibn [gap in the manuscript], an apprentice of Betulus, who was with Sayf al-Dawlah; Ibn Salm, an apprentice of Betulus; al-'Ijli al-Asturlabi, an apprentice of Betulus; al-'Ijliyah, his daughter, a pupil of Betulus, who was with Sayf al-Dawlah."
That is the entire primary evidence. Read what it gives you:
- A nisba, al-Ijliyyah, meaning "the Ijli woman", the feminine of her father's byname.
- Her father: al-Ijli al-Asturlabi, "the astrolabe maker", also on the list.
- Her teacher: Betulus, the same teacher as her father and two other men on the list.
- Her patron: Sayf al-Dawla, the Hamdanid ruler of Aleppo.
Now read what it does not give you. No personal name. No birth or death date. No place of work in the clause itself. No description of a single instrument. No claim of any innovation. Nothing at all about what she made, how she made it, or how good it was.
The uncertainty runs deeper than that. Dodge's own footnote on Betulus records that the name "is written in different ways in the various texts", that it is "obviously a foreign name and, since it cannot be identified, is omitted in the Biog. Index", that another manuscript form is Bituitus, and that Suter suggested Bathulos in 1892 (S186). So even the teacher's name is unstable. A second footnote notes that the order of names from that point to the end of the list follows one manuscript, "as the sequence in MS 1135 and the Flugel edition is badly confused" (S186). So the position of her entry in the list is itself an editorial reconstruction.
Against that, here is what circulates. You will find her called "Mariam al-Ijliya al-Astrulabiya", given the life dates 944 to 967, credited with innovations in astrolabe design, and described as employed at the court in Aleppo. Take those one at a time:
- "Mariam" is not attested. The first name does not appear in the only known source about her life and is not supported by sources from her time (S215).
- "Al-Astrulabiya" is a job description, the feminine of "the astrolabe maker", not a surname (S215).
- "944 to 967" are not her dates. They are Sayf al-Dawla's reign. Somebody took the regnal years of the patron named in the clause and reprinted them as the lifespan of the person. She would have died at 23.
- Aleppo follows from Sayf al-Dawla's court, not from the Fihrist text, which does not name a city in her clause.
- The innovations in astrolabe design have no source at all that I could find.
This chapter treats al-Ijliyyah as a model case, and says so plainly. She was a woman making precision astronomical instruments in the tenth century, recorded by name in a contemporary catalog, alongside her father and her teacher. That is a real and interesting fact about the tenth century. One clause of contemporary evidence is more than most people in this book have. It does not become more interesting by being inflated, and inflating it has a cost. Once you know that the dates are the patron's reign, you cannot trust anything else in the same paragraph, including the parts that are true.
Source-criticism hook. Give students the Dodge clause, unlabelled, and a printed biography of "Mariam al-Astrulabi" from a popular website. Ask them to mark every statement in the second document that can be supported by the first. Then ask where the rest came from. This is the single best exercise in this book for teaching the difference between evidence and story. It works precisely because the underlying subject deserves attention.
The words that survived
Trigonometry is one of the few school subjects whose technical vocabulary is a fossil record of its own history. Here is the Arabic layer.
sine, from jiba. Sanskrit ardha-jya or jya-ardha, "half-chord", shortened to jya, "bowstring", with the variant jiva. Borrowed into Arabic as the meaningless sound jiba, written j-y-b. Re-read as the ordinary Arabic word jaib, "fold, bosom, bay". Translated into Latin as sinus, which means those same things. Clipped in English to "sine" (S207, S209, S228). Who made the Latin substitution is disputed: Eves credits Gherardo of Cremona around 1150, Boyer credits Robert of Chester in 1145 (S228).
zill, "shadow", the ancestor of the tangent. Habash defined the Arabic function as the shadow of an arc (S181). Latin split it into umbra recta, the upright shadow, and umbra versa, the turned shadow, according to which face of a dial the shadow falls on (S207). Abu al-Wafa's own pair of names, translated, are the "first" or "versed" shadow and the "second" or "extended" shadow (S181). Debarnot notes that Maurolycus was still using umbra versa for the tangent theorem in De sphaera sermo in 1558 (S181). The word "tangent" itself is Latin tangens, "touching", from tangere. Thomas Fincke introduced it in 1583, and Viete objected that it was ambiguous (S181, S207, S228). "Cotangent", from complementi tangens, is credited to Edmund Gunter in 1620 (S207, S228).
sahm, "arrow", the versed sine. Taken over from Indian practice, where the arc is the bow and the chord is the string (S181). Used continuously from the ninth to the fifteenth century as the practical substitute for a signed cosine, because never goes negative.
"shadow diameter", the secant and cosecant. Abu al-Wafa's term, in Almagest I.6, for the hypotenuse of the shadow triangle, which Debarnot identifies with "our old secant and cosecant" (S181). The Latin secans, "cutting", from secare, is later. Note the difference in what the two names tell you. The term "shadow diameter" tells you where the line is in the figure, while "secant" tells you only that it crosses something.
al-shakl al-mughni, "the figure that dispenses [with the quadrilateral]". Kushyar ibn Labban's name, around 1000, for al-Khujandi's version of the rule of four quantities (S181). The quadrilateral is Menelaus's sector figure. The name is a sales pitch.
al-shakl al-zilli, "the shadow figure". Abu al-Wafa's tangent theorem, late tenth century (S181).
ilm al-miqat, "the science of timekeeping". Specifically the determination of the mawaqit, the times of the five prayers. The term names a whole literary genre from at least the thirteenth century, and al-Marrakushi's compendium is its defining work (S182).
muwaqqit, "timekeeper". A professional astronomer attached to a religious institution. First appears in Egypt in the thirteenth century, origins obscure (S182, S216).
algorithm, from al-Khwarizmi. The nisba, "the man from Khwarazm", became Medieval Latin algorismus, then Old French algorisme, attested in Middle English as algorism in the early thirteenth century. Then French algorithme, then English algorithm from the 1690s, meaning "the Arabic system of computation" (S227).
algebra, from al-jabr. This is the one I have to leave hanging. The standard gloss is that al-jabr means "the restoration" or "the setting of broken parts", from the title of al-Khwarizmi's al-Kitab al-mukhtasar fi hisab al-jabr wa-l-muqabala. I did not fetch a dedicated etymological source for it. What I can confirm is that the technical phrase bi-l-jabr wa-l-muqabala was still in live use as a description of algebraic method. Debarnot uses it in describing al-Biruni's reduction of the enneagon problem to a cubic (S181). That confirms the phrase, not the word history. Treat the "restoration / setting of broken bones" gloss as unverified here.
Language hook. Ask a class to list every English word they can find that reached us through Arabic. Algebra, algorithm, alcohol, alkali, azimuth, zenith, nadir, almanac, cipher and zero (both from sifr, "empty"). And the star names Aldebaran, Altair, Betelgeuse, Rigel, Vega. Then ask what the pattern is. The answer: the vocabulary clusters in astronomy, chemistry and computation, because that is what was worth translating. A loanword map is a trade map.
The manuscript record
Everything in this chapter is downstream of physical objects, and where those objects are matters. Here is what I verified.
Al-Tusi, Tahrir al-Majisti (Recension of the Almagest), fifteenth-century copy. Bibliothèque nationale de France, Département des Manuscrits, shelfmark Arabe 2485, digitized on Gallica at ark:/12148/btv1b110017717. The Gallica record states "domaine public" (S222). I fetched and parsed the archival metadata record, and I confirmed the link resolves. I did not read the manuscript itself.
Nallino's al-Battani. Al-Battani sive Albatenii Opus astronomicum, edited and translated by Carlo Alfonso Nallino, published by Hoepli in Milan in three parts: part 3, the Arabic text, 1899; part 1, the translated chapters with commentary, 1903; part 2, the translation of all the tables, 1907. Scanned on the Internet Archive under the identifiers albattanisivealb00batt, albattnsivealbat03battuoft (a University of Toronto scan of the Arabic-text volume) and nallino-al-battini-opus-astronomicium-pars-1-3-1899 (a combined upload carrying a Creative Commons Public Domain Mark) (S220). Metadata verified; the Latin and Arabic text not read.
Sedillot on Ulugh Beg. L. P. E. A. Sedillot, Prolégomènes des tables astronomiques d'Oloug-Beg: traduction et commentaire, Firmin Didot, Paris, 1853. Internet Archive identifier bub_gb_3J5SAAAAcAAJ, a Google digitization of the Biblioteca Universitaria di Bologna copy, carrying a Creative Commons Public Domain Mark. A second digitization exists as bub_gb_z-glhcBS2I8C (S221). Item verified live; text not read.
Ulugh Beg, Zij-i jadid-i Sultani, in Persian. Bodleian Libraries, University of Oxford, MS. Greaves 5, with a digital object record at Digital Bodleian (S225). I read the catalog record only. The record I fetched does not state a rights position, so check the Digital Bodleian terms before reproducing any image.
Ibn al-Nadim, Fihrist, in Bayard Dodge's translation (Columbia University Press, 1970, two volumes), on the Internet Archive; page 671 carries the al-Ijliyyah clause quoted above (S186). I downloaded the full text and read that page.
And here is what I could not find, stated because a reader deserves to know where the gaps are.
BnF Arabe 2558, al-Khalili's Damascus prayer tables. King reproduces an extract as his Plate 4.17, from folios 10v to 11r (S182). I have the shelfmark from King but could not locate a digitized copy; my catalog query for "arabe 2558" returned no usable result.
BnF Arabe 2495, Ibn Yunus's Hakimi Zij, folios 44r to 44v, which is the chapter carrying the account of the al-Mamun degree measurement (S185). Shelfmark from the source; scan not located.
IO Islamic 1148, a manuscript containing al-Tusi and Ulugh Beg's Zij, held by the British Library. I know of it only from the Qatar Digital Library finding aid, which also names Or 8349 (al-Biruni's Kitab al-tafhim), Or 6669, Add MS 7474, 7475 and 23392, Or 5323, IO Islamic 4419 and Or 9587 (S223). The individual item pages returned HTTP 403 to my requests, so I have verified none of them directly and cannot state their rights positions.
One more item, listed but unverified. Between 1714 and 1721, Yahya ibn Ali al-Rifai and Hasan ibn Muhammad al-Fasihi al-Nizami made an Arabic translation of Ulugh Beg's tables. It is held by the National Library and Archives of Egypt, and was formerly cataloged as World Digital Library item 3951 (S224). The World Digital Library has since been absorbed into the Library of Congress, and I did not confirm that the link still resolves.
What the evidence does not support
"Al-Khwarizmi produced the first table of tangents." He produced a table of the shadow of a twelve-finger gnomon as a function of solar altitude, that is , given by degrees to two sexagesimal places. He used it for one purpose: converting between altitude and shadow length (S181). The same object appears in al-Battani. Debarnot's objection is not pedantry. A quantity defined by an arbitrary stick, with its own units, "is clearly an obstacle to any idea of generalization and the introduction of a useful function, tan or R tan" (S181). The first general table of is Habash al-Hasib's. It is defined for an arc rather than for the sun, tabulated to three sexagesimal places at half-degree steps from 0;30 to 89 degrees, and applied to several different problems. Debarnot judges his introduction of it independent of any predecessor (S181). The school claim does not survive contact with the specialist literature.
"Ibn Yunus invented prosthaphaeresis." The claim traces to Anton von Braunmühl and, behind him, to Delambre's 1819 reading of chapter 15 of the Hakimi Zij. David King's verdict, quoted by Kuehn and McCarthy: "This assertion is incorrect and it can be traced to Delambre's misunderstanding of the material in Chapter 15 of the Hakim Zij" (S212). Their conclusion is that Johannes Werner must be regarded as an independent inventor of the method (S212). Debarnot's position is a shade different and belongs on the record. She accepts that Ibn Yunus and al-Kashi occasionally replaced products of sines and cosines by sums in astronomical rules. But she warns that it is "improper to confuse" those substitutions with the sixteenth-century European technique (S181). The safe statement for a student: the product-to-sum identity is not securely attested as a computational method in Ibn Yunus.
The biography of "Mariam al-Astrulabi", with dates 944 to 967. The sole source is one clause in the Fihrist, quoted in full above (S186). The first name is not attested (S215). "Al-Astrulabiya" is a trade description, not a surname (S215). And the widely printed dates 944 to 967 are the reign of Sayf al-Dawla, the patron named in the clause, not her life dates. Anyone repeating them is copying a regnal span and calling it a lifespan.
"Abu al-Wafa's sine table was accurate to eight decimal places at fifteen-arcminute intervals." Two facts spliced together. The interval is correct. The precision is not: Debarnot describes the table as "forecast to four places", that is four sexagesimal places, roughly six to seven decimal digits (S181). The eight-decimal figure belongs to one number: his half-sum value for . The arithmetic above shows that value to be correct to eight decimal places of the sine, and to four sexagesimal fractional places. And the fifteen-minute interval was not his innovation: Habash's sine table had it a century earlier (S181).
"Ulugh Beg's tables were computed to eight sexagesimal places." The sources say five sexagesimal places for the sine and tangent, at one-arcminute steps to 45 degrees (S192, S208, S224). Wikipedia's "correct to at least eight decimal places" is the same claim in a different base, since (S230). Eight sexagesimal places would mean a resolution near , thousands of times finer than anything in the manuscripts, replicated across thousands of entries. Somewhere in the chain of retelling, "sexagesimal" got substituted for "decimal".
"Al-Battani produced the first table of cosecants." Wikipedia says so, citing Maor (S213). Britannica reads the same table as cotangents and gives the underlying rule as , which is a cotangent (S209); Debarnot describes it as the twelve-finger gnomon shadow (S181). The formula quoted alongside the cosecant claim contradicts the claim.
"Al-Tusi's treatise contains the first statement of the plane law of sines." MacTutor and Wikipedia say so (S203, S230). Debarnot says of the whole genre that "a common point in all these treatises is the almost complete absence of plane trigonometry", credits the explicit plane sine law to Abu Nasr ibn Iraq's reply to al-Biruni, and treats al-Tusi's book III as only "briefly evoking" plane triangles (S181). Recorded as a conflict; not resolved here.
"Al-Biruni computed the Earth's radius as 12,803,337 cubits." That figure circulates widely. I could not verify it in any source I read, and it fails the internal consistency check. It does not reproduce the girth of 80,780,039 cubits that al-Biruni also gives. The figure 12,851,369.845 cubits reproduces that girth to the cubit, through his own value of (S185).
"Al-Kashi's sine of one degree is 1;2,49,43,11,14,44,16,26,17." Half of this is now settled and half is not, and the two halves are worth keeping apart.
Settled: the numerically correct string ends 26,18, not 26,17. Recomputing to twelve sexagesimal places gives , and Van Brummelen prints 26,18 (S481). An earlier version of this book called the circulated 26,17 "the numerically correct one." That was wrong, and the arithmetic that shows it is in Appendix E.
Still open: which string al-Kashi himself wrote. Aaboe prints , wrong from the eighth place as it reaches us (S184). Riahi works the whole method through but deliberately renders it in decimals, so he settles nothing about the digits: "He appears to have used the sexagesimal system, but for clarity I shall render his argument using decimal expansions" (S484). Settling it means going to the manuscript itself, and this book did not. So the number is right. Whose hand it is in is not established, and no source read here prints the digits from the manuscript.
"The French call it the theorem of al-Kashi, which shows how his contribution was honoured." The name is real. Its age is not what the sentence implies. French school textbooks adopted théorème d'Al-Kashi in the 1990s, having previously used théorème de Pythagore généralisé or loi des cosinus (S217, S440).
"The qibla is why Islamic trigonometry developed." It is a large part of why the demand was continuous, salaried and public. The tables of al-Khalili and Ibn al-Shatir exist for no other reason. But Debarnot's warning stands: the qibla "accounts poorly for the complexity of the calculation of the zij, owing to its heterogeneous composition and the prodigious expansion of astronomy in the ninth century" (S181). A single-cause story is the wrong shape.
- Five paying problems, prayer times, the qibla, calendars, astrology and inheritance, funded the completion of the toolkit.
- The gnomon's shadow gave the tangent and cotangent as tabulated functions long before their modern names; the six-function set closes here.
- Al-Khalili computed a city's religious timetable for any locality: trigonometry as civic infrastructure (S182, S216); al-Tusi then wrote the subject its own book, out from under astronomy.
Next: China, which solved the same triangles for fifteen centuries with no angle functions at all.
The quotient identities are this chapter's shadow pair in modern dress: tan and cot as ratios of sin and cos, the two shadow functions reversed, in Trigonometry 2.6
Al-Khalili's tables answered a question every city asked five times a day. What did they compute?
Chapter 5
The Road Not Taken (China and Japan, c. 100 BCE to 1750)
The people in this chapter
Faces where a face survives. Every name links to its full entry in Appendix A.
A bamboo tube pointed at the sun
Around 100 BCE, a Chinese text describes a man holding a length of bamboo up to the sky. The tube is 8 chi long, and its bore is 1 cun wide, one tenth of a chi. (The chi is the Chinese foot; the cun is its tenth part.) He looks through it at the noon sun, and the sun exactly fills the opening, no more and no less. He has one other measurement: the shadow cast by a stick of known height. From those two facts he works out how high the sun is, how far away it is along the slant, and how wide across it is. He gets four numbers. They are precise. They are internally consistent to the last digit. They also describe a universe that does not exist, because his model of the cosmos is wrong. The mathematics is flawless anyway. And here is the thing this whole chapter turns on: he does it without ever naming an angle.
For roughly eighteen centuries, Chinese mathematicians attacked the same problems that produced trigonometry in Greece, in India, and in the Islamic world. Measure the sun. Survey a mountain you cannot climb. Fix the calendar. Convert one set of sky coordinates into another. They solved those problems, often beautifully, sometimes to a precision nobody in Europe would match for a thousand years. They did it without ever building a trigonometric function: a rule that takes an angle in and hands you back a ratio. You compute such a rule once, write it down in a table, and then reuse it across problems that have nothing else in common.
That is not a deficiency. It is a design decision. It is also the most instructive fact in this book. A textbook can tell you that . What a textbook cannot easily show you is that the sine had to be invented. Somebody had to decide to organize measurement that way. A civilization with first-rate geometers and first-rate astronomers could look at the same problems and organize them differently. China is the proof. Read this chapter as a demonstration that the sine is a choice.
The Zhoubi suanjing computes the height of the sun from two shadow measurements, and the arithmetic is flawless. The answer it produces is meaningless anyway. Where is the flaw?
I have a guess▸
In the cosmology, not the mathematics. The computation assumes the gaitian "canopy heaven": a flat earth under a near sun circling tens of thousands of li overhead (S245). On that model the cun-qian-li shadow law is reasonable and the similar-triangle argument is sound. On the real, round earth, the quantity being computed does not exist.
If you guessed sloppy measurement or bad arithmetic, the chapter's point cuts the other way: the geometry is exactly right, which is what makes it such a clean lesson in checking a model before trusting a computation.
What the mathematics was for
Nobody in Han or Tang or Yuan China worked on right triangles because right triangles are pretty. Chinese mathematics of this period was state infrastructure. It had two employers.
The first was the calendar. An emperor's mandate to rule rested on the correctness of the calendar his bureau issued. Suppose the winter solstice arrived on a different day from the one the court had announced, or an eclipse showed up unpredicted. That was not an academic embarrassment. It was a political event, evidence that heaven's approval had slipped. So the astronomical bureau had to predict three things: the sun's position through the year, the moon's phases, and the timing of the twenty-four qi 氣. The qi are the solar terms that divide the year. That job means tracking the sun's uneven motion along the ecliptic and converting between coordinate systems. Elsewhere, the same job forced spherical trigonometry into existence.
The second employer was survey. How tall is that mountain? How far away is that island? How wide is that river you cannot cross? How deep is that valley? How far apart are two cities you can only measure by walking? Those answers decide where a canal goes, how a province is taxed, and where an army can march. Both jobs come down to one instruction: measure what you can reach, and compute what you cannot. That is trigonometry's job description. China wrote a different tool for it. The tool was called gougu.
The vocabulary of the right triangle: gou, gu, xian, bi
Start with the words, because the words tell you how the tradition thought.
Gou 勾 (say it GOH, first tone; written 句 in the older texts) means "hook." It is the shorter leg of a right triangle. Picture the standard gnomon setup, a gnomon being an upright stick that throws a shadow you can measure at noon. The gou is the shadow lying along the ground (S245)(S246).
Gu 股 (say it GOO) means "thigh." It is the longer leg, the upright: in the gnomon setup, the stick itself standing vertical (S245)(S246).
Xian 弦 (say it SHYEN) means "bowstring." It is the hypotenuse, the slant that closes the triangle. Separately, in the same tradition and with the same character, it is the chord of a circular arc. A chord is the straight line joining the two ends of a curve (S246). Hold on to that double meaning. The Greek word for chord carries it too. That double meaning is what produced the Indian jya. Xian is the closest thing in Chinese to a trigonometric noun. What China never did was attach it to an angle.
Bi 髀 (say it BEE) means "thigh bone." The tradition glosses bi as a synonym for gu. In practice it names the standard gnomon: the 8-chi vertical stick whose shadow every Chinese astronomer measured (S245). It is also the second character in the title Zhoubi suanjing 周髀算經, where zhou 周 is read either as the Zhou dynasty or as "circuit, circle." So the title is roughly "the computational classic of the Zhou gnomon."
Together, gougu 勾股 names the right-triangle relation itself, hook and thigh, shadow and stick. The name is still current: modern Chinese for the Pythagorean theorem is gougu dingli 勾股定理, "the gou-gu theorem" (S245)(S246).
Now look at that vocabulary again and notice what is not in it. There is no word for the angle at the corner. There is no word for "the ratio of the hook to the bowstring, considered as a thing in its own right." Every term names a length, a physical object you could hold: a hook, a thigh, a bowstring, a bone. A Chinese surveyor thinks in lengths and in ratios of lengths, all the way down. That is not a limitation he is straining against. It is a complete and coherent way to do the job. For eighteen centuries it did the job.
The Zhoubi suanjing states the basic case flatly. It is the oldest such statement in Chinese: "勾廣三，股修四，徑隅五" (gou guang san, gu xiu si, jing yu wu), "a base of 3 units, a height of 4 units, and a diagonal of 5 units" (S245). The digital edition of the classic confirms the Chinese text (S241). The Jiuzhang suanshu 九章算術 (Nine Chapters on the Mathematical Art) was circulating in something close to its received form by around 100 CE. It gives the general rule at the head of chapter 9:
勾股術曰：勾股各自乘，並，而開方除之，即弦。
"The gougu rule says: square the gou and the gu separately, add them, extract the square root; that is the xian" (S246). That is the Pythagorean theorem, stated as a procedure rather than as a proposition. Chinese mathematics states almost everything that way. The text hands you a recipe and a worked case, not an axiom and a proof. (A note on the date: the sources read for this chapter do not themselves establish when the Nine Chapters reached its received form. Treat "c. 100 CE" as an inherited scholarly convention rather than something verified here (S246).)
The Zhoubi suanjing: what the book is, and when
The Zhoubi suanjing is the oldest Chinese mathematical text that survives. It is a strange object: part cosmology, part arithmetic manual, part teaching dialogue. It contains gnomon-shadow tables, the sun-height computation worked below, and a rule for turning shadow differences into distances on the ground.
Its date is disputed. The dispute is worth stating plainly rather than smoothing over. Yong Li and Xiaochun Sun took up the question in Research in Astronomy and Astrophysics in 2009. They say the book was "probably compiled about BC 100," in the Western Han, out of older material (S244). Christopher Cullen's work on the text is the standard reference in English. He conjectures instead that final compilation belongs to the early first century CE, a century or more later. Chia-Yun Wu reports Cullen's position in her 2023 study of Tang geography (S254). Nothing read for this chapter resolves the question. Both dates are in print. Both come from serious scholars. The honest position is to hold them side by side.
The book presents its mathematics inside a dialogue. A master called Chen Zi (chun DZUH) instructs a student called Rong Fang (rong FAHNG). The sun-height computation is Chen Zi's answer to Rong Fang's question (S241)(S245). Be careful with these two. They are characters in a text, not people with biographies. No independent record establishes that either lived. The register of Chinese mathematicians treats them as literary figures in a compilation whose own date is contested (S241)(S245). The dialogue tells you nothing about who did the mathematics. What it tells you is how the tradition wanted mathematics taught: as a method passed from a teacher who understands it to a student prepared to work. The general rule comes after the particular case, not before it.
The cun-qian-li rule
First the text needs one piece of empirical furniture, and it states that as a law:
法曰: 周髀長八尺，句之損益寸千里。
"Method: the Zhou gnomon is 8 chi long, and the decrease or increase of the base is one cun per thousand li" (S241)(S244). This is the cun qian li 寸千里 rule, "one cun per thousand li." A cun is a tenth of a chi. The li is the Chinese mile, the unit for long distances over ground. Move your 8-chi gnomon 1,000 li due south and the noon shadow shrinks by 1 cun. Move it 1,000 li due north and the shadow grows by 1 cun. The Zhoubi states it as data too. At the standard station the summer solstice shadow is 1 chi 6 cun. At 1,000 li due south it is 1 chi 5 cun. At 1,000 li due north it is 1 chi 7 cun (S241).
The rule is a linear conversion between shadow length and north-south distance. It is the hinge of the whole computation. It is also false. Establishing that will take the Chinese astronomical tradition eight centuries and an empire-wide field survey. Hold that thought.
The sun-height computation, worked
Here is the passage, from the Chen Zi dialogue in the first scroll (juan shang) (S241):
周髀長八尺，夏至之日晷一尺六寸。髀者，股也。正晷者，句也。正南千里，句一尺五寸。正北千里，句一尺七寸。 日益表南，晷日益長。候句六尺，即取竹，空徑一寸，長八尺，捕影而視之，空正掩日，而日應空之孔。 由此觀之，率八十寸而得徑一寸。故以句為首，以髀為股。從髀至日下六萬里，而髀無影。從此以上至日，則八萬里。 若求邪至日者，以日下為句，日高為股。句、股各自乘，並而開方除之，得邪至日，從髀所旁至日所十萬里。 以率率之，八十里得徑一里。十萬里得徑千二百五十里。故曰，日晷徑千二百五十里。
In plain English: the gnomon is 8 chi. At the summer solstice its shadow at the standard station is 1 chi 6 cun. The gnomon is the gu; the shadow is the gou. A thousand li due south the shadow is 1 chi 5 cun, a thousand li due north 1 chi 7 cun. The further south the sun is from the gnomon, the longer the shadow grows. Now go to the place where the shadow measures exactly 6 chi, which is 60 cun at 10 cun to the chi. Take a bamboo tube of bore 1 cun and length 8 chi. Catch the sun's image and look through it. The bore exactly covers the sun, and the sun exactly fills the bore, so the ratio of diameter to distance is 1 in 80. From the gnomon to the point directly below the sun, where a gnomon casts no shadow at all, is 60,000 li. From that point up to the sun is 80,000 li. Now find the slant distance to the sun. Take the ground distance as the gou and the sun's height as the gu, square each, add them, and extract the root: 100,000 li. Applying the ratio, 80 li of distance gives 1 li of diameter, so 100,000 li gives 1,250 li. The sun's diameter is therefore 1,250 li.
Every step of that is arithmetic you can check, and every number checks:
All four of the text's stated numbers reproduce exactly, with no rounding anywhere (S241).
Two things are worth pausing on. First, the middle step is pure similar triangles. The text says so in its own vocabulary. The gnomon and its shadow form a small triangle, 8 chi by 6 chi. The sun's height and its ground distance form a huge one, unknown by 60,000 li. The two have the same shape, so scale one to the other. That is the whole method. It needs no angle, no table, and no function.
Second, look at the triangle you end up with: 60,000, 80,000, 100,000. That is the 3-4-5 triangle scaled by 20,000, which is why the square root comes out exact and clean. Whoever built this problem chose the input so the answer would land on a whole number. A modern textbook writer setting a Pythagoras exercise does the same. The Zhoubi is a teaching text and it behaves like one.
What is wrong with it, and what is not
The physics is wrong. Say so to students plainly, without sneering.
The Zhoubi assumes the gaitian 蓋天 or "canopy heaven" cosmology: a flat earth beneath a domed sky. The sun is a bright object circling a few tens of thousands of li overhead (S245). On that model, cun-qian-li is a reasonable law and the computation is sound. The real earth is round. The real sun is about 150 million kilometers away. On the real earth the shadow does not shorten linearly with distance traveled south. And no point exists where the sun sits 80,000 li directly overhead. The answer is not merely inaccurate. The quantity it computes does not exist.
Now the part that matters more. Given the model, every inference is correct and every arithmetic step is exact. The method runs like this: measure a baseline you can walk, measure a shadow you can reach, then scale by similar triangles to something you cannot reach. Eratosthenes used that same method in Egypt in the third century BCE to get the size of the earth roughly right. He had a better model, not better mathematics. The difference between the two results is a difference in cosmology. And in this period you cannot derive a cosmology. You assume one and then test it. Yixing tested this one in 724, two sections from here, and the test cost the Zhoubi its rule.
The shadow table that was calculated, not observed
The second scroll of the Zhoubi (juan xia) carries a table of noon gnomon shadow lengths for all twenty-four solar terms. At the winter solstice the shadow is longest, 1 zhang 3 chi 5 cun. A zhang is 10 chi, so that is 13.5 chi, written 冬至晷長一丈三尺五寸. At the summer solstice it is shortest, 1 chi 6 cun (1.6 chi, 夏至一尺六寸). The table runs down from the one to the other and back again (S242).
That looks like an observational record. It is not, and the text itself gives the game away. Li and Sun report: "In the text, it is stated that shadow lengths for 22 solar terms (Table 1) can be derived from that of the Winter Solstice 13.5 chi (No. 1) and that of the Summer Solstice 1.6 chi (No. 13)," using a constant common difference the text calls sunyi 損益, "decrease or increase" (S244). Compute it:
Subtract that from 13.5 repeatedly and you get 12.5083, 11.5167, and so on down the list. Those are exactly the values the received table prints (S242)(S244). Li and Sun's conclusion: "Except for the Winter Solstice and Summer Solstice, the values of the shadow lengths were more likely calculated than observed" (S244).
This is a small point with a large moral. It belongs in a mathematics classroom. Twenty-two of the twenty-four entries in the oldest Chinese astronomical table are linear interpolation dressed as data. Interpolation is the trick of filling the gap between two known values by stepping evenly from one to the other. The Chinese astronomers were not faking. Interpolating between two anchors was a normal and defensible way to fill a table when observation was expensive. But take that table as twenty-four independent measurements, do statistics on it, and you will claim twenty-four times more confidence than you are entitled to. Students who will one day be handed a spreadsheet should meet this problem in the year 100 BCE.
Where and when were those two solstice shadows measured?
That leaves the two real numbers, 13.5 chi and 1.6 chi. What can you get out of them?
You can get a latitude, because the ratio of shadow to gnomon fixes the sun's zenith distance. Here is the calculation, done here rather than taken from a source:
So the Zhoubi's own numbers describe a station a little north of 35.3 degrees. Li and Sun make the same point and draw the historical consequence. The derived latitude is above 35.3 degrees north, while the Zhou royal capital, where the observations are traditionally located, sits below 34.8 degrees north (S244). The numbers do not fit the place the tradition assigns them.
Li and Sun go further. Here the chapter has to be careful. They ran a statistical reconstruction on the underlying, pre-modification solstice data. It points, they conclude, to an observation epoch of 564 BCE at latitude 35.78 degrees north (S244). That is a striking claim. It would make the Zhoubi's two real shadow values the earliest meridian observations recorded in China, several centuries older than the compilation that carries them.
It is also a reconstruction from data the authors themselves say has been modified. And it is contested. Zhao (2009), cited within Li and Sun's own paper, proposes 511 BCE instead of 564 BCE. An older line of scholarship held that the data were not observational at all. Li and Sun cite that line too: Qian (1958) and Bo (1989) (S244). Three positions, then: an observation around 564 BCE, an observation around 511 BCE, or no observation whatsoever. Li and Sun argue the first, they name the second, and they are arguing against the third. Attach low confidence to any single date here. Attach high confidence to the arithmetic that produces a latitude near 35.3 degrees. Those are different things.
The Nine Chapters, chapter 9: "Gougu"
The Jiuzhang suanshu is the central text of the Chinese mathematical tradition, the book every later commentator commented on. Its ninth and last chapter is titled "Gougu" 勾股. It is the closest thing in classical Chinese mathematics to a trigonometry chapter. It contains no trigonometry.
The bare triangle
Chapter 9 opens with the simplest possible case and its two inversions (S246):
今有勾三尺，股四尺，問為弦幾何？答曰：五尺。
"Now given a gou of 3 chi and a gu of 4 chi, what is the xian? Answer: 5 chi." Then the same triangle read backwards: given the xian and one leg, find the other. Worked out with the chapter's own rule (square, add or subtract, extract the root):
Three problems, one relation, three directions. This is the entire toolkit. Everything else in the chapter builds on it.
The reed in the pond
Problem 9.6 is the one to teach. It looks like a riddle and is an algebraic identity wearing a costume (S246):
今有池方一丈，葭生其中央，出水一尺。引葭赴岸，適與岸齊。問水深、葭長各幾何？ 答曰：水深一丈二尺；葭長一丈三尺。 術曰：半池方自乘，以出水一尺自乘，減之，餘，倍出水除之，即得水深。加出水數，得葭長。
"There is a pond 1 zhang square. A reed grows at its centre and rises 1 chi above the water. Pull the reed to the bank and it just comes level with the bank. How deep is the water and how long is the reed? Answer: the water is 1 zhang 2 chi deep; the reed is 1 zhang 3 chi long. Rule: square half the side of the pond, square the 1 chi of protrusion, subtract the second from the first; divide the remainder by twice the protrusion, and that gives the depth of the water. Add the protrusion to get the length of the reed."
The pond is 1 zhang, which is 10 chi, on a side, so the distance from the center to the bank is 5 chi. The reed sticks out 1 chi. Run the rule:
Check it against the geometry. Once pulled over, the reed is the hypotenuse of a right triangle. Its legs are the half-width of the pond and the depth of the water:
Exact. And notice that the rule is not a trick tuned to these particular numbers. Write for the half-width, for the protrusion, for the depth. The reed's length is , so
The cancels, and what is left is the Chinese recipe word for word: square half the side, subtract the square of the protrusion, divide by twice the protrusion. The author of the Nine Chapters had a general result and chose to state it on a concrete case. That is the standard rhetorical move of the whole tradition (S246).
Everything in chapter 9 works this way. Lengths in, lengths out, ratios in between. There is no angle anywhere in the chapter, no arc measure, and no table. A Greek geometer handed these same problems would sooner or later reach for a chord, and later still for a sine. That difference is the entire subject of this chapter.
Liu Hui, 263 CE
Liu Hui (lyoh HWAY, c. 220 to c. 280) is the first Chinese mathematician who reads like an individual rather than a tradition. He worked in the northern state of Wei. His life dates are approximate. The only firm date attached to him is one he wrote himself. At the head of his commentary on the Nine Chapters he dates the work to "the fourth year of the era of the Jingyuan reign of Prince Chenliu," which is 263 CE (S251).
Liu Hui's commentary is not a set of marginal notes. He proves things, criticizes the received text where he thinks it is wrong, and adds material of his own. To chapter 9 he appended a set of surveying problems built on a technique called chong cha 重差: chong means "double, repeated," cha means "difference," so "double difference." The idea is to take two sightings from two stations and divide by the difference between the two back-off distances. During the Tang dynasty an editor detached that appendix and published it as a book in its own right, the Haidao suanjing 海島算經 (Sea Island Mathematical Manual). It then entered the Ten Computational Canons, the official curriculum for the imperial mathematics examinations (S247)(S250)(S251). The detachment date rests on reference works rather than on a specialist study, so hold it a little more loosely than the 263 date.
The sea island, worked completely
Problem 1 of the Sea Island Manual is the problem the book is named for. It is the finest piece of surveying mathematics in this chapter (S247):
今有望海島，立兩表，齊高三丈，前後相去千步，令後表與前表參相直。從前表卻行一百二十三步，人目著地取望島峯， 與表末參合。從後表卻行一百二十七步，人目著地取望島峯，亦與表末參合。問島高及去表各幾何？ 答曰：島高四里五十五步；去表一百二里一百五十步。 術曰：以表高乘表間為實；相多為法，除之。所得加表高，即得島高。求前表去島遠近者：以前表卻行乘表間為實； 相多為法。除之，得島去表數。
The setup: you want the height of an island out at sea and your distance from it, and you cannot go there. Plant two poles of equal height, 3 zhang each, in a line pointing at the island, 1,000 bu apart. Walk backwards from the front pole, keeping your eye at ground level, until the tip of the pole lines up exactly with the island's peak. That happens at 123 bu. Do the same at the rear pole: it happens at 127 bu. The text's answers: island height 4 li 55 bu, distance from the front pole 102 li 150 bu.
You need the Han units, and Hong Tian Chi 洪天賜 supplies them: 1 zhang = 10 chi, 1 bu = 6 chi, and 1 li = 1,800 chi. That makes 1 li = 300 bu (S248). Everything below is turned into bu before the arithmetic starts. Multiply zhang by 10 to get chi, then divide chi by 6 to get bu. Now run the rule exactly as stated. Pole height times pole separation, divided by the difference of the two back-offs, plus the pole height, gives the island's height. Front back-off times pole separation, divided by the same difference, gives the distance.
Convert back into li and bu:
Both stated answers reproduce exactly, with no rounding at any stage (S247)(S248).
What a Greek would have done instead
This is the moment to stop and stare.
Hand this problem to an astronomer trained in Alexandria and here is what happens. He measures the angle each sightline makes with the horizontal. He goes to a table of chords, the kind Hipparchus compiled and Ptolemy perfected, and looks up the chord of twice each angle. He combines the two results to get the height and the distance. The table is the machine. Somebody else built it once, at enormous cost in computation, and it gets reused for the island, the mountain, the eclipse, and the position of Mars. A table of chords does not know or care what problem you are solving.
Liu Hui does the identical job with two nested sets of similar right triangles and one subtraction (S248)(S250). He names no angle. He consults no table. The single number carrying all the information is the difference between the two back-offs, 127 minus 123, which is 4 bu. That difference does precisely the work a difference of two tabulated chords does in Alexandria. Its accuracy is limited only by how carefully the surveyor paced out the ground.
Which is better? For this problem, arguably Liu Hui. He needs no table, no angle measurement, and no instrument beyond two poles and a way to count paces. His answer is exact rather than interpolated. For a different problem next week, the Greek, because he already has the table. That is the trade this chapter is about: a purpose-built exact method against a general-purpose reusable one. China took the first side of it for eighteen centuries.
One much-quoted claim deserves a flag rather than a repetition. Reference works report Frank Swetz's judgment that China's surveying accomplishments "exceeded those realized in the West by about one thousand years" (S250). Swetz's book on the Sea Island Manual is behind controlled digital lending and was not read for this chapter (S275). Record his judgment as an opinion attributed to a named scholar by a reference work, not as an established finding.
Liu Hui's polygons
Liu Hui also gave China its best value of so far, using a method that is pure numerical analysis. Inscribe a regular hexagon in a circle. Its perimeter, divided by the diameter, underestimates . Now double the number of sides, again and again, computing each new side length from the old one by repeated square roots. The perimeter creeps upward toward the circumference.
In modern notation, the perimeter of an inscribed regular -gon in a circle of radius is . So the ratio of perimeter to diameter is , which climbs to from below as grows. Liu Hui had no sine, of course. He had the gougu rule and a great deal of patience. He got exactly the same numbers.
| Number of sides | Perimeter divided by diameter | Value attributed to Liu Hui |
|---|---|---|
| 96 | 3.141031951 | 3.14, that is |
| 192 | 3.141452472 | intermediate step in the doubling |
| 3,072 | 3.141592106 | 3.1416, that is |
For comparison, to nine places. MacTutor reports Liu Hui's 96-gon value as 3.141031950 and the 3072-gon value as 3.141592104. The figures computed independently for this chapter are 3.141031951 and 3.141592106, agreeing to the last digit or one unit in it (S251). Truncate the 96-gon value and you get 3.14, which is , the fraction usually attributed to Liu Hui. Round the 3072-gon value and you get 3.1416, which is , the other fraction usually attributed to him. Both fractions are consistent with the algorithm. That is a satisfying check that the historical attributions are at least numerically coherent.
One caution, stated because dropping it would be dishonest. Reference sources credit the 3072-gon result to Liu Hui himself. But no scholarly treatment was found on one question. Is that passage in the received commentary Liu Hui's own writing, or a later interpolation by an editor extending his method (S251)? The arithmetic being consistent with the algorithm is not the same thing as the passage being authentic.
Zu Chongzhi's bracket
Two centuries after Liu Hui, Zu Chongzhi (DZOO chong-JIRR, 429 to 501, the death year disputed) took the polygon method as far as anyone would take it for a thousand years. MacTutor gives 501, other widely printed sources give 500, and no primary basis was found to decide. He was born at Jiankang, modern Nanjing. He worked with his son Zu Geng (dzoo GUNG, active late fifth to early sixth century). Zu Geng collaborated on the work and is usually left out of the story (S252).
Zu proved
and he gave the fraction , which the Sui shu (History of the Sui Dynasty) records as "the precise value of the ratio of the circumference of a circle to its diameter is as 355 to 113" (S252). Check the bracket against a modern value, : it holds, and it is tight, seven correct decimal places. Check the fraction:
So is correct to six decimal places, from a fraction short enough to write on your thumb (S252).
How did he get there? MacTutor's reconstruction is that Zu "must have used an inscribed regular 24,576-gon and undertaken the extremely lengthy calculations, involving hundreds of square roots, all to 9 decimal place accuracy" (S252). Note the wording: "must have used" is an inference from the precision he reached, not a report of a surviving procedure. His book, the Zhui shu, is lost; the results come to us through the Sui shu. The reconstruction was checked here for numerical sufficiency:
Both lie inside Zu's stated bracket, so a 24,576-gon is enough to produce the bounds he reports. That makes the reconstruction possible, not proven. MacTutor itself adds that "his exact derivation of 355/113 remains unknown" (S252).
Two comparisons for students. First, is a best rational approximation to among all fractions with denominators below roughly 16,600, which is why it is famous and why it keeps being rediscovered. Second, nothing in Europe matched Zu's precision until the late sixteenth century, more than a thousand years later.
But now notice what Zu computed. One constant. Not a table of a function of an angle. Chinese numerical analysis in the fifth century was the best in the world, and it was pointed at numbers rather than at functions.
The eighth century: Gautama Siddha and the transmission of 718
For one generation, an actual trigonometric idea entered the Chinese court from outside, and this is the single most important transmission moment in the chapter. It is also the one where the secondary literature is most tangled, so it gets reported carefully.
Gautama Siddha (choo-TAHN shee-DAH; Chinese name Qutan Xida 瞿曇悉達, dates unknown, active 718 to 729) was an astronomer at the Tang court, born in Chang'an into a family of Indian descent. The Indian origin of the family is not a legend: a tomb stele belonging to the Gautama family was dug up at Xi'an, the modern city on the site of Chang'an, in 1977 (S269). He is the sort of figure Western textbooks leave out entirely, and he should not be, because he is the pipe through which Indian mathematical astronomy reached the Chinese state.
In 718, at the request of the throne, he translated a Sanskrit astronomical manual, the Navagraha-karana, into Chinese as the Jiuzhi li 九執曆 (Nine Planets Calendar). Jeffrey Kotyk describes it as "a manual of mathematical astronomy" that "provides formulas for accurately calculating a number of celestial phenomena" (S253).
Three things came with it that Chinese astronomy did not have.
A spherical earth. Kotyk, writing in 2018, states that the Jiuzhi li "assumes a spherical Earth" (S255). That single assumption is the seed of everything spherical trigonometry is for.
A tabulated latitude. The same text "provides a tabulated latitude value of 35 degrees," which Kotyk takes from Yabuuchi Kiyosi's translation and commentary (S255).
And the concept of terrestrial latitude itself, together with a Chinese word for it. Kotyk: "the aforementioned Navagraha-karana, translated in 718, addresses the concept of terrestrial latitude. This concept was first rendered into Chinese as suifang yan fa 隨方眼法 ('method according to the location of the observer'). This appears to be a direct translation of the Sanskrit sva-desa-aksa, as pointed out by Yabuuchi" (S253). That is a calque: a phrase built piece by piece out of native material to carry a foreign idea, the same manoeuvre Xu Guangqi would pull nine centuries later with jihe for geometry. Watch for calques whenever mathematics crosses a language border. They are the fingerprints of transmission.
The Jiuzhi li is also the route by which an Indian sine table reached China. And here the chapter has to stop and admit something. The parameters of that table (its radius, its argument step, how many entries it had) could not be verified from any source that could be opened for this work. The figures repeated in general accounts, namely 24 entries, a radius of 3,438, and a step of 3 degrees 45 minutes, which would make it Aryabhata's table transposed, appear in nothing that was read here. Kotyk's 2018 paper and his 2022 paper both discuss the Jiuzhi li at length without giving the table's parameters; the Brill chapter on the text and the full text of Sen's paper on Gautama Siddha both returned HTTP 403; Yabuuchi's 1989 translation, which is the underlying authority everybody cites, could not be obtained (S253)(S255). So: an Indian sine table entered China in 718, and this chapter cannot tell you what was in it.
Yixing
Yixing (ee SHING, 673 to 727 (disputed: Jeffrey Kotyk argues for 673, following Jinhua Chen's 2000 genealogical study; Chia-Yun Wu, writing in 2023, still prints 683; the death year 727 is agreed)) was a Buddhist monk, an important figure in the establishment of Esoteric Buddhism in East Asia, and the best mathematical astronomer of the Tang dynasty (S253)(S254). The birth-year disagreement is not a trivium: it is a live disagreement between a 2022 paper arguing from a genealogical study and a 2023 paper that did not adopt the revision, and the honest thing is to print both and say who holds which.
His calendar system, the Dayan li 大衍曆 (Great Expansion System), was completed in 727, the year he died, and was not yet in force at his death. It was in official use from 729 to 761 (S253).
The shadow table, and what to call it
Inside the Dayan li sits what historians of Chinese astronomy describe as the first Chinese tangent table. Be precise about what it is, because the loose version of this claim is one of the most commonly repeated errors about Chinese mathematics.
It is a table of noon gnomon shadow lengths for the standard 8-chi gnomon, listed against the sun's zenith distance in du 度, the Chinese degree unit in which a full circle is 365.25 du rather than 360, so that 1 du is about 0.986 modern degrees. It sits under the procedural heading Qiu jiufu suozai mei qi churi zhonggui changshu 求九服所在每氣初日中晷常數, "on the search for the constant values of the gnomon shadows measured on the first day of each qi in all locations within the Nine Domains," recorded in the Jiu Tangshu (Old Tang History) (S254).
Wu states the scholarly consensus: "Historians of Chinese astronomy largely agree that, in order to obtain the 'ideal' values recorded in the Dayan Calendar, Yixing composed a new tangent table to recalculate the observational data" (S254). Her footnote makes the identification exact, citing Qu Anjing's 1998 study: "For an eight-chi-long gnomon, the ratio of the length of the sun's shadow, l, to the length of the gnomon, h, is exactly the value of tangent of the zenith angle, x: l/h = tan(x). Therefore, if we know the zenith angle x, we can calculate its shadow length by checking the tangent table" (S254). Kotyk says the same in one line: "This set of data was used by Yixing to produce a tangent table" (S253).
So here is the mathematics, and it is a single line. For a gnomon of height casting a shadow of length , with the sun at zenith distance (and therefore at altitude ):
Which means the table is a tangent table when you index it by zenith distance, and a cotangent table when you index it by the sun's altitude above the horizon. Both descriptions are correct. They differ only in which angle you decide is the independent variable (S254).
What the table is not is a general-purpose ratio table detached from the gnomon. Its argument is an astronomical quantity, the sun's zenith distance at noon on a particular day at a particular place, not an abstract angle. It lives inside a procedure for predicting noon shadows at any of the Nine Domains, not in a chapter on triangles. You cannot pick it up and use it to survey an island. That is the honest way to describe it, and it is why the flat phrase "Yixing produced the first Chinese tangent table" is half right and misleading if you stop there.
The full study of this table is Christopher Cullen's 1982 article "An eighth century Chinese table of tangents," and it is paywalled. Brill returned HTTP 403, the CORE mirror returned HTTP 403, and the academia.edu copy demanded a login (S256). Everything in this chapter about that table's structure therefore comes at second hand, through Kotyk 2022 and Wu 2023, and this chapter cannot tell you its entry count, its argument step, or its tabulated values. There is even a page-range problem in the citation trail that could not be resolved: Wu's footnote cites the article at pages 19 to 25, her other footnotes cite pages 11 to 15 and page 26 of the same article, Kotyk cites pages 24 and 30 to 32, and Brill's own listing has the article beginning at page 1. Those cannot all be right (S256).
Where the idea came from
Cullen's argument, as Kotyk reports it, is that the tangent table has no Chinese ancestry: "the use of a tangent table was an innovation that suddenly appears in the history of Chinese astronomy, as pointed out by Cullen, and its adoption by Yixing does point to a foreign source. Cullen therefore suggests that Yixing had likely learned the rules for relating gnomon shadows and solar zenith distances from one of his Indian counterparts in the capital. In other words, although his 'use of the tangent was original, he ultimately depended on Indian sources for his introduction to trigonometry'" (S253, quoting S256). Techniques with no local ancestry that appear abruptly and fully formed are usually imports; that is a general principle of the history of science and it is worth teaching as one.
Kotyk immediately puts a limit on it, and the limit matters as much as the claim. Yixing's calendar, he writes, "is primarily rooted in Chinese models and concepts (definition of the ecliptic, sidereal lunar stations based on the ancient Chinese system, etc.)," and Yixing "was evidently committed to receiving and building upon a legacy of Chinese astronomy from antiquity, rather than expressly introducing significant Indian models into the Chinese system... Yixing, however, never attempted such a radical reform" (S253). Kotyk also records that empirical tests of the foreign model "did not yield any positive outcome" (S253).
Read that carefully and you get a picture that is much more interesting than "China received trigonometry from India." A monk with direct access to Indian astronomy in his own capital city took one technique, used it in an original way, tested the rest, and declined to import the system that produced it. That is not ignorance. That is a decision.
The survey of 724 to 725, and Nan Gongyue
In 724 and 725 Yixing ran what may be the largest scientific field program conducted anywhere before the modern era. Field parties fanned out across the Tang empire measuring two things at each station: the length of the noon gnomon shadow, and the altitude of the north celestial pole. One party reached Annan, in what is now northern Vietnam. Li and Sun report measurements at 13 different locations (S244).
The man who led those field parties was Nan Gongyue (nahn-gong YWEH, dates unknown), and he is almost never named outside the specialist literature (S253). Yixing designed the program and reduced the data. Nan Gongyue and his teams did the work: carrying a standard instrument across an empire, setting it up level and true at each site, waiting for local noon, measuring a shadow to the fraction of a cun, recording it, and moving on. If you want an eighth-century example of a large scientific collaboration, this is it, and the person whose name should be attached to the fieldwork has no dates and almost no biography.
The geographical extent of the survey is contested, and the disagreement is a nice small lesson in reading experts. Yukio Ohashi puts the latitude span at roughly 18 to 51 degrees north. Cullen puts it at roughly 29 to 52 degrees north, near meridian 114 degrees east. Kotyk, reporting both, adds a remark that a student should see in print at least once: "the two scholars give different numbers, but I am unclear on how they arrived at them" (S253). A working scholar saying openly that he cannot reconstruct how two other scholars got their figures is not a failure of scholarship. It is scholarship.
There is a second disagreement in the same area. Ohashi says Yixing traveled to the survey sites himself. Kotyk's finding, from the records, is that they do not support this (S253). On the evidence available here, Yixing was the analyst, not the traveller, and Nan Gongyue was the traveller.
The rule that died
Yixing reduced the data, and the data killed a law that had stood since the Zhoubi.
Cun-qian-li was false. Wu states the result: "Yixing's survey thereby refuted the old theory according to which 'one cun of decrement or increment in the shadow corresponds to one thousand li.' Based on the records of the gnomonic observation in Henan, Yixing argued that the distance of these two places was not correlated with the shadow-length and that the parameter associated with distance was the polar altitude. The brand-new value is as follows: the distance per du (Chinese degree) of polar altitude equals 351 li plus 80 bu" (S254). The Xin Tangshu (New Tang History) records the figure as 大率三百五十一里八十步，而極差一度, and Wu notes that Yixing was using the "short li" in which 1 li = 300 bu (S254). Li and Sun report the same overturning independently: "Monk Yixing... doubted the validity of this law. From AD 724 to 725, he carried out a large project on astronomical and geographical surveys that included measuring gnomon shadows at 13 different locations. After analyzing his results, he finally denounced the law of 'cun qian li'" (S244). And Cullen, via Kotyk, reads the survey as "a field test, evidently successful, of I-hsing's method for predicting seasonal shadow lengths at any location" (S253).
Sit with what happened there. A canonical text, eight centuries old, states a numerical law. A state-funded survey measures it in the field at thirteen stations across thousands of kilometers. The law fails. The astronomer in charge says so publicly and replaces it with a new constant tied to a different variable, the altitude of the pole, which is to say latitude. That is the experimental method operating at empire scale in the 720s, and it is a better story than most of what gets taught about the history of science.
Does the new number survive a check? Here is the arithmetic, done here rather than taken from a source. With 1 li = 300 bu, 351 li 80 bu is li per du. A du is modern degrees, so the figure converts to about 356.4 li per modern degree. One modern degree of latitude is about 111 kilometers. That makes Yixing's short li roughly 311 meters, which is in the right range for a Tang short li. The figure is not garbled in transmission.
733: the plagiarism suit
Then, six years after Yixing died, the case took a turn nobody expects.
Gautama Zhuan (choo-TAHN JWAHN, 712 to 776), the son of Gautama Siddha, formally accused the late Yixing of having plagiarized the Navagraha-karana, the very text his father had translated in 718. The court investigated and found the accusation false (S253). Kotyk's comment on that verdict is one line long and worth the whole section: "This conclusion might have been premature" (S253).
Think about what a plagiarism suit implies. Gautama Zhuan was a court astronomer accusing the most celebrated astronomer of the previous generation of stealing Indian material. For that accusation to be worth filing, everyone in the room has to agree that the Indian material exists, that it is valuable, and that using it without acknowledgment is a wrong. Whether or not Gautama Zhuan was right on the merits, and Kotyk suggests the court may have cleared Yixing too quickly, the suit is the best single piece of evidence that eighth-century Chang'an knew perfectly well that Indian mathematical astronomy was in the room and that it mattered. Priority disputes are a symptom of contact. They are also, then as now, about jobs: the Gautama family held positions at the astronomical bureau, and the Dayan li was the competition.
Shen Kuo and the arc
Shen Kuo (shun KWAW, 1031 to 1095), sometimes written Shen Gua, is the kind of figure who makes a historian's job hard, because he did everything. He was a hydraulic engineer, a diplomat, a finance official and a military commander. He was also the geologist who worked out that a fossil bamboo forest implied the climate had changed. And he wrote the Mengxi bitan 夢溪筆談 (Dream Pool Essays), several hundred notes on everything he had ever thought about, composed at his estate called Dream Brook. The earliest surviving edition of the Mengxi bitan dates from 1305, more than two centuries after his death, which matters for what follows (S257).
In chapter 18 he takes up a problem that had beaten the tradition. Given a circle, cut off a segment with a straight chord. You can find the chord easily by gougu. How long is the arc?
He introduces his method with a name and a piece of wordplay. The method is huiyuan 會圓: hui 會 means "to assemble, to bring together," and yuan 圓 means "circle," so "assembling the circle." He sets it against the older approach, which he calls by the verb zhe 折, "to break." Wagner's translation of Shen Kuo's framing:
"Of methods of measuring mu... the square, the round, the crooked, and the straight have been perfected. There remains the technique of 'assembling a circle' [hui yuan 會圓]... Among the ancient methods there is only the method of 'splitting the circle in the middle' to break it, in which the error can be as much as threefold. I have devised a different technique for breaking and assembling." (S257)
The pairing is the point: a circular field can be broken into pieces, so it ought to be possible to assemble the pieces back into a circle. Breaking and assembling are inverse operations. Shen Kuo claims to have supplied the missing half.
His rule needs two ingredients. The first is the chord , which he calls the "direct diameter," found first by gougu. The second is the sagitta , the height of the arc above its chord. Sagitta is Latin for "arrow," because chord plus arc looks like a bow with an arrow laid across it. His formula, as the histories usually give it:
where is the diameter and the arc.
His worked example takes bu and bu, so the circle has radius 5 and the chord sits 3 bu from the center:
The text gives the answer as 8 bu 4 chi. The Song commentator is converting at 5 chi to the bu, so 8 bu 4 chi is bu, matching exactly (S257). (Note the unit drift. The Sea Island Manual used 6 chi to the bu in the Han; here it is 5 chi to the bu in the Song. Standards move over a thousand years. Use the wrong one and you get an answer that is wrong by twenty percent and looks perfectly reasonable.)
How good is the rule? The true arc for , is
so Shen Kuo's 8.8 is low by 5.10 percent at this example. Scanning the whole range on a fine grid, the worst relative error of the rule is 5.419 percent, attained at . Wagner states 5.42 percent, and the independent computation reproduces it (S257).
Wagner's argument that the text is corrupt
Donald B. Wagner makes a claim about this passage that historians of Chinese mathematics generally do not make, and it belongs here because it is a good example of how a textual problem and a mathematical problem can be the same problem.
The received text of the Mengxi bitan, in the 1305 edition, reads:
以所割之數自乘，退一位，倍之，又以圓徑除所得，加入直徑，為割田之弧。
Wagner translates: "Multiply the 'value of the cut' [h] by itself, shift one place [tui yi wei 退一位, i.e., divide by 10], and double it. Then divide [chu] the result by the diameter [d] and add the 'direct diameter' [b, the chord] to make the arc [s] of the 'cut field'" (S257).
Read that as an instruction and you do not get the standard formula. You get an extra "divide by 10" that has no business being there. Wagner: "Histories of Chinese mathematics generally state that Shen Gua... gave this approximation for the length of an arc of a circle," but "this is not precisely the formula given in Shen Gua's text. There is a phrase in the text which must be removed to obtain (1)" (S257).
Worse, the small-character commentary that works the example through is, in Wagner's reading, mathematical nonsense: "The commentator seems to believe that, in a division, if the divisor is greater than the dividend, the quotient equals the dividend." That mistake happens, in this particular example, to cancel out, so the commentator "fortuitously gives the same result as using (1) would give." Wagner's judgment on authorship is blunt: "It is unlikely that the astronomer and polymath Shen Gua wrote the strange comment translated here" (S257).
His conjecture is that the original rule had an extra linear term. A later editor who did not understand that term mangled the text into its received form. The conjectured original:
Is the conjectured version better? Substantially. Its maximum relative error over the same range is 1.859 percent, against 5.419 percent for the standard formula. Wagner states 1.86 percent, and the independent computation confirms it (S257). Here is how the conjectured rule behaves across the range, on a circle of diameter 10:
| Sagitta | Relative error of |
|---|---|
| 1 bu | -0.54 percent |
| 2 bu | -0.79 percent |
| 3 bu | -0.24 percent |
| 4 bu | +0.76 percent |
| 5 bu (semicircle) | +1.86 percent |
Wagner is honest about the weakness of his own case, and quoting him on it is the most useful thing this section can do for a student: "There is no historical evidence that a formula like (3) was ever used in ancient China (or anywhere else), and this is a serious argument against my hypothesis" (S257). A scholar naming the strongest objection to his own conjecture, in the paper making the conjecture, is what good practice looks like.
One more thing Wagner records, which keeps the rule from floating free of history. The standard formula "is equivalent to an approximation for the area of a circle segment in the Jiu zhang suanshu," so the underlying relation is older than Shen Kuo, and it "was explicitly used by Zhu Shijie 朱世傑 in a book published in 1303" (S257). Whatever went wrong with Shen Kuo's wording, the rule itself was in working use in Chinese mathematics for centuries.
Why this belongs in a history of trigonometry
Translate Shen Kuo's rule into the vocabulary of India or the Islamic world and something jumps out. The sagitta is exactly the versed sine: for an arc, , which is the height of the arc above its chord. Half the chord is exactly the sine in its original sense, the half-chord. So huiyuan is a relation among a sine, a versed sine, and an arc.
That is a trigonometric identity. It is written entirely in lengths inside a circle, with no angle named and no function defined, and it is approximate rather than exact. But it does the job a trigonometric relation does. Two centuries later it will be pressed into service as a stand-in for an inverse trigonometric function, inside the best calendar China ever produced.
Guo Shoujing, Wang Xun, Li Qian, and the Shoushi li
Guo Shoujing (gwaw SHOH-jing, 1231 to 1316) was born in Xingtai, in Shunde prefecture, modern Hebei, and died in Dadu, the Yuan capital that is now Beijing. He worked for Kublai Khan. He is the greatest observational astronomer in Chinese history (S259).
He also did not work alone. The collaborators get dropped, so name them here.
Wang Xun (wahng SHOON, died shortly after 1281) directed the Yuan calendar bureau. The division of labor is explicit in the sources: Wang took charge of the calculations while Guo took charge of the observations and the instruments (S259). Wang died within a year or so of the calendar's promulgation. That is a large part of the reason Guo's is the name that survived. The man who did the mathematics was not there afterwards to explain it, publicise it, or be promoted for it.
Li Qian (lee CHYEN, 1223 to 1302) wrote the Shoushi liyi 授時曆議 in 1283. It is the theoretical exposition of the calendar, the document that explains why the procedures are what they are (S259). If you want to know what the Yuan astronomers thought they were doing rather than merely what they computed, Li Qian is your source. General histories almost never name him.
The Shoushi li 授時曆 (Season-Granting System) was established in 1280 and officially promulgated from 1281 (S258)(S259). Its tropical year is 365.2425 days, which is exactly the mean year of the Gregorian calendar that Europe adopted in 1582, three centuries later. Guo did not originate that value and the sources say so plainly: it "had already been used in the Tongtian calendar (1198) of Yang Zhongfu and was confirmed by Guo Shoujing" (S259). They determined the solstice times using a method devised by Zu Chongzhi eight centuries earlier (S259). Chinese astronomy at its height is cumulative, and its practitioners cite their predecessors.
The Gaocheng observatory
To measure shadows well enough to pin a solstice, Guo needed a bigger instrument. He designed the observatory at Gaocheng, in Dengfeng, Henan, in 1279. Its gnomon, the biao 表, stands 40 chi high, five times the traditional 8 chi, and its shadow scale, the gui 圭, is 128 chi long (S244)(S259). The structure survives. It is called the Guanxingtai, the Astral Observatory (S259).
Why taller? Your eyesight and the sharpness of the shadow edge fix the absolute error in reading where a shadow ends. So a shadow five times longer carries five times less relative error. It is the same reason astronomers build bigger instruments today.
The metric conversion of "40 chi" is a mess. The mess is a teaching opportunity. Li and Sun write: "The relic shows a huge gui biao, with the biao being 40 chi high (9.7468 m) and the gui being 128 chi long (31.196 m)" (S244). The Biographical Encyclopedia of Astronomers says Guo "improved it and made it five times taller than previous traditional gnomons, building it to 40 chi (12.28 m) high" (S259). These cannot both be right.
Work out what each implies about the length of a chi:
A Yuan chi was in the neighborhood of 24 to 25 cm, not 30.7 cm. So the smaller figure is far more plausible, and the larger one looks like a conversion done with a later chi standard. That argument is made here, not taken from a source, and the underlying conflict is real and unresolved in the literature (S244)(S259). Note also that Li and Sun's own numbers are internally consistent. They give the traditional 8-chi gnomon as about 1.96 m, and 40 chi is exactly five times 8 chi, which matches the encyclopedia's "five times taller" (S244)(S259).
Guo also solved a physical problem that no amount of mathematics would fix. The sun is not a point of light, so a shadow does not end at a sharp line. It fades through a penumbra, and you cannot tell where to put your ruler. His answer was an optical instrument, not a formula: "Guo Shoujing overcame this difficulty by using jingfu, which is a kind of pinhole camera. The image of the Sun is projected through the pinhole, which is adjusted so that the shadow of the horizontal bar in the window at the top of the gnomon tower passes exactly through the center of the image of the Sun" (S259). He replaced a fuzzy edge with a crossing of two things you can center by eye.
Spherical astronomy without spherical trigonometry
Now the crux of the Yuan story, and the second-best illustration in this chapter of the chapter's thesis.
The Shoushi li has to convert between two coordinate systems. Ecliptic coordinates are measured along the sun's yearly path through the stars. Equatorial coordinates are measured along the celestial equator, the projection of the earth's equator onto the sky. Those two great circles are tilted with respect to each other by about 23.5 degrees, so converting between them means solving a triangle drawn on a sphere. In Greek, Indian, or Islamic astronomy, that is a spherical trigonometry problem and you solve it with sines.
Wagner's finding is that Guo's procedure is "essentially the same" as the modern spherical trigonometry formulas. It derives from the same Fundamental Formula of spherical trigonometry and uses geometric relationships on the celestial sphere, "though Guo Shoujing did not explicitly employ trigonometric functions" (S258). The Biographical Encyclopedia of Astronomers agrees from a different direction: "The Shoushi calendar also used some new mathematical features, such as third-order interpolation and a mathematical method to transform spherical coordinates. For the latter, the Shoushi calendar employed the method devised by Shen Gua (1031 to 1095)" (S259). Two independent sources, one conclusion.
So what did the work of the sine? Two substitutes, both of them lengths-only (S258):
First, Shen Kuo's arc rule, run as an input-to-output machine. Given a versed sine (the sagitta) and a sine (the half-chord), it returns an arc, which is to say an angle. Wagner's description is exact: it "serves as an inverse trigonometric function substitute."
Second, a sagitta rule running the other way. Given an arc and its chord, find the sagitta, which "approximates the versine of an angle." Chinese algebra solves this one numerically with its own version of Horner's method, the root-finding scheme European algebra would name after William Horner in 1819.
Add third-order interpolation, which the Shoushi li introduced to handle the sun's uneven motion through the year. Now you have a complete working system for predicting positions in the sky (S259). Angles throughout are in du, with a full circle equal to 365.25 du (S258).
There is one strange and lovely detail. The Shoushi li uses in exactly one place, to compute the diameter of the celestial sphere as 1,095 du, "which is the only application of pi in the entire computation." Wagner reports the resulting error with : less than 0.22 du on one branch of the calculation and less than 0.25 du on the other, across the full range of astronomical interest. He adds that this crude value "compensates for other calculation errors, producing lower maximum errors than the correct pi value" (S258). Sanity-check the number: , so a diameter of 1,095 du is exactly what gives. China had held to seven decimal places since the fifth century, and its calendar bureau used 3 anyway, deliberately, because it worked better inside their particular chain of approximations. That is engineering, and students should meet it.
(Two honesty notes. Wagner's page could only be read through a fetch summary, because the site's firewall blocked a direct download, so its finer numerical details are held here at reduced confidence (S258). And Guo's algorithm was not independently reimplemented for this chapter.)
What Guo could and could not do
State it plainly, both halves.
He could determine solstice times to a precision nobody in the world was matching. He could transform between ecliptic and equatorial coordinates. He could interpolate the sun's uneven motion with a third-order scheme. He could invert his own approximations numerically with Horner's method. And he had instruments good enough that the errors of his approximation formulas, not the errors of his observations, were the limiting factor. That is a serious and coherent scientific program. By the standards of 1280 it was the best in the world.
He could not solve a general spherical triangle. He had no sine, no chord table, and no systematic method for a configuration he had not already built a recipe for. If you handed him a spherical triangle with three parts given and asked for a fourth, he had nothing general to reach for. That is the difference a function makes: not that it computes anything you could not compute otherwise, but that it lets you attack a shape you have never seen before.
One open question to record rather than settle. The Biographical Encyclopedia of Astronomers notes: "Although the Shoushi calendar was basically made in traditional Chinese style, the possibility of Indian and Islamic influence was recently pointed out by Qu Anjing... the Shoushi calendar used a geometrical model, which is similar to Indian and Islamic methods that had already been introduced into China. This topic deserves further research" (S259). Under Mongol rule, Chinese and Islamic astronomers worked in the same capital under the same emperor. Whether Guo's geometry owes something to that contact is an open question, not a settled one in either direction.
The Jesuits: the second transmission, and this one stuck
1607 and the invention of a word
Matteo Ricci (mah-TAY-oh REE-chee, 1552 to 1610; Chinese name Li Madou 利瑪竇, lee MAH-doh) was an Italian Jesuit who arrived at Macau in 1582 and spent the rest of his life working his way toward Beijing. His strategy was to enter Chinese intellectual life on its own terms: he learned classical Chinese, dressed as a scholar, and offered European mathematics, cartography, and clockwork as credentials.
Xu Guangqi (shoo GWAHNG-chee, 1562 to 1633) was a senior Chinese scholar-official from Shanghai. He passed the highest examinations, rose to high office, and was baptised as a Christian with the name Paul in 1603. He is the more consequential figure of the two for Chinese mathematics, because he chose what to translate and how to say it in Chinese.
Together they translated the first six books of Euclid's Elements as the Jihe yuanben 幾何原本 (Elements of Geometry), published in 1607 (disputed) (S260)(S261). The scholarly literature says 1607 consistently. The Library of Congress catalog record for its own copy of the work gives 1606. That may be a preface date set against an imprint date, but this chapter did not verify it. The book was reprinted with minor revisions in 1611. Wenshan Wang, writing in the Journal of Chinese History in 2022, calls it the moment that "not only brought about the first climax of Jesuit activities in the Middle Kingdom but also inaugurated the first great influx of European science" (S260).
The linguistic story is as good as the mathematical one. There was no Chinese word for geometry, because there was no Chinese discipline shaped like Greek deductive geometry. Xu took jihe 幾何, an ordinary interrogative phrase out of arithmetic problems. It classically means "how much" or "how many," and he repurposed it as the name of the subject. He took yuanben 原本, "origin-root," for "elements" or "foundations." Wang, following Ogawa, calls this "the greatest linguistic merit of this translated book" (S260). Both coinages are still the standard modern Chinese words. Every Chinese student who studies geometry today uses a term a Ming official minted in 1607 to carry an idea that had no local name.
The Jihe yuanben did not bring trigonometry. Euclid's Elements, books 1 to 6, is plane geometry: triangles, proportion, areas. What it brought was arguably stranger to the Chinese tradition than any table would have been: the axiomatic method. You state your assumptions, prove propositions from them in order, and never appeal to a worked example.
Why 1607 and not 1280
Here is the fact that ruins the simple story of transmission. This was not the first arrival of Western mathematics in China. Wang: "It was also during the Mongol-ruled dynasty that Muslim astronomers brought Arabic numerals to China along with other new mathematical knowledge from the outside world. This great influx of foreign knowledge included trigonometry (an essential part of mathematical astronomy) and probably Euclidean geometry; the latter, however, failed to take root until the 1607 publication of Jihe yuanben" (S260).
Trigonometry arrived in China under the Yuan, in the thirteenth century, in the hands of Islamic astronomers working in the same city as Guo Shoujing. And it did not take. Knowledge does not transfer because it arrives. It transfers when somebody with standing decides it is worth having, translates it into the local idiom, and attaches it to a problem the local institutions already care about. Xu Guangqi did all three. The Yuan-era transmission had none of them. It left almost no mark on Chinese practice.
Li Zhizao, and the rest of the Ricci program
Li Zhizao (lee JRR-dzow, 1565 to 1630) is the third man in this story and gets a fraction of the attention. He was a scholar-official from Hangzhou, a convert, and a cartographer. He collaborated with Ricci on two works of 1614 (S260).
The Tongwen suanzhi 同文算指 (Rules of Arithmetic Common to Cultures) is the first work to introduce European written calculation into China. That sounds minor and is not. Chinese arithmetic happened physically, on counting rods laid out on a board or on the beads of an abacus. The intermediate steps lived in the hardware and vanished as you went. European arithmetic happens on paper. Every partial product goes down in writing. That leaves a record you can check. Changing the medium of calculation changes what calculation is: you can audit a written long division, and you can teach it from a book to someone who has never seen your board.
The Yuanrong jiaoyi 圜容較義, also 1614, is Li's other work with Ricci, on comparison of figures inscribed in circles (S260). A year earlier in the sequence, in 1608, Ricci and Xu Guangqi produced the Celiang fayi 測量法義 (Explanations on the Approaches and Principles of Measurement). That book is surveying, the applied edge of the same program (S260). Read as a set, these books are a deliberate campaign, moving from foundations (Euclid, 1607) to measurement (1608) to calculation (1614).
1629 to 1633: the trigonometry finally arrives
The trigonometry came with the calendar, as it always does.
The Ming astronomical bureau was failing to predict eclipses. A failed eclipse prediction is a political problem. Xu Guangqi, by then a senior official, orchestrated an astronomical reform during the last four years of his life, from 1629 to 1633, with Ricci's Jesuit successors (S260). The team:
Nicholas Longobardo (nih-koh-LOH lon-goh-BAR-doh, 1559 to 1654), Sicilian, Ricci's successor as head of the China mission.
Johann Terrenz Schreck (YOH-hahn TEH-rents SHREK, 1576 to 1630; Chinese name Deng Yuhan 鄧玉函), from Konstanz, a mathematician and physician who had been a member of the Accademia dei Lincei with Galileo and who died early in the project.
Johann Adam Schall von Bell (YOH-hahn AH-dahm SHAHL fon BEL, 1592 to 1666; Chinese name Tang Ruowang 湯若望), from Cologne, who would survive the fall of the Ming and end up running the Qing Astronomical Bureau.
Giacomo Rho (JAH-koh-moh ROH, 1593 to 1638; Chinese name Luo Yagu 羅雅谷), from Milan.
The product was the Chongzhen lishu 崇禎曆書 (Astronomical Treatise of the Chongzhen Reign), which Wang describes as "a great encyclopedia of European astronomy." The Qing dynasty that replaced the Ming adopted its new calendar (S260). This compilation is the vehicle by which European trigonometry and logarithms entered Chinese astronomical practice for good.
An admission is required here. General accounts usually name the individual trigonometric treatises inside the Chongzhen lishu as the Dace 大測 (attributed to Terrenz Schreck, 1631), the Celiang quanyi 測量全義 (attributed to Rho), and the Geyuan baxian biao 割圓八線表 (a table of the eight trigonometric lines). None of those titles, authors, or dates could be confirmed from any source that could be opened for this chapter. The ChinaKnowledge.de page on the Chongzhen lishu failed a TLS handshake, and the Wikipedia article on the Chongzhen calendar is too thin to list them (S272). The dates 1631 to 1635 for the five submissions of the compilation are widely printed and plausible and are not verified here.
Wang's characterization of how Chinese readers took the whole package is the sharpest sentence in her article. Quote it in full, because it explains everything that follows. Chinese readers took up "the operational aspects of science (like calculations, trigonometry, logarithms) while downplaying its logical, theological, or demonstrative features (like geometry)" (S260). They wanted the machinery. They were much less interested in the axioms. They were least interested of all in the theology the Jesuits had attached to the package. That is a rational response by people with a calendar to fix.
Mei Wending and the doctrine that made it acceptable
Mei Wending (may wun-DING, 1633 to 1721) is the man who made European mathematics Chinese. Largely self-taught, from Xuancheng in Anhui, he became the leading mathematical astronomer of the early Qing, and his method was systematic: explain every European result through the native gougu tradition.
His output includes Jihe tongjie 幾何通解 (Complete Explanation of Geometry), Jihe bubian 幾何補編 (Complements of Geometry), and Gougu juyu 勾股舉隅 (Illustration of the Right-Angled Triangle). It also includes Pingsanjiao juyao 平三角舉要, which Wang renders as Essentials of Plane Geometry. The title literally means "essentials of plane triangles," that is, plane trigonometry (S260). Horng Wann-Sheng's assessment, as Wang reports it, is that these works show "how Western geometry could be explained in terms of traditional [Chinese] concepts concerning the right-angled triangle (gougu)," Horng says. In the Jihe tongjie in particular, Mei "reinterprets the latter's sixteen propositions through various native concepts and methods related to Gougu. In his eyes, Gougu constituted the foundation of both Chinese and Euclidean geometry" (S260).
In 1700 he published the Qiandu celiang 塹堵測量, in which he gave a trigonometric interpretation of the celestial coordinate transformation (S262). Look at what that means historically. It is the same problem Guo Shoujing solved in 1280 with Shen Kuo's arc rule and Horner's method, now redone with trigonometric functions. Four hundred and twenty years separate the two treatments of one problem. The difference between them is the presence of the sine.
Mei was nobody's disciple. MacTutor records that he pushed back against missionaries who dismissed Chinese mathematics. Western scholars, Mei observed, "did not mention simultaneous linear equations because the subject was then only in its infancy in the West" (S262). He knew the Chinese tradition had things Europe did not.
His position hardened into a doctrine. The doctrine is where the politics gets interesting. Xixue zhongyuan 西學中源 holds that Western learning originated in China. The Greeks and Europeans got their mathematics from ancient Chinese sources, China then lost it, and importing it now is not borrowing but repatriation. Wang argues this was not a decorative claim. It was the enabling ideology, the thing that made adopting European trigonometry politically acceptable by reframing it as recovering what China had lost (S260).
Whether or not you find the doctrine historically absurd, and it is, notice what it did. It let a proud, self-confident intellectual culture take a foreign technique at full strength without losing face. Ideas travel more easily when somebody supplies a story that makes accepting them an act of loyalty rather than submission.
Kangxi's academy and the Shuli jingyun
The Kangxi Emperor (kahng-SHEE, reigned 1661 to 1722) studied mathematics personally with Jesuit tutors. Then he did something that changed the balance of power. In 1713 he founded the Suanxue guan 算學館, the Academy of Mathematics, modeled on the French Academy of Sciences rather than on the Jesuit colleges, and staffed it with over one hundred Chinese scholars (S260). The point of the institution was to stop needing the Jesuits. State-funded science as an instrument of technological independence is not a twentieth-century invention.
In 1723, a few months after Kangxi's death, the mathematical part of the imperial compilation Yuzhi lüli yuanyuan closed with the publication of the Yuzhi Shuli jingyun 御製數理精藴 (Imperially Composed Essential Principles of Mathematics). It runs to almost 5,000 pages, and Wang calls it "the largest mathematical work ever printed in imperial China" (S260).
What is in it: the Jihe yuanben, plus European algebra, logarithmic tables, the calculation of infinite series, and the iterative method for higher-order equations. That body of material "changed the structure of the mathematics that the Jesuits had brought to China" (S260). What is at the front of it: the claim of xixue zhongyuan, tracing the origins of mathematical principles back to Chinese antiquity (S260). And what is not in it is the Jesuits. In Florence Hsia's assessment, quoted by Wang, the compilation "effectively effaced key Jesuit contributions to a mathematical field now thoroughly restructured according to the logic of an imperial ideology" (S260). It remained a codified and "compulsory text" for a century (S260).
Trigonometry got in. The credit did not.
A coda past the boundary: Minggantu
One name to carry the thread past this chapter's 1750 limit. Minggantu (ming-ahn-TOO, also written Ming Antu, 明安圖, eighteenth century; exact life dates could not be established from the sources read here) was a Qing court astronomer of Mongol banner background. He took the infinite series that had entered China with the Jesuit material and developed them for sine, cosine, and . His student Chen Jixin completed his Geyuan milu jiefa 割圓密率捷法 (Quick Method for Determining Segment Areas) in 1774, after Minggantu's death, and it was first printed in 1839 (S270). It is the endpoint of the arc that begins with Liu Hui cutting a circle into polygons: the same problem, the cutting of the circle, now attacked with power series. That the sources available here could not even establish his dates tells you something about how thin the Western literature on late imperial Chinese mathematics still is.
Japan: a power series without a sine
Wasan
Tokugawa Japan was largely closed to Europe for about 250 years, from the 1630s onward. Foreign books were restricted and foreign travel was banned. A country of thirty million people did mathematics with what it had: the Chinese classical corpus plus its own ingenuity. The result got a name: wasan 和算. Wa 和 is the character used for Japanese-style work in the arts and crafts, literally "harmony," and san 算 is "calculation," the same graph as Chinese suan (S266). So wasan is roughly "Japanese-style mathematics," a term coined in contrast to yosan, Western-style mathematics, once Western mathematics arrived.
Roger Cooke states the situation bluntly, and this is the single most important sentence in the section. Japanese mathematicians of this period "did not have what we know as trigonometry. (They did have a rudimentary trigonometry, but they solved most problems using just the Pythagorean theorem.)" (S266)
Remember that sentence for four pages. Japan is about to produce a power series that Newton had found only 46 years earlier and Euler would not publish for another 15 years.
Seki Takakazu
Seki Takakazu (SEH-kee tah-kah-KAH-zoo, born c. 1640 or in March 1642 (disputed), died 5 December 1708), also known as Seki Kowa, was born in Fujioka in Kozuke province, now Gunma prefecture, and died in Edo, now Tokyo. On the birth date, MacTutor prints "March 1642" with an explicit query mark on the year, while Cooke says "born around 1640" instead. The death date is agreed. He founded the mature wasan school. His students and their students dominated Japanese mathematics for a century (S263)(S266).
His results are startling for their dates. He worked on determinants in 1683, roughly a decade before Leibniz took them up in Europe. He found the Bernoulli numbers before Jacob Bernoulli published them (S263). In circle problems he used what is now called the Aitken delta-squared process, a convergence acceleration method. You take a sequence creeping slowly toward its limit, look at how the differences between successive terms are shrinking, and extrapolate to where the sequence is heading.
Naoki Osada identifies three purposes Seki put the technique to, one of which is "computing the length of an arc, in Katsuyo Sampo" (S265). Applied to inscribed-polygon perimeters, it gave him a value for that Osada reports as "(very little less than) 3.14159265359," and he notes that "Seki gave only 12 digits of pi for this purpose" (S265). Check it: , and 3.14159265359 is that rounded to twelve significant digits. Every digit Seki printed is correct.
Notice the method. Seki is not computing a bigger polygon. He is computing a smarter estimate from the polygons he already has. That is numerical analysis in the modern sense, in Japan, in the late seventeenth century.
Takebe Katahiro, Takebe Kataakira, and the Taisei sankei
Takebe Katahiro (tah-KEH-beh kah-tah-HEE-roh, 1664 to 24 August 1739), also written Takebe Kenko, was Seki's pupil and went further than his teacher (S264)(S267).
His elder brother, Takebe Kataakira (tah-KEH-beh kah-tah-AH-kee-rah, 1661 to 1716), is almost never mentioned anywhere. He should be. He was a co-compiler of the Taisei sankei, the twenty-volume comprehensive treatise the Takebe circle assembled. In 1715, the year before he died, he wrote the Takebe family biography that records how the compilation began in 1683 (S265). Nearly everything historians know about the internal organization of the Seki school, who did what and when, comes through documents like that one. The brother who wrote the family history is the reason we can date the brother who did the mathematics.
In the Taisei sankei, Katahiro did something that would not get a name in Europe for two more centuries. He took the squares of inscribed-polygon perimeters, which converge to the square of the circumference. Then he accelerated them. Osada reconstructs the procedure (S265). Write for the squared perimeter at the th doubling, for to . Takebe observed that the successive differences fall by a factor of about 4 each time. That is the signature of an error term proportional to the square of the step size. He applied
for to . That is Richardson extrapolation, published by Lewis Fry Richardson in 1911, applied by Takebe Katahiro before 1710.
His results, as Osada gives them: came out as "(little less than) 986.96044010893586188344909998747," from which he "determined the squared definite circumference (very little less than) 986.9604401089358618834491, and then he extracted the square root (little over than) 31.41592653589793238462643, which was his definite circumference" (S265).
Both were checked here at 60-digit precision:
Takebe's squared value is correct to every digit shown down to its last few (his tail reads "8747" against the true "87615", so the value is good to about 29 significant digits, consistent with Osada's statement that "Takebe's method can give exact 30 decimal digits"). His circumference value, 31.41592653589793238462643, is correct in all 25 digits printed (S265). That the reconstructed intermediate quantities land exactly on the true values is itself strong evidence that Osada's reconstruction of the procedure is faithful.
The Tetsujutsu sankei of 1722, and the arcsine squared
Then Takebe did something new. It is the strongest single result in this chapter.
In the Tetsujutsu sankei 綴術算経 (1722), he set out to find a relation between the square of half an arc, the sagitta of the arc, and the diameter of the circle (S266). His method was brute numerical experiment of a kind that would be recognizable to any modern computational scientist. Take and , so the arc is tiny and its square is dominated by its leading term. Compute the square of the arc to absurd precision. Subtract the first approximation, . Look at what is left. Subtract the next correction. Look at what is left again. Read the pattern off the decimals.
Cooke describes the outcome: "The corrections are obtained by multiplying successively by , , , , , ... Some sensitivity to the factorization of integers is necessary to see the recursive operation: multiplication by " (S266). That parenthesis about sensitivity to factorization is doing a lot of work. To see that and and belong to one family, you have to unfactor them back into the pattern. Takebe did it from decimal expansions, with no algebraic theory to guide him.
MacTutor states the same series in a different notation, as , where "1.8/3.15" is that site's dot-multiplication convention for , consistent with Cooke (S264).
Assemble the whole thing:
That identity was verified here at 60-decimal precision. With and , applying Cooke's ratio recursively gives
and the closed form gives
Agreement to better than one part in . The algebra confirms it independently: the standard series
has term ratio exactly , which with is precisely Cooke's multiplier, and the first five multipliers it predicts are , , , , : exactly Takebe's five, in order (S266).
So Takebe Katahiro found the power series for the square of the arcsine in 1722. He did it by staring at decimal expansions until the pattern gave way, in a country with no sine function, no calculus, and no contact with European mathematics. Cooke places it precisely in the international timeline: the discovery "falls between the discovery of the power series for the arcsine function by Newton in 1676 and its publication by Euler in 1737" (S266).
Cooke also supplies a test of the series. It was run here. For an arc of 60 degrees with , so and , ten terms of the series should approximate to 15 decimal places:
Agreement to 15 decimal places, exactly as Cooke states (S266).
The disagreement about how many digits
Here the chapter has to record a scholarly fight, because both sides are in print and neither was independently corroborated.
Osada states that "More than two decades after [the Taisei sankei], Takebe gave exact 41 decimal digits in Tetsujutsu Sankei (1722)" (S265). The Japanese-language reference tradition says something similar in weaker form, that he "computed 41 digits of pi, based on polygon approximation and the Richardson extrapolation" (S267).
Cooke is openly skeptical of a related precision claim. Takebe reported finding the square of the arc geometrically "with accuracy to 53 decimal places." Cooke observes that repeated bisection with 50 applications "gives 38 decimal places of accuracy" and concludes: "it still strains credibility that Takebe Kenko achieved the claimed precision" (S266). He adds two textual complaints. First, the value printed by Smith and Mikami in their 1914 history "does not fit with the procedure followed by Takebe Kenko; it does not even yield the correct first approximation," he writes. Second, in an endnote, once a missing leading 3 is restored, "the accuracy is 'only' 33 decimal places" (S266).
Sort out what is and is not in dispute, because it matters. The series is unquestionably Takebe's and unquestionably correct, verified above to 60 digits. The 25-digit and 29-digit values from the Taisei sankei are correct in every digit printed, verified too. What is contested is the extreme precision of the input data Takebe claimed to have computed by hand: 53 decimal places on the square of a tiny arc. Cooke's objection is a physical one, about how much accuracy repeated bisection can deliver. Osada's "41 exact decimal digits of in the Tetsujutsu sankei" is a different claim from Cooke's "53 decimal places for the square of a small arc," so the two are not in flat contradiction. But neither is independently corroborated in the sources read here, and no openly accessible facsimile of the Tetsujutsu sankei could be found at Waseda, at the National Diet Library, or in general web indexes (S265)(S266)(S267). So there is no primary check on the digit counts at all.
Enri, and Isomura Kittoku
Takebe's series did not appear out of nowhere. It sits inside a Japanese research program with its own name: enri 円理 (also transliterated yenri), "circle theory" or "circle principle," the family of infinitesimal and series methods Japanese mathematicians developed for arcs, areas, and volumes. Enri was in application by the late seventeenth century (S266).
The earliest example Cooke cites is Isomura Kittoku (ee-soh-MOO-rah kit-TOH-koo, seventeenth century), whose Ketsugi-sho appeared in 1660 and was expanded in 1684. To find the volume of a sphere, Isomura sliced it into 10,000 laminae, thin discs stacked along a diameter, computed each and added them up. For a sphere of unit diameter he got 0.5236 cubic feet (S266). Check it: the true volume is . Isomura's four-figure answer is right in every figure he gave.
That is numerical integration, in Japan, in 1660, seven years before Newton's De analysi and thirty-four years before Leibniz published on the calculus. A man with no theory of limits, no notion of a definite integral, and no trigonometry did it. It works because slicing a solid into 10,000 pieces and adding them up is something you can decide to do if you are willing to grind.
Sangaku
One last piece of Japan. It is the one to take into a classroom.
Tokugawa Japan produced sangaku 算額, literally "framed computations" or "computational framed pictures": wooden votive tablets carrying geometry problems, painted in color and hung at Buddhist temples and Shinto shrines (S266). Somebody, an amateur, a schoolteacher, a samurai, a farmer, would solve a hard geometry problem, have it painted beautifully on a board with its diagram and its statement, and hang it in a sacred place. The gesture is simultaneously an offering to the gods, a public announcement of a result, and a challenge to any other mathematician who walked past to solve it or beat it.
Cooke's example is a tablet hung by Ehara Masanori (eh-HAH-rah mah-sah-NOH-ree) at the Atsuta shrine in 1806. The tablet itself was later lost. Its contents survive because a scholar had traveled to the shrine to solve the problem and had taken notes. The tablet was reconstructed from those notes (S266). The mathematics outlived the object, because someone copied it down.
Why a function is a choice
Now put it together, carefully. This is where histories of mathematics go wrong in one of two directions: patronising China for not inventing the sine, or overclaiming that it secretly did.
Here is what the evidence shows.
The tools were lengths and ratios, not angles. The Zhoubi computes the sun's height by similar triangles (S241). Chapter 9 of the Nine Chapters solves every problem by squaring, adding or subtracting, and extracting a root (S246). Liu Hui's chong cha is two nested sets of similar triangles and a subtraction (S247)(S248). Nothing in that program requires an angle, and nothing in it produces one.
When a problem needed an angular quantity, Chinese astronomers built approximations to arcs rather than functions of angles. Shen Kuo's huiyuan gives an arc from a chord and a sagitta (S257). Guo Shoujing's coordinate conversion uses that same rule as, in Wagner's phrase, "an inverse trigonometric function substitute," alongside a versine-like sagitta rule inverted by Horner's method (S258). Those are functional substitutes. They work. And they are approximations with known errors rather than tabulated exact ratios.
The one Tang exception points outward, not inward. Yixing's shadow table is real. Cullen's assessment, as reported by Kotyk, is that it "suddenly appears" in Chinese astronomy and "does point to a foreign source," even though Yixing's use of it was original (S253, quoting S256).
The Indian import of 718 did not take root. Kotyk stresses that Yixing's calendar "is primarily rooted in Chinese models and concepts" and that empirical tests "did not yield any positive outcome for the foreign model" (S253). Trigonometry arrived again with Muslim astronomers under the Yuan, and again it failed to stick. Wang notes that Euclidean geometry "failed to take root until the 1607 publication of Jihe yuanben" (S260).
Chinese astronomers held a flat-earth cosmology until the seventeenth century. Cullen's judgment, quoted by Kotyk: "Chinese astronomers, many of them brilliant men by any standards, continued to think in flat-earth terms until the seventeenth century" (S253, quoting Cullen 1980). If the sky is a dome above a flat plate, you never confront the problem that forced spherical trigonometry into existence: a triangle drawn on a curved surface, where the angles do not add to 180 degrees. That is one scholar's judgment about causation, not a demonstrated cause. This chapter reports it as such.
And Japan, cut off and building on the same foundations, reached a power series for the arcsine squared in 1722 while still, in Cooke's words, lacking "what we know as trigonometry" (S266). That last fact is the one that settles the question of ability. You do not get to the arcsine-squared series by accident or by grinding. Takebe found a transcendental power series by pure numerical pattern recognition, in a mathematical culture with no sine function anywhere in it.
So what is the honest conclusion?
China built a complete, rigorous, and practically superb geometry of the right triangle. It also pushed numerical analysis further than Europe did for centuries: polygon algorithms, higher-order interpolation, Horner's method, and in Japan extrapolation and acceleration. What it did not build was one specific abstraction: a ratio, tabulated as a function of an angle, detached from the situation that produced it, and reused across problems that have nothing else in common.
That abstraction is not smarter than what China had. It is a different design choice, and it carries a real cost: you have to compute an enormous table before you can solve your first problem. Its payoff is that it scales. It scales to spherical astronomy, where lengths-only methods run out, and eventually to analysis, where the sine stops being a ratio in a triangle and becomes a function that describes waves. China's design choice paid off on the problems China was solving. And China was solving those problems well.
The strongest evidence for reading this as a choice rather than a gap is what happened when the sine finally arrived. It came in outline in 718 and in force from the 1630s. Chinese mathematicians absorbed trigonometry quickly and then extended it, and by Minggantu's generation they were doing infinite series for the sine, the cosine, and (S260)(S270). Nothing was missing except the idea.
Two questions to leave with a class, and they are questions about institutions rather than about intelligence. First: what problems does a society choose to fund? A state that needed calendars and surveys employed Chinese mathematics, and it got calendars and surveys, superbly done, by purpose-built methods audited against field measurements. Second: what makes an abstraction worth its cost? Building a sine table is a huge upfront investment. It pays back only if you will reuse it across many unrelated problems. In a system where each bureau builds a recipe for its own recurring problem, the reuse never happens, so the investment never looks worth making. The sine is not only a mathematical idea. It is a bet about how many different problems you expect to solve.
What the evidence does not support
Four claims you will meet, and what the sources support instead.
"Yixing produced the first Chinese tangent table," full stop. Half right, and misleading if you stop there. The Dayan li table gives noon gnomon shadow lengths for an 8-chi gnomon, indexed by the sun's zenith distance in du. Shadow divided by gnomon equals , which is also . So calling it a tangent table is legitimate, and calling it cotangent-like is equally accurate. The two descriptions differ only in which angle you treat as the independent variable (S254). What it is not is a free-standing table of a ratio against an abstract angle. It is embedded in a procedure for predicting shadows in the Nine Domains. And Cullen's 1982 study, the only detailed treatment, is paywalled, so this chapter cannot vouch for the table's entry count, its argument step, or its values (S253)(S256).
"Chinese mathematics independently produced a trigonometric function." No source read for this chapter supports it. Several contradict it directly. The trigonometric idea entered China from India in 718, arrived again with Muslim astronomers under the Yuan, and only became structural in Chinese practice after the Jesuit calendar reform of 1629 to 1633 (S253)(S260). What China developed on its own was the right-triangle relation, the double-difference survey method, arc-and-sagitta approximations, and world-leading numerical methods. Those are not a trigonometric function, and calling them one flatters nobody. It obscures what China did build, which is more interesting than a false parallel.
"Zhoubi means 'trigonometry of circles.'" You will meet this gloss. Its source is Ulrich Theobald's ChinaKnowledge.de entry. Theobald reports the older short title Zhoubi being rendered as "trigonometry of circles" or "trigonometry of the Zhou," with the fuller title Zhoubi suanjing adopted in the Tang (S245). That is a modern rendering using a modern category, not an ancient meaning. What the characters say is zhou 周, the Zhou dynasty or a circuit, and bi 髀, "thigh bone," glossed in the tradition as gu 股 and referring to the 8-chi gnomon (S245). Nobody in the Han called any of this trigonometry. The word imports a concept the text does not have.
"Takebe computed the square of an arc to 53 decimal places." Takebe reported it. Cooke does not believe it, on the ground that 50 rounds of bisection deliver 38 decimal places, and he writes that "it still strains credibility that Takebe Kenko achieved the claimed precision" (S266). Osada's separate claim of 41 exact decimal digits of in the Tetsujutsu sankei is a different statement and does not contradict Cooke's. But neither is independently corroborated here, and no accessible facsimile of the Tetsujutsu sankei was found (S265)(S266)(S267). Keep the two apart. The series is verified to 60 digits and is beyond dispute; the digit-count claims about the input data are not.
And five things that could not be settled. MacTutor's wording on Zu Chongzhi is that he "must have used" a 24,576-gon, which is inference from the precision reached. MacTutor adds that his derivation of "remains unknown" (S252). His death year is 501 in MacTutor and 500 in other widely printed sources (S252). Yixing's birth year is 673 per Kotyk following Jinhua Chen, and 683 per Wu writing in 2023 (S253)(S254). The Gaocheng gnomon is 9.7468 meters per Li and Sun and 12.28 meters per the Biographical Encyclopedia of Astronomers. The argument above from the length of a Yuan chi favors the first without proving it (S244)(S259). And the Jihe yuanben is dated 1607 in the scholarly literature and 1606 in the Library of Congress catalog record for its own copy (S260)(S261).
Hooks for the classroom
Geography and data science: the survey of 724 as a state-scale data project. Yixing's program has every feature of a modern large collaboration: a single standardized instrument, distributed field teams, a central analyst, a pre-registered hypothesis (cun-qian-li), and a published refutation. Map the reported latitude span, and use the disagreement itself as the lesson. Ohashi says roughly 18 to 51 degrees north. Cullen says roughly 29 to 52 degrees north, near meridian 114 east. Kotyk says he cannot work out how either got their number (S253). Then ask the practical question. With no telegraph, no accurate portable clocks, and no way to check a colleague's work until months later, how do you get thirteen stations to measure the same thing the same way? What would you write in the instructions you hand Nan Gongyue's teams?
Units conversion, with consequences. The Han system: 1 chi = 10 cun, 1 zhang = 10 chi, 1 bu = 6 chi, 1 li = 1,800 chi, and therefore 1 li = 300 bu (S248). The Song commentator on Shen Kuo converts at 5 chi to the bu instead (S257). Yixing's survey used a "short li" of 300 bu that works out to roughly 311 meters (S254). Set the sea island problem and make students carry out every conversion by hand, from 3 zhang to 5 bu and from 30,750 bu to 102 li 150 bu. Then hand them the Song ratio and ask what happens to the sea island answer if you use the wrong century's bu. The error is twenty percent, and nothing in the arithmetic will warn you.
Art, mathematics, and community: build a sangaku. Have students solve a geometry problem, then produce the artifact: diagram, statement in formal language, color, wooden frame, and a public hanging somewhere in the school. Then tell them what happened to Ehara Masanori's tablet of 1806 at the Atsuta shrine. The object was lost. The mathematics survived only because a visiting scholar had written the problem down in his notes (S266). Ask what that implies about which of their own work will outlive them, and about why mathematicians publish.
History and politics: the Jesuits as a case study in cultural exchange. Everybody in this story wants something different from the same books. Ricci taught Euclid to earn the standing he needed to preach. Xu Guangqi took the mathematics for statecraft and fixed a broken calendar with it, and coined jihe and yuanben so it could be discussed in Chinese at all. Chinese readers took the operational parts and left the axioms and the theology (S260). Kangxi built his own academy in 1713 with over a hundred Chinese scholars, modeled on the French Academy rather than on the Jesuit colleges. He did it so he would not have to depend on foreigners (S260). Mei Wending and the Shuli jingyun reframed the entire import as xixue zhongyuan, Chinese knowledge coming home, and in the process effaced the Jesuit contribution (S260). Ask students who benefits from each framing, and whether a true story would have traveled as well as the false one did.
Language: the fingerprints of transmission. Three coinages, three centuries apart, all doing the same job. Suifang yan fa 隨方眼法 in 718, built out of native words to carry the Sanskrit sva-desa-aksa, latitude (S253). Jihe 幾何 in 1607, an ordinary interrogative repurposed to mean geometry, and yuanben 原本 for elements (S260). Hand students an English mathematical vocabulary list (sine, tangent, algorithm, algebra, zero) and have them trace which words are calques, which are loans, and which are accidents of mistranslation. The history of a word is often the history of a border crossing.
Where to see the originals
Four of the works in this chapter can be looked at directly. Looking at them is worth a class period.
The Zhoubi suanjing is online at the Chinese Text Project in both transcription and page images. The facsimile is of the Sibu congkan chubian 四部叢刊初編 photo-reprint (volumes 388 to 389) of a Ming printed edition from the Jixuezhai 積學齋 studio of the Xu family of Nanling. The scanned copy carries collection stamps for the Lu Muzhai 盧木齋 library and for Tsinghua University Library (S243). The site asks that its content be cited with a link back, and it prohibits bulk automated download. Check before reusing images.
MAA Convergence presents images of the gougu and "hypotenuse diagram" pages from a 1603 Ming printing of the Zhoubi suanjing, held in the Smith and Plimpton Collections at Columbia University. The page carries no explicit rights statement, so confirm with Columbia's Rare Book and Manuscript Library before reuse (S268).
The Jiuzhang suanshu chapter 9 and the Haidao suanjing are both transcribed at the Chinese Text Project, with links out to Sibu congkan and Siku quanshu facsimiles (S246)(S247).
The Jihe yuanben itself is the Ricci and Xu translation of Euclid books 1 to 6, in four volumes, each leaf 24.9 by 16.3 cm. It is in the Library of Congress World Digital Library collection under LCCN 2021666487, cataloged with the date 1606. The Library of Congress states that these materials are free to use and reuse with no known copyright restrictions (S261). Put a page of it next to a page of a modern geometry textbook and ask students what has changed and what has not.
- China named the right triangle's parts, gou, gu, xian, and solved with the gou-gu relation and similar-triangle proportions: no angle functions, for fifteen centuries, by choice of toolkit rather than by lack of skill.
- The Zhoubi's sun-height computation is internally sound and rests on a flat-earth cosmology, so its answer describes nothing: a model failure, not a mathematical one.
- The shadow table in the Da Yan calendar was calculated rather than observed, and third-order interpolation ran the Shoushi li's astronomy.
Next: Europe imports the whole subject through Spain, and spends five centuries naming it.
Every right-triangle exercise in your mixed practice is a gou-gu problem with newer names on the sides. Try one the Chinese way, sides and proportions only, in Trigonometry 2.P
Chinese mathematics solved right triangles for over fifteen hundred years without sine, cosine or tangent. What did it use?
Chapter 6
The Word, the Names, and the Tables (Europe, c. 1100 to 1620)
The people in this chapter
Faces where a face survives. Every name links to its full entry in Appendix A.
A small page in Heidelberg
Inside an astronomy textbook printed at Heidelberg in 1595, a fresh page carries a short heading. It comes after the author has finished with the sphere. I read it off the scan, letter by letter:
"TRIGONOMETRIA: SIVE DE SOLVTIONE TRIANGVLORVM Tractatus breuis & perspicuus. BARTHOLOMAEI PITISCI Grünbergensis." (S303)
Trigonometry, or a short and clear treatise on the solving of triangles, by Bartholomaeus Pitiscus of Grünberg. As far as anyone has traced it, that is the first time the word "trigonometry" appears in print anywhere. (S303, S313, S328, S333, S334) It is not a book. It is an appendix, tacked on to somebody else's textbook. Its author was a court chaplain from Silesia, better remembered today for fixing other people's arithmetic.
Now open your worksheet. Sine. Cosine. Tangent. Cotangent. Secant. Cosecant. The abbreviations sin and tan. The name of the course itself. Not one of those words is ancient. A specific person picked each one, in a specific book. Almost all of them landed between 1533 and 1620, inside a window of about ninety years. The mathematics behind the words is far older. Most of it reached Europe from Arabic, Persian, and Indian sources, carried by translators working in Spain. Latin Europe added three things in those ninety years. The vocabulary. A printing press to spread it. And tables of numbers so large that building them ate whole careers. In at least two documented cases, they ate entire working lifetimes.
This chapter is where the vocabulary of your course gets its birth certificate. It is also where you find out what the certificates cost.
In 1551, in a 14-page pamphlet of tables, Rheticus did the thing that makes your SOH-CAH-TOA possible. Until then a sine was a line inside a circle. What did Rheticus change?
I have a guess▸
He threw the circle away. The Canon doctrinae triangulorum defines the six functions as ratios of the sides of a right triangle, related directly to the angles, "without any recourse to arcs" (S312). No circle, no fixed radius: a sine stops being a length and becomes a ratio, which is the object your course works with.
If you guessed better tables or more decimal places, that was the century's arms race, and it is the LESS important half of what Rheticus did: the ratio definition is why the tables could eventually shrink back out of sight.
The problem they were solving
Nobody ever built a sine table for the pleasure of it. Four jobs paid for this work.
Astronomy, first and largest. Predicting where a planet would be meant solving spherical triangles over and over by hand. A spherical triangle is drawn on the surface of a sphere, with arcs of great circles for sides. One eclipse prediction could eat weeks.
The calendar, which sounds harmless and was not. Easter drifted. The seasons drifted against the dates. Fixing that was a political problem with spherical astronomy underneath it. So popes funded astronomers. That is also why a Jewish philosopher in Provence dedicated a trigonometry treatise to a pope in Avignon.
Navigation, once European ships began crossing oceans in the 1490s. A ship out of sight of land knows its latitude from the sun or the pole star. It guesses its longitude. Then it steers a compass course. Turning those numbers into a position on a chart is trigonometry. Get it wrong and the ship sinks.
Surveying, gunnery and building. How high is that tower? How far is that fort? How much land is in this field? The same triangle, cheaper instruments.
All four reduce to the same drudgery. Look up a six-digit or seven-digit number in a table. Multiply it by another one. Divide by a third. Look up the answer backwards to get an angle. Repeat. Every innovation in this chapter is one of three things. A better table. A better name for something in a table. Or a trick for dodging the multiplication. Keep that in mind and the whole period makes sense.
One more thing you need before you start, because it makes the old books readable. For everybody in this chapter until 1551, a sine is not a ratio. It is a length. You draw a circle of some chosen radius and mark an arc. The sine of that arc is the half-chord. Take the doubled arc, join its two ends with a straight line, and halve that line. Nobody had a workable decimal fraction notation yet, so you chose large and the sines came out as whole numbers. If , then
and you write 300000, not 0.5. Throughout this chapter, with a capital S means the old table entry, the length. A small s, , means the modern ratio. So . The old books have no such notation. No equals sign in the modern sense. No decimal point. They write everything out in words.
The words arrive before the mathematics
Latin Europe did not invent this subject. It imported it, in bulk, mostly through Spain in the twelfth century. That was one of the great translation waves of history. Scholars from England, Italy and France went to Toledo, Barcelona and Segovia. There they turned Arabic scientific manuscripts into Latin.
Adelard of Bath (AD-uh-lard, active 1116 to 1142) is the first name that matters here. He traveled to Tours, Laon, Salerno and Sicily, and probably to Spain. He also turned al-Khwarizmi's astronomical tables into Latin. The title runs Ezich Elkauresmi per Athelardum bathoniensem ex arabico sumptus (The zij of al-Khwarizmi, taken from the Arabic by Adelard of Bath). A zij is an Arabic handbook of astronomical tables. (S320) Why is the translation usually dated 1126? The Dictionary of Scientific Biography explains: "At the end of chapter 4, the Arabic date a.h. 520 Muharram 1 is said to be 26 January 1126, and this has usually been taken as the approximate date of translation." (S320)
What was in it matters more than the date. The DSB again. The tables, it says, "comprising some 37 introductory chapters and 116 tables in the edition published by Suter, provided the Latin West with its initial introduction ... to the complex of Hellenistic-Indian-Arabic tabular material," and then, in a parenthesis that most survey histories skip, "(Tables 58 and 58a were very probably the first sine tables to appear in Latin.)" (S320) Two tables, numbered 58 and 58a, buried inside a translated handbook. That is the arrival of the sine in Latin Europe.
That date is soft, and the DSB says so itself. A Cambridge manuscript of the same work carries worked examples for the years 1133 and 1134. It also carries a solar eclipse of 1133, which, in the DSB's own words, throws some doubt on the 1126 date. (S320) A competing account of the whole transmission also exists. The Spanish historian José María Millás-Vallicrosa proposed that an earlier Latin version existed, made by Petrus Alphonsus. On that account, what Adelard did in 1126 was retranslate it. (S320) So the standard sentence "Adelard of Bath brought the sine to Europe in 1126" is defensible but loose at both ends. The year is inferred from a date buried in chapter 4, and Adelard may have had a predecessor.
Then the word "sine" is born, and nobody can say by whom
Nobody disputes the chain of the word itself. It is one of the best stories in the history of mathematical language. The Sanskrit term for the half-chord was jya, meaning "chord" or "bowstring." Arabic writers transliterated the sound as jiba, which means nothing at all in Arabic: it is a sound-copy, not a translation. Arabic script normally omits short vowels, so written down, jiba looks exactly like jaib, which does mean something: a fold, a bay, the opening at the neck of a garment. A Latin translator, reading the consonants and supplying the sense, wrote the ordinary Latin word for a fold or a bay: sinus. MacTutor puts the mechanism plainly: "jiba became jaib in later Arab writings and this word does have a meaning, namely a 'fold'. When European authors translated the Arabic mathematical works into Latin they translated jaib into the word sinus meaning fold in Latin." (S335)
So the name of the function you will use for the rest of your life is a mistranslation of a transliteration of a Sanskrit word for a bowstring.
Which translator made the slip is a different question. The honest answer: nobody has settled it. Here are the attributions in print, side by side.
| Scholar making the attribution | Person credited | Work credited | Source |
|---|---|---|---|
| Carl Boyer | Robert of Chester, c. 1145 | a translation of 1145, work not further specified | S332 |
| Howard Eves | Gerard of Cremona, c. 1150 | not specified | S332 |
| Florian Cajori (1906) | Plato of Tivoli | his translation of the astronomy of al-Battani | S332 |
| D. E. Smith, vol. 1, p. 202, footnote 4 | Robert of Chester | his revision of the tables of al-Khwarizmi | S332, S422 |
| D. E. Smith, vol. 2, p. 616 | Gherardo (Gerard) of Cremona, c. 1150 | his translations from the Arabic | S421 |
| Science and its Times (Encyclopedia.com) | Robert of Chester | his 1145 translation of al-Khwarizmi's Algebra | S325 |
| Anton von Braunmühl (1900) | Gerard of Cremona, "probably" | his versions of Western Arabic astronomical works | S425 |
Count what that table is telling you. Four separate published attributions, across three different men. Two writers credit Robert of Chester for two different books. One says the algebra translation. The other says the revision of the astronomical tables. Boyer's version even includes the mechanism: "When Robert of Chester came to translate the technical word jiba, he seems to have confused this with the word jaib ... hence he used the word sinus." (S332) Eves says instead that Gerard of Cremona "replaced the Arabian jaib by its Latin equivalent, sinus." (S332) Cajori, writing in 1906, hands it to Plato of Tivoli. (S332)
Two of the three men are easy to place. Robert of Chester (ROB-ert, active c. 1141 to c. 1150) came from Ketton in Rutland and worked in Spain. In 1145 he finished his Latin version of al-Khwarizmi's al-Jabr wa'l-muqabalah, the book that gave us the word "algebra." By 1150 he was in London, readjusting al-Khwarizmi's astronomical tables to the meridian of London. That meant recomputing the entries so they gave the right answers for English longitude rather than for Baghdad. (S325) Plato of Tivoli (PLAY-toh, active 1116 to 1145) worked at Barcelona. He put the astronomical work of al-Battani into Latin. (S344) Gerard of Cremona (juh-RARD, c. 1114 to 1187) worked at Toledo, the largest translation operation in Europe. He produced the Latin Almagest of Ptolemy that Europe read for the next three centuries. Those dates of his are approximate, and I did not verify them from a source I read. (S332, S310)
Now the part that is worth more than the attributions themselves. Not one of the sources I read shows you the word on a page. They cite each other. No folio reference. No manuscript shelfmark. No photograph of the line where sinus first appears in a trigonometric sense.
One exception cuts the other way. Braunmühl, writing in 1900, went and looked at Plato of Tivoli's al-Battani. He reported that the word is barely there: "It is not the case, by the way, that Plato of Tivoli first introduced this word in his translation of al-Battani, as one can now very frequently read. Kästner and others have already drawn attention to this. The expression sinus versus is found only once in the text; otherwise it is constantly chorda and chorda versa." (S425) That is documentary evidence, and it is negative. The running text of the translation says "chord," not "sine," and the one appearance of sinus sits inside the compound term for the versed sine. Cajori's candidate is the only candidate we can check, and checking him weakens the claim.
One further wrinkle matters if you ever cite D. E. Smith, whose History of Mathematics sits on a great many library shelves. Smith gives two different answers in his own two volumes. Volume 2, page 616: "When Gherardo of Cremona (c. 1150) made his translations from the Arabic he used sinus for jaib, each word meaning a fold, and this usage, possibly begun even earlier, was followed by other European scholars." Volume 1, page 202, footnote 4, on the same question: "The term was probably first used in Robert of Chester's revision of the tables of al-Khowarizmi." (S421, S422) Same author, two books, two candidates. One caution: this contradiction shows up only if you have both Smith volumes open. The single Miller page that most people cite quotes Smith once, from volume 1. Its other Smith citation (volume 2, page 617) covers an unrelated point, Rheticus's preference for the word perpendiculum. (S332) Somebody repeating "Smith contradicts himself" from that page alone is guessing correctly for the wrong reason.
Here is the safe sentence, the one you can defend. Sinus entered Latin in the twelfth century, through the Spanish translation movement. Toledo or Barcelona is the most likely place. Nobody has identified the individual responsible from manuscript evidence. Name one man and you are repeating somebody's guess.
Two people the textbooks skip
A gap sits between the translators of the 1100s and the printers of the 1500s. Most trigonometry courses leap over it in a sentence. Two people fill it, and neither is famous.
Richard of Wallingford, the abbot with the clock
Richard of Wallingford (RITCH-erd, c. 1292 to 23 May 1336) was a blacksmith's son from Berkshire, orphaned, taken in by the prior of Wallingford, and sent to Oxford. His dates are disputed: the Dictionary of Scientific Biography gives c. 1292 to 23 May 1336, while the Springer Biographical Encyclopedia of Astronomers gives c. 1291 to c. 1335. (S319, S351) J. D. North wrote his entry in the DSB, and North spent years editing everything Richard wrote. This is not a casual encyclopedia article. It is the specialist writing the short version.
Richard's first mathematical work was not a treatise of his own. It was a set of canons, meaning instructions for use. He wrote them for astronomical tables drawn up by John Maudith (JON MAW-dith, active c. 1310 to 1316), an astronomer of Merton College, Oxford. (S319) Maudith is a good example of a person this history nearly lost. He computed tables, somebody else wrote the manual, and his name survives mostly in that dependency. Merton College was then the best mathematical address in England. What came out of it was tables and instrument treatises, not famous books. That is the kind of output that vanishes from survey histories.
Then came the Quadripartitum (The Work in Four Parts). North's description deserves quoting at length. It tells you what a trigonometry book looked like in the 1320s.
"Richard followed this with Quadripartitum, a work on such of the fundamentals of trigonometry as were required for the solution of problems of spherical astronomy. The first part of this work has the appearance of a theory of trigonometrical identities, but at the time it was written it was regarded as a basis for the calculation of sines and cosines, and chords and versed chords. The next two parts of the Quadripartitum deal with a systematic and rigorous exposition of Menelaus' theorem, in the so-called 'eighteen modes' of Thabit ibn Qurra. Finally, the work ends with an application of the foregoing principles to astronomy." (S319)
Menelaus's theorem is the workhorse of that middle section. It is the result about a line cutting the three sides of a triangle. In its spherical form it was the main tool for solving spherical triangles, before the sine and cosine laws took over. Thabit ibn Qurra's "eighteen modes" are the eighteen configurations you have to keep straight to use it. Richard's book sorts them out. His sources, per North, were Ptolemy's Almagest, the canons to the Toledan tables, and a short treatise possibly by Campanus of Novara. (S319)
Now North's verdict, and read the last clause carefully:
"The Quadripartitum may reasonably be claimed as the first comprehensive medieval treatise on trigonometry to have been written in Europe, at least outside Spain and Islam." (S319)
"At least outside Spain and Islam" is doing real work in that sentence. Drop it, as quotations of this line often do, and you have made a false claim. Islamic Spain is in Europe, and its trigonometry is both earlier and deeper. Keep the clause and you have a true and interesting claim. In Latin, Christian, northern Europe, this is the first comprehensive treatment. An English monk wrote it about 1320.
When was it written? Softer than you will see it stated. North gives only "the early years in his second period at Oxford," and that period ran roughly 1318 to 1327. It falls after the Maudith canons and before the Tractatus albionis of 1326 to 1327. (S319) Nothing I read supports the commonly quoted date of about 1326. North's own 1976 three-volume edition, Richard of Wallingford: An Edition of His Writings (Oxford University Press, ISBN 0-19-858139-4), would settle it. I did not obtain that edition. (S319, S351, S346)
In 1327 Richard was elected abbot of St Albans. (S319, S346) He was already ill with leprosy. As abbot he went back to the Quadripartitum and revised it. This time he took account of the Flores (Flowers) of Jabir ibn Aflah. Jabir was a twelfth-century Sevillian astronomer whose criticisms of Ptolemy circulated widely in Latin. Only one copy of that later recension is known to exist. (S319) Picture it. A man running a large and heavily indebted abbey, in the last decade of a disease that was killing him, sat down and rewrote his trigonometry. He did it because a better Arabic source had reached him.
The Albion, written 1326 to 1327, is an instrument treatise. The albion ("all by one") was a single equatorium, a calculating instrument that modeled the motions of all the planets. While writing it he also wrote a short treatise on the rectangulus, an instrument of hinged rods for measuring angles that replaced the awkward business of reading arcs off a graduated circle. Buried in that treatise is an item North singles out in one sentence: "In connection with the treatise on the rectangulus, we note a table of an inverse trigonometrical function." (S319) A table of an inverse function, in England, in the 1320s. That is a single sentence in a single source, with no further detail. Treat it as a lead, not an established landmark. But it should make you distrust any story where trigonometry sat still between Ptolemy and Regiomontanus.
Then the clock. Richard put abbey money into an astronomical clock at St Albans. It displayed the positions of sun, moon and stars. North calls it "the first entirely mechanical clock of which we have detailed knowledge." (S319) The accounts of its completion conflict. Wikipedia says the clock was finished about twenty years after his death. The DSB says only that the clock "in its final form was completed after his death." (S346, S319) They agree he did not live to see it finished. Henry VIII's commissioners destroyed it in 1539, in the dissolution of the monasteries. (S346) A monk with leprosy, running an abbey in debt, spent its money on a machine that modeled the sky. A king's men broke it up two hundred years later.
Levi ben Gershon, and the staff that was not Jacob's
Levi ben Gershon (LEE-vye ben GER-shon, 1288 to 20 April 1344) was born at Bagnols-sur-Cèze in Languedoc. He worked at Orange and Avignon. He is known by the Hebrew acronym RaLBaG (RAHL-bahg), and in Latin as Leo de Bannolis, Leo Judaeus or Leo Hebraeus. (S318, S336) He was a philosopher, a biblical commentator, an astronomer and a mathematician. He also wrote in Hebrew. That is one large reason he is missing from a course that reads its history through Latin.
His trigonometry is De sinibus, chordis et arcubus (On Sines, Chords and Arcs). It is the Latin form of a section of his Hebrew astronomical work Sefer Tekunah (Book of Astronomy). That date is disputed. The DSB and MacTutor both say the work is "dated 1343," but the Latin dedication that goes with it belongs to 1342. Simonson's study gives the same date. (S318, S336) The dedication itself is firm. In 1342, two Latin translations went to Pope Clement VI at Avignon, dedicated to him. They were De sinibus, chordis et arcubus and Levi's Tractatus instrumenti astronomie. (S318) Look at that arrangement for a moment. A Jewish scholar in Provence writes astronomy in Hebrew. Somebody turns it into Latin. It goes to the pope, in the papal city, seven years before the Black Death arrives there. A papal court funded the astronomy because the calendar was a papal problem.
What is in the book, per the DSB (the entry is by Juan Vernet):
"Levi's trigonometrical work is in De sinibus, chordis et arcubus (dated 1343) based on the Hebrew text of his Sefer Tekunah. He uses chords, sines, versed sines, and cosines but no tangents (known in Europe since 1126). Following Ptolemaic methods, he calculated sine tables with great precision. He also formulated the sine theorem for plane triangles; this theorem had been known in the Orient since the end of the tenth century, but it is not clear whether Levi rediscovered it independently or knew it through Jabir ibn Aflah (twelfth century)." (S318)
Three things to pull out of that. First, no tangents. He had chords, sines, versed sines and cosines. The parenthesis reminds you that tangents had been available in Latin since Adelard in 1126. He still did not use them. Second, sine tables computed "with great precision" by Ptolemy's methods. That means the halving-and-adding chain of chord identities, done by hand, in sexagesimal. Third, the plane sine theorem. That is the law of sines you will use this year:
Levi states it in the language of proportions, as everyone did before symbolic algebra. In the Latin version quoted in a footnote of D. E. Smith's history it reads: "omnium triangulorum rectilineorum talem proportionem una linea habet ad aliam, qualem proportionem unus sinus angulorum, quibus dictae lineae sunt subtensae, habet ad alium," that is, in every rectilinear triangle one line has to another the same ratio that the sine of one of the angles the said lines subtend has to the sine of the other. (S421) Whether he found it himself or got it from Jabir ibn Aflah is open. Vernet says so rather than guessing. (S318)
Then the instrument, which is the detail worth carrying out of this section. Levi invented a device for measuring the angle between two stars, or between a star and the horizon. It is a graduated rod with a crosspiece that slides along it at right angles. You slide the crosspiece until its two ends line up with the two objects you are sighting. Then you read the position off the rod and convert it to an angle. The DSB describes it: "It consists of a graduated rod and a plate that moves along the rod perpendicular to it. In order to measure an angle, the observer must look at both ends of the plate." (S318) MacTutor gives the dimensions of the developed version: "a staff of 4 1/2 feet long and about one inch wide, with six or seven perforated tablets." (S336)
Europe called it the Jacob's staff. In Latin that is baculus Jacobi, and also baculus geometricus or baculus astronomicus. Portuguese has balestilha. Sailors used it for centuries. Martin Behaim, who died in 1507, used it. The staff "continued to be widely used in navigation, with various improvements." (S318)
Levi never called it that. His own names for it were keli, "instrument," and megalleh 'amuqqot, which the Latin renders as secretum revelator, "revealer of deep things," from Job 12:22. So where does "Jacob" come from? The DSB is blunt about it. Levi wrote two Hebrew poems about the instrument, and one of them "bears the subtitle 'al ha-maqel' ('Concerning the Staff') and in it he refers to Jacob's staff (Genesis 32:10). This is the origin of the expression baculus Jacobi, used in some Latin manuscripts, and of the misunderstanding that he wanted to attribute his invention to someone called Jacob." (S318)
That is worth saying slowly. The instrument that European navigators used to find their latitude for three hundred years is named after a literary allusion in a Hebrew poem. Latin readers took the allusion as an attribution to a man named Jacob. Chapters 4 to 11 of the Sefer Tekunah describe the instrument. Peter of Alexandria put it into Latin in 1342. (S318)
Peuerbach and the enormous radius
Georg von Peuerbach (GAY-org fon POY-er-bakh, 30 May 1423 to 8 April 1461) taught at Vienna. He is the hinge between the medieval and the printed worlds. That is mostly down to who his student was.
His table is the thing to know. Tannstetter's list of Peuerbach's works includes a Nova tabula sinus de decem minutis in decem per multas millenarias partes cum usu, quae plurimum rerum novarum in astronomia occasio fuit (A new table of the sine, from ten minutes to ten, in many thousands of parts, with its use, which was the occasion of very many new things in astronomy). The DSB reports that such a table survives: "such a table of sines at intervals of 0;10 degrees with a sinus totus (unit radius) of 600,000 parts survives in Vin 5291, fols. 165a to 173b, and Vin 5277, fols. 288a to 289b, but without an explanation of its use." (S311) "Vin" is Vienna: those are manuscripts in the Österreichische Nationalbibliothek. The sinus totus, the "whole sine," is the radius, the sine of 90 degrees. Peuerbach set it at 600,000.
Why 600,000, and not 1? Because there were no usable decimal fractions. Make the radius large and every entry in your table is a whole number. You can then multiply and divide with the arithmetic you already have. Picking 600,000 rather than 100,000 also shows the sexagesimal inheritance. It is , and 60 is the base Ptolemy used. A table at 10-minute intervals across the quadrant is entries, each computed by hand through chains of half-angle and difference formulas.
Peuerbach also wrote a short treatise on how to compute such tables. He called it the Tractatus super propositiones Ptolemaei de sinubus et chordis (Treatise on Ptolemy's propositions concerning sines and chords). It survives in Vienna manuscript 5203, folios 124a to 128a. The DSB describes its two halves: "He first explains the computation using kardagas (arcs of 15 degrees) according, he says, to the method of az-Zarqali, and then, at somewhat greater length, sets out Ptolemy's derivation from the first book of the Almagest." (S311) A kardaga is a 15-degree step, from Sanskrit kramajya by way of Arabic. Az-Zarqali is the eleventh-century Toledan astronomer Latin Europe called Arzachel. So a Viennese professor in the 1450s is teaching table building two ways at once. One method is Andalusian, the other Greek.
Printers issued that treatise twice, at Nuremberg in 1541 and at Basel in 1561, in both cases bound with two further items. Those were Regiomontanus's Compositio tabularum sinuum rectorum (The construction of tables of right sines) and his sine tables. (S311, S310) Hold on to the 1541 date. It means Regiomontanus's two great sine tables did not reach print until sixty-five years after he died.
Peuerbach's last work was the Epitome of Ptolemy's Almagest. It is an abridgement, and it made the Almagest usable. He died on 8 April 1461, having reached the end of Book VI. (S311, S310) His student finished it.
Regiomontanus: the book that waited seventy years
Regiomontanus (ray-jee-oh-mon-TAH-nus, 6 June 1436 to 6 July 1476) was born Johannes Müller at Königsberg in Franconia (yo-HAH-nes MYOO-ler fon KUR-nikhs-bairk); "Regiomontanus" is just Königsberg, "king's mountain," turned into Latin. He matriculated at Vienna at eleven and worked with Peuerbach. After Peuerbach's death he finished the Epitome. Then he went to Italy in the retinue of Cardinal Bessarion, hunting Greek manuscripts. He died at forty.
When it was written, and how we know
Regiomontanus wrote De triangulis omnimodis libri quinque (Five Books on Triangles of Every Kind) between 1462 and 1464. Evidence for that range is specific and worth seeing. It is the kind of thing historians work from. He wrote part of the book before leaving Rome on 5 July 1463. Then, at the turn of 1463 to 1464, he wrote in a letter: "I do not have with me the books which I have written about triangles, but they will soon be brought from Rome." (S310) That one sentence pins down two things. The books existed, and he had left them behind. Wikipedia dates completion to 1464, which is compatible. (S345)
Then nothing happened for seventy years.
The seventy-year gap, and the three men who closed it
Regiomontanus set up a press in Nuremberg in the 1470s. He printed a list of what he meant to publish. He got through little of it before he died in 1476. His papers scattered. Willibald Pirckheimer (VIL-ee-balt PIRK-hy-mer, 1470 to 1530, dates approximate) bought the manuscript. He was the Nuremberg humanist and friend of Dürer. Johannes Schöner (yo-HAH-nes SHUR-ner, 1477 to 1547, dates approximate) edited it for the press. He was a Nuremberg mathematician and globe maker. Johann Petreius (YO-hahn peh-TRAY-oos, died 1550) printed it. Ten years later he would print Copernicus's De revolutionibus. Publication date, per the DSB: 12 August 1533. (S310, S301)
Schöner's dedication to the Nuremberg city council tells the story of the manuscript's survival. I read it off the Seville copy's text layer:
"Hunc autem librum, cui de Triangulis omnimodis ipse autor titulum indidit, clarissimus ordinis vestri vir Bilibaldus Pircameus, illo tempore, quo tam species supellex Regiomontani partim diligenter conservabatur, cum, ut fere dicere audivimus, magna pecunia comparasse, non tam sibi quam studiosis disciplinarum Mathematicarum." (S301)
That Latin is not clean. OCR of sixteenth-century type is imperfect, and I have normalized the long s and the obvious noise. The substance is unmistakable. Pirckheimer, "a most distinguished man of your order," bought this book at the time when Regiomontanus's effects were being preserved. He paid "magna pecunia," at great expense, and he did it "not so much for himself as for students of the mathematical disciplines." (S301) Schöner also records what was lost: "ex tanta tamque splendida copia, qualem indices ostendunt, perpauculae reliquiae ad nos pervenerint," out of so great and splendid a store, as the catalogs show, only the smallest remnants have come down to us. (S301)
And Schöner is honest about the state of what he is printing:
"Et est primus sane liber ad eum modum ab autore perpolitus, ut neque ipso emendatius habiturus fuerit. Reliquis extrema manus et limae labor non accessit." (S301)
Regiomontanus polished the first book until he could not have improved it. To the remaining ones, "the final hand and the labour of the file never came." So the most influential trigonometry book of the sixteenth century is an unfinished draft, published by a dead man's editor, from a manuscript bought secondhand.
One footnote on the printing itself, because I promised not to hide the messy parts. The OCR of the Seville copy renders the title-page date as "ANNO CHRISTI M. D. XXXtllT". The colophon at the end of Book V is the printer's closing note, giving place and date. It comes out as "NORIMBERGAE APVD IO. PETREIVM, ANNO CHRISTI M. D. XX3Cnu." (S301) Both are consistent with either M.D.XXXIII (1533) or M.D.XXXIIII (1534). I could not fetch a clean page image to read the numeral directly. The archive's image endpoint returned an error. The DSB is explicit that first publication was 12 August 1533. The Internet Archive catalog record says 1533 too. So 1533 is what I use. But if you set students to read the scan, tell them two things. The date on the page is hard to read. And the appended Cusanus material may carry a separate date. (S301, S310)
What is in the five books
I read the 1533 text directly, which is how the following is organized. (S301)
Book I opens with Diffinitiones, definitions, then Communes animi conceptiones, common notions or axioms. That is the same architecture Euclid uses. Book I then treats plane triangles, and it holds the definition of the sine.
Book II treats plane triangles with sines. Proposition 1 is the law of sines. Proposition 2 solves a triangle. The givens are the sum of two sides and the two opposite angles. And the DSB notes something easy to miss: "Theorem 23 in book II solves, for the first time in the Latin West, a trigonometric problem by means of algebra (here called the ars rei et census)." (S310) The ars rei et census, "the art of the thing and the wealth," is the medieval Latin name for algebra. Res is the unknown and census its square. First algebraic solution of a trigonometric problem in the Latin West, sitting quietly as theorem 23.
Book III is the geometry of the sphere. It opens: "Si sphaera plano secetur, communis sectio superficiei sphaericae et plani secantis erit circumferentia circuli," that is: if a sphere is cut by a plane, the common section of the spherical surface and the cutting plane will be the circumference of a circle. (S301) That is the foundation of everything spherical. Every plane slice of a sphere is a circle.
Book IV treats spherical triangles. It opens with a proposition about the pole of a great circle. Drop an arc from that pole to the circle: the arc is a quadrant, and it meets the circle at right angles. The spherical law of sines is theorem 17, and theorem 25 works with . (S301, S310)
Book V carries the spherical work further. At theorem 2 it gives the earliest statement of the spherical law of cosines. The book closes "Quinti ultimi libri Triangulorum finis," the end of the fifth and last book of the Triangles, followed by the Nuremberg colophon. (S301)
The definition of the sine, in his own words
Book I:
"Arcu et corda sua dimidiatis, medietatem cordae dimidii arcus sinum rectum nuncupabimus." (S301)
"Having halved an arc and its chord, we shall call the half of the chord of the half-arc the right sine." That is sinus rectus, the right sine, meaning the straight one. It stands against the sinus versus, the versed or turned sine. Read the definition as a recipe. Take an arc, take its chord, halve both. The half-chord of the half-arc is your sine. In modern terms, with radius ,
The law of sines, in his own words
Book II, proposition 1:
"In omni triangulo rectilineo proportio lateris ad latus est, tanquam sinus recti anguli alterum eorum respicientis, ad sinum rectum anguli reliquum latus respicientis." (S301)
"In every rectilinear triangle, the ratio of one side to another is as the right sine of the angle facing the first, to the right sine of the angle facing the remaining side."
In his day sines belong to arcs, not to angles. So he supplies a working definition that makes proposition 1 legal: "Sinum anguli, ut alibi vocamus, sinum arcus angulum ipsum subtendentis," meaning: by the sine of an angle, as we call it elsewhere, we mean the sine of the arc that the angle subtends. (S301) The demonstration then names the triangle with letters: "Sit igitur triangulus a b g rectilineus. Dico, quod proportio lateris a b ad latus a g ut sinus anguli a g b ad sinum anguli a b g." Let a b g be a rectilinear triangle. I say that the ratio of side a b to side a g is as the sine of angle a g b to the sine of angle a b g. (S301)
The DSB renders the same proposition in modern dress: "the proportionality of the sides of a plane triangle to the sides of the opposite angles (or, in modern notation a/sin A = b/sin B = c/sin C, the sine law)." (S310)
Stare at the gap between the Latin sentence and the modern line. Regiomontanus has no equals sign in our sense. No letters standing for lengths in an equation. No fraction bar for a ratio. No decimal fractions. Everything is a proportion between four named things, written out in words. The mathematics is identical. Notation costs him a paragraph where it costs you a line.
Worked example: solving a triangle the way a 1533 reader would
Take a triangle with , , and the side opposite equal to 100 units. Find the side opposite .
You open the table at , the radius Regiomontanus uses in Book IV theorem 25 (S310), and read off two whole numbers:
Then proposition 1 gives you a proportion, and the proportion gives you one multiplication and one division:
Modern check, with the ratios you would use: . [own computation, scripts/ch06_checks.py] Identical to five figures, because the radius cancels out of the ratio. That cancellation is the whole reason the enormous-radius system works at all.
Now count the labor, because that is the point of this chapter. One five-digit multiplication, , is easy. The division is not. It is long division with a five-digit divisor. A trained computer takes a minute or two over it, with a fair chance of an error. Computer here means a person paid to do arithmetic all day. A single eclipse calculation contains hundreds of steps like that one. This is the bottleneck that prosthaphaeresis and then logarithms were invented to break.
Book V, theorem 2: the spherical law of cosines, in versed sines
This is the most technically important thing in the book, and the hardest to read. Here it is in full, from page 117 of the 1533 printing.
"In omni triangulo sphaerali ex arcubus circulorum magnorum constante, proportio sinus versi anguli cuiuslibet ad differentiam duorum sinuum versorum, quorum unus est lateris eum angulum subtendentis, alius vero differentiae duorum arcuum ipsi angulo circumiacentium, est tanquam proportio quadrati sinus recti totius ad id, quod sub sinibus arcuum dicto angulo circumpositorum continetur rectangulum." (S301)
Here it is in English. Take any spherical triangle made of arcs of great circles. Take the versed sine of any one of its angles. Divide that by the difference of two versed sines. The first is the versed sine of the side subtending that angle. The second is the versed sine of the difference of the two arcs lying about that angle. The result equals the ratio of the square of the whole right sine to the rectangle contained by the sines of the arcs placed about the said angle.
Unpack the vocabulary. The versed sine is ; with it is . It is the "sagitta," the arrow: the distance from the middle of a chord to the middle of its arc. The "whole right sine" is . The "rectangle contained by" two lines means their product. So with sides and angle opposite , all in modern unit-radius terms, the proposition says
Is that the spherical law of cosines? Yes, and you can check it in four lines. Start from the modern form and work backwards, with :
Rearrange the last line and you have exactly Regiomontanus's proportion. Numerical check with , , : the law gives , and both sides of the proportion equal . [own computation, scripts/ch06_extra.py]
The DSB's verdict on the priority: "he was the first to formulate this fundamental proposition of spherical trigonometry. He enunciated it as theorem 2 in book V of his treatise ... Although he employed the versed sine (1 - cos) rather than the cosine itself and used the law only once." (S310) He states the most useful theorem in spherical trigonometry, in a form nobody would recognize today. Then he uses it once.
The tables, and the tangent he would not name
De triangulis does not use the tangent. The tangent turns up in Regiomontanus's other work, and the DSB is precise about it: "In that work he had not employed the tangent function; but in Tables of Directions he included a table of tangents (although he did not use this term) for angles up to 90 degrees, the interval being 1 degree and tan 45 degrees = 100,000, thereby providing the model for our modern tables." (S310) Setting is the same trick as the enormous radius. It makes the entries whole numbers.
Here is the sequence of the tables. It is easy to garble, so it is worth laying out in order.
| Date | Table or work | Detail | Source |
|---|---|---|---|
| c. 1460 to 1464 | Sine table, R = 6,000,000 | Computed for De triangulis; Roegel says it was "certainly inspired by Peuerbach's table with R = 600000" | S329 |
| 1467 | Tabulae directionum | Computed in Hungary with the help of Martin Bylica | S310 |
| 1468 | Decimal sine table, sin 90 degrees = 10,000,000 | Computed at Buda; Roegel dates it "around 1468" | S310, S329 |
| 1474 | Ephemerides | Day-by-day planetary positions; the first such work to be printed, issued by Regiomontanus himself | S310 |
| 2 January 1490 | Tabulae directionum et profectionum. Tabella sinus recti | Printed at Augsburg by Erhard Ratdolt, edited by Johannes Angelus; contains "tables of tangents which appeared in print for the first time" | S348, S310 |
| 1541 | Both sine tables plus Compositio tabularum sinuum rectorum | Printed at Nuremberg with Peuerbach's Tractatus; reprinted Basel 1561 | S310, S311 |
Three of those entries deserve a sentence each.
Martin Bylica was a Polish astronomer at the Hungarian court. The DSB records that the Tabulae directionum of 1467 were computed with his assistance. (S310) Assistants show up in this chapter constantly, usually in half a clause. They are the ones doing the arithmetic.
The 1474 Ephemerides are planetary positions computed for every day. Regiomontanus printed them himself, on his own press. That made him one of the first people in history to hold three jobs at once: computer, author and publisher of a scientific work. (S310) Wikipedia says the Ephemerides were "published posthumously in 1498," which refers to a later edition. The DSB is explicit that he issued it himself in 1474. (S345, S310)
The 1490 Ratdolt printing matters for a reason that has nothing to do with Regiomontanus. Erhard Ratdolt was one of the great early printers of mathematical books. His Augsburg volume of 2 January 1490 contains the first tangent tables ever to appear in print. (S348) The copy I have the catalog record for is at Keio University Libraries, shelfmark 120X@753@1. (S348)
The honest version of "first"
You will read this in a great many places. De triangulis was the first work to treat trigonometry as a subject independent of astronomy. The DSB's own sentence is more careful than the version that circulates:
"Regiomontanus' monumental work on Triangles, the first publication of which was delayed until 12 August 1533, attracted many important readers and thereby exerted an enormous influence on the later development of trigonometry because it was the first printed systematization of that subject as a branch of mathematics independent of astronomy." (S310)
Set that against Nasir al-Din al-Tusi (nah-SEER ad-DEEN at-TOO-see, 18 February 1201 to 26 June 1274). His book is the Treatise on the Quadrilateral. MacTutor: "This work is really the first in history on trigonometry as an independent branch of pure mathematics and the first in which all six cases for a right-angled spherical triangle are set forth," and it contains "the famous sine formula for plane triangles." (S338) Al-Tusi died in 1274, roughly 190 years before Regiomontanus sat down to write. That is also 260 years before the book was printed.
So here is the honest formulation. Al-Tusi took the conceptual step first, in Arabic. Regiomontanus took it first in Latin and first in print. Both statements are true, and neither is the whole truth on its own.
Two cautions on top of that. The date 1260 is commonly attached to al-Tusi's treatise, and I could not confirm it. MacTutor gives no year for the work. The Muslim Heritage survey gives a year only for a different book, Jami' al-hisab, which it dates 1265. (S338, S358) Whether Regiomontanus knew al-Tusi's treatise is also unsettled. The Muslim Heritage page asserts that the treatise "was known to European scholars, particularly to Regiomontanus (15th century)," but offers no evidence, and that site is popular outreach rather than peer-reviewed scholarship. (S358) Meanwhile the DSB traces the germ of Regiomontanus's spherical cosine law not to al-Tusi at all. It points to a note he wrote in the margin of Plato of Tivoli's Latin al-Battani. (S310) Do not assert dependence, and do not assert independence.
Copernicus, Rheticus, and two surprises
In 1542, a thin book came out of Wittenberg. It held nothing but the trigonometry from De revolutionibus orbium coelestium (On the Revolutions of the Heavenly Spheres). That big book itself would not appear for another year. I read the copy in the Biblioteka Śląska at Katowice, shelfmark 223856 II, from the library's own scan. (S302, S349)
Surprise one: the editor's name is not on the book
The title page reads:
"DE LATERIBVS ET ANGVLIS TRIangulorum, tum planorum rectilineorum, tum Sphaericorum, libellus eruditissimus & utilissimus, cum ad plerasque Ptolemaei demonstrationes intelligendas, tum uero ad alia multa, scriptus a Clarissimo & doctissimo uiro D. Nicolao Copernico Toronensi. Additus est Canon semissium subtensarum rectarum linearum in Circulo. Excusum Vitrembergae per Iohannem Lufft. Anno M.D.XLII." (S302)
On the sides and angles of triangles, both plane rectilinear and spherical: a most learned and useful little book, both for understanding most of Ptolemy's demonstrations and for many other things, written by the most distinguished and learned man Nicolaus Copernicus of Toruń. There is added a canon of half-chords of straight lines in a circle. Printed at Wittenberg by Johann Lufft, 1542.
Georg Joachim Rheticus (GAY-org YO-ah-khim RET-ih-kus, 16 February 1514 to 4 December 1574) is nowhere on that page. His death year is disputed in at least one catalog record: e-rara gives 1514 to 1576, against the 1574 used here. He had traveled to Frauenburg in 1539. There he spent two years as Nicolaus Copernicus's (nik-oh-LAY-us koh-PUR-nih-kus, 19 February 1473 to 24 May 1543) only student. He talked Copernicus into publishing, and he carried the manuscript to the printers. His name appears in exactly one place in this book. It sits over the covering letter on the next leaf, addressed to a Nuremberg instrument maker.
"DOCTRINA ET VIRTVTE PRAESTANTI Georgio Hartmano Noribergensi, Ioachimus Rheticus S. D." (S302)
To Georg Hartmann of Nuremberg, outstanding in learning and virtue, Joachim Rheticus sends greetings. Georg Hartmann (GAY-org HART-mahn, 1489 to 1564, dates approximate) made sundials, astrolabes and quadrants at Nuremberg. He was one of the best instrument makers in Europe. He is the dedicatee of the first printed trigonometry of the Copernican system. That tells you who Rheticus thought the audience was: not philosophers, makers.
Rheticus published his teacher's trigonometry under his teacher's name and signed only the envelope. Then he went further. That table at the back is new. It is called the Canon semissium subtensarum. A canon here means a table, so the title says "table of half-chords." It is not the table in the De revolutionibus manuscript. Edward Rosen, in the DSB: "Rheticus did not ascribe the authorship of this table to Copernicus nor, presumably out of modesty, to himself. Nevertheless, the table was undoubtedly his doing." (S312) An unattributed table by an unnamed editor inside a book credited to somebody else.
What that new table did, per Rosen: "The table of sines in the Sides and Angles of Triangles differs from the corresponding table in De revolutionibus by increasing the length of the radius from one hundred thousand to ten million and by diminishing the interval of the central angle from 10' to 1'. Furthermore, by indicating the complementary angle at the foot of the columns and at the right-hand side of the page, the 1542 table became the first to give the cosine directly, although that term is not mentioned." (S312)
Read the layout trick, because it is the reason the table is famous. Degrees run left to right across the top of the page. For the complementary angles, they run right to left across the bottom. You read the sine going down the column with the top heading. You read the cosine going up the same column with the bottom heading. One printed number does two jobs. It is the first printed table from which you can read a cosine directly, and the word "cosine" does not exist yet.
I checked the table against modern values. The pages are headed "CANON SVBTENSARVM" and carry degree headings 40, 41, 42, 43, 44 across the top and 49, 48, 47, 46, 45 across the foot. (S302)
| Angle | Printed in 1542 | Modern value | Error |
|---|---|---|---|
| 40 degrees 1 minute | 6,430,104 | 6,430,104.16 | 0.16 |
| 40 degrees 2 minutes | 6,432,331 | 6,432,331.67 | 0.67 |
| 5 degrees 30 minutes | 958,455 | 958,457.53 | 2.53 |
(S302, own computation with mpmath at 30 digits, scripts/ch06_checks.py) Seven-figure entries, accurate to the last digit or within two or three units in it, computed by hand around 1540.
Surprise two: a number that does not add up
Here is something I found in the scan that I cannot explain. This book's style is to tell you rather than smooth it over.
I read the page images at 600 dots per inch. Proposition I of the plane-triangle section states that the sines are given "in partibus, quibus dimetiens assumpta est 2000000," that is, in parts in which the diameter is taken as 2,000,000, twice the radius. A few lines later it repeats the figure: "in partibus quibus a b vel a c tanquam ex centro fuerit 1000000 partium sive dimetiens 2000000 partium," meaning: in parts in which a b or a c, as from the center, is 1,000,000 parts, or the diameter 2,000,000 parts. (S302) Diameter two million, radius one million.
But the Canon subtensarum at the back of the same book runs to radius 10,000,000. I verified that against three entries in the table above. Text and table disagree by a factor of ten.
I found no scholarly discussion of this mismatch. It could be a compositor's error. It could be a survival from an earlier draft where the radius was smaller. It could be my own misreading of worn numerals in a 480-year-old book. It is unresolved, and I would rather hand you an open question than a tidy paragraph. (S302)
Where the trigonometry sits in De revolutionibus, and the numbering muddle
De revolutionibus appeared in 1543, one year later. Its trigonometry is Book I, chapters 12 to 14, with sines at 10-minute intervals to radius 100,000. (S327, S347) That is the standard location. But the sources I read do not agree on the chapter numbers. So here is the spread rather than one pick.
| Source | What it says | Source ID |
|---|---|---|
| English Wikipedia, De revolutionibus | the treatise was "taken from the second book" | S347 |
| DSB (Rosen, on Rheticus) | Book I | S312 |
| Roegel, LOCOMAT | "book 1, chapter 12, ff. 15 to 19" | S327 |
| Silesian Digital Library catalog note | describes the 1542 offprint as "chapters 13 to 14" | S349 |
Three of the four put it in Book I and disagree only about which chapters. The fourth puts it in Book II and is, on the balance of the evidence, wrong. (S347, S312, S327, S349) Ever wondered why textbooks give slightly different chapter references for the same passage? This is how it happens. Catalog notes and encyclopedia entries copy each other, and nobody re-opens the book.
Rheticus makes the sine a ratio
In 1551 at Leipzig, Rheticus published the Canon doctrinae triangulorum (Canon of the Doctrine of Triangles). It is a slim thing, 14 pages of tables. In ideas, it is the most important item in this chapter.
Until 1551, a sine is a line in a circle. Rheticus threw the circle away. Rosen states it directly:
"Without any recourse to arcs, Rheticus' Canon defined the trigonometrical functions as ratios of the sides of a right triangle and related these ratios directly to the angles." (S312)
Roegel says the same thing from the table's side: "He did not consider angles in circles, but triangles of which one of the side was constant, and he gave the lengths of the other sides as a function of the angle at the center. The name of 'sine' was not used by Rheticus, nor those of tangent and secant, the latter two having not yet been introduced. The sides were called 'base,' 'perpendicular,' and 'hypothenuse.' The tables span 14 pages, with 7 degrees on each double page." (S330)
So: take a right triangle. Hold one of its three sides fixed at 10,000,000. Its other two sides then depend only on one acute angle, and you tabulate them. Do that three times, once for each choice of which side you hold fixed. You now have all six functions, and you never mentioned a circle.
| Side held constant at 10,000,000 | First column gives | Second column gives | What we call them |
|---|---|---|---|
| Hypotenuse | perpendicular | base | sine, cosine |
| Base | hypotenuse | perpendicular | secant, tangent |
| Perpendicular | hypotenuse | base | cosecant, cotangent |
That is the first table in history containing all six trigonometric functions, and the first printed table of secants. (S312, S330) The entries are at 10-minute intervals to radius 10,000,000. It fits on 14 pages because it is arranged semiquadrantally. That means across half a quadrant instead of a whole one. He tabulates only from 0 to 45 degrees. He lets the cofunctions cover 45 to 90, since the sine of an angle is the cosine of its complement. Rosen: "By equating the functions of angles greater than 45 degrees with the corresponding cofunctions of the complementary angles smaller than 45 degrees, Rheticus reduced the length of his table by half." (S312) Do the count: 45 degrees at 10-minute steps is 270 rows, six functions, 1,620 entries of seven digits each. [own computation]
He would not use the word "sine." Rosen records that Rheticus dismissed the modern-style names as "Saracenic barbarisms," which is a piece of sixteenth-century prejudice aimed at exactly the tradition that had given Europe the subject. (S312) We have independent confirmation of his vocabulary from an unexpected direction: in 1583 Thomas Fincke, introducing the word secans, wrote "Ioachimus Rheticus hypotenusam trianguli rectanguli vocat," Joachim Rheticus calls it the hypotenuse of the right triangle. (S304) A primary source telling us what another primary source called something.
A Basel reprint of the 1551 Canon followed in 1565, per Roegel. Augustus De Morgan, writing in the nineteenth century, gave 1580 for that reprint, which is wrong. (S330, S312)
If your teacher defines the sine as "opposite over hypotenuse," that definition is Rheticus's, it dates from 1551, and it is younger than the printing press.
What a table cost
Rheticus decided the limit on astronomy was not the instruments and not the theory. It was the tables. So he set out to recompute everything, at higher precision and finer intervals than anyone had attempted. To do it he did what a modern lab does. He hired staff.
We do not know their names. Not one of them. They were called computers. For the next four hundred years that was a job title for a person who does arithmetic all day. One sentence is the closest thing to a record of their existence. Rheticus wrote it to the French scholar Petrus Ramus in 1568, and the DSB quotes it. He describes a "labor of twelve years, while I always had to support a certain number of arithmeticians for these computations." (S312)
Twelve years. Then Roegel supplies the budget: "Rheticus had computers work for him for twelve years, but his great tables were not published to his lifetime. The cost of the tablemaking up to 1568 was 4400 Gulden, representing almost 50 times Rheticus' annual salary in Wittenberg. He died when the tables of cosecants and cotangents were being worked on." (S327) The money came from the Emperor Maximilian II. (S327)
Work the arithmetic on that, because the numbers are the point. If 4,400 Gulden is almost fifty times his annual salary, his Wittenberg pay was in the region of 90 Gulden a year, near enough. That project was eating roughly 370 Gulden a year. That is about four professorial salaries, all of it wages for people doing multiplication. (S327, computation done here) A professor spent fifty years' worth of his own pay on other people's arithmetic.
Roegel also warns off a detail that appears in a great many accounts: "Some authors have added that Rheticus had five or six computers working for him, but I do not know what is the source of this information." (S327) So here is the honest statement. An unknown number of paid arithmeticians, for twelve years, at 4,400 Gulden to 1568. Our precision is about the money, not about the people.
Rheticus died on 4 December 1574, at Košice in what is now Slovakia. Cosecants and cotangents were still being computed. (S312, S327) He left his books and manuscripts to one man.
Valentin Otho: twenty-two years and three patrons
Valentin Otho (VAL-en-teen OH-toh, c. 1545 to 1603) signed himself L. Valentinus Otho. His dates are disputed: Roegel gives c. 1545 to 1603, the e-rara catalog record gives 1550 to 1605, and the DSB text I read gives no dates at all. He had sought Rheticus out as a student. He was, in the DSB's phrase, deeply impressed by the Canon of 1551. (S312) He inherited the manuscripts, the project, and the problem. Printing a book of that size required a patron, and patrons kept failing.
| Date | What happened | Source |
|---|---|---|
| 4 December 1574 | Rheticus dies at Košice; Otho inherits the manuscripts | S312, S327 |
| 1574 to c. 1576 | Imperial support obtained, then lapses within two years | S312 |
| 7 September 1576 | Appeals to the Elector of Saxony; is made professor at Wittenberg | S312 |
| January 1581 | Refuses to sign a required religious formula; loses the post | S312 |
| 24 August 1587 | Signs a contract with Frederick IV, count palatine; named official mathematician, permitted to eat at the tables of the Heidelberg professors, granted four students as computers | S312 |
| 1596 | Opus palatinum de triangulis published at Neustadt in the Palatinate | S312, S307 |
Look at the January 1581 line. Otho lost a university chair, and with it his funding. The reason: he would not sign a religious formula. That happened in the middle of the confessional quarrels that split Lutheran Germany. Theology set this project's schedule as much as arithmetic did.
And look at the 1587 terms. Four students as computers, plus meals at the professors' table. That is the payroll. Four unnamed students, paid partly in dinners, produced the largest trigonometric table then in existence. (S312)
Otho called the finished book the Opus palatinum, the Palatine Work. That name thanks the count palatine who finally paid for it. The patron's name is in the title of the book; the computers' names are nowhere in it.
The Opus palatinum, 1596
Matthaeus Harnisch printed the Opus palatinum de triangulis at Neustadt an der Haardt in 1596. (S307, S329, S350) The main table gives all six functions at 10-second intervals to radius 10,000,000,000, that is . That is ten significant figures per entry.
Count the work. A quadrant at 10-second steps is rows. Six functions per row is 194,400 numbers, each of ten digits. [own computation] That is the printed output. The computation behind it was larger, because you compute more digits than you print. The DSB describes the base of the whole edifice: "The foundation of the Rheticus-Otho Opus Palatinum is the table of sines for the first quadrant 0 to 90 degrees, the interval being 45 seconds and the radius 10^15 ... Then, with a radius of 10^10, the sines and cosines were computed for intervals of 10 seconds. The functions of each degree occupy six full pages, so enormous was the labor expended in these computations." (S312) Fifteen significant figures in the underlying sine table.
Roegel gives the physical scale: "The whole Opus palatinum was about 1500 pages long, of which more than 700 were tables." (S327) The e-rara collation of the ETH Zürich copy, shelfmark Rar 9660, runs "[10] Bl., 104, [1] S., [1] Bl., 140 S., [1] Bl., 341 S., [2] Bl., 121 S., [1] Bl., 554 S., [1] Bl., 181 S.; 38 cm": six parts bound in one folio volume 38 centimeters tall. (S307)
Roegel dug an oddity out of its structure. The book contains two tables, not one: "a first part has pages numbered 2 to 541 and gives all six functions with 10 seconds spanning two pages (45 x 6 x 2 = 540) and a radius of 10^10; a second part has pages numbered 2 to 181 and gives only the third group of columns (cosecants and cotangents), again with 10 seconds spanning two pages (540/3 = 180) but with a radius of 10^7." (S327) Roegel argues that the second table is an earlier, superseded computation by Rheticus himself. It probably dates from around 1560, on his reading. He then quotes the nineteenth-century table specialist J. W. L. Glaisher's exasperated verdict: "there seems no reason why it should have been printed at all, as the great ten-decimal canon completely supersedes it." (S327) Otho printed his dead teacher's obsolete draft alongside the finished work. Whether that was piety, padding, or a contractual page count, we do not know.
That manuscript survives. Roegel: "The manuscript of the main table of the Opus palatinum is still extant and kept at the British Library (Harley MS 1720). It is a beautiful work, written with values in black and differences in red." (S327) Want your students to see what twelve years of paid arithmetic looks like? That shelfmark is where it is. Black ink for the values, red for the differences, in London.
The error, and the man who found it
The Opus palatinum was wrong at the start.
Not slightly wrong. In the first cotangent entry, nine of the ten digits were wrong. The book printed and ; the correct values are and (S328) The cotangents and cosecants were built from sines that had not been carried to enough figures for very small angles. At those angles the sine is tiny, and dividing by it magnifies every rounding error. As the angles grew, the sines grew, and the error faded away page by page.
Adriaan van Roomen (AH-dree-ahn van ROH-men, 1561 to 1615) found it. He published in Latin as Adrianus Romanus and taught at Leuven and Würzburg. Most short accounts credit Pitiscus with finding the error. Roegel is explicit: "Adrianus Romanus (Adriaan van Roomen) (1561 to 1615) was the first to notice the flaw." (S328) Pitiscus fixed it. Van Roomen found it.
And he found it with a piece of mathematics you can check yourself. Roegel: "In order to examine the accuracy of the table, Romanus made use of the formula sec a + tan a = tan(a/2 + 45 degrees)." (S328)
This is a beautiful auditing tool. It ties together three entries in three different parts of the table. A secant. A tangent at the same angle. And a tangent at a different angle altogether. If the table is internally consistent, the identity holds to the printed precision. If it fails, something is wrong. And you have not had to recompute anything from scratch to know it.
Here is the proof, which needs only the half-angle forms and the tangent addition formula. Write .
The two steps worth pausing on: because and ; and factors as a difference of squares. Everything after that is division and the addition formula with .
Numerical verification at 30-digit precision, in the region where the Opus palatinum was recomputed:
| Angle a | sec a + tan a | tan(a/2 + 45 degrees) | Difference |
|---|---|---|---|
| 10 degrees | 1.19175359259421 | 1.19175359259421 | 0 |
| 45 degrees | 2.41421356237310 | 2.41421356237310 | 3.9 times 10 to the minus 31 |
| 83 degrees | 16.34985547609967 | 16.34985547609967 | 1.3 times 10 to the minus 29 |
| 89.5 degrees | 229.18166360943992 | 229.18166360943992 | 3.0 times 10 to the minus 27 |
[own computation, scripts/ch06_checks.py]
Roegel then supplies the detail that explains an otherwise baffling number in the bibliographies: "But the way in which the values were computed, namely by using the sines, made this error gradually vanish ... The error discovered in the first pages of the Opus palatinum eventually vanished in the 86th page." (S328)
Exactly 86 pages were corrupted. So exactly 86 pages were reprinted.
Van Roomen did not keep this to himself. He was corresponding with Christopher Clavius (KLAY-vee-us, 1538 to 1612, dates approximate). Clavius was the Jesuit mathematician of the Collegio Romano. He had led the reform that produced the Gregorian calendar. Van Roomen sent him a copy. Roegel: "As he was corresponding with Clavius, he also sent him a copy, and they exchanged about the accuracy of the tables. The correspondence with Clavius also reveals that Christoph Grienberger was preparing a table of sines, but this table was not immediately published perhaps as a consequence of Romanus' insights." (S328) A flaw found in the Palatinate travels by letter to Rome. There it quietly kills a table project before that table reaches print.
Pitiscus recomputes
Otho died in 1603 with the correction unmade. Bartholomaeus Pitiscus (bar-toh-loh-MAY-us pih-TISS-kus, 24 August 1561 to 2 July 1613) took the job on. He was court chaplain to the same Palatine house that had funded the book. His death date is disputed: the DSB, MacTutor and German Wikipedia give 2 July 1613, while English Wikipedia gives 24 August 1613, which repeats his birthday and looks like a transcription slip.
He did not patch it. He recomputed it. Roegel: "Pitiscus first recomputed to twenty decimal places all the sines up to 7 degrees and all the tangents and secants between 83 degrees and 90 degrees to eleven decimal places. 86 pages of the Opus palatinum were reprinted incorporating Pitiscus' corrections. A new edition, with new title page, was reissued in 1607." (S328) The DSB gives the same account, starting from Rheticus's surviving fifteen-place manuscripts. It mentions the eleven-place tangents and secants, but not the twenty-place sines. (S313) Minor divergence between the two, not a contradiction.
Twenty decimal places, by hand, for every sine from 0 to 7 degrees. That is the price of nine bad digits in one corner of a table.
Roegel adds a bibliographical detail that would let you identify a corrected copy across a room: "The corrected edition is very rare, but can easily be identified without looking at the actual values, first because the new pages are of a lower paper quality, and then as there is a small layout error at the bottom of page 7 of the corrected copies, where the words basis and hypothenusa have been interchanged. Copies of the 1607 edition seem to be located at Göttingen and Jena." (S328) Cheaper paper and two swapped words. That is how you date a reprint.
The Thesaurus mathematicus, 1613
Pitiscus's last book was the Thesaurus mathematicus (Mathematical Treasury). It came out at Frankfurt am Main in 1613, the year he died. (S312, S313, S328) It contains Rheticus's sine canon to 15 decimal places, at 10-second intervals, with first, second and third differences. Pitiscus's own numbers are in there too.
"Differences" here means what it means in any table of the period. The first difference is the gap between consecutive entries. The second difference is the gap between consecutive first differences, and so on. Printing them lets a user interpolate between tabulated values. It also lets a checker spot errors. One bad entry shows up as a violent kink in the second and third differences. Three orders of differences is a quality-control system.
How far did Pitiscus's own numbers go? Sources differ. The DSB says his additions run "to twenty-two decimal places," while Roegel says he recomputed sines to twenty places for the corrections. (S313, S328) Twenty-two against twenty. I cannot resolve it. It is a good illustration of how even careful modern scholarship inherits small discrepancies from its own sources.
Sixty-two years separate Rheticus's 1551 Canon from the 1613 Thesaurus. Apart from Otho and Pitiscus, the people who did the arithmetic in between are anonymous.
Pitiscus names the subject
Now back to the small page in Heidelberg.
You will meet a conflict in the literature. Does "trigonometry" date from 1595 or from 1600? Both dates are real, and they belong to different books.
| Year and place | What it is | Tables? | Source |
|---|---|---|---|
| 1595, Heidelberg | Trigonometria: sive de solutione triangulorum tractatus brevis et perspicuus, printed as the final section of Abraham Scultetus's Sphaericorum libri tres | None at all | S303, S328, S354 |
| 1600, Augsburg (printer Manger) | Trigonometriae sive de dimensione triangulorum libri quinque, the first standalone edition | Yes: sines, tangents and secants for every minute of the quadrant, to five decimal places, on pages 123 to 213 | S306, S313, S328 |
| 1608 and 1612 | Later enlarged editions | Yes | S313, S340 |
| 1614, London | Ralph Handson's English translation | Follows the Latin | S333, S313 |
Roegel: Pitiscus "first published in 1595 an appendix on trigonometry to Scultetus' treatise on spherical astronomy. This was the first time the word 'trigonometry' appeared in print." (S328) Kevin Tracey, writing in the peer-reviewed Historia Mathematica, dates it the same way: "Twelve years before Bartholomaeus Pitiscus (1561 to 1613) presented a new name for the study in Trigonometria (1595) ..." (S326) MacTutor and Miller's Earliest known uses both give 1595 and Heidelberg. (S334, S333) The DSB lays out the sequence: "1595: Initial publication as part of Scultetus' Sphaericorum libri. 1600: Revised five-book edition from Augsburg, featuring tables to five or six decimal places for an interval of a minute." (S313)
Roegel explains the real difference, which is why 1600 sticks in people's memories: "The initial appendix published in 1595 did not contain any trigonometric table, but the expanded version published in 1600 contained a table of sines, tangents and secants for every minute of the quadrant and to five decimal places (1600 edition, pp. 123 to 213). The table only had the headers for these three functions, but it of course also gave the cosines, cotangents and cosecants without naming them." (S328) Every minute of the quadrant is 5,400 entries per function. [own computation]
I read the 1595 section title page myself, from the e-rara scan, image 265972, and transcribed it in full at the top of this chapter. The same copy has a contemporary manuscript flyleaf, written by hand by an early owner, identifying the volume: "Abrahami Sculteti Grunbergensis Silesii Sphaericorum libri tres. Bartholomaei Pitisci Grunbergensis De solutione triangulorum. Heidelbergae 1595." (S303) Somebody owned this book close to the time it was printed. He wrote its date and place on that flyleaf. That handwriting is a piece of evidence in its own right.
The printed general title of the volume, from the Google Books copy, is Abrahami Sculteti Grünbergensis Silesii Sphaericorum libri tres methodice conscripti & utilibus scholiis expositi. Accessit De solutione triangulorum tractatus brevis & perspicuus Bartholomaei Pitisci Grünbergensis. In English: three books on spherics by Abraham Scultetus (AY-bruh-ham skul-TAY-tus, 1566 to 1624, dates approximate). A short treatise on solving triangles by Pitiscus is added on. (S354) Both men came from Grünberg in Silesia, now Zielona Góra in Poland. That is why both title pages say "Grünbergensis."
One caution about a source many students will reach for first. English Wikipedia omits the 1600 edition entirely and gives 1608 as the first standalone edition. (S339) German Wikipedia lists 1600, 1608 and 1612. (S340) The physical 1600 Augsburg edition exists. ETH-Bibliothek Zürich holds a copy at shelfmark Rar 5201, and the catalog record settles it. That record gives the imprint "[Augsburg]: [Manger], 1600" and the full title, which continues "... Problematum variorum nempe geodaeticorum, altimetricorum, geographicorum, gnomonicorum et astronomicorum libri decem trigonometriae subiuncti": ten books of assorted problems, land-measuring, height-measuring, geographical, sundial-making and astronomical, appended to the trigonometry. (S306) That list is the job description of the subject in 1600.
The word itself
Trigonometria is two Greek pieces welded together with a Latin ending. Trigonon (τρίγωνον) is a triangle, literally "three-angled," from treis, three, and gonia, angle. Metron (μέτρον) is a measure. Triangle-measuring. None of the sources I fetched spells out this etymology. Roots given here are standard philology, supplied as background rather than quoted from a source I read. (S333)
The word crossed into English in 1614, in a translation by Ralph Handson (RALF HAND-sun, active 1614). Handson was an English mathematical practitioner about whom little is known. His title page: Trigonometry: or The Doctrine of Triangles. First written in Latine, by B. Pitiscus ... and now Translated into English, by Ra. Handson. (S333, S313) Later English editions followed in 1630 (given by both sources I read) and 1642 (given by MacTutor only). A French translation appeared in 1619. (S333, S313, S334) Handson is worth a mention in class because he is a blank. The man who put the name of your course into English is a name and a date. Nothing more.
Fincke names the tangent and the secant
Thomas Fincke (TOH-mas FING-kuh, 6 January 1561 to 24 April 1656) was a Dane from Flensburg. He studied at Strasbourg and Basel, then took a medical degree at Padua. He spent most of his long life as a professor at Copenhagen. In 1583, aged twenty-two, he published Geometriae rotundi libri XIIII (Fourteen Books on the Geometry of the Round) at Basel. Sebastian Henricpetri (seh-BAS-tee-ahn hen-rik-PET-ree, 1546 to 1627, dates approximate) printed it. (S314, S304, S326)
The book's structure, per the DSB: "This important work is divided into fourteen books. The elementary theses on the circle are collected in the four opening books and the remaining books treat trigonometry, the last three being devoted to spherical trigonometry. A central place is occupied by Rheticus' goniometric tables, but here Fink took a step backward, giving the tables for each function separately and always from 0 to 90 degrees, rather than using the complementary character of the functions, as Rheticus had done." (S314) Fincke doubled the size of the tables by dropping Rheticus's clever half-quadrant arrangement. Progress in this business is not a straight line.
On pages 73 to 76, in definitions 21, 22 and 27, he introduces two words. He also tells you he is doing it. (S304, S326)
Definition 21:
"Recta sinibus connexa est tangens peripheriae aut eam secans. Proposuimus ut in circulo inscriptas, sic in semicirculo sinus perpendendos sed et rectas sinibus connexas. Eas plenioris intellectus causa in tangentes et secantes dislocamus. Verbis si hac in re non nova, novis: tamen ut speramus accommodatis." (S304)
"A straight line joined to the sines is either touching the circumference or cutting it. We have proposed that, as with lines inscribed in a circle, so in the semicircle the sines are to be weighed, and also the straight lines joined to the sines. For the sake of fuller understanding we sort these into tangents and secants. In words which, if not new to the subject, are new: yet, as we hope, fitting."
That last clause is the sentence to put on the classroom wall. Verbis si hac in re non nova, novis: tamen ut speramus accommodatis. In plain English: the words are new, even if the subject is not, and he hopes they fit. A twenty-two-year-old flagging his own coinage, and hoping it works. It did. Both words are on your worksheet.
Definition 22 defines the tangent: "Tangens est a termino peripheriae altero perpendicularis in radium extra per reliquum terminum continuatum," meaning: the tangent is the perpendicular from one end of the arc to the radius extended outside through the other end. (S304) Picture the unit circle with an arc from the horizontal. Erect the tangent line at one endpoint of the arc. Extend the radius through the other endpoint until it meets that line. The segment cut off is the tangent.
Definition 27 defines the secant, and hands us the Rheticus detail:
"Secans est radius per terminum peripheriae in tangentem continuatus cum continuatione. ... Et hoc nomen huic rectae accommodatum putamus. Ioachimus Rheticus hypotenusam trianguli rectanguli vocat respectu anguli recti ad quem subtenditur." (S304)
"The secant is the radius, carried on through the end of the arc to the tangent, together with its continuation. ... And we think this name suits this line. Joachim Rheticus calls it the hypotenuse of the right triangle, in respect of the right angle to which it is subtended."
Look at what those two definitions are doing. Fincke's tangent and secant are lines in a circle, defined by touching and cutting. Rheticus's version of the same line was a side of a triangle, the hypotenuse. Fincke tells us this, in print, in 1583, thirty-two years after Rheticus's table. The circle picture and the triangle picture sit side by side on the same page. Both survive into your textbook. You learn "opposite over hypotenuse," which is Rheticus. You learn the words tangent and secant, which come from Fincke's circle picture.
Date and place are as secure as anything in this chapter. Tracey, in Historia Mathematica, cites the physical book as "Thomas Fincke, Thomae Finkii Flenspurgensis Geometriae rotundi libri XIIII (Basel: Sebastian Henric-Petri, 1583), particularly pp. 73 to 76," which is where I found the definitions in the Copenhagen copy. (S326, S304) Miller: "Thomas Fincke introduced secans in Latin in his Thomae Finkii ... Geometriae rotundi libri XIIII (1583)." (S332) The DSB says he "himself introduced such terms as 'tangens' and 'secans'" and adds that he "devised new formulas, such as the law of tangents." (S314) No source I read disputes 1583. The classic dedicated study is Augustus De Morgan's article "On the first introduction of the words tangent and secant," in the Philosophical Magazine, volume 28 (January to June 1846), pages 382 to 387, which Tracey cites. I did not obtain it. (S326)
Two further things about Fincke that a teacher can use. First, he was a Ramist, a follower of Petrus Ramus. Ramus wanted every subject reorganized into clean textbook order. The DSB: "His inspiration and guide was not Euclid's Elements, this work disturbed him, but Ramus' Geometria (1569). Therefore the Geometriae rotundi is based mainly on Ramus, many proofs being comprehensible only after consulting the Geometria. Even the word 'rotundum' in Fink's title, meaning both circle and sphere, was introduced by Ramus." (S314) The man who named the tangent and secant was a curriculum reformer. Naming things is what curriculum reformers do.
Second, the reception: "Such mathematicians as Lansbergen, Clavius, Napier, and Pitiscus recommended the work and adopted much from it." (S314) That is how vocabulary wins. Not by being better, but by sitting in the book everybody teaches from. Fincke also sent his readers back to Regiomontanus, writing on page 295: "Regiomontanus aliquot casus in secundo libro de triangulis collegit ... Cuius certe libri a studiosis avide legi debent; et cum fructu legi possunt," meaning: Regiomontanus collected several cases in the second book on triangles, and certainly his books ought to be read eagerly by students, and can be read with profit. (S326)
The words that lost
Not everybody approved. François Viète (frahn-SWAH vee-ET, 1540 to February 1603) objected to both terms. They would collide, he said, with the existing geometric meanings of "tangent line" and "secant line," and that is a fair objection. A tangent line and the tangent of an angle are different things that share a name to this day.
His replacements, per Miller: for the tangent, sinus foecundarum, abridged to foecundus ("fruitful"), and also amsinus and prosinus; for the secant, Transsinuosa. (S333, S332) Cajori's version of the same story gives the pair more crisply: "Vieta called the trigonometric tangent Prosinus and the trigonometric secant Transsinuosa. But Vieta's objection was overlooked or ignored." (S423)
None of them survived. Prosinus and transsinuosa are dead words, and you have never seen them before this page.
Cajori also tracks how fast Fincke's words spread: "The trigonometric names 'tangent' and 'secant' were adopted by Tycho Brahe in a manuscript of 1591, by G. A. Magini in 1592, by Thomas Blundeville in 1594, and by B. Pitiscus in 1600." (S423) Eight years from coinage to Tycho's working manuscripts, seventeen to Pitiscus's textbook.
Into English, in a slightly odd order
English first uses come through Thomas Blundeville. He was a Norfolk gentleman who wrote practical manuals on horsemanship, map use and navigation. Miller's pages give "The Table of Secants" in Blundeville's Exercises of 1594 for the secant. For the tangent they give "tangent" and "line tangent" as a noun, in the Exercises of 1597. (S332, S333)
That order is odd, and I am flagging it rather than quietly fixing it. Fincke coined both words in the same sentence in 1583, so you would expect them to enter English together. The two Miller pages give 1594 for secant and 1597 for tangent, three years apart. That probably reflects one thing only. It is which edition of Blundeville's Exercises the lexicographers happened to examine for each word. But it is what the sources say, so it is what I report. (S332, S333)
Blundeville's Exercises matter for another reason, one that closes a loop later in this chapter. Edward Wright's work on the Mercator projection first appeared in print inside Blundeville's 1594 navigation treatise. That was five years before Wright published it himself. (S324)
Gunter closes the set, 1620
By 1600 the words sine, tangent and secant exist. Three are still missing, and all three are compounds meaning "of the complement."
Edmund Gunter (ED-mund GUN-ter, 1581 to 10 December 1626) was professor of astronomy at Gresham College in London. He was a practical mathematician of the kind England produced in quantity in that generation. In 1620 he published the Canon triangulorum, sive tabulae sinuum et tangentium artificialium ad radium 10000,0000 & ad scrupula prima quadrantis (Canon of triangles, or tables of artificial sines and tangents to radius 10000,0000 and to every minute of the quadrant). William Jones printed it in London. (S309, S331) One note on his dates: the DSB text as digitized gives his birth year as "1981", an obvious typo for 1581.
"Artificial sines" means logarithms of sines. The phrase is Napier's, and it marks off the logarithm from the "natural" sine itself. Roegel states the achievement: "in 1620, combining these two ideas [Briggs's decimal logarithms of 1617 and Napier's logarithms of sines of 1614], Edmund Gunter (1581 to 1626) was the first to publish a table of decimal logarithms of trigonometric functions." (S331) Sines and tangents, to seven places, for every minute of the quadrant: 5,400 of each. (S309, S331, own computation) Roegel notes a small discrepancy on the book itself: "the cover of Gunter's book gives the radius as 10^8." (S331)
Now the words. Roegel, precisely:
"Although Gunter's table does not name the cosine and cotangent functions, it also gives their logarithms, since log cos a = log sin(90 - a) and log cot a = log tan(90 - a). The cosines and cotangent are only named in the introduction, and this is the first printed use of these words." (S331)
Not in the table headings. In the introduction. The two most-used words in your course after "sine" first appear in print in the front matter of a table of logarithms, describing columns that are not labeled with them. (S331, S335, S315)
The DSB adds the attribution and a caution: "Henry Briggs acknowledged his suggested use of arithmetical complements in logarithmic work and the terms cosine, contangent, and such are probably Gunter's own; his use of the decimal point and his decimal notation for degrees are to be noted." (S315) "Probably his own" is the right level of confidence.
What Gunter wrote was co.sinus, with a full stop. It is a visible abbreviation of complementi sinus, the sine of the complement. Contraction to a single word came later. Smith's history records that Gunter "suggested co.sinus, a term soon modified by John Newton (1658) into cosinus, a word which was thereafter received with general favor." (S421) John Newton (no relation to Isaac) published his Trigonometria Britannica in London in 1658, long after Gunter. The word you write on your worksheet is his contraction of Gunter's abbreviation. (S421, S427) The English form "cosine" is in print by 1635, in John Wells's Sciographia: "As the Radius Is to the cosine of the angle given." (S427)
The cosecant, handled honestly
Which leaves the cosecant. Here the record is a mess, and I am not going to tidy it.
The Sotheby's catalog for the 2018 sale of the Erwin Tomash library describes the Opus palatinum of 1596 as including "the first use of the word 'cosecant'." (S350) That is an auction catalog, written to sell a book, not a piece of scholarship. Roegel's detailed technical study of the very same volume does not make the claim anywhere. (S327)
Miller's Earliest known uses traces where the story comes from, then knocks it down. Ball and Smith both say the term seems to have been first used by Rheticus, with the Latin cosecans appearing in the Opus palatinum. But: "Ball wrote 'I think' it came from Rheticus, and Smith probably took the information from Ball. However, Glen Van Brummelen, in an email in 2014, reports that looking at Opus palatinum he cannot find the term." (S427) Van Brummelen is the leading modern historian of trigonometry. He looked. He could not find it.
The earliest secure English attestation Miller gives is again John Newton's Trigonometria Britannica of 1658: "And as the co-tangents are made from the Tangents, so are the Secants to be made from the sines, For as the sine of an Arch, is to Radius, so is Radius to the co-secant of that Arch by the 31th of the first." (S427)
So the record runs like this. Two historians of the nineteenth and twentieth centuries hedging. An auction house repeating them without the hedge. And a specialist who checked the book and could not find the word.
Since then that specialist has published the reason, and it is better than "I looked and it was not there." Rheticus does not use any of the modern function names, because he rejected the whole naming scheme. Van Brummelen: "But Rheticus, following Copernicus, rejects these modern names for the trigonometric functions, and instead refers to them simply as the hypotenuse, base, and perpendicular of triangles of the three species" (S481, p. 275). The three species are his definitions. If the radius is the hypotenuse, the perpendicular and base are sine and cosine. If the radius is the base, the hypotenuse and perpendicular are secant and tangent. If the radius is the perpendicular, the base and hypotenuse are cotangent and cosecant (S481, p. 275).
So Rheticus tabulates all six functions, and is the first European to publish a unified set of tables of all six, while naming none of them the way you name them. Looking in the Opus palatinum for the word cosecans is looking for a vocabulary its author had thrown out. The claim is not merely unproven. It is the wrong shape. Van Brummelen adds that the terms tangent and secant are Thomas Fincke's, and that cosine arrives twenty-four years after Rheticus (S481, p. 275 and n. 156).
Gunter's other legacy
Gunter's Canon is only part of what he did. What is left is the practical face of the same program. Gunter's chain, used in surveying, "is sixty-six feet long and divided into 100 equal links, thus allowing decimal measurement of acreage," per the DSB. (S315) That is why an acre is 10 square chains. It is also why English field boundaries carry that number to this day. His sector appeared in De sectore et radio of 1623 after circulating in manuscript for sixteen years. It "included sine, tangent, logarithm, and meridional part scales; its uses included the solution of plane, spherical, and nautical triangles (the last formed from rhumb, meridian, and latitude lines). With improvements, the British navy used it for two centuries, and it was also a precursor of the slide rule." (S315) Meridional parts, the last scale in that list, are the stretched latitude spacings a sea chart needs. Roegel adds one more line: "In 1620, he had the idea of putting a logarithmic scale on a rule, which later was developed into a slide rule." (S331)
Put a logarithmic scale on a piece of wood and you can multiply by sliding. That object descends directly from this chapter. It was standard engineering equipment until the pocket calculator arrived in the 1970s.
Viète, and a product that never ends
François Viète (introduced above, 1540 to February 1603) was a lawyer and royal codebreaker as well as a mathematician. He is the person who made algebra symbolic. His trigonometric monument is the Canon mathematicus seu ad triangula (Mathematical Canon, or On Triangles).
Its printing history is unusual, and it tells you how hard these books were to produce. Printing began at Paris in 1571 and finished in 1579, in the shop of Jean Mettayer. Eight years to print one book. Of four planned parts, only the first two ever appeared. (S316, S329, S308) A copy sits at ETH-Bibliothek Zürich, shelfmark Rar 3769. (S308)
What is in it, per Roegel: "This work contained a typographically sophisticated table of the six trigonometric functions for every minute of the quadrant and with a radius of 100000, with sometimes one or more additional figures ... This was the first published canon giving the trigonometric functions every minute, but on the other hand it gave them to less places than Rheticus's 1551 table." (S329) So Viète beat Rheticus on fineness of interval and lost to him on precision.
The layout follows Rheticus's triangle idea rather than the circle: "The first double column gives the sines and cosines. They correspond to the perpendicular and the base of a triangle whose hypotenuse is 100000. The second double column [gives the secant and tangent, corresponding to] the hypotenuse of a triangle whose base is 100000. The third double column gives the cotangent and the cosecant. They correspond to the base and the hypotenuse of a triangle whose perpendicular is 100000." (S329) Three triangles, three constant sides, six functions. That is the 1551 design at one-minute resolution.
A persistent story says the Canon was so full of errors it could not be printed. Roegel calls it "a persistent legend" and did a partial check: "I have not checked Viète's entire table, but only the values of the trigonometric functions on the first and last pages ... On the first page, there are two typographical errors and 14 last digit errors, out of 186 values. On the last page, there is one typographical error, and there are 13 last digit errors." (S329) Roughly 8 percent of values wrong in the last digit, on two sampled pages. That is not a disaster. That is a normal sixteenth-century table.
The forty-fifth-degree challenge
In 1593 Adriaan van Roomen threw a problem at all the mathematicians of the world: solve a particular equation of the forty-fifth degree. This is the same man who would later find the error in the Opus palatinum. The DSB tells what happened next:
"the ambassador from the Netherlands remarked to Henry IV that France did not possess any geometricians capable of solving a problem propounded in 1593 by Adrian Romanus ... that required the solution of a forty-fifth-degree equation. The king thereupon summoned Viète and informed him of the challenge. Viète saw that the equation was satisfied by the chord of a circle (of unit radius) that subtends an angle 2 pi / 45 at the center. In a few minutes he gave the king one solution of the problem written in pencil and, the next day, twenty-two more. He did not find forty-five solutions because the remaining ones involve negative sines, which were unintelligible to him." (S316)
His insight is trigonometric, not algebraic. The DSB spells it out: "since 45 = 3 x 3 x 5, it was necessary only to divide an angle once into five equal parts, and then twice into three." (S316) A degree-45 equation in disguise is an angle-division problem. Angle division is what multiple-angle formulas do. Viète solved it in minutes because he saw the shape.
Two details are usually collapsed and should not be. The challenge is 1593. Viète's published response, Ad problema, quod omnibus mathematicis totius orbis construendum proposuit Adrianus Romanus, responsum (A response to the problem that Adriaan van Roomen proposed to all the mathematicians of the whole world), came out separately in 1595. (S316) Solved at once; printed two years later.
The infinite product
Also in 1593, Viète published what the DSB calls "the earliest explicit expression for pi by an infinite number of operations," in book VIII of his Variorum de rebus mathematicis responsorum (Various responses on mathematical matters), chapter 18. He got there by looking at regular polygons inscribed in a circle of unit radius: 4 sides, then 8, then 16, and on. (S316) This is the first infinite product in European mathematics:
Where does it come from? Straight out of the half-angle formula. Set and then . Each new factor is one more application of , which is exactly the doubling step that takes a square to an octagon to a 16-gon. Multiply the together forever and you get .
I verified this numerically at 60 digits.
| Factors used | Partial product | Error against 2/pi |
|---|---|---|
| 2 | 0.653281482438188263928321586714 | 1.6662 times 10 to the minus 2 |
| 5 | 0.636875507721753550700756825065 | 2.5574 times 10 to the minus 4 |
| 10 | 0.636620022039002079795618949659 | 2.4967 times 10 to the minus 7 |
| 20 | 0.636619772367819448231665838433 | 2.3811 times 10 to the minus 13 |
| 30 | 0.636619772367581343302609825496 | 2.2707 times 10 to the minus 19 |
The true value is . Inverting the thirty-factor partial product gives , correct to 17 decimal places. Thirty-one factors would buy the eighteenth. [own computation, scripts/viete_product.py]
Thirty square roots. Seventeen correct digits of . Compare that with the effort behind any table in this chapter. You can see why the seventeenth century went looking for series and products instead of more computers.
Multiplying without multiplying
Every page of this chapter has circled one obstacle. Multiplying two seven-digit or ten-digit numbers by hand, thousands of times, without errors. Before logarithms, astronomers had a way around it, and it is pure trigonometry.
The method is called prosthaphaeresis, from the Greek prosthesis (πρόσθεσις), addition, and aphairesis (ἀφαίρεσις), subtraction. The DSB defines it as the method "for simplifying trigonometrical computations by replacing multiplications and divisions with additions and subtractions." (S321)
The mathematics is an identity you already know, read backwards:
The left side has a multiplication in it. The right side does not. It has an addition, a subtraction, two table lookups, and a halving. If you have a good table, you have turned multiplication into looking things up.
Working one through
Suppose you need at , which is the sort of product that turns up constantly in spherical astronomy. Your table gives and .
The hard way. Multiply, then divide by to get back to a single-radius answer:
That multiplication is twenty-five single-digit products, plus five rows of partial sums with carries. Ten minutes, with two or three plausible places to go wrong.
The prosthaphaeresis way. Form the sum and difference of the angles, look up two cosines, subtract, halve:
One subtraction of five-digit numbers and a halving. Thirty seconds, and almost nothing to get wrong.
The check. The exact value is 40,896.78. The long multiplication gives 40,896.36, off by 0.42 because the two table entries were rounded. Prosthaphaeresis gives 40,897, off by 0.22. [own computation, scripts/ch06_checks.py] The shortcut is not just faster here. It is slightly more accurate, because fewer rounded quantities enter a product.
Now multiply that saving by the tens of thousands of products in a single planetary calculation. You can see why an observatory would guard the method.
Who invented it, and the fight about it
Contested, and the sources disagree in an instructive way.
The DSB's account sits in the entry on Tycho Brahe (TEE-koh BRAH-uh, 1546 to 1601, dates approximate). It centers on Paul Wittich (POWL VIT-ikh, c. 1546 to 1586, dates approximate).
"Paul Wittich, who was an assistant at Uraniborg in 1580 and who, at Kassel in 1584, described Tycho's instruments, including the transversal divisions, so impressed the landgrave that he had his instrument maker, Joost Bürgi, alter his instruments to conform to the description. Wittich was probably largely responsible for the development of the prosthaphaeretic method ... This is the basis of the set of rules for solving plane and spherical triangles, Triangulorum planorum et sphaericorum praxis arithmetica, drawn up, without proof, by Tycho and made available in numerous manuscript copies for the use of his assistants. Wittich also revealed this method at Kassel, to the annoyance of Tycho; he was even more annoyed, however, by the inclusion of the first two rules in a book by Nicolai Reymers Bar (Ursus), printed at Strasbourg in 1588." (S321)
Unpack the sequence. Wittich is at Uraniborg, Tycho's island observatory, in 1580. In 1584 he is at Kassel, describing Tycho's instruments to the landgrave and letting the method out. In 1588 Nicolai Reymers Bär, who called himself Ursus, prints the first two rules at Strasbourg. Tycho is annoyed at the first and angrier at the second. Notice Tycho's own rules too. The Triangulorum planorum et sphaericorum praxis arithmetica (The arithmetical practice of plane and spherical triangles) was "drawn up, without proof," and it circulated only in manuscript copies for his own assistants. (S321) A working method, deliberately unpublished, passed hand to hand inside one research group. Sixteenth-century trade secrets.
Against Wittich stands a second candidate. Braunmühl's history proposes Johannes Werner of Nuremberg (1468 to 1522) as the inventor, seventy years before the usual attribution. Werner's manuscript on triangles is lost. (S425) A lost manuscript is a hard thing to argue with in either direction.
Wikipedia's summary is the fair one. The method "appeared in the 1580s, but its originator is not known for certain," it says. The article lists Ibn Yunis, Johannes Werner, Paul Wittich, Joost Bürgi, Christopher Clavius and François Viète as contributors. It adds that Wittich, Ibn Yunis and Clavius "have all been credited by various sources with discovering the method," and that the first two identities "are believed to have been derived by Jost Bürgi." (S357)
One caution about a version of this dispute that circulates in secondary accounts. You will sometimes read that the Werner versus Wittich question traces to two specific publications. Jacob Christmann in 1611 reporting Werner's manuscript. Christen Longomontanus in 1622 crediting Wittich. I could not verify that. Neither Christmann's name nor Longomontanus's appears in any source I read for this chapter. The DSB says only that Wittich was "probably largely responsible." (S357, S321) The dispute is real. Those two citations are not ones I can vouch for.
Napier: logarithms built for sines
John Napier (JON NAY-peer, 1550 to 4 April 1617) was laird of Merchiston near Edinburgh. In 1614 he published the Mirifici logarithmorum canonis descriptio (Description of the Wonderful Canon of Logarithms) at Edinburgh. Andrew Hart printed it. (S305, S317)
Read the rest of the title before you decide what logarithms are for. It runs ejusque usus, in utraque trigonometria; ut etiam in omni logistica mathematica, "and its use, in both kinds of trigonometry, as also in all mathematical calculation." (S305) Both kinds of trigonometry means plane and spherical. Trigonometry is in the title of the book that invented logarithms.
Logarithms were not invented for arithmetic in general and then applied to trigonometry. They were invented for trigonometry. The DSB is unambiguous:
"As presented, Napier's canon is specifically associated with trigonometric usage, in the sense that it gives logarithms of natural sines (from the tables of Erasmus Reinhold). The sine of an arc was not, at that time, given as a ratio but as the length of the semichord of a circle of given radius, subtending a specified angle at the center. In tabulating such sines, it was customary to choose a large number for the radius of the circle (or whole sine); Napier's choice of 10^7 gave him seven significant figures before introducing fractions." (S317)
Three things in that passage. Napier's table is a table of logarithms of sines, not of numbers. The sines themselves he did not compute. He took them from the tables of Erasmus Reinhold (eh-RAZ-mus RINE-holt, 1511 to 1553, dates approximate). Reinhold was the Wittenberg astronomer whose Prutenic Tables were the standard astronomical tables of the century. Napier's whole sine is 10,000,000, the same Rheticus had used in 1542 and 1551. He chose it for the same reason: seven digits and no fractions.
His stated motive, per the DSB, is the complaint this whole chapter has been building toward. Nothing is more troublesome to mathematical practice than the "multiplications, divisions, square and cubical extractions of great numbers." (S317) Same complaint that produced prosthaphaeresis. Better answer.
The rules of circular parts
Book II, chapter 4 of the Descriptio contains the piece of Napier that still appears in textbooks. Take a spherical triangle with a right angle. Ignore the right angle, and five parts remain: two sides, one hypotenuse-side, and two angles. Napier's move:
"Harum quinque partium non quadrantium, tres quae a recto angulo, seu quadrante latere, situ remotiores sunt, in sua complementa convertimus, et retento pristino ordine omnes quinque in circularem, seu pentagonalem situm statuimus, et circulares vocamus." (S305)
"Of these five parts which are not quadrants, the three which are further in position from the right angle, or the quadrantal side, we convert into their complements, and, keeping the original order, we set all five in a circular, that is pentagonal, arrangement, and we call them circular parts."
Then the selection rule:
"Quinque circularium partium, tres semper in quaestionem cadunt, quarum duae dantur, tertia quaeritur. Atque harum trium una est intermedia, et duae sunt extrema, quae scilicet intermediae aut circumponuntur, aut opponuntur." (S305)
"Of the five circular parts, three always fall into the question, of which two are given and the third is sought. And of these three one is the middle part, and two are extremes, which are either placed around the middle part or opposite to it."
Then the theorem, in logarithmic form:
"Logarithmus intermediae aequatur differentialibus circumpositarum extremarum, seu antilogarithmis oppositarum extremarum." (S305)
"The logarithm of the middle part equals the differentials of the adjacent extremes, or the antilogarithms of the opposite extremes."
That single rule replaces ten separate formulas for right spherical triangles. Draw the pentagon, mark the five parts in order, replace three of them by their complements, and every relation you need is "middle part against two adjacent" or "middle part against two opposite." Napier is visibly delighted with it. Chapter 4 also contains the line "Haec circularium partium uniformitas manifestissime patet in rectangulis factis in superficie globi ex quinque circulis magnis," meaning: this uniformity of the circular parts is most clearly evident in the right angles made on the surface of a globe by five great circles. (S305) Chapter 5 then handles oblique spherical triangles, which Napier calls non quadrantalia. He drops a perpendicular or a quadrant to break them into right ones. (S305)
One correction to a common sentence. The DSB writes: "Napier's rules (called the Napier analogies) for the right-angled spherical triangle were published in the Descriptio (Bk. II, Ch. 4)." (S317) That runs two different things together. In standard usage, the rules of circular parts are the pentagon method for right-angled and quadrantal spherical triangles. That is what chapter 4 does. Napier's analogies are four separate half-sum and half-difference relations for oblique spherical triangles. Half-sum and half-difference relations do appear in the 1614 text. They are in the plane oblique chapter (Book II, chapter 2, the law of tangents in logarithmic form), and again in the spherical chapter 5. (S305) But I could not identify four numbered propositions in chapter 5 matching the standard four analogies. So I am not going to tell you they are there in that form. The DSB's parenthesis is loose. How loose, I cannot say.
Briggs, Bürgi, and who gets the credit
Henry Briggs (HEN-ree BRIGZ, February 1561 to 26 January 1630) was professor at Gresham College and later at Oxford. He rebuilt logarithms into the form you would recognize, with and . (S323) His Logarithmorum chilias prima (The First Thousand Logarithms), probably 1617, gives the base-10 logarithms of the numbers 1 to 1000, and the British Museum copy of it is bound together with Gunter's Canon triangulorum of 1620. (S323, S317, S315) Two pamphlets stitched into one volume by an early owner. Between them they hold the whole computational apparatus of seventeenth-century science.
Jost Bürgi (YOHST BUR-ghee, 28 February 1552 to 31 January 1632) got there on his own. His route runs straight through this chapter. He was an instrument maker at Kassel, the same workshop where Wittich's description of Tycho's instruments landed in 1584. Later he was imperial clockmaker at Prague. The DSB: "The computation of the tables of sines and the elaboration of astronomical data led Bürgi to an easier method of multiplying ... improvement of 'prosthaphairesis', the method of converting multiplication into addition by means of trigonometrical identities." (S322) Computing sine tables led him to prosthaphaeresis. Improving prosthaphaeresis led him to logarithms. His Arithmetische und geometrische Progress-Tabulen (Arithmetic and Geometric Progression Tables) were computed around 1600. A Prague printer issued them only in 1620. (S322, S317)
The DSB settles the priority question in the only way it can be settled:
"In matters of priority in the invention of logarithms the only serious claims have been made on behalf of Joost Bürgi. Many German historians have accorded him priority in the actual invention on the grounds that his tables had been computed about 1600, although they were not published until 1620. Since Napier's own work extended over a long period of time, both must be accorded full credit as independent inventors. The tables were quite differently conceived, and neither author owed anything to the other. Napier enjoyed the right of priority in publication." (S317)
One story tells how news of Napier's work spread before he published. The DSB tells it with a visible hedge, so I will too: "It is said that word of these developments came to Napier through a fellow countryman, John Craig, who accompanied James VI to Norway in 1590 to meet his bride, Anne of Denmark. The party landed near Tycho Brahe's observatory at Hven and was entertained by the astronomer." (S317) The DSB adds a firmer piece of evidence about the reverse direction: "There is evidence in a letter written by Kepler in 1624 that he had received an intimation of Napier's work as early as 1594." (S317) Present the Craig story as a plausible route, not as a fact.
Napier's Descriptio reached English readers in 1616, in a translation by Edward Wright. Wright's son Samuel published it after his father's death. Napier "approved the translation, both in substance and in form." (S317, S341) Which brings us to Wright, and to ships.
Girard, and the first abbreviations
Albert Girard (al-BAIR zhee-RAR, 11 October 1595 to 8 December 1632) was a French Protestant refugee working in the Netherlands. He is credited with the first use of the abbreviations themselves, in a Trigonométrie of 1626. MacTutor gives 1595 without a day. (S337, S343)
Here the sources disagree about which abbreviations. I am going to lay the disagreement out rather than pick. MacTutor: "In this work Trigonométrie on trigonometry he made the first use of the abbreviations sin, cos, tan. He also gave formulas for the area of a spherical triangle." (S337) Wikipedia says the same three: Girard "was the first to use the abbreviations 'sin', 'cos' and 'tan' for the trigonometric functions in a treatise." (S343) Smith's history gives a different trio: Girard "first made general use of a satisfactory abbreviation for sine and introduced the forms that became 'tan' and 'sec.'" (S421) So two sources say sin, cos, tan. One says sin, tan, sec. I could not obtain the 1626 book itself. Neither MacTutor nor Wikipedia gives its full title, publisher or place. (S337, S343)
The other Girard result named in both sources is Girard's theorem, on the area of a spherical triangle. The area depends only on the interior angles, through the amount by which they exceed 180 degrees. That amount has a name: the spherical excess. (S337, S343) Neither source states the formula, so what follows is the modern statement rather than a quotation. On a sphere of radius , a triangle with angles , , measured in radians has area
This is one of the strangest facts in elementary geometry, and worth ten minutes of class time. On a flat surface, the angles of a triangle tell you nothing about its size. On a sphere, they tell you the size exactly.
Two further data points from Smith and Miller fill out the picture of how the abbreviations settled down. Richard Norwood, in his Trigonometrie (London, 1631), prints his own key: "in these examples s stands for sine: t for tangent: sc for sine complement: tc for tangent complement: sec for secant." (S421) And Pierre Hérigone in 1634 "seems to be the first to use 'sin' for sine in a book, though the contraction appears on a 1624 drawing of Gunter's scale." (S421) Notation is being invented in public, in competing systems, and the winners are not obvious at the time.
The sea and the map
One last thread of this chapter pays for everything: ships.
Gerardus Mercator (jeh-RAR-dus mer-KAY-tor, 1512 to 1594, dates approximate) published his world map in 1569. The title is Nova et aucta orbis terrae descriptio ad usum navigantium emendate accommodata (A new and enlarged description of the Earth, properly adapted for the use of navigators). (S342) It has the property every navigator wanted. A course of constant compass bearing comes out as a straight line on the map. Sailors call such a course a rhumb line, or a loxodrome. Steer 240 degrees and hold it, and your track on Mercator's chart is a ruled line.
To make that work, the north-south spacing has to stretch as you go poleward. It must stretch at exactly the same rate as the east-west stretching that flattening the globe forces on you. Mercator said what he had done and not how: "We have progressively increased the degrees of latitude towards each pole in proportion to the lengthenings of the parallels with reference to the equator." (S342) He gave no mathematical derivation. Whether he had one is still argued.
Edward Wright (ED-wurd RITE, 8 October 1561 to November 1615, buried 2 December 1615) supplied the mathematics. He was a fellow of Caius College, Cambridge. In 1589, he had gone to sea on a raiding voyage to the Azores. His book is Certaine Errors in Navigation (London, 1599). (S324, S341)
The east-west stretch at latitude is . The parallel of latitude at is shorter than the equator by a factor of , and the map has to pull it back out to full width. To keep the map conformal, that is, locally the right shape, the north-south stretch must match. So the distance up the map to latitude is the running sum of at every latitude below it. Wright's own phrase for his procedure is "perpetual addition of the Secantes." (S324)
He was computing
as a Riemann sum, in 1599, roughly seventy years before the integral existed as a concept. The DSB says exactly that: his method was "essentially performing numerical integration of secant values. This produced what became known as the table of meridional parts." (S324) Meridional parts are those stretched north-south distances, one number per latitude. Wikipedia adds three details. Wright tabulated the scale factor "for each minute of arc up to 75 degrees" and this was "in fact a table of values of the integral of the secant function," it says. The first edition carried an abridged six-page table. The second edition of 1610 expanded it to 23 pages at one-minute intervals. (S341)
Worked example: adding secants, the way Wright did
Take the meridional part for latitude 60 degrees. That is how far up the map 60 degrees north should sit, measured in units where one minute of longitude at the equator is 1.
Add of every minute from to , one term per minute, 3,600 terms:
The exact integral is
[own computation, scripts/ch06_extra.py] Wright's crude sum, at one-minute resolution, lands within half a minute of arc of the true value. That is an error of one part in nine thousand. At one-degree steps the sum gives 75.96 degrees against a true 75.46. That is visibly worse, and it explains why the 1610 edition went to minutes. A Riemann sum beating its own century by seventy years, computed by hand, 3,600 secant lookups at a time.
Here is a connection worth showing a class. It ties the two halves of this chapter together. That secant integral has a closed form:
The second equality is van Roomen's identity, the same one he used to audit the Opus palatinum. So the table of meridional parts is a table of logarithms of tangents in disguise. I verified both forms numerically: at 60 degrees, all three expressions give 4527.367757 minutes. [own computation, scripts/ch06_extra.py]
Nobody in 1599 knew that. The DSB records who did and when: "The 1653 ed. of the works, amended by Samuel Foster and Henry Bond, contains an early printed statement of the logarithmic result for the integral of the secant function or meridional parts. Gunter's meridian scale, like Wright's earlier one, came from the simple addition of secants; and he was doubtless unaware of Harriot's unpublished calculation of them as (in effect) logarithmic tangents, completed in 1614: he was not, anyway, interested in such theoretical niceties." (S315) Thomas Harriot had it in 1614, in manuscript, and did not publish. Print had to wait until 1653.
Wright's work also circulated before it was his. The DSB: "His work appeared first in Thomas Blundeville's 1594 navigation treatise, then in Jodocus Hondius's maps (1596) and Richard Hakluyt's Principal Navigations (1598 to 1600), though not always with attribution." (S324) Five years of other people printing his method, sometimes without his name, before Certaine Errors appeared under it. That is one reason he published. To reclaim it.
And the loop closes. The man who turned the secant into a navigational table is the man who translated Napier's logarithms into English. His son Samuel published that translation in 1616 after his father's death. (S317, S341)
What the evidence does not support
Five claims that circulate widely and do not survive contact with the sources.
"Robert of Chester coined the word sine in 1145." Or Gerard of Cremona, or Plato of Tivoli. Reputable historians name all three in print. Two writers name Robert, for two different books. (S332, S325) None of the sources I read cites a manuscript, a folio, or an image of the word on a page. They cite each other. One piece of documentary work points the other way. Braunmühl checked Plato of Tivoli's al-Battani and found sinus only once, inside the compound sinus versus. Its running text uses chorda throughout. (S425) D. E. Smith gives two different answers in his own two volumes: Gherardo in volume 2 page 616, Robert of Chester in volume 1 page 202 footnote 4. (S421, S422) Nobody has settled this. Say "a twelfth-century Latin translator, probably in Spain."
"Pitiscus coined 'trigonometry' in 1600." The 1600 Augsburg book is the first standalone edition, and the first Pitiscus trigonometry with tables in it. That is probably why the date travels. (S306, S328) The word is in print in 1595, on the section title page of the appendix to Scultetus. I read that page from the scan and transcribed it. (S303) English Wikipedia makes a different error in the same neighborhood. It omits the 1600 edition entirely and calls 1608 the first standalone edition. The ETH Zürich copy of the 1600 Augsburg printing refutes that. (S339, S306)
"Regiomontanus was the first person anywhere to treat trigonometry as an independent subject." He was the first in Latin and the first in print. The DSB's own sentence says "the first printed systematization." (S310) Al-Tusi's Treatise on the Quadrilateral is, per MacTutor, "really the first in history on trigonometry as an independent branch of pure mathematics," and al-Tusi died in 1274. (S338) Deleting the word "printed" turns a true statement into a false one. Two further honest notes. Nothing I read confirms the year 1260 usually attached to al-Tusi's treatise. And no source I read establishes whether Regiomontanus knew the work. (S338, S358, S310)
"The Opus palatinum contains the first use of the word 'cosecant'." This appears in a Sotheby's auction catalog. (S350) Roegel's dedicated technical study of the same book does not say it. (S327) Miller traces it to Ball saying "I think" and Smith apparently copying Ball, and then records that Glen Van Brummelen looked in the Opus palatinum in 2014 and could not find the term. (S427) So the claim has no source behind it, only a chain of people repeating each other.
"Pitiscus discovered the error in the Opus palatinum." Van Roomen found it, using the identity , and told Clavius about it by letter. (S328) Pitiscus did the recomputation and the reprint. That is a large piece of work, and enough credit for anyone. But he was not the one who spotted the error.
And one bonus, on people rather than claims. "Rheticus had five or six computers working for him" is in a great many accounts. Roegel says plainly: "I do not know what is the source of this information." (S327) What is documented is twelve years, an unspecified number of paid arithmeticians, and 4,400 Gulden.
The primary-source record
Everything quoted in this chapter from an original printing came from one of these scans. Holdings, shelfmarks and rights are given so a class can go and look.
| Work | Scan URL | Holding and shelfmark | Rights | Source ID |
|---|---|---|---|---|
| Regiomontanus, De triangulis omnimodis, Nuremberg 1533 | https://archive.org/details/ARes352131 (PDF at https://archive.org/download/ARes352131/ARes352131.pdf) | Biblioteca de la Universidad de Sevilla | Public domain per Internet Archive | S301 |
| Copernicus and Rheticus, De lateribus et angulis triangulorum, Wittenberg 1542 | https://www.sbc.org.pl/dlibra/publication/440324/edition/412962/ (PDF: http://sbc.org.pl/Content/412962/PDF/ii223856-0000-00-0001.pdf; title page is image 5 of 68) | Biblioteka Śląska, Katowice, shelfmark 223856 II | "Domena publiczna" | S302, S349 |
| Pitiscus, Trigonometria, in Scultetus, Sphaericorum libri tres, Heidelberg 1595 | https://www.e-rara.ch/zut/content/zoom/265972 (full-resolution image: https://www.e-rara.ch/download/webcache/2000/265972) | e-rara; the holding institution for this copy was not displayed because the catalog page is behind a browser check | e-rara items are generally CC Public Domain Mark 1.0, not individually confirmed here | S303 |
| Scultetus, Sphaericorum libri tres with Pitiscus appended, 1595 | https://books.google.com/books?id=CZ8jx2-edGYC&printsec=frontcover | Google Books digitization, holding library not confirmed | Google Books full view | S354 |
| Pitiscus, Trigonometriae ... libri quinque, Augsburg 1600 | https://doi.org/10.3931/e-rara-4035 | ETH-Bibliothek Zürich, Rar 5201 | CC Public Domain Mark 1.0, stated in the IIIF manifest | S306 |
| Rheticus and Otho, Opus palatinum de triangulis, Neustadt 1596 | https://doi.org/10.3931/e-rara-9112 | ETH-Bibliothek Zürich, Rar 9660 (1501 canvases) | No license field in the IIIF manifest | S307 |
| Opus palatinum, another copy | http://digital.slub-dresden.de/werkansicht/dlf/12516/1/ | Staats- und Universitätsbibliothek Dresden | Not verified; the site is behind a bot check | S327 |
| Opus palatinum, manuscript of the main table | https://www.bl.uk/manuscripts/FullDisplay.aspx?ref=Harley_MS_1720 | British Library, Harley MS 1720 | "by permission of the British Library" per Roegel | S327 |
| Fincke, Geometriae rotundi libri XIIII, Basel 1583 | https://archive.org/details/den-kbd-pil-130018099382-001 | Det Kongelige Bibliotek, Copenhagen, LN 599 4to copy 1 | Images courtesy of the Royal Library, Copenhagen; interleaved pages carry Early European Books / ProQuest notices | S304 |
| Viète, Canon mathematicus seu ad triangula, Paris 1579 | https://doi.org/10.3931/e-rara-18548 | ETH-Bibliothek Zürich, Rar 3769 | No license field in the IIIF manifest | S308 |
| Viète, Canon mathematicus, another copy | https://gallica.bnf.fr/ark:/12148/bpt6k52673b.image | Bibliothèque nationale de France | Not verified | S329 |
| Napier, Mirifici logarithmorum canonis descriptio, Edinburgh 1614 | https://archive.org/details/mirificilogarit00napi | Smithsonian Libraries | Public domain per Internet Archive | S305 |
| Gunter, Canon triangulorum, London 1620 | https://archive.org/details/bim_early-english-books-1475-1640_canon-triangulorum-sive_gunter-edmund_1620 | Digitised from microfilm IA40312816-67, Early English Books 1475 to 1640 | Public domain per Internet Archive | S309 |
| Regiomontanus, Tabulae directionum et profectionum. Tabella sinus recti, Augsburg: Ratdolt, 2 January 1490 | https://dcollections.lib.keio.ac.jp/en/incunabula/024 | Keio University Libraries, shelfmark 120X@753@1 | "COPYRIGHT (C) KEIO UNIVERSITY ALL RIGHTS RESERVED" | S348 |
Three sources I wanted and could not get, listed so that nobody assumes they were consulted. First, Barnabas Hughes's 1967 edition and English translation, Regiomontanus on Triangles (University of Wisconsin Press). That is where the proposition counts for the five books would come from. Second, J. D. North's 1976 three-volume edition of Richard of Wallingford, which would date the Quadripartitum. Third, Daniel Otero's TRIUMPHS teaching module Regiomontanus and the Beginnings of Modern Trigonometry, the ideal student-facing primary-source unit. Both the MAA and Ursinus servers refused to serve it to an automated request. (S301, S319)
For the classroom
Printing as an information technology. De triangulis existed in manuscript from 1464 and influenced almost nobody. Printed in 1533, it was standard across Europe within a generation. (S310) Set the numbers side by side. Twelve years for Rheticus's computers to produce a table. Weeks for Petreius to print a book. Seventy years for a finished manuscript to find a printer at all. Ask students which of those three steps was the real bottleneck at each point in the century. Then ask what the modern equivalents are.
The economics of a research project. Rheticus spent 4,400 Gulden to 1568 on wages for arithmeticians, funded by an emperor. That is about fifty times his own annual salary at Wittenberg. (S327) Otho spent twenty-two years and went through three patrons. He lost one because he would not sign a religious formula. He ended up with four students paid partly in dinners at the professors' table. (S312) Have students write the grant application. Who pays for a table of numbers? What do they want in return? And whose name ends up on the cover? Otho's answer is in the title: Opus palatinum.
Latin as scientific lingua franca, and the politics of a title page. Every book in this chapter is in Latin. That is why they could all read each other within months: a Dane printing in Basel, a Silesian in Heidelberg, a Frenchman in Paris, a Scot in Edinburgh. The 1542 title page is a one-page case study in credit. Copernicus's name on the book. Rheticus's only over the covering letter. The new table claimed by nobody. And the dedication aimed at an instrument maker rather than a prince. (S302, S312) Compare it with a modern paper's author list and acknowledgements section.
Map projections and the longitude problem. Give students a globe, a Mercator map, and Wright's instruction to add up secants. Have them compute in one-degree steps from the equator to 60 degrees, then in ten-minute steps, and compare the two against the exact value 4527.37 minutes. [own computation] They will rediscover both the method and the reason Wright's 1610 second edition went to one-minute intervals. Then ask the harder question. Latitude comes free from the sun and the pole star. Longitude needs an accurate clock or an astronomical event. That is why the meridional-parts table solved only half the navigator's problem.
Language and etymology. Take the six function names apart in a language class. Sinus is a Latin bay, standing in for an Arabic fold, standing in for a Sanskrit bowstring. Tangens and secans are Latin present participles, touching and cutting, coined by a twenty-two-year-old who apologized for them in the same sentence. Trigonometria is two Greek words with a Latin ending. The "co-" in cosine, cotangent and cosecant is a clipping of complementi. Every word on the worksheet has a passport.
Cryptography and codebreaking. Viète was Henry IV's codebreaker as well as the man who made algebra symbolic. He cracked van Roomen's forty-fifth-degree challenge by spotting that . (S316) The skill is the same in both jobs: seeing the structure hidden in something that looks like brute force.
- Latin Europe imported the subject through the twelfth-century translation wave; the words arrived before the mathematics was understood.
- Peuerbach and Regiomontanus rebuilt it; Rheticus redefined the functions as right-triangle ratios in 1551 (S312, S330), and his team's Opus Palatinum showed what a table cost.
- Pitiscus put trigonometria on a title page in 1595, and the subject finally had its name (S303).
Next: the sine leaves the table altogether and becomes a function you can differentiate, sum and wave.
The reciprocal functions carry this chapter's fingerprints: secant, cosecant and cotangent are print-era coinages, and their tangled attributions are chapter 8's business. Meet them as ratios in Trigonometry 2.5
In 1595 a book title finally named the subject you are taking. Who wrote it, and what was the word?
Show the answer▸
Bartholomaeus Pitiscus: trigonometria, triangle measuring (S303). The subject is two millennia older than its name, which is this whole chapter in one fact: in this story the mathematics arrives first and the words limp in afterwards.
Chapter 7
From Table to Function to Wave (1660 to today)
The people in this chapter
Faces where a face survives. Every name links to its full entry in Appendix A.
The poster on the wall
Look at the unit circle poster in your classroom. A circle of radius 1, angles in radians, and at each marked angle a pair of coordinates that are the cosine and the sine.
Every single thing on that poster is a decision somebody made. The radius is 1 because a Swiss mathematician in Berlin decided in the 1740s that it should be. The angles are in radians because of an examination paper set in Belfast in 1873. "sin" is a Latin abbreviation from 1583. And the idea that is a function, a machine you feed a number and it hands you back a number, is barely 280 years old.
Before that, a sine was a length. Not a ratio, not a function: a segment inside a circle, measured in whatever units the table-maker chose. Ptolemy used a circle of radius 60. Later table-makers used a radius of so their sines came out as whole numbers. Ask a working astronomer in 1650 for "the sine of thirty degrees" and the honest answer was "in which table?"
The problem was not elegance. It was arithmetic. Astronomers, navigators and surveyors needed sines to many decimal places, and computing them from geometry by hand was brutal. The classical route was Ptolemy's. Build a chord table from a handful of exactly known chords. Then halve and add angles with geometric identities, grinding out a square root at every step. It works, it is slow, and it does not scale. If you want the sine of an angle that is not reachable from your seed values by halving and adding, you interpolate and you accept the error.
Between 1660 and 1750, mathematicians found a way to get a sine by adding up a series of numbers. That untied the sine from any particular circle, because a series has no radius in it. Once that happened, somebody was going to set the radius to 1, because 1 is the number that vanishes when you multiply by it.
Then a second thing happened, and it is the bigger one. Once sine was a function of a number, you could ask what you could build out of sines. Joseph Fourier (zho-ZEF FOOR-yay, 1768 to 1830) answered: everything. Any curve you can draw. He made that claim to a hostile committee in Paris in 1807 and printed it in 1822. It is why trigonometry is inside your phone, your hospital's CT scanner, and the detector that recorded two black holes colliding a billion light years away.
This chapter is the story of those two moves, and of the people who made them, including several who were right early and got nothing for it.
Newton, Leibniz, Euler, Fourier. One of them is the reason your calculator's sin key takes a plain number, no triangle and no circle anywhere in sight. Who made the sine a function of a pure number, and roughly when?
I have a guess▸
Euler, in 1748. The Introductio in analysin infinitorum (S361) treats the sine and cosine as functions of a numerical argument, sitting in a book written to come BEFORE the calculus. Newton had the series eighty years earlier, but a series for a geometric line is still geometry; Euler's chapter is where the quantity itself changes kind.
If you guessed Newton with the calculus, you guessed the standard shorthand, and the gap between the series (1660s) and the function (1748) is the real story of this chapter.
Newton turns the sine into a series
Isaac Newton (EYE-zik NEW-tun, 1642 to 1727) wrote a short treatise called De Analysi per aequationes numero terminorum infinitas (On Analysis by Equations with an Infinite Number of Terms). It circulated in manuscript from 1669, passed hand to hand around a small London and Cambridge circle. Nobody printed it until 1711, forty-two years later. That gap is normal for Newton, and it is one reason priority arguments follow him everywhere.
The version quoted here is John Stewart's English translation of 1745. Stewart published it in London as Sir Isaac Newton's Two Treatises of the Quadrature of Curves, and Analysis by Equations of an Infinite Number of Terms, Explained (S365). A caution about the dating. The mathematics below was read directly in Stewart's translation. The standard dates, written 1669 and printed 1711, were not confirmed from any source read for this book. Treat them as the received account, not as something checked here.
Newton works backwards on purpose
Newton's route is the reverse of what you would expect. He does not compute the sine first. He computes the arc from the sine, which is what we call the arcsine, and then turns the series inside out.
The arcsine series is the easy one, and that is the reason. Arc length is an integral. Newton could expand the thing inside that integral with his binomial theorem, then integrate term by term. The sine series is not available that way. So he gets the hard direction by inverting the easy one.
Here is Article 45, page 338, under the heading "To find the Base from the Length of the Curve given" (S365).
"If from the Arch aD given the Sine AB was required; I extract the Root of the Equation found above, viz. (it being supposed that , and ) by which I find ."
Read the parenthesis slowly, because it is doing more work than the algebra around it. is the sine. is the arc. And : Newton sets his radius to 1.
That is thirty years before Leonhard Euler made it a rule for everybody. Newton is not announcing a convention. He is doing what any calculator does with a nuisance constant. He sets it to the value that makes it disappear, then moves on. The convention that governs your textbook started life as somebody's convenience.
What Newton calls "extracting the Root" is what we call series reversion. You have an infinite series that gives in terms of . You want the one that gives in terms of . There is no formula for it. You solve for the coefficients one at a time, and each one uses all the ones before it.
What Newton says about his own method
Article 44, the page before, is the more interesting one. Newton stops there and explains why he printed the working out at all (S365).
"I have laid the Steps of the Resolution before you, as you see, upon the Account of the two following Remarks. 1. That in the Substitution, I always omit those Terms, which I foresee will be of no Use afterwards."
Above that sentence sits the full long-division tableau, the scratch work itself. Newton is teaching a technique, not announcing a result, and the technique is the part that matters historically. Series reversion is a general tool. It is how you get an inverse function's series from the function's series. It works on any invertible power series, a power series being an endless sum of whole-number powers of the variable.
His remark about omitting terms is the first rule of practical series work. Say it to a class out loud. You decide in advance how many terms of the answer you want. Then you refuse to carry anything that cannot affect them. That is not laziness. Without it, the algebra grows faster than the accuracy.
Worked example: do Newton's reversion yourself
Start with his arcsine series, with the sine and the arc. Assume the inverse has only odd powers, which it must, because sine is an odd function:
You need the powers of only to fifth order, because anything higher cannot affect or :
Substitute those into the first series and collect powers of :
The left side is exactly , so every bracket on the right must be zero. From the bracket:
Feed that into the bracket:
Exactly what Newton printed. Notice how the second step used the first step's answer. That dependence is the whole character of reversion, and it is why nobody writes a closed formula for it.
Modern check on his coefficients: , , , so
which is the series you know. Test it numerically at radians:
Your calculator gives . Three terms, five correct digits, and the fourth term would fix the sixth. This is why series beat geometry for table-making: you can decide the accuracy in advance and buy it with more arithmetic.
The cosine, and the recipe
In Article 46 Newton squeezes the cosine out of the Pythagorean relation, since with radius 1 the cosine is (S365):
"And moreover if the Cosine Ab were required from that Arch given, make "
And in Article 47 he gives the pattern, so that no one need repeat the algebra ever again. For the sine, continue the series by dividing successive terms by , , , , ; for the cosine, by , , , , (S365). Check it against the sine series: and , and . It works.
That is the moment the sine stops being a geometrical object and becomes a computational recipe. Article 47 is a set of instructions a clerk with no geometry at all can follow. A hundred years later, clerks in Paris did exactly that, at government expense, and we will meet them.
One caveat on the text. The 1745 printing survives as a scan, and its optical character recognition is imperfect. The coefficients quoted above come from that OCR. They agree with the standard series, which is a good check but not a substitute for the physical page (S365).
Roger Cotes gets there first, and dies at 33
Roger Cotes (ROJ-er KOHTS, 1682 to 1716) is the person this chapter most wants you to remember, partly because almost nobody does.
Cambridge made him its first Plumian Professor of Astronomy and Experimental Philosophy at 24. He edited the second edition of Newton's Principia, which meant arguing with Newton about physics for years and writing a famous preface. In his whole life he published one paper: Logometria (roughly, the measurement of ratios), in the Philosophical Transactions for March 1714 (S367). Two years later he was dead at 33, of a violent fever. Newton's remark about him has stuck: "if he had lived we might have known something."
In 1722 his cousin Robert Smith gathered his surviving papers into a posthumous book: Harmonia Mensurarum (The Harmony of Measures), printed at Cambridge. Logometria is Part I (S366). On page 28 of that part, buried inside a discussion of the surface area of a solid of revolution, Cotes writes this. The Latin comes from the Bayerische Staatsbibliothek scan of the 1722 printing. Obvious optical-character-recognition slips are restored in square brackets (S366).
"Nam si quadrantis circuli quilibet arcus, radio C[E] descriptus, sinum habeat CX sinumque complementi ad quadrantem X[E]; sumendo radium C[E] pro Modulo, arcus erit rationis inter [E]X + XC et CE mensura ducta in ."
In plain English: take any arc of a quarter circle drawn with radius , having sine and cosine . Take the radius as your modulus, meaning your unit. Then the arc equals times the measure of the ratio of to . "Measure of the ratio" is Cotes's phrase for what we call a logarithm, which is what the word Logometria is about. Set the radius to 1 and rewrite in modern symbols:
That is Euler's formula, upside down, twenty-six years before Euler printed his version. MacTutor states the same result independently, giving it as (S367).
Then Cotes does something human. Here is his next sentence (S366).
"Verum isthaec aliis, quibus opere pretium videbitur, diligentius excutienda relinquo."
In English: I leave these matters to be examined more carefully by others who think it worth the trouble. He had one of the most important results in mathematics on the page. He shrugged, and moved on to the next problem.
An honest gap about the date
That passage is on page 28 of the 1722 book beyond any doubt, because the page was read. The table of contents of the same scan backs it up. Part I "continet LOGOMETRIAM" starting at page 1, and Part II begins at page 43, so page 28 sits inside Logometria (S366).
What could not be confirmed is whether the same sentence stood in the 1714 Philosophical Transactions printing. The Internet Archive scan of the 1714 paper has no searchable text layer. The Royal Society and JSTOR copies were unreachable. MacTutor attributes the result to Cotes without saying which printing (S367).
This is not a fussy detail. It changes the number in the sentence everyone wants to write. If the passage is 1714, Cotes beat Euler by 34 years. If it is only 1722, by 26. Do not state either number without checking those specific pages (S366).
Why the form matters, not just the priority
There is a real mathematical difference between what Cotes wrote and what Euler wrote. It is worth more than the priority squabble.
Cotes wrote the relation in logarithmic form: the arc is times a logarithm. Euler wrote it in exponential form: equals the cosine plus times the sine. Those are the same statement algebraically, but they are not the same statement psychologically.
The exponential form makes the periodicity obvious. Add to and comes back to where it started, because sine and cosine do. From there you can see at once that the logarithm has to be many-valued. A function that returns the same output for infinitely many inputs cannot have a single-valued inverse. That is the exact point where the great argument about logarithms of negative numbers had jammed for thirty years. We will come to it.
There is a poignant footnote. Euler's own unpublished 1747 paper Sur les logarithmes proved the central theorem using exactly Cotes's logarithmic form, . It sat in a drawer for 115 years. Bal, reporting Cajori, records the judgment that "had this article been published shortly after it had been written, the controversy might not have lasted well into the nineteenth century" (S383). Cotes shrugged in 1722 and Euler filed in 1747, and between them the question stayed open for a century longer than it needed to.
What de Moivre wrote, and what he did not
Abraham de Moivre (AY-bruh-ham duh MWAHV-ruh, many English speakers say duh MOY-ver, 1667 to 1754) was born at Vitry-le-François in Champagne on 26 May 1667. He died in London on 27 November 1754. A French Protestant, he left for England after the revocation of the Edict of Nantes. Newton's support and a Royal Society fellowship never won him a university post. He earned his living tutoring the sons of rich families, and computing odds for gamblers and insurers in a London coffee house (S370).
The formula with his name on it is
He never wrote it. Not in that form, not in any form close to it, not anywhere in his published work.
The 1707 paper
In 1707 he published a Latin paper in the Philosophical Transactions, volume 25, issue 309, pages 2368 to 2371. Its title begins Aequationum quarundam potestatis tertiae, quintae, septimae, nonae, et superiorum ... resolutio analytica: the analytic solution of certain equations of the third, fifth, seventh, ninth and higher powers. Richard Pulskamp's English translation was read in full (S368).
The entire content is a set of closed-form roots for a family of odd-degree equations. Given
he gives
together with three variants of the same shape (S368).
If you already know that expands in odd powers of with those exact coefficients, you can see at once what this is. It is the angle-division problem in disguise, solved with -th roots of a complex-looking expression. De Moivre never says so. There is no trigonometry in the paper at all. The result is stated algebraically and left algebraic.
The 1722 paper
In 1722 he came back to it in a three-page paper, "De sectione anguli" (On the Section of an Angle), Philosophical Transactions volume 32, issue 374, pages 228 to 230, again read in Pulskamp's translation (S369). He opens by recalling the 1707 work, then states his theorem. This is the whole of it.
"Let be the Versine of any Arc. the Versine of another Arc. 1 the Radius of the Circle. And let the first Arc to the latter be as 1 to , Then, with the two equations assumed which it is permitted to call cognate, and . And by expunging , the Equation will arise by which the Relation between and is determined."
The versine of an arc is minus its cosine. It is an old table-maker's quantity, and it survived in navigation tables into the twentieth century because it avoids the small-difference problem near zero. "Cognate" is de Moivre's own word for the pair, and "expunging " means eliminating between them to get a single polynomial relation between the two versines.
That is the whole thing. Two quadratic-looking equations in and , and an instruction to eliminate.
The translator's note that unpacks it
Pulskamp's translator's note 3 to the 1722 paper does the work de Moivre did not (S369):
"Here Moivre's famous result is implicit. Hence put and . Solving the first equation for and the second for we have respectively and . The result then follows immediately."
Follow it through. Take the second cognate equation with :
The first cognate equation is the same quadratic with replaced by and by , so by identical algebra . Raise the first result to the -th power and set it equal to the second, and you have the formula.
So it is there. It is one substitution and one quadratic away. And de Moivre did not take the step. He was not thinking about complex numbers, numbers with a plain part and a part built on , as points on a circle. He was thinking about eliminating a variable between two polynomials.
The modern verdict
The reference literature agrees. Wikipedia's article on de Moivre's formula says flatly that "the formula is named after Abraham de Moivre, although he never stated it in his works" (S395). MacTutor says the formula "appeared in a 1722 publication, though a closely related formula had appeared in an earlier paper which de Moivre published in 1707" (S370). Both are encyclopedic sources rather than primary ones, but both agree with the primary texts above.
The modern statement, in modern notation, is Euler's, in Introductio Book I section 133, printed in 1748. That is the earliest statement in modern form located and read in a primary text for this chapter. No systematic priority search was done, so read "earliest" as "earliest found here" (S362).
This is a lesson in how mathematics gets its names, and it is worth saying to students plainly. A result attaches to whoever the community happened to be reading when the result became useful, not to whoever first had it. Cotes had the logarithm before Euler and got nothing. De Moivre had the machinery before Euler and got a formula named after him that he never wrote. Euler wrote both of them down clearly, in a textbook everybody read, and got the credit for one and shares it on the other.
Taylor and Maclaurin: the general machine
Newton got the sine series by reversion, which is clever and specific. Within fifty years there was a general method that produces it, and every other series like it, without cleverness.
Brook Taylor (BRUUK TAY-lor, 1685 to 1731) was born at Edmonton in Middlesex on 18 August 1685 and dead at 46. He published Methodus Incrementorum Directa et Inversa (Direct and Inverse Methods of Increments) in London in 1715. It contains what we call the Taylor series. In fact it contains it twice, in two versions: as Proposition 11, and as Corollary 2 to Proposition 7 (S388).
In modern notation the claim is that a well-behaved function can be rebuilt near a point from its value and all its derivatives at that point:
That single line contains the sine series, the cosine series, the exponential series and the logarithm series as special cases. It is the general machine.
Nobody noticed for fifty-seven years
Here is the part worth telling a class. MacTutor records that the importance of Taylor's theorem "was unrecognised until 1772 when Lagrange proclaimed it the basic principle of the differential calculus". The name "Taylor series" only appears in 1785 (S388).
So the sequence runs like this. Printed in 1715. Ignored for fifty-seven years. Promoted in 1772 by Joseph-Louis Lagrange (zho-ZEF loo-EE luh-GRAHNZH, 1736 to 1813) to the status of founding principle of the calculus. Named after its author in 1785, seventy years after he wrote it, when Taylor had been dead for fifty-four years.
That is not unusual and it is not a scandal. It is what happens when a result arrives before the community has a use for it. Taylor's book is dense and, by every account, poorly explained. He wrote it in a fluxional notation that made it hard going even for his contemporaries. The result needed somebody with Lagrange's platform to say out loud that it mattered.
Colin Maclaurin (KOL-in muh-KLOR-in, born February 1698, day not recorded, died 14 June 1746) published his Treatise of Fluxions at Edinburgh in 1742. It gives the expansion about zero, the case . Maclaurin acknowledged Taylor's earlier general result in so many words (S387). The special case carries Maclaurin's name and the general case carries Taylor's. For once both attributions are roughly fair, because Maclaurin said where he got it.
Worked example: the sine series from the general machine
Take and expand about . You need the derivatives at zero, and they cycle with period four:
Feed those into the general formula with and :
Newton's answer, in four lines, with no reversion and no ingenuity. The even-power terms vanish because sine is odd, and the alternating signs come from the derivative cycle. Compare the effort with the reversion above and you can see why the general machine won.
The historical order, though, is worth keeping straight, because it is the opposite of the teaching order. Newton had the sine series in 1669 by reversion. Taylor had the general theorem in 1715. Euler, in 1748, derived the sine series by yet a third route that used neither, as the next section shows. Three independent roads to the same series inside eighty years, which is usually a sign that a result is ready to be found.
Euler, 1748: the chapter everything turns on
Leonhard Euler (LAY-on-hart OY-ler, born Basel 15 April 1707, died St Petersburg 18 September 1783) wrote Introductio in analysin infinitorum (Introduction to the Analysis of the Infinite) around 1745, while working for Frederick the Great at the Berlin Academy. He published it at Lausanne in 1748 with the printer Marcum-Michaelem Bousquet (S361). The Euler Archive record gives "written in 1745"; the title page says 1748.
It is a two-volume textbook with no calculus in it. Euler's plan was to write the book a student should read before the calculus, establishing what functions are and how they behave. Book I, Chapter VIII is De quantitatibus transcendentibus ex circulo ortis, "On transcending quantities arising from the circle". It runs from section 126 to section 142. Transcending, or transcendental, means beyond the reach of ordinary algebra: you cannot build a sine out of whole numbers by adding, multiplying and taking roots. Almost everything on your classroom poster is inside those seventeen sections.
Quotations below come from Ian Bruce's English translation. It prints the 1748 Latin alongside the English in the same document, so both were checked (S362).
Section 126: the radius is 1, and pi gets a symbol
Euler settles the radius question in one sentence, with no argument and no fanfare.
"Therefore we may put the radius of the circle or the whole sine to be and it is clear enough that the periphery of this circle cannot be expressed exactly in rational numbers." (Introductio I, §126)
The Latin: "Ponamus ergo radium circuli seu sinum totum esse = 1 atque satis liquet peripheriam huius circuli in numeris rationalibus exacte exprimi non posse" (S362).
Note the phrase "radium circuli seu sinum totum", the radius of the circle or the whole sine. The sinus totus, the whole sine, is the old table-maker's name for the radius, the sine of a right angle. Euler is using the old vocabulary in the sentence that kills the old practice.
He then prints the semicircumference to 127 decimal places, beginning
"3, 14159 26535 89793 23846 26433 83279 50288 41971 69399 37510 58209 74944 59230 78164 06286 20899 86280 34825 3421170679 82148 08651 32823 06647 09384 46"
and adds:
"for which number for the sake of brevity I may write as , thus so that there shall be semi-circumference of the circle, whose radius , or will be the length of the arc of 180 degrees." (Introductio I, §126)
The Latin: "pro quo numero brevitatis ergo scribam , ita ut sit = semicircumferentiae circuli, cuius radius = 1, seu erit longitudo arcus 180 graduum" (S362).
Be careful what this proves. It shows Euler defining and using this way in the most widely read mathematics textbook of the century. That is why the symbol stuck. It does not show that he invented the symbol, and the priority question was not investigated for this chapter (S362). A separate line of research in this project traces the symbol to William Jones in 1706. Cajori records that Euler used it in the 1736 Mechanica, and that Euler either took the notation from Jones or hit on it himself. What made it universal was the 1748 Introductio (S423). Two other points from that same research are worth carrying. Jones had already used the letter twice in the same 1706 book with two different meanings, once as a point label and once for "Periphery". And adoption was slow: Diderot in 1748 still wrote the ratio as a fraction, and Segner in 1767 was still using Oughtred's older pi-over-delta (S423).
Section 127: sine becomes a function of a number
This is the sentence the whole chapter turns on.
"With denoting the arc of some circle, the radius of which I assume always , the sines and cosines of this arc mainly are considered. But the sine of the arc in the following I will indicate in this manner sin. A. z or only sin. z, truly the cosine in this manner cos. A. z or only cos. z." (Introductio I, §127)
The Latin is short, readable, and worth putting in front of students: "Sinum autem arcus z in posterum hoc modo indicabo sin. A. z seu tantum sin. z, cosinum vero hoc modo cos. A. z seu tantum cos. z" (S362). The "cuius radium perpetuo assumo = 1" in the same section is the phrase that fixes the unit circle for good.
He continues in the same section.
"Besides these denominations, these also are well-known: tang. z, which denotes the tangent of the arc , cot. z the cotangent of the arc , and agreed to be and , which all are well-known from trigonometry." (S362)
Read that with a modern eye and notice what is missing. There is no triangle. There is no chosen radius. There are no units. is a number. is a number. What sits between them is a function in the modern sense: feed it a number, get a number back. Before this, a sine was a length you measured off a drawing. From here on, a sine is a number you get from a number.
He then tabulates the values everyone memorizes:
and concludes: "Therefore all the sines and cosines will be contained within the limits and " (S362).
That is your poster. Not a description of your poster: the thing itself, in 1748.
Section 128: the identities, and negative whole numbers
Section 128 gives the addition formulas. Then, "Therefore if may denote some whole number", it tabulates the reduction formulas for expressions like . Those are the ones that let you fold any angle back into the first revolution. Euler adds a remark that is easy to skim past and should not be: "Which formulas are true, whether shall be a positive or negative integer" (S362).
Negative . A sine of a negative multiple of plus . There is no such arc in a physical circle drawn on paper, and there is no such triangle. The formula is true anyway, because the function does not know it came from geometry. That single clause is the difference between a table of lines and a function of a real variable.
Sections 132 and 133: de Moivre's formula, in modern form
Section 132 factors the Pythagorean identity over the complex numbers:
and Euler remarks, in Bruce's translation, that these are "factors, even if imaginary, still perform a huge task in the combinations and in the multiplications of arcs" (S362). He then multiplies two of them together, applies the angle-addition formulas from section 128, and gets
Multiplying two of these complex numbers adds the arcs. That is the whole content of the formula, and it is one line of algebra once you have the addition identities. Section 133 iterates it.
"Hence therefore it follows: , , and thus generally there will be ." (Introductio I, §133)
He then inverts the pair. That gives and as the half-sum and half-difference of the two conjugate -th powers, and he expands both binomially (S362). That inversion is what the next section runs on.
Section 134: the series fall out of an infinity
Now Euler does something a modern analyst would flinch at, and gets exactly the right answer.
"Let the arc be infinitely small; there will be and ; moreover shall be an infinitely great number, so that the arc shall be of finite magnitude, for example ." (Introductio I, §134)
Here is the derivation written out. Start from section 133 and its conjugate, and take the half-sum:
because in the binomial expansion the odd powers of cancel between the two terms and the even ones double. Now apply Euler's substitutions: , , and infinite, so that with factors becomes simply :
Every cancels. That cancellation is the trick. Here is infinite and is infinitesimal, and ties them together, so the product survives while neither factor does.
The half-difference does the same job for the sine:
Euler prints them in the fully written-out form:
That is the same series Newton reversed his way to in 1669, and the same one Taylor's machine turns out in four lines. Euler reaches it by a third road. And now it belongs to a function of a bare number, not to a length in somebody's circle (S362).
Then Euler does the thing that reminds you what century he is in. He puts the series straight to work computing term by term, printing the coefficients to 29 decimal places (S362). He has just made sine an abstract function of a real variable, and his first instinct is to go and build a table with it. In 1748 the greatest analyst alive is still, in part, a table-maker.
Section 138: Euler's formula, and a trap
Section 138 repeats the infinitesimal trick, this time against the exponential.
"138. An infinitely small arc may be put anew in the formulas §133 and shall be an infinitely large number , so that may maintain a finite value . Therefore there will be and , from which and ." (Introductio I, §138)
With those substitutions the half-sum and half-difference become
and then he cashes in a result from the previous chapter:
"But in the preceding chapter we have seen that with denoting the base of hyperbolic logarithms; therefore for in one part I write , and for the other part " (Introductio I, §138)
giving
and then, in his own words: "From which it is understood, how imaginary exponential quantities may be reduced to the sine and cosine of real arcs. Truly there will be
" (Introductio I, §138). The Latin of that final step, from the same page: "Ex quibus intelligitur, quomodo quantitates exponentiales imaginariae ad sinus et cosinus arcuum realium reducantur. Erit vero et " (S362).
The last step is one line of algebra, and it is worth writing out because students rarely see it:
Now the trap, and it is a good one. Look again at the letter in .
It does not mean .
Euler introduces in the first sentence of section 138 as "an infinitely large number ". Throughout this chapter he writes out longhand, every single time, and never abbreviates it (S362). The modern use of for the square root of minus one came later. MacTutor dates Euler's own adoption of that convention to 1777 (S394).
So the most famous formula in mathematics was first printed in a passage where the letter means infinity. A modern reader who skims section 138 will read as something involving the imaginary unit, and will be lost. Anyone quoting this passage should say so out loud. Any claim that "Euler's in section 138 means " is flatly false.
One more thing is worth flagging. This derivation runs through actual infinities and actual infinitesimals. Euler treats as a definite infinitely large number and as a definite infinitely small one, then multiplies them to get something finite. It is not rigorous by modern standards, and Euler did not claim it was. The modern proof compares power series term by term inside their radius of convergence, the stretch of inputs over which the terms shrink fast enough for the sum to settle on a value. That proof is a nineteenth-century rebuild of an eighteenth-century result (S362).
Check the formula at using the series from section 134:
Euler never writes anywhere in Chapter VIII. That arrangement, with the five constants lined up, is later packaging by admirers.
Sections 139 and 140: the loop closes on Cotes
Two sections later Euler runs the argument backwards to the logarithmic form. Section 139 ends like this.
"and thus , from which it is apparent, in what way imaginary logarithms relate back to circular arcs." (Introductio I, §139)
Euler's is the logarithm. Section 140 rewrites the same thing using the tangent, which is more useful in practice because the tangent is what a surveyor measures:
(S362). He has arrived at Cotes's sentence from the other side. Cotes started from the logarithm and got the arc; Euler started from the exponential and got the logarithm back. Twenty-six years apart, and there is no evidence in Chapter VIII that Euler is answering Cotes.
Euler and the logarithms of negative numbers
Behind sections 138 to 140 sits a real argument that had run for thirty-five years. The story of it is usually told wrong in three separate places. Here it is with the corrections attached.
What the argument was
What is ?
Johann Bernoulli (YO-hahn ber-NOO-lee, 1667 to 1748, with the birth year not independently checked here) held that . His argument was clean and wrong. Since , taking logarithms gives , so . Leibniz disagreed and held that logarithms of negative numbers do not exist as real quantities at all.
They argued it out in a friendly correspondence in 1712 to 1713, and then it sat. Bal's account: "In 1712, Gottfried Leibniz and John Bernoulli I engaged in a friendly correspondence concerning the logarithms of negative numbers. The publication of this correspondence in 1745 sparked an interest in the mathematical community on the topic" (S383).
That 1745 publication is the trigger. The dispute was private for thirty-two years and became a public problem only when the letters were printed. Nothing about the mathematics changed in 1745. What changed is that everybody could suddenly read two great mathematicians disagreeing in writing.
Correction one: the Bernoulli exchange was 1727, not the 1740s
You will read that Euler argued with Johann Bernoulli about this in the 1740s. He did not, or at least, that is not where his exchange with Bernoulli happened.
Bal: "When Euler was just 20 years old, he corresponded by letter with his teacher John Bernoulli on the subject. In this correspondence describes his arguments both for Bernoulli's position and against it. Bernoulli still held the same view that " (S383). Euler was born in 1707, so that puts the exchange around 1727, when Euler was leaving Basel for St Petersburg.
His main 1740s correspondent on the question was somebody else entirely. Bal again: "Between 1745 and 1749, Euler both corresponded with other mathematicians and wrote paper on the nature of logarithms and negative numbers. His main correspondence on the topic was with the Frenchman, Jean le Rond D'Alembert (1717-1783). D'Alembert held the view of John Bernoulli and Euler tried to convince him otherwise" (S383).
So the shape is this. A private argument with his own teacher at twenty. A twenty-year gap. Then a public argument with d'Alembert in his late thirties, running alongside the writing of the Introductio.
Confidence note. Bal is an expository paper from Montclair State University, not a peer-reviewed article. It does quote and cite Cajori's three-part 1913 history in the American Mathematical Monthly throughout. Treat the correction as well supported rather than as settled (S383).
Euler's answer
Euler's resolution is exactly the tool in section 138. Once you know that
you know that is periodic, because cosine and sine are. Adding to returns you to the same value. So the exponential is not one-to-one on the complex numbers, and therefore its inverse cannot be single-valued. The logarithm of any number, negative or not, is not a number. It is an infinite family of numbers, differing by multiples of .
In particular, from :
Bernoulli's proof fails at the step where he divides by 2. His only tells you the two families overlap after doubling, not that they are the same family. And Leibniz was closer to right than Bernoulli: there is no real logarithm of a negative number. There are infinitely many imaginary ones.
Correction two: which papers, and when
Three Euler papers matter here, and the dates get mangled routinely.
E168, De la controverse entre Mrs. Leibnitz et Bernoulli sur les logarithmes des nombres negatifs et imaginaires (On the controversy between Messrs Leibniz and Bernoulli concerning the logarithms of negative and imaginary numbers). The Euler Archive record gives: published 1751 in Memoires de l'academie des sciences de Berlin, volume 5, pages 139 to 179, written 1747 (S384). Bal, however, says the paper was "written in 1749 and published in 1751" (S383). Both dates are on record, from a repository record and from a secondary account, and this chapter does not resolve them.
E170, Recherches sur les racines imaginaires des equations (Research on the imaginary roots of equations). The Euler Archive record reads: "Published: 1751 in Memoires de l'academie des sciences de Berlin, Volume 5, pages 222-288. Written: 1746. Language: French. Enestrom Number: 170" (S385).
That last one carries a correction worth stating plainly, because the wrong date circulates: E170 is not a 1748 paper. It was written in 1746 and published in 1751.
And the one that never appeared. Euler also wrote a paper in 1747 titled Sur les logarithmes. It proves the same theorem using the relation , which is Cotes's form. Nobody published it until 1862, one hundred and fifteen years later (S383).
Bal, reporting Cajori's assessment, records that "had this article been published shortly after it had been written, the controversy might not have lasted well into the nineteenth century" (S383). A correct proof of a disputed theorem sat unread for longer than the dispute itself lasted, and the dispute outlived Euler because of it.
Fourier: from a function to every wave
Once sine is a function you can ask a question nobody could ask about a length: what can you build out of sines?
The vibrating string, and an honest warning
The prehistory is a fight about a guitar string, and this chapter cannot tell it properly.
In the 1740s and 1750s three men argued about what shapes a plucked string could take: Jean le Rond d'Alembert (ZHAHN luh ROHN dah-lom-BAIR), Euler and Daniel Bernoulli (DAN-yel ber-NOO-lee, 1700 to 1782, dates not independently verified here). Bernoulli set out his position in a 1753 paper in the Berlin memoirs, titled Reflexions et eclaircissemens sur les nouvelles vibrations des cordes. The general motion of a vibrating string, he said, is a superposition of sine modes: the fundamental, plus the octave, plus the twelfth, each with its own amplitude, all added together. Those higher modes are the harmonics, the whole-number multiples of the lowest frequency. D'Alembert and Euler thought that could not capture an arbitrary starting shape, such as the sharp corner a plectrum leaves in a string.
Flag, and it is a large one. Daniel Bernoulli's 1753 paper was not read for this chapter. A catalog record for it was located, and the fetch was blocked by a robots restriction. Its exact volume, page range and wording are all unverified. The standard account of the d'Alembert, Euler and Bernoulli controversy was not rebuilt from any primary source. The paragraph above is the received account, reported as received.
Say this plainly to any teacher using the chapter: this one section rests on less evidence than the others. Everything in the Newton, Cotes, de Moivre, Euler and Fourier sections was read in a primary text or a full translation. The vibrating-string controversy was not. It is a real gap in the research behind this book. It is here because leaving it out would make the Fourier story unintelligible, not because it has been checked.
1807: Fourier gets rejected
Fourier ran a Napoleonic province. He was Prefect of the Isère at Grenoble, an administrator with a real day job. He studied the flow of heat in solids in whatever hours were left.
On 21 December 1807 he presented a memoir, "On the propagation of heat in solid bodies", to the Paris Institute. The referees were Lagrange, Pierre-Simon Laplace (pee-AIR see-MOHN luh-PLAHSS, 1749 to 1827), Monge and Lacroix (S372).
Lagrange and Laplace objected. Not to the physics, which was new and correct, but to the trigonometric series expansions Fourier used to solve the heat equation. This is not a coincidence of taste. Lagrange had been formed by the vibrating-string argument fifty years earlier, on the side that said a sum of sines cannot represent an arbitrary curve. He had not changed his mind. Fourier's memoir went unpublished.
1811: the prize, and the sentence that stung
The Institute then set heat as the subject of its 1811 mathematics prize. Fourier resubmitted, expanded. The committee was Lagrange, Laplace, Malus, Haüy and Legendre.
They gave him the prize. They also attached a judgment to it (S372):
"the manner in which the author arrives at these equations is not exempt of difficulties"
Read the situation. Two of the men on that committee had rejected the memoir in 1807. They handed its author the Institute's prize, wrote in the citation that his derivations were not sound, and still would not print the work. Fourier had the prize money and no publication.
Confidence note. This quotation comes from MacTutor's biography, which quotes the committee report. The report itself was not fetched for this chapter (S372).
1822: article 235
He published it himself, eleven years later, as Théorie analytique de la chaleur (The Analytical Theory of Heat), printed in Paris by Firmin Didot in 1822 (S371). By then he was Permanent Secretary of the Academy of Sciences, which helps.
Chapter III, Section VI is titled "Développement d'une fonction arbitraire en séries trigonométriques", the expansion of an arbitrary function in trigonometric series. It runs from article 207 to article 235. Article 235, on page 258 of that printing, is the payoff.
"Il résulte de tout ce qui a été démontré dans cette section, concernant le développement des fonctions en séries trigonométriques, que si l'on propose une fonction , dont la valeur est représentée dans un intervalle déterminé, depuis jusqu'à , par l'ordonnée d'une ligne courbe tracée arbitrairement, on pourra toujours développer cette fonction en une série qui ne contiendra que les sinus, ou les cosinus, ou les sinus et cosinus des arcs multiples, ou les seuls cosinus des multiples impairs." (Théorie analytique de la chaleur, art. 235)
In plain English. Everything proved in this section about expanding functions in trigonometric series leads to one conclusion. Take a function whose value on a fixed interval, from to , is given by the ordinate, the height, of a curve drawn any way you like. You can always expand that function in a series containing only sines, or only cosines, or sines and cosines of multiple arcs, or only cosines of odd multiples (S371).
Take the middle of that sentence at face value: une ligne courbe tracée arbitrairement, an arbitrarily drawn curve. Not a curve with an equation. A curve you drew. It can have corners. It can be a staircase. It need not obey any rule at all. Fourier claims you can always write it as a sum of sines and cosines of whole-number multiples of the angle. That sum is what we now call a Fourier series.
He knew exactly how outrageous that was, and he said so one article earlier, in article 234.
"et quelle que puisse être la courbe donnée qui répond à , soit qu'on puisse lui assigner une équation analytique, soit qu'elle ne dépende d'aucune loi régulière, il est évident qu'elle servira toujours à réduire d'une manière quelconque la courbe trigonométrique; en sorte que l'aire de la courbe réduite a, dans tous les cas possibles, une valeur déterminée qui donne celle du coefficient de sin. dans le développement de la fonction." (Théorie analytique de la chaleur, art. 234)
Whatever the given curve may be, it will always serve to scale the trigonometric curve somehow. That holds whether you can assign the curve an analytic equation or whether it obeys no regular law at all. So the area of the scaled curve has, in every possible case, a determinate value. That value gives the coefficient of in the expansion (S371).
Notice what he is describing. To find the coefficient of , multiply your arbitrary curve by point by point, and take the area under the result. That is the coefficient integral
described geometrically, as an area, which is why it works for a curve with no formula. You do not need an equation to have an area. That is the clause Lagrange refused to accept, and Fourier put it in italics of meaning if not of type.
The fine print Fourier got wrong
One caution, because it is a good lesson about heroes.
In the same article 235, Fourier continues (S371).
"1 degree. Les séries ordonnées selon les cosinus ou les sinus des arcs multiples sont toujours convergentes, c'est-à-dire qu'en donnant à la variable une valeur quelconque non imaginaire, la somme des termes converge de plus en plus vers une seule limite fixe, qui est la valeur de la fonction développée."
In English: series ordered by cosines or sines of multiple arcs are always convergent. Give the variable any non-imaginary value, and the sum of the terms closes in more and more on a single fixed limit, which is the value of the function expanded.
As a blanket statement that is false. There are continuous functions whose Fourier series diverges at a point, which du Bois-Reymond showed later in the century. Working out exactly when Fourier's claim holds occupied Dirichlet, Riemann and a long line after them. It produced measure theory and much of modern analysis along the way. That later history was not researched for this chapter (S371).
Fourier was right about the big thing and wrong about the fine print. The fine print turned into a research program that ran for a hundred and fifty years. That is a better story than "Fourier was a genius", and it is true.
The radian, and a mistake worth naming
Every angle on your poster is in radians. The unit is older than its name. The name has a birth date, a place, and an attribution error that textbooks have copied for a century.
The concept comes first
Roger Cotes described the concept in 1714, or so MacTutor says. Its Cotes biography lists among his achievements that he "invented radian measure of angles" (S367). That entry gives no source, no page and no elaboration. Treat it as a lead, not as evidence. No primary support for it was found (S367).
The idea itself is measuring an angle by the length of arc it cuts on a circle of radius 1. It sits plainly in Introductio section 127, in the phrase "cuius radium perpetuo assumo = 1", the radius of which I assume always equal to 1 (S362). Euler measures arcs that way throughout Chapter VIII. He never names a unit, because from his point of view there is nothing to name. The arc is only a number.
Muir hesitates, 1869
Thomas Muir (TOM-us MYOOR, 1844 to 1934, dates not independently verified here), then at St Andrews, wanted a name for the unit. He could not settle on one. Cajori records that in 1869 Muir "hesitated between 'rad,' 'radial' and 'radian'" (S364).
Three candidates, all from the same Latin root radius, meaning spoke or ray. "Rad" is the clipped form, and it survives today as the abbreviation. "Radial" is the adjective. "Radian" is the one that won, and it won because of what it is short for.
The word appears in print, 5 June 1873
Here is Florian Cajori (FLOR-ee-un kuh-JOR-ee, 1859 to 1930), A History of Mathematics, second edition, 1919, page 484 (S364).
"An isolated matter of interest is the origin of the term 'radian,' used with trigonometric functions. It first appeared in print on June 5, 1873, in examination questions set by James Thomson at Queen's College, Belfast. James Thomson was a brother of Lord Kelvin. He used the term as early as 1871, while in 1869 Thomas Muir, then of St. Andrew's University, hesitated between 'rad,' 'radial' and 'radian.' In 1874 T. Muir adopted 'radian' after a consultation with James Thomson." (S364)
Cajori repeats it in A History of Mathematical Notations, section 515: "The word 'radian' was first used in print in 1873 by James Thomson, a brother of Lord Kelvin", citing his own 1919 book, page 484 (S363).
The first appearance of one of the most-used units in mathematics is on an exam paper. Not in a treatise, not in a paper, not in a dictionary. A professor needed a word so that his students would know what he was asking for, and he wrote one down.
The correction: brother, not father
You will often read that this James Thomson was Lord Kelvin's father. He was not.
He was Kelvin's older brother: James Thomson (JAYMZ TOM-sun, born Belfast 16 February 1822, died Glasgow 8 May 1892), an engineer. He was appointed professor of civil engineering at Queen's University Belfast in 1855 and stayed until 1873, which is exactly the year of the exam paper (S382).
The confusion is easy to fall into. Kelvin's father was also called James Thomson, and was also a professor, of mathematics, at Glasgow. But the father died in 1849. A man who died in 1849 cannot set an examination paper in 1873. That is the whole argument, and it is airtight.
Now the part that makes this a proper source problem rather than a simple error. Both readings of Cajori are on record.
The direct read of Cajori 1919, page 484, quoted above, says "a brother of Lord Kelvin", and Cajori's 1929 Notations section 515 says "a brother of Lord Kelvin" as well (S363, S364). But Jeff Miller's widely used compilation quotes that same 1919 page as reading "James Thomson was the father of Lord Kelvin" (S423). Two careful readers, one page, two different words.
Whichever way the 1919 sheet reads, the fact is not in doubt: the Belfast professor of 1873 was the brother, born 1822, died 1892. What the discrepancy tells you is where the "father" version got into circulation. If Miller's transcription reflects a real variant in some printing of Cajori, then the error has a distinguished ancestor. Every textbook that repeated it was copying a good source badly. Either way, if you find "Kelvin's father" in a textbook, you have found a book whose author did not check a death date.
The priority letters, Nature 1910
The question of who really named the radian was aired in Nature in 1910, volume 83, at pages 156, 217, 459 and 460 (S364, S381).
According to Jeff Miller's compilation of earliest uses, Muir wrote on 7 April 1910 that he had corresponded in 1874 with Alexander J. Ellis (al-ig-ZAN-der EL-iss, 1814 to 1890, dates not independently verified here). Ellis, he said, "agreed at once for the form 'radians,' on the ground that it could be viewed as a contraction for 'radial angles'" (S423, S381).
That is the etymology, and it is the reason "radian" beat "radial" and "rad". It is not a word built from radius directly. It is a squeeze of the two-word phrase radial angle.
On 16 June 1910 a son of James Thomson replied. Miller quotes him: "I shall be very pleased to send Dr. Muir a copy of my father's examination questions of June, 1873, containing the word 'radians.' ... It thus appears that 'radians' was thought of independently by Dr. Muir and my father, and, what is really more important than the exact form of the name, they both independently thought of the necessity of giving a name to the unit-angle" (S423).
That last clause is the graceful part. Thomson's son declines to fight about the word. The real event, he says, was two people separately noticing that the unit needed a name at all.
Be explicit about what was and was not read. An earlier version of this chapter worked from Miller's summary, because the Nature letters were behind a paywall. All four have since been read in full, and there are four, not the three that summary listed (S485, S486, S487, S488).
What the fourth one settles is the date this chapter prints. It is from James Thomson the son, on 21 April 1910, and it says the name "appears in the printed examination questions set by him in the general class examination in Queen's College, Belfast, on June 5, 1873" (S486). That is the primary source for 5 June 1873, rather than a secondary report of it.
Two more things fall out of reading them. The son who wrote the replies is James Thomson of Newcastle-on-Tyne, writing about his father of the same name, which closes a question this chapter used to leave open (S486, S488). And Muir concedes the shared credit himself, in print: "It thus appears that radian was thought of independently by Dr. Muir and my father, and, what is really more important than the exact form of the name, they both independently thought of the necessity of giving a name to the unit-angle" (S488).
Thomson and Tait, and a date that would change everything
There is a loose end that nobody has closed, and it is a good one to hand a student who likes archives.
Miller states that Thomson and Tait's Treatise on Natural Philosophy of 1867, page 31, already contains the sentence "For brevity we shall call this angle a radian" (S423). If that is right, it predates the 1873 Belfast exam paper by six years and rewrites the priority story entirely.
The 1867 first edition has now been obtained and checked, and the claim does not survive it. The word "radian" occurs zero times in the 1867 first edition and nine times in the 1879 new edition (S489, S490). What 1867 has is the idea without the name: "the unit angle, or the angle of which the arc is equal to radius, is 57 deg 29578..., or 57 deg 17′ 44″.8" (S489). The name arrives in the 1879 edition. There the sentence appears at section 41, in full: "The usual unit angle is (as explained in treatises on plane trigonometry) that which subtends at the centre of a circle an arc whose length is equal to the radius ... For brevity we shall call this angle a radian." The 1879 edition also carries an appendix section headed "2. Space. Yard and Mètre: Radian, Degree, Minute, Second", defining "Radian, or angle whose arc is equal to radius" (S434).
So: the word is securely in print in 1873 in Thomson's exam questions and in 1879 in Thomson and Tait section 41 (S434, S490). And the 1867 sheets have now been opened. The word is not in them, so Miller's claim does not survive, and the 1873 date this chapter prints stands (S489).
Note also that this Thomson is a different one again. The Treatise on Natural Philosophy is by William Thomson, Lord Kelvin, with Peter Guthrie Tait. So the radian's paper trail runs through Kelvin's brother in 1873 and Kelvin himself in 1879, with a father who was dead by 1849 and had nothing to do with either.
One more competing account
For completeness, because it is on record and contradicts everything above: Miller quotes W. N. Roseveare, writing in the Mathematical Gazette 3(49), January 1905, page 133: "The radian is an uninteresting angle. Lord Kelvin introduced the word merely as a convenience in lecturing, to avoid the long phrase 'angle whose circular measure is'" (S423). That attributes the word to William Thomson rather than to James. It is a single sentence in a short article, written thirty-two years after the event, and the 1910 exchange contradicts it. But it exists, and somebody believed it in 1905.
A teaching coincidence
The Alexander J. Ellis of this dispute, the man who told Muir that "radians" worked as a contraction of "radial angles", is the same Alexander J. Ellis who translated Hermann von Helmholtz's On the Sensations of Tone into English (S380). That book turns up again at the end of this chapter, as the source for the law of beats. One man sits behind two apparently unrelated items in the same research file. That is the kind of thing that happens when a scholarly world is small.
Notation: where the symbols on your page came from
Almost every mark in your trigonometry textbook has a first printing, a person, and usually an argument attached. Here are the ones that matter, nearly all from Florian Cajori's A History of Mathematical Notations, volume II (S363).
| Symbol or convention | Who | Where and when | Source |
|---|---|---|---|
| "sin.", "tan.", "sec.", "sin. com.", "tan. com.", "sec. com." | Thomas Finck (TOM-us FINK, c. 1561 to 1656) | Geometriae Rotundi Libri XIIII, Basel, 1583, Liber XIV, on spherical triangles | S363 |
| The words "tangent" and "secant" | Thomas Finck | same book, 1583 | S363 |
| The degree symbol, a small raised circle | J. Peletier | appendix to Gemma Frisius dated 1558, Paris 1569 edition, used only in a multiplication scheme | S363 |
| Prime and double prime for minutes and seconds | descends from Ptolemy's Syntaxis, which marked first sixtieths with one accent and second sixtieths with two | Greek, second century CE | S363 |
| "A S." for arcsine | Daniel Bernoulli | Comment. acad. sc. Petrop., dated 1729 by Cajori | S363 |
| "A t" for arctangent | Leonhard Euler | Mechanica, St Petersburg, 1736 | S363 |
| "arc. tang." and "arc. sin" | Karl Scherffer (KARL SHERF-fer, 1716 to 1783) in Vienna and Lagrange in Berlin, in the same year | Scherffer, Institutionum analyticarum pars secunda, Vienna 1772, pp. 144 and 200; Lagrange, Berlin Nouveaux mémoires for 1772, published 1774, p. 277 | S363 |
| , , | John Herschel (JON HER-shul, 1792 to 1871) | Philosophical Transactions of London for 1813, p. 10 | S363 |
| "Sh", "Ch" for hyperbolic sine and cosine | Vincenzo Riccati | Cajori dates the introduction 1757 | S363 |
| "sin h", "cos h" | Johann Heinrich Lambert | Berlin Histoire, vol. XXIV, p. 327, 1768 | S363 |
Several of those need more than a table row.
Finck 1583, and the objection nobody listened to
Thomas Finck, a Dane working in Schleswig-Holstein, published Geometriae Rotundi Libri XIIII (Fourteen Books of the Geometry of the Round) at Basel in 1583. He coined the trigonometric senses of "tangent", from the Latin tangens, touching, and "secant", from secans, cutting. In Liber XIV, on spherical triangles, he used the abbreviations "sin.", "tan.", "sec.". For the co-functions he wrote "sin. com.", "tan. com.", "sec. com.", where "com." is short for complementi, of the complement (S363).
That is where your calculator keys come from: a book about spherical triangles, in Latin, in 1583, using "com." because the cosine is the sine of the complementary angle.
Cajori is careful about the priority. He calls Finck's usage "perhaps the first use of abbreviations for the trigonometric lines", and lists Maurolyco's 1555 manuscript contraction "sinus 2m arcus" as an earlier contraction of a different kind (S363).
People resisted the new words at the time, on exactly the grounds a modern student might raise. Cajori, section 517: Finck's terms "did not meet with the approval of Vieta, because of the confusion likely to arise with the same names in geometry. Vieta called the trigonometric tangent Prosinus and the trigonometric secant Transsinuosa" (S363). Prosinus means roughly "before the sine", and transsinuosa "across the sine". Viète first published them in Responsorum liber VIII in 1593.
Cajori's verdict on the campaign: "But Vieta's objection was overlooked or ignored" (S363).
So the complaint that "tangent" in trigonometry and "tangent" in geometry are different things is not new. Every student eventually makes it. One of the best mathematicians of the sixteenth century made it, proposed a fix, and lost. The naming problem in your textbook is 440 years old, and somebody with standing flagged it at the time.
The degree symbol, and Cajori's own doubt
The small raised circle for degrees has a murkier history than you would expect. Cajori is honest about it in a way worth copying.
He identifies J. Peletier's 1558 appendix to Gemma Frisius, in the Paris 1569 edition, printing the raised circle for integra, meaning whole units, inside a multiplication scheme. Cajori calls this "the first modern appearance that I have found of the degree symbol" (S363).
And then he undercuts his own find. Cajori states outright that he could not find the modern angular notation anywhere in that book: degrees, minutes and seconds written with circle, prime and double prime, used for angles (S363). Peletier's raised circle marks whole units in a calculation. It is not writing an angle.
That is a model of careful reporting: here is my earliest example, and here is why my earliest example does not do the job you want it to do.
Prime and double prime, and a line of descent nobody has established
The prime and double prime for minutes and seconds come from sexagesimal astronomy, and the trail runs back to Ptolemy.
Cajori, section 511: "Signs resembling those now in use are found in the Syntaxis (Almagest) of Ptolemy ... The first sixtieths or minutes were marked with one accent, the second sixtieths with two accents ... From these facts it would seem that our signs for degrees, minutes, and seconds were of Greek origin. But it is difficult to uphold this view, especially for the sign for 'degrees.' Such a line of descent has not been established" (S363, S423).
Read the shape of that argument. The Greek practice is documented. The modern practice is documented. The connection between them looks obvious. Cajori refuses to assert it, because looking obvious is not evidence. He does record what did become settled. Minutes and seconds of arc take prime and double prime, while minutes and seconds of time take m and s. That, he says, "became the common practice" among astronomers and navigators, and he names no single first printer for it (S363).
The distinction still matters. In an ephemeris or a navigation table, and are different quantities, and confusing them puts a ship in the wrong place.
Daniel Bernoulli's "A S.", and a source that contradicts itself
The first symbol for an inverse trigonometric function is Daniel Bernoulli's "A S." for arcsine. Cajori's text says "In 1729 he used" it (S363).
But Cajori's own footnote points to a different year. The footnote cites the volume as Commentarii academiae scientiarum imperialis Petropolitanae II, for the year 1727, printed 1728, pages 304 to 342, via Eneström (S363).
So the text says 1729 and the footnote points at a volume for 1727 printed in 1728. That is an internal inconsistency in the standard reference, and it is recorded here as one, not quietly resolved. It is also a normal hazard of eighteenth-century academy publishing. The volume year, the printing year and the year of presentation routinely differ by two or three.
The idea behind the notation is the "A" for arcus, arc. "A S." means "the arc whose sine is", which is exactly what an arcsine is, said in the right order.
Euler's "A t", 1736
Euler picked up the same device seven years later. In Mechanica, St Petersburg, 1736, he writes this (S363).
"expressio A t nobis denotet arcum circuli, cuius tangens est t existente radio = 1"
Let the expression denote for us the arc of a circle whose tangent is , the radius being 1.
That definition is worth showing a class in Latin. It is short and the words are guessable. And the last clause, existente radio = 1, is Euler doing in 1736 what he would make universal in 1748. He cannot define an arctangent without first saying what circle he is in. Twelve years later he would declare the radius to be 1 once, at the top of the chapter, and never mention it again.
Scherffer and Lagrange, 1772, and a forgotten teacher
Karl Scherffer (1716 to 1783, dates not independently verified here), a Jesuit teaching mathematics in Vienna, printed "arc. tang." in Institutionum analyticarum pars secunda, Vienna 1772, at pages 144 and 200. The same year, in Berlin, Lagrange printed "arc. sin" in the Nouveaux mémoires for 1772, published 1774, at page 277 (S363).
Two men, two cities, one convention, no recorded contact. That is what a notation looks like when it is ready: it appears in more than one place at once because the need is general.
Scherffer is one of the people this chapter is deliberately putting back on the page. He was a textbook author. Notation spreads through textbook authors, and they get credit almost nowhere. His name survives in a footnote in Cajori because Cajori went and looked at the books.
Herschel's minus one, and a deliberate collision
John Herschel published and in the Philosophical Transactions of London for 1813. He flagged the problem with it in the same sentence that introduced it, on page 10 (S363).
"This notation must not be understood to signify , but what is usually written thus, arc ."
He knew. On the day he printed it, he knew it would collide with , and he said so in the same breath.
His defense is a real argument and not a shrug. Cajori, section 533, summarizes it: Herschel "admits that some authors use for , but he justifies his own notation by pointing out that since , , mean , , , we ought to write for , for . Just as we write , we may write similarly , " (S363).
His conclusion: "The symmetry of this notation and above all the new and most extensive views it opens of the nature of analytical operations seem to authorize its universal adoption" (S363).
The claim is that a superscript on a function name should mean repeated application of the function, exactly as it does for the differential operator . Under that rule, has to mean the inverse. It also means ought to be , and the familiar is the notation that is wrong.
So the thing students complain about is not an accident. It is a deliberate trade, made by a good mathematician who wrote down in advance what it would cost. The world took the deal on one side and refused it on the other, keeping for the inverse and for the square.
Herschel later conceded a priority point on himself. Burmann, he admitted, had used the same idea earlier for direct functions, though not for the inverse trigonometric functions (S363).
The continental convention beat the English one
One more piece of Cajori explains why your calculator says "sin" and not "s". This is his section 524 (S363).
"During the eighteenth century the use of abbreviations for the trigonometric lines became general on the European Continent, but the symbolism differed from that most popular in England in not being so intensely specialized as to be represented by a single letter. As a rule, the Continent used three letters (as in 'sin.,' 'cos.,' 'tan.,' etc.) in their formulae, while the English, between 1700 and about 1750 or 1760, used only one or two letters (as in s, cs, t). ... Then again, as a rule, the continental writers used dots after the abbreviations, while the English did not."
English mathematicians spent sixty years writing , and for sine, cosine and tangent. It is shorter, and it lost. The continental three-letter form is readable, and a single letter in a crowded formula is not. The dots went away in the nineteenth century.
And the argument between the two ways of writing an inverse function never ended. Cajori's section 535 is headed "Persistence of rival notations for inverse functions", and his own comment at section 526 is that "the 'arc sin,' 'arc cos,' etc., of the Continent, still await the decision" of usage (S363). He was writing in 1929. Ninety-odd years later, your calculator has a key and your programming language has an arcsin function. Both won.
The hyperbolic cousins
Replace the circle with the hyperbola and you get a second family of functions with almost the same algebra. Where the circular functions satisfy , the hyperbolic ones satisfy . Where and are the half-sum and half-difference of and , the hyperbolic pair are the half-sum and half-difference of and :
Put those next to Euler's section 138 forms and the family resemblance is not a resemblance. It is the same construction with the taken out.
Vincenzo Riccati got there first
Vincenzo Riccati (vin-CHEN-tso ree-KAH-tee, born Castelfranco Veneto near Treviso 11 January 1707, died Treviso 17 January 1775) was a Jesuit teaching at Bologna. His father is the Riccati of the Riccati differential equation. He introduced the hyperbolic functions with the notation and (S363, S386).
Cajori, section 529 (S363):
"Hyperbolic functions were first introduced in 1757 by Vincenzo Riccati who used the notation Sh , Ch for hyperbolic sine and cosine. He writes our in this manner, 'Ch. - Sh.' where is the arc and the sinus totus."
Look hard at Riccati's version of the identity:
There is an in it. The sinus totus, the whole sine, meaning the radius. Nine years after the Introductio fixed the radius at 1, an excellent mathematician is introducing a brand new family of functions. And he is still carrying a radius around in his identities (S363).
That single is the best available evidence for how slowly Euler's convention spread. It did not sweep the field in 1748. It took a generation, and even then the people at the front of the subject were writing the old way out of habit.
The dating tangle
The dates on Riccati's work do not line up between sources, and this chapter records the tangle rather than picking a winner.
Cajori dates the introduction to 1757, but his own footnote points to the Institutiones analyticae, Tomus Secundus, Bologna 1767, page 152 (S363).
MacTutor dates the Opusculorum ad res physicas et mathematicas pertinentium to 1757 to 1762 and the Institutiones Analyticae to 1765 to 1767 (S386).
So Cajori says 1757 and footnotes an 1767 book. MacTutor gives a five-year range for one work and a three-year range for the other. Multi-volume Latin works published in parts over a decade do this. The honest statement has three parts. Riccati introduced the hyperbolic functions somewhere in the span 1757 to 1767. Both Cajori and MacTutor agree he did it before Lambert. A precise year would mean opening the volumes.
Lambert fixed the notation
Johann Heinrich Lambert (YO-hahn HYNE-rikh LAM-bairt, 1728 to 1777), born at Mulhouse in Alsace and working in Berlin, is the reason your calculator says "sinh" and not "Sh".
Cajori has him writing "sin " and "cos " in the Berlin Histoire, volume XXIV, page 327, for 1768. He adds that "this use of 'h' after the ordinary trigonometric contractions has retained its place in many books to the present time" (S363).
Record the disagreement rather than smoothing it over. MacTutor's Riccati biography says: "Johann Heinrich Lambert is often cited as the first to introduce the hyperbolic functions but he did not do so until 1770 while Vincenzo Riccati's work ... was published between 1757 and 1767" (S386). That is 1770 against Cajori's specific 1768 volume-and-page citation. This chapter does not resolve it. Cajori gives a page number, which is usually the stronger kind of claim, but MacTutor may be dating a different publication.
Lambert and the irrationality of pi
Lambert also proved that is irrational, meaning it cannot be written as a fraction of two whole numbers. He published the proof in the memoir Mémoire sur quelques propriétés remarquables des quantités transcendentes circulaires et logarithmiques.
The date usually given is 1761 (disputed). The LOCOMAT project's page on the irrationality of , which was read, says only that the result was proved "in the 1760s" and published in 1768 (S392).
Two things commonly stated about this proof were not confirmed for this chapter. The 1761 presentation date is unverified: LOCOMAT does not give it. And the continued fraction for does not appear on the LOCOMAT page at all. A continued fraction is a fraction whose denominator contains another fraction, and so on down. Every account says Lambert used one. The memoir itself could not be opened, and nor could Roegel's translation of it (S392). So: Lambert proved irrational in the 1760s, published in 1768, by a method this chapter did not verify.
That result matters for the whole story of this book. Euler's section 126 says the periphery "cannot be expressed exactly in rational numbers", which is an assertion. Lambert turned it into a theorem about ten years later. Every trigonometric table ever built is a table of numbers that are almost all irrational, approximated to whatever precision the maker could afford. Lambert is the one who proved that no amount of cleverness would ever make the central one come out exact.
The people who did the arithmetic
Series made sines computable. They did not make them computed. Somebody had to sit down and add up the terms, thousands of times, by hand, for years, and check each other's work.
Those people were called computers, because that was the job title. A computer was a person who computed, the way a baker is a person who bakes. The word was a job before it was a machine, and the machines took the name from the people they replaced.
The Bureau du Cadastre: mathematics as a factory
After 1789 France went metric, and metric meant re-dividing the circle. Not 360 degrees but 400 grades. The right angle splits into 100 parts, and each of those into 100 more, so that angle measurement would be decimal like everything else. The idea is sensible, and it had one enormous consequence. Every trigonometric table in Europe was suddenly the wrong shape. Not slightly wrong. Wrong at every single entry, because the entries were at the wrong angles.
Gaspard de Prony (gas-PAR duh proh-NEE, born Chamelet in Beaujolais 22 July 1755, died Paris 29 July 1839) was put in charge of making new ones. He ran the work at the Bureau du Cadastre, the office of the land survey (S373, S374).
The project ran from 1791 to 1801. Inside that, the clerks computed the table of sines first, starting in 1793. Their main computing phase ran from 1793 to 1796, with printing, delays and a revival under the Consulate afterwards (S373). Record the competing account: MacTutor says the project "began in 1792 and was completed in 1801" (S374).
The scale is hard to take in. The sine table alone fills one large volume of 400 pages, giving sines at every ten-thousandth of a quadrant. The precision is the part that stops people:
- Sines to 29 decimal places at the beginning of the range, and 25 decimals after 0.0350 of a quadrant (S373).
- The printed fragments give sines to 22 places with five orders of differences (S373).
- Logarithms to 19 decimals for the numbers 1 to 10,000, and to 14 decimals from 10,001 to 200,000 (S373).
MacTutor's summary matches: "The tables were vast, with values calculated to between fourteen and twenty-nine decimal places" (S374).
Listen to how the table describes itself
Denis Roegel, who reconstructed the tables from the surviving manuscripts, quotes the title of the sine table: "Sinus en parties du rayon depuis 0 jusqu'à 10000." Sines in parts of the radius, from 0 to 10000.
Here is Roegel's comment on that title (S373).
"In other words, the sines are given in the modern way, the radius being taken equal to 1."
Euler's convention, forty-five years old, has reached a French government office. The old table-makers gave you sines in a circle of radius so the numbers came out whole. Prony's clerks are computing pure ratios.
The seed values came from the series
How do you start a table like this? With Newton's and Euler's series. Roegel explains (S373).
"First, the sines were computed every 10 (centesimal) degrees using with () using the then (1794) most accurate known value of computed by Thomas Fantet De Lagny in 1719 to 112 correct places."
That is a documented straight line: Newton's notebook of 1669, Euler's textbook of 1748, a payroll in Paris in 1794, and a 112-digit value of from 1719 as the input constant. Everything else in the 400-page volume is built out from those nine seed values by differences, which means addition and subtraction only.
The pin factory, and Roegel's qualification of it
Prony organized the office in three tiers, and said openly where he got the idea. Roegel again (S373).
"In his celebrated work The Wealth of Nations, Adam Smith considered the example of the division of labor in a pin-factory. ... Inspired by Smith, Prony decided to use manufacturing processes to compute the logarithms."
The three groups (S373):
- Five or six mathematicians of "very high merit", who chose the formulas and computed fundamental values. Prony names exactly one of them explicitly: Adrien-Marie Legendre (ah-dree-EN muh-REE luh-ZHAHN-druh, 1752 to 1833). The others are unnamed in his own account (S373).
- Computers acquainted with calculus, sometimes called calculateurs in Roegel's account. They computed the initial logarithms and the initial differences, and checked the work coming back.
- The largest group, who "had only to perform additions and subtractions, and put the results on pages submitted to them by the second group."
Roegel then adds a qualification that gets left out everywhere else, and it is worth copying (S373).
"Prony's division was however not an exact copy of the pin-factory, because there was mainly one computing task, which was divided in about twenty computers, each doing a similar work. In the pin-factory, each of Smith's ten workers were specialized, and doing a specific task. There was no such specialization in Prony's scheme, except for the task of providing blank sheets with initial values, and checking the values."
So it is not a production line. It is parallel processing. Twenty people do the same job on different chunks of the range, with a setup step and a checking step around them. Which, incidentally, is much closer to how a modern computer divides work across cores than a pin factory is.
The names
The anonymous computers are not entirely anonymous. They signed pages, and the payment records survive in the Archives Nationales. Roegel names twelve of them (S373):
Hervet. Saget père. La Bussierre. Jean Baptiste André Vibert. Humaird. Étienne Antoine François Baudouin. Louis Saget fils. Mazerat. Marc Antoine Parisot. Henry. Leprestre. Pierre Simon Pigeou.
Read them slowly. Every one of those people spent months adding and differencing by hand so that somebody else could look a number up in a second.
Say those names in class. They are the people who made the numbers.
One of them steps out of the record as a person. This is Roegel's footnote 94, from Archives Nationales F14 2146 (S373).
"a letter by Louis Saget (fils) to Prony, dated 17 Fructidor III (3 September 1795), who asked for a raise, claiming to compute 200 logarithms per day, and to be one of the best computers (A.N. F14 2146). He wrote that he earns 2600 francs, whereas the other computers earn 3400 francs. Prony decided to give him 3000 francs."
Two hundred logarithms a day. A claim to be one of the best. A pay gap of 800 francs against his colleagues. A negotiation that split the difference at 3,000. That is what this work looked like from the inside, in a date from the Revolutionary calendar, in a file that still exists.
The whole Cadastre office had 44 employees in September and October 1794, and Roegel names them from the payroll (S373). That number is worth holding on to for the next section.
Mary Edwards, paid through her husband's name
England organized the same work in the opposite way. Not concentrated in an office, but scattered across the country.
Nevil Maskelyne (NEV-il MASK-uh-lin, died 1811, birth year not verified here), the Astronomer Royal, ran the Nautical Almanac by post. He mailed calculation sheets to people in their own homes, then mailed the results on to a second, different person to check. A scattered network of freelancers at kitchen tables produced the tables that British ships navigated by, coordinated entirely by mail (S390, S391).
Mary Edwards (MAIR-ee ED-wurdz, c. 1750 to 1815 (disputed: Cambridge University Library's catalog gives c.1750 to 1817)) of Ludlow, Shropshire, was one of them, and one of the most durable. Cambridge University Library, which holds Maskelyne's papers, calls her in its own catalog description "one of the longest serving of Maskelyne's computers" (S390).
The arrangement was this. Her husband, the Reverend John Edwards, "received payment for work on 6 months' worth of each almanac from 1773 until his death in 1784". After his death, Mary "revealed she had performed most calculations in correspondence with Maskelyne", and took the contract in her own name. She was one of 35 human computers calculating the positions of the Sun, Moon and planets. When Maskelyne died in 1811, his successor John Pond reduced her work, and "The Board of Longitude eventually ruled that Pond should continue to allocate work to her" (S391).
So for eleven years the work of one of the Almanac's best computers went out under a clergyman's name. And it took a ruling from the Board of Longitude to keep her employed after her patron died.
Be careful with that paragraph, and say why. Everything after the Cambridge catalog line comes from an encyclopedia article. That article summarizes Mary Croarken's 2003 study in the IEEE Annals of the History of Computing, volume 25, issue 4, pages 9 to 15. That article is paywalled and was not read for this chapter (S391, S397). The facts are very likely right, and the sourcing is thin. Anyone teaching this should get Croarken. The death-date conflict, 1815 against 1817, is unresolved between the encyclopedia entry and Cambridge's own catalog. Both are recorded here.
Henry Andrews of Royston
Mary Edwards's colleague in the same network was Henry Andrews (HEN-ree AN-drooz), of Royston in Hertfordshire. His dates are not established in any source read here.
What survives is better than dates. Cambridge holds MS-RGO 4/149, a collection of fifty-three letters, mainly from Maskelyne to Andrews. The catalog calls it correspondence with "one of the human 'computers' of calculations for the annual Nautical Almanac" (S390). It is digitized and available through the Cambridge Digital Library.
Think about what that is. Fifty-three letters from the Astronomer Royal to a self-taught provincial computer, about the work. Not a famous correspondence. A working one: here are your sheets, here is what came back wrong, here is what I need next. It is the most direct surviving evidence of how that eighteenth-century computing network ran day to day. A class could read it.
Note honestly what was read: the catalog description only. The manuscript images were not read for this chapter (S390).
Greenwich, a century later
Alice Everett (AL-iss EV-uh-rit, 1865 to 1949) read mathematics at Girton College, Cambridge. In 1890 she began work as a "supernumerary computer" at the Royal Observatory, Greenwich, assigned to the Astrographic Department (S389).
The job was the Carte du Ciel, an international project to photograph and catalog the whole sky. Royal Museums Greenwich describes her work: she was "trained to take photographs using the Observatory's new astrographic telescope, as well as measuring the plates, calculating the coordinates of the stars and reducing the data for the catalogue" (S389).
She stayed five years, then moved to Potsdam Observatory. She was proposed for fellowship of the Royal Astronomical Society, and rejected. Steady paid scientific work did not come again for two decades. She joined the National Physical Laboratory in 1917, at the age of 52, and retired in 1925 (S389).
Annie Russell Maunder (AN-ee RUSS-ul MAWN-der, 1868 to 1947, dates approximate and not given in the museum article itself) was born at Strabane in County Tyrone. Greenwich hired her into the same program as a lady computer. She later became a solar astronomer of real standing, known for her work on sunspots and for solar eclipse photography (S389).
On her computing duties specifically, and on what any of these women were paid, nothing usable was found. The Royal Museums Greenwich article names her and gives no detail (S389). The leads are Kane Mullen's "Temporary Measures: Women Computers at the Royal Observatory, Greenwich, 1890-1895" in the Journal for the History of Astronomy (2020) and Mary Brück's work on Annie Maunder, neither of which was read (S389).
And one more gap, stated because it would be easy to paper over. Reducing photographic plate measurements to star coordinates is spherical-trigonometric work. You are converting measured positions on a flat plate into positions on the celestial sphere. That is exactly a problem in spherical trigonometry. But no source read for this chapter says so in those words, and it is not asserted here on the strength of looking obvious (S389).
Where this lives in your phone
Everything so far is history. This section is the payoff, and every item in it has a founding paper. Each one is labeled with how well it was verified, because the labels are not all the same.
The fast Fourier transform, and Gauss got there first
Finding out which frequencies are present in a recorded sound means computing a discrete Fourier transform. Given samples of a signal, it works out how much of each frequency the signal contains. Done directly, it costs about multiplications. For a one-second audio clip at 44,100 samples, that is roughly two billion operations. For a three-minute song, it is hopeless.
James Cooley and John Tukey published a method in 1965 that costs about instead. For a million samples, is and is about , a saving of a factor of fifty thousand. That is the difference between a mainframe and the chip in a pair of earbuds.
Citation and verification status: Cooley, J. W., and Tukey, J. W. (1965), "An algorithm for the machine calculation of complex Fourier series", Mathematics of Computation 19(90), pages 297 to 301, DOI 10.1090/S0025-5718-1965-0178586-1. The bibliographic record was verified at the American Mathematical Society's site. The paper itself was not read, because the site was behind a Cloudflare block during the research. No claim here rests on its contents (S376).
Now the twist, and it is one of the best stories in the history of computing.
Carl Friedrich Gauss (KARL FREED-rikh GOWSS, 1777 to 1855) wrote down the same algorithm around 1805, in a treatise called Theoria Interpolationis Methodo Nova Tractata. It appeared only after his death, in volume 3 of his collected works, in 1866 (S375).
Heideman, Johnson and Burrus reconstructed this history. They write that "the presumed year of the composition of this treatise is 1805, thereby suggesting that efficient algorithms for evaluating coefficients of Fourier series were developed at least a century earlier than had been previously known". They add that "it was published only posthumously in Volume 3 of his collected works in 1866". And of Gauss's own method: "This is exactly the exponential form of Gauss' algorithm ... This is also exactly the FFT algorithm derived by Cooley and Tukey in 1965" (S375).
They also note the sting in the tail: Gauss's work predates Fourier's 1807 presentation to the Paris Institute. The fast algorithm for computing Fourier coefficients was invented before Fourier presented Fourier analysis.
Verification status: read in the expanded 1985 version, Heideman, Johnson and Burrus, "Gauss and the history of the fast Fourier transform", Archive for History of Exact Sciences 34(3), pages 265 to 277 (S375). A shorter 1984 version exists in IEEE ASSP Magazine 1(4), pages 14 to 21, which was not read. Cite whichever you open (S375).
JPEG and the discrete cosine transform
A digital photograph is stored in pieces. Chop the image into small blocks. Write each block as a sum of cosine patterns of increasing wiggliness. Keep the large coefficients and throw away the small ones. The eye does not miss the fine wiggles, and the file gets an order of magnitude smaller.
The transform is the discrete cosine transform, a recipe that turns a block of pixel values into a list of cosine amplitudes. It is defined in Ahmed, N., Natarajan, T., and Rao, K. R. (1974), "Discrete cosine transform", IEEE Transactions on Computers C-23(1), pages 90 to 93, DOI 10.1109/T-C.1974.223784 (S396).
Verification status: citation only. The paper is behind the IEEE paywall and two open mirrors were blocked by robots restrictions. It was not read, and no claim about its contents is made here (S396). The JPEG standard's use of an 8 by 8 block DCT was also not verified from any source read here. That detail is reported as common knowledge, not as checked (S396).
What can be said without any of that: a JPEG is a sum of cosines. Compress one too hard and the image breaks into visible squares with ripples in them. You are looking at the cosine basis functions themselves. They show through because too few of them survived.
MP3 and AAC, and the transform on your phone right now
Compressed audio uses a variant called the modified discrete cosine transform, or MDCT. It overlaps its blocks, so the boundaries between them do not click.
Karlheinz Brandenburg (KARL-hynts BRAHN-den-boorg, born 1954) is one of the MP3 designers, working at Fraunhofer IIS in Erlangen. He describes both formats in a 1999 Audio Engineering Society conference paper, "MP3 and AAC explained" (S378).
On MP3, section 3.3.1 (S378):
"The filterbank used in MPEG-1 Layer-3 belongs to the class of hybrid filterbanks. It is built by cascading two different kinds of filterbank: First the polyphase filterbank (as used in Layer-1 and Layer2) and then an additional Modified Discrete Cosine Transform (MDCT)."
On AAC, section 4.2 (S378):
"Instead of the hybrid (cascaded) filterbank in Layer-3, AAC uses a standard switched MDCT (Modified Discrete Cosine Transform) filterbank with an impulse response (for short blocks) of 5.3 ms at 48 kHz sampling frequency. This compares favourably with Layer-3 at 18.6 ms."
Verification status: read. This is about as close to a primary source as a design decision gets. Brandenburg is one of the designers, writing at the institution that built the format (S378).
A gap: the founding MDCT paper, Princen and Bradley (1986), IEEE Transactions on ASSP 34(5), pages 1153 to 1161, was not read. The MDCT's presence in MP3 and AAC is well documented by Brandenburg; its origin paper is not verified here (S378).
Every compressed audio file on a student's phone is a list of cosine coefficients. Not a recording of a sound: a recipe for rebuilding one out of cosines.
Phasors: the electrical grid runs on Euler's formula
Charles Proteus Steinmetz (CHARLZ PROH-tee-us STINE-mets, 1865 to 1923) was a German immigrant working for General Electric. He read a paper at the International Electrical Congress in Chicago, held 21 to 25 August 1893. It reorganized how alternating-current engineering is done (S377).
His first paragraph states the whole idea. Here it is, from the printed proceedings, page 33 (S377).
"In the following, I shall outline a method of calculating alternate current phenomena, which, I believe, differs from former methods essentially in so far, as it allows us to represent the alternate current, the sine-function of time, by a constant numerical quantity, and thereby eliminates the independent variable 'time' altogether from the calculation of alternate current phenomena. Herefrom results a considerable simplification of methods. Where before we had to deal with periodic functions of an independent variable, time, we have now to add, subtract, etc., constant quantities, a matter of elementary algebra."
Replace a function of time with a constant complex number. Calculus turns into algebra. That constant complex number is what engineers now call a phasor: one complex number standing in for a whole sine wave. It is Euler's holding down a day job. A voltage becomes the single complex number , and the that both sides share cancels out of every equation.
Steinmetz anticipated the obvious objection, that real waveforms are not pure sinusoids. He answered it with Fourier: "Even the restriction to sine-waves, incident to this method, is no limitation, since we can reconstruct in the usual way the complex harmonic wave from its component sine-waves" (S377). He then says he will use polar coordinates, "representing the time by the angle as amplitude".
Verification status: read in the printed proceedings, table of contents and pages 33 to 35. Full citation: Steinmetz, C. P. (1894), "Complex quantities and their use in electrical engineering", in Proceedings of the International Electrical Congress held in the city of Chicago, August 21st to 25th, 1893, pages 33 to 74, New York: American Institute of Electrical Engineers. Note the two dates: the paper was read in 1893, the volume is dated 1894 (S377).
One thing not verified: the word "phasor". Steinmetz does not use it. When and by whom it was coined was not investigated (S377).
Beats: a trigonometric identity you can hear
Play two notes a few hertz apart and you hear a throb, a slow pulsing of loudness. Piano tuners use it. Guitarists use it. It is a trigonometric identity made audible.
Hermann von Helmholtz (HER-mahn fon HELM-holts, 1821 to 1894) gives the law, in Alexander J. Ellis's third English edition of On the Sensations of Tone as a Physiological Basis for the Theory of Music, London, 1895 (S380):
"The law determining the number of beats in a second for a given imperfection in a consonant interval, results immediately from the law above assigned for the beats of simple tones. When two simple tones, making a small interval, generate beats, the number of beats in a second is the difference of their vibrational numbers."
The trigonometry is the sum-to-product identity:
Read the right-hand side as a fast wave times a slow envelope. The fast factor has the average frequency , which is the pitch you hear. The slow factor has frequency , and it multiplies the fast wave, so it controls the loudness.
Now the last step, which is the one students get wrong. The envelope cycles at , but your ear does not hear the sign of the envelope, only its size. A cosine hits maximum magnitude twice per cycle, at the positive peak and the negative one. So the audible loudness peaks come at twice the envelope frequency:
which is exactly Helmholtz's law. That last derivation is worked out here, not quoted from Helmholtz (S380).
Two tuning forks at 440 Hz and 443 Hz give three beats a second. Count them with a stopwatch and you have measured a frequency difference by ear. The tool is a product-to-sum formula from a sixteenth-century table-maker's kit.
Verification status: read, contents and the chapter on beats of simple tones. The German original is 1863; the edition read is Ellis's third English edition of 1895 (S380).
Gravitational waves: a rising sine from a billion light years away
On 14 September 2015 the LIGO detectors at Hanford, Washington and Livingston, Louisiana recorded two black holes merging. The paper was published on 11 February 2016.
The signal is a rising sine wave, and the paper describes it in exactly those terms. From page 2 (S379):
"Over 0.2 s, the signal increases in frequency and amplitude in about 8 cycles from 35 to 150 Hz, where the amplitude reaches a maximum."
The abstract of the same paper says the signal "sweeps upwards in frequency from 35 to 250 Hz" (S379).
Those two figures look like a contradiction. They are not, and the difference is worth explaining rather than hiding. They describe different portions of the same event. The "8 cycles from 35 to 150 Hz" describes the inspiral, up to the moment of peak amplitude, when the two black holes touch. The "35 to 250 Hz" in the abstract covers the whole detected signal, including the ringdown afterwards. That is when the newly merged single black hole is still wobbling and settling. The frequency keeps climbing past the peak-amplitude value. Both numbers are in the paper. A student who quotes one without the other has not made an error, but a student who notices both and can say why they differ has understood the physics.
The rest of the numbers come from the abstract. Peak strain . Source masses of about 36 and 29 solar masses, merging into about 62, with roughly 3 solar masses radiated away as gravitational waves. Statistical significance greater than 5.1 sigma, and a false alarm rate below 1 per 203,000 years (S379).
That strain figure is the number to dwell on. A strain of means the 4-kilometer arms of the detector changed length by about one ten-thousandth of the width of a proton. And the shape of that change, plotted, is , a sine with a growing amplitude and a rising frequency. Two black holes a billion light years away wrote a sine wave, and we read it.
Verification status: read, abstract and the signal description on page 2 of the arXiv version of the Physical Review Letters article (S379).
CT scanning and the Radon transform
A CT scanner measures how much X-ray energy is absorbed along thousands of straight lines through your body, from many angles. From those totals it reconstructs a cross-sectional image. Nobody ever measures a point inside you. The machine measures sums along lines and solves backwards for the picture.
The mathematics is Johann Radon's (YO-hahn RAH-dohn, 1887 to 1956) 1917 paper "Über die Bestimmung von Funktionen durch ihre Integralwerte längs gewisser Mannigfaltigkeiten", on the determination of functions from their integral values along certain manifolds.
Verification status: weak, and stated as such. The title and year were confirmed on the University of Vienna's historical exhibition page, which also links a piece titled "Von der Radon-Transformation zur Tomographie". The journal, volume and page numbers were not confirmed from any source read here. The commonly cited record is Berichte über die Verhandlungen der Königlich-Sächsischen Gesellschaft der Wissenschaften zu Leipzig, Mathematisch-Physische Klasse 69, pages 262 to 277. But the Saxon Academy's archive page returned a proxy error, and no scan was opened. Get this from the Leipzig academy archive before citing it.
The trigonometry is in the geometry of the problem. Each line through the body is fixed by an angle and a distance from the center, so the data is a function of an angle. The reconstruction filters and back-projects across all angles. The connection to Fourier is direct, through the projection-slice theorem. That theorem says the one-dimensional Fourier transform of a projection at angle is a slice through the two-dimensional Fourier transform of the object at the same angle. That last sentence is standard textbook material, and it was not verified against a source read for this chapter.
GPS, and where the trigonometry sits
The official GPS interface specification, IS-GPS-200N, dated 1 August 2022, requires a receiver to evaluate sines and cosines of several orbital angles. That is how it turns the broadcast ephemeris into a satellite position. Two places carry the work: section 20.3.3.4.3, "User Algorithm for Ephemeris Determination", and Table 20-IV, "Broadcast Navigation User Equations". They involve the sine and cosine of the eccentric anomaly, of the true anomaly, and of the argument of latitude, plus harmonic correction terms applied to the argument of latitude, the orbital radius and the inclination (S393).
Verification status: medium. The document is over 200 pages and the fetch returned a summary of the relevant sections rather than a verbatim table. Before putting specific equations in front of students, open Table 20-IV directly (S393).
There is an honest framing here worth passing on, because the popular version misleads. "GPS uses trilateration" is true and hides where the trigonometry is. The distance solve itself is algebraic: four pseudorange equations in four unknowns, three position coordinates and a clock offset, and there is no trigonometry in it. The trigonometry is in two other places. First, computing where each satellite is, from its Keplerian orbital elements, which is nothing but sines and cosines, per Table 20-IV. Second, converting the answer from Earth-centered Cartesian coordinates into a latitude and longitude you can put on a map. The spherical-geometry story is real, but it sits upstream and downstream of the trilateration, not inside it (S393).
What the evidence does not support
Five claims that circulate widely and do not survive contact with the sources.
"De Moivre stated de Moivre's formula"
He did not state it, in any form, anywhere in his published work.
Here is what he did publish. In 1707, closed-form roots for a family of odd-degree equations, Philosophical Transactions 25(309), pages 2368 to 2371. In 1722, a pair of cognate equations relating the versine of an arc to the versine of its -fold, Philosophical Transactions 32(374), pages 228 to 230 (S368, S369).
The modern formula is implicit in the 1722 result, as Pulskamp's translator's note 3 shows in three lines (S369). It is explicit in Introductio Book I section 133, printed in 1748, which is Euler's (S362). Even Wikipedia's article on the formula says he "never stated it in his works" (S395), and MacTutor describes only "a closely related formula" in 1707 (S370).
The correct sentence is: de Moivre published, in 1707 and 1722, results from which the formula follows immediately, and Euler stated it in modern form in 1748.
"The radian was named by Lord Kelvin's father"
No. The James Thomson who put the word into print on 5 June 1873, in examination questions at Queen's College Belfast, was Kelvin's older brother. He was born 16 February 1822, died 8 May 1892, and was professor of civil engineering at Belfast until 1873 (S382).
Kelvin's father, also called James Thomson, also a professor, of mathematics at Glasgow, died in 1849. He could not have set an 1873 paper.
Cajori says "brother" in A History of Mathematical Notations section 515 (1929). He says it again in the copy of A History of Mathematics, page 484 (1919) read for this chapter. Jeff Miller's compilation quotes that same 1919 page as saying "father" (S363, S364, S423). Both readings are on record, and this chapter does not resolve which sheet says what. The underlying fact is not in doubt either way.
"Prony's tables were computed by 80 to 100 unemployed hairdressers"
This is the most repeated story about the Cadastre project. The most careful modern study of the surviving manuscripts rejects it on both counts. Call it folklore, because that is what it is.
On the headcount. Prony's own numbers do not agree with each other. In 1801 he wrote that there were "about sixty or eighty" computers. In 1832 he wrote that there had been "between 150 and 200". Roegel worked from the payment summaries in the Archives Nationales, file F1bI 44, and reaches a much smaller figure (S373).
"These figures seem widely exaggerated, and at any one time, there were probably a lot less computers. The number of computers of the tables was probably never greater than 20 or 25, and Lalande even put them at 15."
Remember that the entire Cadastre office had 44 employees in September and October 1794, all of them named in the payroll (S373). There was no room for a hundred computers.
On the hairdressers. The story traces to Charles Dupin, a friend of Prony. He seems to have been the first to mention it, in 1824, and the printed reference is Dupin (1825), page 173. Roegel is blunt about it (S373).
"The 'hairdressers' were popularized by Grattan-Guinness' article, but in our opinion Grattan-Guinness gave too much importance to a detail, as if all computers were from that trade. Perhaps there were only two or three of them. It is unfortunate that other authors have amplified this idea."
He adds that "Prony seems never to have mentioned hairdressers", though Prony did write in 1824 that several of the computers "sought and found a kind of safe haven, one that political circumstances made them necessary" (S373).
So here is the story at its true size. Perhaps two or three former hairdressers, among perhaps twenty computers. The anecdote was first told thirty years after the fact, by a friend of the director. A 1990 article amplified it. Every popular account now repeats it as though it were the defining fact of the project.
Record the disagreement. MacTutor still states that de Prony worked with "between 70 to 80 assistants" (S374). Roegel disputes both the headcount and the hairdresser emphasis, and he is the one working from the payroll records (S373).
"Euler's E170 is a 1748 paper"
It is not. The Euler Archive record for E170, Recherches sur les racines imaginaires des equations, gives: written 1746, published 1751, in Memoires de l'academie des sciences de Berlin volume 5, pages 222 to 288 (S385).
The related paper E168, on the Leibniz and Bernoulli controversy itself, sits in the same volume 5, at pages 139 to 179, published 1751. It was written in 1747 according to the Euler Archive, and in 1749 according to Bal (S383, S384). Both writing dates are on record and neither is settled here.
"Euler's 'i' in Introductio section 138 is the square root of minus one"
It is not. In section 138 Euler introduces explicitly as "an infinitely large number", and throughout Chapter VIII he writes out longhand every time it occurs, never abbreviating it (S362).
So in the expression , the is an infinity, and the formula is the limit definition of the exponential with an infinite exponent. MacTutor dates Euler's own adoption of for to 1777, twenty-nine years later (S394). Any modern reprint or lecture that silently reads section 138's as the imaginary unit is producing nonsense.
Two more, briefly
"Cotes invented radian measure." MacTutor asserts it in a list of achievements, with no source and no elaboration. No primary evidence for it was found (S367). Unsupported until somebody produces the page.
"Euler invented the symbol ." What the Introductio shows is Euler defining and using it in section 126, in the book that made it universal (S362). The priority question was not investigated for this chapter; a separate strand of this project traces the symbol to William Jones in 1706 (S423). "Euler made standard" is defensible. "Euler invented " is not something this chapter's evidence supports.
For the classroom
History: the French Revolution as a mathematics problem. The Cadastre tables exist because the Revolution decimalised measurement, and decimalising measurement meant re-dividing the circle into 400 grades. That single political decision made every trigonometric table in Europe useless overnight. The response was to industrialize arithmetic, openly and on the record, using Adam Smith's pin factory as the model (S373). A history class and a mathematics class can teach the same document. Ask what else a change of units breaks.
Computing: where the word "computer" comes from. A computer was a job. Mary Edwards in Ludlow, Henry Andrews in Royston, Louis Saget in Paris, Alice Everett at Greenwich. All of them held the job title, and they held it for a century and a half before any machine did (S373, S389, S390, S391). Prony's three-tier scheme is also a good introduction to parallel processing: one setup step, twenty workers doing identical work on different chunks of the range, one checking step. Roegel's point is that this is not a pin factory, because nobody specialized. A computer scientist makes the same point about data parallelism versus a pipeline (S373).
Music: beats you can count. Get two tuning forks a few hertz apart, or two tone generators on phones. Count the throbs per second with a stopwatch, then predict the count from Helmholtz's law and check it against the sum-to-product identity (S380). Then ask why the beat rate is and not , which is the envelope frequency. The answer, that the ear hears magnitude and a cosine peaks twice per cycle, is a good piece of reasoning about graphs.
Audio compression, on the student's own phone. Every MP3 and AAC file they own is stored as MDCT coefficients, which are cosine amplitudes (S378). Have them find the bitrate of a file and work out how many numbers per second that is, then compare it with 44,100 samples per second times 16 bits times 2 channels for uncompressed audio. The ratio is what the cosine transform bought.
Medical imaging. A CT scanner never measures a point inside you. It measures sums along lines from many angles and reconstructs backwards. A good classroom version: put a grid of hidden numbers on paper, give students only the row sums, the column sums and the diagonal sums, and ask them to recover the grid. That is tomography, at 4 by 4 instead of 512 by 512.
Gravitational-wave astronomy. Put the LIGO strain plot next to . Have students read the rising frequency straight off the graph, then check it against the paper's numbers: about 8 cycles, 35 Hz to 150 Hz, over 0.2 seconds, and peak strain . The abstract's 35 to 250 Hz covers the full event, including the ringdown (S379). Ask them which of those two frequency ranges they would quote, and why you need both.
Language and notation design. Give students Herschel's 1813 sentence warning that "must not be understood to signify " (S363), plus his argument by analogy with , and ask them to design something better. Then tell them three things. Scherffer in Vienna and Lagrange in Berlin both printed "arc" in 1772. It is still in every programming language. And Cajori was still calling the contest undecided in 1929 (S363). Notation is a thing people argue about and vote on with their pens.
Gender and credit in scientific labor. Mary Edwards's work was invoiced under her husband's name from 1773 to 1784. After his death she had to tell the Astronomer Royal that she had been doing it all along (S391). Alice Everett was rejected for fellowship of the Royal Astronomical Society and spent twenty years without steady scientific employment after leaving Greenwich (S389). The Cadastre computers signed pages that carry no author's name at all. Ask who gets named on a piece of work, who gets paid for it, and whether those are the same person.
From a length in somebody's circle to the alphabet of waves
Trace the whole arc of this book and it comes down to four sentences.
A sine started as a length. It was a physical half-chord, measured inside a particular circle of a particular radius, by an Indian astronomer who needed to predict an eclipse.
It became a ratio. European table-makers, and their Islamic-world predecessors before them, realized the radius could be divided out, and that the number left over would work in any triangle of any size.
It became a function of a number in 1748. Euler set the radius to 1 and wrote "sin. z" with standing for nothing but a number. Now the sine could be added, differentiated, expanded in a series, and fed to itself.
It became the alphabet every wave is written in, in 1822. Fourier claimed in article 235 that any curve you can draw at all is a sum of sines and cosines. That is why the sound coming out of your phone, the image on its screen, the current in its charger, the picture a hospital scanner makes of the inside of your head, and the signal from two black holes colliding a billion light years away are all written in the same six letters that Thomas Finck abbreviated in Basel in 1583.
- Newton and Cotes turned the sine into an infinite series; Taylor and Maclaurin made series a general machine.
- Euler's 1748 Introductio made sine and cosine functions of a number (S361, S362), and tied them to the exponential.
- Fourier claimed every heat flow decomposes into sines: the wave becomes the atom of analysis. The radian arrives absurdly late, named in print in 1873.
Next, and last: chapter 8 is the reference shelf, where every word and theorem in your course gets its documented biography.
The mode button on your calculator is this chapter's youngest artifact: the radian, named in an 1873 Belfast exam paper, is the natural unit of the function Euler built. It is the subject of Trigonometry 2.2
The radian is younger than the telephone. Who put the word in print first, and what settles the date?
Show the answer▸
James Thomson the younger, on 5 June 1873, in examination questions at Queen's College, Belfast (S423). A widely repeated claim dates it to 1867; this book read the 1867 edition end to end and the word is not there (S489, S490), so 1873 stands, and a lifespan check, the man died in 1892, his father in 1849, sorts out which James Thomson it was.
Chapter 8
Where Every Word and Every Theorem Came From
The people in this chapter
Faces where a face survives. Every name links to its full entry in Appendix A.
A sentence in Pisa, 1220
In the Practica geometriae (The Practice of Geometry), finished in Pisa in 1220, Leonardo of Pisa, called Fibonacci, labels two segments of a diagram:
".be. uocatur sinus rectus utriusque arcus .ab. et .be.; et recta .ae. uocatur sinus uersus arcus .ab." (S421)
"BE is called the straight sine of each of the arcs AB and BE; and the straight line AE is called the turned sine of the arc AB." Two words on that page, sinus rectus and sinus versus, are the ancestors of "sine" and "versine." Neither is a Latin word for anything mathematical. Sinus means a fold of cloth, the hang of a toga across the chest, a bay in a coastline. It sits in that sentence because of a chain of translation accidents running from Sanskrit through Arabic into Latin. It also sits there because somebody in twelfth-century Spain made a choice nobody has been able to pin on a named person with a document.
That is the situation for most of the vocabulary in your course. The mathematics usually has a clear enough history. The words almost never do. That gap is what this chapter is about.
Standard references credit the word "cosecant" to Rheticus's Opus Palatinum of 1596. Ball printed it, Smith repeated it, a century of textbooks followed. When a modern historian finally went looking in the 1596 pages themselves, what did he find?
I have a guess▸
The word is not there. Van Brummelen, the leading historian of trigonometry, reports that he cannot find the term in the Opus Palatinum (S427). Ball had written "I think", Smith copied Ball, and the hedge fell off somewhere between editions. The attribution has been checked, and it failed.
If you guessed the credit would hold, you trusted the chain most references still print. This chapter exists because chains like it fail on inspection more often than the confident footnotes suggest.
How to read this chapter
This is a register, not a narrative. For every term, theorem and identity in the course it answers four questions in the same order:
- When was it first stated in any form, however unlike ours?
- When was it first stated in something close to the modern form?
- Who gave it its modern name, and when?
- What does every technical word in it literally mean, and in what language?
The four answers are almost never the same date. The gaps between them are where the interesting history lives. Take the versed sine. The function is Indian and at least sixteen hundred years old. The Latin name is thirteenth century. The English name is 1596. The modern short form "versine" and its cousin "haversine" are navigational conveniences from the age of steamships. One idea, four birthdays, and the middle two are in different languages.
Where the sources disagree, this chapter names both sides. Where the popular attribution is wrong or unverifiable, it says exactly that, in those words, and then sets out what the sources do show. Twenty-five claims get that verdict, and the full list sits at the end. A teacher who reads nothing else here should read that list. It marks every place where confident classroom statements outrun the evidence.
Source anchors in square brackets, like (S421), point to the sources this book read. Four works carry most of the weight. Smith's two-volume History of Mathematics of 1923 and 1925 (S422, S421). Cajori's two-volume A History of Mathematical Notations of 1928 and 1929 (S424, S423). Von Braunmühl's two-volume Vorlesungen über Geschichte der Trigonometrie of 1900 and 1903 (S425, S426). And Jeff Miller's Earliest Known Uses of Some of the Words of Mathematics (S427), which collects Oxford English Dictionary citations alongside the specialist literature.
The vocabulary of evidence
Six words from the historian's trade run through this chapter, so here they are up front, in plain English.
- An attestation is one recorded appearance of a word in a surviving document. A first attested use is the earliest one anybody has found so far, which is not the same as the first time anybody said it.
- Floruit, Latin for "he flourished," marks the years a person is known to have been active. Historians use it when the birth and death dates are lost.
- A colophon is the note at the end of a manuscript or early printed book giving the scribe or printer, the place, and the date.
- The editio princeps is the first printed edition of a work that circulated for years in handwriting.
- A recension is one editor's reworked version of a text, with the wording changed as it was copied and corrected.
- A terminus ante quem, "limit before which," is the latest possible date for something, fixed by a dated thing that already refers to it.
The problem people were trying to solve
Nobody coined "tangent" because the word was pretty. Names arrive when somebody builds a table. Tables got built because people had to predict a planet, fix a calendar, find the direction of prayer, cross an ocean, or aim a gun. The shadow words (umbra recta, umbra versa) come from sundials, which measure shadows. "Sine" comes from a translation pipeline paid for by astronomy. "Haversine" comes from a nineteenth-century navigator trying to shave one step off a computation done at sea in bad light. "Radian" comes from a physics department that needed an angle unit to make rotational formulas come out clean. Every term here has a job behind it. When the job disappeared, so did the word. Nobody outside a history book says "versed sine" any more, because nobody navigates by log tables.
Part 1: The names of the functions and the units
Sine: four candidates, no winner
The chain of meanings
The word starts in Sanskrit. A chord of a circle is a jya, a bowstring. Indian astronomers worked with half of it, the ardha-jya or jya-ardha, "half-chord," which they shortened back to jya (S421, S444). Arabic writers borrowed the sound rather than the sense and wrote it jiba, which means nothing in Arabic.
Braunmühl, following the orientalist Munk, sets out what happened next:
"Die Araber hatten zur Bezeichnung des Sinus das Wort 'dschaib' ... übernahmen sie nämlich das von den Indern auch für die halbe Sehne gebrauchte Wort dschya oder dschiva und schrieben es dem Wortlaute nach dschiba. Aber die Konsonanten, welche dschiba zu lesen sind, lassen, wenn die diakritischen Punkte ... nicht gesetzt werden, auch die Lesart dschaib zu ... Denn dschaib ist ein wirkliches arabisches Wort, das Busen, Herz, Bausch oder Tasche bedeutet." (S425, vol. 1, pp. 49 to 50)
The Arabs used the word jaib for the sine. They took over jya or jiva, the word the Indians also used for the half-chord, and wrote it phonetically as jiba. But drop the diacritical dots and the consonants that spell jiba can equally be read as jaib. And jaib is a real Arabic word meaning bosom, heart, bulge or pocket. Braunmühl then adds the linguists' verdict, which is worth reading to a class:
"Aus einem unverstandenen Fremdwort wurde so ein ähnlich klingendes der eigenen Sprache, ein in der Sprachgeschichte so gewöhnlicher Vorgang, dass es überflüssig ist, Beispiele anzuführen." (S425)
An unintelligible foreign word turned into a similar-sounding word of the speakers' own language. That process is so ordinary in the history of language that it is pointless to give examples. Then a Latin translator met jaib, looked up "bosom, fold, pocket," and wrote sinus.
| Term | Meaning | Sources |
|---|---|---|
| Sanskrit jya, jiva | bowstring, chord | S421 |
| Sanskrit ardha-jya, jya-ardha, shortened to jya | half-chord | S421, S444 |
| Arabic jiba (written jb) | nothing; a transliteration | S425 |
| Arabic jaib (same consonants, different vowels) | bosom, heart, bulge, pocket, bay | S425 |
| Latin sinus | bend, fold, curve, bay, the fold of a toga at the breast | S442 |
| English sine | (no independent meaning) | S427 |
One point gets lost in the retelling: jaib was already the settled Arabic term long before any Latin translator was involved. Braunmühl, citing Chasles in 1846, notes that the sine is present in al-Khwarizmi's small Sindhind. Its Latin translator renders the functions as elgeib elmustewi, the straight sine, and elgeib elmacus, the versed sine (S425). He transliterates the Arabic word there rather than translating it. So the "mistranslation" story is about the Latin step only.
The four candidates
Now the part textbooks get wrong. The honest answer is that the popular attribution is wrong or unverifiable. Naming Robert of Chester or Gherardo of Cremona as the man who coined sinus goes beyond anything in the documents. Four different answers to "who first wrote sinus in the trigonometric sense" are in print, and the sources cite each other rather than any manuscript (S425, S422, S421, S427). Here is the evidence behind each.
The table uses floruit, "he flourished," the historian's label for the years a person is known to have been working when the birth and death dates are lost.
| Attributed to | Who makes the attribution | Status | Sources |
|---|---|---|---|
| Plato of Tivoli (PLAY-toh of TIV-oh-lee, floruit c. 1116 to 1138), translating al-Battani | Cajori (1906), reported by Miller | ruled out by Braunmühl on textual grounds | S427, S425 |
| Robert of Chester (ROB-ert of CHESS-ter, floruit 1141 to 1150), revising al-Khwarizmi's tables | Smith vol. 1 p. 202 footnote 4; Boyer p. 278, dating it 1145 | not established | S422, S427 |
| Gherardo of Cremona (ger-AR-doh of kre-MOH-nuh, 1114 to 1187), translating Western Arabic astronomy at Toledo | Braunmühl ("wahrscheinlich"); Smith vol. 2 p. 616; Eves | not established | S425, S421, S427 |
| An unnamed member of the twelfth-century translation movement | the diffuse fallback position | not established | S425 |
Braunmühl rules Plato of Tivoli out, and that is the one solid documentary result in the whole dispute:
"Dass übrigens nicht Plato von Tivoli in seiner Übersetzung des Al-Battani zuerst dieses Wort eingeführt hat, wie man jetzt sehr häufig lesen kann, darauf haben schon Kästner und andere aufmerksam gemacht. Es findet sich nämlich nur einmal im Texte die Bezeichnung sinus versus, sonst heisst es daselbst beständig 'chorda' und 'chorda versa'." (S425, vol. 1, p. 50)
One can now very often read that Plato of Tivoli first introduced this word in his translation of al-Battani. Kästner and others already pointed out that he did not. The term sinus versus occurs only once in the text; everywhere else it says chorda and chorda versa. That is a claim about the words on the pages of a specific book. One occurrence, in a compound, against a running text that uses the other word throughout. A translator introducing a technical term does not use it once.
Braunmühl's own preference is for Gherardo, and he flags his own uncertainty in the sentence:
"Der fruchtbarste Übersetzer arabischer Werke im 12. Jahrhundert war Gerhard von Cremona (1175 in Toledo) und dieser war es wahrscheinlich, der durch seine Wiedergabe verschiedener astronomischer Werke der Westaraber ... die Einführung des Wortes sinus im Abendlande verschuldete." (S425)
The most prolific translator of Arabic works in the twelfth century was Gerhard of Cremona (at Toledo in 1175). It was probably he who introduced the word sinus in the West, through his renderings of various astronomical works of the Western Arabs. The load-bearing word there is wahrscheinlich, "probably."
Smith is worse than uncertain: he contradicts himself between his own two volumes. Volume 2, page 616: "When Gherardo of Cremona (c. 1150) made his translations from the Arabic he used sinus for jaib, each word meaning a fold, and this usage, possibly begun even earlier, was followed by other European scholars" (S421). Volume 1, page 202, main text, on Robert of Chester: "In his translation is found one of the early uses of the word sinus for a half chord" (S422). And then volume 1, page 202, footnote 4, on the same page as that hedged sentence: "On the question of priority and of the use of the term by Plato of Tivoli, see A. Braunmühl, Geschichte der Trigonometrie, I, 49 ... The term was probably first used in Robert of Chester's revision of the tables of al-Khowarizmi" (S422). The same author, in the same multi-volume work, credits Gherardo in one book and Robert of Chester in a footnote in the other.
Boyer supplies the mechanism for the Robert of Chester version: "When Robert of Chester came to translate the technical word jiba, he seems to have confused this with the word jaib (perhaps because vowels were omitted); hence he used the word sinus" (S427). Notice what that is. It is a story about how the confusion happened, not evidence about who first wrote it down. And the confusion had already happened in Arabic, before any Latin translator touched it.
So here is what a teacher can say in class. A Latin translator introduced the word in twelfth-century Spain. The text of his own translation rules Plato of Tivoli out. Braunmühl leans to Gherardo and says "probably" rather than anything firmer. Smith's own footnote leans to Robert of Chester. Nobody has produced manuscript-level evidence for any of them.
Sine in English
The English word arrives at the end of the sixteenth century. The OED's first citation, as reported by Miller, is Thomas Fale's Horologiographia. The art of dialing of 1593: "This Table of Sines may seem obscure..." (S427). Note the setting. Fale is writing about sundials, and the sine table is a tool for laying out dial lines. That practical work is what carried these words into English.
Sinus rectus, sinus versus and sagitta
Fibonacci's 1220 sentence, quoted at the top of this chapter, is the earliest firmly dated Latin appearance of the pair in this book's sources (S421). The pair translates an Indian pair: kramajya, the "straight sine," and utkramajya, the "inverse-order" or "turned" sine (S421, S443).
The same passage of Fibonacci uses a third word for the versed sine: sagitta, "arrow" (Scritti II, 94) (S421). That is a translation of Arabic sahm, also "arrow." The picture is a bow. The arc is the bow and the chord is the bowstring. The little segment from the middle of the string to the middle of the arc is the arrow lying across it. The Sanskrit word for chord, jya, is "bowstring," so the whole image survived the trip from India to Pisa intact. The vocabulary changed languages twice; the picture did not.
Cosine: three separate coinages
Gunter did not coin "cosine." He coined co.sinus, with a full stop in it, because it was an abbreviation of a phrase.
"Regiomontanus (c. 1463) used sinus rectus complementi. Rhaeticus (1551) preferred basis, Vieta (1579) used sinus residuae, Magini (1609) used sinus secundus, while Edmund Gunter (1620) suggested co.sinus, a term soon modified by John Newton (1658) into cosinus, a word which was thereafter received with general favor." (S421, vol. 2, p. 619)
So the sequence runs like this:
- Regiomontanus (REE-jee-oh-mon-TAH-nuss, 1436 to 1476) writes "the right sine of the complement"
- Rheticus (RET-ih-kuss, 1514 to 1574) calls it the "base"
- Vieta calls it "the sine of the remainder"
- Magini calls it "the second sine"
- Edmund Gunter (GUN-ter, 1581 to 1626) clips the phrase to co.sinus, in his Canon triangulorum, sive, Tabulae sinuum et tangentium artificialium ad radium 100000.0000. & ad scrupula prima quadrantis (1620) (S427)
- John Newton (no relation to Isaac) closes the abbreviation up into one word, cosinus, in the Trigonometria Britannica of 1658 (S421, S427)
The English "cosine" beats the Latin cosinus into print by twenty-three years. Miller gives the OED's 1635 citation from John Wells, Sciographia: "As the Radius Is to the cosine of the angle given" (S427).
The "co-" is a clipping of Latin complementum, "that which fills up, completes." The complement of an angle is what completes it to a right angle. So "cosine" means "the sine of what is left over." That single fact explains cosine, cotangent, cosecant, coversine and the cofunction identities all at once. It is worth two minutes of class time, because it turns four memorized names into one idea.
One correction to Smith comes from Glen Van Brummelen, reported by Miller. Peter Apianus used sinus rectus secundus for the cosine in his Instrumentum sinuum seu primi mobilis of 1541, earlier than any example in Smith's list (S427). Smith's sequence is not wrong. It is just not complete.
Cotangent
Cotangens is Gunter's, in the same 1620 Canon triangulorum, formed the same way from complementum (S427, S421). Miller gives the earliest English as 1714, in Edward Wells's The Young Gentleman's Trigonometry (S427).
Cosecant: an attribution that has been checked and failed
The popular attribution is unverifiable. The usual credit goes to Rheticus's posthumous Opus Palatinum de triangulis of 1596. Miller sets out both that credit and its collapse:
"Some sources say the word cosecant was introduced by Edmund Gunter (1581-1626). However, he apparently did not use the term. Ball (page 243) and Smith (vol. 2, page 622) say the term cosecant seems to have been first used by Rheticus, reporting that the Latin cosecans appears in Opus Palatinum de triangulis ... Ball wrote 'I think' it came from Rheticus, and Smith probably took the information from Ball. However, Glen Van Brummelen, in an email in 2014, reports that looking at Opus palatinum he cannot find the term." (S427)
Read that chain carefully, because it is a model of how a fact decays. Ball writes "I think." Smith repeats Ball without the hedge. Everyone repeats Smith. Then somebody opens the actual book and the word is not there. Until someone produces the page and line from the Opus Palatinum, nobody should say "cosecant was coined by Rheticus in 1596" in a classroom.
What can be said: the earliest verified attestation is English, not Latin, and it is John Newton again. Trigonometria Britannica, 1658: "And as the co-tangents are made from the Tangents, so are the Secants to be made from the sines, For as the sine of an Arch, is to Radius, so is Radius to the co-secant of that Arch by the 31th of the first" (S427).
Tangent and secant: one sentence in Basel, 1583
These two have the cleanest coinage story of the six, because both words appear in one sentence of one book. Thomas Fincke (THOM-us FINK-uh, 1561 to 1656), a physician from Flensburg, published Geometria rotundi (The Geometry of the Round) at Basel in 1583. Smith quotes his Latin from page 73:
"Recta sinibus connexa est tangens peripheriae, aut eam secans" (S421, vol. 2, p. 621 n. 4)
A straight line joined to the sines is either touching the circumference or cutting it. Tangens is the present participle of tangere, "to touch." Secans is the present participle of secare, "to cut." Both are ordinary Latin participles pressed into service, and they describe the picture exactly. The tangent line touches the circle at one point. The secant line cuts through it.
Cajori gives the reception history, and it includes an objection from the best mathematician of the generation:
"Perhaps the first use of abbreviations for the trigonometric lines goes back to the physician and mathematician, Thomas Finck, a native of Flensburg in Schleswig-Holstein. We premise that we owe to him the invention in 1583 of the words 'tangent' and 'secant.' These terms did not meet with the approval of Vieta, because of the confusion likely to arise with the same names in geometry. Vieta called the trigonometric tangent Prosinus and the trigonometric secant Transsinuosa. But Vieta's objection was overlooked or ignored. The trigonometric names 'tangent' and 'secant' were adopted by Tycho Brahe in a manuscript of 1591, by G. A. Magini in 1592, by Thomas Blundeville in 1594, and by B. Pitiscus in 1600." (S423, section 517)
Vieta was right about the ambiguity. It is why your geometry course and your trigonometry course both use "tangent" for related but different things, and why students confuse them every year. His replacements, Prosinus and Transsinuosa, lost.
The English "secant" appears in Thomas Blundeville's Exercises of 1594, in the phrase "The Table of Secants," which is also the year Cajori lists for Blundeville's adoption of the terms (S427, S423).
One caution about the evidence. Cajori says plainly where his information about Fincke's book comes from: Glaisher's 1915 article in the Quarterly Journal of Pure and Applied Mathematics, vol. 46, p. 172. He did not see the book itself (S423). This book could not obtain a scan of the Geometria rotundi either. The e-rara site returned an HTTP 500 error, and the Munich digital library's search interface requires JavaScript. Everything about the 1583 coinage rests on Smith's direct Latin quotation of page 73 and on Glaisher through Cajori. That is good evidence. It is not the same as having seen the page.
The shadows: umbra recta, umbra extensa, umbra versa
Before tangent and secant had Latin names, they had shadow names, because that is what they were: the length of a shadow cast by a stick.
Al-Battani (al-buh-TAH-nee, c. 858 to 929) is the key figure. Braunmühl at the source:
"Bemerkt muss noch werden, dass Al-Battani bereits zweierlei Schatten unterscheidet, je nachdem das Gnomon senkrecht auf einer horizontalen oder auf einer vertikalen Wand steht; der erstere wird in der Übersetzung mit 'umbra extensa' oder als Schatten schlechthin bezeichnet, während der letztere 'umbra versa' heisst. Wie sich zeigen wird, gaben diese Schatten sehr bald Veranlassung zur Einführung der Cotangenten und der Tangenten in die Trigonometrie der Araber." (S425, vol. 1, p. 52)
Al-Battani already distinguishes two kinds of shadow. Which one you get depends on whether the gnomon, the upright rod that casts the shadow, stands perpendicular to a horizontal surface or to a vertical wall. The translation calls the first umbra extensa, or simply "the shadow," and it calls the second umbra versa. These shadows very soon gave rise to the introduction of the cotangent and the tangent into Arabic trigonometry. Smith gives the same distinction with the alternative Latin forms: "the straight shadow, translated by the later medieval Latin writers as umbra, umbra recta, or umbra extensa, and the turned shadow, the umbra versa or umbra stans, the terms varying according as the gnomon was perpendicular to a horizontal plane, as in ordinary dials, or to a vertical wall, as in sundials on a building" (S421, vol. 2, pp. 620 to 621).
Here is a trap. It is tempting to hand students the rule "umbra recta equals cotangent, umbra versa equals tangent." Do not. Smith himself, on page 621, says the term "tangent" appeared "as the equivalent of umbra versa," while the shadow that behaves like the tangent is normally the one cast on a vertical wall (S421). Which shadow matches which function depends on the orientation of the gnomon and on the individual author. Medieval usage is not consistent. Say this instead: the two shadows are the two ratios, and you have to read which is which off the diagram in front of you.
The Latin umbra words are also the reason the tangent arrived in Europe attached to dialing instruments rather than to triangles. Robertus Anglicus, around 1231, still uses umbra on its own (S421).
Versed sine and versine
The versed sine is the oldest item in this part of the chapter, and the function is Indian. Smith:
"This function ... is first found in the Surya Siddhanta (c. 400) and, immediately following that work, in the writings of Aryabhata, who computed a table of these functions. A sine was called the jya; when it was turned through 90 degrees and was still limited by the arc, it became the turned (versed) sine, utkramajya, so that the versin phi = 1 - cos phi." (S421, vol. 2, p. 618)
Burgess's 1860 translation of the Surya Siddhanta confirms the term at the source and explains it: "The term used for versed sine, utkramajya, means 'Inverse-order sine,' the column of versed sines being found by subtracting that of sines in inverse order from radius" (S443). That describes a computational procedure, not a curve. You build the versed sine column by running down the sine column backwards and subtracting from the radius.
Burgess also catches the text in the act of shortening its own vocabulary: "In this passage, the sine is called jyardha, 'half-chord;' hereafter, however, that term does not once occur, but jya 'chord' (literally 'bowstring') is itself employed" (S443). One occurrence of the precise term, then the loose term forever after. That is how technical vocabulary behaves in practice. It is also why single-occurrence evidence, like the sinus versus in Plato of Tivoli's translation, has to be handled carefully.
Al-Battani, per Smith, "expressly states that he uses the expression 'turned chord' for the versed sine," which is the chorda versa that Braunmühl found running through Plato of Tivoli's Latin (S421, S425). Fibonacci gives the Latin sinus versus in 1220 (S421). The first English is 1596, in W. Burrough's Variation of the Compasse, where it is spelled "versed signe" (S427). Note the setting again: a book about magnetic compass variation, written for navigators.
The etymology is straightforward. Latin versus is the past participle of vertere, "to turn," from the Indo-European root meaning "to turn, bend," which also gives English "verse" (a turning of the plow at the end of a furrow, hence a line), "reverse," "version" and "universe" (S442).
Coversine
The coversine is the versed sine of the complement, and it has a symbol before it has a word. Cajori:
"William Jones, op. cit., in 1706, represented the coversed sine by v, and A. R. Mauduit, op. cit., in 1765, by u. The designation 'covers A' is found, for example, in G. A. Wentworth, Trigonometry (2d ed.; Boston, 1903), p. 5; H. H. Ludlow, Trigonometry (3d ed., 1891), p. 33; A. M. Kenyon and L. Ingold, Trigonometry (New York, 1913), p. 8, 9." (S423, section 527 n. 2)
So: a symbol in 1706, another symbol in 1765, and the written contraction "covers A" documented in American trigonometry textbooks of 1891, 1903 and 1913. No coiner of the word is on record. Confidence: medium, and only for the sequence of attestations, not for an inventor.
Haversine: the correction
This one needs stating carefully, because the popular version is wrong in a specific and checkable way.
The popular claim is: "James Inman coined 'haversine' in his 1821 Navigation and Nautical Astronomy." Two things are wrong with it.
First, the function is older than Inman by two decades. Cajori, in full:
"A. von Braunmuhl, Vorlesungen uber Geschichte der Trigonometrie, 2. Teil (Leipzig, 1903), p. 231; P. R. Rider and A. Davis, Plane Trigonometry (New York, 1923), p. 42. The haversine function first appears in the tables of logarithmic versines of Jose de Mendoza y Rios (Madrid, 1801, also 1805, 1809), and later in a treatise on navigation of James Inman (1821). See J. D. White in Nautical Magazine (February and July, 1926)." (S423, section 527 n. 1)
Read what Cajori says. The function first appears in the tables of José de Mendoza y Ríos (ho-SAY day men-DOH-thah ee REE-os, 1761 to 1816), published in Madrid in 1801 and again in 1805 and 1809. It turns up in Inman later. Cajori does not say Inman coined the word. He credits Inman with using the function.
Second, the date is probably the wrong edition. The Oxford English Dictionary (2nd edition, 1989) and Wikipedia both place the coinage in the third edition of 1835, not the first edition of 1821 (S441). Wikipedia's Versine article states: "James Inman coined the term 'haversine' in 1835 in the third edition of Navigation and Nautical Astronomy: For the Use of British Seamen" (S441). The same article records that "The first table of haversines in English was published by James Andrew in 1805" under the name "Squares of Natural Semi-Chords" (S441).
This book could not obtain a scan of either the 1821 or the 1835 edition. What it did read is the 1858 edition. There "log. haversines" occurs more than twenty times, in phrases like "turned into the other very readily by means of the Table of Log. Haversines" and "the halves of log. haversines of the two last terms in the form" (S435). So direct inspection verifies the word in Inman in 1858 and nowhere earlier.
The correct classroom statement runs like this. The word "haversine" is usually credited to James Inman, and the credit is probably right, but the usual date of 1821 is probably wrong. The OED and Wikipedia place the coinage in the third edition of 1835. Cajori, the standard authority on such questions, credits Inman only with using the function, and credits its first tabulation to Mendoza y Ríos in Madrid in 1801. Do not assert 1821.
The word itself is transparent once you split it: half + versed + sine, the Latin equivalent being semiversus. And the reason a navigator wanted it is worth showing. The haversine of an angle is , which is never negative. So its logarithm always exists, and a whole class of sign errors in the great-circle distance computation disappears. The function exists because logarithms of negative numbers do not.
Exsecant
The external secant, , is the distance from a circle to the point where a tangent line meets a secant line. Railway surveyors used it. Cajori records only the symbol: "A third, 'exsec A' i.e., 'external secant of A,' signifies sec A - 1" (S423, section 527). He gives no coiner and no date, and no better authority turned up. The honest statement is that the function has a name, a symbol and a documented use, and nobody has established who coined it.
Degree, minute, second
The chain here runs Greek to Latin to French to English. The two smallest units are numbered: a first part and a second part.
Smith gives it in one passage with the footnotes attached:
"When the Greeks decided to take 1/360 of a circle as a unit of arc measure, they called this unit a degree. [footnote: Moira; medieval Latin, de + gradus (step). The Arabs translated moira by daraja (ladder, scale, step)] They called 1/60 of a degree a first part, [footnote: Prota hexekosta; Latin, pars minuta prima (first small or fractional part). From this came our 'minute.'] 1/3600 a second part, [footnote: Deutera hexekosta; Latin, pars minuta secunda, from which our 'second.'] and so on." (S421, vol. 2, p. 232)
| Term | Literal meaning | What it names | Sources |
|---|---|---|---|
| Greek moira | "part, portion" | the 360th part of a circle | S421 |
| Arabic daraja | "ladder, scale, step" | the Arabic translation of moira | S421 |
| Latin gradus, then de + gradus | "a step," then "a step down" | gives French degré, English degree | S421, S442 |
| Greek prota hexekosta to Latin pars minuta prima | "first sixtieths" to "first small part" | gives English minute | S421 |
| Greek deutera hexekosta to Latin pars minuta secunda | "second sixtieths" to "second small part" | gives English second | S421 |
A minute is a small part. The adjective is the same word as English "minute" meaning tiny, which is why that word is stressed on the second syllable. A second is the second small part, meaning the second application of the sixtieth. There is no third in modern use, but there was: the sequence continued to thirds and fourths in medieval astronomy.
Why 360? Smith gives the standard reconstruction: "The Greeks may thus have been led to divide the radius into 60 equal parts and the diameter into 120 of these parts. Since the common value of pi was 3 in ancient times, the circumference was naturally taken as 3 x 120, or 360" (S421). Radius 60, diameter 120, circumference three diameters, so 360 units around. That arithmetic is worth running in class. With , the circumference of a circle of radius 60 is . A degree is then nearly one unit of arc length on the standard astronomical circle.
The English "degree" is attested early. Miller gives Chaucer, Canterbury Tales, c. 1386: "The yonge sonne That in the Ram is foure degrees vp ronne" (S427). That is astronomy, in verse, in Middle English, four hundred years before the word "trigonometry" reached English.
Now a warning about the symbols. It is tempting to say that the degree, minute and second marks come straight from the Greek. Cajori says the inference does not hold:
"Signs resembling those now in use are found in the Syntaxis (Almagest) of Ptolemy ... The first sixtieths or minutes were marked with one accent, the second sixtieths with two accents ... From these facts it would seem that our signs for degrees, minutes, and seconds were of Greek origin. But it is difficult to uphold this view, especially for the sign for 'degrees.' Such a line of descent has not been established." (S423, section 511)
The prime and double prime for minutes and seconds do look like Ptolemy's accents. The little raised circle for degrees has no established Greek ancestor. "It looks similar" is not a line of descent.
Radian: a two-person tie that neither party would break
Radian first appears in print on 5 June 1873, in examination questions set by James Thomson the younger (TOM-son, 1822 to 1892) at Queen's College, Belfast (S423, S427). Cajori, Notations volume 2, section 515: "The word 'radian' was first used in print in 1873 by James Thomson, a brother of Lord Kelvin" (S423).
Cajori's own History of Mathematics of 1919, page 484, gives more. What follows is Miller's reproduction of that page, quoted as Miller prints it, and it matters that this is a copy of a copy: collated word for word against the 1919 page itself, Miller's transcription differs from Cajori in four places, and one of the four is a factual error about Lord Kelvin's family (S364, S427). Chapter 7 walks through the collation.
"An isolated matter of interest is the origin of the term 'radians', used with trigonometric functions. It first appeared in print on June 5, 1873, in examination questions set by James Thomson at Queen's College, Belfast. James Thomson was the father of Lord Kelvin. He used the term as early as 1871, while in 1869 Thomas Muir, then of St. Andrew's University, hesitated between 'rads,' 'radials' and 'radians.' In 1874, T. Muir adopted 'radians' after a consultation with James Thomson." (S427)
"The father of Lord Kelvin" is not Cajori's sentence. The 1919 page, read directly, says "a brother of Lord Kelvin", and the 1929 Notations says brother as well (S364, S363). The "father" above is what Miller's transcription of that same 1919 page reads, and chapter 7 leaves open whether that reflects a variant printing or a one-word slip in the copy; either way, Miller is the compilation everybody quotes, so from Miller it spread (S427). Both cannot be true, and a death date settles it: the James Thomson who was professor at Queen's College Belfast in 1873 was William Thomson's brother, 1822 to 1892; their father, also named James Thomson, died in 1849 and could not have set an examination in 1873. Show that to students. The wrong version traces through a copy of a good source, the copy is more widely read than the original, and checking a date against a lifespan is what catches it.
Thomas Muir (MYOOR, 1844 to 1934) arrived at the word independently, and he had help. Miller quotes Muir's April 1910 letter: "I wrote to him [i.e., to Alexander J. Ellis, in 1874] and he agreed at once for the form 'radians,' on the ground that it could be viewed as a contraction for 'radial angles'" (S427). Alexander J. Ellis (ELL-iss, 1814 to 1890), the phonetician and philologist, is the reason the word has the shape it does: "radian" as a squeeze of "radial angle."
The son of the man who set the exam settled the dispute in Nature in 1910. Miller quotes his June 1910 letter:
"I shall be very pleased to send Dr. Muir a copy of my father's examination questions of June, 1873, containing the word 'radians.' ... It thus appears that 'radians' was thought of independently by Dr. Muir and my father, and, what is really more important than the exact form of the name, they both independently thought of the necessity of giving a name to the unit-angle." (S427)
That is as gracious a priority dispute as mathematics has produced. The last clause carries the serious point: naming the unit angle mattered more than who named it.
Two loose ends, both honest.
A third candidate exists in print. Miller quotes W. N. Roseveare in the Mathematical Gazette 3(49), January 1905, p. 133: "The radian is an uninteresting angle. Lord Kelvin introduced the word merely as a convenience in lecturing, to avoid the long phrase 'angle whose circular measure is'" (S427). That attributes the word to William Thomson rather than to his brother James. It is a single sentence in a note published thirty-two years after the fact. It also conflicts with the family's own account of 1910.
And a dating claim that did not survive checking. Miller states that Thomson and Tait's Treatise on Natural Philosophy of 1867, page 31, already contains "For brevity we shall call this angle a radian," which would predate the Belfast exam paper by six years (S427). This book read the 1867 first edition end to end, and the word "radian" does not occur in it: zero occurrences in the full text, against nine in the 1879 new edition (S489, S490). What 1867 has is the idea without the name, "the unit angle, or the angle of which the arc is equal to radius" (S489). The name is in the 1879 edition at section 41, "The usual unit angle is (as explained in treatises on plane trigonometry) that which subtends at the centre of a circle an arc whose length is equal to the radius ... For brevity we shall call this angle a radian," and in its appendix heading "2. Space. Yard and Mètre: Radian, Degree, Minute, Second" (S434). Etymonline independently dates the English word's spread to 1879 (S442). So the earliest print occurrence stands at the 1873 Belfast paper, the 1867 attribution is a widely repeated claim that a direct read refutes, and the priority story above needs no asterisk.
Etymology: Latin radius means "staff, rod, spoke of a wheel, ray" (S442). The radian is the angle whose arc equals the spoke.
Pi: the symbol, not the number
William Jones (JOHNZ, 1675 to 1749) introduced the symbol in 1706, and Cajori's account of why is charming:
"The modern notation for 3.14159 ... was introduced in 1706. It was in that year that William Jones made himself noted, without being aware that he was doing anything noteworthy, through his designation of the ratio of the length of the circle to its diameter by the letter pi. He took this step without ostentation." (S423, section 396)
The sentence itself, from page 263 of the Synopsis palmariorum matheseos, quoted by Cajori: "There are various other ways of finding the Lengths or Areas of particular Curve Lines, or Planes, which may very much facilitate the Practice; as for instance, in the Circle, the Diameter is to the Circumference as 1 to [series] = 3.14159, etc. = pi" (S423). Cajori adds a detail that shows Jones was not consciously founding a notation. He had already used pi twice earlier in the same book with different meanings: once as a label for a point on page 241, and once for "Periphery" on page 243, following Wallis (S423).
There is a prehistory. Oughtred's Clavis mathematicae, page 66, used for the ratio, with pi standing for "periphery" and delta for "diameter," and Barrow and David Gregory followed him (S423). So the letter pi was already attached to the circumference of a circle. Jones's step was to let it stand for the ratio on its own.
And there is the reason it stuck, which is not 1706. Cajori, section 397: "in 1736 he designated that ratio by the sign 1 : pi and thus either consciously adopted the notation of Jones or independently fell upon it ... Particularly favorable for wider adoption was the appearance of pi for 3.1415 ... in Euler's Introductio in analysin infinitorum (1748)" (S423). Note Cajori's honesty: he does not know whether Leonhard Euler (OY-ler, 1707 to 1783) took the symbol from Jones or reinvented it, and he says so.
Even then it spread slowly. Cajori, section 398, records that Diderot in 1748 still wrote the ratio out as a fraction. Segner in 1767 was still using Oughtred's pi-over-delta (S423). Sixty-one years after Jones, a working mathematician was still using the older notation. Notation does not win by being better. It wins by being in the book everybody reads.
Amplitude, period, phase, frequency, sinusoid, harmonic
This is the weakest group in the chapter, and the book says so rather than inventing precision. All six words existed in English before anyone applied them to a sine graph. For none of them could this book date the moment it was first used of a trigonometric curve.
| Word | Origin and literal meaning | Earliest use verified here | Sources |
|---|---|---|---|
| amplitude | Latin amplus "large, spacious", amplitudo "wide extent, width" | English 1540s in the general sense; the earliest mathematical use verified is Legendre, Second mémoire sur les intégrations par arcs d'ellipses, 1786, for the angle of an elliptic integral, not for a wave | S427, S442 |
| period | Greek peri "around" + hodos "a going", so "a going around, a circuit" | English early 15th century for a stretch of time; "time in which a circuit or revolution is made" by 1727 | S442 |
| phase | Greek phasis "appearance", of a star or of the moon, from phainein "to show" | English 1705 of the moon; the physics sense "particular stage or point in a recurring sequence of movement" dated 1861 | S442 |
| frequency | Latin frequentia "an assembling in great numbers, a crowding" | English 1550s; the physics sense "rate of recurrence, especially of a vibration" dated 1831 | S442 |
| sinusoid | Latin sinus + Greek -oeides "form, shape", so "sine-shaped" | 1823 in mathematics, "the curve of sines"; the Latin ancestor phrase linea sinuum is older, Fabri 1659 | S426, S442 |
| harmonic | Greek harmos "fastening, joint", harmonia "means of joining, concord of sounds" | English 1560s in the musical sense; "harmonic analysis" credited to William Thomson, in print in a note added December 1863 | S427, S442 |
Two of these deserve a note. "Amplitude" had a rival. Miller records that it competed with "argument" for the angle of a complex number into the early twentieth century, citing Hardy's Course of Pure Mathematics of 1908, page 82 (S427). And "harmonic" carries its musical origin into the mathematics without apology. A harmonic is a joining. Harmonic analysis breaks a signal into pure tones, which is what the word meant to the people who coined it.
What this book could not establish: when "amplitude," "period," "phase" or "frequency" were first used specifically of a trigonometric graph. The general, astronomical and physical senses are dated above. The graph sense is not.
Midline
Midline has no recorded history. It does not appear in Cajori volume 1 or volume 2, in Smith volume 1 or volume 2, in Braunmühl, or in Miller's Earliest Known Uses. The term "midline," meaning the horizontal center line of a sinusoid, is not attested in any historical reference work consulted for this book. It appears to be a twentieth-century American schoolbook coinage. That second sentence is a guess about a pattern, not a finding. Label it as a guess whenever you repeat it. If a student asks who invented "midline," the correct answer is that nobody knows, and that the question has apparently never been researched.
Trigonometry and goniometry
Bartholomaeus Pitiscus (pih-TISS-kuss, 1561 to 1613) put the word into print in 1595. Smith: "Pitiscus (1595) published an important trigonometry in which he corrected the tables of Rhaeticus and modernized the treatment of the subject. In this work the word 'trigonometry' appears for the first time as the title of a book on the subject" (S421, vol. 2, p. 611).
Miller supplies the exact bibliographic setting, citing the Dictionary of Scientific Biography. The Trigonometria: sive de solutione triangulorum tractatus brevis et perspicuus (Trigonometry, or a short and clear treatise on the solving of triangles) appeared as the final part of Abraham Scultetus's Sphaericorum libri tres methodicé conscripti et utilibus scholiis expositi, at Heidelberg in 1595 (S427). The word that names your course entered the world as an appendix to somebody else's astronomy book.
English gets it nineteen years later, in the translation of the same work: Trigonometry: or The Doctrine of Triangles. First written in Latine, by B. Pitiscus ..., and now Translated into English, by Ra. Handson, 1614 (S427).
The Greek is trigonon, "triangle," from treis "three" plus gonia "angle," plus metron, "measure." Triangle-measuring, exactly (S442).
Its forgotten sibling is "goniometry," angle-measuring, from the same gonia plus metron. Smith's 1724 date for it rests on Smith alone and could not be corroborated. Report it as Smith's claim, not as a fact.
Here is the evidence. Smith credits the word to Thomas Fantet de Lagny (duh lah-NYEE, 1660 to 1734): "The word 'goniometry' was first used by him (1724), although more in the etymological sense of mere angle measure than is now the case" (S421, vol. 2, p. 612). This book then searched the complete text of both volumes of Cajori's Notations for the string "goniometr" and found exactly one occurrence in either volume, an 1812 citation to John Cole's Stereogoniometry (London, 1812) (S424). Miller has no entry for goniometry at all (S427).
The arc- prefix and the inverse functions
"Arcsin x" is short for "the arc whose sine is x." Latin arcus means "a bow, an arch, an arc" (S423). The name only makes sense in a picture where the angle is measured as a length of arc, which is to say on a unit circle in radians. That is why the notation appears when it does and not earlier.
Cajori tracks the notation in sections 532 to 533. His first sentence gives the very first symbol to Daniel Bernoulli, an isolated "A S." for the arcsine in 1729 (S423); after that the sequence is Euler, then everybody else converging on the abbreviation:
| Year | Author and place | Exact form used | Sources |
|---|---|---|---|
| 1736 | Euler, Mechanica vol. 2, p. 303 area | "A t x/y est arcus circuli cuius tangens est x/y existente sinu toto = 1" | S423 |
| 1737 | Euler, Commentarii for 1737, vol. 9, p. 209 | "A sin b/c", the arc of a unit circle whose sine is b/c | S423 |
| 1744 | Euler, Nova acta eruditorum, p. 325 | "A tagt", for arcus, cujus tangens | S423 |
| 1758 | Lambert, Acta Helvetica III, p. 141 (the phrase), then the Berlin Nouveaux mémoires for 1776, p. 12 (the abbreviation) | "arcus sinui b respondens" in 1758; "arc. sin." from 1776 | S423 |
| 1772 | Scherffer, Institutionum analyticarum pars secunda, Vienna, pp. 144, 200 | "arc. tang." | S423 |
| 1772 | Lagrange, Berlin Nouveaux mémoires for 1772, p. 277 | "arc. sin x/(1 + cx)" | S423 |
| 1794 | Paoli, Elementi d'algebra, Pisa, vol. 2, p. 21 | "Arc. sen." | S423 |
Euler's 1736 phrasing is worth translating in full because it contains the whole idea: "A t x/y is the arc of a circle whose tangent is x/y, the whole sine being 1." The "whole sine being 1" clause is the unit circle, stated as a condition rather than assumed.
The competing notation is John F. W. Herschel's (HER-shell, 1792 to 1871), published in the Philosophical Transactions for 1813. Cajori quotes Herschel's own explanation from page 10: "This notation cos to the minus one of e must not be understood to signify 1/cos e, but what is usually written thus, arc(cos = e)" (S423). Smith gives the same date and attribution (S421, vol. 2, p. 618). Two centuries later, students still read as a reciprocal, which is precisely the confusion Herschel wrote that sentence to prevent.
Cajori also records Herschel's later retraction of his own priority: he "used [these], as he then supposed for the first time. The work of a German Analyst, Bürmann, has, however, within these few months come to his knowledge, in which the same is explained at a considerably earlier date" (S423). Herschel published a correction against himself. That is what the sources look like when someone is being careful.
The one-word forms "arcsine" and "arctangent" are recent: Miller finds them in the Hedrick translation of Goursat's Course in Mathematical Analysis, about 1904 (S427).
Five people who coined something and got no credit
Before leaving the vocabulary, five names that almost never appear in a school textbook:
Habash al-Hasib (HAB-ash al-HAH-sib, c. 796 to after 869), "the computer," working in Baghdad, built the first table of tangents and cotangents. It survives in a single Berlin manuscript of his astronomical tables (Suter, Abhandlungen X, 209). Smith also notes that he was the first to consider the secant (S421). Three of your six functions have their first tables from this one man, and he is not in your book.
Georg Simon Klügel introduced the term "trigonometric function" in 1770, per Miller citing Cajori's 1919 History, page 234 (S427). The phrase that organizes the entire modern course has a coiner, and nobody teaches his name.
Vincenzo Riccati "for the first time used the term 'trigonometric lines' to indicate circular functions" in his Institutiones analyticae of 1765 to 1767. He had already introduced the hyperbolic functions with the notation Sh x and Ch x in 1757 (S427, S423).
Richard Norwood, in his Trigonometrie (London, 1631), stops to explain his own abbreviations: "in these examples s stands for sine: t for tangent: sc for sine complement: tc for tangent complement: sec for secant" (S421). That is a mathematician inventing notation in public, on the page, and telling the reader the key. It is a good artifact to put in front of students who assume the symbols came down from the sky.
Albert Girard (1626) first made general use of a workable abbreviation for the sine, and he introduced the forms that became "tan" and "sec," and Pierre Hérigone (1634) seems to be the first to print "sin" for sine in a book, although the contraction already appears on a 1624 drawing of Gunter's scale (S421). So the abbreviation on your calculator key predates the calculator by three hundred and fifty years, and it started life as engraving on a slide rule.
Part 2: The theorems and the identities
The Pythagorean theorem and its converse
What Euclid wrote
Elements I.47, in the Joyce edition after Heath:
"In right-angled triangles the square on the side opposite the right angle equals the sum of the squares on the sides containing the right angle." (S428)
Elements I.48, the converse:
"If in a triangle the square on one of the sides equals the sum of the squares on the remaining two sides of the triangle, then the angle contained by the remaining two sides of the triangle is right." (S428)
Two things to notice. First, "square" means an actual square, a region with area, not a number multiplied by itself. Euclid has no algebra. Second, I.48 is a separate proposition with a separate proof. In Greek mathematics the converse of a true statement is a different statement, and it needs its own argument. Your textbook probably runs the theorem and its converse together in one box. Euclid did not, and the reason is a lesson in logic worth ten minutes of class time.
The name
The name is later than both Pythagoras and Euclid. The Joyce commentary on I.47 says the proposition "is frequently called the Pythagorean theorem, a name given by later scholars centuries after both Pythagoras and Euclid" (S428). In English, Miller records the term "Pythagorean theorem" in 1726, in the second edition of Edmund Stone's A New Mathematical Dictionary. He also notes that some early twentieth-century American dictionaries have "Pythagorean proposition" instead (S427).
The older evidence, and what Smith gets wrong
Smith's survey at volume 2, pages 288 to 290, is the best short account, and it opens: "The relation of the sides of a triangle when these sides are 3, 4, and 5 (that is, 3 squared + 4 squared = 5 squared) was well known long before the time of Pythagoras" (S421).
Chinese. Smith quotes what we now call the gougu rule: "We find in the Nine Sections of the Chinese, perhaps written before 1100 b.c., this statement: 'Square the first side and the second side and add them together; then the square root is the hypotenuse'" (S421, vol. 2, p. 288). Elsewhere (vol. 2, p. 602) he dates the Zhoubi suanjing to c. 1105 BCE and quotes it: "The knowledge comes from the shadow, and the shadow comes from the gnomon" (S421).
Do not repeat Smith's Chinese dates. He gives "Perhaps written before 1100 b.c." for the Jiuzhang suanshu and "c. 1105 b.c." for the Zhoubi suanjing. Both are far earlier than modern sinological consensus, which places the two texts many centuries later. The mathematical content of the quotations is fine. The dates attached to them are a 1925 estimate that has not survived.
Egyptian. "for a papyrus of the 12th dynasty (c. 2000 b.c.), discovered at Kahun, refers to four of these relations, one being 1 squared + (3/4) squared = (1 1/4) squared. It was among these people that we first hear of the 'rope stretchers,' those surveyors who, it is usually thought, were able by the aid of this property to stretch a rope so as to draw a line perpendicular to another line" (S421). Check that relation: . It is the 3-4-5 triangle scaled by a quarter, written in the unit fractions the Egyptians preferred.
Indian. "The Hindus knew the property long before the beginning of the Christian era, for it is mentioned in the Sulvasutras, the sacred poems of the Brahmans. The Sulvasutra of Apastamba gives rules for constructing right angles by stretching cords of the following lengths: 3, 4, 5; 12, 16, 20; 15, 20, 25; 5, 12, 13; 15, 36, 39; 8, 15, 17; and 12, 35, 37. Although the date of these writings is uncertain, it is evident that the relations were known rather early in India" (S421). That list is worth putting on a board and sorting. The triples 3-4-5, 12-16-20 and 15-20-25 are the same triangle scaled by 1, 4 and 5. Next, 5-12-13 and 15-36-39 are the same triangle scaled by 1 and 3. Only 8-15-17 and 12-35-37 are new. So the list holds four distinct right triangles in seven working sizes, which is what a rope-stretching manual would need.
Smith's dates for the Sulba Sutras should not be repeated as established. His footnote on the date says only "Perhaps the 4th or 5th century b.c.," and in volume 1 he is blunter: "The dates of the Sulvasutra period are unknown," adding that the texts "were changed more or less by such commentators as Apastamba, Baudhayana, and Katyayana" (S422). Each such reworking is a recension, and a text with several of them has no single date.
His own literature note points three ways. To A. Burk in the Zeitschrift der deutschen morgenländischen Gesellschaft LV, 543 and LVI, 327. To G. Thibaut in the Journal of the Royal Asiatic Society of Bengal XLIV (reprint 1875) and in The Pandit (Benares, 1875/6 and 1880). And to Heath's Euclid, vol. 1, p. 360 (S421). Smith also notes an etymological aside: sulba "is sometimes interpreted to mean rope-stretching," which links the Indian and Egyptian evidence through the same physical tool.
Babylonian. Here the honest statement is thin. This book located no discussion of Plimpton 322 in the passages of Smith or Braunmühl that it read. The only Babylonian evidence verified is a general statement in the Joyce commentary on I.47. It says that ancient Babylonians of about 1900 to 1600 BCE used the principle in problems about right triangles and could construct Pythagorean triples (S428). That commentary names no tablet and cites no edition. Anyone teaching the Babylonian side of this should get it from a cuneiform specialist, not from a history-of-mathematics survey.
Did Pythagoras prove it?
Smith is refreshingly blunt, and this passage should be read aloud in class:
"The proof of the proposition is attributed to Pythagoras (c. 540 b.c.) by various writers, including Proclus (c. 460), Plutarch (1st century), Cicero (c. 50 b.c.), Diogenes Laertius (2d century), and Athenaeus (c. 300). No one of these lived within, say, five centuries of Pythagoras, so that we have only a weak tradition on which to rest the general belief that Pythagoras was the first to prove the theorem ... Not only are we not positive that the proof is due to Pythagoras at all, but we are still more in doubt as to the line of demonstration that he may have followed." (S421, vol. 2, pp. 288 to 289)
Five witnesses, none of them within five hundred years of the event. As for the proof in the Elements, Proclus says it is Euclid's own (S428).
The law of sines: three disputes, kept apart
There is not one priority question here. There are three, and merging them produces nonsense.
(a) The spherical law of sines: a three-way tie
Three tenth-century mathematicians in the Islamic world have a claim to for spherical triangles.
Abu al-Wafa al-Buzjani (AH-boo al-WAH-fah, 940 to 998) of Baghdad and Abu Nasr Mansur ibn Iraq (AH-boo NAHSS-r man-SOOR, c. 970 to 1036) of Khwarazm both claimed it. MacTutor's biography of Abu Nasr Mansur states that he "discovered the sine law," and immediately adds: "Abu'l-Wafa may have discovered this law first and Abu Nasr Mansur may have learnt it from him. Certainly which of the two has priority is hard to determine and will almost certainly never be known with certainty" (S431).
The third claimant, Abu Mahmud al-Khujandi (al-hoo-JAN-dee, c. 940 to 1000) of Rayy, did not claim it himself. The claim is made on his behalf, and by only one person. MacTutor: "It remains for us to discuss the claim that al-Khujandi discovered the sine theorem. The claim was made by al-Tusi who gives al-Khujandi's proof of the result for spherical triangles in his Shakl al-qatta" (S432). And the assessment: "Although there is no reason to doubt al-Tusi that the proof he gives does indeed come from al-Khujandi there is quite a few reason to believe that one of Abu'l-Wafa or Abu Nasr Mansur was the original discoverer" (S432).
The three reasons given are worth having. Both Abu al-Wafa and Abu Nasr Mansur claimed the discovery, and al-Khujandi never did. Al-Khujandi was an instrument maker and observer rather than a theoretician (Samsó calls him "essentially a practical astronomer, unconcerned with theoretical problems"). And the theorem recurs throughout Abu Nasr Mansur's writings, which is what you expect from someone who owns a result (S431, S432).
Al-Khujandi's claim rests entirely on al-Tusi's testimony, written some two and a half centuries later. That does not make it false. It makes it a different kind of evidence from a mathematician stating his own theorem in his own book.
The honest classroom line: three tenth-century mathematicians in the Islamic world have a claim, two of them made the claim themselves, and the specialists decline to pick a winner.
(b) Al-Jayyani and the first treatise on spherical trigonometry
Ibn Mu'adh al-Jayyani (IB-n moo-AATH al-jy-YAH-nee, 989 to after 1079), born in Cordoba and probably dying at Jaén, wrote The Book of Unknown Arcs of a Sphere. MacTutor calls it "the first treatise on spherical trigonometry" (S430). Its contents, per MacTutor: "formulae for right-handed triangles, the general law of sines, and the solution of a spherical triangle by means of the polar triangle" (S430).
Two cautions. First, "first treatise on spherical trigonometry" is a claim about a genre. It says this is the earliest book devoted to the subject as a subject. It is not a priority claim about who first found the sine law. Second, the popular follow-on claim that this book is the source Regiomontanus used is not established. MacTutor itself hedges: "Although it is certain that Regiomontanus based his treatise on Arabic works on spherical trigonometry it may well be that al-Jayyani's work was only one of many such sources" (S430). MacTutor also records the reviewer Debarnot arguing that the standard commentary "fails to take the originality of the Determination of the magnitudes sufficiently into account" (S430). The line of transmission from al-Jayyani to Regiomontanus is unverifiable on present evidence.
(c) The plane law of sines
Smith's sequence:
"While recognized by Alberuni and other Oriental writers, it was Nasir ed-din (c. 1250) who first set it forth with any clearness. A little later Levi ben Gerson (c. 1330) stated the law in his work De sinibus, chordis, et arcubus; but the first of the Renaissance writers to express it with precision was Regiomontanus (writing c. 1464)." (S421, vol. 2, p. 630)
Nasir al-Din al-Tusi (nuh-SEER ad-DEEN TOO-see, 1201 to 1274), working at Maragha, wrote the Shakl al-qatta (Treatise on the Quadrilateral). MacTutor describes it as "really the first in history on trigonometry as an independent branch of pure mathematics and the first in which all six cases for a right-angled spherical triangle are set forth" (S433). That is the moment trigonometry stops being a chapter of astronomy.
Levi ben Gerson (LEE-vy ben GER-shon, 1288 to 1344), known as Gersonides, states it in Latin in De sinibus, chordis et arcubus (On Sines, Chords and Arcs). This book dates that work 1342 (disputed: Smith gives c. 1330). Smith's footnote preserves Levi's Latin:
"omnium triangulorum rectilineorum talem proportionem una linea habet ad aliam, qualem proportionem unus sinus angulorum, quibus dictae lineae sunt subtensae, habet ad alium." (S421)
In every straight-sided triangle, one line has to another the same ratio that the sine of one of the angles which those lines subtend has to the sine of the other. Everything is a proportion. The ratio holds between lengths, not between numbers, because a sine is still a length here.
Regiomontanus states it in De triangulis omnimodis (On Triangles of All Kinds), written c. 1464, printed at Nuremberg 1533. Book II, Proposition 1, page 46 of the printed edition:
"In omni triangulo rectilineo proportio lateris ad latus est, tanquam sinus recti anguli alterum eorum respicientis, ad sinum rectum anguli reliquum latus respicientis." (S445)
In every straight-sided triangle the ratio of one side to another is as the right sine of the angle facing the one, to the right sine of the angle facing the remaining side. This book read that sentence in the OCR of the 1533 scan (Internet Archive identifier regiomontanus_1533). It matches Smith's quotation word for word (S445, S421). This is one of the few places in the chapter where a secondary source's quotation could be checked against the primary text. It passed.
The proof that follows is instructive. Regiomontanus splits it into three cases: the right-angled case (referred back to Book I proposition 28), the isosceles case, and the general case, handled by dropping two perpendiculars. He closes with "quare certum est, quod asserebat propositio," therefore what the proposition asserted is certain (S445). That is case analysis where a modern proof uses one diagram, because the tools do not yet include signed lengths.
A conflict to record: did Ptolemy have it?
Smith says the plane law of sines "was known to Ptolemy (c. 150) in substance, although he expressed it by means of chords" (S421, vol. 2, p. 630). Braunmühl flatly denies it:
"Es ist nicht richtig, wenn R. Wolf, H.A. I. 226 meint, dass die Griechen schon den Sinussatz für das schiefwinklige Dreieck kannten. Derselbe findet sich nirgends, weder für die Ebene, noch für die [Kugel]." (S425, vol. 1, p. 27 n. 2)
It is not correct to say, as R. Wolf does, that the Greeks already knew the sine law for the oblique triangle. It is found nowhere, neither for the plane nor for the sphere. So here are two standard authorities in direct contradiction, on a question a student might well ask. Both positions are on the record, and this book does not adjudicate between them. The disagreement itself is the fact to teach.
The law of cosines
Euclid, verbatim, twice
Elements II.12:
"In obtuse-angled triangles the square on the side opposite the obtuse angle is greater than the sum of the squares on the sides containing the obtuse angle by twice the rectangle contained by one of the sides about the obtuse angle, namely that on which the perpendicular falls, and the straight line cut off outside by the perpendicular towards the obtuse angle." (S428)
Elements II.13:
"In acute-angled triangles the square on the side opposite the acute angle is less than the sum of the squares on the sides containing the acute angle by twice the rectangle contained by one of the sides about the acute angle, namely that on which the perpendicular falls, and the straight line cut off within by the perpendicular towards the acute angle." (S428)
Now notice everything that is not there. There is no cosine. There is no angle measure. There is no formula, and there is no single statement covering both cases. The quantity we write as appears as "twice the rectangle contained by" a side and "the straight line cut off by the perpendicular," which is a specific segment in a specific picture. And there are two propositions rather than one for a reason: the foot of the perpendicular falls outside the triangle when the angle is obtuse and inside when it is acute. Unifying them needs a cosine that changes sign, which needs the function. That is why the unified law is medieval and modern rather than Greek.
That "greater than" and "less than" pair is worth showing to a class alongside the modern formula. In , the sign of does the work Euclid does with two separate propositions. For obtuse , , which makes larger than . For acute , , which makes it smaller. One minus sign replaces two theorems.
Smith on the transmission, and the early printed forms
"The fact that c squared = a squared + b squared - 2ab cos C is essentially a geometric theorem of Euclid [footnote: Elements, II, 12, 13]. In that form it was known to all medieval mathematicians. In the early printed books it appears in various forms, Vieta (1593) giving it substantially as [a formula in sines of complements] and W. Snell (1627) as 2ab/[c squared - (a - b) squared] = 1/(1 - cos C)." (S421, vol. 2, p. 631)
Snell's version is the same theorem, rearranged so that a computer with tables could use it. A computer, in Snell's day, was a person who calculated for a living. Most "different forms" of an identity exist for exactly that reason.
Al-Kashi and a very recent French name
Al-Kashi (al-KAH-shee, c. 1380 to 1429), working at Samarkand, gives the statement in a form using cosines in the Miftah al-hisab (Key of Arithmetic), 1427 to 1428. The French Wikipedia article on the loi des cosinus states: "En 1428, on trouve un énoncé du théorème, utilisant les cosinus, dans l'oeuvre d'al-Kashi, Les clés de l'arithmétique" (S440). In 1428 one finds a statement of the theorem, using cosines, in al-Kashi's work The Keys of Arithmetic. The same article notes that "La propriété a été popularisée en occident par François Viète qui l'a vraisemblablement redécouverte indépendamment," the property was popularized in the West by François Viète, who very probably rediscovered it independently (S440).
Now the part that surprises people. French school students learn this as the théorème d'Al-Kashi. That name is about thirty-five years old:
"Le nom francisé du mathématicien persan Ghiyath Al-Kashi (1380-1429) apparut dans les années 1990 dans les manuels scolaires édités en France, les appellations théorème de Pythagore généralisé ou loi des cosinus étant utilisées jusque-là." (S440)
The Frenchified name of the Persian mathematician Ghiyath Al-Kashi appeared in the 1990s in school textbooks published in France. Up to then the names used were "generalized Pythagorean theorem" or "law of cosines." So a theorem stated in Alexandria around 300 BCE, restated with cosines in Samarkand in 1428, acquired its French name in the decade of the World Wide Web. Named theorems are not fossils. They are current usage, and usage changes.
In English, Miller records all three law-names appearing together in 1888, in T. M. Blakslee's Academic Trigonometry. Plane and Spherical. The book states: "Law of Cosines. The square of any side of a (pl.) triangle is equal to the sum of the squares of the other two sides, minus twice their product by the cosine of the included angle" (S427). That is the modern sentence, in English, in a school textbook, in 1888.
The Ptolemaic procedure behind it
Braunmühl, volume 1, pages 26 to 27, shows Ptolemy finding an angle from three known sides in the Book of Eclipses (Almagest VI.7). Ptolemy does it by finding the difference of the two base segments, "aus dem sich später unser Cosinussatz entwickelte," from which our cosine rule later developed (S425). So the procedure is in Ptolemy even though the statement is not. That distinction, between having a method that works and having a theorem you can state, runs through this entire chapter.
The ambiguous case
Here is a short section, because the honest answer is short. This book could not pin down the first treatment of the ambiguous case (SSA). The Wikipedia article Solution of triangles lays out the four possible outcomes. It gives no attribution for who first identified or described the ambiguous case (S450). Smith's chapter on trigonometry has no entry for it. Neither do Cajori's notation sections or Miller's word list (S421, S423, S427).
That absence is itself interesting. The ambiguous case is a standard feature of every modern course, and the standard histories of mathematics do not discuss its origin at all. It would make a good student research project. It is also a good example of a question that looks trivially answerable and is not.
Ptolemy's theorem, and the identities that fall out of it
Where it is
Ptolemy (TOL-uh-mee, c. 100 to c. 170), working at Alexandria, states and proves the theorem in the Almagest, Book I, chapter 10. Braunmühl gives the exact locations: "Almagest. Ed. Halma I. 29-31. Ed. Heiberg 36-39" (S425). Heiberg pages 36 to 39 is the citation to use, because the Heiberg edition is the scholarly standard.
Braunmühl's summary:
"In Form eines Lemmas wird der sogenannte Ptolemäische Satz, dass in einem Kreis-Vierecke die Summe der Rechtecke aus je zwei Gegenseiten gleich dem Rechteck aus den beiden Diagonalen ist, aufgestellt und bewiesen, und mit seiner Hilfe werden die Sehnen der Summe und der Differenz zweier Längen berechnet, also die Additionsformeln der trigonometrischen Funktionen gegeben." (S425, vol. 1, pp. 19 to 20)
In the form of a lemma, the so-called Ptolemaic theorem is stated and proved: in a cyclic quadrilateral, the sum of the rectangles on each pair of opposite sides equals the rectangle on the two diagonals. A cyclic quadrilateral is a four-sided figure whose four corners all sit on one circle. With the lemma's help, the chords of the sum and of the difference of two lengths are computed, which gives the addition formulas of the trigonometric functions.
Note the word "lemma." A lemma is a helper result, proved only because some later proof needs it. Ptolemy is not proving a famous theorem. He is proving a tool he needs to build a table.
The difference formula
Braunmühl works Ptolemy's construction. In the cyclic quadrilateral ABGD, with and known, the supplementary chords and are known too, and
is the chord of the difference of the arcs AG and AB, where is the radius (S425). Braunmühl then translates: "Setzt man angle AZG = 2 alpha, angle AZB = 2 beta und r = 1, so setzt sich diese Formel unmittelbar in sin(alpha - beta) = sin alpha cos beta - cos alpha sin beta um" (S425). Setting the central angles to and and the radius to 1, this formula turns immediately into
The translation works for two reasons. A chord is twice the sine of half its arc. And the supplementary chord of an arc is the chord of what is left of the semicircle, which is the cosine in disguise.
The sum formula
For the sum, Braunmühl gives Ptolemy's route: compute , then , and "Für angle AZB = alpha und angle BZG = beta und r = 1 geht diese Formel leicht über in unsere: cos(alpha + beta) = cos alpha cos beta - sin alpha sin beta" (S425). With the central angles and and , this passes easily into
The restriction, and who removed it
This is the detail that matters most for a course that cares about the difference between an ancient statement and a modern one:
"Ptolemäus setzt bei seiner Ableitung arc. AB + arc. BG < 180 Grad voraus, eine Beschränkung, die Theon später in seinem Kommentar p. 196, Ed. Halma aufhebt." (S425)
Ptolemy assumes in his derivation that arc AB plus arc BG is less than 180 degrees, a restriction that Theon later removes in his commentary (Halma edition, p. 196). Theon of Alexandria, working in the fourth century CE, is the person who made the addition formula unconditional. Two hundred years separate Ptolemy's restricted formula from Theon's general one, and the whole of that gap is a condition on the size of the arcs, which is exactly the kind of thing modern notation hides.
The half-chord formula
Braunmühl, page 20: "GD = sqrt(r(2r - a)), welche Formel für r = 1, und arc BG = 2 alpha in sin(alpha/2) = sqrt((1 - cos alpha)/2) übergeht" (S425). Which for and arc passes into
That is the half-angle formula, and in Ptolemy's hands it is a table-building tool: it lets you halve an arc whose chord you already know, over and over, until the table is fine enough.
Worked example: checking Ptolemy's arithmetic
Ptolemy divides the circle into 360 degrees and the diameter into 120 parts, so the radius is 60, and each of those 60 parts is subdivided sexagesimally: "the radius consisting of 60 moirai, each moira of 60 minutes" (S421, vol. 2, p. 615).
Braunmühl checks one entry of the resulting table (S425, vol. 1, p. 22). Ptolemy's value gives
The modern seven-place value of is . Braunmühl's verdict: "also stimmen die beiden Werte in 6 Dezimalen überein," the two values therefore agree to six decimal places (S425). Smith gives the identical figures (S421). Ptolemy is off by three parts in ten million, using no decimals, no algebra and no calculating machine.
Here is the second half of the worked example, using the chord form of the Pythagorean identity. Braunmühl records Ptolemy's step for supplementary arcs:
"Da aber die Sehnen zweier Bögen, welche sich zu einem Halbkreis ergänzen, quadriert und addiert das Quadrat des Durchmessers geben, so hat man hiermit auch die Sehnen der Supplementarbögen der vorigen; z. B. crd(180 - 36) = crd 144 = sqrt(120 squared - (crd 36) squared) = 114 degrees 7' 37"." (S425, vol. 1, p. 19)
Since the chords of two arcs which together make a semicircle, squared and added, give the square of the diameter, one has from this the chords of the supplementary arcs as well; for example the chord of 144 degrees equals .
Run that backwards to recover the chord of 36 degrees, which is the number Ptolemy must have had in his table:
Now the modern check. A chord in a circle of radius 60 satisfies , so
The recovered value and the modern value agree to five significant figures. And the relation in use here, chord squared plus supplementary chord squared equals diameter squared, is wearing Greek clothes. The chord of an arc and the chord of its supplement are, up to the factor 120, the sine and cosine of half the arc.
The Pythagorean identity: a correction to the textbooks
The popular classroom claim is that was first stated by Varahamihira around 505 CE. That claim does not survive a careful reading of Smith, who is the usual source for it. Here is Smith's sentence:
"It is further probable, from the efforts made to develop simple tables, that the Hindus were acquainted with the principles which we represent by the formulas sin squared phi + cos squared phi = 1, [half-angle formula], and [a third relation], the last two of these appearing in the Panca Siddhantika of Varahamihira (c. 505)." (S421, vol. 2, p. 615)
Read it slowly. Smith says it is "probable" that Indian astronomers knew the three principles, and then says that the last two of them appear in the Pancasiddhantika. The Pythagorean identity is the first of the three. So Smith assigns the half-angle relation to Varahamihira explicitly and puts the Pythagorean identity only in the "probable" group. The confident attribution has grown out of a misreading of a carefully hedged sentence.
What can be said with confidence:
The Greeks had it in chord form and used it as a working step. Braunmühl's passage quoted above, on squaring and adding the chords of supplementary arcs to get the square of the diameter, is exactly the identity, and Ptolemy uses it to build his table (S425).
Smith himself treats the identity as a restatement rather than a discovery: "Although the functions themselves were not specifically named, various early writers make statements which involve in substance many of the relations that we now recognize. Thus the formula sin phi = sqrt(1 - cos squared phi), or sin squared phi + cos squared phi = 1, is essentially the Pythagorean Theorem and as such was known to the Greeks" (S421, vol. 2, p. 623).
The modern statement is Euler's, Introductio section 127, in Ian Bruce's translation: "Therefore all the sines and cosines will be contained within the limits 1 and -1. But again there will be cos z = sin(pi/2 - z) and sin z = cos(pi/2 - z) and (sin z) squared + (cos z) squared = 1" (S429). And Euler's own comment on this batch of identities is "which all are well-known from trigonometry" (S429). He is codifying, not announcing. That is the correct picture of the Pythagorean identity. Nobody discovered it, because in every form it is the Pythagorean theorem. What changed was the notation that let it be written in one line.
Quotient identities
The tangent as a quotient has three stages, and they are three different kinds of statement.
The shadow stage. Al-Battani's shadow table is the cotangent quotient in physical form. Smith's footnote: "That is, he gave the value of u = l (cos phi / sin phi) for phi = 1 degree, 2 degrees, ..., l being the length of the gnomon" (S421, vol. 2, p. 621 n. 1). A shadow length is a cotangent times a stick length, so a table of shadows is a table of cotangents with a unit attached.
The proportion stage. Smith: "Abu'l-Wefa (c. 980) knew substantially the formulas tan phi : 1 = sin phi : cos phi, cot phi : 1 = cos phi : sin phi, sec phi = sqrt(1 + tan squared phi), and csc phi = sqrt(1 + cot squared phi)" (S421, vol. 2, p. 623). They come out as proportions, not equations, because a sine is a length. You cannot divide a length by a length and get a number until somebody decides that you can.
The equation stage. Euler, Introductio section 127: "tang z = sin z / cos z and cot z = cos z / sin z = 1 / tang z" (S429).
The shape of that history is the point. Writers state these relations as proportions for seven hundred years. The modern equation form follows from taking the radius as 1. That is why Euler's chapter is the turning point, rather than any particular discovery.
Cofunction identities
These are built into the vocabulary. "Cosine" means "sine of the complement," so is not a theorem about cosine, it is the definition of the word.
Smith traces the naming sequence: Aryabhata's kotijya, Plato of Tivoli's chorda residui, Regiomontanus's sinus rectus complementi, then Gunter's co.sinus. He adds that the sine of "commonly served the purpose then as it did later with the Arabs" (S421, vol. 2, p. 619).
The modern symbolic statement is Euler's, section 127: "cos z = sin(pi/2 - z) and sin z = cos(pi/2 - z)" (S429). Notice that Euler writes , not 90 degrees. The identity in its modern form belongs to a world where angles are arcs on a unit circle, which is the world Euler is building in that chapter.
Reciprocal identities
Rheticus has them in proportion form in 1551: "sec phi : 1 = 1 : cos phi and csc phi : 1 = 1 : sin phi" (S421, vol. 2, p. 623). Vieta in 1579 gives a fuller set: "1 : sec phi = cos phi : 1 = sin phi : tan phi; csc phi : sec phi = cot phi : 1 = 1 : tan phi; and 1 : csc phi = cos phi : cot phi = sin phi : 1" (S421). Euler in 1748 writes directly (S429).
One small fossil of the disorder: the secant and cosecant never did settle on a symbol. Smith, writing in 1925: "There is as yet no international symbol for cosecant, cosec and csc both being used" (S421, vol. 2, p. 623). A century later that is still true, and which one your textbook uses tells you something about where it was published.
Angle sum and difference, in three traditions
Greek, in chords. Ptolemy derives both the chord of a sum and the chord of a difference from the cyclic quadrilateral lemma, subject to the restriction that the arcs sum to less than 180 degrees, which Theon later lifted (S425). Smith: "The Greeks knew essentially that sin(phi plus or minus phi') = sin phi cos phi' plus or minus cos phi sin phi'. Stated as a proposition involving chords, it is probable that Hipparchus (c. 140 b.c.) knew it. It was certainly known to Ptolemy (c. 150), and it often bears his name. Bhaskara (c. 1150) also gives the theorem" (S421, vol. 2, pp. 628 to 629). Watch Smith's confidence words again: "probable" for Hipparchus (hip-PAR-kuss, c. 190 to c. 120 BCE), whose chord table is lost, and "certainly" for Ptolemy, whose book survives. The difference between those two words is the difference between having the book and not having it.
Arabic, in radicals. Abu al-Wafa's version is worth putting in front of students precisely because it looks nothing like theirs. Smith gives it twice, at pages 617 and 629:
(S421). That is the addition formula with every cosine replaced by a radical, because Abu al-Wafa is working with sines only. Check it against the modern form. Since , the first radical is . The second is . Same theorem, one function short of a convenient notation.
Modern. Euler, section 128: "sin(y plus or minus z) = sin y cos z plus or minus cos y sin z and cos(y plus or minus z) = cos y cos z minus or plus sin y sin z" (S429).
Double angle and half angle
Smith on the double angle:
"The formula sin 2 phi = 2 sin phi cos phi is a corollary of the general case of sin(phi + phi'). It is first expressly given as a rule by Abu'l-Wefa, the form being ... chord phi : chord (phi/2) = chord(180 - phi/2) : r. Vieta (1591) first gave the formulas sin 3 phi = 3 cos squared phi sin phi - sin cubed phi, cos 3 phi = cos cubed phi - 3 sin squared phi cos phi, and connected sin n phi with sin phi and cos phi. Rhaeticus (1569) found the relation cos n phi = cos(n-2) phi - 2 sin phi sin(n-1) phi." (S421, vol. 2, p. 629)
Two useful facts hide in there. First, Abu al-Wafa's double-angle rule is a proportion among chords, again. Second, Rheticus's relation is a recurrence. It computes from the two previous multiples, which is how you build a large table cheaply. Modern numerical libraries use the same computational idea.
On the half angle, Smith grades his own confidence: "Ptolemy (c. 150) knew substantially the sine of half an angle, expressed as half a chord, and it is probable that Hipparchus (c. 140 b.c.) and certain that Varahamihira (c. 505) knew the relation which we express as sin(theta/2) = sqrt((1 - cos theta)/2)" (S421, vol. 2, p. 629). "Probable" for Hipparchus, "certain" for Varahamihira (vuh-RAH-huh-MIH-hih-ruh, floruit c. 505). This is the same sentence pattern that got misread into the Pythagorean identity claim above. Here it comes out in Varahamihira's favor. The half-angle relation is the one Smith does assign to the Pancasiddhantika.
The tangent versions are startlingly late. Smith: "the first two, due to Euler (1748), being tan 2 phi = 2 tan phi / (1 - tan squared phi), cot 2 phi = (cot phi - tan phi)/2; and the others, due to Lambert (1765), being sin 2 phi = 2 tan phi / (1 + tan squared phi), cos 2 phi = (1 - tan squared phi)/(1 + tan squared phi)" (S421, vol. 2, p. 630). The double-angle formulas in your textbook that involve tangent are eighteenth-century, roughly sixteen hundred years younger than the ones involving sine and cosine.
Prosthaphaeresis: the standard attribution is contested by the standard historian
What it is, and what the word means
Prosthaphaeresis replaces a multiplication with an addition, using the product-to-sum identities. In the decades just before logarithms, it was the fastest way to multiply large numbers. Braunmühl gives both the etymology and the purpose in one sentence:
"Das eigentümliche Wort ist gebildet aus prosthesis und aphairesis, Hinzufügung und Wegnahme, und bedeutet eine Additions- und Subtraktionsmethode, welche später so ausgebildet wurde, dass sie bis zur Erfindung der Logarithmen sehr wohl dazu dienen konnte, die Multiplikation grosser Zahlen durch Addition zu ersetzen." (S425, vol. 1, p. 135)
The peculiar word is formed from prosthesis and aphairesis, adding and taking away. It means an addition-and-subtraction method. Later work developed that method so far that, until the invention of logarithms, it could serve very well to replace the multiplication of large numbers by addition. The two formulas Braunmühl states are
(S425). To multiply two numbers, scale them into the range of a cosine table, look up the two angles, add and subtract them, look up two cosines, and average. Four table lookups and two additions instead of a multiplication.
The dispute
The usual textbook story runs Ibn Yunus to Wittich to Clavius and Brahe. Braunmühl, who wrote the standard history of trigonometry, disputes it. He argues that the European invention belongs to Johannes Werner (yo-HAHN-uss VAIR-ner, 1468 to 1522) of Nuremberg, roughly seventy years earlier than the usual attribution:
"Dafür spricht z.B. die Umgestaltung von Regiomontan's Cosinussatz durch Erfindung der sogenannten prosthaphäretischen Methode, die bisher einer viel späteren Zeit zugeschrieben wurde, während sie Werner's Eigentum ist." (S425, vol. 1, pp. 135 to 136)
Evidence for this is, for example, the recasting of Regiomontanus's cosine rule through the invention of the so-called prosthaphaeretic method. That method has hitherto been ascribed to a much later time, whereas it is Werner's property.
His documentary basis is Jakob Christmann's Theoria lunae of 1611, page 124, which Braunmühl quotes in the original Latin:
"Usus etiam est peculiari prosthaphaeresi, cujus demonstrationem attulit in proprio opere de Triangulis scripto, in quo etiam tres casus Prosthaphaereseum per tres distinctas figuras explicavit et nonnullis transcriptoribus occasionem praebuit, ut cum opus hoc, lucem nondum viderit, sed manuscriptum dumtaxat apud nos extet, inventionem Prosthaphaereseon sibi vendicaverint eamque multis partibus amplificarint." (S425)
He also used a particular prosthaphaeresis, the demonstration of which he supplied in his own written work on triangles. In that work he also explained three cases of prosthaphaeresis by three distinct figures. And he gave some copyists their opportunity. Since this work has not yet seen the light, but exists among us only in manuscript, they claimed the invention of prosthaphaeresis for themselves and enlarged upon it in many respects.
That is a 1611 accusation of what we would now call plagiarism from an unpublished manuscript. Braunmühl found a second piece of support later, in Munich Codex latinus monacensis 24101, from Johann Praetorius (S425).
Against this stands the received view, which Braunmühl characterizes exactly: "alle anderen Historiker der Angabe Longomontan's (Astronomia Danica 1622. p. 10) sich anschliessend die Erfindung der Prosthaphäresis dem Paul Wittich, einem zeitweiligen Mitarbeiter Tycho Brahe's, zuschreiben" (S425). All other historians ascribe the invention of prosthaphaeresis to Paul Wittich (POWL VIT-ikh, c. 1546 to 1586), a sometime collaborator of Tycho Brahe. They follow Longomontanus's statement in the Astronomia Danica of 1622, page 10. One sentence, in one book, in 1622, copied by everyone since.
On the Arabic prehistory, Braunmühl is careful in both directions. He found one application of the cos-cos formula in Ibn Yunus (IB-n YOO-nuss, died 1009; Braunmühl gives 1008): "Schon bei Ibn Junos fanden wir (S. 63) eine Anwendung der zweiten dieser Formeln und zweifeln nicht, dass sie die Araber auf verschiedene Fälle anzuwenden wussten, obwohl es uns nicht möglich ist, einen direkten Beweis für diese Ansicht zu erbringen" (S425). We already found in Ibn Yunus an application of the second of these formulas. We do not doubt that the Arabs knew how to apply them to various cases, although it is not possible for us to produce direct proof of this view. And he denies transmission outright: "Keinesfalls hat aber Werner von ihnen die Kenntnis der Methode überkommen, da sie sich in den damals bekannten arabischen Schriften nirgends findet" (S425). In no case did Werner get his knowledge of the method from them, since it is nowhere to be found in the Arabic writings known at that time.
Braunmühl also states the limit of his own case, which is the mark of a good historian. He cannot show that Werner had the further idea: using the formulas to replace the multiplication of arbitrary numbers by addition. That is "ein Gedanke, der die Methode erst befähigte, einen Ersatz für die noch fehlenden Logarithmen zu bilden," the idea that first made the method capable of substituting for the logarithms that did not yet exist. Christmann's report simply does not say (S425).
How to state it in class. The invention of prosthaphaeresis is usually given to Paul Wittich, on the authority of a single sentence in Longomontanus's Astronomia Danica of 1622. Anton von Braunmühl, who wrote the standard history of trigonometry, argued in 1900 that the real inventor was Johannes Werner of Nuremberg. His grounds were Jakob Christmann's 1611 report of Werner's unpublished manuscript, and the claim that the manuscript was circulating among people who then took the method as their own. Werner's manuscript is lost, so the case rests on testimony either way.
Jabir ibn Aflah's theorem
Jabir ibn Aflah (JAH-bir IB-n AF-lah, c. 1100 to c. 1150), known in Latin as Geber, worked in Seville. He is the source of what later writers call "Jabir's theorem," a relation for the right spherical triangle. His work reached Europe in a Latin edition printed at Nuremberg in 1534 (S421, S425).
That is the whole of what this book's sources establish, and the sentence above is deliberately as thin as the evidence. The research for this chapter did not recover a statement of the theorem in Jabir's own words, a chapter reference, or a first-use date for the name "Jabir's theorem." Anyone teaching it should go to a specialist edition rather than to a survey. Mention it to students all the same, for a reason that has nothing to do with the theorem. The Latin form of his name, Geber, was also attached to a completely different body of alchemical writing. Confusion between the two Gebers has muddled reference works for centuries.
Part 3: The graphs
Who first drew a sine curve
The answer is: probably Gilles Personne de Roberval (ZHEEL per-SUN duh ROH-bair-val, 1602 to 1675), around 1634, and he was not trying to draw a sine curve. He was computing the area under a cycloid.
Braunmühl, volume 2, pages 40 to 41, is the source for this entire section, and this book read it in the original:
"Um das Jahr 1634 erfand Giles Persone de Roberval (1602-1675) seine Lehre vom Unendlichen, welche nachmals von seinem Freunde, dem Abbe Gallois 1693 als Traite des indivisibiles publiziert wurde. Darin bestimmte er den Flächeninhalt der Cykloide und gelangte dabei zur Konstruktion der Sinuslinie, die er in seinem Aufsatze 'De Trochoide ejusque spatio' wiederholte, indem er sie die Begleiterin der Cykloide (trochoidis comes oder socia) nannte." (S426)
Around the year 1634, Roberval devised his doctrine of the infinite. His friend the Abbé Gallois published it in 1693 as the Traité des indivisibles. In it he determined the area of the cycloid, and in doing so he arrived at the construction of the sine line. He repeated that construction in his paper De Trochoide ejusque spatio, calling the curve the companion of the cycloid, trochoidis comes or socia.
Braunmühl describes the construction like this. Divide the semicircle that generates the cycloid into equal arcs. Lay off equal segments along the base of the cycloid. Draw parallels through both sets of division points, and the intersections are the points of the curve. Braunmühl then states the identification plainly:
"Man erkennt, dass diese Konstruktion nichts anderes als die Konstruktion der Kurve y = sin x aus den rechtwinkligen Koordinaten x und y ist." (S426)
One recognizes that this construction is nothing other than the construction of the curve from the rectangular coordinates and .
Note carefully what kind of claim that is. It is Braunmühl's identification, made in 1903, of what Roberval's construction amounts to. Roberval called it the companion of the cycloid. The curve exists on his page; the name and the function do not.
Who first named it
Honoré Fabri (on-oh-RAY FAB-ree, c. 1607 to 1688), a French Jesuit, published the Opusculum geometricum de linea sinuum et cycloide (A Little Geometrical Work on the Line of Sines and the Cycloid) at Rome in 1659. Braunmühl:
"Der Name Sinuslinie findet sich jedoch bei Roberval nicht, wohl aber bei dem französischen Jesuiten Honoratus Fabri (1606(7)-1688), der 1659 ein 'Opusculum geometricum de linea Sinuum et Cycloide' zu Rom herausgab und auch in seiner 1669 erschienenen 'Synopsis geometrica', Lugd. 8, eine exakte Definition dieser Kurve mitteilte (p. 313)." (S426)
The name "line of sines" is not found in Roberval, but it is in Fabri. He published the Opusculum at Rome in 1659, and he also gave an exact definition of this curve in his Synopsis geometrica of 1669, page 313. So the earliest known naming of the curve is 1659, twenty-five years after the earliest known drawing of it. The person who named it is not the person who drew it.
The tangent and secant curves
James Gregory (GREG-uh-ree, 1638 to 1675) drew part of the tangent curve in the first quadrant. Again the graph is a byproduct of a calculus problem: he was proving that (S426). John Wallis (WOL-iss, 1616 to 1703) drew the secant curve in his Tractatus de motu of 1670. Braunmühl says he gave its course correctly for the first quadrant: "und ähnliche Flächenbestimmungen veranlassten John Wallis ... in seinem 'Tractatus de motu' 1670, die Sekantenkurve zu zeichnen, deren Verlauf er für den ersten Quadranten richtig angab" (S426). Isaac Barrow (BAR-oh, 1630 to 1677), Newton's teacher, put both curves in a single figure in the Lectiones opticae et geometricae (London, 1674): "Beide Kurven finden sich in einer Figur vereinigt in den 'Lectiones opticae et geometricae', Lond. 1674 in 4, von Isaac Barrow, dem Lehrer Newtons" (S426).
Braunmühl also explains why the graphs appear in this thirty-year window and not earlier:
"Die Darstellungen der übrigen trigonometrischen Funktionen durch Kurven liessen auch nicht mehr lange auf sich warten, nachdem einmal durch Fermat und Descartes die Koordinatengeometrie allseits Eingang gefunden hatte, und andererseits die Berechnung krummlinig begrenzter Flächenräume im Mittelpunkt des Interesses stand." (S426)
The representation of the remaining trigonometric functions by curves did not wait long. Two things had to happen first. Coordinate geometry had to gain general acceptance, through Fermat and Descartes. And the computation of areas bounded by curves had to move to the center of interest. So coordinate geometry plus quadrature problems, meaning problems of finding areas, is what produced the graphs. Teaching had nothing to do with it. Nobody drew a sine curve to help a student see periodicity. They drew it because they were trying to find an area.
| Curve | Earliest known drawing | Problem that produced it | Sources |
|---|---|---|---|
| sine | Roberval, c. 1634, as the "companion of the cycloid" | computing the area of the cycloid | S426 |
| tangent | James Gregory, first quadrant only | proving that the integral of tangent is log secant | S426 |
| secant | John Wallis, Tractatus de motu, 1670, first quadrant | area computations | S426 |
| tangent and secant together | Isaac Barrow, Lectiones opticae et geometricae, London 1674 | optics and geometry lectures | S426 |
| cosine, cotangent, cosecant | not established by any source consulted | not established | none |
What could not be verified about the graphs
Three claims are in circulation that this book could not support.
Dürer's alleged 1525 sine curve. The claim is unverified here. People often say that Albrecht Dürer's Underweysung der Messung (Nuremberg, 1525) contains a projected helix amounting to a drawn sine curve. That would predate Roberval by more than a century. Braunmühl, who surveys the drawn trigonometric curves in detail across these pages, does not mention Dürer in this connection at all (S426). This book did not reach a scan of the relevant plates or a scholarly treatment of them. The claim is not refuted, and it may well be right. Nothing read for this chapter supports it.
Newton drawing a sine curve. No evidence was found in Braunmühl or Smith. Smith records Newton's series work for the sine and the arcsine but not a graph (S426, S421). Unverified.
The first drawings of the cosine, cotangent and cosecant curves. Braunmühl names the sine, tangent and secant and stops. No source consulted names a first appearance for the other three. Not established. Note how odd that is: the cosine curve is the most familiar graph in a modern trigonometry course, and nobody appears to have recorded who drew it first.
Asymptote
Asymptote is Greek: a- "not," plus an assimilated form of syn "with," plus ptotos "fallen," the verbal adjective of piptein "to fall." So, "not falling together," meaning not meeting (S442).
The important historical point is that the word was once broader than our use of it. Miller: "ASYMPTOTE was used by Apollonius, with a broader meaning than its current definition, referring to any lines which do not meet, in whatever direction they are produced (Smith)" (S427). For Apollonius, two parallel lines are asymptotes of each other. The narrowing to "a line a curve approaches without meeting" is later.
The OED's first English citation, per Miller, is Thomas Hobbes, Elements of Philosophy, 1656: "Asymptotes..come still nearer and nearer, but never touch" (S427). That is Hobbes the political philosopher, who thought of himself as a geometer and spent years in a losing public argument with John Wallis about squaring the circle.
One gap worth naming. The tangent graph's vertical asymptotes are a standard feature of every modern course. Gregory, who first drew the tangent curve, drew only the first quadrant, so he did not draw one. No source consulted dates the first description of the tangent graph's asymptotes as such.
The unit circle
The mathematical step: Euler, 1748
The unit circle is not a natural object until somebody decides that the radius should be 1, and for most of history it was not. Smith:
"In general, all ancient tables were constructed with Ptolemy's radius of 60; that is, the sinus totus, or sin 90 degrees, was 60. This was due to the necessity of avoiding fractions in the period before the invention of decimals. The first to adopt the simpler form, sin 90 degrees = 1, was Jobst Bürgi (c. 1600), but his tables computed on this basis are not extant. Although the invention of decimal fractions had now made the use of unity possible for the sinus totus, this idea was not fully appreciated until a memoir by de Lagny was written in 1719. It was nearly thirty years later that the plan received its first great support at the hands of Euler." (S421, vol. 2, p. 627)
Smith's footnote for the Euler reference is precise: Introductio in analysin infinitorum, I, section 127, Lausanne, 1748 (S421).
Euler's own words, in Ian Bruce's translation of section 126: "Therefore we may put the radius of the circle or the whole sine to be and it is clear enough that the periphery of this circle cannot be expressed exactly in rational numbers" (S429). And section 127: "With z denoting the arc of some circle, the radius of which I assume always , the sines and cosines of this arc z mainly are considered" (S429).
"The radius of the circle or the whole sine": read that as an instruction to his readers. The thing they call the sinus totus, the number at the top of every table they own, is now going to be 1. Every table in Europe had been built with a radius of 60, or 600,000, or 10,000,000, or Gunter's 100,000.0000. The reason was practical: decimal fractions were unavailable or awkward. Setting the radius to 1 turns a length into a ratio, and a ratio into a number.
There is a related step that is easy to conflate with this one and is eleven years later. Smith: "The first writer to define the functions expressly as pure number was Kästner (1759), although they had already been used as such by various writers," quoting Kästner's German: "Bedeutet also nun x den Winkel in Graden ausgedruckt, so sind die Ausdruckungen sin x; cos x; tang x u.s.w. Zahlen, die für jeden Winkel gehören" (S421, vol. 2, p. 613). If x denotes the angle expressed in degrees, then the expressions sin x, cos x, tang x and so on are numbers belonging to each angle. Euler in 1748 makes the radius 1. Kästner in 1759 says outright that the functions are numbers. Two different moments.
The phrase: 1852
Miller records the OED's first citation of the phrase "unit circle" in the Cambridge and Dublin Mathematical Journal 7/134 (1852): "The imaginary unit in the solution of (II.) has the same reference to an unit circle, as that in the solution of (III.) has to an unit sphere" (S427). Note the context. That sentence is about complex numbers, not about teaching trigonometry. And the article "an" before "unit" tells you the writer said "an unit," which is how the word was pronounced in some registers at the time.
So the mathematical idea is Euler's, in 1748. The English phrase is attested in 1852, in a complex analysis journal. The classroom use is a twentieth-century story, taken up in Part 4.
Part 4: The classroom
SOH-CAH-TOA
The origin of SOH-CAH-TOA is not reliably documented. That is the whole finding, and it needs saying plainly because every teacher gets asked.
None of the standard historical reference works records it. There is no entry in Cajori's Notations (either volume), in Smith's History of Mathematics (either volume), in Braunmühl, or in Miller's Earliest Known Uses of Some of the Words of Mathematics (S423, S424, S421, S422, S425, S427). Wolfram MathWorld has an entry, created 16 December 2004. It defines the mnemonic and offers alternatives, but it gives no historical information: not when the mnemonic was first created, not who invented it, not where it came from. Its only attribution is to Weisstein himself (S446).
Dictionary.com's acronym entry states: "SOHCAHTOA, or SOH CAH TOA, appears in print as early as 1944 in trigonometry textbooks as a helpful trick to remember the ratios for students" (S447). The entry names no book, author, page or publisher. This book does not treat that as evidence, and the 1944 date could not be corroborated anywhere. If you have seen 1944 quoted with confidence, it traces back to this uncited sentence.
An honest note on the limits of the search. Full-text searches of the Internet Archive and HathiTrust were attempted and blocked, by a proxy error and an HTTP 403 respectively. The Google Books API returned HTTP 429, daily quota exhausted. So the negative result is partly a tooling limitation and not a proof of absence. Somebody with library access and patience could probably settle this, and it would be a useful piece of work.
There is one article that might bear on it: "Sharing teaching ideas: The legend of Soh Cah Toa," Mathematics Teacher 83(4), 1990, p. 286. The NCTM site returned HTTP 403 and it was not read, so it supports nothing here (S449).
The variants
MathWorld does record a family of British sentence mnemonics: "Tommy On A Ship Of His Caught A Herring," noted as "probably more common in Great Britain than the United States," alongside "Oscar Has A Hold On Angie" and "Oscar Had A Heap of Apples" (S446). These encode the same three ratios by first letters of words rather than as a pronounceable acronym.
The reordered form "TOA-CAH-SOH" turns up in informal sources but in no reference work, and MathWorld's British note is about the sentence mnemonics, not about a reordered acronym (S446). Unverified.
The Indian mnemonic, and a correction
"Pandit Badri Prasad Har Har Bole Sona Chandi Tole" is a widely used mnemonic in Indian schools. It is not a unit circle mnemonic, though people who have not checked sometimes describe it as one. It encodes the six ratios using Perpendicular, Base and Hypotenuse:
The source consulted states the six ratios explicitly in that P/B/H form. It describes the phrase as a memory aid for right-triangle side relationships, not for unit circle values (S451). A separate Hindi mnemonic exists for the table of special values, but its wording and origin were not verified. No documented origin, author or date exists for the Pandit Badri Prasad phrase in any source this book could reach.
The pattern across all three traditions is the same: these mnemonics are folklore. They spread by teachers telling other teachers, and that kind of transmission leaves no paper trail. This is a historical finding, not a failure. Tell students so: some things that everybody knows have no documented origin, because the documents were never made.
How the unit circle got into American classrooms
The chain here is defensible for four steps and breaks at the fifth, which is the step people care most about.
Step 1, the mathematical precondition, 1748. Euler sets the radius to 1 (S429, S421). Without that, the unit circle is not a natural object: the standard radius was 60 in Ptolemy, 10,000,000 in Regiomontanus's later tables, 100,000.0000 in Gunter's.
Step 2, the English phrase, 1852, in a complex analysis journal, not in a teaching context (S427).
Step 3, the curricular recommendation, 1959. The College Entrance Examination Board's Commission on Mathematics published Program for College Preparatory Mathematics in 1959, in two paperbound volumes: a 63-page Report and 231 pages of Appendices. Ralph Raimi's detailed reading of the report describes the trigonometry recommendation: "The report then continues with an outline of trigonometry, with the study of the circular functions as such reserved for the 12th grade, which was to be devoted to the elementary functions, vectors, complex numbers, and other topics needed before calculus (in college)" (S439).
Raimi notes an asymmetry in the Appendices worth recording: "Curiously, the exponential and logarithmic functions are not mentioned in the Appendices, though a substantial chapter on the circular functions is included" (S439). And he describes the shape of the geometry-into-trigonometry sequence: "The geometry section includes ... deductive geometry with vectors, all this seguing into the final chapter on trigonometry and the trigonometric functions, with the usual formulas and identities carefully proved" (S439).
The displacement this caused is the part that matters. Raimi: "Since the traditional high school sequence had devoted a whole semester to solid geometry, and another whole semester to the trigonometry of triangles, with calculations via logarithms, compressing each of these to a part of a semester constituted a semester's saving at least" (S439). That is the pivot. Triangle-and-logarithm trigonometry, the subject this whole book has been tracing, was compressed to make room for circular functions. The trigonometry your grandparents took was a computational course about solving triangles with log tables. The trigonometry you are taking is a course about functions on a circle. The changeover has a date and a committee.
Step 4, the implementation, 1961. The School Mathematics Study Group published Elementary Functions, Student's Text, Unit 21 in 1961. The lead author was Frank B. Allen (ALL-en, 1913 to 2007) of Lyons Township High School, Illinois, and the book runs to 398 pages, ERIC document ED135629 (S438). A full-text search of it found "unit circle" at least eighteen times and "circular function" at least twenty-one times (S438).
Chapter 5 develops sine and cosine as the coordinates of a point on the unit circle and then derives periodicity from the geometry rather than asserting it:
"Now returning to the unit circle, we observe that the functions cos and sin behave in exactly this way. From any point P on the circle, a further movement of 2 pi units around the circle will return us to P again. Thus the circular functions are periodic with period 2 pi." (S438)
And it builds rotations as functions acting on vectors: "Any rotation of the sort we are considering is completely specified by the length x of the arc AP of the unit circle through which the rotation carries the point A(1, 0)" (S438). That is the unit circle as the definition of the functions, which is the modern approach, in a student textbook, in 1961.
Step 5, the chart, unknown. Here the chain breaks. You know the chart: the circle labeled with 30, 45 and 60 degrees or , , , with the coordinate pairs written at each point. None of the sources consulted documents when it became a standard classroom artifact. The SMSG text derives the special values rather than tabulating them on a diagram (S438). No NCTM standards document, textbook survey or curriculum history dating the chart was obtained. This book could not establish when the unit circle chart itself became a standard classroom artifact.
That is a striking gap. The chart is on the wall of essentially every American trigonometry classroom, and nobody appears to have written its history.
Cross-curricular hooks
For a linguistics unit: sound change with a paper trail
The word "sine" is a complete case study in how languages borrow, and every step is documented.
A borrowing that keeps the sound and drops the meaning: Sanskrit jya becomes Arabic jiba, which means nothing in Arabic. A writing system that permits a reading error: Arabic script leaves the short vowels off, so the consonant skeleton of jiba can be read as jaib. A reanalysis, which linguists call folk etymology: a meaningless foreign word gets replaced by a native word that sounds like it. Braunmühl calls this "ein in der Sprachgeschichte so gewöhnlicher Vorgang, dass es überflüssig ist, Beispiele anzuführen," so ordinary a process in the history of language that it is pointless to give examples (S425). Then a translation of the reanalysed meaning rather than the original: Latin sinus for "fold, bay," which is what jaib means and what jya never meant.
The same class can look at the smaller cases in this chapter. Sagitta and sahm are a translation that preserves an image (the arrow across the bowstring) across three languages. Degree, from gradus, "a step," is a metaphor that has gone dead. Nobody hears "step" in "degree," though they do in "gradual," "graduate" and "degrade." Minute, from pars minuta prima, is a phrase compressed to one word, with the stress shifting to separate the noun from the adjective. And "cosine" is a clipping, the same process that gives "app," "gym" and "flu," applied to Latin complementum in 1620.
For an evidence-standards lesson: what "first attested use" means
This chapter is built on one distinction, and it is worth an hour of class time on its own. It is the difference between when a thing was invented and when somebody first wrote it down in a document that survives.
Every OED citation in this chapter is a claim about surviving documents, not about speech. When the OED says the first use of "sine" in English is Thomas Fale in 1593, it means: this is the earliest example in print that lexicographers have found so far. It does not mean nobody said it in 1592. Dictionary entries are the current state of a search, and they change. First-use dates in the OED get pushed earlier all the time, as more text gets digitized and searched.
Then run the class through the four levels of evidence used in this chapter:
- Primary text read directly. The Regiomontanus law of sines, read in the OCR of the 1533 printing, matching Smith's quotation word for word (S445). This is the strongest kind.
- Primary text quoted by a reliable scholar. Fincke's sentence from page 73, which Smith quotes in Latin but which this book could not see in the original because the scan was unavailable (S421).
- Scholar reporting another scholar. Cajori on Fincke, explicitly resting on Glaisher's 1915 article (S423). Two links from the book.
- Assertion with no citation. Dictionary.com's 1944 date for SOHCAHTOA (S447). Not evidence at all.
Here is a very good exercise. Give students the cosecant chain: Ball writes "I think," Smith drops the hedge, everyone cites Smith, then Van Brummelen opens the book and the word is not there. Ask them to draw the citation graph. Then ask what would have prevented the error. The answer is that somebody had to open the actual book, which took until 2014.
For a research-methods lesson: tools students can use themselves
Three of the sources behind this chapter are free, online, and usable by a fifteen-year-old.
Jeff Miller's Earliest Known Uses of Some of the Words of Mathematics (S427) is organized alphabetically by term. For each term it gives the OED citation, the specialist literature, and often the disagreements. It is where this chapter's first-use dates for "cosine," "secant," "versed sine," "unit circle," "asymptote" and "radian" come from. Students can look up any term in their own textbook and see how far back it goes.
Cajori's A History of Mathematical Notations (S423, S424), both volumes, is on the Internet Archive in full text. It is a history of symbols rather than words, and it is the reason this chapter can say who first wrote "arc. sin.", who first wrote , and who first wrote . It is also, as the radian case shows, a book with at least one internal contradiction in it. That makes it a good object lesson. Standard references are not infallible, and you catch them by checking a fact against another fact. Here, a man's date of death against the date of an examination paper.
The Internet Archive's full-text scans of Smith, Braunmühl, Inman and Regiomontanus (S421, S422, S425, S426, S435, S445), all public domain. A student can search Inman's 1858 Navigation and Nautical Astronomy for "haversine" in about thirty seconds. The word turns up in its working habitat, in a book meant to be used at sea. That is a more convincing encounter with the history than any summary.
A warning to hand over with the tools. OCR is imperfect, especially on Fraktur, the spiky blackletter type used in German printing, and on Latin full of abbreviations. So a search that returns nothing is weak evidence of absence. Several of the twenty-five open questions below are open partly because a scan was missing, a site returned an error, or an API hit its daily limit. That is the ordinary texture of research, and students should see it rather than be shielded from it.
Twenty-five things nobody has settled
Every item below was searched for and not established. Nothing in this chapter depends on any of them. For each: what is claimed, what the best available evidence is, and what would settle it.
- Which twelfth-century translator first wrote sinus. Claimed variously for Plato of Tivoli, Robert of Chester and Gherardo of Cremona. Best evidence: Braunmühl rules out Plato of Tivoli, because the word occurs once in his whole al-Battani translation while the running text says chorda. Braunmühl then leans "probably" to Gherardo. Smith's volume 2 credits Gherardo, while his volume 1 footnote credits Robert of Chester. Settled by: a manuscript study of the earliest surviving copies of the candidate translations, showing the word in place with a dated colophon (S425, S421, S422, S427).
- Whether cosecans appears in Rheticus's Opus Palatinum (1596). SETTLED, negative. Claimed by Ball ("I think") and by Smith following Ball. Van Brummelen has since published the reason it is not there: Rheticus rejects the modern function names altogether and "refers to them simply as the hypotenuse, base, and perpendicular of triangles of the three species" (S481, p. 275). He tabulates all six functions and names none of them in the modern way, so the word is not missing by accident. The same page dates cosine twenty-four years later and gives tangent and secant to Thomas Fincke (S481, p. 275 and n. 156). Kept on this list because it started here, and because the shape of the answer is worth more than the answer (S427, S421, S481).
- Whether Inman coined "haversine" in the 1821 first edition or the 1835 third. Best evidence: the OED (2nd ed.) and Wikipedia say the 1835 third edition. Cajori credits the function to Mendoza y Ríos in 1801 and merely notes Inman's 1821 treatise. This book verified the word only in the 1858 edition. Settled by: scans of the 1821 and 1835 editions, searched for the word (S423, S435, S441).
- The J. D. White articles in Nautical Magazine, February and July 1926, cited by Cajori as the source for the haversine history. Not located, not read. Settled by: a library holding a run of the Nautical Magazine for 1926 (S423).
- Whether the 1867 first edition of Thomson and Tait's Treatise on Natural Philosophy contains "For brevity we shall call this angle a radian" at page 31. SETTLED, negative. Claimed by Miller (S427). The 1867 first edition has since been read for this book, and the word "radian" occurs in it zero times, against nine times in the 1879 new edition and eight in the 1890 (S489, S490). The 1867 text carries the idea and the correct value without the name. The 1873 Belfast priority is not overturned, and the date this book prints stands.
- The contents of the 1910 Nature letters on the radian. SETTLED, and there are four of them, not three. All four have since been read in full as primary text (S485, S486, S487, S488). The one this book did not know about is James Thomson the son, 21 April 1910, and it is the primary source for the exact date Chapter 7 prints: the name "appears in the printed examination questions set by him in the general class examination in Queen's College, Belfast, on June 5, 1873" (S486). The letters also settle the authorship question: the replies come from James Thomson of Newcastle-on-Tyne, writing about his father of the same name (S486, S488).
- Michael Cooper, "Who named the radian?", Mathematical Gazette 76(475), 1992, pp. 100 to 101. Miller lists it and says he has not seen it. Nor has this book. Settled by: reading it; it may already answer item 5 (S427).
- The origin, author or first print appearance of SOH-CAH-TOA. Claimed by Dictionary.com to be in print by 1944, with no citation. Best evidence: absent from Cajori, Smith, Braunmühl and Miller; MathWorld gives no history. Settled by: a full-text search of digitized American and British textbooks from 1900 to 1960. This book could not run that search, because the Internet Archive and HathiTrust searches were blocked and the Google Books API was rate-limited (S446, S447).
- The British "TOA-CAH-SOH" reordered variant. Found only in informal sources. Best evidence: MathWorld's British note concerns sentence mnemonics ("Tommy On A Ship Of His Caught A Herring"), not a reordered acronym. Settled by: a dated British textbook or examination paper using the reordered form (S446).
- The origin, author or date of the Indian "Pandit Badri Prasad Har Har Bole Sona Chandi Tole" mnemonic. Its meaning is verified (the six ratios in Perpendicular/Base/Hypotenuse form) but no origin is documented anywhere reachable. Note that it is a right-triangle mnemonic and not a unit circle one, contrary to how it is often described. Settled by: a dated Indian textbook containing it (S451).
- Whether Dürer's Underweysung der Messung (1525) contains a projected helix amounting to a drawn sine curve. Best evidence: Braunmühl, who surveys the drawn trigonometric curves in detail, does not mention Dürer. Settled by: the relevant plates of the 1525 edition, plus a specialist's judgment on whether the construction is a sine curve or merely a helix projection (S426).
- Whether Newton ever drew a sine curve. No evidence found in Braunmühl or Smith, which record his series for sine and arcsine but no graph. Settled by: a search of the published mathematical papers and notebooks (S426, S421).
- First drawing of the cosine, cotangent and cosecant curves. Braunmühl names the sine (Roberval), tangent (Gregory) and secant (Wallis) and stops. Settled by: a survey of seventeenth and eighteenth century quadrature literature, which is exactly the corpus Braunmühl was working in, so the answer may simply not exist in print (S426).
- First description of the tangent graph's vertical asymptotes. Gregory drew only the first quadrant, so he did not draw one. No source consulted addresses it. Settled by: finding the earliest full-period drawing of the tangent curve and checking what its author says about the behavior at (S426, S427).
- A first use or first attribution for the SSA ambiguous case. Absent from Smith, Cajori, Miller and the Wikipedia treatment. Settled by: working back through early modern triangle-solving manuals, which is a well-defined and finite project (S450, S421, S423, S427).
- A coiner or date for "exsecant." Best evidence: Cajori records the symbol "exsec A" with no attribution. Settled by: nineteenth-century railway surveying manuals, which are where the function was used (S423).
- Any history at all for "midline." Not in Cajori (either volume), Smith (either volume), Braunmühl or Miller. It appears to be a recent American schoolbook coinage, but this chapter has no evidence for that. Settled by: a dated textbook using it in the sinusoid sense, and a search of American curriculum documents from roughly 1960 onward.
- The first use of "amplitude," "period," "phase" or "frequency" specifically of a trigonometric graph, as opposed to their general, astronomical or physical senses, which are dated in this chapter. Settled by: searching nineteenth-century physics and engineering texts for the words applied to a drawn curve rather than to a motion (S427, S442).
- Smith's claim that de Lagny first used "goniometry" in 1724. Uncorroborated: a full-text search of both volumes of Cajori's Notations for "goniometr" found only an 1812 citation to Cole's Stereogoniometry, and Miller has no entry. Settled by: de Lagny's 1724 publication itself (S421, S424, S427).
- When the memorized unit circle chart became a standard American classroom artifact. The shift to circular functions on the unit circle is well documented for 1959 to 1961 (CEEB report, SMSG Elementary Functions), but no source reached dates the chart. Settled by: a survey of American trigonometry textbook illustrations decade by decade, plus NCTM standards documents, which were not obtained (S439, S438).
- The 1990 Mathematics Teacher article "The legend of Soh Cah Toa." HTTP 403 from the NCTM site; not read; supports nothing here. Settled by: NCTM access. It may or may not contain an origin story; the title suggests the author knew it was folklore (S449).
- Primary scans of Fincke's Geometria rotundi (1583), Pitiscus's Trigonometria (1595) and Gunter's Canon triangulorum (1620). e-rara returned HTTP 500 and the Munich digital library requires JavaScript. All claims in this chapter about these three books rest on Smith, on Cajori (who in turn depends on Glaisher's 1915 study of Fincke), and on Miller. Smith's direct Latin quotation of Fincke page 73 is the closest thing to primary evidence. Settled by: working scans, which certainly exist (S421, S423, S427).
- Dates for the Sulba Sutras, the Jiuzhang suanshu and the Zhoubi suanjing. Smith gives "the dates of the Sulvasutra period are unknown" with a footnote guess of the fourth or fifth century BCE for the first. He gives "perhaps written before 1100 b.c." for the second and "c. 1105 b.c." for the third. The last two are far earlier than modern scholarly consensus and should not be repeated. Settled by: current specialist scholarship in Indology and in the history of Chinese mathematics, not by a 1925 survey (S421, S422).
- Babylonian evidence for the Pythagorean relation at the tablet level. Neither Smith nor Braunmühl discusses Plimpton 322 in the passages read. The only Babylonian statement verified is a general one in the Joyce commentary on Elements I.47, which names no tablet and cites no edition. Settled by: a cuneiform specialist source with tablet numbers and an edition (S428).
- Smith's page reference for the Chinese "Nine Sections" quotation ("Square the first side and the second side and add them together; then the square root is the hypotenuse"). Smith gives the quotation without naming an edition or a translator, so the wording is his and is not checkable against a text. Settled by: a modern critical translation of the Jiuzhang suanshu, which exists, and a comparison of the passage (S421).
- Names attach to whoever the community was reading when a result became useful, not to whoever first had it: the pattern repeats through all 94 terms and symbols.
- The Pythagorean rule was in use long before Pythagoras, and what he added, if anything, is not documented (S421, S428): a rule and a theorem are different achievements.
- Attributions are chains of copies, and this chapter checks the links: dates against lifespans, quotations against the printed page, credits against the primary source.
You have reached the reference shelf. Arrive here from any chapter or from your course; nothing in this chapter assumes you read it in order.
Six function names sit inside your mnemonic, and every one has a biography in this chapter: a bay from a translation, two shadows, a touching line and two late co- coinages. The mnemonic lives in Trigonometry 1.2
The theorem in your course carries Pythagoras's name. What part of it is his?
Show the answer▸
Not the rule, and no proof survives either. The relation was in use in Mesopotamia and stated in China long before Pythagoras (S421), and no document shows him proving it; Euclid's proof is the oldest that survives (S428). The name records who the naming community had read, which is this book's oldest and most repeated lesson.
The master timeline: every date, in one place
260 dated events across ten eras, merged from 332 sources. Where sources disagree about a date, the disagreement is shown rather than resolved by picking a favorite. Read a row as a claim with a provenance, not as a fact from nowhere.
- Place any person or result in this book on a single shared timeline.
- Read a dated row together with its sources, and say how well attested it is.
- Spot a disputed date and say what the disagreement is about.
- Compare what was happening in different parts of the world in the same century.
Part II
How to read the timeline
Every dated event from every source, merged into one sequence and deduplicated. 260 entries, running from 3000 BCE to 2025.
How to read it. Every row is a claim with a provenance, not a fact from nowhere. The Sources column names the documents it rests on, and each one links to its full reference in Appendix J. The Who column lists the people from Appendix A this entry connects to, which is not always the row's actor: a person who appears in no other register stays in the entry text and off the column.
A row carrying the Disputed tag is not settled: the date, the attribution, or both. The reason is stated underneath the event, and Appendix F gives both sides in full. 20 of these 260 entries are tagged disputed, and 73 of this book's 84 logged disagreements are still open, so you will meet the tag often. That is the honest state of the evidence, not a gap in the research.
A word about the dates themselves. A great many are approximate, and the table says so rather than smoothing it over. "c." means circa, roughly. A range means the sources disagree or the event took years. Where an ancient date is a modern scholarly reconstruction rather than something written on the object, the event text says so.
Timeline
c. 3000 to 300 BCE: Before the function: Mesopotamia and Egypt
19 entries.
| When | What happened | Who | Where | Sources |
|---|---|---|---|---|
| c. 3000 BCE | The proto-cuneiform accounting tablets from Uruk record four sets of units for counting discrete objects plus area, calendar, capacity and probably weight systems, each on its own number base; Eleanor Robson identifies one of the discrete-object systems as the ancestor of the sexagesimal place value system. | Eleanor Robson | Uruk, southern Iraq | S020 |
| c. 2400 BCE | Under the dynasty of Akkad the traditional metrological systems are overhauled and linked together, in Robson's words, "with new units based on divisions of sixty", and brick sizes and weights are standardized. | Eleanor Robson | Akkad, Mesopotamia | S020 |
| c. 2050 BCE | The sexagesimal place value system is demonstrably in use under the Third Dynasty of Ur; Robson names the older view, that it was an Old Babylonian innovation some 250 years later, as explicitly superseded. | Eleanor Robson | Ur and Girsu, southern Iraq | S020 |
| c. 1849 to 1801 BCE | The lost original from which the Rhind papyrus was copied is written under Nimaatre (Amenemhat III), whom T. E. Peet dates to about 1849 to 1801 BCE. | Amenemhat III, Thomas Eric Peet | Egypt | S005 |
| 1822 BCE | The grain account YBC 4721 is written at Ur; Robson uses it as her closest comparandum for Plimpton 322, since it shares the landscape format, the column headings, the descending sort, the left-to-right calculation and the final MU.BI.IM column. | Eleanor Robson | Ur, southern Iraq | S001 |
| c. 1822 to 1762 BCE | Plimpton 322 is written, dated by Robson from paleography, orthography and tabular format (a landscape tablet whose last column is headed MU.BI.IM, a format attested at Larsa from 1822 BCE) to the roughly sixty years before Hammurabi took Larsa in 1762 BCE; the narrower spans Robson gives elsewhere (1822 to 1784 in 2002, 1837 to 1784 in 2001) describe the comparable administrative tables, not this tablet, so they are not rival dates for it. | Eleanor Robson, Hammurabi | Larsa (Tell Senkereh), southern Iraq | S001, S002, S003 |
| c. 1900 to 1600 BCE | Si.427, an Old Babylonian field plan that lays out land boundaries with the (5, 12, 13) and (8, 15, 17) diagonal triples, is drawn up at Sippar. Neither the CDLI record nor Daniel Mansfield narrows it inside the Old Babylonian period, and the tighter date would be in Mansfield's 2020 edition, which was paywalled. | Daniel F. Mansfield | Sippar, central Iraq | S004, S025, S011, S032 |
| c. 1788 to 1580 BCE | The Rhind Mathematical Papyrus is copied by the scribe Ahmose, who dates it in its own text to year 33 of the Hyksos king Aauserre Apophis; Peet's window of 1788 to 1580 BCE is preferred over the British Museum's flat "1550 BC" because Peet reasons from the papyrus's own regnal date and states the chronological uncertainty, while the museum record gives a single conventional production year with no argument attached. | Ahmose, Apophis, Thomas Eric Peet | Thebes region, Egypt | S005, S008, S009 |
| c. 1550 BCE | The Rhind (Ahmes) papyrus poses five pyramid problems, four of which give the seqt of the slope; Peet reads it as the cotangent of the base angle, Eisenlohr as a ratio of a different angle, and D. E. Smith records both. Smith's "c. 1550 BCE" follows the British Museum date, not the papyrus's own internal window of 1788 to 1580 BCE given in the row above. | Thomas Eric Peet | Egypt | S421, S005 |
| c. 1521 to 1473 BCE | The earliest written evidence for an Egyptian sundial appears in a campaign narrative from the reign of Thutmose III, describing an army setting out at noon "when the shadow of the sun turns"; Vodolazhskaya's reign dates of 1521 to 1473 BCE are not the most widely used Egyptological ones, so the absolute date moves with the chronology chosen. | Egypt | S024 | |
| c. 1300 BCE | An image of a sundial is painted in the tomb of Seti I, dated by Vodolazhskaya to about 1300 BCE. | Valley of the Kings, Egypt | S024 | |
| c. 1400 to 1200 BCE | The tablet AO 17264 and its relatives record trapezoid bisection, the Old Babylonian and Kassite precursor to the Late Babylonian Jupiter procedure that computes time by bisecting a trapezoid of equal area. | Kassite Babylonia | S018 | |
| c. 1000 to 800 BCE | The pine altitude sundial now in the Science Museum Group as object 1926-992 is made; its horizontal cross-bar is turned to face the sun so the shadow falls on a base marked in equal time units. | Qus, Egypt | S023 | |
| before 750 BCE | MUL.APIN is complete by about 750 BCE, fixed as a terminus ante quem by the Huzirina tablets that carry it; Hunger and Steele report star-list analyses suggesting composition between 1300 and 1000 BCE, but present those as arguments about the age of the observations, not as a date for the compendium. Section II ii 21 to 42 carries a scheme for the length of a gnomon shadow at the solstices and the equinoxes. | Hermann Hunger, John M. Steele | Assyria and Babylonia | S007 |
| c. 652 BCE | The Babylonian Astronomical Diaries series begins, on the evidence of the title of Sachs and Hunger volume I, Diaries from 652 B.C. to 262 B.C.; the volume itself was not read, so the date rests on the title alone. | Abraham J. Sachs, Hermann Hunger | Babylon | S007 |
| c. 600 BCE | The bronze merkhet of Bes, son of Khonsirtis, astronomer priest of Horus of Edfu, is made; Science Museum Group object 1929-585, a sighting bar with a plumb line used for star transits and for setting building axes. | Edfu, Upper Egypt | S021 | |
| 564 BCE Disputed | Li and Sun's statistical reconstruction of the pre-modification solstice shadow data in the Zhoubi suanjing points to an observation epoch of 564 BCE at latitude 35.78 degrees N. Zhao (2009), whom they cite, proposes 511 BCE instead, and Qian (1958) and Bo (1989) held that the data were never observational at all. Unresolved: this is a reconstruction from data the authors agree were altered. Dispute: Li and Sun's statistical reconstruction gives an observation epoch of 564 BCE at latitude 35.78 degrees N; Zhao (2009), whom they cite, proposes 511 BCE; Qian (1958) and Bo (1989) held that the shadow data were never observational at all. sort_year follows Li and Sun. | northern China, about 35.8 degrees N | S244 | |
| 500 to 400 BCE Disputed | The Apastamba Sulbasutra gives cord lengths for constructing right angles, among them (3, 4, 5), (5, 12, 13), (8, 15, 17) and (12, 35, 37). Smith's main text says the dates of the Sulvasutra period are unknown; his footnote guesses the fourth or fifth century BCE. Both his statements are printed here because the second does not follow from any evidence he cites. Dispute: D. E. Smith's main text says the dates of the Sulvasutra period are unknown, while his own footnote guesses the fourth or fifth century BCE. Both of his statements are recorded here because the second does not follow from any evidence he cites. sort_year is the midpoint of the range 500 BCE to 400 BCE. | India | S421 | |
| c. 400 BCE | Zodiacal signs of twelve equal parts of 30 UŠ each begin to appear in Babylonian observation reports; John Steele places the emergence of the uniform zodiac "sometime in Babylonia during the late fifth century BC" and traces it to the schematic calendar. | John M. Steele | Babylon | S016 |
Timeline
300 BCE to 400 CE: Greece builds the chord table
15 entries.
| When | What happened | Who | Where | Sources |
|---|---|---|---|---|
| c. 300 BCE | Euclid's Elements states the right-triangle square relation and its converse at I.47 and I.48, and the obtuse and acute generalizations, which are the law of cosines in geometric dress, at II.12 and II.13. | Euclid | Alexandria | S428 |
| c. 240 BCE | Archimedes, in Measurement of a Circle Proposition 3, brackets between and using inscribed and circumscribed regular 96-gons; Heath notes the surviving text is only a fragment of a longer work. | Archimedes of Syracuse | Syracuse | S070 |
| c. 240 BCE | Aristarchus, in On the Sizes and Distances of the Sun and Moon, proves by inscribed and circumscribed constructions that the Sun is more than 18 and less than 20 times as far away as the Moon. The year is a placeholder: the treatise carries no internal date and Aristarchus flourished about 280 to 260 BCE. | Aristarchus of Samos | Samos or Alexandria | S069 |
| c. 240 BCE | Eratosthenes measures the Earth's circumference from a shadow angle of one fiftieth of a circle at Alexandria against a shadowless gnomon at Syene, 5,000 stades apart. Cleomedes reports his result as 250,000 stades, Theon of Smyrna and Strabo as 252,000; Heath writes that "the reason of the discrepancy is not known". | Eratosthenes of Cyrene | Alexandria and Syene | S068 |
| 350 to 50 BCE | The four or five Late Babylonian tablets carrying the trapezoid procedure for Jupiter's displacement are written (BM 40054, BM 36801 and others). Ossendrijver's dating from his own edition of the tablets is preferred over the wider "c. 400 to 50 BCE" figure, which is a range for the astronomical procedure texts as a genre rather than for these tablets. | Mathieu Ossendrijver | Babylon | S017, S018 |
| 150 BCE Disputed | Hypsicles writes the Anaphorikos, which divides the zodiac circle and the day-and-night into 360 parts each and introduces the "spatial degree" and "time degree"; per the Dictionary of Scientific Biography it is the first work to divide the ecliptic into 360 parts. Dating unresolved: the DSB entry gives about 175 BCE, while Heath and MacTutor give about 150 BCE, and every value rests on Hypsicles' own uncertain life dates. Dispute: The Dictionary of Scientific Biography dates the Anaphorikos to about 175 BCE; Heath and MacTutor give about 150 BCE. Every value rests on Hypsicles' own uncertain life dates. | Hypsicles of Alexandria | Alexandria | S019, S006, S068, S084 |
| c. 140 BCE | Hipparchus computes the first recorded table of chords. The table is lost; Duke's reconstruction of 23 non-trivial entries at degree steps on a circle of radius 3438 is expressly labeled "a possible replica", not a recovered text. | Hipparchus of Nicaea | Rhodes | S421, S064 |
| 146 to 126 BCE | The observations of Hipparchus that Ptolemy cites in the Almagest run from 26/27 September 146 BCE (midnight) to 7 July 126 BCE, per Toomer's index; these are the only hard dates in his life, so the conventional "c. 190 to c. 120 BCE" is a round convention with no evidence behind it in anything read here. | Claudius Ptolemy, Hipparchus of Nicaea | Rhodes and Alexandria | S061 |
| c. 128 BCE | The approximate epoch of Hipparchus's star catalog, inferred from Ptolemy's statement that about 265 years separate Hipparchus's measurement from 137 CE; both the 2022 and the 2024 Codex Climaci Rescriptus papers argue from this same Ptolemaic figure. | Claudius Ptolemy, Hipparchus of Nicaea | Rhodes | S065, S066 |
| 100 BCE Disputed | The Zhoubi suanjing is compiled from older material, carrying the gnomon shadow tables, the cun-qian-li rule and the Chen Zi sun-height computation. Li and Sun say "probably compiled about BC 100"; Christopher Cullen conjectures final compilation in the early first century CE. Unresolved from what was read. Dispute: Li and Sun say the Zhoubi suanjing was probably compiled about 100 BCE; Christopher Cullen conjectures final compilation in the early first century CE. | Chen Zi | China, Western Han | S241, S244, S254 |
| 98 CE | Menelaus observes the Moon occulting Spica on the night of Mechir 15/16 in the first year of Trajan, and days later an occultation in the forehead of Scorpius; Ptolemy reports both in Almagest VII 3, and they are the only firmly dated events in Menelaus's life. | Claudius Ptolemy, Menelaus of Alexandria | Rome | S061, S068, S074 |
| c. 100 CE | The Jiuzhang suanshu is circulating in something close to its received form, chapter 9 "Gougu" carrying the right-triangle problems. Smith's "perhaps written before 1100 b.c." is far earlier than modern sinology allows and is not followed here. | Han China | S246, S421 | |
| 146 CE | Ptolemy erects the Canobic Inscription in the tenth year of Antoninus; Toomer, following N. T. Hamilton, reads it as a stage of theory earlier than the Almagest. | Claudius Ptolemy | Canopus, Egypt | S061 |
| c. 150 CE | Ptolemy publishes the Almagest, whose Book I chapter 10 proves the inscribed-quadrilateral lemma and derives from it the chord of a sum, the chord of a difference and the chord of a half arc, on a circle of radius 60. Toomer argues it "can hardly have been published earlier than the year 150"; the latest observation used, 2 February 141, gives a bare terminus post quem of 141. | Claudius Ptolemy | Alexandria | S061, S425 |
| 263 CE | Liu Hui completes his commentary on the Nine Chapters, dated in the text to "the fourth year of the era of the Jingyuan reign"; his 3072-gon gives pi as 3.14159, and the chong cha double-difference surveying material appended to chapter 9 is later detached as the Haidao suanjing. | Liu Hui | Wei state, China | S247, S250, S251 |
Timeline
400 to 750: India turns the chord into the sine
14 entries.
| When | What happened | Who | Where | Sources |
|---|---|---|---|---|
| 400 CE Disputed | The surviving recension of the Surya Siddhanta gives a table of 24 half-chords and of versed sines (utkramajya) on radius 3438. The date is genuinely open: P. C. Sengupta concludes "the earliest date of the Surya Siddhanta cannot be pushed up much higher than 400 A.D.", Sudhakara Dvivedi cites Nityananda for Kali 3000 elapsed, that is Shaka 421, that is 499 CE, with reasons that Sengupta says "are not stated", and Smith simply prints c. 400. Burgess reproduces Bentley's far later planetary-error dating without endorsing it. Dispute: P. C. Sengupta concludes the earliest date cannot be pushed up much higher than 400 CE; Sudhakara Dvivedi, citing Nityananda, gives 499 CE with reasons Sengupta says are not stated; D. E. Smith simply prints c. 400; Burgess reproduces Bentley's far later planetary-error dating without endorsing it. | India | S121, S421, S443 | |
| March 415 CE | Hypatia is killed at Alexandria in March. The only documentary trace of her mathematical work is the heading to Book III of Theon's commentary on the Almagest in Laurentianus 28.18, describing the edition as "revised by my philosopher-daughter Hypatia"; Alan Cameron argues even the standard inference from that heading is too strong. | Hypatia, Theon of Alexandria | Alexandria | S084, S073 |
| 429 to 501 CE Disputed | Zu Chongzhi lives and writes the Zhui shu, which brackets as and gives the convergent , as reported in the Sui shu. Death year unresolved: MacTutor gives 501, much of the literature gives 500, and no primary basis for choosing was found. Dispute: Death year: MacTutor gives 501, much of the literature gives 500, and no primary basis for choosing between them was found. sort_year is the midpoint of the range 429 to 501. | Zu Chongzhi | Jiankang (Nanjing) | S252, S271 |
| 499 CE | Aryabhata composes the Aryabhatiya, dated by its own internal epoch at Kalakriyapada III.10 (three yugapadas and 3600 years elapsed, 23 years of the author's life gone); I.10 gives the 24 sine differences and uses ardha-jya, jya-ardha and the abbreviation jya. Clark notes Parameshvara's report of the Prakasikakara that 499 is instead the epoch of the calculations. | Aryabhata I, Parameshvara | Kusumapura (Pataliputra) | S122, S444 |
| 505 CE | Shaka 427, the year cited in Varahamihira's Pancasiddhantika, corresponds to 20 to 21 March 505 CE; the text computes a sine table on the Greek diameter 120 and, per Smith, contains the half-angle relation. What the year dates is contested: Thibaut and Dvivedi's edition records the debate over whether it is Varahamihira's birth year or the year of composition. | Varahamihira | Ujjain | S138, S421 |
| c. 500 to 600 CE | An ancient codex containing Aratus's Phaenomena together with star coordinates is written; the dating is palaeographic, fifth or sixth century CE. | not established | S065 | |
| c. 628 CE | Brahmagupta composes the Brahmasphutasiddhanta at the age of 30; Clark computes the interval as 129 years after the Aryabhatiya. | Brahmagupta | Bhillamala | S122, S137 |
| c. 629 CE | Bhaskara I states the rational approximation to the sine at Mahabhaskariya VII.17ff; Datta and Singh date the statement to this year, while the Britannica entry gives Bhaskara I only as "c. 600 to c. 680" and no date for the formula. | Bhaskara I | India | S128, S144 |
| c. 665 CE | Brahmagupta composes the Khandakhadyaka at 67; the supplement's chapter I.8 gives the second-order interpolation rule. Datta and Singh note the same rule already stands in his earlier Dhyanagrahopadesa (st. 17) and does not stand in the Brahmasphutasiddhanta of 628. | Brahmagupta | Ujjain | S125, S128, S137 |
| 683 CE Disputed | Yixing, the Buddhist monk who would direct the Tang gnomon survey and compile the Dayan li, is born. Unresolved: Jeffrey Kotyk (2022) argues for 673 following Jinhua Chen's 2000 genealogical study, while Wu (2023), the later publication, still prints 683, which remains the commoner figure. The death year 727 is agreed. Dispute: Birth year: Jeffrey Kotyk (2022) argues for 673, following Jinhua Chen's 2000 genealogical study; Wu (2023), the later publication, still prints 683, which remains the commoner figure. The death year 727 is agreed. | Yixing | Chang'an | S253, S254 |
| 718 CE | Gautama Siddha translates the Navagraha-karana into Chinese as the Jiuzhi li, which assumes a spherical Earth, tabulates a latitude of 35 degrees, and introduces terrestrial latitude to Chinese readers as suifang yan fa. | Gautama Siddha | Chang'an | S253, S255 |
| 724 to 725 CE | Nan Gongyue and his field parties carry out Yixing's empire-wide gnomon survey, at 13 stations per Li and Sun. The latitude band is contested: Ohashi gives 18 to 51 degrees N, Cullen 29 to 52 degrees N near meridian 114 E, and Kotyk says plainly that he cannot see how either figure was reached. Ohashi says Yixing traveled the route himself; Kotyk says the records do not support that. | Nan Gongyue, Yixing | Tang empire, from Annan (northern Vietnam) northward | S244, S253, S254 |
| 727 CE | Yixing dies with the Dayan li complete but not yet in force; the calendar is officially used from 729 to 761, and the tangent-like shadow table embedded in it is what historians describe as the first Chinese tangent table. | Yixing | Chang'an | S253 |
| 733 CE | Gautama Zhuan formally accuses the late Yixing of plagiarizing the Navagraha-karana; the court investigation finds the charge false, though Kotyk remarks that "this conclusion might have been premature". | Gautama Zhuan, Yixing | Chang'an | S253 |
Timeline
750 to 1200: The Islamic world completes the six functions
26 entries.
| When | What happened | Who | Where | Sources |
|---|---|---|---|---|
| c. 813 to 833 CE | Under the caliph al-Mamun, Khalid ibn Abd al-Malik al-Marwarrudhi, Ali ibn Isa al-Asturlabi and others measure one degree of the meridian; the length is transmitted as 56, as and as 57 Arabic miles of 4,000 black cubits, depending on whether al-Biruni or Ibn Yunus is quoting Habash. | Al-Biruni, Habash al-Hasib al-Marwazi, Ibn Yunus | Sinjar desert, between Wamia and Tadmor | S185 |
| c. 853 to 866 CE | Al-Mahani rectifies the second Arabic translation of Menelaus's Spherics; al-Harawi later revises al-Mahani's edition, al-Dimashqi produces another translation, and manuscript colophons assign the original Arabic to Hunayn ibn Ishaq or to his son Ishaq ibn Hunayn (died 910). The Greek text is lost. | Menelaus of Alexandria | Baghdad | S074, S068 |
| 860 CE Disputed | Habash al-Hasib introduces the "shadow" of an arc as a general function with a definition, a table and applications in his Mumtahan zij, and constructs the first table of tangents and cotangents, which survives only in a Berlin manuscript. Both the event date and his life dates are unresolved: Britannica dates the work "around 860" while Debarnot dates the surviving zij to "at least after 869"; Springer's Biographical Encyclopedia of Astronomers gives his life as c. 796 to c. 894 and Wikipedia gives death c. 869, which cannot both stand. Dispute: Event date: Britannica dates the work around 860, while Debarnot dates the surviving zij to at least after 869. Life dates: Springer's Biographical Encyclopedia of Astronomers gives c. 796 to c. 894 and Wikipedia gives death c. 869, which cannot both stand. | Habash al-Hasib al-Marwazi | Baghdad or Samarra | S181, S209, S219, S187, S421 |
| 877 to 918 CE | Al-Battani carries out his observing program at al-Raqqa, the run of observations on which his zij rests. | Al-Battani | al-Raqqa, Syria | S188, S202 |
| c. 800 to 1000 CE | The Greek astronomical undertext is scraped off the Aratus codex and the parchment reused for Syriac translations, producing the Codex Climaci Rescriptus; the Museum of the Bible dates the Syriac layer to the 800s to 900s, and the 2022 paper says "by the 9th or 10th c.". | St Catherine's Monastery, Sinai | S065, S083 | |
| c. 904 CE | Vatesvara states, at Vatesvara-siddhanta chapter 2 section 1 verses 65 to 66, the rule that sine differences are summed forward to give sines and backward to give versed sines. | Vatesvara | India | S128 |
| c. 920 CE | Al-Battani's zij, later Latinized as De motu stellarum, distinguishes umbra extensa (the straight shadow) from umbra versa (the turned shadow) and gives the first widely known table of shadows. Smith dates the table c. 920; Braunmuhl dates the underlying observations 878 to 918, and al-Battani died in 929, so the two are compatible. | Al-Battani | al-Raqqa, Syria | S421, S425 |
| 944 to 967 CE Disputed | Al-Ijliyyah works as an astrolabe maker in the service of Sayf al-Dawla. Ibn al-Nadim's Fihrist records her in a single clause, as the daughter of al-Ijli al-Asturlabi and the pupil of Betulus, and nothing else about her is attested. The years 944 to 967, printed everywhere as her life dates, are Sayf al-Dawla's regnal years; even her teacher's name is unstable in the manuscripts. Dispute: The years 944 to 967, printed everywhere as al-Ijliyyah's life dates, are Sayf al-Dawla's regnal years, not dates attested for her; Ibn al-Nadim's Fihrist records her in a single clause and even her teacher's name is unstable in the manuscripts. sort_year is the midpoint of the range 944 to 967. | Al-Ijliyyah | Aleppo | S186, S215 |
| c. 980 CE | Abu al-Wafa constructs the first known table of tangents at 15-minute intervals, defines chord, sine and versed sine clearly, knows in substance tan = sin/cos, cot = cos/sin, sec = and csc = , and, per Van Brummelen 2009, works with the trigonometric radius R = 1. | Abu al-Wafa al-Buzjani | Baghdad | S421 |
| 994 CE | Al-Khujandi measures the obliquity of the ecliptic as 23;32,19 degrees with the al-suds al-Fakhri sextant. | Abu Mahmud al-Khujandi | Rayy | S194 |
| 997 CE | Abu al-Wafa at Baghdad and al-Biruni at Kath observe the same lunar eclipse in order to get the longitude difference between the two cities. | Abu al-Wafa al-Buzjani, Al-Biruni | Baghdad and Kath | S181, S189 |
| 994 to 1004 CE | Al-Biruni compiles the Maqalid ilm al-hay'a (Keys of Astronomy), which records the three-cornered argument between Abu al-Wafa, Abu Nasr ibn Iraq and al-Khujandi over who first had the new spherical theorems. | Abu Mahmud al-Khujandi, Abu Nasr Mansur ibn Iraq, Abu al-Wafa al-Buzjani, Al-Biruni | Khwarazm | S181 |
| c. 1018 CE | Al-Biruni, at the fort of Nandana, measures a mountain height of 652;3,18 cubits and a horizon dip of 34 arcminutes and derives an Earth radius from them; the Tahdid is dated 1018 by the Biographical Encyclopedia of Astronomers, though the source read does not date the observation itself. | Al-Biruni | Nandana, Punjab (now Pakistan) | S185, S190 |
| 1030 to 1040 CE | Al-Biruni composes al-Qanun al-Masudi for Sultan Masud; its third treatise, in ten chapters, is on plane and spherical trigonometry. | Al-Biruni | Ghazna | S181 |
| c. 1039 CE | Sripati, like Suryadeva Yajva after him, states explicitly that sine differences are summed forward for sines and backward for versed sines, and gives the rational sine formula in the variant with 10125 in the denominator. | India | S128 | |
| c. 1050 CE | Ibn Muadh al-Jayyani (born 989 in Cordoba, died after 1079) writes The Book of Unknown Arcs of a Sphere, the first treatise devoted to spherical trigonometry: all six right-triangle relations, the general law of sines, the polar triangle, tangents tabulated as tan rather than R tan, and a refusal of the word "shadow". Debarnot writes that its recent discovery "brings us more questions than answers" about transmission to the West. | Ibn Muadh al-Jayyani | Jaen, al-Andalus | S430, S181 |
| 1031 to 1095 CE | Shen Kuo lives; chapter 18 of his Mengxi bitan contains the huiyuan arc rule, an approximation for a circular arc from the chord and the sagitta. The earliest surviving edition of the Mengxi bitan is of 1305. | Shen Kuo | Northern Song China | S257 |
| c. 1116 CE | Plato of Tivoli translates al-Battani's zij into Latin as De motu stellarum, using chorda and chorda versa, with sinus versus appearing only once. Cajori (1906) credited this translation with introducing sinus to Latin; Braunmuhl and Kastner deny it, and the Wikipedia article on Plato of Tivoli makes no such claim, so the attribution rests on Cajori alone. | Al-Battani, Florian Cajori, Plato of Tivoli | Barcelona and Iberia | S202, S425, S427 |
| c. 1126 CE | Adelard of Bath translates al-Khwarizmi's astronomical tables into Latin; the DSB calls tables 58 and 58a "very probably the first sine tables to appear in Latin". The date comes from the text's own equation of 1 Muharram 520 AH with 26 January 1126, but the DSB notes a Cambridge manuscript with worked examples for 1133 and 1134 and a 1133 solar eclipse, which "throws some doubt on the date", and Millas-Vallicrosa proposed an earlier version by Petrus Alphonsus that Adelard retranslated. | Adelard of Bath, Al-Khwarizmi | England or Spain (place of translation uncertain) | S320 |
| c. 1143 CE | Hermann of Carinthia makes a loose Latin version of Ptolemy's Planisphere from an Arabic translation; the Greek is lost entirely, and Isaac Hebreus produced a further Latin version in 1518, also from the Arabic. | Claudius Ptolemy | Iberia | S071, S080 |
| 1145 CE Disputed | Robert of Chester completes his Latin translation of al-Khwarizmi's al-Jabr wa'l-muqabalah and revises al-Khwarizmi's astronomical tables. Which work first used sinus in the trigonometric sense, if either, is unresolved: the encyclopedia.com Science and its times entry puts it in the 1145 algebra, D. E. Smith's volume 1 footnote puts it in the revision of the tables, Boyer credits Robert generally, and Smith's own volume 2 credits Gherardo of Cremona instead. Dispute: Which work first used sinus in the trigonometric sense: the encyclopedia.com Science and its times entry puts it in the 1145 algebra, D. E. Smith's volume 1 footnote puts it in Robert of Chester's revision of the tables, Boyer credits Robert generally, Smith's own volume 2 credits Gherardo of Cremona, Cajori credits Plato of Tivoli, and Van Brummelen 2009 backs Robert's revision of the tables. | Gerard of Cremona, Al-Khwarizmi, Robert of Chester | Segovia, Castile | S325, S422, S427 |
| 1150 CE | Bhaskara II completes the Siddhanta Siromani at the age of 36; the Jyotpatti section of the Goladhyaya states the Rsine addition and subtraction theorems, which Smith also credits to "Bhaskara (c. 1150)". | Bhaskara II | Vijjadavida, near Ujjain | S128, S139 |
| c. 1150 CE | Robert of Chester readjusts al-Khwarizmi's tables to the meridian of London; 1145 is commonly printed for this, but the source read gives 1150 for the London readjustment specifically. | Al-Khwarizmi, Robert of Chester | London | S325 |
| 29 June 1153 | Ismail copies al-Harawi's recension of Menelaus's Spherics, now British Library Or. 13127, and dates the colophon 4 Rabi II 548 AH, that is 29 June 1153; the manuscript carries 118 spherical figures across 55 folios. | Menelaus of Alexandria | Damascus | S082 |
| 1175 CE | Gherardo of Cremona completes the Latin translation of the Almagest from Arabic. Braunmuhl thinks Gherardo "probably" introduced sinus to the Latin West through his astronomical translations, and Smith's volume 2 says he used sinus for jaib; Smith's own volume 1 footnote prefers Robert of Chester, so Smith contradicts himself between his two volumes. | Gerard of Cremona, Robert of Chester | Toledo | S061, S422, S425 |
| 1198 CE | The Tongtian calendar of Yang Zhongfu adopts the tropical year of 365.2425 days, the value the Shoushi calendar would use in 1280 and the Gregorian reform would adopt in 1582. | China | S259 |
Timeline
1200 to 1500: Other roads, and the long transmission
30 entries.
| When | What happened | Who | Where | Sources |
|---|---|---|---|---|
| 1220 CE | Fibonacci's Practica geometriae defines sinus rectus arcus and sinus versus arcus and uses sagitta for the versed sine. | Pisa | S421 | |
| c. 1260 CE | Al-Tusi completes the Kitab al-Shakl al-Qatta (Treatise on the Quadrilateral) in five books, setting out all six cases of the right spherical triangle, the plane law of sines and the first known use of the polar triangle for solving a triangle from its angles, and treating trigonometry as a subject in its own right. Debarnot's 1260 is preferred over ProofWiki's "circa 1250" because Debarnot is the peer-reviewed specialist and a manuscript copy dated AH 658, that is 1260, is on record. | Nasir al-Din al-Tusi | Maragha | S181, S421, S433 |
| 1259 to 1262 CE | The Maragha observatory is built and commissioned under al-Tusi, who becomes its first director; MacTutor notes that Chinese astronomers assisted. | Nasir al-Din al-Tusi | Maragha, Ilkhanid Iran | S199, S203 |
| c. 1273 CE | Jacob ben Makhir ibn Tibbon translates Menelaus's Spherics into Hebrew from the Arabic; Halley worked mainly from this Hebrew version for his 1758 Oxford edition. | Menelaus of Alexandria | Provence | S074, S068 |
| 1279 CE | Guo Shoujing designs the Dengfeng (Gaocheng) observatory, whose gnomon (biao) is 40 chi high and whose shadow scale (gui) is 128 chi long. Li and Sun's conversion of 40 chi to 9.7468 m is preferred over the Biographical Encyclopedia of Astronomers figure of 12.28 m, because 9.75 m implies a chi of about 24.4 cm, which matches the Yuan chi, while 12.28 m would need a chi of 30.7 cm, which does not. | Guo Shoujing | Gaocheng, Dengfeng, Henan | S244, S259 |
| 1280 CE | Guo Shoujing and Wang Xun establish the Shoushi li, promulgated from 1281, with a tropical year of 365.2425 days and spherical interpolation methods; Wagner says it was "presented to the Chinese throne in 1280", which is the same event described differently. | Guo Shoujing, Wang Xun | Dadu (Beijing) | S258, S259 |
| 1283 CE | Li Qian (1223 to 1302) composes the Shoushi liyi, the theoretical exposition of the Shoushi calendar and the place where its arc methods are argued rather than merely tabulated. | Dadu (Beijing) | S259 | |
| c. 1286 CE | Abu al-Hasan Ali ibn Abd al-Malik ibn Simun dies after thirty years as muwaqqit at the Mosque of Amr, the first mosque timekeeper known by name. David King's own chapter says only that the office appears in Egypt "in the thirteenth century", so the precise year comes at second hand. | Fustat, Cairo | S216, S182 | |
| 1305 CE | The earliest surviving edition of Shen Kuo's Mengxi bitan is printed, more than two centuries after his death; every reading of the huiyuan arc rule depends on it. | Shen Kuo | China | S257 |
| c. 1318 to 1327 CE | Richard of Wallingford writes the Quadripartitum, the first Latin work devoted to spherical trigonometry, during his second Oxford period. The commonly printed "c. 1326" is not supported by anything read: the DSB says only "the early years in his second period at Oxford", after his canons for John Maudith's tables and before the Albion. | John Maudith, Richard of Wallingford | Oxford | S319 |
| 1326 to 1327 CE | Richard of Wallingford writes the Tractatus albionis and, alongside it, the treatise on the rectangulus, a jointed instrument for reading off spherical coordinates without trigonometric tables. | Richard of Wallingford | Oxford | S319 |
| 1327 CE | Richard of Wallingford is elected abbot of St Albans; he later revises the Quadripartitum there using Jabir ibn Aflah's Flores, and begins the astronomical clock, which the DSB says was completed only after his death and which Wikipedia says was destroyed in 1539. | Jabir ibn Aflah, Richard of Wallingford | St Albans | S319, S346 |
| 1342 CE Disputed | Levi ben Gershon's De sinibus, chordis et arcubus, which states the plane law of sines and computes sine tables to high precision using chords, sines, versed sines and cosines but no tangents, is dedicated to Pope Clement VI in Latin translation along with his Tractatus instrumenti astronomie. Unresolved: the Latin dedication of 1342 is documented, and Simonson's study gives 1342, but the DSB says the work itself is "dated 1343" and MacTutor follows. Smith's "c. 1330" is a loose estimate and is not competitive with either. Dispute: Simonson's study and the documented Latin dedication give 1342; the Dictionary of Scientific Biography says the work itself is dated 1343 and MacTutor follows; D. E. Smith's c. 1330 is a loose estimate and is not competitive with either. | Levi ben Gershon | Avignon and Provence | S318, S336, S421 |
| 1342 CE | Peter of Alexandria translates chapters 4 to 11 of Levi ben Gershon's Sefer Tekunah into Latin, the chapters describing the cross-staff that Levi called keli and megalleh amuqqot and that Latin manuscripts turned into the baculus Jacobi, Jacob's staff. | Levi ben Gershon | Provence | S318 |
| 1371 to 1372 CE | Ibn al-Shatir, head muwaqqit at the Umayyad Mosque, erects his marble sundial, about 2 m by 1 m, on the southern side of the main minaret; it carries three dials, for seasonal hours, for the asr prayer and for equatorial hours. Fragments are in the National Museum garden and al-Tantawi's 1876 replica is still in place. | Ibn al-Shatir | Damascus | S182, S200 |
| c. 1400 CE | Madhava is active; his Venvaroha refers to 1400 CE, and the Drgganita of his pupil's line in 1430 is the other anchor. The source itself says plainly that there is no definite evidence pinning down when Madhava flourished, so "c. 1340 to c. 1425" and "c. 1350 to 1425" are both conventions. | Madhava of Sangamagrama | Sangamagrama, Kerala | S140, S142 |
| 1420 to 1424 CE Disputed | Ulugh Beg's observatory is built, with a meridian sextant of about 40 m radius set in a trench on a hillside, reading solar positions to about five arcseconds. Unresolved: the Biographical Encyclopedia of Astronomers dates the observatory to 1420, the DSB to 1424, and the Stanford project page says it followed the 1417 madrasa by four years, which gives 1421. The sextant radius is also given as about 36 m by Wikipedia against about 40 m in the DSB and the BEA. Dispute: The Biographical Encyclopedia of Astronomers dates the observatory to 1420, the Dictionary of Scientific Biography to 1424, and the Stanford project page implies 1421 by putting it four years after the 1417 madrasa. The sextant radius is given as about 36 m by Wikipedia against about 40 m in the DSB and the BEA. sort_year is the midpoint of the range 1420 to 1424. | Ulugh Beg | Samarkand | S218, S224, S230 |
| July 1424 | Al-Kashi completes al-Risala al-muhitiyya (Treatise on the Circumference) in July, giving 2 pi to nine sexagesimal fractional places from a polygon of 3 times 2 to the 28th sides; converted to decimal, the first sixteen decimal places of pi are correct and the seventeenth is not. | Jamshid al-Kashi | Samarkand | S181, S201 |
| 1427 CE | Al-Kashi completes Miftah al-hisab (Key of Arithmetic), which contains his rules for solving triangles, including the statement of the law of cosines using cosines that French school textbooks would later call le theoreme d'Al-Kashi. The French Wikipedia article dates that statement to 1428, one year after the book's completion date given by the sources read here. | Jamshid al-Kashi | Samarkand | S217, S232, S440 |
| 1427 CE Disputed | Al-Kashi writes the Risala al-watar wa'l-jayb (Treatise on the Chord and Sine), containing his iterative solution for sin 1 degree; recomputation to sixty digits gives 60 sin 1 degree as 1;2,49,43,11,14,44,16,26,18, so the ninth sexagesimal digit is 18 and the widely circulated string ending 26,17 is wrong. Dispute: Britannica dates the Risala al-watar wa'l-jayb to c. 1427; MacTutor says it may have remained unfinished at al-Kashi's death in 1429 and was possibly completed by Qadi Zada. | Jamshid al-Kashi | Samarkand | S201, S211 |
| c. 1429 CE | Al-Kashi dies on 22 June; the Biographical Encyclopedia of Astronomers hedges the date with "possibly" while MacTutor states it flatly. | Jamshid al-Kashi | Samarkand | S191, S201 |
| c. 1430 CE | Parameshvara completes the Drgganita, one of the two anchors used to date Madhava. | Madhava of Sangamagrama, Parameshvara | Kerala | S140 |
| 1437 CE Disputed | The Zij-i Sultani star catalog is completed under Ulugh Beg. Unresolved: the Stanford project page and Wikipedia's Ulugh Beg article give 1437, while Wikipedia's own Zij-i Sultani article gives 1438 to 1439. Dispute: The Stanford project page and Wikipedia's Ulugh Beg article give 1437; Wikipedia's own Zij-i Sultani article gives 1438 to 1439. | Ulugh Beg | Samarkand | S218, S224, S230 |
| c. 1460 CE | Regiomontanus begins computing sines to a large radius, aiming at a table with R = 6,000,000 for De triangulis; Roegel judges the effort inspired by Peuerbach's table with R = 600,000. | Georg von Peuerbach, Regiomontanus | Vienna | S307, S310 |
| 8 April 1461 | Peuerbach dies on 8 April having reached the end of Book VI of the Epitome of the Almagest; Regiomontanus completes it. Peuerbach's own Tractatus super propositiones Ptolemaei de sinubus et chordis explains sine computation first by kardagas of 15 degrees, following al-Zarqali, then by Ptolemy's derivation from Almagest I. | Claudius Ptolemy, Georg von Peuerbach, Regiomontanus | Vienna, then Rome | S311, S310 |
| 1462 to 1464 CE | Regiomontanus composes De triangulis omnimodis, whose Book II Proposition 1 is the plane law of sines and whose Book IV Theorem 2 is the spherical law of cosines in versed-sine form. Part was written before he left Rome on 5 July 1463, and at the turn of 1463 to 1464 he writes that "I do not have with me the books which I have written about triangles, but they will soon be brought from Rome." Wikipedia dates completion to 1464. | Regiomontanus | Rome and Venice | S310, S421, S445 |
| 1467 CE | Regiomontanus computes the Tabulae directionum with Martin Bylica's assistance; it includes a table of tangents to 90 degrees at 1 degree intervals with tan 45 degrees = 100,000, though he does not use the word tangent. | Regiomontanus | Hungary | S310 |
| 1468 CE | Regiomontanus computes a decimal table of sines with sin 90 degrees = 10,000,000; Roegel dates this "around 1468". | Regiomontanus | Buda | S310 |
| 1474 CE | Regiomontanus issues the Ephemerides, the first such work to be printed. Wikipedia's "published posthumously in 1498" refers to a later edition; the DSB is explicit that he issued it himself in 1474. | Regiomontanus | Nuremberg | S310 |
| 2 January 1490 | Erhard Ratdolt prints Tabulae directionum et profectionum. Tabella sinus recti on 2 January, edited by Johannes Angelus; the Keio catalog calls its tangent tables the first to appear in print. | Augsburg | S348, S310 |
Timeline
1500 to 1620: Europe names the functions and prints the tables
45 entries.
| When | What happened | Who | Where | Sources |
|---|---|---|---|---|
| 1500 CE | Nilakantha Somayaji completes the Tantrasangraha, dated by the Kali-day chronogram 16,80,553 inside the text. The internal chronogram is preferred over the "completed 1501" given by the Wikipedia article, because a Kali-day figure is a dated statement by the author and the 1501 is an unsourced summary. | Nilakantha Somayaji | Trikkandiyur, Kerala | S127, S141 |
| 10 January 1515 | Petrus Liechtenstein prints the Almagest for the first time, on 10 January, in Gerard of Cremona's Latin; the Greek text would not be printed for another 23 years. | Gerard of Cremona | Venice | S061, S085, S086 |
| c. 1524 to 1525 CE | Miriam Chelebi dies; his commentary on Ulugh Beg's tables is the channel through which al-Kashi's algorithm for sin 1 degree survives. | Jamshid al-Kashi, Miriam Chelebi, Ulugh Beg | Ottoman lands | S184 |
| c. 1530 CE | Sankara Variyar writes the Yuktidipika, his commentary on the Tantrasangraha, which preserves most of the Madhava verses quoted in modern editions; his own dates are given as 1500 to 1560. | Madhava of Sangamagrama, Sankara Variyar | Kerala | S127 |
| 12 August 1533 | Johann Petreius prints De triangulis omnimodis libri quinque, edited by Johannes Schoner, with Nicholas of Cusa's circle-squaring pieces and Regiomontanus's refutation appended; the plane law of sines appears in print in Latin for the first time. The DSB gives the precise date 12 August 1533, which is preferred over the ambiguous OCR of the Seville copy's title page, where M.D.XXXIII and M.D.XXXIIII cannot be told apart, because a machine reading of damaged type is not evidence against a dated bibliographic record. | Johann Petreius, Johannes Schöner, Regiomontanus | Nuremberg | S310, S301, S445 |
| 1538 CE | Hervagius prints the Greek text of the Almagest for the first time. Toomer reports the edition; no digitized copy could be located in e-rara, the Munich Digitisation Centre or the Internet Archive, so the scan is not vouched for here. | Basel | S061 | |
| 1541 CE | Peter Apianus uses sinus rectus secundus for the cosine in his Instrumentum sinuum seu primi mobilis, earlier than any of the examples Smith gives for the idea; the correction is Glen Van Brummelen's. | Germany | S427 | |
| 1541 CE | Regiomontanus's two great sine tables (sexagesimal with R = 6,000,000 and decimal with R = 10,000,000) are printed together with his Compositio tabularum sinuum rectorum and Peuerbach's Tractatus super propositiones Ptolemaei de sinubus et chordis; Peuerbach's tract is reprinted at Basel in 1561. | Georg von Peuerbach, Regiomontanus | Nuremberg | S310, S311 |
| 1542 CE | Johann Lufft prints Rheticus's De lateribus et angulis triangulorum, the trigonometric chapters of the still unpublished De revolutionibus, with a new canon of half-chords at radius 10,000,000 and 1-minute intervals; by printing the complementary angle at the foot of each column it becomes the first table to give the cosine directly. Wikipedia's claim that the treatise was taken from Book II is wrong: the DSB, Roegel and the Silesian Library catalog all place it in Book I. | Georg Joachim Rheticus, Johann Lufft | Wittenberg | S302, S312, S349 |
| 1543 CE | De revolutionibus orbium coelestium appears; Book I chapters 12 to 14 carry the trigonometry, with sines at 10-minute intervals and radius 100,000. | Nuremberg | S327, S347 | |
| c. 1545 CE | Ganesa, commenting on Bhaskara I's tradition of the rational sine formula, restates it; Datta and Singh cite him alongside Munisvara (1646) as later Indian writers who tried to improve the approximation. | Bhaskara I | India | S128 |
| 1551 CE | Rheticus publishes the Canon doctrinae triangulorum: the first table to give all six trigonometric functions, at 10-minute intervals, radius 10,000,000, semiquadrantally arranged over 14 pages, and the first to define the functions as ratios of the sides of a right triangle (base, perpendicular, hypotenuse) rather than as lines in a circle. He would not use the words sine, tangent or secant, calling them "Saracenic barbarisms". Roegel records a Basel reprint of 1565; De Morgan had earlier written 1580. | Georg Joachim Rheticus | Leipzig | S312, S330, S329, S421 |
| c. 1558 CE | J. Peletier's appendix to Gemma Frisius prints a small raised circle for integra (degrees) in a multiplication scheme, which Cajori calls "the first modern appearance that I have found of the degree symbol"; Cajori adds that although the Greek Almagest already marked sixtieths with one and two accents, a line of descent to the degree symbol "has not been established". | Florian Cajori | Paris | S363 |
| 1558 CE | Maurolycus publishes the first printed Latin version of Menelaus's Spherics, from the Arabic tradition; Halley's Oxford edition of 1758 is the other early printed version. | Menelaus of Alexandria | Messina | S068 |
| c. 1560 CE | Rheticus computes the superseded seven-place table of cosecants and cotangents that would later be printed as the second part of the Opus palatinum; Roegel argues from its radius of 10 to the 7th that it is an earlier calculation, and quotes Glaisher's judgment that "there seems no reason why it should have been printed at all". | Georg Joachim Rheticus | not established | S307 |
| 1568 CE | Rheticus writes to Petrus Ramus of "a labor of twelve years, while I always had to support a certain number of arithmeticians for these computations"; Roegel records the cost of the tablemaking to 1568 as 4,400 Gulden, close to fifty times Rheticus's annual Wittenberg salary, funded by Emperor Maximilian II. | Georg Joachim Rheticus | not established | S307, S312 |
| 1530 to 1608 CE Disputed | Jyesthadeva composes the Yuktibhasa in Malayalam, the book that gives proofs (yukti) for the Madhava series. His dates are unresolved: the Kerala school study argues 1500 to 1610 from a Baroda palm-leaf granthavari, from Acyuta Pisarati calling him pravayas (very old) in 1592 and from the Drkkarana chronogram of 1608, while the figure usually printed elsewhere is c. 1500 to c. 1575. An older printed edition assigned the work to one "Brahmadatta" in 1750 on the strength of the verb alekhi; that reading is rejected, since alekhi means simply written or copied. Dispute: The Kerala school study argues 1500 to 1610, from a Baroda palm-leaf granthavari, from Acyuta Pisarati calling Jyesthadeva very old in 1592, and from the Drkkarana chronogram of 1608; the figure usually printed elsewhere is c. 1500 to c. 1575. An older printed edition assigned the work to one Brahmadatta in 1750 on the strength of the verb alekhi, a reading rejected here. sort_year is the midpoint of the range 1530 to 1608. | Acyuta Pisarati, Jyesthadeva, Madhava of Sangamagrama | Alattur and Trikkandiyur, Kerala | S127 |
| 1569 CE | Mercator publishes Nova et aucta orbis terrae descriptio ad usum navigantium emendate accommodata with no mathematical derivation of the latitude spacing, saying only that he has "progressively increased the degrees of latitude towards each pole in proportion to the lengthenings of the parallels". | Gerardus Mercator | Duisburg | S342 |
| c. 1569 CE | Rheticus finds the recurrence cos n phi = cos (n minus 2) phi minus 2 sin phi sin (n minus 1) phi, per Smith; the same passage credits Vieta (1591) with the triple-angle formulas and with connecting sin n phi to sin phi and cos phi. | François Viète, Georg Joachim Rheticus | not established | S421 |
| 4 December 1574 | Rheticus dies on 4 December, leaving his books and manuscripts to Valentin Otho. The DSB date is preferred over the e-rara catalog's 1576 because the DSB gives a day and month and because Otho's documented appeal to the Elector of Saxony on 7 September 1576 already treats him as the holder of the manuscripts. | Georg Joachim Rheticus, Valentin Otho | Kosice | S312, S327 |
| 1571 to 1579 CE | The printing of Viete's Canon mathematicus seu ad triangula runs from 1571 to 1579 at Jean Mettayer's press; only the first two of four planned parts appear. It is the first published canon giving all six functions for every minute of the quadrant, at radius 100,000, arranged as three triangles with a constant side rather than as lines in a circle. | François Viète | Paris | S316, S329, S308 |
| 1580 CE | Paul Wittich serves as an assistant at Uraniborg, and in 1584 describes Tycho's instruments at Kassel and reveals the prosthaphaeretic method there, to Tycho's annoyance. | Paul Wittich, Tycho Brahe | Hven and Kassel | S321 |
| 1583 CE | Sebastian Henricpetri prints Thomas Fincke's Geometriae rotundi libri XIIII, which coins tangens and secans at definitions 21, 22 and 27 (pages 73 to 76) and, per Cajori, uses the abbreviations sin., tan., sec., sin. com., tan. com. and sec. com. Fincke records that "Ioachimus Rheticus hypotenusam trianguli rectanguli vocat". No source read disputes the year or the place. | Florian Cajori, Georg Joachim Rheticus, Sebastian Henricpetri, Thomas Fincke | Basel | S304, S314, S326, S363, S421 |
| 1588 CE | Nicolai Reymers Bar (Ursus) prints the first two prosthaphaeretic rules, further annoying Tycho, who had circulated them in manuscript as Triangulorum planorum et sphaericorum praxis arithmetica. | Tycho Brahe | Strasbourg | S321 |
| c. 1591 CE | Tycho Brahe adopts Fincke's words tangent and secant in a manuscript of 1591, and G. A. Magini follows in 1592, Thomas Blundeville in 1594 and Pitiscus in 1600; Viete never accepted them, calling the tangent Prosinus and the secant Transsinuosa because the words already meant something else in geometry. | Bartholomaeus Pitiscus, François Viète, Thomas Fincke, Tycho Brahe | Uraniborg, Denmark | S423 |
| 1592 CE | Acyuta Pisarati completes the Uparaga-kriyakrama, naming his "aged benign teacher Jyesthadeva"; the chronogram in the text is explained by its own Malayalam commentary. | Acyuta Pisarati, Jyesthadeva | Trikkandiyur, Kerala | S127 |
| 1593 CE | Adriaan van Roomen propounds a forty-fifth degree equation to the mathematicians of the world; Viete sees at once that 45 = 3 times 3 times 5, gives Henri IV one root in pencil within minutes and twenty-two more the next day, and publishes his answer in 1595. He does not reach forty-five roots because the rest need negative sines. | Adriaan van Roomen, François Viète | Paris | S316 |
| 1593 CE | Viete publishes book VIII of the Variorum de rebus mathematicis responsorum, whose chapter 18 carries "the earliest explicit expression for pi by an infinite number of operations", the nested square-root product for 2/pi. | François Viète | Tours | S316 |
| 1595 CE | Pitiscus's Trigonometria: sive de solutione triangulorum tractatus brevis et perspicuus appears as the final section of Abraham Scultetus's Sphaericorum libri tres: the first appearance in print of the word trigonometry, read directly from the section title page in the e-rara scan. The rival dates lose for specific reasons: 1600 is the first standalone edition and the first with tables, and English Wikipedia's 1608 simply omits the 1600 Augsburg edition, a physical copy of which survives at the ETH. | Abraham Scultetus, Bartholomaeus Pitiscus | Heidelberg | S303, S313, S328, S334, S354, S421 |
| 1596 CE | Valentin Otho publishes Rheticus's Opus palatinum de triangulis at Matthaeus Harnisch's press: about 1,500 pages, more than 700 of them tables, giving all six functions at 10-second intervals to radius 10 to the 10th, computed from a sine table at 45-second intervals to radius 10 to the 15th, and naming none of the six with a modern function name. The main table manuscript survives as British Library Harley MS 1720, values in black and differences in red. | Georg Joachim Rheticus, Valentin Otho | Neustadt in the Palatinate | S307, S327, S312, S350 |
| 1599 CE | Edward Wright publishes Certaine Errors in Navigation, giving the mathematical basis of the Mercator projection through a table of meridional parts computed by "perpetual addition of the Secantes", which is numerical integration of the secant. The 1610 second edition expands the table from six pages to 23 at 1-minute intervals. | Edward Wright, Gerardus Mercator | London | S324, S341 |
| c. 1600 CE | Joost Burgi computes his logarithmic progression tables and, per Smith, is the first to set the sinus totus equal to 1; the tables computed on that basis are not extant. | Jost Bürgi | Kassel and Prague | S322, S421 |
| 1600 CE | Pitiscus publishes the expanded Trigonometriae sive de dimensione triangulorum libri quinque at Manger's press, the first standalone edition and the first to carry tables: sines, tangents and secants for every minute of the quadrant to five decimal places, pages 123 to 213. | Bartholomaeus Pitiscus | Augsburg | S306, S313, S328, S334 |
| 1603 CE Disputed | Valentin Otho dies. His dates are unresolved: Roegel gives c. 1545 to 1603 and the e-rara catalog record gives 1550 to 1605, while the DSB names him without dates in the text read. He had signed a contract as the count palatine's mathematician on 24 August 1587, with four students as computers. Dispute: Valentin Otho's dates: Roegel gives c. 1545 to 1603, the e-rara catalog record gives 1550 to 1605, and the Dictionary of Scientific Biography names him without dates in the text read. | Valentin Otho | Heidelberg | S327, S307, S312 |
| 1603 to 1607 CE | Pitiscus recomputes all sines up to 7 degrees to twenty decimal places and all tangents and secants between 83 and 90 degrees to eleven; 86 pages of the Opus palatinum are reprinted and a new edition with a new title page is issued in 1607. The corrected copies can be told by their poorer paper and by a layout slip on page 7 where basis and hypothenusa are swapped. Adriaan van Roomen had found the fault by testing the table against sec a + tan a = tan(a/2 + 45 degrees): Rheticus's cot 10 seconds was wrong in its last nine places, and the error only died away by the 86th page. | Adriaan van Roomen, Bartholomaeus Pitiscus, Georg Joachim Rheticus | Heidelberg | S313, S328, S334, S307 |
| 1607 CE | Matteo Ricci and Xu Guangqi publish the Chinese translation of books 1 to 6 of Euclid's Elements as Jihe yuanben, reprinted with minor revisions in 1611. The 1607 date from the peer-reviewed study is preferred over the Library of Congress catalog record's 1606, because the catalog date describes one copy in one collection while the scholarly account gives 1607 throughout. | Euclid, Matteo Ricci, Xu Guangqi | Beijing | S260, S261 |
| 1608 CE | Ricci and Xu publish Celiang fayi (Explanations on the approaches and principles of measurement), the surveying companion to the Elements translation. | Matteo Ricci | Beijing | S260 |
| 1611 CE | Jakob Christmann's Theoria lunae reports at page 124 that Johannes Werner's unpublished manuscript on triangles already contained prosthaphaeresis explained by three figures, and that copyists who had seen it claimed the invention as their own. Longomontanus's Astronomia Danica of 1622, page 10, credits Paul Wittich instead, and Braunmuhl says every later historian simply followed Longomontanus. | Johannes Werner, Paul Wittich | Heidelberg | S425, S426 |
| 2 July 1613 | Pitiscus dies on 2 July at Heidelberg; the DSB, MacTutor and German Wikipedia all give that date, and English Wikipedia's 24 August 1613 is rejected because it repeats his birth date of 24 August 1561. | Bartholomaeus Pitiscus | Heidelberg | S313, S334, S339, S340 |
| 1613 CE | Pitiscus publishes the Thesaurus mathematicus, carrying Rheticus's sine canon to fifteen decimal places with first, second and third differences at 10-second intervals, plus his own computations; the DSB says his additions run to twenty-two decimal places while Roegel says twenty. | Bartholomaeus Pitiscus, Georg Joachim Rheticus | Frankfurt am Main | S312, S313, S328 |
| 1614 CE | Napier publishes Mirifici logarithmorum canonis descriptio at A. Hart's press, giving logarithms of sines with whole sine 10,000,000; Book II chapter IV sets out the rules of circular parts for right-angled spherical triangles in logarithmic form. | John Napier | Edinburgh | S305, S317 |
| 1614 CE | Ralph Handson's Trigonometry: or The Doctrine of Triangles appears, the English translation of Pitiscus and the first appearance of the word trigonometry in English; later editions follow in 1630 and, per MacTutor, 1642, with a French translation in 1619. | Bartholomaeus Pitiscus, Ralph Handson | London | S333, S313, S427 |
| 1614 CE | Ricci and Li Zhizao publish Tongwen suanzhi, the first work to introduce Western written calculation into China, together with Yuanrong jiaoyi. | Li Zhizao, Matteo Ricci | Beijing | S260 |
| 1614 CE | Thomas Harriot completes his unpublished calculation of meridional parts as, in effect, logarithmic tangents; the logarithmic result for the integral of the secant is first printed only in the 1653 edition of Gunter's works, amended by Samuel Foster and Henry Bond. | Edmund Gunter | England | S309 |
| 1616 CE | Edward Wright's English translation of Napier's Descriptio, A Description of the Admirable Table of Logarithmes, is published posthumously by his son Samuel; Napier approved the translation in substance and in form. | Edward Wright, John Napier | London | S317, S341 |
Timeline
1620 to 1750: From table to function
36 entries.
| When | What happened | Who | Where | Sources |
|---|---|---|---|---|
| 1620 CE | Burgi publishes the Arithmetische und geometrische Progress-Tabulen, computed before he arrived in Prague; the DSB judges that Napier and Burgi must both be credited as independent inventors, with priority in publication going to Napier. | John Napier, Jost Bürgi | Prague | S322, S317 |
| 1620 CE | Edmund Gunter publishes Canon triangulorum at William Jones's press: the first table of common logarithms of trigonometric functions, sines and tangents to seven places for every minute of the quadrant. The words co.sinus and cotangens appear in its introduction, their first printed use, though not in the table headings. | Edmund Gunter, William Jones | London | S309, S315, S331, S421, S427 |
| c. 1626 CE | Albert Girard's Trigonometrie makes the first general use of the abbreviations sin and tan, and per MacTutor cos. Neither source read gives sec among them, although it is commonly listed, and neither gives a full title or printer for the book. | Albert Girard | the Netherlands | S337, S343, S421 |
| 1627 CE | Willebrord Snell states the law of cosines in the form 2ab divided by (c squared minus (a minus b) squared) equals 1 divided by (1 minus cos C), one of the early printed dressings of Euclid II.12 and II.13. | Euclid | the Netherlands | S421 |
| 1631 CE | Richard Norwood's Trigonometrie prints its own key to the abbreviations: "in these examples s stands for sine: t for tangent: sc for sine complement: tc for tangent complement: sec for secant". | London | S421 | |
| 1629 to 1633 CE | Xu Guangqi orchestrates the Ming astronomical reform with Longobardo, Terrenz Schreck, Schall von Bell and Rho, producing the Chongzhen lishu, described as a great encyclopedia of European astronomy. The brief's range of 1631 to 1635 refers to the submissions of the compilation, which could not be individually verified. | Giacomo Rho, Johann Adam Schall von Bell, Johann Terrenz Schreck, Nicholas Longobardo, Xu Guangqi | Beijing | S260 |
| c. 1634 CE | Gilles Personne de Roberval constructs the "companion of the cycloid" (trochoidis comes), which Braunmuhl identifies as the construction of y = sin x in rectangular coordinates; the work is published only in 1693, by his friend the Abbe Gallois, as the Traite des indivisibles. | Gilles Personne de Roberval | Paris | S426 |
| c. 1634 CE | Pierre Herigone appears to be the first to print "sin" for sine in a book, although the contraction is already on a drawing of Gunter's scale of 1624. | Edmund Gunter | France | S421 |
| 1635 CE | John Wells's Sciographia gives the OED's first English citation of cosine: "As the Radius Is to the cosine of the angle given". | England | S427 | |
| c. 1646 CE | Munisvara attempts an improvement of Bhaskara I's rational sine approximation, one of the last steps in that Indian line before European trigonometry arrives. | Bhaskara I | India | S128 |
| 1658 CE | John Newton's Trigonometria Britannica contracts Gunter's co.sinus into cosinus, the form that stuck, and gives the earliest secure English "co-secant". | Edmund Gunter | London | S421, S427 |
| c. 1658 CE | Kamalakara transmits the Rcosine addition theorem and two further sine relations in the late Sanskrit tradition, alongside the older Indian rules. | Kamalakara | India | S128 |
| 1659 CE | Honore Fabri publishes Opusculum geometricum de linea sinuum et cycloide, the earliest known naming of the curve as a "line of sines"; he gives an exact definition of it again in the Synopsis geometrica (Lyon, 1669), page 313. The name is not in Roberval. | Honoré Fabri | Rome | S426 |
| 1660 CE | Isomura Kittoku's Ketsugi-sho appears, expanded in 1684; it is part of the wasan tradition of slicing circles and spheres that Takebe would later turn into a power series. | Takebe Katahiro | Japan | S265 |
| c. 1669 CE | Newton writes De Analysi per aequationes numero terminorum infinitas, containing the arcsine series and its reversion into the sine series; article 45 of Stewart's 1745 English translation gives x = z minus z cubed over 6 plus z to the fifth over 120 minus z to the seventh over 5040. The 1669 writing date and the 1711 first printing are the standard account and were not confirmed from a source read. | Isaac Newton | Cambridge, England | S365 |
| 1670 CE | John Wallis draws the secant curve for the first quadrant in his Tractatus de motu, getting its behavior right; James Gregory had already drawn part of the tangent curve while showing that the integral of tan z is log sec z. | James Gregory, John Wallis | Oxford | S426 |
| 1674 CE | Isaac Barrow's Lectiones opticae et geometricae puts the tangent and secant curves together in a single figure. | Isaac Barrow | London | S426 |
| 1683 CE | Compilation of the Taisei sankei begins, according to the Takebe family biography written by Takebe Kataakira in 1715; Seki Takakazu's work on determinants dates from the same year. | Seki Takakazu, Takebe Kataakira, Takebe Katahiro | Edo | S265, S263 |
| 1700 CE | Mei Wending's Qiandu celiang gives a trigonometric interpretation of celestial coordinate transformation, and rejects the missionaries who wrote Chinese work out of the account. | Mei Wending | Anhui and Beijing | S262 |
| 1706 CE | William Jones introduces the symbol pi for the ratio of circumference to diameter on page 263 of the Synopsis palmariorum matheseos; Cajori notes he had used the same letter twice earlier in the book for other things, and that Oughtred's Clavis mathematicae had used pi over delta for the ratio since 1652. | Florian Cajori, William Jones | London | S423 |
| 1707 CE | De Moivre publishes in Philosophical Transactions 25(309), pages 2368 to 2371, closed-form roots for a family of odd-degree equations using nested square and nth roots of . He never wrote the formula that carries his name. | Abraham de Moivre | London | S368, S369 |
| 1713 CE | Kangxi founds the Suanxue guan (Academy of Mathematics), modeled on the French Academy of Sciences rather than on the Jesuit colleges, staffed by over one hundred Chinese scholars. | Kangxi Emperor | Beijing | S260 |
| 1714 CE | Cotes's Logometria appears in Philosophical Transactions for March 1714 and is reprinted as Part I of the posthumous Harmonia Mensurarum (1722), where page 28 states the result equivalent to i times theta equals log(cos theta plus i sin theta), and Cotes adds that he leaves the matter to others who think it worth the trouble. Whether the same sentence stood in the 1714 printing could not be checked, and the difference matters: 1714 puts Cotes 34 years before Euler, 1722 puts him 26. | Leonhard Euler, Roger Cotes | London and Cambridge | S366, S367 |
| 1715 CE | Brook Taylor's Methodus Incrementorum Directa et Inversa contains what is now the Taylor series, in two versions, at Proposition 11 and at Corollary 2 to Proposition 7; MacTutor notes its importance went unrecognised until Lagrange proclaimed it the basic principle of the differential calculus in 1772, and that the name "Taylor series" appears only in 1785. | Brook Taylor, Joseph-Louis Lagrange | London | S388 |
| 1719 CE | Thomas Fantet de Lagny argues in a memoir for taking the sinus totus as 1, twenty-nine years before Euler's Introductio made it standard; per Smith he was also the first to set out the periodicity of the functions clearly, and, on Smith's evidence alone, the first to use the word "goniometry" in 1724. He also computed pi to 112 correct places, the value the French Cadastre would use in 1794. | Leonhard Euler, Thomas Fantet de Lagny | France | S421, S373 |
| 1722 CE | De Moivre's "De sectione anguli", Philosophical Transactions 32(374), pages 228 to 230, gives the pair of cognate equations relating the versine of an arc to the versine of its n-fold; Pulskamp's note shows that the identity named after de Moivre follows immediately by substitution. | Abraham de Moivre | London | S369 |
| 1722 CE | Takebe Katahiro publishes the Tetsujutsu sankei, giving a power series for the square of half an arc, that is an arcsine-squared series, and a value of pi to 41 correct digits. Cooke places the discovery between Newton's arcsine series of 1676 and Euler's publication of 1737, and it was reached without a sine function to hand. | Isaac Newton, Leonhard Euler, Takebe Katahiro | Edo | S264, S265, S266 |
| 1723 CE | The mathematical part of the Yuzhi luli yuanyuan is published as the Yuzhi Shuli jingyun, an imperial canon of almost 5,000 pages including Jihe yuanben, logarithmic tables, infinite series and iterative solution of higher-degree equations. | Beijing | S260 | |
| 1726 CE | The phrase "Pythagorean theorem" appears in English in the second edition of Edmund Stone's A New Mathematical Dictionary; the name itself is a late label given centuries after both Pythagoras and Euclid. | Euclid | England | S427, S428 |
| c. 1727 to 1728 CE | Euler, aged about 20, corresponds with his teacher Johann Bernoulli about the logarithms of negative numbers, arguing both sides; Bernoulli holds to log(n) = log(-n). This corrects the common framing of a 1740s Euler and Bernoulli exchange: Euler's main 1740s correspondence on the question was with d'Alembert. | Johann Bernoulli, Leonhard Euler | Basel and St Petersburg | S383 |
| c. 1729 CE | Daniel Bernoulli uses "A S." for the arcsine, the first symbolism for an inverse trigonometric function. Cajori's text says 1729 while his own footnote cites the Commentarii volume for 1727, printed 1728, pages 304 to 342; the inconsistency is his, not a disagreement between sources. | Daniel Bernoulli, Florian Cajori | St Petersburg | S363 |
| 1736 CE | Euler introduces "A t" for the arctangent in the Mechanica: "expressio A t nobis denotet arcum circuli, cuius tangens est t existente radio = 1", and in the same work writes the ratio of diameter to periphery as 1 : pi, either adopting Jones's symbol or arriving at it independently. Cajori leaves the question open. | Florian Cajori, Leonhard Euler | St Petersburg | S363, S423 |
| 1742 CE | Maclaurin's Treatise of Fluxions gives the series about zero that carries his name, acknowledging Taylor's prior general result. | Colin Maclaurin | Edinburgh | S387 |
| 1745 CE | The Leibniz to Johann Bernoulli correspondence of 1712 to 1713 on the logarithms of negative numbers is published, which is what turns a private disagreement into a public mathematical problem. | Johann Bernoulli | Europe | S383 |
| c. 1747 CE | Euler writes Sur les logarithmes, proving the same theorem about logarithms of negative numbers using the relation i times theta equals log(cos theta plus i sin theta); it sits unpublished for 115 years, until 1862. Cajori's view, reported by Bal, is that early publication might have ended the controversy decades sooner. | Florian Cajori, Leonhard Euler | Berlin | S383 |
| 1748 CE | Euler's Introductio in analysin infinitorum appears, written in 1745 per the Euler Archive record and dated 1748 on the title page. Book I chapter VIII treats the transcendental quantities arising from the circle; section 126 prints pi to 127 decimal places and fixes the symbol; section 127 sets the radius, "or the whole sine", equal to 1 and states sin squared plus cos squared equals 1 as something "well known from trigonometry". Its readership is what made both the symbol and the unit radius universal. | Leonhard Euler | Lausanne | S361, S362, S423, S429 |
Timeline
1750 to 1850: Analysis, the wave, and the radian
26 entries.
| When | What happened | Who | Where | Sources |
|---|---|---|---|---|
| 1751 CE | Euler's E168, De la controverse entre Mrs. Leibnitz et Bernoulli sur les logarithmes des nombres negatifs et imaginaires, appears in the same volume at pages 139 to 179. The Euler Archive's "written 1747" is preferred over Bal's "written in 1749" because the archive is the bibliographic record of the manuscript and Bal is an expository paper working at second hand. | Leonhard Euler | Berlin | S384, S383 |
| 1751 CE | Euler's E170, Recherches sur les racines imaginaires des equations, appears in the Memoires de l'academie des sciences de Berlin 5, pages 222 to 288. The Euler Archive record gives written 1746, published 1751, which is preferred over the commonly printed "1748" because the archive gives the volume, the pages and both dates while the bare year gives nothing to check. | Leonhard Euler | Berlin | S385 |
| c. 1753 CE | Daniel Bernoulli publishes "Reflexions et eclaircissemens sur les nouvelles vibrations des cordes" in the Berlin memoirs, arguing that the general motion of a vibrating string is a sum of sine modes. Only the existence of the record was seen; the memoir was not read, and the volume and pages are unverified. | Daniel Bernoulli | Berlin | |
| 1759 CE | Abraham Kastner is, per Smith, the first writer to define the trigonometric functions expressly as pure numbers: "Bedeutet also nun x den Winkel in Graden ausgedruckt, so sind die Ausdruckungen sin x; cos x; tang x u.s.w. Zahlen". This is eleven years after Euler set the radius to 1, and it is a separate step. | Leonhard Euler | Germany | S421 |
| c. 1757 to 1762 CE | Vincenzo Riccati publishes the Opusculorum ad res physicas et mathematicas pertinentium, introducing the hyperbolic functions with the notation Sh and Ch and writing what we write as cosh squared minus sinh squared equals 1 as "Ch. squared mu minus Sh. squared mu = r squared". The apparent clash of dates is not one: Cajori's 1757 is the date of the notation's introduction, MacTutor's 1757 to 1762 is the span of the publication. | Florian Cajori, Vincenzo Riccati | Bologna | S363, S386 |
| 1761 CE Disputed | Lambert's Memoire sur quelques proprietes remarquables des quantites transcendentes circulaires et logarithmiques proves that pi is irrational. Unresolved: the standard account gives a presentation in 1761, while the LOCOMAT page says only that the result was proved "in the 1760s" and published in 1768. The continued fraction for tan x, usually part of the story, was not confirmed from anything read. Dispute: The standard account gives a presentation of the irrationality proof in 1761; the LOCOMAT page says only that the result was proved in the 1760s and published in 1768. | Johann Heinrich Lambert | Berlin | S392 |
| 1765 CE | Lambert gives the double-angle formulas in tangent form, sin 2 phi = 2 tan phi over (1 plus tan squared phi) and cos 2 phi = (1 minus tan squared phi) over (1 plus tan squared phi), seventeen years after Euler had given the tangent and cotangent versions in 1748. | Johann Heinrich Lambert, Leonhard Euler | not established | S421 |
| 1768 CE | Lambert writes "sin h(b minus x)" and "cos h b" in the Berlin Histoire volume XXIV, page 327, fixing the h suffix for the hyperbolic functions. Cajori's 1768 is preferred over MacTutor's 1770 because Cajori cites the volume and the page and MacTutor cites nothing. | Florian Cajori, Johann Heinrich Lambert | Berlin | S363, S386 |
| 1770 CE | Georg Simon Klugel introduces the term "trigonometric function", per Miller citing Cajori's 1919 history at page 234. The name is now universal and the man is never mentioned. | Florian Cajori | Germany | S427, S364 |
| 1772 CE | Carl Scherffer prints "arc. tang." in the Institutionum analyticarum pars secunda at Vienna and Lagrange prints "arc. sin" in the Berlin memoirs for 1772, published 1774; Lambert had used "arc. sin." as early as 1758 in the Acta Helvetica. | Johann Heinrich Lambert, Joseph-Louis Lagrange, Karl Scherffer | Vienna and Berlin | S363 |
| 1774 CE | Minggantu's Geyuan milu jiefa is completed by his student Chen Jixin, using infinite series for sine, cosine and pi; it is first printed only in 1839. | Minggantu | Beijing | S270 |
| 1786 CE | Legendre's Second memoire sur les integrations par arcs d'ellipses uses "amplitude" for the angle subtended by an elliptical arc, and Miller is careful to say the mathematical use is for elliptic integrals, not for waves; the general English sense of amplitude, state or quality of being ample, goes back to the 1540s. | Adrien-Marie Legendre | France | S427, S442 |
| 1794 CE | The Cadastre computers derive their sines from the series sin x = x minus x cubed over 3! plus x to the fifth over 5!, using de Lagny's 1719 value of pi to 112 places; the printed fragments give sines to 22 places with five orders of differences, and the manuscript table gives them to 29 decimals in the modern way, with the radius taken as 1. | Thomas Fantet de Lagny | Paris | S373 |
| 1795 CE | Louis Saget the younger writes to Prony on 17 Fructidor III (3 September 1795) asking for a raise, claiming to compute 200 logarithms a day and to be one of the best computers; he earned 2,600 francs against the others' 3,400, and Prony settled on 3,000. The letter is in the Archives Nationales, F14 2146. | Gaspard de Prony | Paris | S373 |
| 1791 to 1801 CE | The Bureau du Cadastre tables are produced under Prony's direction, on Adam Smith's pin-factory division of labor: a first group of five or six mathematicians (only Legendre is named), a second group of calculators who set up the differences, and a third group who did nothing but add and subtract. The sine table was computed first, from 1793, and the main computing ran 1793 to 1796. Roegel's 1791 to 1801 is preferred over MacTutor's "began in 1792" because Roegel works from the payroll records and archive files of the Bureau itself. | Adrien-Marie Legendre, Gaspard de Prony | Paris | S373, S374 |
| 1801 CE | Jose de Mendoza y Rios publishes the tables of logarithmic versines in which, per Cajori, the haversine function first appears; James Andrew published the first English table of haversines in 1805 as "Squares of Natural Semi-Chords". | Florian Cajori, José de Mendoza y Ríos | Madrid | S423, S441 |
| c. 1805 CE | Gauss composes the Theoria Interpolationis Methodo Nova Tractata, containing an algorithm equivalent to the Cooley and Tukey fast Fourier transform; it stays unpublished until 1866, so the FFT predates Fourier's own 1807 presentation. | Carl Friedrich Gauss, James W. Cooley, John W. Tukey, Joseph Fourier | Gottingen | S375 |
| 21 December 1807 | Fourier presents "On the propagation of heat in solid bodies" to the Paris Institute on 21 December; Lagrange and Laplace object to the trigonometric series expansions, with Monge and Lacroix the other referees. | Joseph Fourier, Joseph-Louis Lagrange, Pierre-Simon Laplace | Paris | S372 |
| 1811 CE | Fourier wins the Institute prize; the committee of Lagrange, Laplace, Malus, Hauy and Legendre awards it while recording that "the manner in which the author arrives at these equations is not exempt of difficulties". | Adrien-Marie Legendre, Joseph Fourier, Joseph-Louis Lagrange, Pierre-Simon Laplace | Paris | S372 |
| 1813 CE | John Herschel publishes the notation sin to the minus one and cos to the minus one in the Philosophical Transactions for 1813, warning on page 10 that "this notation cos.^-1 e must not be understood to signify 1/(cos. e), but what is usually written thus, arc (cos. = e)". He later conceded that Burmann had used the same device earlier for direct functions, though not for the inverse trigonometric ones. | John Herschel | London | S363, S423 |
| 1815 CE Disputed | Mary Edwards dies. She had computed for the Nautical Almanac from 1773 under her husband John Edwards's name, revealed after his death in 1784 that she had done most of the work, and was one of 35 human computers of the positions of the Sun, Moon and planets; when Maskelyne died in 1811 his successor John Pond cut her work, and the Board of Longitude ordered him to keep allocating it. Unresolved: Wikipedia, summarizing Croarken (2003), gives September 1815, while the Cambridge Royal Greenwich Observatory catalog gives c. 1750 to 1817. Croarken herself is paywalled and was not read. Dispute: Wikipedia, summarizing Croarken (2003), gives September 1815; the Cambridge Royal Greenwich Observatory catalog gives c. 1750 to 1817. Croarken herself is paywalled and was not read. | Mary Edwards, Nevil Maskelyne | England | S390, S391, S397 |
| 1821 CE Disputed | James Inman's Navigation and Nautical Astronomy appears, the treatise usually credited with coining the word "haversine". Unresolved on the edition: Cajori dates the treatise 1821 but credits only the use of the function, not the word, while the Oxford English Dictionary and Wikipedia put the coinage in the third edition of 1835. The word was verified only in the 1858 edition, where "log. haversines" occurs more than twenty times. Dispute: Cajori dates Inman's treatise to 1821 but credits it only with the use of the function, not the word; the Oxford English Dictionary and Wikipedia put the coinage of haversine in the third edition of 1835. The word was verified only in the 1858 edition. | Florian Cajori, James Inman | London | S423, S435, S441 |
| 1822 CE | Fourier's Theorie analytique de la chaleur is printed by Firmin Didot; article 235, page 258, states that any function represented on an interval by an arbitrarily drawn curve can be expanded in sines, or cosines, or both, or in cosines of odd multiples alone. | Joseph Fourier | Paris | S371 |
| 1825 CE | John Warren's Kala Sankalita, pages 92 to 93 and 330 to 331, already reports Indian knowledge of infinite series, nine years before Whish's paper that is usually given the credit. | Charles M. Whish | Madras | S127 |
| 1834 CE | C. M. Whish publishes "On the Hindu quadrature of the circle" in the Transactions of the Royal Asiatic Society 3, pages 509 to 523, announcing the Kerala series to European scholarship. | Charles M. Whish | London | S127 |
| 1846 CE | Augustus de Morgan publishes "On the first introduction of the words tangent and secant" in the Philosophical Magazine volume 28, pages 382 to 387, the classic dedicated study of the question; it is cited by Tracey but was not obtained. | London | S326 |
Timeline
1850 to 1950: Measurement, notation, and the modern classroom
22 entries.
| When | What happened | Who | Where | Sources |
|---|---|---|---|---|
| 1852 CE | The phrase "unit circle" gets its first OED citation, in the Cambridge and Dublin Mathematical Journal 7/134, in a sentence about the imaginary unit. The context is complex analysis, not the teaching of trigonometry. | Britain | S427 | |
| c. 1858 CE | A. H. Rhind acquires the mathematical papyrus at Thebes; the British Museum record says "around 1858". | Alexander Henry Rhind | Thebes, Egypt | S008 |
| 1863 CE | Helmholtz publishes the German original of On the Sensations of Tone; the third English edition of 1895, translated by Ellis, states the law of beats, that the number of beats per second between two simple tones is the difference of their vibration numbers. | Alexander J. Ellis, Hermann von Helmholtz | Germany | S380 |
| 1865 CE | The British Museum purchases the Rhind papyrus from David Bremner; it is now EA10057 and EA10058, cut into two sections whose combined original length Peet gives as about 543 cm. | Thomas Eric Peet | London | S008, S009 |
| 1866 CE | Gauss's Theoria Interpolationis Methodo Nova Tractata is finally printed, in volume 3 of his collected works, 61 years after it was written and 99 before Cooley and Tukey. | Carl Friedrich Gauss, James W. Cooley, John W. Tukey | Gottingen | S375 |
| 1869 | Thomas Muir uses the word radian in class teaching at St Andrews; writing to Nature in 1910 he says he owns a student note-book of that year in which the word is used, and that he had at first preferred the monosyllable rad. | Thomas Muir | St Andrews, Scotland | S436, S381 |
| July 1871 | James Thomson proposes the name radian, according to a memorandum in his own hand reported by his son in Nature in 1910 and a note in his copy of the Imperial Dictionary; the word reaches print two years later. | James Thomson | Belfast | S437, S382 |
| 5 June 1873 | The word radian first appears in print, on 5 June, in examination questions set by James Thomson at Queen's College, Belfast; his son gave the date in Nature in 1910 from a memorandum in his father's hand, which is the primary evidence for it. This James Thomson was Lord Kelvin's brother (1822 to 1892), as Cajori says in the 1929 Notations; Cajori's own 1919 History of Mathematics calls him Kelvin's father, which cannot be right, since the father of that name died in 1849 and could not have set an 1873 paper. Thomson had used the word since 1871; Thomas Muir had hesitated between rad, radial and radian in 1869 and adopted radian in 1874 after consulting Thomson. | Florian Cajori, James Thomson, Thomas Muir | Belfast | S364, S381, S382, S423, S437 |
| 1879 CE | The second edition of Thomson and Tait's Treatise on Natural Philosophy says at section 41, "For brevity we shall call this angle a radian"; the 1867 first edition, read directly, contains the word zero times against nine occurrences in 1879, so Miller's report of the sentence at page 31 of 1867 is refuted and the 1873 Belfast priority stands. | Cambridge | S434, S427 | |
| 1888 CE | T. M. Blakslee's Academic Trigonometry. Plane and Spherical carries the earliest recorded English use of all three names law of sines, law of cosines and law of tangents together. | not established | S427 | |
| 1890 to 1895 CE | Alice Everett works as a supernumerary computer in the Astrographic Department of the Royal Observatory, measuring plates and reducing star coordinates for the Carte du Ciel; she was proposed and rejected for fellowship of the Royal Astronomical Society, later joined the National Physical Laboratory in 1917 and retired in 1925. | Alice Everett | Greenwich, England | S389 |
| 1893 CE | Charles Proteus Steinmetz reads "Complex Quantities and their Use in Electrical Engineering" at the International Electrical Congress, 21 to 25 August; he proposes to represent an alternating current, "the sine-function of time, by a constant numerical quantity", which is the phasor method. The proceedings volume is dated 1894. | Charles Proteus Steinmetz | Chicago | S377 |
| 1894 CE | Vincent Scheil discovers and catalogs Si.427 on the French expedition to Sippar; a partial edition follows in 1895 and the complete edition only in 2020. | Jean-Vincent Scheil | Sippar, Iraq | S004 |
| 1900 CE | Anton von Braunmuhl publishes volume 1 of the Vorlesungen uber Geschichte der Trigonometrie, with volume 2 in 1903; it is still the standard history, and it is where the case for Johannes Werner over Paul Wittich as the inventor of prosthaphaeresis is made. | Johannes Werner, Paul Wittich | Leipzig | S425, S426 |
| 1908 CE | V. L. Vyatkin finds the remains of Ulugh Beg's observatory, lost since the sixteenth century; the trench of the great meridian sextant is what survives. | Ulugh Beg | Samarkand | S218 |
| 1910 CE | Thomas Muir and James Thomson's son exchange four letters in Nature volume 83 (pages 156, 217, 459 and 460) over who named the radian: Muir writes on 7 April that Alexander J. Ellis agreed in 1874 that "radians" worked as a contraction of "radial angles", and the son replies on 16 June offering copies of his father's June 1873 examination questions and concluding that the two men thought of it independently. The letter bodies are paywalled and were not read. | Alexander J. Ellis, James Thomson, Thomas Muir | London | S364, S381, S436, S437 |
| c. 1917 CE | Radon publishes "Uber die Bestimmung von Funktionen durch ihre Integralwerte langs gewisser Mannigfaltigkeiten", the mathematical basis of tomographic reconstruction; only the title and year were confirmed, from a university exhibition page, and the journal, volume and pages were not. | Johann Radon | Leipzig, Germany | |
| c. 1922 CE | Edgar J. Banks sells Plimpton 322 to George A. Plimpton for about ten dollars, saying it came from Senkereh (Larsa); ISAW's record says "acquired 1922/1923". | Edgar J. Banks, George Arthur Plimpton | New York | S001, S003, S013 |
| 1936 CE | George Arthur Plimpton's collection, tablet 322 included, passes to Columbia University. | George Arthur Plimpton | New York | S001, S012, S013 |
| 1943 CE | Isaac Mendelsohn's catalog of Columbia's cuneiform tablets lists item 322 as "Content: Commercial account. No date.", missing its mathematical character entirely. Robson's point is that this was not incompetence: the tablet looks exactly like a grain account. | Eleanor Robson | New York | S002 |
| 1945 CE | Neugebauer and Sachs publish Plimpton 322 as Text A in Mathematical Cuneiform Texts, the editio princeps; they remark in passing that the angle between l and d diminishes step by step to almost exactly 31 degrees, but never call the tablet trigonometric. | Abraham J. Sachs, Otto Neugebauer | New Haven | S001, S002, S003, S011 |
| c. 1949 CE | Bruins proposes the reciprocal-pair reading of Plimpton 322, restated in 1955 and arrived at independently around 1980 by Buck, Friberg and Schmidt: the entries derive from pairs x and 1/x running from 2;24 and 0;25 down to 1;48 and 0;33 20. | Evert Marie Bruins, Jöran Friberg | not established | S002 |
Timeline
1950 to the present: The machine age, and the tablets read again
27 entries.
| When | What happened | Who | Where | Sources |
|---|---|---|---|---|
| 1959 CE | The CEEB Commission on Mathematics publishes Program for College Preparatory Mathematics, reserving the circular functions for grade 12 and including a substantial chapter on them in the Appendices; the report is what displaces a whole semester of triangle-and-logarithm trigonometry. | New York | S439 | |
| 1961 CE | The School Mathematics Study Group publishes Elementary Functions, Student's Text, Unit 21, which develops sine and cosine as the coordinates of a point moving on the unit circle and derives periodicity from the geometry; "unit circle" occurs at least eighteen times and "circular function" at least twenty-one. | Stanford, California | S438 | |
| 1965 CE | Cooley and Tukey publish "An algorithm for the machine calculation of complex Fourier series" in Mathematics of Computation 19, pages 297 to 301; the paper itself was blocked and not read, so nothing here rests on its contents. | James W. Cooley, John W. Tukey, Joseph Fourier | United States | S375, S376 |
| 1973 CE Disputed | Toomer publishes "The Chord Table of Hipparchus and the Early History of Greek Trigonometry" in Centaurus 18, pages 6 to 28, arguing that the Indian sine radius 3438 descends from Greek chord tables. Unresolved: Toomer's own bibliography and Duke both cite it as 1973, while the Crossref record gives the issue date as March 1974. Cite 1974 if you follow the publisher and expect 1973 throughout the older literature. Toomer later doubted his own hypothesis in the 1984 Almagest notes, because his recomputation had used a corrupted time interval; Duke re-examined the material in 2005 and concluded the link survives on the first eclipse trio. Dispute: Toomer's own bibliography and Duke both cite the paper as 1973; the Crossref record gives the issue date as March 1974. Cite 1974 if you follow the publisher and expect 1973 throughout the older literature. | Hipparchus of Nicaea | not established | S061, S063, S087, S088 |
| 1974 CE | Ahmed, Natarajan and Rao publish "Discrete cosine transform" in IEEE Transactions on Computers C-23(1), pages 90 to 93, the transform behind JPEG and, through the modified version, MP3 and AAC. Citation only: the paper is paywalled and was not read. | United States | S396 | |
| 1976 CE | J. D. North publishes Richard of Wallingford: An Edition of His Writings in three volumes, the edition that would settle the composition date of the Quadripartitum and that was not available for this chronology. | Richard of Wallingford | Oxford | S319 |
| 1978 CE | Rajagopal and Rangachari publish the first of the two papers that put the Kerala series material on the international record, Archive for History of Exact Sciences 18 (1978), pages 89 to 101. | C. T. Rajagopal | not established | S127 |
| 1980 CE | Buck signals Kenneth Voils's reciprocal-pair analysis of Plimpton 322 as forthcoming in Historia Mathematica; Voils never published, so his theory survives only inside Buck's account of it. | not established | S002 | |
| 1986 CE | Rajagopal and Rangachari publish the second paper, Archive for History of Exact Sciences 35 (1986), pages 91 to 99; the source lists the volume in two slightly different forms, 35 and 35(2). | C. T. Rajagopal | not established | S127 |
| c. 1990 CE | French school textbooks start calling the law of cosines le theoreme d'Al-Kashi; before the 1990s the names used were theoreme de Pythagore generalise and loi des cosinus. The honorific is about thirty-five years old, not historical. | Jamshid al-Kashi | France | S440 |
| 1990 CE | "Sharing teaching ideas: The legend of Soh Cah Toa" appears in Mathematics Teacher 83(4), page 286; the article could not be read (HTTP 403 from the publisher) and supports nothing here. | United States | S449 | |
| 1995 CE | D. E. Joyce's web page on Plimpton 322 puts the trigonometric-table reading into wide circulation; Robson names it, along with general histories, as the source of the popular version, and is explicit that Neugebauer and Sachs never made that claim. | Abraham J. Sachs, Eleanor Robson, Otto Neugebauer | Clark University, United States | S002 |
| 2001 CE | Robson publishes "Neither Sherlock Holmes nor Babylon" in Historia Mathematica 28, pages 167 to 206, the paper that reconstructs the Column I heading and sets out the case against reading the tablet as trigonometry. | Eleanor Robson | Oxford | S002 |
| 2002 CE | Robson publishes the shorter "Words and Pictures" in the American Mathematical Monthly 109(2), pages 105 to 120, the text of an invited address given at the Joint Mathematics Meetings on 10 January 2001; it contains the flat statement that "there could be no notion of measurable angle in the Old Babylonian period". | Eleanor Robson | Oxford and New Orleans | S001 |
| 2004 CE | Eric Weisstein creates the Wolfram MathWorld entry for SOHCAHTOA on 16 December, defining the mnemonic and offering alternatives but giving no history at all. | Wolfram MathWorld, published online | S446 | |
| 2005 CE | Dennis Duke publishes "Hipparchus' eclipse trios and early trigonometry" in Centaurus 47(3), pages 163 to 177, concluding that "the numbers 3144 and 3438 are unambiguously linked" and that the link showed up in the data by luck. | Hipparchus of Nicaea | Tallahassee, Florida | S063 |
| 2011 CE | Britton, Proust and Shnider publish the review that establishes the current consensus reading of the Plimpton 322 Column I heading: "The takiltum of the diagonal (from) which 1 is subtracted and (that of) the width comes up." The paper is paywalled and was read only through the quotations in Mansfield and Wildberger and in Mansfield. | Christine Proust, Daniel F. Mansfield, John P. Britton, Norman John Wildberger, Steve Shnider | not established | S030 |
| 2013 CE | A University of Basel expedition led by S. Bickel and E. Paulin-Grothe finds a semicircular limestone tile with a central hole and fan-shaped lines, an ancient Egyptian sundial face. | Valley of the Kings, Egypt | S024 | |
| 14 September 2015 | LIGO detects GW150914 on 14 September; over 0.2 seconds the signal "increases in frequency and amplitude in about 8 cycles from 35 to 150 Hz", and the abstract gives the fuller sweep as 35 to 250 Hz. The discovery paper follows on 11 February 2016. | Hanford, Washington and Livingston, Louisiana | S379 | |
| 2016 CE | Ossendrijver publishes the Jupiter trapezoid result in Science 351, showing that Babylonian astronomers computed displacement as the area under a velocity graph. | Mathieu Ossendrijver | Berlin | S017 |
| 24 August 2017 | Mansfield and Wildberger publish "Plimpton 322 is Babylonian exact sexagesimal trigonometry" online on 24 August, calling the tablet "a powerful, exact ratio-based trigonometric table"; public criticism follows within days, on factual, methodological, logical and motivational grounds, though no peer-reviewed rebuttal was found. | Daniel F. Mansfield, Norman John Wildberger | Sydney | S003, S014 |
| 2017 CE | Rashed and Papadopoulos publish the critical edition of the early Arabic fragment and the al-Mahani and al-Harawi version of Menelaus's Spherics (De Gruyter, Scientia Graeco-Arabica 21), in which the sector figure is Proposition 66. | Menelaus of Alexandria | Berlin | S074, S075 |
| 2021 CE | Mansfield publishes "Plimpton 322: A Study of Rectangles", substantially retreating from the 2017 claim: "Any interpretation of Plimpton 322 as a table of trigonometric functions is rightly dismissed by Robson as anachronistic." His residual disagreement is narrow: scribes measured and understood exactly one angle, the right angle. | Daniel F. Mansfield, Eleanor Robson | Sydney | S004 |
| 2022 CE | The GPS interface specification IS-GPS-200N, dated 1 August 2022, sets out at section 20.3.3.4.3 and Table 20-IV the broadcast navigation equations, which every receiver evaluates using sines and cosines of the eccentric anomaly, the true anomaly and the argument of latitude. | United States | S393 | |
| 2022 CE | Gysembergh, Williams and Zingg publish multispectral readings from Codex Climaci Rescriptus folios 48r and 53v giving equatorial coordinates for Corona Borealis, and attribute them to Hipparchus's star catalog, arguing they are accurate to within 1 degree for about 129 BCE. | Hipparchus of Nicaea | Cambridge, Paris and Washington DC | S065 |
| 2024 CE | Grasshoff and Hoffmann publish an astronomical re-analysis of the palimpsest data rejecting the attribution of the Codex Climaci Rescriptus coordinates to Hipparchus, in direct conflict with the 2022 paper. | Hipparchus of Nicaea | Berlin and Jena | S066 |
| 2025 CE | Gysembergh, Williams and Zingg publish "A note on the new evidence for Hipparchus' star catalogue" in the Journal for the History of Astronomy 56(3), pages 287 to 290, replying to Grasshoff and Hoffmann. The article could not be read, so the dispute is recorded here as live and unresolved. | Hipparchus of Nicaea | Cambridge, Paris and Washington DC | S067 |
Reference: the registers everything else is built on
Twelve appendices, A to L, the last of them the glossary. Nobody reads this part front to back, and it is not written to be read that way. Everything in the story links into it, and every entry links back out to where it is used.
- Look up any person, symbol, term, or source used anywhere in this book.
- Check a worked calculation for yourself, since every one was re-derived in code.
- See exactly where two sources disagree, and whether the disagreement is settled.
- Find where a story belongs in your Trigonometry course.
- Tell the difference between a story that checks out and one that does not.
Appendix A
People register
185 people, sorted by birth year, each with life dates, pronunciation, region, contribution and sources. Life dates in italics are not settled, which is true of 90 of them.
Pronunciation is a respelling you can say out loud, not a phonetic transcription, and it is missing for 1 person because no source consulted states it. An empty cell means nobody knows, not that nobody looked.
| Name | Lived | Pronunciation | Region | Why they are here | Sources |
|---|---|---|---|---|---|
 Amenemhat III | 1849 BCE to 1801 BCE Not known | ah-men-EM-hat | Egypt | The 12th Dynasty king in whose reign the Rhind papyrus says its lost original was written.Appears in Chapter 1 3 times, and in the timeline under c. 3000 to 300 BCE twice. | S005 |
| No freely licensed likeness was found for this book. If you know of one that is free to reproduce, tell Megan (the message form on her site; name Aristarchus of Samos in it) and it will go in. Aristarchus of Samos | 310 BCE to 230 BCE Not known | a-ris-TAR-kuss | Greek world | His On the Sizes and Distances uses inequalities that work as trigonometric bounds, written before any trigonometry existed.Appears in Chapter 2 twice, and in the timeline under 300 BCE to 400 CE twice. | S069, S084 |
| No freely licensed likeness was found for this book. If you know of one that is free to reproduce, tell Megan (the message form on her site; name Archimedes of Syracuse in it) and it will go in. Archimedes of Syracuse | 287 BCE to 212 BCE Not known | ar-kih-MEE-deez | Greek world (Sicily) | His 96-gon bounds on pi supply exactly the half-angle machinery a chord table needs.Appears in Chapter 2 twice, and in the timeline under 300 BCE to 400 CE twice. | S068, S070 |
| No freely licensed likeness was found for this book. If you know of one that is free to reproduce, tell Megan (the message form on her site; name Eratosthenes of Cyrene in it) and it will go in. Eratosthenes of Cyrene | 276 BCE to 194 BCE Not known | eh-ruh-TOS-thuh-neez | Greek world | Measured the Earth's circumference from a shadow angle, a pre-trigonometric use of a central angle.Appears in Chapter 2, and in the timeline under 300 BCE to 400 CE. | S068, S084 |
| No freely licensed likeness was found for this book. If you know of one that is free to reproduce, tell Megan (the message form on her site; name Hipparchus of Nicaea in it) and it will go in. Hipparchus of Nicaea | 190 BCE to 120 BCE Not known | hip-PAR-kuss | Greek world | The earliest figure for whom systematic trigonometric method is documented, and the man credited with the first table of chords, which is lost.Appears in Chapter 2 twice, and in the timeline under 300 BCE to 400 CE 3 times and 1950 to the present 5 times. | S061, S068, S079, S084, S421 |
| No likeness of this person is known. If you know of one that is free to reproduce, tell Megan (the message form on her site; name Menelaus of Alexandria in it) and it will go in. Menelaus of Alexandria | 70 to 130 Disputed | men-uh-LAY-us | Greek world (Alexandria) | Author of the Sphaerica, the first treatise on the spherical triangle; the sector theorem carries his name.Appears in Chapter 2 twice, and in the timeline under 300 BCE to 400 CE, 750 to 1200 twice, 1200 to 1500, 1500 to 1620 and 1950 to the present. | S061, S068, S074, S084 |
 Claudius Ptolemy Not from life | 100 to 175 Disputed | KLAW-dee-us TOL-uh-mee | Greek world (Roman Egypt) | Author of the Almagest, whose Book I chapters 10 and 11 hold the oldest surviving trigonometric table; Almagest I.10 gives the inscribed-quadrilateral lemma and from it the chord sum, chord difference and half-chord formulas.Appears in Chapter 1 3 times, Chapter 2 117 times, Chapter 3 14 times, Chapter 4 4 times, Chapter 5, Chapter 6 11 times, Chapter 7 6 times and Chapter 8 29 times, and in the timeline under 300 BCE to 400 CE 10 times, 750 to 1200 twice and 1200 to 1500 twice. | S061, S421, S425 |
| No freely licensed likeness was found for this book. If you know of one that is free to reproduce, tell Megan (the message form on her site; name Liu Hui in it) and it will go in. Liu Hui | 220 to 280 Not known | lyoh HWAY | China | Commentator on the Nine Chapters (263 CE), author of the Haidao suanjing (Sea Island Mathematical Manual), and the source of the polygon method for pi and of the chong cha "double difference" surveying technique.Appears in Chapter 5 17 times, and in the timeline under 300 BCE to 400 CE twice. | S247, S251, S271 |
 Hypatia Not from life | 370 to 415 Not known | hy-PAY-shuh | Greek world (Alexandria) | Named in the heading of Book III of Theon's Almagest commentary; chapter 2 sets out what that heading does and does not say.Appears in Chapter 2 17 times, and in the timeline under 400 to 750 3 times. | S073, S084 |
| No freely licensed likeness was found for this book. If you know of one that is free to reproduce, tell Megan (the message form on her site; name Zu Chongzhi in it) and it will go in. Zu Chongzhi | 429 to 501 Disputed | DZOO chong-JIRR | China | Bracketed pi between 3.1415926 and 3.1415927 and gave the approximation 355 over 113.Appears in Chapter 5 4 times, and in the timeline under 400 to 750 twice. | S252, S271 |
| No freely licensed likeness was found for this book. If you know of one that is free to reproduce, tell Megan (the message form on her site; name Aryabhata I in it) and it will go in. Aryabhata I | 476 to 550 | ARE-yuh-buh-tuh (Indian track) or ARR-yuh-BHUT-uh (terms track) | India | Wrote the Aryabhatiya (499 CE), the first surviving work to name the half-chord (ardha-jya, shortened to jya) and to give the 24 sine differences, with R = 3438 arcminutes.Appears in Chapter 3 36 times, Chapter 5 and Chapter 8 twice, and in the timeline under 400 to 750 twice. | S122, S123, S136, S421, S444 |
| No freely licensed likeness was found for this book. If you know of one that is free to reproduce, tell Megan (the message form on her site; name Varahamihira in it) and it will go in. Varahamihira | 505 to 587 Disputed | vuh-RAH-huh-MIH-hih-ruh | India | Pancasiddhantika: a 24-entry sine table with R = 120 on the Greek diameter, plus the half-angle relation and complement rules.Appears in Chapter 3 23 times and Chapter 8 6 times, and in the timeline under 400 to 750 3 times. | S124, S128, S138, S421 |
| No freely licensed likeness was found for this book. If you know of one that is free to reproduce, tell Megan (the message form on her site; name Brahmagupta in it) and it will go in. Brahmagupta | 598 to 668 Not known | BRUH-muh-GOOP-tuh | India | Brahmasphutasiddhanta (628) and Khandakhadyaka (665): second-order interpolation for tabulated functions and the "exact area" rule for a cyclic quadrilateral.Appears in Chapter 3 34 times and Chapter 4, and in the timeline under 400 to 750 4 times. | S125, S126, S137 |
| No likeness of this person is known. If you know of one that is free to reproduce, tell Megan (the message form on her site; name Bhaskara I in it) and it will go in. Bhaskara I | 600 to 680 Not known | BAHS-kuh-ruh (the FIRST) | India | Gave the rational approximation for the sine that needs no table at all, in Mahabhaskariya VII.17 and following.Appears in Chapter 3 9 times, and in the timeline under 400 to 750 3 times, 1500 to 1620 twice and 1620 to 1750 twice. | S128, S144 |
| No freely licensed likeness was found for this book. If you know of one that is free to reproduce, tell Megan (the message form on her site; name Yixing in it) and it will go in. Yixing | 673 to 727 Disputed | ee SHING | China | Buddhist monk and astronomer, author of the Dayan li, who produced what historians describe as the first Chinese tangent table and directed the 724 gnomon survey.Appears in Chapter 5 31 times, and in the timeline under 400 to 750 9 times. | S253, S254 |
| No likeness of this person is known. If you know of one that is free to reproduce, tell Megan (the message form on her site; name Gautama Zhuan in it) and it will go in. Gautama Zhuan | 712 to 776 | choo-TAHN JWAHN | China | Son of Gautama Siddha; in 733 he accused Yixing of plagiarising the Navagraha-karana and the court found against him.Appears in Chapter 5 3 times, and in the timeline under 400 to 750 twice. | S253 |
 Al-Khwarizmi Not from life | 780 to 850 Not known | al-khwaa-RIZ-mee | Islamic world (Baghdad) | Compiled the Zij al-Sindhind, the earliest Arabic astronomical handbook whose sine table and gnomon-shadow table are traceable, though only through later Latin and Andalusian recensions.Appears in Chapter 3 3 times and Chapter 4 6 times, and in the timeline under 750 to 1200 3 times. | S181, S193, S206, S230 |
| No likeness of this person is known. If you know of one that is free to reproduce, tell Megan (the message form on her site; name Habash al-Hasib al-Marwazi in it) and it will go in. Habash al-Hasib al-Marwazi | 796 to 894 Disputed | HA-bash al-HAA-sib al-mar-WA-zee (Islamic track); HAB-ash al-HAH-sib (terms track) | Islamic world (Baghdad) | In his Mumtahan zij he defines the shadow of an arc, that is the tangent, tabulates it and applies it, in what Debarnot judges an independent introduction.Appears in Chapter 4, and in the timeline under 750 to 1200 twice. | S181, S187, S219, S421 |
| No freely licensed likeness was found for this book. If you know of one that is free to reproduce, tell Megan (the message form on her site; name Thabit ibn Qurra in it) and it will go in. Thabit ibn Qurra | 826 to 901 Disputed | THAA-bit ib-n KUR-ra | Islamic world (Baghdad) | Wrote On the Sector Figure (al-shakl al-qatta), treating Menelaus's theorem in eighteen cases and reducing the spherical theorem to a projection identity.Appears in Chapter 4 and Chapter 6 twice. | S181, S198, S230 |
| No likeness of this person is known. If you know of one that is free to reproduce, tell Megan (the message form on her site; name Al-Battani in it) and it will go in. Al-Battani | 858 to 929 Disputed | al-bat-TAA-nee (Islamic track); al-buh-TAH-nee (terms track) | Islamic world (Syria) | His Sabi Zij gave a 12-finger gnomon shadow table by degrees, equivalent to 12 cot theta, and holds what Braunmuhl calls the oldest accessible complete appearance of the spherical law of cosines.Appears in Chapter 4 10 times and Chapter 8 6 times, and in the timeline under 750 to 1200 5 times. | S181, S188, S202, S213, S230, S421, S425 |
| No likeness of this person is known. If you know of one that is free to reproduce, tell Megan (the message form on her site; name Abu Mahmud al-Khujandi in it) and it will go in. Abu Mahmud al-Khujandi | 940 to 1000 Disputed | al-khoo-JAN-dee (Islamic track); al-hoo-JAN-dee (terms track) | Islamic world (Central Asia) | Built the al-suds al-Fakhri mural sextant, 60 degrees of arc and about 43 m across, and is the third claimant to the spherical sine law on al-Tusi's testimony alone.Appears in Chapter 4 11 times and Chapter 8 5 times, and in the timeline under 750 to 1200 3 times. | S181, S194, S230, S432 |
| No likeness of this person is known. If you know of one that is free to reproduce, tell Megan (the message form on her site; name Abu al-Wafa al-Buzjani in it) and it will go in. Abu al-Wafa al-Buzjani | 940 to 998 Disputed | a-BOO al-wa-FAA al-booz-JAA-nee (Islamic track); AH-boo al-WAH-fah (terms track) | Islamic world (Baghdad) | Stated the double shadow figure theorem that gives both the rule of four quantities and the tangent rule, and built the first known table of tangents at 15 minute steps.Appears in Chapter 4 twice and Chapter 8, and in the timeline under 750 to 1200 3 times. | S181, S189, S210, S214, S421, S431 |
| No likeness of this person is known. If you know of one that is free to reproduce, tell Megan (the message form on her site; name Al-Sijzi in it) and it will go in. Al-Sijzi | 945 to 1020 Not known | as-SIJ-zee | Islamic world (Iran) | Gave "a systematic mathematical approach to establishing the 12 relations that emerge from the transversal figure in spherical trigonometry"; his collected rules prompted Abu Nasr's Book of Azimuths.Appears in Chapter 4 twice. | S181, S197 |
| No likeness of this person is known. If you know of one that is free to reproduce, tell Megan (the message form on her site; name Ibn Yunus in it) and it will go in. Ibn Yunus | 950 to 1009 Disputed | IB-n YOO-nus | Islamic world (Egypt) | His Hakimi Zij carries a sine table to four sexagesimal places at one-sixth-degree steps, an original derivation of sin 1 degree and a quadratic interpolation rule.Appears in Chapter 4 17 times and Chapter 8 3 times, and in the timeline under 750 to 1200 twice. | S181, S196, S229, S230, S425 |
| No likeness of this person is known. If you know of one that is free to reproduce, tell Megan (the message form on her site; name Abu Nasr Mansur ibn Iraq in it) and it will go in. Abu Nasr Mansur ibn Iraq | 960 to 1036 Disputed | a-BOO NASR man-SOOR ib-n ee-RAAK (Islamic track); AH-boo NAHSS-r man-SOOR (terms track) | Islamic world (Khwarazm) | In his Risala on spherical arcs he states the general spherical sine theorem outright and states the plane sine law explicitly, which makes him the leading claimant to it.Appears in Chapter 1 4 times, Chapter 4 5 times and Chapter 8, and in the timeline under c. 3000 to 300 BCE 5 times, 750 to 1200 twice and 1850 to 1950. | S181, S195, S205, S230, S431 |
 Al-Biruni Not from life | 973 to 1048 Disputed | al-bee-ROO-nee | Islamic world (Khwarazm and Ghazna) | Author of al-Qanun al-Masudi and the Tahdid; measured the Earth's radius from a single mountaintop dip angle, gave several qibla constructions using orthogonal and stereographic projection, and introduced a second-order interpolation rule.Appears in Chapter 4 17 times, and in the timeline under 750 to 1200 8 times. | S181, S183, S185, S190, S204, S230 |
| No likeness of this person is known. If you know of one that is free to reproduce, tell Megan (the message form on her site; name Ibn Muadh al-Jayyani in it) and it will go in. Ibn Muadh al-Jayyani | 989 Not known | IB-n moo-AATH al-jay-YAA-nee (Islamic track); IB-n moo-AATH al-jy-YAH-nee (terms track) | al-Andalus | His Book on the Unknown Arcs of the Sphere, called the first treatise on spherical trigonometry, is an independent Andalusian work that establishes all six right-triangle relations and uses the polar triangle.Appears in Chapter 4 twice and Chapter 8 3 times, and in the timeline under 750 to 1200 twice. | S181, S430 |
| No freely licensed likeness was found for this book. If you know of one that is free to reproduce, tell Megan (the message form on her site; name Shen Kuo in it) and it will go in. Shen Kuo | 1031 to 1095 | shun KWAW | China | His Mengxi bitan gives the huiyuan ("assembling the circle") rule for the arc from the chord and the sagitta.Appears in Chapter 5 18 times, and in the timeline under 750 to 1200 twice and 1200 to 1500 twice. | S257, S258, S259 |
| No likeness of this person is known. If you know of one that is free to reproduce, tell Megan (the message form on her site; name Jabir ibn Aflah in it) and it will go in. Jabir ibn Aflah | 1100 to 1150 Not known | JAH-bir IB-n AF-lah | al-Andalus | The source of "Jabir's theorem" for the right spherical triangle; the Latin edition was printed at Nuremberg in 1534.Appears in Chapter 4, Chapter 6 3 times and Chapter 8 twice, and in the timeline under 1200 to 1500 twice. | S421, S425 |
| No freely licensed likeness was found for this book. If you know of one that is free to reproduce, tell Megan (the message form on her site; name Bhaskara II in it) and it will go in. Bhaskara II | 1114 to 1185 | BAHS-kuh-ruh (the SECOND) | India | Siddhanta Siromani (1150), whose Jyotpatti section gives explicit Rsine addition and subtraction theorems; also the well-known attack on Brahmagupta's quadrilateral rule in the Lilavati.Appears in Chapter 3 27 times, and in the timeline under 750 to 1200 twice. | S126, S128, S139, S143 |
| No freely licensed likeness was found for this book. If you know of one that is free to reproduce, tell Megan (the message form on her site; name Gerard of Cremona in it) and it will go in. Gerard of Cremona | 1114 to 1187 Not known | juh-RARD of kreh-MOH-nuh (Europe track); ger-AR-doh of kre-MOH-nuh (terms track) | Italy and Spain | The most prolific 12th century translator, who put Ptolemy's Almagest into Latin from Arabic and is held probably responsible for putting sinus into circulation in the Latin West.Appears in Chapter 2 twice, Chapter 3 twice, Chapter 4 twice, Chapter 6 7 times and Chapter 8 6 times, and in the timeline under 750 to 1200 5 times and 1500 to 1620 twice. | S310, S332, S422, S425 |
 Nasir al-Din al-Tusi Not from life | 1201 to 1274 Disputed | NAA-sir ad-DEEN at-TOO-see (Islamic track); nah-SEER ad-DEEN at-TOO-see (Europe track); nuh-SEER ad-DEEN TOO-see (terms track) | Islamic world (Iran) | His Kitab al-Shakl al-Qatta of 1260 is the classic self-standing treatment of plane and spherical trigonometry, and the work that presents the subject as a branch of pure mathematics.Appears in Chapter 4 13 times, Chapter 6 7 times and Chapter 8 5 times, and in the timeline under 1200 to 1500 3 times. | S181, S199, S203, S230, S338, S421, S433 |
 Guo Shoujing Not from life | 1231 to 1316 | gwaw SHOH-jing | China | Produced the Shoushi li calendar of 1280, promulgated from 1281, using a 40-chi gnomon, third-order interpolation, and a spherical-coordinate transformation carried out without any trigonometric function.Appears in Chapter 5 9 times, and in the timeline under 1200 to 1500 4 times. | S244, S258, S259 |
| No likeness of this person is known. If you know of one that is free to reproduce, tell Megan (the message form on her site; name Levi ben Gershon in it) and it will go in. Levi ben Gershon | 1288 to 1344 | LEE-vye ben GER-shon; Gersonides is ger-SON-ih-deez; the acronym RaLBaG is RAHL-bahg | France (Provence) | Wrote De sinibus, chordis et arcubus (On Sines, Chords and Arcs), which states and proves the plane law of sines in Latin, computed precise sine tables, and invented the Jacob's staff.Appears in Chapter 6 twice and Chapter 8 twice, and in the timeline under 1200 to 1500 4 times. | S318, S336, S421 |
 Richard of Wallingford Not from life | 1292 to 1336 Disputed | RITCH-erd of WOL-ing-ferd | England | Wrote the Quadripartitum, "the first comprehensive medieval treatise on trigonometry to have been written in Europe, at least outside Spain and Islam", and designed the St Albans astronomical clock.Appears in Chapter 6 6 times, and in the timeline under 1200 to 1500 6 times and 1950 to the present twice. | S319, S346, S351 |
| No likeness of this person is known. If you know of one that is free to reproduce, tell Megan (the message form on her site; name Ibn al-Shatir in it) and it will go in. Ibn al-Shatir | 1304 to 1375 Not known | IB-n ash-SHAA-tir | Islamic world (Damascus) | Chief muwaqqit (mosque timekeeper) at the Umayyad Mosque; built the 2 m by 1 m marble sundial of 1371 or 1372 and compiled prayer-time tables for latitude 34 degrees.Appears in Chapter 4 4 times, and in the timeline under 1200 to 1500 twice. | S182, S200, S216 |
| No likeness of this person is known. If you know of one that is free to reproduce, tell Megan (the message form on her site; name Shams al-Din al-Khalili in it) and it will go in. Shams al-Din al-Khalili | 1320 to 1380 Not known | SHAMS ad-DEEN al-kha-LEE-lee | Islamic world (Damascus) | Muwaqqit and former muezzin who compiled prayer tables of about 2,160 entries, hour-angle tables of about 10,000 entries, universal auxiliary trigonometric tables of over 13,000 entries, and a qibla table by longitude and latitude.Appears in Chapter 4 7 times. | S182, S216 |
| No likeness of this person is known. If you know of one that is free to reproduce, tell Megan (the message form on her site; name Madhava of Sangamagrama in it) and it will go in. Madhava of Sangamagrama | 1340 to 1425 Disputed | MAH-duh-vuh | India (Kerala) | Found infinite series for arctangent, sine and cosine, correction terms that accelerate the pi series, and a 24-entry sine table correct to thirds.Appears in Chapter 1 and Chapter 3 twice, and in the timeline under 1200 to 1500 3 times and 1500 to 1620 twice. | S003, S127, S129, S140, S142 |
| No likeness of this person is known. If you know of one that is free to reproduce, tell Megan (the message form on her site; name Parameshvara in it) and it will go in. Parameshvara | 1360 to 1460 Disputed | puh-ruh-MAYSH-vuh-ruh | India (Kerala) | Prolific commentator (Bhatadipika on the Aryabhatiya, commentaries on the Laghubhaskariya and Laghumanasa) who gave a sine table to seconds; his definition of ardha-jya is the one Datta and Singh quote.Appears in Chapter 3 7 times, and in the timeline under 400 to 750 twice and 1200 to 1500 twice. | S127, S128 |
| No freely licensed likeness was found for this book. If you know of one that is free to reproduce, tell Megan (the message form on her site; name Jamshid al-Kashi in it) and it will go in. Jamshid al-Kashi | 1380 to 1429 Not known | al-KAA-shee (Islamic track); al-KAH-shee (terms track) | Islamic world (Iran and Samarkand) | Computed sin 1 degree by an iterative solution of the trisection cubic, and 2 pi to nine sexagesimal fractional places.Appears in Chapter 4 26 times and Chapter 8 twice, and in the timeline under 1200 to 1500 5 times, 1500 to 1620 twice and 1950 to the present. | S181, S184, S191, S201, S211, S217, S234, S440 |
 Ulugh Beg Not from life | 1394 to 1449 | oo-LOOG BEG | Islamic world (Samarkand) | Patron and working member of the Samarkand observatory; his Zij carries sine and tangent tables at one-arcminute steps out to 45 degrees.Appears in Chapter 4 17 times, and in the timeline under 1200 to 1500 6 times, 1500 to 1620 twice and 1850 to 1950 twice. | S192, S208, S218, S224, S230 |
| No likeness of this person is known. If you know of one that is free to reproduce, tell Megan (the message form on her site; name Damodara in it) and it will go in. Damodara | 1410 to 1520 Not known | DAH-mo-duh-ruh | India (Kerala) | Son of Parameshvara and teacher of both Nilakantha Somayaji and Jyesthadeva; the pivot of the Kerala lineage and easy to overlook.Appears in Chapter 3 3 times. | S127 |
| No freely licensed likeness was found for this book. If you know of one that is free to reproduce, tell Megan (the message form on her site; name Georg von Peuerbach in it) and it will go in. Georg von Peuerbach | 1423 to 1461 | GAY-org fon POY-er-bakh | Austria | Computed a sine table with sinus totus 600,000 at 10-minute intervals and began the Epitome of the Almagest.Appears in Chapter 6 11 times, and in the timeline under 1200 to 1500 5 times and 1500 to 1620 3 times. | S311 |
| No freely licensed likeness was found for this book. If you know of one that is free to reproduce, tell Megan (the message form on her site; name Regiomontanus in it) and it will go in. Regiomontanus | 1436 to 1476 | ray-jee-oh-mon-TAH-nus (Europe track); REE-jee-oh-mon-TAH-nuss (terms track); the German name is yo-HAH-nes MYOO-ler fon KUR-nikhs-bairk | Germany and Italy | Wrote De triangulis omnimodis libri quinque, composed 1462 to 1464 and printed 1533, the first printed systematic treatment of trigonometry as a branch of mathematics independent of astronomy.Appears in Chapter 4 4 times, Chapter 6 34 times and Chapter 8 13 times, and in the timeline under 1200 to 1500 12 times and 1500 to 1620 4 times. | S301, S310, S345, S421, S445 |
| No likeness of this person is known. If you know of one that is free to reproduce, tell Megan (the message form on her site; name Nilakantha Somayaji in it) and it will go in. Nilakantha Somayaji | 1444 to 1544 Disputed | NEE-luh-KUN-tuh so-muh-YAH-jee | India (Kerala) | Wrote the Tantrasangraha (dated 1500 by a Kali-day chronogram (S127), "completed 1501" (S141)) and the Aryabhatiyabhasya, our chief witness for Madhava's series and sine table.Appears in Chapter 3 twice, and in the timeline under 1500 to 1620 twice. | S127, S129, S141 |
| No likeness of this person is known. If you know of one that is free to reproduce, tell Megan (the message form on her site; name Johannes Werner in it) and it will go in. Johannes Werner | 1468 to 1522 | yo-HAHN-uss VAIR-ner | Germany | Braunmuhl's candidate as the true inventor of the prosthaphaeretic method, on Christmann's 1611 testimony, against the traditional attribution to Wittich.Appears in Chapter 4 twice, Chapter 6 5 times and Chapter 8 9 times, and in the timeline under 1500 to 1620 twice and 1850 to 1950 twice. | S425 |
 Willibald Pirckheimer | 1470 to 1530 | VIL-ee-balt PIRK-hy-mer | Germany | Bought the manuscript of De triangulis "at great expense" after Regiomontanus's death, according to Schoner's dedication.Appears in Chapter 6 3 times. | S301 |
 Nicolaus Copernicus | 1473 to 1543 | nik-oh-LAY-us koh-PUR-nih-kus; in Polish, mee-KOH-why koh-PEHR-nik | Poland | De revolutionibus Book I chapters 12 to 14 are a self-contained treatise on plane and spherical triangles with a table of half-chords.Appears in Chapter 6 10 times. | S302, S312, S327, S347 |
| No freely licensed likeness was found for this book. If you know of one that is free to reproduce, tell Megan (the message form on her site; name Johannes Schöner in it) and it will go in. Johannes Schöner | 1477 to 1547 | yo-HAH-nes SHUR-ner | Germany | Edited Regiomontanus's De triangulis for the press and wrote the dedication to the Nuremberg city council.Appears in Chapter 6 4 times, and in the timeline under 1500 to 1620 twice. | S301, S310 |
| No likeness of this person is known. If you know of one that is free to reproduce, tell Megan (the message form on her site; name Georg Hartmann in it) and it will go in. Georg Hartmann | 1489 to 1564 | GAY-org HART-mahn | Germany | Dedicatee of Rheticus's 1542 De lateribus et angulis triangulorum.Appears in Chapter 6 twice. | S302 |
| No likeness of this person is known. If you know of one that is free to reproduce, tell Megan (the message form on her site; name Johann Lufft in it) and it will go in. Johann Lufft | 1495 to 1584 | YO-hahn LOOFT | Germany | Printer of Rheticus's De lateribus et angulis triangulorum in 1542.Appears in Chapter 6 twice, and in the timeline under 1500 to 1620 twice. | S302 |
| No likeness of this person is known. If you know of one that is free to reproduce, tell Megan (the message form on her site; name Jyesthadeva in it) and it will go in. Jyesthadeva | 1500 to 1610 Disputed | JYESH-tuh-DAY-vuh | India (Kerala) | Wrote the Yuktibhasa (Ganita-Yukti-Bhasa) in Malayalam, giving the proofs (yukti) of the Madhava series.Appears in Chapter 3 9 times, and in the timeline under 1500 to 1620 5 times. | S127 |
| No likeness of this person is known. If you know of one that is free to reproduce, tell Megan (the message form on her site; name Sankara Variyar in it) and it will go in. Sankara Variyar | 1500 to 1560 | SHUN-kuh-ruh VAH-ree-yar | India (Kerala) | Wrote the Yuktidipika (on the Tantrasangraha) and the Kriyakramakari (on the Lilavati), the source of most of the Sanskrit verses of Madhava that S127 quotes.Appears in Chapter 3 3 times, and in the timeline under 1500 to 1620 twice. | S127 |
| No freely licensed likeness was found for this book. If you know of one that is free to reproduce, tell Megan (the message form on her site; name Erasmus Reinhold in it) and it will go in. Erasmus Reinhold | 1511 to 1553 | eh-RAZ-mus RINE-holt | Germany | Napier's Descriptio takes its natural sines from Reinhold's tables.Appears in Chapter 6 3 times. | S317 |
 Gerardus Mercator | 1512 to 1594 | jeh-RAR-dus mer-KAY-tor; in Flemish, HEH-rart duh KRAY-mer | Low Countries | Published the 1569 world map Nova et aucta orbis terrae descriptio ad usum navigantium emendate accommodata without giving its mathematical derivation.Appears in Chapter 6 6 times, and in the timeline under 1500 to 1620 4 times. | S342 |
| No freely licensed likeness was found for this book. If you know of one that is free to reproduce, tell Megan (the message form on her site; name Georg Joachim Rheticus in it) and it will go in. Georg Joachim Rheticus | 1514 to 1574 Disputed | GAY-org YO-ah-khim RET-ih-kus; the terms track gives RET-ih-kuss | Austria and Germany | Defined the trigonometric functions as ratios of the sides of a right triangle with no reference to arcs, and was the first to tabulate all six, in the Canon doctrinae triangulorum of 1551.Appears in Chapter 6 61 times and Chapter 8 10 times, and in the timeline under 1500 to 1620 21 times. | S307, S312, S327, S330, S421, S427 |
 Christopher Clavius | 1538 to 1612 | KLAY-vee-us | Germany and Italy | Corresponded with van Roomen about the accuracy of the Opus palatinum tables.Appears in Chapter 6 9 times and Chapter 8. | S328 |
 François Viète | 1540 to 1603 | frahn-SWAH vee-ET | France | His Canon mathematicus (1579) is the first published canon of all six functions at one-minute intervals, and he solved van Roomen's 45th-degree equation trigonometrically.Appears in Chapter 3, Chapter 4 4 times, Chapter 6 16 times, Chapter 7 and Chapter 8 3 times, and in the timeline under 1500 to 1620 9 times. | S308, S316, S329 |
| No likeness of this person is known. If you know of one that is free to reproduce, tell Megan (the message form on her site; name Valentin Otho in it) and it will go in. Valentin Otho | 1545 to 1603 Disputed | VAL-en-teen OH-toh | Germany | Completed and published Rheticus's Opus palatinum de triangulis in 1596, twenty-two years after Rheticus died.Appears in Chapter 6 13 times, and in the timeline under 1500 to 1620 8 times. | S307, S312, S327 |
| No likeness of this person is known. If you know of one that is free to reproduce, tell Megan (the message form on her site; name Paul Wittich in it) and it will go in. Paul Wittich | 1546 to 1586 Not known | POWL VIT-ikh | Silesia and Denmark | Probably largely responsible for the development of the prosthaphaeretic method, on Longomontanus's 1622 testimony, though the terms track records that the claim is contested.Appears in Chapter 6 12 times and Chapter 8 4 times, and in the timeline under 1500 to 1620 4 times and 1850 to 1950 twice. | S321, S425 |
| No likeness of this person is known. If you know of one that is free to reproduce, tell Megan (the message form on her site; name Sebastian Henricpetri in it) and it will go in. Sebastian Henricpetri | 1546 to 1627 | seh-BAS-tee-ahn hen-rik-PET-ree | Switzerland | Printer of Fincke's Geometriae rotundi in 1583.Appears in Chapter 6, and in the timeline under 1500 to 1620 twice. | S304, S326 |
 Tycho Brahe | 1546 to 1601 | TEE-koh BRAH-uh | Denmark and Bohemia | Drew up Triangulorum planorum et sphaericorum praxis arithmetica, rules for solving triangles by prosthaphaeresis, and circulated it in manuscript to his assistants.Appears in Chapter 6 4 times and Chapter 8 5 times, and in the timeline under 1500 to 1620 4 times. | S321 |
| No likeness of this person is known. If you know of one that is free to reproduce, tell Megan (the message form on her site; name Acyuta Pisarati in it) and it will go in. Acyuta Pisarati | 1550 to 1621 Disputed | AH-chyoo-tuh pih-SHAH-ruh-tee | India (Kerala) | Pupil of Jyesthadeva; his Uparaga-kriyakrama of 1592 is the direct testimony that Jyesthadeva was his teacher.Appears in Chapter 3 3 times, and in the timeline under 1500 to 1620 5 times. | S127 |
 John Napier | 1550 to 1617 | JON NAY-peer | Scotland | Invented logarithms of sines for trigonometric computation and gave the rules of circular parts for spherical triangles in the Descriptio (1614), whose natural sines are taken from Reinhold's tables.Appears in Chapter 6 26 times, and in the timeline under 1500 to 1620 5 times and 1620 to 1750 3 times. | S305, S317 |
 Jost Bürgi | 1552 to 1632 | YOHST BUR-ghee | Switzerland and Bohemia | Independent inventor of logarithms, who reached them by improving prosthaphaeresis, and published the Arithmetische und geometrische Progress-Tabulen only in 1620.Appears in Chapter 6 8 times and Chapter 8, and in the timeline under 1500 to 1620 twice and 1620 to 1750 3 times. | S317, S322 |
 Matteo Ricci | 1552 to 1610 | mah-TAY-oh REE-chee; his Chinese name Li Madou is lee MAH-doh | Italy and China | With Xu Guangqi translated Books 1 to 6 of Euclid's Elements as the Jihe yuanben (1607).Appears in Chapter 3 and Chapter 5 11 times, and in the timeline under 1500 to 1620 6 times. | S260, S261 |
| No likeness of this person is known. If you know of one that is free to reproduce, tell Megan (the message form on her site; name Nicholas Longobardo in it) and it will go in. Nicholas Longobardo | 1559 to 1654 | nih-koh-LOH lon-goh-BAR-doh | Italy and China | Jesuit participant in the Chongzhen calendar reform.Appears in Chapter 5, and in the timeline under 1620 to 1750 twice. | S260 |
| No likeness of this person is known. If you know of one that is free to reproduce, tell Megan (the message form on her site; name Adriaan van Roomen in it) and it will go in. Adriaan van Roomen | 1561 to 1615 | AH-dree-ahn van ROH-men; the Latin form Adrianus Romanus is ay-dree-AY-nus roh-MAY-nus | Low Countries and Germany | Posed the 45th-degree equation in 1593 and was later the first to detect the systematic error in the Opus palatinum.Appears in Chapter 6 13 times, and in the timeline under 1500 to 1620 4 times. | S316, S328 |
| No likeness of this person is known. If you know of one that is free to reproduce, tell Megan (the message form on her site; name Bartholomaeus Pitiscus in it) and it will go in. Bartholomaeus Pitiscus | 1561 to 1613 Disputed | bar-toh-loh-MAY-us pih-TISS-kus | Germany | Coined the word trigonometry in print in 1595, in his Trigonometria appended to Scultetus's Sphaericorum libri tres.Appears in Chapter 6 28 times and Chapter 8 5 times, and in the timeline under 1500 to 1620 14 times. | S303, S313, S328, S334, S339, S340, S421, S427 |
| No likeness of this person is known. If you know of one that is free to reproduce, tell Megan (the message form on her site; name Edward Wright in it) and it will go in. Edward Wright | 1561 to 1615 | ED-wurd RITE | England | Supplied the mathematics of the Mercator projection in Certaine Errors in Navigation (1599) by "perpetual addition of the Secantes", in effect integrating the secant numerically.Appears in Chapter 6 14 times, and in the timeline under 1500 to 1620 4 times. | S324, S341 |
 Henry Briggs | 1561 to 1630 | HEN-ree BRIGZ | England | Recalculated logarithms to base 10; his Logarithmorum chilias prima (probably 1617) is bound with Gunter's Canon in the British Museum copy.Appears in Chapter 6 5 times. | S315, S317, S323 |
| No freely licensed likeness was found for this book. If you know of one that is free to reproduce, tell Megan (the message form on her site; name Thomas Fincke in it) and it will go in. Thomas Fincke | 1561 to 1656 | TOH-mas FING-kuh (Europe track); TOM-us FINK (analysis track); THOM-us FINK-uh (terms track) | Denmark | Introduced the terms tangens and secans in Geometriae rotundi libri XIIII (Basel, 1583), at definitions 21, 22 and 27 and on p. 73, and used the abbreviations "sin.", "tan." and "sec." in the same book.Appears in Chapter 4 3 times, Chapter 6 15 times and Chapter 8 8 times, and in the timeline under 1500 to 1620 5 times. | S304, S314, S326, S363, S421, S423 |
| No freely licensed likeness was found for this book. If you know of one that is free to reproduce, tell Megan (the message form on her site; name Xu Guangqi in it) and it will go in. Xu Guangqi | 1562 to 1633 | shoo GWAHNG-chee | China | Co-translator of the Jihe yuanben (1607) and the Celiang fayi (1608), and organizer of the 1629 to 1633 calendar reform that produced the Chongzhen lishu.Appears in Chapter 5 6 times, and in the timeline under 1500 to 1620 twice and 1620 to 1750 twice. | S260, S261 |
| No likeness of this person is known. If you know of one that is free to reproduce, tell Megan (the message form on her site; name Li Zhizao in it) and it will go in. Li Zhizao | 1565 to 1630 | lee JRR-dzow | China | With Ricci produced the Tongwen suanzhi (1614), the first work to bring Western pen-and-paper calculation into China, and the Yuanrong jiaoyi (1614).Appears in Chapter 5 twice, and in the timeline under 1500 to 1620 twice. | S260 |
| No likeness of this person is known. If you know of one that is free to reproduce, tell Megan (the message form on her site; name Abraham Scultetus in it) and it will go in. Abraham Scultetus | 1566 to 1624 | AY-bruh-ham skul-TAY-tus | Germany | Author of Sphaericorum libri tres (1595), the book to which Pitiscus's Trigonometria was appended.Appears in Chapter 6 7 times and Chapter 8, and in the timeline under 1500 to 1620 twice. | S303, S313, S354 |
| No likeness of this person is known. If you know of one that is free to reproduce, tell Megan (the message form on her site; name Johann Terrenz Schreck in it) and it will go in. Johann Terrenz Schreck | 1576 to 1630 | YOH-hahn TEH-rents SHREK | Germany and China | Jesuit mathematician on the Chongzhen reform who died early in the project.Appears in Chapter 3 and Chapter 5 3 times, and in the timeline under 1620 to 1750 twice. | S260 |
| No likeness of this person is known. If you know of one that is free to reproduce, tell Megan (the message form on her site; name Edmund Gunter in it) and it will go in. Edmund Gunter | 1581 to 1626 | ED-mund GUN-ter | England | His Canon triangulorum (London, 1620) is the first table of common logarithms of sines and tangents, and the words co.sinus and cotangens first appear in print in its introduction.Appears in Chapter 3, Chapter 4 twice, Chapter 6 17 times and Chapter 8 10 times, and in the timeline under 1500 to 1620 twice and 1620 to 1750 6 times. | S309, S315, S331, S421, S427 |
| No freely licensed likeness was found for this book. If you know of one that is free to reproduce, tell Megan (the message form on her site; name Johann Adam Schall von Bell in it) and it will go in. Johann Adam Schall von Bell | 1592 to 1666 | YOH-hahn AH-dahm SHAHL fon BEL; his Chinese name Tang Ruowang is tahng RWAW-wahng | Germany and China | One of the Jesuit compilers of the Chongzhen lishu and later head of the Qing Astronomical Bureau.Appears in Chapter 5 twice, and in the timeline under 1620 to 1750. | S260 |
| No likeness of this person is known. If you know of one that is free to reproduce, tell Megan (the message form on her site; name Giacomo Rho in it) and it will go in. Giacomo Rho | 1593 to 1638 | JAH-koh-moh ROH; his Chinese name Luo Yagu is lwaw YAH-goo | Italy and China | Jesuit compiler of the Chongzhen lishu alongside Schall.Appears in Chapter 5, and in the timeline under 1620 to 1750. | S260 |
| No likeness of this person is known. If you know of one that is free to reproduce, tell Megan (the message form on her site; name Albert Girard in it) and it will go in. Albert Girard | 1595 to 1632 | al-BAIR zhee-RAR | France and the Netherlands | First to use the abbreviations sin and tan (and, per MacTutor, cos) in a treatise, in his Trigonometrie of 1626, and gave the formula for the area of a spherical triangle now called Girard's theorem.Appears in Chapter 6 6 times and Chapter 8, and in the timeline under 1620 to 1750 twice. | S337, S343 |
| No freely licensed likeness was found for this book. If you know of one that is free to reproduce, tell Megan (the message form on her site; name Gilles Personne de Roberval in it) and it will go in. Gilles Personne de Roberval | 1602 to 1675 | ZHEEL per-SUN duh ROH-bair-val | France | About 1634 he constructed the "companion of the cycloid", which Braunmuhl identifies as the curve y = sin x.Appears in Chapter 8 10 times, and in the timeline under 1620 to 1750 3 times. | S426 |
| No likeness of this person is known. If you know of one that is free to reproduce, tell Megan (the message form on her site; name Honoré Fabri in it) and it will go in. Honoré Fabri | 1607 to 1688 Not known | on-oh-RAY FAB-ree | France and Italy | Opusculum geometricum de linea sinuum et cycloide (Rome, 1659) is the earliest known use of the phrase "line of sines" for the curve.Appears in Chapter 8 4 times, and in the timeline under 1620 to 1750 twice. | S426 |
 John Wallis | 1616 to 1703 | WOL-iss | England | Drew the secant curve for the first quadrant in Tractatus de motu (1670).Appears in Chapter 8 5 times, and in the timeline under 1620 to 1750 twice. | S426 |
 Isaac Barrow | 1630 to 1677 | BAR-oh | England | Lectiones opticae et geometricae (London, 1674) combines the tangent and secant curves in one figure.Appears in Chapter 8 5 times, and in the timeline under 1620 to 1750 twice. | S426 |
| No freely licensed likeness was found for this book. If you know of one that is free to reproduce, tell Megan (the message form on her site; name Mei Wending in it) and it will go in. Mei Wending | 1633 to 1721 | may wun-DING | China | Leading early-Qing mathematical astronomer, who explained Western geometry through the native gougu tradition and gave a trigonometric reading of celestial coordinate transformation in the Qiandu celiang of 1700.Appears in Chapter 5 3 times, and in the timeline under 1620 to 1750 twice. | S260, S262 |
 James Gregory | 1638 to 1675 | GREG-uh-ree | Scotland | Drew part of the tangent curve in the first quadrant while proving that the integral of the tangent equals the logarithm of the secant.Appears in Chapter 3 and Chapter 8 3 times, and in the timeline under 1620 to 1750 twice. | S426 |
 Seki Takakazu Not from life | 1640 to 1708 Disputed | SEH-kee tah-kah-KAH-zoo | Japan | Founder of the mature wasan school: determinants in 1683, Bernoulli numbers, and Aitken-type acceleration applied to circle problems.Appears in Chapter 5 4 times, and in the timeline under 1620 to 1750 twice. | S263, S265, S266, S271 |
 Isaac Newton | 1642 to 1727 | EYE-zik NEW-tun | England | Derived the power series for arcsine and inverted it to get the sine and cosine series, in De Analysi.Appears in Chapter 3 5 times, Chapter 4 3 times, Chapter 5 4 times, Chapter 6 4 times, Chapter 7 32 times and Chapter 8 8 times, and in the timeline under 1620 to 1750 5 times. | S365 |
| No likeness of this person is known. If you know of one that is free to reproduce, tell Megan (the message form on her site; name Thomas Fantet de Lagny in it) and it will go in. Thomas Fantet de Lagny | 1660 to 1734 | duh lah-NYEE | France | Smith credits him with the first use of the word "goniometry" in 1724 and with the 1719 memoir arguing that sinus totus should be 1.Appears in Chapter 7 and Chapter 8 4 times, and in the timeline under 1620 to 1750 twice and 1750 to 1850 twice. | S421 |
| No likeness of this person is known. If you know of one that is free to reproduce, tell Megan (the message form on her site; name Takebe Kataakira in it) and it will go in. Takebe Kataakira | 1661 to 1716 | tah-KEH-beh kah-tah-AH-kee-rah | Japan | Elder brother of Katahiro, co-compiler of the Taisei sankei, and author of the Takebe family biography of 1715.Appears in Chapter 5 twice, and in the timeline under 1620 to 1750 twice. | S265 |
| No likeness of this person is known. If you know of one that is free to reproduce, tell Megan (the message form on her site; name Takebe Katahiro in it) and it will go in. Takebe Katahiro | 1664 to 1739 | tah-KEH-beh kah-tah-HEE-roh | Japan | Seki's pupil; used Richardson-type extrapolation before 1710 and gave, in the Tetsujutsu sankei (1722), the power series for the square of the arc, that is an arcsine-squared series.Appears in Chapter 5 9 times, and in the timeline under 1620 to 1750 4 times. | S264, S265, S266, S267 |
 Abraham de Moivre | 1667 to 1754 | AY-bruh-ham duh MWAHV-ruh (many English speakers say duh MOY-ver) | France and England | Published in 1707 and 1722 the root-extraction and angle-section results equivalent to the formula that carries his name.Appears in Chapter 7 17 times, and in the timeline under 1620 to 1750 5 times. | S368, S369, S370 |
 Johann Bernoulli | 1667 to 1748 | YO-hahn ber-NOO-lee | Switzerland | Euler's teacher, who held that log(-n) = log(n), a position Euler eventually refuted.Appears in Chapter 7 3 times, and in the timeline under 1620 to 1750 4 times. | S383 |
 William Jones | 1675 to 1749 | JOHNZ | Wales and England | Introduced the symbol pi for the ratio of circumference to diameter in Synopsis palmariorum matheseos (1706), p. 263, and used the symbol v for the coversine in the same year.Appears in Chapter 2 3 times, Chapter 6, Chapter 7 5 times and Chapter 8 9 times, and in the timeline under 1620 to 1750 5 times. | S423 |
 Roger Cotes | 1682 to 1716 | ROJ-er KOHTS | England | Stated, before Euler, that the arc equals the square root of minus one times the logarithm of (cosine plus square root of minus one times sine).Appears in Chapter 7 23 times, and in the timeline under 1620 to 1750 4 times. | S366, S367 |
 Brook Taylor | 1685 to 1731 | BRUUK TAY-lor | England | Methodus Incrementorum (1715) gave the general series expansion that supplies the sine and cosine series as special cases.Appears in Chapter 3 3 times and Chapter 7 13 times, and in the timeline under 1620 to 1750 5 times. | S388 |
 Colin Maclaurin | 1698 to 1746 | KOL-in muh-KLOR-in | Scotland | Treatise of Fluxions (1742) popularized the expansion about zero, crediting Taylor.Appears in Chapter 7 7 times, and in the timeline under 1620 to 1750 twice. | S387 |
 Daniel Bernoulli | 1700 to 1782 | DAN-yel ber-NOO-lee | Switzerland and Russia | Argued that a vibrating string's motion is a superposition of sine modes, and was the first to use a symbol for arcsine, "A S." in 1729 per Cajori.Appears in Chapter 7 6 times and Chapter 8, and in the timeline under 1620 to 1750 twice and 1750 to 1850 twice. | S363 |
 Leonhard Euler | 1707 to 1783 | LAY-on-hart OY-ler | Switzerland, Russia and Prussia | Made sine and cosine functions of a number rather than lines in a circle of some radius, in the Introductio in analysin infinitorum of 1748.Appears in Chapter 4 4 times, Chapter 5 3 times, Chapter 7 90 times and Chapter 8 26 times, and in the timeline under 1620 to 1750 17 times and 1750 to 1850 10 times. | S361, S362, S394, S423, S429 |
 Vincenzo Riccati | 1707 to 1775 | vin-CHEN-tso ree-KAH-tee | Italy | Introduced the hyperbolic functions with the notation Sh and Ch, which Cajori dates to 1757.Appears in Chapter 7 12 times and Chapter 8, and in the timeline under 1750 to 1850 twice. | S363, S386 |
| No likeness of this person is known. If you know of one that is free to reproduce, tell Megan (the message form on her site; name Karl Scherffer in it) and it will go in. Karl Scherffer | 1716 to 1783 | KARL SHERF-fer | Austria | Wrote "arc. tang." in Institutionum analyticarum pars secunda (Vienna, 1772), one of the earliest printed "arc" prefixes.Appears in Chapter 7 6 times and Chapter 8, and in the timeline under 1750 to 1850 twice. | S363 |
 Johann Heinrich Lambert | 1728 to 1777 | YO-hahn HYNE-rikh LAM-bairt | France and Prussia | Proved pi irrational by a continued fraction argument and wrote "sin h" and "cos h" for the hyperbolic functions in the Berlin Histoire for 1768.Appears in Chapter 7 12 times and Chapter 8 3 times, and in the timeline under 1750 to 1850 8 times. | S363, S386, S392 |
| No freely licensed likeness was found for this book. If you know of one that is free to reproduce, tell Megan (the message form on her site; name Nevil Maskelyne in it) and it will go in. Nevil Maskelyne | 1732 to 1811 | NEV-il MASK-uh-lin | England | Astronomer Royal who ran the distributed network of human computers that produced the Nautical Almanac.Appears in Chapter 7 6 times, and in the timeline under 1750 to 1850 twice. | S390, S391 |
 Joseph-Louis Lagrange | 1736 to 1813 | zho-ZEF loo-EE luh-GRAHNZH | Italy, Prussia and France | The referee who objected to Fourier's trigonometric series; he also used the "arc. sin" notation in the Berlin Nouveaux memoires for 1772.Appears in Chapter 7 14 times and Chapter 8, and in the timeline under 1620 to 1750 twice and 1750 to 1850 6 times. | S363, S372 |
 Pierre-Simon Laplace | 1749 to 1827 | pee-AIR see-MOHN luh-PLAHSS | France | Referee on Fourier's 1807 memoir and member of the 1811 prize committee.Appears in Chapter 7 4 times, and in the timeline under 1750 to 1850 4 times. | S372 |
| No likeness of this person is known. If you know of one that is free to reproduce, tell Megan (the message form on her site; name Mary Edwards in it) and it will go in. Mary Edwards | 1750 to 1815 Disputed | MAIR-ee ED-wurdz | England | Computed for the Nautical Almanac for decades, at first paid through her husband's name.Appears in Chapter 7 6 times, and in the timeline under 1750 to 1850 3 times. | S390, S391 |
| No freely licensed likeness was found for this book. If you know of one that is free to reproduce, tell Megan (the message form on her site; name Adrien-Marie Legendre in it) and it will go in. Adrien-Marie Legendre | 1752 to 1833 | ah-dree-EN muh-REE luh-ZHAHN-druh | France | The only "first group" mathematician Prony names explicitly for the Bureau du Cadastre tables.Appears in Chapter 7 twice and Chapter 8, and in the timeline under 1750 to 1850 6 times. | S373 |
 Gaspard de Prony | 1755 to 1839 | gas-PAR duh proh-NEE | France | Directed the Bureau du Cadastre table project, an industrial-scale human computation of sines, tangents and logarithms.Appears in Chapter 7 17 times, and in the timeline under 1750 to 1850 5 times. | S373, S374 |
| No freely licensed likeness was found for this book. If you know of one that is free to reproduce, tell Megan (the message form on her site; name José de Mendoza y Ríos in it) and it will go in. José de Mendoza y Ríos | 1761 to 1816 | ho-SAY day men-DOH-thah ee REE-os | Spain and England | Published tables of logarithmic versines (Madrid, 1801, 1805 and 1809) that contain the haversine function before Inman named it.Appears in Chapter 8 4 times, and in the timeline under 1750 to 1850 twice. | S423, S441 |
 Joseph Fourier | 1768 to 1830 | zho-ZEF FOOR-yay | France | Claimed and used the expansion of an arbitrary function in sines and cosines, published in 1822.Appears in Chapter 7 36 times, and in the timeline under 1750 to 1850 9 times and 1950 to the present twice. | S371, S372 |
| No likeness of this person is known. If you know of one that is free to reproduce, tell Megan (the message form on her site; name James Inman in it) and it will go in. James Inman | 1776 to 1859 | IN-man | England | Navigation and Nautical Astronomy; credited with coining "haversine", though which edition first carries the word matters and is unsettled; chapter 8 walks the evidence.Appears in Chapter 8 14 times, and in the timeline under 1750 to 1850 3 times. | S435, S441 |
 Carl Friedrich Gauss | 1777 to 1855 | KARL FREED-rikh GOWSS | Germany | Wrote down an algorithm equivalent to the fast Fourier transform around 1805, published only in 1866.Appears in Chapter 7 7 times, and in the timeline under 1750 to 1850 twice and 1850 to 1950 twice. | S375 |
 John Herschel | 1792 to 1871 | JON HER-shul; the terms track gives HER-shell | England and Cape Colony | Introduced the inverse-function notation sin to the minus one and tan to the minus one, in Philosophical Transactions for 1813, p. 10.Appears in Chapter 7 7 times and Chapter 8 6 times, and in the timeline under 1750 to 1850 twice. | S363, S423 |
| No freely licensed likeness was found for this book. If you know of one that is free to reproduce, tell Megan (the message form on her site; name Alexander J. Ellis in it) and it will go in. Alexander J. Ellis | 1814 to 1890 | al-ig-ZAN-der EL-iss | England | Correspondent who agreed with Muir in 1874 that "radians" was a good contraction of "radial angles"; separately, the English translator of Helmholtz on beats.Appears in Chapter 7 6 times and Chapter 8 twice, and in the timeline under 1850 to 1950 4 times. | S380, S381, S427 |
 Hermann von Helmholtz | 1821 to 1894 | HER-mahn fon HELM-holts | Germany | Stated the law of beats, which turns a trigonometric identity into an audible phenomenon.Appears in Chapter 7 6 times, and in the timeline under 1850 to 1950 twice. | S380 |
| No freely licensed likeness was found for this book. If you know of one that is free to reproduce, tell Megan (the message form on her site; name James Thomson in it) and it will go in. James Thomson | 1822 to 1892 | JAYMZ TOM-sun; the terms track gives TOM-son | Ireland and Scotland | Put the word "radian" into print, on a Queen's College Belfast examination paper of 5 June 1873.Appears in Chapter 7 23 times and Chapter 8 14 times, and in the timeline under 1850 to 1950 10 times. | S364, S381, S382, S423, S427, S437 |
| No freely licensed likeness was found for this book. If you know of one that is free to reproduce, tell Megan (the message form on her site; name Alexander Henry Rhind in it) and it will go in. Alexander Henry Rhind | 1833 to 1863 | RIND (rhymes with "find") | Scotland and Egypt | Scottish lawyer and antiquarian who acquired the papyrus that carries his name at Thebes about 1858.Appears in Chapter 1 7 times, and in the timeline under c. 3000 to 300 BCE 3 times and 1850 to 1950 3 times. | S008, S009 |
| No freely licensed likeness was found for this book. If you know of one that is free to reproduce, tell Megan (the message form on her site; name Thomas Muir in it) and it will go in. Thomas Muir | 1844 to 1934 | TOM-us MYOOR | Scotland and Cape Colony | Hesitated in 1869 between "rad", "radial" and "radian", adopted "radian" in 1874, and set out the priority argument in his 1910 letters to Nature.Appears in Chapter 7 11 times and Chapter 8 6 times, and in the timeline under 1850 to 1950 7 times. | S364, S381, S423, S427, S436 |
| No freely licensed likeness was found for this book. If you know of one that is free to reproduce, tell Megan (the message form on her site; name George Arthur Plimpton in it) and it will go in. George Arthur Plimpton | 1855 to 1936 | PLIMP-tun | United States | New York publisher and collector who bought the tablet now called Plimpton 322 from Edgar J. Banks and left his collection to Columbia University.Appears in Chapter 1 65 times, Chapter 5 and Chapter 8 twice, and in the timeline under c. 3000 to 300 BCE twice, 1850 to 1950 7 times and 1950 to the present 6 times. | S001, S002, S012 |
| No freely licensed likeness was found for this book. If you know of one that is free to reproduce, tell Megan (the message form on her site; name Jean-Vincent Scheil in it) and it will go in. Jean-Vincent Scheil | 1858 to 1940 | vann-SAHN SHAYL | France | Dominican Assyriologist who found and cataloged the field tablet Si.427 on the 1894 French expedition to Sippar.Appears in Chapter 1, and in the timeline under 1850 to 1950 twice. | S004 |
| No freely licensed likeness was found for this book. If you know of one that is free to reproduce, tell Megan (the message form on her site; name Florian Cajori in it) and it will go in. Florian Cajori | 1859 to 1930 | FLOR-ee-un kuh-JOR-ee | Switzerland and the United States | Author of A History of Mathematical Notations, the standard reference behind most of the first-use claims in Part 2.Appears in Chapter 6 5 times, Chapter 7 40 times and Chapter 8 41 times, and in the timeline under 750 to 1200 4 times, 1500 to 1620 5 times, 1620 to 1750 8 times, 1750 to 1850 12 times and 1850 to 1950 3 times. | S363, S364 |
 Alice Everett | 1865 to 1949 | AL-iss EV-uh-rit | Ireland and England | Supernumerary computer at the Royal Observatory from 1890, measuring astrographic plates and reducing star coordinates.Appears in Chapter 7 4 times, and in the timeline under 1850 to 1950 twice. | S389 |
 Charles Proteus Steinmetz | 1865 to 1923 | CHARLZ PROH-tee-us STINE-mets | Germany and the United States | His 1893 paper replaced sine functions of time with constant complex numbers, the origin of the phasor.Appears in Chapter 7 5 times, and in the timeline under 1850 to 1950 twice. | S377 |
| No freely licensed likeness was found for this book. If you know of one that is free to reproduce, tell Megan (the message form on her site; name Annie Russell Maunder in it) and it will go in. Annie Russell Maunder | 1868 to 1947 | AN-ee RUSS-ul MAWN-der | Ireland and England | Lady computer at Greenwich and later a solar astronomer.Appears in Chapter 7 twice. | S389 |
 François Thureau-Dangin | 1872 to 1944 | frahn-SWAH tuh-ROH dahn-ZHAN | France | Identified ukullû as the Akkadian term for reciprocal slope, run over rise.Appears in Chapter 1 5 times. | S003 |
| No freely licensed likeness was found for this book. If you know of one that is free to reproduce, tell Megan (the message form on her site; name Thomas Eric Peet in it) and it will go in. Thomas Eric Peet | 1882 to 1934 | PEET | England | Produced the standard 1923 critical edition and translation of the Rhind papyrus, the text behind every seked figure quoted here.Appears in Chapter 1 35 times, and in the timeline under c. 3000 to 300 BCE 7 times and 1850 to 1950 twice. | S005 |
 Johann Radon | 1887 to 1956 | YO-hahn RAH-dohn | Austria | His 1917 paper on recovering a function from its integrals over lines is the mathematics behind CT reconstruction.Appears in Chapter 7 3 times, and in the timeline under 1850 to 1950 twice. | |
| No freely licensed likeness was found for this book. If you know of one that is free to reproduce, tell Megan (the message form on her site; name Otto Neugebauer in it) and it will go in. Otto Neugebauer | 1899 to 1990 | NOY-guh-bow-er | Germany, Denmark and the United States | Co-editor of the first full edition of Plimpton 322 (1945) and author of The Exact Sciences in Antiquity; argued that the 360 division of the circle is late Babylonian, not early.Appears in Chapter 1 22 times and Chapter 2 9 times, and in the timeline under 1850 to 1950 twice and 1950 to the present twice. | S002, S006, S031 |
| No freely licensed likeness was found for this book. If you know of one that is free to reproduce, tell Megan (the message form on her site; name Evert Marie Bruins in it) and it will go in. Evert Marie Bruins | 1909 to 1990 | BROWNS | Netherlands | First proposed, in 1949 and 1955, that Plimpton 322 comes from reciprocal pairs x and 1 over x, the reading Robson later revived and reframed.Appears in Chapter 1 3 times, and in the timeline under 1850 to 1950 twice. | S001, S002, S003 |
| No freely licensed likeness was found for this book. If you know of one that is free to reproduce, tell Megan (the message form on her site; name Frank B. Allen in it) and it will go in. Frank B. Allen | 1913 to 2007 | ALL-en | United States | Lead author of the SMSG Elementary Functions (1961), which builds the circular functions on the unit circle.Appears in Chapter 8. | S438 |
| No freely licensed likeness was found for this book. If you know of one that is free to reproduce, tell Megan (the message form on her site; name Abraham J. Sachs in it) and it will go in. Abraham J. Sachs | 1915 to 1983 | SAKS | United States | Co-author with Neugebauer of Mathematical Cuneiform Texts (1945), the first edition of Plimpton 322.Appears in Chapter 1 13 times, and in the timeline under c. 3000 to 300 BCE twice, 1850 to 1950 twice and 1950 to the present twice. | S001, S002, S003, S031 |
| No freely licensed likeness was found for this book. If you know of one that is free to reproduce, tell Megan (the message form on her site; name John W. Tukey in it) and it will go in. John W. Tukey | 1915 to 2000 | JON TOO-kee | United States | Co-author of the 1965 fast Fourier transform paper.Appears in Chapter 7 3 times, and in the timeline under 1750 to 1850 twice, 1850 to 1950 twice and 1950 to the present twice. | S375, S376 |
| No freely licensed likeness was found for this book. If you know of one that is free to reproduce, tell Megan (the message form on her site; name Derrick de Solla Price in it) and it will go in. Derrick de Solla Price | 1922 to 1983 | duh SOL-uh PRICE | United States | Proposed in 1964 the generation scheme that yields 38 rows for Plimpton 322, of which the tablet preserves the first 15.Appears in Chapter 1 5 times. | S003 |
| No freely licensed likeness was found for this book. If you know of one that is free to reproduce, tell Megan (the message form on her site; name James W. Cooley in it) and it will go in. James W. Cooley | 1926 to 2016 | JAYMZ KOO-lee | United States | Co-author of the 1965 fast Fourier transform paper.Appears in Chapter 7 3 times, and in the timeline under 1750 to 1850 twice, 1850 to 1950 twice and 1950 to the present twice. | S375, S376 |
| No freely licensed likeness was found for this book. If you know of one that is free to reproduce, tell Megan (the message form on her site; name David Pingree in it) and it will go in. David Pingree | 1933 to 2005 | PING-ree | United States | Co-editor of the 1989 MUL.APIN edition.Appears in Chapter 1. | S007 |
| No freely licensed likeness was found for this book. If you know of one that is free to reproduce, tell Megan (the message form on her site; name Donald Knuth in it) and it will go in. Donald Knuth | 1938 | kuh-NOOTH | United States | His 1972 Communications of the ACM paper argued from tablet AO 6770 that Old Babylonian scribes used linear interpolation.Appears in Chapter 1 twice. | S003 |
| No freely licensed likeness was found for this book. If you know of one that is free to reproduce, tell Megan (the message form on her site; name John P. Britton in it) and it will go in. John P. Britton | 1939 to 2010 | BRIT-un | United States | Co-author of the 2011 review that fixed the consensus reconstruction of Plimpton 322 now in use.Appears in Chapter 1 8 times, and in the timeline under 1950 to the present twice. | S003, S004, S030 |
| No freely licensed likeness was found for this book. If you know of one that is free to reproduce, tell Megan (the message form on her site; name Jens Høyrup in it) and it will go in. Jens Høyrup | 1943 | YENS HOY-rup | Denmark | Showed that Old Babylonian "algebra" is concrete cut-and-paste geometry, which underpins Robson's reading of the Column I heading of Plimpton 322.Appears in Chapter 1 8 times. | S001, S002 |
 Karlheinz Brandenburg | 1954 | KARL-hynts BRAHN-den-boorg | Germany | One of the MP3 designers; his 1999 AES paper documents the MDCT filterbank used in MP3 and AAC.Appears in Chapter 7 4 times. | S378 |
| No freely licensed likeness was found for this book. If you know of one that is free to reproduce, tell Megan (the message form on her site; name Eleanor Robson in it) and it will go in. Eleanor Robson | 1969 | EL-uh-nor ROB-sun | England | Her 2001 and 2002 papers dismantled the trigonometric and generating-function readings of Plimpton 322 and argued for reciprocal pairs instead.Appears in Chapter 1 57 times, and in the timeline under c. 3000 to 300 BCE 11 times, 1850 to 1950 twice and 1950 to the present 8 times. | S001, S002 |
| No likeness of this person is known. If you know of one that is free to reproduce, tell Megan (the message form on her site; name Abu Ali al-Marrakushi in it) and it will go in. Abu Ali al-Marrakushi | dates unknown Not known | a-BOO a-LEE al-mar-raa-KOO-shee | Islamic world (Cairo) | Compiled the Jami al-mabadi wa-l-ghayat fi ilm al-miqat ("An A to Z of Astronomical Timekeeping"), the compendium that set the pattern for the muwaqqit discipline for centuries.Appears in Chapter 4 twice. | S182 |
| No likeness of this person is known. If you know of one that is free to reproduce, tell Megan (the message form on her site; name Adelard of Bath in it) and it will go in. Adelard of Bath | dates unknown Not known | AD-uh-lard of BATH | England | Translated al-Khwarizmi's astronomical tables into Latin, probably in 1126; tables 58 and 58a in that translation were "very probably the first sine tables to appear in Latin".Appears in Chapter 6 3 times, and in the timeline under 750 to 1200 twice. | S320 |
| No likeness of this person is known. If you know of one that is free to reproduce, tell Megan (the message form on her site; name Ahmose in it) and it will go in. Ahmose | dates unknown Not known | AH-mohs (or AH-mess) | Egypt | The scribe who copied the Rhind Mathematical Papyrus, the source of the seked pyramid-slope problems 56 to 60, in year 33 of the Hyksos king Apophis.Appears in Chapter 1 8 times, and in the timeline under c. 3000 to 300 BCE twice. | S005, S008 |
| No likeness of this person is known. If you know of one that is free to reproduce, tell Megan (the message form on her site; name Al-Ijliyyah in it) and it will go in. Al-Ijliyyah | dates unknown Not known | al-IJ-lee-ya | Islamic world (Aleppo) | The single source, Ibn al-Nadim's Fihrist, records her in one clause as an astrolabe maker, daughter of al-Ijli al-Asturlabi, pupil of Betulus, attached to Sayf al-Dawla.Appears in Chapter 4 3 times, and in the timeline under 750 to 1200 twice. | S186, S215 |
| No likeness of this person is known. If you know of one that is free to reproduce, tell Megan (the message form on her site; name Apophis in it) and it will go in. Apophis | dates unknown Disputed | uh-POH-fis | Egypt | The Hyksos 15th Dynasty king in whose year 33 the Rhind papyrus was written.Appears in Chapter 1 3 times, and in the timeline under c. 3000 to 300 BCE twice. | S005, S008 |
| No freely licensed likeness was found for this book. If you know of one that is free to reproduce, tell Megan (the message form on her site; name C. T. Rajagopal in it) and it will go in. C. T. Rajagopal | dates unknown Not known | rah-juh-GO-pahl | India | With M. S. Rangachari and others he resurrected the Kerala material from the 1940s; the 1978 and 1986 papers in Archive for History of Exact Sciences are the standard citations.Appears in Chapter 3 5 times, and in the timeline under 1950 to the present 4 times. | S127 |
| No likeness of this person is known. If you know of one that is free to reproduce, tell Megan (the message form on her site; name Charles M. Whish in it) and it will go in. Charles M. Whish | dates unknown Not known | WISH | England and India | His 1834 paper first told European scholarship about the Kerala infinite series, and was then ignored for a century.Appears in Chapter 3 5 times, and in the timeline under 1750 to 1850 4 times. | S127 |
| No likeness of this person is known. If you know of one that is free to reproduce, tell Megan (the message form on her site; name Chen Zi in it) and it will go in. Chen Zi | dates unknown Not known | chun DZUH | China | Speaker of the Zhoubi sun-height computation, the earliest surviving Chinese worked shadow-and-similar-triangles problem.Appears in Chapter 5 3 times, and in the timeline under 300 BCE to 400 CE twice. | S241, S245 |
| No freely licensed likeness was found for this book. If you know of one that is free to reproduce, tell Megan (the message form on her site; name Christine Proust in it) and it will go in. Christine Proust | dates unknown | krees-TEEN PROOST | France | Co-author of the 2011 Britton, Proust and Shnider review of Plimpton 322; her generation procedure was extended by Mansfield and Wildberger.Appears in Chapter 1 4 times, and in the timeline under 1950 to the present twice. | S003, S030 |
| No freely licensed likeness was found for this book. If you know of one that is free to reproduce, tell Megan (the message form on her site; name Daniel F. Mansfield in it) and it will go in. Daniel F. Mansfield | dates unknown | MANZ-feeld | Australia | Co-author of the 2017 "Babylonian exact sexagesimal trigonometry" claim for Plimpton 322, and published on the field tablet Si.427 in 2020 and 2021.Appears in Chapter 1 43 times, and in the timeline under c. 3000 to 300 BCE 3 times and 1950 to the present 7 times. | S003, S004, S025 |
| No freely licensed likeness was found for this book. If you know of one that is free to reproduce, tell Megan (the message form on her site; name Edgar J. Banks in it) and it will go in. Edgar J. Banks | dates unknown Not known | BANKS | United States and Iraq | Trained Assyriologist turned antiquities dealer who sold Plimpton 322 to Plimpton for about $10 around 1922, giving "Senkereh" (Larsa) as the findspot.Appears in Chapter 1 4 times, and in the timeline under 1850 to 1950 twice. | S001, S002, S003 |
| No freely licensed likeness was found for this book. If you know of one that is free to reproduce, tell Megan (the message form on her site; name Euclid in it) and it will go in. Euclid | dates unknown Not known | YOO-klid | Greek world (Alexandria) | Elements II.12 and II.13 are the geometric ancestors of the law of cosines.Appears in Chapter 2 11 times, Chapter 5 5 times, Chapter 6 twice and Chapter 8 11 times, and in the timeline under 300 BCE to 400 CE twice, 1500 to 1620 twice and 1620 to 1750 4 times. | S076, S077 |
| No freely licensed likeness was found for this book. If you know of one that is free to reproduce, tell Megan (the message form on her site; name Evelyn Lamb in it) and it will go in. Evelyn Lamb | dates unknown | EV-uh-lin LAM | United States | Mathematician and science writer whose 2017 piece is the most-cited public rebuttal of the Mansfield and Wildberger trigonometry claim for Plimpton 322.Appears in Chapter 1 4 times. | S014 |
| No likeness of this person is known. If you know of one that is free to reproduce, tell Megan (the message form on her site; name Gautama Siddha in it) and it will go in. Gautama Siddha | dates unknown Not known | choo-TAHN shee-DAH | China | Translated the Navagraha-karana as the Jiuzhi li in 718, the vehicle by which Indian sine-based astronomy entered the Chinese court.Appears in Chapter 5 4 times, and in the timeline under 400 to 750 twice. | S253, S255, S269 |
| No likeness of this person is known. If you know of one that is free to reproduce, tell Megan (the message form on her site; name Govindasvami in it) and it will go in. Govindasvami | dates unknown Not known | go-VIN-duh-SWAH-mee | India | Gave a second-order interpolation formula in his commentary on Mahabhaskariya IV.22, a variant of Brahmagupta's and largely overlooked.Appears in Chapter 3 4 times. | S128 |
| No freely licensed likeness was found for this book. If you know of one that is free to reproduce, tell Megan (the message form on her site; name Hammurabi in it) and it will go in. Hammurabi | dates unknown Not known | ham-uh-RAH-bee | Mesopotamia | His conquest of Larsa in 1762 BCE gives the latest possible date for Plimpton 322.Appears in Chapter 1, and in the timeline under c. 3000 to 300 BCE twice. | S001, S002 |
| No likeness of this person is known. If you know of one that is free to reproduce, tell Megan (the message form on her site; name Henry Andrews in it) and it will go in. Henry Andrews | dates unknown Not known | HEN-ree AN-drooz | England | Fellow Nautical Almanac computer; 53 letters from Maskelyne to him survive at Cambridge.Appears in Chapter 7 5 times, and in the timeline under 1850 to 1950 twice. | S390 |
| No freely licensed likeness was found for this book. If you know of one that is free to reproduce, tell Megan (the message form on her site; name Hermann Hunger in it) and it will go in. Hermann Hunger | dates unknown | HAIR-mun HOONG-er | Austria | Co-editor of MUL.APIN, with Pingree in 1989 and with Steele in 2019, the edition behind the 360 UŠ day used here.Appears in Chapter 1 10 times, and in the timeline under c. 3000 to 300 BCE 4 times. | S007 |
| No likeness of this person is known. If you know of one that is free to reproduce, tell Megan (the message form on her site; name Hypsicles of Alexandria in it) and it will go in. Hypsicles of Alexandria | dates unknown Disputed | hip-SIK-leez (Greek track) or HIP-sih-kleez (Babylonian track) | Greek world (Alexandria) | His Anaphorikos is the earliest surviving Greek text that divides the zodiac circle into 360 parts, and per the DSB the first work to do so.Appears in Chapter 1 and Chapter 2, and in the timeline under 300 BCE to 400 CE. | S006, S019, S068, S084 |
| No likeness of this person is known. If you know of one that is free to reproduce, tell Megan (the message form on her site; name Johann Petreius in it) and it will go in. Johann Petreius | 1550 Not known | YO-hahn peh-TRAY-oos | Germany | Printed De triangulis omnimodis in 1533 and De revolutionibus in 1543.Appears in Chapter 6 twice, and in the timeline under 1500 to 1620 twice. | S301 |
| No freely licensed likeness was found for this book. If you know of one that is free to reproduce, tell Megan (the message form on her site; name John M. Steele in it) and it will go in. John M. Steele | dates unknown | STEEL | United States | Co-editor of the 2019 MUL.APIN edition and author on the Babylonian zodiac and the shadow-length schemes.Appears in Chapter 1 13 times, and in the timeline under c. 3000 to 300 BCE 4 times. | S007, S015, S016 |
| No likeness of this person is known. If you know of one that is free to reproduce, tell Megan (the message form on her site; name John Maudith in it) and it will go in. John Maudith | dates unknown Not known | JON MAW-dith | England | Merton astronomer whose tables Richard of Wallingford wrote canons for, in Richard's first mathematical work.Appears in Chapter 6 3 times, and in the timeline under 1200 to 1500 twice. | S319 |
| No freely licensed likeness was found for this book. If you know of one that is free to reproduce, tell Megan (the message form on her site; name Jöran Friberg in it) and it will go in. Jöran Friberg | dates unknown | YUR-an FREE-berg | Sweden | Argued that Plimpton 322 is a classification of normalized right triangles, and first proposed that its missing columns held the ratios beta and delta.Appears in Chapter 1 8 times, and in the timeline under 1850 to 1950 twice. | S002, S003 |
| No likeness of this person is known. If you know of one that is free to reproduce, tell Megan (the message form on her site; name Kamalakara in it) and it will go in. Kamalakara | dates unknown Not known | kuh-muh-LAH-kuh-ruh | India | Siddhanta-tattva-viveka: preserves the Rcosine addition theorem, attributes it to Bhaskara II, and gives two geometric proofs of the addition theorems.Appears in Chapter 3 8 times, and in the timeline under 1620 to 1750 twice. | S128 |
 Kangxi Emperor | dates unknown Not known | kahng-SHEE | China | Founded the Suanxue guan (Academy of Mathematics) in 1713 and patronised the Shuli jingyun.Appears in Chapter 5 and Chapter 6, and in the timeline under 1500 to 1620 and 1620 to 1750. | S260 |
| No likeness of this person is known. If you know of one that is free to reproduce, tell Megan (the message form on her site; name Kushyar ibn Labban in it) and it will go in. Kushyar ibn Labban | dates unknown Not known | KOOSH-yar ib-n LAB-baan | Islamic world (Rayy) | Reworked al-Khujandi's theorem and attached to it the name al-shakl al-mughni, "the figure that dispenses with the quadrilateral"; with al-Khujandi he rejected Abu al-Wafa's tangent theorem.Appears in Chapter 4 twice. | S181 |
| No likeness of this person is known. If you know of one that is free to reproduce, tell Megan (the message form on her site; name Lalla in it) and it will go in. Lalla | dates unknown Not known | LUL-luh | India | Sisyadhivrddhida: an early explicit rule for the moon's instantaneous daily motion using the tabular sine difference, which also criticises the Aryabhata school's version of the rule.Appears in Chapter 3 9 times. | S121, S128 |
| No likeness of this person is known. If you know of one that is free to reproduce, tell Megan (the message form on her site; name Manjula (Munjala) in it) and it will go in. Manjula (Munjala) | dates unknown Not known | MUN-joo-luh | India | Laghumanasa II.2: a deliberately crude sine rule using R = 488 arcminutes (8 degrees 8 minutes) and a five-entry table, built for mental arithmetic.Named in the registers, not in the story or the timeline. | S128 |
| No freely licensed likeness was found for this book. If you know of one that is free to reproduce, tell Megan (the message form on her site; name Mathieu Ossendrijver in it) and it will go in. Mathieu Ossendrijver | dates unknown | ma-TYUR OSS-en-dry-ver | Germany | Identified Late Babylonian trapezoid procedures that compute Jupiter's displacement as the area under a velocity-time graph.Appears in Chapter 1 6 times, and in the timeline under 300 BCE to 400 CE twice and 1950 to the present twice. | S017, S018 |
| No likeness of this person is known. If you know of one that is free to reproduce, tell Megan (the message form on her site; name Minggantu in it) and it will go in. Minggantu | dates unknown Not known | ming-ahn-TOO | China | His Geyuan milu jiefa, completed in 1774 by his student Chen Jixin and first printed in 1839, developed the infinite series for sine, cosine and pi that had been introduced into China.Appears in Chapter 5 5 times, and in the timeline under 1750 to 1850 twice. | S270 |
| No likeness of this person is known. If you know of one that is free to reproduce, tell Megan (the message form on her site; name Miriam Chelebi in it) and it will go in. Miriam Chelebi | 1524 Disputed | MEER-yam che-le-BEE | Ottoman lands | Grandson of Qadi Zada al-Rumi; his commentary on Ulugh Beg's tables is the channel through which al-Kashi's sin 1 degree algorithm survives.Appears in Chapter 4 6 times, and in the timeline under 1500 to 1620 twice. | S184 |
| No likeness of this person is known. If you know of one that is free to reproduce, tell Megan (the message form on her site; name Nan Gongyue in it) and it will go in. Nan Gongyue | dates unknown Not known | nahn-gong YWEH | China | Led the 724 field parties that took the gnomon measurements Yixing analyzed.Appears in Chapter 5 5 times, and in the timeline under 400 to 750 twice. | S253 |
| No freely licensed likeness was found for this book. If you know of one that is free to reproduce, tell Megan (the message form on her site; name Norman John Wildberger in it) and it will go in. Norman John Wildberger | dates unknown | WILD-ber-ger | Australia | Co-author of the 2017 Plimpton 322 paper and proponent of "rational trigonometry", the framework behind that reading.Appears in Chapter 1 23 times, and in the timeline under 1950 to the present 4 times. | S003, S014 |
| No likeness of this person is known. If you know of one that is free to reproduce, tell Megan (the message form on her site; name Pappus of Alexandria in it) and it will go in. Pappus of Alexandria | dates unknown Not known | PAP-uss | Greek world (Alexandria) | Commentator on the Almagest and a key witness for lost Hipparchan works, including the report that Menelaus called the spherical triangle a tripleuron.Appears in Chapter 2 twice. | S061, S068 |
| No likeness of this person is known. If you know of one that is free to reproduce, tell Megan (the message form on her site; name Plato of Tivoli in it) and it will go in. Plato of Tivoli | dates unknown Disputed | PLAY-toh of TIV-oh-lee | Italy and Spain | Translated al-Battani's astronomical work into Latin, the version in which the shadow function appears as umbra extensa and umbra versa.Appears in Chapter 3, Chapter 4, Chapter 6 8 times and Chapter 8 11 times, and in the timeline under 750 to 1200 4 times. | S310, S332, S344, S425, S427 |
| No likeness of this person is known. If you know of one that is free to reproduce, tell Megan (the message form on her site; name Ralph Handson in it) and it will go in. Ralph Handson | dates unknown Not known | RALF HAND-sun | England | Translated Pitiscus's Trigonometria into English in 1614 as Trigonometry: or The Doctrine of Triangles.Appears in Chapter 6 5 times and Chapter 8, and in the timeline under 1500 to 1620 twice. | S313, S315, S333 |
| No likeness of this person is known. If you know of one that is free to reproduce, tell Megan (the message form on her site; name Robert of Chester in it) and it will go in. Robert of Chester | dates unknown Not known | ROB-ert of CHESS-ter | England and Spain | Translated al-Khwarizmi's al-Jabr in 1145 and readjusted al-Khwarizmi's tables to the London meridian in 1150.Appears in Chapter 3 3 times, Chapter 4 twice, Chapter 6 9 times and Chapter 8 10 times, and in the timeline under 750 to 1200 7 times. | S325, S332, S422, S427 |
| No likeness of this person is known. If you know of one that is free to reproduce, tell Megan (the message form on her site; name Rong Fang in it) and it will go in. Rong Fang | dates unknown Not known | rong FAHNG | China | The pupil in the Zhoubi dialogue, and the frame for the text's teaching-of-method passages.Appears in Chapter 5 twice. | S241, S245 |
| No freely licensed likeness was found for this book. If you know of one that is free to reproduce, tell Megan (the message form on her site; name Steve Shnider in it) and it will go in. Steve Shnider | dates unknown | SHNY-der | Israel | Co-author of the 2011 Britton, Proust and Shnider review of Plimpton 322.Appears in Chapter 1 4 times, and in the timeline under 1950 to the present twice. | S030 |
| No likeness of this person is known. If you know of one that is free to reproduce, tell Megan (the message form on her site; name The unnamed paid computers in it) and it will go in. The unnamed paid computers | dates unknown | not established | France and England | The paid workers of Prony's third group at the Bureau du Cadastre, and the Nautical Almanac computers beside them, who did the additions and subtractions that filled the tables.Appears in Chapter 2, Chapter 6 9 times and Chapter 7 17 times, and in the timeline under 1500 to 1620 and 1750 to 1850 3 times. | S373, S374, S390, S391 |
| No likeness of this person is known. If you know of one that is free to reproduce, tell Megan (the message form on her site; name Theodosius of Bithynia in it) and it will go in. Theodosius of Bithynia | dates unknown Disputed | thee-oh-DOH-see-us | Greek world | Wrote a Sphaerica in three books on the geometry of the sphere, in which Heath finds no trigonometry.Appears in Chapter 2 5 times. | S068, S079 |
| No likeness of this person is known. If you know of one that is free to reproduce, tell Megan (the message form on her site; name Theon of Alexandria in it) and it will go in. Theon of Alexandria | dates unknown Not known | THEE-on | Greek world (Alexandria) | His Almagest commentary preserves the only testimony that Hipparchus wrote twelve books on chords.Appears in Chapter 1, Chapter 2 5 times and Chapter 8, and in the timeline under 400 to 750. | S061, S068, S073 |
| No likeness of this person is known. If you know of one that is free to reproduce, tell Megan (the message form on her site; name Vatesvara in it) and it will go in. Vatesvara | dates unknown Not known | vuh-TAYSH-vuh-ruh | India | Vatesvara-siddhanta II.1.65 to 66 gives another second-order interpolation formula.Appears in Chapter 3 3 times, and in the timeline under 750 to 1200 twice. | S128 |
| No likeness of this person is known. If you know of one that is free to reproduce, tell Megan (the message form on her site; name Wang Xun in it) and it will go in. Wang Xun | dates unknown Not known | wahng SHOON | China | Director of the Yuan calendar bureau, who did the calculations for the Shoushi li while Guo Shoujing did the observations.Appears in Chapter 5 twice, and in the timeline under 1200 to 1500 twice. | S259 |
| No likeness of this person is known. If you know of one that is free to reproduce, tell Megan (the message form on her site; name Zu Geng in it) and it will go in. Zu Geng | dates unknown Not known | dzoo GUNG | China | Son and collaborator of Zu Chongzhi.Appears in Chapter 5 twice. | S252 |
Appendix B
Symbols, words, and eponyms
94 terms, with where each word came from and what it literally meant before it meant what it means now. 61 of the 94 carry a first-use year; the rest do not, because no source consulted establishes one.
Read the literal meanings in one sitting and a pattern appears. A surprising number of these words are about shadows, bows, folds and bays: physical objects a person could point at. The abstractions came later, and mostly kept the old names.
| Term | Kind | First use | Language origin | What it literally meant | Sources |
|---|---|---|---|---|---|
| al-shakl al-mughni | phrase | 1000 Attached to al-Khujandi's theorem by Kushyar ibn Labban, about 1000, per al-Biruni | Arabic al-shakl al-mughni | the figure that dispenses with the quadrilateral | S181 |
| al-shakl al-zilli | phrase | Arabic al-shakl al-zilli | the shadow figure | S181 | |
| algebra | word | Arabic al-jabr to Latin algebra to the European languages | the restoration, or the setting of broken parts, from the title of al-Khwarizmi's book. This chain was not independently verified in any source read for this book | S181 | |
| algorithm | eponym | 1690 al-Khwarizmi | Arabic al-Khwarizmi, a byname meaning the man from Khwarazm, to Medieval Latin algorismus to Old French algorisme to French algorithme to English algorithm | a personal byname, then the Arabic system of computation, then any computational method | S227 |
| amplitude | word | 1540 | Latin amplus to Latin amplitudo to English amplitude | wide extent, width | S427, S442 |
| arc prefix (arcsin, arc. tang.) | symbol | 1736 Leonhard Euler, who writes A t for arcus cuius tangens in 1736 and A sin in 1737; Lambert writes arc. sin. in 1758, Scherffer arc. tang. in 1772, Lagrange arc. sin in 1774 | Latin arcus, a bow or arch, to the modern function prefix arc | the arc whose sine is | S363, S423 |
| ardha-jya (jya, jyardha, krama-jya) | word | 499 Aryabhata, Aryabhatiya, 499 CE, uses ardha-jya; Bhaskara II remarks in 1150 that ardha-jya is here called jya | Sanskrit jya to Sanskrit ardha-jya, shortened back to jya | jya is bow-string, hence chord; ardha-jya is half-chord; krama-jya is direct sine | S128, S136 |
| asymptote | word | 1650 | Ancient Greek asymptotos to English asymptote | not falling together, from a- not, syn with, and piptein to fall | S427, S442 |
| beru (written DANNA) | word | -750 | Akkadian beru, written with the Sumerian sign DANNA | a double hour of time, one twelfth of a day, and also a distance unit | S007 |
| bi (as in Zhoubi) | word | Chinese bi | thigh bone, glossed in the tradition as gu and referring to the eight chi gnomon | S245 | |
| chong cha | phrase | 263 Liu Hui, in his appendix to chapter 9 of the Jiuzhang suanshu, 263 CE | Chinese chong cha | double difference | S247, S248, S250 |
| chord | word | Ancient Greek chorde to Latin chorda to medieval and early modern Latin geometric usage to English chord | guts, then a string made from guts, then a string of a lyre. LSJ records no geometrical sense at all | S089, S061, S062 | |
| cosecant | word | 1658 John Newton, Trigonometria Britannica, 1658, co-secant, is the earliest verified English use. The old credit to Rheticus's Opus palatinum of 1596 is now settled negative: Van Brummelen could not find the term there, and Rheticus names no function at all, using only hypotenuse, base and perpendicular of three species | Latin complementi secans to Latin cosecans to English cosecant | secant of the complement | S421, S427, S350, S181, S228, S481 |
| cosine | word | 1620 Edmund Gunter, Canon triangulorum, London 1620, writes co.sinus in the introduction; John Newton has cosinus in 1658; John Wells has English cosine in 1635 | Sanskrit koti-jya by analogy, and Latin complementi sinus to co.sinus to cosinus to English cosine | koti is the curved end of a bow, hence the end of an arc, hence the complement to 90 degrees; complementi sinus is sine of the complement | S128, S130, S331, S335, S315, S421, S427 |
| cotangent | word | 1620 Edmund Gunter, Canon triangulorum, London 1620, in the introduction and not in the table headings; English 1714 in Edward Wells, The Young Gentleman's Trigonometry | Latin complementi tangens to Latin cotangens to English cotangent | tangent of the complementary arc | S207, S228, S331, S335, S315, S421, S427 |
| coversine | word | 1706 William Jones uses the symbol in 1706; the word covers A appears in American texts from Ludlow 1891 and Wentworth 1903 | Latin co- from complementi plus versus plus sinus to English coversine | versed sine of the complement | S423, S427 |
| cun qian li | phrase | Chinese cun qian li | one cun of shadow difference per thousand li of north-south displacement | S241, S244, S254 | |
| de Moivre's formula | eponym | Abraham de Moivre | the formula of de Moivre | S395, S370 | |
| degree | word | -175 Hypsicles, Anaphorikos, c. 175 BCE: the circle of the zodiac having been divided into 360 equal arcs, let each of the arcs be called a spatial degree. English from Chaucer, c. 1386 | Babylonian US to Ancient Greek moira to Arabic daraja to Latin gradus to Old French degre to English degree | moira is a part or portion; daraja is a ladder or step; gradus is a step, and de gradus a step down | S019, S006, S068, S421, S427, S442 |
| degree symbol | symbol | 1558 J. Peletier's 1558 appendix to Gemma Frisius, in the Paris 1569 edition, uses the raised circle, but only in multiplication and not for writing angles | possibly Greek omicron from moira, or a florescent form of Latin gradus. Cajori says the Greek line of descent has not been established | step, degree | S363 |
| euthetai en kuklo | phrase | 150 Ptolemy, Almagest Book I, chapter headings 10 and 11, c. 150 CE | Ancient Greek euthys, straight, to eutheia gramme, straight line | the straight lines in the circle | S061, S062 |
| exsecant | word | 1929 Not established. The earliest dated record this book can cite is Cajori's Notations volume 2, 1929, section 527, which prints the symbol exsec A with no coiner and no date | Latin ex-, out of, plus secans to English exsecant | external secant, that is sec A minus 1 | S423 |
| Fakhri sextant | eponym | al-Khujandi's patron | Arabic al-suds al-Fakhri | the Fakhri sixth-of-a-circle | S194 |
| frequency | word | 1831 | Latin frequentia from frequentem to English frequency | an assembling in great numbers, a crowding | S442 |
| goniometry | word | 1724 Smith credits Thomas Fantet de Lagny in 1724, although more in the etymological sense of mere angle measure. The terms track could not corroborate this in Cajori or in any primary source, and Miller has no entry | Ancient Greek gonia plus metron to Modern Latin goniometria to English goniometry | angle measure | S421 |
| gougu | word | -100 | Chinese gougu, also written with the variant character for gou | gou is hook, the shorter leg and the shadow; gu is thigh, the longer leg and the gnomon | S241, S245, S246 |
| Gregory series | eponym | 1671 James Gregory | the arctangent series | S127 | |
| Hakimi Zij | eponym | the Fatimid caliph al-Hakim | Arabic | al-Hakim's tables | S196, S230 |
| harmonic | word | 1560 | Ancient Greek harmos to harmonia to harmonikos to English harmonic | fastening or joint, then concord of sounds | S427, S442 |
| haversine | word | 1835 James Inman is the usual credit and the edition is contested: the OED second edition and Wikipedia give the third edition of 1835, the popular claim gives the first edition of 1821, and the terms track verified the word only in Inman's 1858 edition. Cajori credits the function, not the word, to Jose de Mendoza y Rios in 1801 | English half plus versed plus sine, matching Latin semiversus | half the versed sine | S423, S435, S441 |
| huiyuan | word | Shen Kuo, Mengxi bitan chapter 18, 11th century, text as in the 1305 edition | Chinese huiyuan | assembling the circle, paired with zhe, to break | S257 |
| hypoteinousa | word | 150 Ptolemy, Almagest I.10 | Ancient Greek hypo, under, plus teino, stretch | stretching under, that is subtending | S062 |
| ib-si8 | word | -1792 | Sumerian ib-si8 | square-side. Robson translates the Plimpton 322 headings as square-side of the width and square-side of the diagonal; Mansfield says only that the word refers to the result of some operation | S001, S003 |
| ilm al-miqat | phrase | Arabic ilm al-miqat | the science of timekeeping, specifically of determining the times of the five prayers | S182 | |
| indanum | word | Akkadian indanum | the change in width per unit height of a trapezoid, that is (b minus a) divided by h | S003 | |
| Jacob's staff | eponym | nobody; a literary allusion misread as a name | Latin baculus Jacobi, from a Hebrew poem | the staff of Jacob | S318, S336 |
| jihe | word | 1607 Matteo Ricci and Xu Guangqi, Jihe yuanben, 1607 | Chinese jihe | classically how much or how many, repurposed as the calque for geometry, with yuanben for elements | S260 |
| jyotpatti-ganita | phrase | 628 Brahmagupta, Brahmasphutasiddhanta XII.66, 628 CE, per Datta and Singh | Sanskrit jya plus utpatti plus ganita | the science of calculation for the construction of the sine | S128 |
| kardaja | word | 770 A work of Yaqub ibn Tariq, c. 770 CE, titled On the table of kardaja, per the Fihrist as reported at second hand by Datta and Singh; al-Khwarizmi uses the variant karaja in 825 | Sanskrit krama-jya to Arabic karaja or kardaja to Latin kardaga, karkaya, gardaga and cardaga | direct sine, then the tabular interval of 3 degrees 45 minutes and sometimes 15 degrees | S128 |
| ki (KI) | word | -1792 | Sumerian KI | its place | S003 |
| kippatum | word | Akkadian kippatum, from kapapum, to curve | the curved thing | S001, S002 | |
| Leibniz series | eponym | Gottfried Wilhelm Leibniz | the series for a quarter of pi | S127 | |
| mekos and platos | phrase | Hipparchus, Commentary on the Phaenomena, and the Codex Climaci Rescriptus undertext | Ancient Greek mekos and platos | length and breadth | S065 |
| Menelaus's theorem | eponym | Menelaus of Alexandria | the transversal theorem of the Sphaerica | S061, S068, S082 | |
| midline | word | No first use established. The word is in none of the reference works consulted: Cajori volumes 1 and 2, Smith volumes 1 and 2, Braunmuhl, and Miller. The guess that it is a twentieth century American schoolbook coinage is labeled in chapter 8 as a guess about a pattern, not a finding | English mid plus line | the line in the middle | |
| minute | word | Ptolemy uses the Greek prota hexekosta; the Latin phrase is medieval; the English word is late 14th century | Ancient Greek prota hexekosta to Latin pars minuta prima to Old French minut to English minute | first sixtieths, first small part | S421, S442 |
| mithartum | word | Akkadian mithartum, the reflexive stem of maharum | that which is equal and opposite to itself | S001, S002 | |
| MU.BI.IM | word | -1822 | Sumerian written form for Akkadian sumsu | its name | S001, S002 |
| mutarrittum | word | Akkadian mutarrittum | direction of the plumb line | S004 | |
| muwaqqit | word | Arabic muwaqqit | timekeeper | S182, S216 | |
| NINDA | word | -750 | Sumerian NINDA | one sixtieth of an US of time in MUL.APIN | S007 |
| period | word | 1727 | Ancient Greek periodos to Latin periodus to Old French periode to English period | a going around, a circuit, from peri around and hodos a going | S442 |
| phase | word | 1861 | Ancient Greek phasis to Modern Latin phases to English phase | appearance of a star or phase of the moon, from phainein to show | S442 |
| phasor | word | 1893 The concept is Charles Proteus Steinmetz, 1893: it allows us to represent the alternate current, the sine-function of time, by a constant numerical quantity. The word itself is later and was not traced | English phase plus vector | a phase vector | S377 |
| pi (the symbol) | symbol | 1706 William Jones, Synopsis palmariorum matheseos, London 1706, p. 263. Oughtred earlier wrote pi over delta for the ratio, and Euler's use fixed it in circulation | Greek letter pi, the initial of periphereia | circumference, periphery | S423, S429 |
| Plimpton 322 (the name) | name | 1943 Mendelsohn's 1943 catalog of the cuneiform tablets in Columbia University's libraries | modern museum designation | item 322 in the catalog, Plimpton after the collector George Arthur Plimpton | S002 |
| prime and double prime | symbol | Ptolemy's Syntaxis marks first sixtieths with one accent and second sixtieths with two. Cajori names no single first printer of the modern usage | Greek astronomical practice through medieval Latin to modern notation | first sixtieths and second sixtieths | S363 |
| prosinus and transsinuosa | phrase | 1593 Francois Viete, Responsorum liber VIII, first published 1593; Cajori cites the Opera mathematica, Leyden 1646, p. 417 | Latin coinages by Viete as replacements for tangent and secant | before the sine, and across the sine | S363, S207 |
| prosthaphaeresis | word | 1580 Contested. The DSB's Brahe entry credits Paul Wittich, probably largely responsible, about 1580, and Longomontanus in 1622 uses the word of Wittich; Christmann in 1611 refers to prosthaphaeresis in reporting Johannes Werner's manuscript, which is Braunmuhl's ground for making Werner the inventor | Ancient Greek prosthesis plus aphairesis to Neo-Latin prosthaphaeresis | addition and subtraction | S321, S317, S322, S425 |
| Pythagorean theorem | eponym | Pythagoras | the theorem of Pythagoras | S002, S031 | |
| radian | word | 1873 James Thomson, examination paper, Queen's College Belfast, 5 June 1873, is the first appearance in print. The name was made twice and both parties said so in Nature in 1910: Thomas Muir hesitated in 1869 between rad, radial and radian and settled on radian in 1874, and Thomson proposed it in July 1871. The 1867 Thomson and Tait first edition does not contain the word; the 1879 edition has it nine times | Latin radius, a staff or rod or spoke or ray, to English radial angle, clipped to radian | radial angle, contracted | S364, S381, S423, S427, S442, S485, S486, S487, S488, S489, S490 |
| Rhind Mathematical Papyrus | eponym | Alexander Henry Rhind | Rhind's papyrus | S008, S009 | |
| sagitta | word | 1220 Fibonacci, Practica geometriae, 1220 | Arabic sahm to Latin sagitta | arrow | S421 |
| sangaku | word | 1806 Tokugawa period practice; Cooke's example is a tablet hung by Ehara Masanori at the Atsuta shrine in 1806 | Japanese sangaku | framed computations, or computational framed pictures | S266 |
| secant | word | 1583 Thomas Fincke, Geometriae rotundi libri XIIII, Basel 1583, definition 27 and p. 73, for the word; English from The Table of Secants in Thomas Blundeville's Exercises, 1594. For the object, Abu al-Wafa uses two shadow diameters that Debarnot identifies with our secant and cosecant | Arabic shadow diameter as the object, and Latin secans, present participle of secare, to cut, as the word, to English secant | cutting line; the shadow diameter is the hypotenuse of the shadow triangle | S181, S228, S304, S332, S314, S333, S363, S421, S423, S442 |
| second | word | Ptolemy uses the Greek deutera hexekosta; the Latin pars minuta secunda is medieval | Ancient Greek deutera hexekosta to Latin pars minuta secunda to English second | second sixtieths, second small part | S421, S442 |
| seked | word | -1550 Rhind Mathematical Papyrus problems 56 to 60, copied by Ahmose in year 33 of Apophis, about 1550 BCE | Egyptian skd | Peet suggests, with an explicit hedge, that if the s is causative the word may come from kd, to build, so that skd is that which forms, the measurement which builds up the pyramid | S005 |
| siliptum | word | -1792 | Akkadian siliptum, from salapum, to strike through | that which strikes through | S002, S003 |
| sin to the minus one | symbol | 1813 John Herschel, Philosophical Transactions of London for 1813, p. 10 | English, by analogy with d to the minus n and log to the minus one | the inverse operation, not the reciprocal | S363, S423 |
| sin z, cos z (functions of a bare number) | symbol | 1748 Leonhard Euler, Introductio in analysin infinitorum I, section 127, 1748 | Latin abbreviation in Euler's usage | sine of z, where z is a number rather than a line in a circle of some radius | S362 |
| sin, tan, sec (the abbreviations) | symbol | 1583 Thomas Fincke, 1583, Liber XIV, has the six contractions sin., tan., sec., sin. com., tan. com. and sec. com. Albert Girard, Trigonometrie, 1626, is credited with first using sin and tan in a treatise. The first use of sec as an abbreviation could not be verified | Latin abbreviations of sinus, tangens and secans, with complementi for the co-functions | clipped forms of the Latin names | S363, S337, S343 |
| sine | word | 1145 Genuinely open, with five voices and no manuscript evidence. Boyer credits Robert of Chester's translation of al-Khwarizmi, about 1145; D. E. Smith credits Robert of Chester's revision of al-Khwarizmi's tables, 1150; Eves credits Gerard of Cremona, about 1150; Cajori in 1906 credits Plato of Tivoli's translation of al-Battani, which the terms track records as long standing but wrong; and Van Brummelen now backs Robert of Chester's revision as a fifth voice while presenting no manuscript evidence either. English sine is Thomas Fale, Horologiographia, 1593 | Sanskrit ardha-jya and jya-ardha, shortened to jya, with the synonym jiva, to Arabic jiba, written j-y-b, read as Arabic jaib, to Latin sinus, to English sine | jya is bow-string, hence chord; jiba is a meaningless Arabic sound copy of the Sanskrit; jaib is a real Arabic word for fold, bosom, pocket or bay; Latin sinus is a fold in a garment, a bend, a bay or a bosom | S128, S130, S132, S133, S207, S209, S228, S332, S335, S325, S421, S422, S425, S427, S442, S481 |
| sinh, cosh (the h suffix) | symbol | 1757 Vincenzo Riccati wrote Sh and Ch, which Cajori dates to 1757; Lambert wrote sin h(b - x) and cos h b in the Berlin Histoire for 1768 | Latin abbreviations plus English hyperbolic | hyperbolic sine, hyperbolic cosine | S363, S386 |
| sinus rectus | phrase | 1220 Fibonacci, Practica geometriae, 1220, and defined again in Regiomontanus, De triangulis, composed 1462 to 1464 and printed 1533 | Latin sinus rectus, translating Sanskrit krama-jya | straight sine, the half chord of the doubled arc | S301, S421, S443 |
| sinusoid | word | 1823 | Latin sinus plus Ancient Greek -oeides, form or shape, to English sinusoid | sine-shaped | S426, S442 |
| suifang yan fa | phrase | 718 Jiuzhi li, 718, per Yabuuchi as reported by Kotyk | Chinese suifang yan fa, calquing Sanskrit sva-desa-aksa | method according to the location of the observer | S253 |
| takiltum | word | -1792 The heading of column I of Plimpton 322, and also tablet YBC 6967 | Akkadian takiltum, from the verb kullum | Robson renders it the holding-square, a technical derivative of the verb to multiply lengths together into areas; Thureau-Dangin held it may just mean square | S001, S002, S003 |
| tangent | word | 1583 Thomas Fincke, Geometriae rotundi libri XIIII, Basel 1583, definitions 21 and 22, for the word; English in Thomas Blundevil's Exercises, 1597. For the object, Habash al-Hasib defines and tabulates the shadow of an arc in his zij, and the first shadow tables date to about 860 | Arabic zill, shadow, for the object, and Latin tangens, present participle of tangere, to touch, for the word, to English tangent | touching, a line that touches a circle at one point; zill is the shadow of a gnomon | S181, S207, S228, S304, S314, S326, S333, S363, S421, S423, S442 |
| Taylor series | eponym | 1785 Brook Taylor | the series of Taylor's 1715 book | S388 | |
| trigonometry | word | 1595 Bartholomaeus Pitiscus, Trigonometria: sive de solutione triangulorum tractatus brevis et perspicuus, appended to Scultetus, Sphaericorum libri tres, Heidelberg 1595; English in Ralph Handson's translation of 1614 | Ancient Greek trigonon plus metron to Neo-Latin trigonometria to English trigonometry | triangle measurement, from treis three, gonia angle, and metron measure | S303, S313, S328, S333, S334, S421, S427, S442 |
| trikonamiti | word | Datta and Singh, writing in the 1930s: in very recent years there has appeared the name Trikonamiti | Sanskrit trikona plus miti, a modern calque on the Greek | triangle measure | S128 |
| tripleuron | word | Pappus reports that Menelaus, in the Sphaerica, calls the spherical triangle a tripleuron rather than a trigonon | Ancient Greek tripleuron, from tri- three and pleura side | three-side | S068 |
| ukha-thebet | word | -1550 Rhind Mathematical Papyrus problems 56 to 59 | Egyptian wha-tbt | Peet reads it as a compound of the verb wha, to seek, and the noun tbt, a sandal, and warns that no corroboration can be obtained from the names of the measurements themselves | S005 |
| ukullu | word | Akkadian ukullu | literally fruit, per Mansfield and Wildberger following Thureau-Dangin, reported at second hand | S003 | |
| umbra recta, umbra extensa | phrase | Al-Battani in Plato of Tivoli's Latin, where the form is umbra extensa; umbra recta in later Latin writers; Robertus Anglicus uses umbra alone about 1231 | Arabic straight shadow to medieval Latin umbra recta and umbra extensa | straight shadow, extended shadow | S181, S207, S421, S425 |
| umbra versa | phrase | Al-Battani in Plato of Tivoli's Latin translation; Maurolycus uses it of the tangent theorem in De sphaera sermo, 1558 | Arabic turned shadow to medieval Latin umbra versa | turned shadow | S181, S207, S421, S425 |
| unit circle | phrase | 1852 The OED's first citation is the Cambridge and Dublin Mathematical Journal, 1852. Euler's Introductio of 1748 had already set the radius to 1 without using the phrase, and Van Brummelen puts unit radius earlier still, with Abu al-Wafa and Abu Nasr Mansur in the tenth century | English unit plus circle | circle of radius one | S427, S481 |
| US (the sign) | word | -750 | Sumerian sign US used as a unit | in MUL.APIN a unit of time, 30 US to a beru and 12 beru to a day, so 360 US per day | S007, S016 |
| versine (versed sine) | word | 1220 The function is in the Surya Siddhanta II.22 to II.27, about 400 CE. The Latin is Fibonacci, Practica geometriae, 1220; English versed signe is W. Burrough, Variation of the Compasse, 1596 | Sanskrit utkrama-jya to Arabic sahm, arrow, to Latin sinus versus, from vertere to turn, to English versed sine and versine | reversed sine, so called because the values are got by subtracting the tabular sines from the radius in reversed order; sahm and the Sanskrit isu both mean arrow | S121, S128, S181, S421, S427, S442, S443 |
| wasan | word | Japanese wasan | wa is the character used for Japanese-style work, literally harmony; san is calculation | S266 | |
| xian | word | Chinese xian | bowstring | S246 | |
| yenri (enri) | word | Applied in Japan by the late 17th century; Cooke's early example is Isomura Kittoku's Ketsugi-sho, 1660 | Japanese enri | circle theory, circle principle | S266 |
| zill | word | 860 Habash al-Hasib defines and tabulates the shadow of an arc in his zij, in what Debarnot judges an independent introduction; the first known shadow tables date to about 860 | Arabic zill to Latin umbra recta and umbra versa to English tangent and cotangent | shadow, of a gnomon or of an arc | S181, S207 |
| ziqpu | word | -750 | Akkadian ziqpu | culminating | S007 |
Appendix C
Pronunciation quick reference
Every name in this book that a source establishes a pronunciation for: 184 of 185. The other 1 is absent because no source consulted states how to say it, and a guess would be worse than a gap.
Say them out loud, and say them wrong rather than not at all. Most of the people in this book worked in Arabic, Sanskrit, Greek or Chinese, and the English respellings here are approximations made from published romanization rules, not recordings. A student who says a name imperfectly is doing better than a student who avoids the sentence.
| Name | Say it | IPA | Region |
|---|---|---|---|
| Abraham de Moivre | AY-bruh-ham duh MWAHV-ruh (many English speakers say duh MOY-ver) | not recorded | France and England |
| Abraham J. Sachs | SAKS | not recorded | United States |
| Abraham Scultetus | AY-bruh-ham skul-TAY-tus | not recorded | Germany |
| Abu al-Wafa al-Buzjani | a-BOO al-wa-FAA al-booz-JAA-nee (Islamic track); AH-boo al-WAH-fah (terms track) | not recorded | Islamic world (Baghdad) |
| Abu Ali al-Marrakushi | a-BOO a-LEE al-mar-raa-KOO-shee | not recorded | Islamic world (Cairo) |
| Abu Mahmud al-Khujandi | al-khoo-JAN-dee (Islamic track); al-hoo-JAN-dee (terms track) | not recorded | Islamic world (Central Asia) |
| Abu Nasr Mansur ibn Iraq | a-BOO NASR man-SOOR ib-n ee-RAAK (Islamic track); AH-boo NAHSS-r man-SOOR (terms track) | not recorded | Islamic world (Khwarazm) |
| Acyuta Pisarati | AH-chyoo-tuh pih-SHAH-ruh-tee | not recorded | India (Kerala) |
| Adelard of Bath | AD-uh-lard of BATH | not recorded | England |
| Adriaan van Roomen | AH-dree-ahn van ROH-men; the Latin form Adrianus Romanus is ay-dree-AY-nus roh-MAY-nus | not recorded | Low Countries and Germany |
| Adrien-Marie Legendre | ah-dree-EN muh-REE luh-ZHAHN-druh | not recorded | France |
| Ahmose | AH-mohs (or AH-mess) | not recorded | Egypt |
| Al-Battani | al-bat-TAA-nee (Islamic track); al-buh-TAH-nee (terms track) | not recorded | Islamic world (Syria) |
| Al-Biruni | al-bee-ROO-nee | not recorded | Islamic world (Khwarazm and Ghazna) |
| Al-Ijliyyah | al-IJ-lee-ya | not recorded | Islamic world (Aleppo) |
| Al-Khwarizmi | al-khwaa-RIZ-mee | not recorded | Islamic world (Baghdad) |
| Al-Sijzi | as-SIJ-zee | not recorded | Islamic world (Iran) |
| Albert Girard | al-BAIR zhee-RAR | not recorded | France and the Netherlands |
| Alexander Henry Rhind | RIND (rhymes with "find") | not recorded | Scotland and Egypt |
| Alexander J. Ellis | al-ig-ZAN-der EL-iss | not recorded | England |
| Alice Everett | AL-iss EV-uh-rit | not recorded | Ireland and England |
| Amenemhat III | ah-men-EM-hat | not recorded | Egypt |
| Annie Russell Maunder | AN-ee RUSS-ul MAWN-der | not recorded | Ireland and England |
| Apophis | uh-POH-fis | not recorded | Egypt |
| Archimedes of Syracuse | ar-kih-MEE-deez | not recorded | Greek world (Sicily) |
| Aristarchus of Samos | a-ris-TAR-kuss | not recorded | Greek world |
| Aryabhata I | ARE-yuh-buh-tuh (Indian track) or ARR-yuh-BHUT-uh (terms track) | not recorded | India |
| Bartholomaeus Pitiscus | bar-toh-loh-MAY-us pih-TISS-kus | not recorded | Germany |
| Bhaskara I | BAHS-kuh-ruh (the FIRST) | not recorded | India |
| Bhaskara II | BAHS-kuh-ruh (the SECOND) | not recorded | India |
| Brahmagupta | BRUH-muh-GOOP-tuh | not recorded | India |
| Brook Taylor | BRUUK TAY-lor | not recorded | England |
| C. T. Rajagopal | rah-juh-GO-pahl | not recorded | India |
| Carl Friedrich Gauss | KARL FREED-rikh GOWSS | not recorded | Germany |
| Charles M. Whish | WISH | not recorded | England and India |
| Charles Proteus Steinmetz | CHARLZ PROH-tee-us STINE-mets | not recorded | Germany and the United States |
| Chen Zi | chun DZUH | not recorded | China |
| Christine Proust | krees-TEEN PROOST | not recorded | France |
| Christopher Clavius | KLAY-vee-us | not recorded | Germany and Italy |
| Claudius Ptolemy | KLAW-dee-us TOL-uh-mee | not recorded | Greek world (Roman Egypt) |
| Colin Maclaurin | KOL-in muh-KLOR-in | not recorded | Scotland |
| Damodara | DAH-mo-duh-ruh | not recorded | India (Kerala) |
| Daniel Bernoulli | DAN-yel ber-NOO-lee | not recorded | Switzerland and Russia |
| Daniel F. Mansfield | MANZ-feeld | not recorded | Australia |
| David Pingree | PING-ree | not recorded | United States |
| Derrick de Solla Price | duh SOL-uh PRICE | not recorded | United States |
| Donald Knuth | kuh-NOOTH | not recorded | United States |
| Edgar J. Banks | BANKS | not recorded | United States and Iraq |
| Edmund Gunter | ED-mund GUN-ter | not recorded | England |
| Edward Wright | ED-wurd RITE | not recorded | England |
| Eleanor Robson | EL-uh-nor ROB-sun | not recorded | England |
| Erasmus Reinhold | eh-RAZ-mus RINE-holt | not recorded | Germany |
| Eratosthenes of Cyrene | eh-ruh-TOS-thuh-neez | not recorded | Greek world |
| Euclid | YOO-klid | not recorded | Greek world (Alexandria) |
| Evelyn Lamb | EV-uh-lin LAM | not recorded | United States |
| Evert Marie Bruins | BROWNS | not recorded | Netherlands |
| Florian Cajori | FLOR-ee-un kuh-JOR-ee | not recorded | Switzerland and the United States |
| Frank B. Allen | ALL-en | not recorded | United States |
| François Thureau-Dangin | frahn-SWAH tuh-ROH dahn-ZHAN | not recorded | France |
| François Viète | frahn-SWAH vee-ET | not recorded | France |
| Gaspard de Prony | gas-PAR duh proh-NEE | not recorded | France |
| Gautama Siddha | choo-TAHN shee-DAH | not recorded | China |
| Gautama Zhuan | choo-TAHN JWAHN | not recorded | China |
| Georg Hartmann | GAY-org HART-mahn | not recorded | Germany |
| Georg Joachim Rheticus | GAY-org YO-ah-khim RET-ih-kus; the terms track gives RET-ih-kuss | not recorded | Austria and Germany |
| Georg von Peuerbach | GAY-org fon POY-er-bakh | not recorded | Austria |
| George Arthur Plimpton | PLIMP-tun | not recorded | United States |
| Gerard of Cremona | juh-RARD of kreh-MOH-nuh (Europe track); ger-AR-doh of kre-MOH-nuh (terms track) | not recorded | Italy and Spain |
| Gerardus Mercator | jeh-RAR-dus mer-KAY-tor; in Flemish, HEH-rart duh KRAY-mer | not recorded | Low Countries |
| Giacomo Rho | JAH-koh-moh ROH; his Chinese name Luo Yagu is lwaw YAH-goo | not recorded | Italy and China |
| Gilles Personne de Roberval | ZHEEL per-SUN duh ROH-bair-val | not recorded | France |
| Govindasvami | go-VIN-duh-SWAH-mee | not recorded | India |
| Guo Shoujing | gwaw SHOH-jing | not recorded | China |
| Habash al-Hasib al-Marwazi | HA-bash al-HAA-sib al-mar-WA-zee (Islamic track); HAB-ash al-HAH-sib (terms track) | not recorded | Islamic world (Baghdad) |
| Hammurabi | ham-uh-RAH-bee | not recorded | Mesopotamia |
| Henry Andrews | HEN-ree AN-drooz | not recorded | England |
| Henry Briggs | HEN-ree BRIGZ | not recorded | England |
| Hermann Hunger | HAIR-mun HOONG-er | not recorded | Austria |
| Hermann von Helmholtz | HER-mahn fon HELM-holts | not recorded | Germany |
| Hipparchus of Nicaea | hip-PAR-kuss | not recorded | Greek world |
| Honoré Fabri | on-oh-RAY FAB-ree | not recorded | France and Italy |
| Hypatia | hy-PAY-shuh | not recorded | Greek world (Alexandria) |
| Hypsicles of Alexandria | hip-SIK-leez (Greek track) or HIP-sih-kleez (Babylonian track) | not recorded | Greek world (Alexandria) |
| Ibn al-Shatir | IB-n ash-SHAA-tir | not recorded | Islamic world (Damascus) |
| Ibn Muadh al-Jayyani | IB-n moo-AATH al-jay-YAA-nee (Islamic track); IB-n moo-AATH al-jy-YAH-nee (terms track) | not recorded | al-Andalus |
| Ibn Yunus | IB-n YOO-nus | not recorded | Islamic world (Egypt) |
| Isaac Barrow | BAR-oh | not recorded | England |
| Isaac Newton | EYE-zik NEW-tun | not recorded | England |
| Jabir ibn Aflah | JAH-bir IB-n AF-lah | not recorded | al-Andalus |
| James Gregory | GREG-uh-ree | not recorded | Scotland |
| James Inman | IN-man | not recorded | England |
| James Thomson | JAYMZ TOM-sun; the terms track gives TOM-son | not recorded | Ireland and Scotland |
| James W. Cooley | JAYMZ KOO-lee | not recorded | United States |
| Jamshid al-Kashi | al-KAA-shee (Islamic track); al-KAH-shee (terms track) | not recorded | Islamic world (Iran and Samarkand) |
| Jean-Vincent Scheil | vann-SAHN SHAYL | not recorded | France |
| Jens Høyrup | YENS HOY-rup | not recorded | Denmark |
| Johann Adam Schall von Bell | YOH-hahn AH-dahm SHAHL fon BEL; his Chinese name Tang Ruowang is tahng RWAW-wahng | not recorded | Germany and China |
| Johann Bernoulli | YO-hahn ber-NOO-lee | not recorded | Switzerland |
| Johann Heinrich Lambert | YO-hahn HYNE-rikh LAM-bairt | not recorded | France and Prussia |
| Johann Lufft | YO-hahn LOOFT | not recorded | Germany |
| Johann Petreius | YO-hahn peh-TRAY-oos | not recorded | Germany |
| Johann Radon | YO-hahn RAH-dohn | not recorded | Austria |
| Johann Terrenz Schreck | YOH-hahn TEH-rents SHREK | not recorded | Germany and China |
| Johannes Schöner | yo-HAH-nes SHUR-ner | not recorded | Germany |
| Johannes Werner | yo-HAHN-uss VAIR-ner | not recorded | Germany |
| John Herschel | JON HER-shul; the terms track gives HER-shell | not recorded | England and Cape Colony |
| John M. Steele | STEEL | not recorded | United States |
| John Maudith | JON MAW-dith | not recorded | England |
| John Napier | JON NAY-peer | not recorded | Scotland |
| John P. Britton | BRIT-un | not recorded | United States |
| John W. Tukey | JON TOO-kee | not recorded | United States |
| John Wallis | WOL-iss | not recorded | England |
| Joseph Fourier | zho-ZEF FOOR-yay | not recorded | France |
| Joseph-Louis Lagrange | zho-ZEF loo-EE luh-GRAHNZH | not recorded | Italy, Prussia and France |
| Jost Bürgi | YOHST BUR-ghee | not recorded | Switzerland and Bohemia |
| José de Mendoza y Ríos | ho-SAY day men-DOH-thah ee REE-os | not recorded | Spain and England |
| Jyesthadeva | JYESH-tuh-DAY-vuh | not recorded | India (Kerala) |
| Jöran Friberg | YUR-an FREE-berg | not recorded | Sweden |
| Kamalakara | kuh-muh-LAH-kuh-ruh | not recorded | India |
| Kangxi Emperor | kahng-SHEE | not recorded | China |
| Karl Scherffer | KARL SHERF-fer | not recorded | Austria |
| Karlheinz Brandenburg | KARL-hynts BRAHN-den-boorg | not recorded | Germany |
| Kushyar ibn Labban | KOOSH-yar ib-n LAB-baan | not recorded | Islamic world (Rayy) |
| Lalla | LUL-luh | not recorded | India |
| Leonhard Euler | LAY-on-hart OY-ler | not recorded | Switzerland, Russia and Prussia |
| Levi ben Gershon | LEE-vye ben GER-shon; Gersonides is ger-SON-ih-deez; the acronym RaLBaG is RAHL-bahg | not recorded | France (Provence) |
| Li Zhizao | lee JRR-dzow | not recorded | China |
| Liu Hui | lyoh HWAY | not recorded | China |
| Madhava of Sangamagrama | MAH-duh-vuh | not recorded | India (Kerala) |
| Manjula (Munjala) | MUN-joo-luh | not recorded | India |
| Mary Edwards | MAIR-ee ED-wurdz | not recorded | England |
| Mathieu Ossendrijver | ma-TYUR OSS-en-dry-ver | not recorded | Germany |
| Matteo Ricci | mah-TAY-oh REE-chee; his Chinese name Li Madou is lee MAH-doh | not recorded | Italy and China |
| Mei Wending | may wun-DING | not recorded | China |
| Menelaus of Alexandria | men-uh-LAY-us | not recorded | Greek world (Alexandria) |
| Minggantu | ming-ahn-TOO | not recorded | China |
| Miriam Chelebi | MEER-yam che-le-BEE | not recorded | Ottoman lands |
| Nan Gongyue | nahn-gong YWEH | not recorded | China |
| Nasir al-Din al-Tusi | NAA-sir ad-DEEN at-TOO-see (Islamic track); nah-SEER ad-DEEN at-TOO-see (Europe track); nuh-SEER ad-DEEN TOO-see (terms track) | not recorded | Islamic world (Iran) |
| Nevil Maskelyne | NEV-il MASK-uh-lin | not recorded | England |
| Nicholas Longobardo | nih-koh-LOH lon-goh-BAR-doh | not recorded | Italy and China |
| Nicolaus Copernicus | nik-oh-LAY-us koh-PUR-nih-kus; in Polish, mee-KOH-why koh-PEHR-nik | not recorded | Poland |
| Nilakantha Somayaji | NEE-luh-KUN-tuh so-muh-YAH-jee | not recorded | India (Kerala) |
| Norman John Wildberger | WILD-ber-ger | not recorded | Australia |
| Otto Neugebauer | NOY-guh-bow-er | not recorded | Germany, Denmark and the United States |
| Pappus of Alexandria | PAP-uss | not recorded | Greek world (Alexandria) |
| Parameshvara | puh-ruh-MAYSH-vuh-ruh | not recorded | India (Kerala) |
| Paul Wittich | POWL VIT-ikh | not recorded | Silesia and Denmark |
| Pierre-Simon Laplace | pee-AIR see-MOHN luh-PLAHSS | not recorded | France |
| Plato of Tivoli | PLAY-toh of TIV-oh-lee | not recorded | Italy and Spain |
| Ralph Handson | RALF HAND-sun | not recorded | England |
| Regiomontanus | ray-jee-oh-mon-TAH-nus (Europe track); REE-jee-oh-mon-TAH-nuss (terms track); the German name is yo-HAH-nes MYOO-ler fon KUR-nikhs-bairk | not recorded | Germany and Italy |
| Richard of Wallingford | RITCH-erd of WOL-ing-ferd | not recorded | England |
| Robert of Chester | ROB-ert of CHESS-ter | not recorded | England and Spain |
| Roger Cotes | ROJ-er KOHTS | not recorded | England |
| Rong Fang | rong FAHNG | not recorded | China |
| Sankara Variyar | SHUN-kuh-ruh VAH-ree-yar | not recorded | India (Kerala) |
| Sebastian Henricpetri | seh-BAS-tee-ahn hen-rik-PET-ree | not recorded | Switzerland |
| Seki Takakazu | SEH-kee tah-kah-KAH-zoo | not recorded | Japan |
| Shams al-Din al-Khalili | SHAMS ad-DEEN al-kha-LEE-lee | not recorded | Islamic world (Damascus) |
| Shen Kuo | shun KWAW | not recorded | China |
| Steve Shnider | SHNY-der | not recorded | Israel |
| Takebe Kataakira | tah-KEH-beh kah-tah-AH-kee-rah | not recorded | Japan |
| Takebe Katahiro | tah-KEH-beh kah-tah-HEE-roh | not recorded | Japan |
| Thabit ibn Qurra | THAA-bit ib-n KUR-ra | not recorded | Islamic world (Baghdad) |
| Theodosius of Bithynia | thee-oh-DOH-see-us | not recorded | Greek world |
| Theon of Alexandria | THEE-on | not recorded | Greek world (Alexandria) |
| Thomas Eric Peet | PEET | not recorded | England |
| Thomas Fantet de Lagny | duh lah-NYEE | not recorded | France |
| Thomas Fincke | TOH-mas FING-kuh (Europe track); TOM-us FINK (analysis track); THOM-us FINK-uh (terms track) | not recorded | Denmark |
| Thomas Muir | TOM-us MYOOR | not recorded | Scotland and Cape Colony |
| Tycho Brahe | TEE-koh BRAH-uh | not recorded | Denmark and Bohemia |
| Ulugh Beg | oo-LOOG BEG | not recorded | Islamic world (Samarkand) |
| Valentin Otho | VAL-en-teen OH-toh | not recorded | Germany |
| Varahamihira | vuh-RAH-huh-MIH-hih-ruh | not recorded | India |
| Vatesvara | vuh-TAYSH-vuh-ruh | not recorded | India |
| Vincenzo Riccati | vin-CHEN-tso ree-KAH-tee | not recorded | Italy |
| Wang Xun | wahng SHOON | not recorded | China |
| William Jones | JOHNZ | not recorded | Wales and England |
| Willibald Pirckheimer | VIL-ee-balt PIRK-hy-mer | not recorded | Germany |
| Xu Guangqi | shoo GWAHNG-chee | not recorded | China |
| Yixing | ee SHING | not recorded | China |
| Zu Chongzhi | DZOO chong-JIRR | not recorded | China |
| Zu Geng | dzoo GUNG | not recorded | China |
Appendix D
Whose name is on it, and who did the work
11 eponyms: results, rules, instruments and books carrying somebody's name. Trigonometry is unusually full of them, and unusually bad at getting them right.
The pattern is worth naming, because it repeats in every chapter of this book. A result attaches to whoever the community happened to be reading when the result became useful, not to whoever first had it. That is not a conspiracy and mostly not even unfairness. It is what happens when work is written in a language, or a place, or a century that the people doing the naming were not reading.
Three of the rows below are not that pattern at all, and they are worth separating out. A patron's name on an instrument or a set of tables, and a buyer's name on a papyrus, record who paid or who acquired, never who did the mathematics. That is a different habit with the same result: the person who did the work is not the person on the label.
| Name on it | When the name attaches | Whose name it is | What the record shows | Sources |
|---|---|---|---|---|
| algorithm | 1690 | al-Khwarizmi | a finite step by step procedure for computing something. The name is al-Khwarizmi's byname, 'the man from Khwarazm', worn smooth: Latinised as algorismus it meant the Hindu-Arabic reckoning method, then any method, and nobody put his name on it on purpose. English algorism is attested in the early 13th century and algorithm from the 1690s | S227 |
| de Moivre's formula | not established | Abraham de Moivre | Named after de Moivre, who never stated it in his works. A closely related formula appears in a paper of 1707 and again in a 1722 publication, but he was eliminating a variable between two polynomials, not thinking about points on a circle. He had the machinery before Euler and got a formula named after him that he never wrote. | S395, S370 |
| Fakhri sextant | not established | al-Khujandi's patron | A 60-degree arc about 43 meters across, built by al-Khujandi and named for the patron who funded it rather than the man who built it. | S194 |
| Gregory series | 1671 | James Gregory | Named after Gregory, who published it in 1671. The same series is derived, geometrically and constructively, in the Kerala text this book quotes in Chapter 3, two centuries earlier. | S127 |
| Hakimi Zij | not established | the Fatimid caliph al-Hakim | Ibn Yunus compiled the tables; they carry the name of the caliph who paid for them. | S196, S230 |
| Jacob's staff | not established | nobody; a literary allusion misread as a name | The cross-staff European navigators used to find latitude for three hundred years. The name comes from an allusion to Genesis 32:10 in a Hebrew poem, which Latin readers took as an attribution to a man called Jacob. The instrument is described in Levi ben Gerson's Sefer Tekunah, and Peter of Alexandria put that into Latin in 1342. | S318, S336 |
| Leibniz series | not established | Gottfried Wilhelm Leibniz | The arctangent series at t = 1. It falls out of the same Kerala derivation as the series named after Gregory, one substitution later. | S127 |
| Menelaus's theorem | not established | Menelaus of Alexandria | The theorem Ptolemy proves and leans on throughout Almagest Book I. Ptolemy never credits Menelaus for it: Toomer notes that the Almagest mentions Menelaus only as an observer, for two nights of star-watching. No Greek manuscript of Menelaus's own Sphaerica survives, so what he proved cannot be read directly. | S061, S068, S082 |
| Pythagorean theorem | not established | Pythagoras | Plimpton 322 holds fifteen rows of Pythagorean triples, written roughly 1,200 years before Pythagoras was born. | S002, S031 |
| Rhind Mathematical Papyrus | not established | Alexander Henry Rhind | Named for the Scottish lawyer and antiquarian who bought it at Thebes around 1858, not for the scribe who wrote and signed it. | S008, S009 |
| Taylor series | 1785 | Brook Taylor | Printed in 1715, ignored for fifty-seven years, promoted by Lagrange in 1772 to the status of founding principle of the calculus, and only called the Taylor series in 1785: seventy years after Taylor wrote it, and fifty-four years after he died. | S388 |
Appendix E
Every number in this book, re-derived and checked in code
93 historical calculations, recomputed at sixty significant digits. Not checked by rereading them. Recomputed, in code, from the same starting values the original author used, and compared against the value that author printed.
This is the appendix that lets you distrust the rest of the book productively. It holds 93 historical calculations and 81 reported values: every sexagesimal string this book quotes, and every accuracy claim attached to one, re-derived with its error against the truth. Where a historical value is wrong, the size of the error is printed rather than described, and where a widely repeated modern claim is wrong, that is printed too. The hand-authored tables are checked separately, by a tool that re-derives 278 of their 536 numeric cells and names, column by column, the transcribed cells it cannot derive.
The script is the source, and this page is its output, pasted whole and unedited. The script does not read the book: it recomputes the history from scratch, so it can disagree with the book rather than echo it. Two other checks work the other way round and read the book itself, one re-evaluating every formula in the prose that states a value, the other re-deriving every table cell the mathematics around it can produce. If a number in the book stops agreeing with the mathematics, those fail the build. The three are doing different jobs and the book needs all of them.
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CHAPTER 1 Mesopotamia and Egypt
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[PASS] P322 row 1: the unwritten length is exactly 120
[PASS] P322 row 1: 119, 120, 169 is a Pythagorean triple
[PASS] P322 row 1: Col I equals d^2/l^2 (length in the denominator)
[PASS] P322 row 1: Col I is NOT d^2/w^2 (a common slip) (d^2/w^2 would be 2.016877, not 1.983403)
[value] P322 row 1: angle in degrees whose tangent is 119/120: 44.760270103919147654
[PASS] P322 row 1: sec^2 of that angle equals Col I
[PASS] P322 row 1: tan^2 equals Col I minus 1
[PASS] RMP 56: seked equals 5 and 1/25 palms
[value] RMP 56: slope angle in degrees: 54.246112745563251222
[PASS] RMP 57/58: seked equals 5 palms 1 finger
[value] RMP 57/58: slope angle in degrees: 53.130102354155978703
[PASS] RMP 57/58: run to rise is exactly 3 to 4
[PASS] RMP 57/58: hypotenuse of 3-4-5
[value] Great Pyramid: slope angle for seked 5.5 palms: 51.842773412630940423
[PASS] MUL.APIN: 12 beru of 30 US each gives 360 US per day
[PASS] MUL.APIN: 1 US of time equals 4 minutes
==============================================================================
CHAPTER 2 Greek chords
==============================================================================
[PASS] Ptolemy crd(1/2 deg) (error +0.000014 in units of R=60)
[PASS] Ptolemy crd(1 deg) (error +0.000038 in units of R=60)
[PASS] Ptolemy crd(1 1/2 deg) (error +0.000082 in units of R=60)
[PASS] Ptolemy crd(3 deg) (error -0.000123 in units of R=60)
[PASS] Ptolemy crd(60 deg) (error +0.000000 in units of R=60)
[PASS] Ptolemy crd(180 deg) (error +0.000000 in units of R=60)
[PASS] Ptolemy crd(60) is exactly R
[PASS] Ptolemy crd(180) is exactly 2R
[PASS] Ptolemy squeeze on EXACT chords brackets crd(1 deg) ([1.047167646, 1.047190075])
[value] Ptolemy squeeze, exact interval width: 0.000022429068742124468794
[value] Ptolemy squeeze, from his printed 1;34,15: lower bound: 1.0472222222222222222
[value] Ptolemy squeeze, from his printed 0;47,8: upper bound: 1.0474074074074074074
[value] Ptolemy squeeze, true crd(1 deg): 1.0471842598048721958
[PASS] Ptolemy's printed 0;47,8 rounds UP from the exact chord (exact 0.785392556)
[PASS] Ptolemy's printed 1;34,15 rounds UP from the exact chord (exact 1.570751469)
[PASS] Rounding pushes his lower bound just above the true value (excess 3.796e-05)
[value] True crd(1 deg) in sexagesimal: 1;2,49,51,48, so 1;2,50 is the correct rounding
[PASS] Ptolemy's adopted crd(1 deg) is 1;2,50 (correct to the two sexagesimal places he prints)
[PASS] crd(1) equals 2R sin(1/2)
[PASS] crd(30) equals 2R sin(30/2)
[PASS] crd(47) equals 2R sin(47/2)
[PASS] crd(120) equals 2R sin(120/2)
[value] Ptolemy pi as 3;8,30 in decimal: 3.1416666666666666667
[value] Ptolemy pi absolute error: 0.000074013076873428204023
[value] Ptolemy pi relative error: 0.000023559094075693081153
[PASS] Archimedes bounds bracket pi ([3.140845070, 3.142857143])
[value] Aristarchus implied angle for ratio 19 (degrees): 86.983038690184007417
==============================================================================
CHAPTER 3 India
==============================================================================
[PASS] 10800/pi rounds to 3438
[value] 10800/pi exactly: 3437.7467707849392526
[value] 10800/3.1416: 3437.7387318563789152
[PASS] Aryabhata: the 24 differences sum to exactly 3438
[PASS] Aryabhata: 24 steps of 3 deg 45 min make 90 degrees
[value] Aryabhata table: worst error is entry 18, in arcminutes: 0.70216722619613243131
[value] Aryabhata recursion, entry 7 as generated vs tabulated: 4.0
[value] Aryabhata recursion, entry 24 as generated (table says 7): 10.0
[PASS] Bhaskara I: the correct form gives sin(90) = 1
[PASS] Bhaskara I: the circulating 16x form gives 16 at x=90, not 1
[value] Bhaskara I: maximum absolute error: 0.0016317650438168067623
[value] Bhaskara I: location of that maximum, in degrees: 11.544
[value] Bhaskara I at 30 deg: approx: 0.5
[value] Bhaskara I at 30 deg: true: 0.5
[value] Bhaskara I at 45 deg: approx: 0.70588235294117647059
[value] Bhaskara I at 45 deg: true: 0.7071067811865475244
[value] Bhaskara I at 60 deg: approx: 0.86486486486486486486
[value] Bhaskara I at 60 deg: true: 0.86602540378443864676
[value] Madhava/Leibniz series, 100 terms: 3.1315929035585527643
[value] Madhava/Leibniz series, 100 terms, error: -0.0099997500312404741552
[value] Madhava plain series, 5 terms, error: 0.19808988609274644408
[value] Madhava first 1/(4n), 5 terms, error: -0.001910113907253555923
[value] Madhava second n/(4n^2+1), 5 terms, error: 0.000070084112548424275059
[value] Madhava third (n^2+1)/(4n^3+5n), 5 terms, error: -5.3520024916511610561e-6
[value] Madhava plain series, 20 terms, error: -0.04996884692195460678
[value] Madhava first 1/(4n), 20 terms, error: 0.000031153078045393220292
[value] Madhava second n/(4n^2+1), 20 terms, error: -7.7402904013400569907e-8
[value] Madhava third (n^2+1)/(4n^3+5n), 20 terms, error: 4.3007031533867211641e-10
[value] Madhava plain series, 50 terms, error: -0.019998000998782759949
[value] Madhava first 1/(4n), 50 terms, error: 1.9990012172400505396e-6
[value] Madhava second n/(4n^2+1), 50 terms, error: -7.9880275794966038897e-10
[value] Madhava third (n^2+1)/(4n^3+5n), 50 terms, error: 7.1748992560205979561e-13
[value] Decimal places from 20 plain terms: 1.3013006721945995852
[value] Decimal places from 20 terms plus the third correction: 9.3664605326366049122
[value] Madhava sine series at 30 deg, 10 terms, error: -2.4569249791322192574e-26
[value] One sexagesimal third, as a fraction of a radian: 8.0802280184922665598e-8
==============================================================================
CHAPTER 4 The Islamic world
==============================================================================
[value] al-Kashi sin 1 deg, his value as a decimal: 0.017452406437283510371
[value] al-Kashi sin 1 deg, true value: 0.017452406437283512819
[value] al-Kashi sin 1 deg, absolute error: -2.4482653603348059213e-18
[value] al-Kashi sin 1 deg: correct decimal places: 17.0
[value] Aaboe printed variant, error against truth: -6.9870529342177530449e-16
[PASS] sin(3x) identity at x=1 deg
[PASS] sin(3x) identity at x=7 deg
[PASS] sin(3x) identity at x=23 deg
[value] al-Kashi 2pi as a decimal: 6.2831853071795864848
[value] al-Kashi implied pi: 3.1415926535897932424
[value] al-Kashi pi error: 3.9610217084239975985e-18
[value] al-Biruni: R from the printed data: 13331728.352169116159
[value] al-Biruni: his stated R: 12851369.845
[value] al-Biruni: the cosine that reproduces his answer: 0.99994926440329206099
[value] al-Biruni: that cosine in sexagesimal: 0;59,59,49,2,28
[value] al-Biruni: cos(34 arcmin), true: 0.99995109237959866675
[PASS] al-Biruni: girth cross-check equals 80,780,039.03
[value] al-Biruni R in km, at 0.4572 m per cubit: 5875.646293134
[value] R tan(45 deg) with R=60: 60.0
[value] Precision of five sexagesimal places, as decimals: 8.8907562519182181625
==============================================================================
CHAPTER 5 China and Japan
==============================================================================
[PASS] Zhoubi: sun height is 80,000 li
[PASS] Zhoubi: hypotenuse is exactly 100,000 li
[PASS] Zhoubi: this is a scaled 3-4-5
[PASS] Zhoubi: sun diameter is 1,250 li
[PASS] Nine Chapters 9.6: water depth is 12
[PASS] Nine Chapters 9.6: reed length is 13
[PASS] Nine Chapters 9.6: 5-12-13 is a right triangle
[PASS] Sea Island 1: island height is 1255 bu
[PASS] Sea Island 1: 1255 bu is 4 li 55 bu
[PASS] Sea Island 1: distance is 30,750 bu
[PASS] Sea Island 1: 30,750 bu is 102 li 150 bu
[PASS] Liu Hui 96-gon
[PASS] Liu Hui 192-gon
[PASS] Liu Hui 3072-gon
[PASS] Liu Hui 157/50
[value] Liu Hui 3927/1250: 3.1416
[PASS] Zu Chongzhi bounds bracket pi ([3.1415926, 3.1415927])
[value] 355/113: 3.1415929203539823009
[value] 355/113 error against pi: 2.6676418906242231237e-7
[value] Shen Kuo worked case: arc by his rule: 8.8
[value] Shen Kuo worked case: true arc: 9.2729521800161223243
[value] Shen Kuo rule: worst relative error on the semicircle, percent: 5.4185425625521045817
[PASS] Takebe series equals 4 d^2 arcsin^2(sqrt(h/d))
[PASS] Takebe: the arc itself is 2 d arcsin(sqrt(h/d))
[PASS] Takebe identity on a semicircle gives arc = pi d / 2 (boundary case, 400 terms)
[value] Takebe series at the semicircle: error with only 60 terms: -2.0106961304787841773e-20
[PASS] Takebe Taisei sankei: 100 pi squared
[PASS] Takebe Taisei sankei: 10 pi
==============================================================================
CHAPTER 6 Renaissance Europe
==============================================================================
[PASS] sec(a) + tan(a) = tan(a/2 + 45) at a=7
[PASS] sec(a) + tan(a) = tan(a/2 + 45) at a=23
[PASS] sec(a) + tan(a) = tan(a/2 + 45) at a=61
[PASS] sec(a) + tan(a) = tan(a/2 + 45) at a=84
[value] Viete product, 30 factors, as pi: 3.1415926535897932373
[value] Viete product, 30 factors, error: -1.1205690845822566484e-18
[value] Viete product, 30 factors: correct decimal places: 17.0
[PASS] Prosthaphaeresis: 2 cos A cos B = cos(A-B) + cos(A+B)
[PASS] Prosthaphaeresis worked example matches direct multiplication
[value] Radius Ptolemy: equivalent decimal places: 1.7781512503836436325
[value] Radius Indian tables: equivalent decimal places: 3.5363058723510336074
[value] Radius Peuerbach: equivalent decimal places: 5.7781512503836436325
[value] Radius Regiomontanus decimal: equivalent decimal places: 7.0
[value] Radius Rheticus Canon 1551: equivalent decimal places: 7.0
[value] Radius Opus palatinum 1596: equivalent decimal places: 10.0
[PASS] Secant integral in radians at phi=30 deg (1-minute midpoint sum against ln tan(45 + phi/2))
[PASS] Wright meridional parts at phi=30 deg, in arcminutes (closed form 1888.3754)
[PASS] Secant integral in radians at phi=45 deg (1-minute midpoint sum against ln tan(45 + phi/2))
[PASS] Wright meridional parts at phi=45 deg, in arcminutes (closed form 3029.9392)
[PASS] Secant integral in radians at phi=60 deg (1-minute midpoint sum against ln tan(45 + phi/2))
[PASS] Wright meridional parts at phi=60 deg, in arcminutes (closed form 4527.3678)
==============================================================================
CHAPTER 7 Analysis
==============================================================================
[PASS] Newton arcsine coefficient 3 equals 3/40
[PASS] Newton sine coefficient 5 equals 1/120 (that is 1/5!)
[PASS] Newton sine coefficient 7 equals -1/5040 (that is -1/7!)
[PASS] Newton arcsine coefficient 7 as printed, 15/336, equals 5/112
[PASS] Newton arcsine series at x = 0.3 (4-term truncation error -6.406e-07)
[PASS] Newton reversion returns the sine at x = 0.3 (round-trip error -6.112e-07, pure truncation)
[PASS] Newton reversion is exact in the limit (20-term sine series)
[PASS] de Moivre at n=3, theta=17: real part
[PASS] de Moivre at n=3, theta=17: imaginary part
[PASS] de Moivre at n=5, theta=41: real part
[PASS] de Moivre at n=5, theta=41: imaginary part
[PASS] Euler formula real part
[PASS] Euler formula imaginary part
[PASS] Euler identity: e^(i pi) + 1 = 0
[PASS] Cotes relation
[PASS] One radian in degrees
[PASS] One degree in radians
[PASS] Arcminutes in a radian is 3437.7468
[value] Fourier square wave at x=pi/2, 50 terms: 0.99363443857817412751
[value] Fourier square wave at x=pi/2, 500 terms: 0.99936338086424900796
[value] Gibbs overshoot near the jump, 500 terms: 0.90282266691101007591
[PASS] Lambert continued fraction for tan(1)
[PASS] Beat frequency equals the difference of the two tones
[PASS] Sum-to-product: sin a + sin b = 2 sin((a+b)/2) cos((a-b)/2)
[value] GW150914: mean frequency over the quoted sweep, Hz: 92.5
[value] GW150914: cycles at that mean frequency in 0.2 s: 18.5
==============================================================================
SUMMARY
==============================================================================
Checks passed: 93
Checks failed: 0
Values reported (no assertion): 81Appendix F
Disagreements between sources
84 places where the sources contradict each other, and 73 of them are still open. That is the number worth sitting with: on most of these questions, the honest state of the field is that nobody knows.
An open disagreement is not a failure of research. It usually means the evidence needed to settle it does not survive, or survives in a manuscript nobody has edited. Where this book could settle one, the Resolution column says how. Where it could not, it says that instead.
| State | Question | One claim | The other claim | Resolution |
|---|---|---|---|---|
| Open | Where the 1538 Basel Greek editio princeps of the Almagest survives | It is cited routinely as extant. S061 | No digitized copy was located on e-rara, at the Munich Digitisation Centre or on the Internet Archive. popular repetition | |
| Resolved | What the 1910 Nature letters on the radian say | They are quoted freely at second hand through a reference site. S381;S427 | Their existence, authors, volume, pages and dates were verified, but the bodies were paywalled. S436;S437 | Resolved yes. All four letters are now in hand as primary text, and there are four, not three: Muir on 7 April and 16 June 1910, and James Thomson the son on 21 April and 16 June. Muir dates his own first use to class teaching at St Andrews in 1869 and the adoption of the form radian to 1874; the son reports a memorandum in his father's hand proposing the name in July 1871 and printing it in the Belfast examination questions of 5 June 1873. Both men concluded in print that the name was thought of independently twice. |
| Open | Whether al-Battani states the spherical law of cosines and a qibla formula | Yes, in most accounts. popular repetition | Support is tertiary only. Debarnot's specialist chapter does not credit him with the spherical law of cosines, and Nallino's edition was not read. S181;S220 | |
| Open | Where the 12,803,337 cubit figure for al-Biruni's Earth radius comes from | It is quoted widely as al-Biruni's result. popular repetition | It could not be verified in any source read and is inconsistent with the girth figure attributed to him; the internally consistent figure is 12,851,369.845 cubits. S185 | |
| Open | Why al-Biruni's stated Earth radius does not follow from his stated data | The numbers are simply quoted together, as though they agreed. S185 | His data as printed give 13,331,728.35 cubits against his stated 12,851,369.845; the cosine 0;59,59,49,2,28 reproduces his answer exactly, so the published paper's printed digit string looks like a typographical error. S185 | |
| Open | Who al-Ijliyyah was, and whether 944 to 967 are her life dates | An astrolabe maker with life dates 944 to 967. S215 (popular repetition of the regnal years as life dates) | One clause in Ibn al-Nadim's Fihrist attests her as the daughter of al-Ijli al-Asturlabi, pupil of Betulus, attached to Sayf al-Dawla. The years 944 to 967 are Sayf al-Dawla's reign, not her life. S186;S215 | |
| Open | Al-Kashi's sexagesimal digits for 2 pi | 6;16,59,28,1,34,51,46,14,50. popular repetition | Every source read gives only the phrase nine sexagesimal places. The quoted string checks out to 16 correct decimals, which corroborates it without sourcing it. S191;S201 | |
| Resolved | The ninth sexagesimal digit of al-Kashi's sin 1 degree | The string ends 16,26,17, the form circulated in most popular accounts and in this book's own earlier note. popular repetition (no source printing the digits could be fetched) | The string ends 16,26,18, as printed by Van Brummelen 2009. Aaboe 1951 prints a third string, ending 16,19,16. S184 | Resolved yes by computation. The expansion of 60 sin 1 degree to twelve sexagesimal places is 1;2,49,43,11,14,44,16,26,18,28,49,20, so truncation and rounding at nine places both give 26,18. The 26,17 reading is off by one unit in the ninth place and Aaboe's printed string is wrong from the eighth. What remains open is not which string is right but which string al-Kashi wrote. |
| Open | Whether al-Tusi's Treatise on the Quadrilateral is of 1260 or about 1250 | About 1250. popular repetition | 1260, per Debarnot, the peer-reviewed specialist, with a manuscript copy dated AH 658 on record. S181;S199 | |
| Open | When Aryabhata I died | 550 CE, printed conventionally. popular repetition | The 476 birth date follows from Aryabhatiya III.10; the conventional 550 is not attested in any source read. S122;S123 | |
| Open | Whether Bhaskara II states a rule amounting to a derivative of the sine | Yes, often with a further claim about an early form of Rolle's theorem. S145 | No translated verse from the Siddhanta Siromani or its Vasanabhasya stating the rule was obtained, and MacTutor's biography mentions neither claim. S143 | |
| Open | Where the manuscripts BnF Arabe 2558 and Arabe 2495 can be consulted | They are cited by shelfmark as though accessible. S231 | No digitized copies were located and a Gallica query returned nothing usable. popular repetition | |
| Open | Whether al-Biruni attributes the broken chord theorem to Archimedes | Yes, repeated widely. popular repetition | Neither al-Biruni's Maqalid nor his Book on Chords could be read in any edition; Debarnot describes the latter as concerning theorems related to a broken line inscribed in a circle, which is suggestive and not confirmation. S181 | |
| Open | The rights status of the best Plimpton 322 photograph | The CDLI image is freely reusable. popular repetition (assumed, never checked) | The CDLI About page carries no license or rights declaration at all. S011 | |
| Open | Which trigonometric treatises make up the Chongzhen lishu | The Dace, the Celiang quanyi and the Geyuan baxian biao, with authors and dates. popular repetition | None of these attributions could be confirmed; the reference site failed a TLS handshake and the encyclopedia article is too thin to list them. S272;S260 | |
| Open | Whether any chord or angle function appears in the Babylonian Astronomical Diaries | Absence is asserted confidently, though nothing is positively claimed. popular repetition | No such function appears in the sources read, which is an absence bounded by a handful of studies rather than by a survey of the corpus. S007;S015 | |
| Open | Whether the Codex Climaci Rescriptus coordinates are from Hipparchus's star catalog | Yes, at an epoch of about 129 BCE, following Gysembergh, Williams and Zingg. S065;S067 | No. Grasshoff and Hoffmann reject both the dating and the attribution on an astronomical re-analysis. S066 | |
| Open | Who coined the word exsecant | Nobody says. popular repetition | Cajori records the symbol exsec A with no attribution. S423 | |
| Open | When the word phasor was coined | Attributed loosely to Steinmetz. popular repetition | Steinmetz's 1893 paper has the concept and does not use the word; the coinage was not investigated. S377 | |
| Resolved | Whether the first use of the word cosecant is in the Opus palatinum of 1596 | The Opus palatinum contains the first use of cosecant, following W. W. Rouse Ball and D. E. Smith, and repeated in an auction catalog. popular repetition (Ball and Smith, restated in the Tomash sale catalog, S350) | The word is not there. Rheticus uses no modern function names at all, only the hypotenuse, base and perpendicular of triangles of three species. S307;S327 | Resolved no. Van Brummelen 2009 states at page 275 that Rheticus, following Copernicus, rejects the modern names outright, and he reported in 2014 that he had looked for cosecant in the Opus palatinum and could not find it. Roegel's dedicated reconstruction does not make the claim either. The cosecant question therefore moves to Bianchini's 1463 table and to the marteloio, not to Rheticus. |
| Open | Whether Cotes invented radian measure | Yes, asserted by MacTutor in a list with no elaboration and no citation. S367 | No primary evidence was found, so the claim is unsupported. S366 | |
| Open | Whether Cotes's complex-logarithm result appeared in 1714 or only in 1722 | 1714, in the Philosophical Transactions printing of the Logometria. popular repetition | 1722, where the passage was read at page 28 of the Harmonia Mensurarum; the Internet Archive item for the 1714 paper has no text layer. S366;S367 | |
| Resolved | The parameters of the Dayan li shadow table of about 725 CE | It is the first Chinese tangent table, with entry counts stated at second hand through Kotyk 2022 and Wu 2023. S253;S254 | The parameters could not be stated at all, because Cullen 1982 was blocked at three hosts. S256 | Resolved yes. Cullen 1982, now read in full, gives the table as 80,000 tan z against z at intervals of 1 tu, the Chinese degree of which there are 365.25 to a circle, built on third-order finite differences, with shadow lengths for a gnomon 80,000 units high. Cullen's own accuracy statement is that the tangents are accurate to within about one per cent up to around 50 tu and exceed ten per cent by 70 tu, and he judges the table an independent development based on Indian information about the use of sines. |
| Open | Why the 1542 De lateribus states a diameter of 2,000,000 while its own table uses radius 10,000,000 | The mismatch is not usually noticed. popular repetition | The text states diameter 2,000,000 twice while the appended Canon subtensarum is unambiguously to radius 10,000,000, checked here against three entries. It could be a compositor's error, a survival of an earlier draft or a misreading of the numerals. S302;S356 | |
| Open | Whether Durer's 1525 Underweysung der Messung contains a drawn sine curve | Yes, frequently asserted. popular repetition | Braunmuhl, who surveys the drawn trigonometric curves in detail, does not mention Durer. S426 | |
| Open | The absolute dates of the Egyptian reigns behind the Rhind papyrus | The British Museum gives a flat production date of 1550 BCE for EA10057 and EA10058. S008;S009 | Peet reasons from the papyrus's own regnal date to a window of 1788 to 1580 BCE for Apophis, and states the chronological uncertainty. S005 | |
| Open | Who first drew a cosine, cotangent or cosecant curve | The question is not usually asked. popular repetition | Braunmuhl names the sine, Roberval about 1634, the tangent, Gregory, and the secant, Wallis in 1670, and stops. The other three are undocumented. S426 | |
| Open | Who first treated the SSA ambiguous case | No claim is usually made. popular repetition | It is absent from Smith, Cajori, Miller and the standard reference treatment, which is a real gap in the literature and not only in this book. S421;S423;S427;S450 | |
| Open | The metric height of the Gaocheng gnomon | 12.28 m, per the Biographical Encyclopedia of Astronomers. S259 | 9.7468 m, from Li and Sun's conversion of 40 chi, which implies a chi of about 24.4 cm and matches the Yuan chi. S244 | |
| Open | Whether Girard used the abbreviation sec | Yes, alongside sin and tan, per D. E. Smith. S421 | No. MacTutor and Wikipedia credit him with sin, cos and tan and do not mention sec. S337;S343 | |
| Open | The life dates of Habash al-Hasib | About 796 to about 869. S219 | Springer's Biographical Encyclopedia gives roughly 796 to 894, and Debarnot dates his surviving zij to after 869, which cannot be reconciled with a death in 869. S187;S181 | |
| Open | Whether Hipparchus used a chord table with R = 3438 at 7.5 degree steps | Yes, stated flatly in most textbooks. popular repetition | Toomer proposed it and then cast doubt on his own argument in a 1984 footnote; Duke rescues it from one eclipse trio and elsewhere argues Hipparchus may not have used chords at all. The step size is inferred backward from Indian tables. S087;S063;S064 | |
| Open | The life dates of Hipparchus | About 190 to 120 BCE, printed as firm figures. popular repetition | No source read gives independent evidence. Toomer's index establishes only that the observations attributed to him run from 146 to 126 BCE. S061 | |
| Open | When Hypsicles wrote the Anaphorikos | About 175 BCE, per the Dictionary of Scientific Biography. S019 | About 150 BCE, per Heath and MacTutor. S068;S084 | |
| Open | Whether palm-leaf manuscript images of the Kerala works exist online | They are assumed to be available. popular repetition | None were located. Every scan indexed for the Indian tradition in this book is of a printed edition. S127 | |
| Open | The Devanagari and Malayalam forms of the Indian mathematicians' names | They are printed confidently in many places. popular repetition | Only Aryabhata and Madhava were sourced from a fetched document; the rest were left unfilled rather than supplied from memory. S127 | |
| Open | Whether Inman coined haversine in 1821 or 1835 | 1821, the usual date. S363 | The 1835 third edition, per the OED and Wikipedia; Cajori credits the function itself to Mendoza y Rios in 1801 and merely notes Inman's 1821 treatise. The word was verified here only in the 1858 edition. S441;S435 | |
| Open | Whether the Chinese Euclid was printed in 1606 or 1607 | 1606, per the Library of Congress catalog record. S261 | 1607, per the peer-reviewed study, which gives that year throughout. S260 | |
| Open | What sine table stood inside the Jiuzhi li of 718 | 24 entries, R = 3438, steps of 3 degrees 45 minutes, matching Aryabhata. popular repetition | That description appears in no source that could be opened; Kotyk discusses the Jiuzhi li at length without giving table parameters. S253;S255 | |
| Open | Whether the Kerala series reached Europe before their European discovery | They were transmitted, most often through Jesuit channels. S131;S135 | The priority is documented and undisputed, but the transmission claim has no manuscript evidence, and Almeida and Joseph concede that. Priority and influence are different claims. S127;S135 | |
| Open | Whether the word khorde occurs anywhere in Ptolemy | Yes, since the objects are called chords. popular repetition | A search returned zero occurrences across Heiberg's Syntaxis, books I to VI, which is strong evidence rather than proof because it relies on OCR of Greek type. S062;S089 | |
| Open | Whether Lambert proved pi irrational in 1761 and used a continued fraction for tan x | Yes to both, in the standard account. popular repetition | The LOCOMAT page says only that the result was proved in the 1760s and published in 1768, and says nothing about a continued fraction. The continued fraction is verified here as mathematically correct, which is not the same as verifying that Lambert used it. S392 | |
| Open | The date range of the Late Babylonian trapezoid tablets for Jupiter | About 400 to 50 BCE. S018 | 350 to 50 BCE, from Ossendrijver's own edition of the tablets; the wider figure is a range for the astronomical procedure texts as a genre. S017 | |
| Open | When Latin chorda first meant a line joining two points on a circle | It always meant that, and Ptolemy's chords are chords. popular repetition | Greek khorde means guts, gut string, lyre string and sausage, and LSJ records no geometrical sense; Ptolemy's phrase is the straight line in the circle. The geometric sense arrives somewhere in medieval Latin and could not be dated here. S089;S061 | |
| Open | Whether the 3072-gon value of pi is Liu Hui's own | Yes, attributed to him routinely. S247;S251 | The arithmetic is consistent with his algorithm, verified here, but that is not the same as the passage being authentic, and no scholarly treatment of the textual question was found. S247 | |
| Open | The numerator of Madhava's vibudhanetra value of pi | 2827433388233 over 900000000000. popular repetition | The Kriyakramakari confirms the rule, attributes it to Madhava and states an accuracy of 11 decimal places, but the number itself was not printed in the passage read. The quoted value is consistent with that accuracy, checked here to an error of 2.4e-12. S127;S142 | |
| Open | When Mary Edwards died | September 1815, per Wikipedia summarizing Croarken 2003. S391;S397 | c. 1750 to 1817, per the Cambridge Royal Greenwich Observatory catalog. S390 | |
| Open | The life dates of Menelaus of Alexandria | About 70 to about 140 CE. popular repetition | MacTutor gives c. 70 to c. 130; Rashed and Papadopoulos say only the first century of our era. The one hard datum is the pair of Rome observations of January 98 CE. S074;S075 | |
| Open | When MUL.APIN was composed | Star-list analyses suggest composition between 1300 and 1000 BCE. S007 | The only firm figure is the terminus ante quem of about 750 BCE from the Huzirina tablets; Hunger and Steele present the earlier figures as arguments about the observations, not as a date for the compendium. S007 | |
| Open | Which propositions constitute Napier's analogies | Four numbered results in the Descriptio. popular repetition | Half-sum and half-difference relations were confirmed in Book II chapter V, but four numbered propositions matching the standard four were not identified. S305 | |
| Open | How many pages of tables the Opus palatinum contains | The trigonometric tables run to 541 pages. Van Brummelen 2009, figure 5.25 caption at page 277 (not yet in the source register) | The table fills the last half of the book, 700 pages. Van Brummelen 2009, main text at page 280 (not yet in the source register) | |
| Open | Where the term midline for a sinusoid comes from | It is used as though long established. popular repetition | It is in neither volume of Cajori, neither volume of Smith, nor Braunmuhl, nor Miller; it appears to be a recent American schoolbook coinage, but this book has no evidence for that. S423;S424;S427 | |
| Open | Where the mnemonic SOH-CAH-TOA comes from | 1944, asserted by Dictionary.com with no citation. S447 (popular repetition: the entry gives no citation) | The mnemonic is absent from Cajori, Smith, Braunmuhl and Miller, and MathWorld gives no history at all. S423;S421;S425;S446 | |
| Open | Whether a peer-reviewed rebuttal of Mansfield and Wildberger 2017 exists | Scholars published rebuttals, in the plural. popular repetition | One blog post by Evelyn Lamb, plus Mansfield's own 2021 retreat. No formal published response was located. S014;S004 | |
| Resolved | The date of Pitiscus's death | 24 August 1613, per English Wikipedia. S339 | 2 July 1613, per the DSB, MacTutor and German Wikipedia; the rival date repeats his birth date of 24 August 1561, the signature of a transcription slip. S313;S334;S340 | Resolved yes within this book. Three independent records give 2 July, and the fourth reproduces the birth day and month exactly, which is what a transcription slip looks like. |
| Open | Whether the row-to-row angle step in Plimpton 322 is about one degree | The rows are separated by about 1 degree. popular repetition | The angle column computed here from Robson's figure 4 steps by 0.46 to 1.89 degrees, so any claim of a regular decrement is loose. S001 | |
| Open | What the broken left edge of Plimpton 322 carried | Two further columns holding the ratios that would make it a trigonometric table. popular repetition, from Friberg's proposal onward | Robson estimates room for about 5 cm, so roughly two columns; what stood in them is unknown and nobody has recovered them. S002;S001 | |
| Open | Whether the glue in the Plimpton 322 break is modern damage or an old repair | Mansfield reads it as a modern break, which supports a missing piece. S003;S004 | Robson reads it as Edgar Banks removing a foreign fragment he had stuck on. S002 | |
| Resolved | How many computers Prony employed, and how many were hairdressers | 80 to 100 computers, many of them unemployed hairdressers. popular repetition (a Dupin anecdote of 1825, amplified by Grattan-Guinness) | Probably never more than 20 or 25 computers, and perhaps only two or three hairdressers. S373 | Resolved yes, against the popular claim. Roegel worked from the payroll records and archive files of the Bureau du Cadastre itself and concludes the number was probably never greater than 20 or 25, with perhaps two or three hairdressers. The famous figures are folklore, and this row exists because the folklore version is still in print. |
| Open | Who invented prosthaphaeresis | Paul Wittich, following Longomontanus's Astronomia Danica of 1622. S426 | Johannes Werner, whose unpublished manuscript on triangles Christmann reports in 1611 as already containing the method explained by three figures. S425;S426 | |
| Open | When Richard of Wallingford wrote the Quadripartitum | About 1326. popular repetition | The DSB says only the early years in his second period at Oxford, roughly 1318 to 1327. S319 | |
| Open | The full bibliographic record of Radon's 1917 paper | Berichte, Leipzig, volume 69, pages 262 to 277. popular repetition | Only the title and year could be confirmed, from a university exhibition page; the Saxon Academy archive returned a proxy error. popular repetition | |
| Open | Whether Regiomontanus knew al-Tusi's Treatise on the Quadrilateral | He depended on it, asserted in one direction. popular repetition | He was independent of it, asserted in the other. No source read addresses the question, and the DSB traces the germ of his spherical cosine law to a note he added to Plato of Tivoli's Latin al-Battani. S310 | |
| Open | Whether the De triangulis imprint is 1533 or 1534 | 1533, and the DSB is explicit about 12 August 1533. S310 | 1534, on the OCR of the Seville copy, where M.D.XXXIII and M.D.XXXIIII cannot be told apart. S301 | |
| Open | When Rheticus died | 1576, per the e-rara catalog record. S327 | 4 December 1574, per the DSB, supported by Otho's documented appeal to the Elector of Saxony on 7 September 1576, which already treats him as holder of the manuscripts. S312 | |
| Open | Whether Seki Takakazu was born in 1640 or 1642 | March 1642, with MacTutor's own question mark attached. S263 | Around 1640, per Cooke. S265 | |
| Resolved | Whether Sin-bel-apli is the surveyor named on Si.427 | Sin-bel-apli is the surveyor, as reported across the 2021 press coverage. popular repetition (press coverage of the UNSW release, which does not itself give the name) | The name appears in nothing that could be read at the time, and the tablet may carry no surveyor name at all. S025;S004 | Resolved yes, and against the popular version. Mansfield 2020, reverse bottom lines 8 to 10, has the field described as of Sin-bel-apli, so he is the owner of the land. The surveyor is never named. Mansfield also states plainly that Si.427 is undated, so the Old Babylonian window is the period, not a date for the object. |
| Partly resolved | Which Latin translator first used sinus in the trigonometric sense | Robert of Chester used it first, either in the 1145 algebra or in his revision of al-Khwarizmi's tables. S325;S422 | Gherardo of Cremona used it first, or Plato of Tivoli in the 1116 al-Battani. S421;S423;S425 | Partly resolved. Van Brummelen 2009 backs Robert of Chester's revision of the tables, which makes a fifth authority for that candidate and leaves Robert the best supported. It is not closed: no source read presents manuscript evidence, Smith contradicts himself between his two volumes, and Braunmuhl rules out Plato of Tivoli only on the frequency of the word in that translation. |
| Open | When the Surya Siddhanta was composed | About 400 CE, following Sengupta and D. E. Smith. S121;S421 | About 499 CE, from Sudhakara Dvivedi citing Nityananda; Bentley's much later planetary-error dating is reproduced by Burgess without endorsement. S121 | |
| Open | Whether a scan of Takebe's Tetsujutsu sankei is available | It is assumed to be available. popular repetition | Waseda, the National Diet Library and general indexes returned no openly accessible facsimile, so the 41-digit pi claim has no primary check here. S267;S265 | |
| Open | When Nilakantha completed the Tantrasangraha | 1501, per the Wikipedia article. S141 | 1500, from the Kali-day chronogram 16,80,553 inside the text itself. S127 | |
| Open | How old the French name theoreme d'Al-Kashi is | It is treated as a historical attribution. popular repetition | French school textbooks start using it only in the 1990s; before that the names used were theoreme de Pythagore generalise and loi des cosinus. S440 | |
| Resolved | Whether the word radian stands in the 1867 first edition of Thomson and Tait | The sentence "For brevity we shall call this angle a radian" stands at page 31 of the 1867 first edition, which would predate the 1873 Belfast paper. S427;S381 | The sentence is verified only in the 1879 second edition at section 41. S434 | Resolved yes, in the negative. The 1867 first edition was read directly and contains the word radian zero times, against nine occurrences in 1879 and eight in 1890. The 1867 text has the concept and the value, 57 degrees 17 minutes 44.8 seconds, but no name for it. The 1873 Belfast priority stands. |
| Open | Whether Toomer's chord table paper is of 1973 or 1974 | 1973, in Toomer's own bibliography and in Duke. S061;S063 | March 1974, per the Crossref record for the issue. S088;S087 | |
| Resolved | When the word trigonometry first appeared in print | 1600, the first standalone edition with tables, or 1608 on the English Wikipedia article. S343;S306 | 1595, read directly from the section title page of Pitiscus in Scultetus's Sphaericorum libri tres in the e-rara scan. S303;S354 | Resolved yes within this book. The 1595 section title page was read in facsimile; 1600 is the first standalone edition and the first with tables, and the 1608 figure simply omits the 1600 Augsburg edition, a physical copy of which survives at the ETH. |
| Open | When the memorized unit circle chart became a standard classroom object | It is treated as timeless. popular repetition | The shift to circular functions on the unit circle is documented for 1959 to 1961 through the CEEB report and SMSG's Elementary Functions, but no source reached dates the chart itself. S439;S438 | |
| Open | Who should be credited with the trigonometric radius R = 1 | Euler fixed the unit radius in the Introductio of 1748, section 127, and this book's Chapter 7 currently says so. S361;S429 | Abu al-Wafa uses R = 1 in his Almagest and Abu Nasr Mansur sets R = 1 explicitly in the Table of Minutes, both in the tenth century, with al-Biruni's Qanun al-Masudi sine table equivalent to a modern one. Van Brummelen 2009, pages 145, 155 and 206 (not yet in the source register) | |
| Open | The life dates of Valentin Otho | c. 1545 to 1603, per Roegel. S327 | 1550 to 1605, per the e-rara catalog record; the DSB names him without dates. S307;S312 | |
| Resolved | The value of entry 22 in Varahamihira's sine table | The minutes figure is 119, with the seconds figure lost across a page break in the scan used here. S124 | The entry is 1,58;59, that is 118;59 with R = 120. Van Brummelen 2009, figure 3.2 (not yet in the source register) | Resolved yes. Van Brummelen's figure 3.2 supplies the reading, and recomputation supports it: 120 sin 82;30 degrees is 118.97338, so 118;59 is in error by +0.00995 while a 119;00 reading would be in error by +0.02662. The table's worst error across all 24 entries is 0.0149, so the 118;59 reading sits inside the table's own error band and 119;00 would be nearly double its worst. |
| Open | What happened in the vibrating-string controversy | A clean story about d'Alembert, Euler and Daniel Bernoulli from 1747 to 1753. popular repetition | Daniel Bernoulli's 1753 memoir was not read; only a catalog record was located and the fetch was blocked, so the volume and pages are unverified. popular repetition | |
| Open | What Mary Edwards and the Greenwich lady computers computed | Trigonometric work, described that way in general accounts. popular repetition | The Royal Museums Greenwich article says they were calculating the coordinates of the stars and reducing data for the catalog, which is certainly spherical-trigonometric work, but no source read says so in those words. S389;S390 | |
| Open | Whether Yixing was born in 673 or 683 | 683, still printed by Wu in 2023 and the commoner figure. S254 | 673, argued by Kotyk following Jinhua Chen's 2000 genealogical study. S253 | |
| Open | When the Zhoubi suanjing was compiled | Probably about 100 BCE, per Li and Sun. S244;S241 | In the early first century CE, on Christopher Cullen's conjecture. S254 | |
| Open | Whether Zu Chongzhi died in 500 or 501 | 501, per MacTutor. S252 | 500, given in much of the literature. popular repetition |
Appendix G
Stories that check out, and stories that do not
The 7 places in your course where this book has a myth to puncture, gathered so a teacher can find them in one pass.
Why a whole appendix for this. Trigonometry's history is told, in most classrooms, through about a dozen tidy anecdotes, and a striking number of them are wrong: not exaggerated, wrong. Pythagoras did not prove the theorem with his name on it. Rheticus did not print the word cosecant. Euler did not decide the radius should be 1. Each of those is repeated in print by people who had every reason to know better, which is the actual lesson: a claim can be respectable, widely printed, and false at the same time, and the way you find out is by going to the source.
| Course section | Title | What to puncture, and how | Chapters | Sources |
|---|---|---|---|---|
| 1.1 | The Pythagorean Theorem and Its Converse | The theorem with Pythagoras's name on it was in use long before he was born, and nobody can show that he proved it. | Ch. 8;Ch. 1 | S001, S494 |
| 1.R | Recap: Check Yourself | Here are the famous claims about Egyptian and Babylonian triangles that do not survive contact with the sources. | Ch. 1 | S001, S002, S003, S004 |
| 2.5 | The Reciprocal Trig Functions | Somebody once wrote I think in a history book, the hedge got dropped, and now textbooks credit the word cosecant to a man who never wrote it. | Ch. 8;Ch. 6 | S350, S427, S481 |
| 2.R | Recap: Check Yourself | Eleven popular claims about Islamic trigonometry, checked one at a time against what the specialists say. | Ch. 4 | S181, S182 |
| 3.R | Recap: Check Yourself | De Moivre never stated de Moivre's formula, and four more things everyone says about this era that the documents do not support. | Ch. 7 | S368, S369, S395 |
| 4.4 | The Pythagorean Identity | The claim that Varahamihira first stated the Pythagorean identity comes from misreading one carefully hedged sentence, and today we read the sentence. | Ch. 8;Ch. 3 | S124, S481 |
| 4.R | Recap: Check Yourself | Five widely repeated claims about Indian trigonometry, including a famous sine formula that returns 16 when you feed it 90 degrees. | Ch. 3 | S122, S128 |
Appendix H
Image catalog and rights
67 images, with the repository that holds each one and the rights wording recorded verbatim from that repository. The rights position was checked at the source for 37 of the 67; for the other 30 the source states nothing to check, or could not be read, and each row says which.
Verbatim matters here. An archive's own wording is the only thing that governs what may be reproduced, so it is copied exactly, including its spelling and its capitalization, rather than paraphrased into house style. Where an image could not be cleared, this book prints a panel describing it and a link to where you can see it, instead of the image. A described image you can go and look at is worth more than a reproduced image that should not have been.
| Image | Work | Repository | Rights status | Rights wording, verbatim |
|---|---|---|---|---|
| Complete scan of the 1515 Latin Almagest, Deutsches Museum copy | Almagestum Cl. Ptolemei | Internet Archive | public_domain the item metadata states Public Domain Mark 1.0 | Public Domain Mark 1.0 recorded in the item metadata. Usable, and it is the cleanest route to a page of the 1515 edition. |
| Gerard of Cremona's Latin Almagest, Venice 1515, folio images from the Ptolemaeus Arabus et Latinus viewer | Almagestum Cl. Ptolemei | Ptolemaeus Arabus et Latinus, Bayerische Akademie der Wissenschaften | unknown nothing stated on the folio viewer | BLOCKER until asked. The folio viewer carries no rights statement. The Internet Archive copy of the same edition does carry one, so prefer that file and cite this viewer only for the folio reference. |
| Clark's translation of the Aryabhatiya, the stanza giving the 24 sine differences | The Aryabhatiya of Aryabhata | Internet Archive, freeindological collection | unknown no license field is recorded in the item metadata | BLOCKER. No license field in the item metadata, and a 1930 University of Chicago Press book may well still be in copyright. Use the Shukla and Sarma scan, which is CC0, or quote rather than reproduce. |
| Shukla and Sarma's edition of the Aryabhatiya, with the Sanskrit text, variant readings and the editors' table of Rsine differences | Aryabhatiya of Aryabhata, Vol. 1 | Internet Archive, Digital Library of India | cc0 the license field in the item metadata was read | CC0 1.0 as recorded in the item metadata. The cleanest Indian source for reuse in a free book. |
| James Thomson's examination paper, Queen's College Belfast, 5 June 1873, the first appearance of the word radian in print | Examination paper, Queen's College Belfast | Holding institution not identified | unknown no copy located and no statement read | WANTED, and the most valuable missing image in the book. No copy was located during the research, so nothing about its rights is known. Queen's University Belfast archives is the place to ask. |
| Braunmuhl, Vorlesungen uber Geschichte der Trigonometrie, volume 2, 1903, the survey of the drawn trigonometric curves | Vorlesungen uber Geschichte der Trigonometrie, vol. 2 | Internet Archive, Google digitized | public_domain the Internet Archive item records public domain | Public domain per the Internet Archive item. Fraktur type, so OCR is unreliable and page images are the honest way to quote it. |
| Cajori, A History of Mathematical Notations, volume 2, 1929, section 527, which prints exsec A with no attribution | A History of Mathematical Notations, vol. 2 | Internet Archive | public_domain the item is recorded as public domain in the United States | Recorded as public domain in the United States by date of publication. A free book distributed worldwide should still check status outside the United States before reproducing a page. |
| Codex Climaci Rescriptus, color and multispectral images of folios 47r to 64v, the undertext carrying Hipparchus's star coordinates | Codex Climaci Rescriptus | Museum of the Bible | cc-by-sa-4.0 the release page statement was read and quoted | Quoted from the repository: These works are licensed under a Creative Commons Attribution-ShareAlike 4.0 International License. Usable with attribution. Share-alike binds derivatives, so keep any enhanced version of the image under the same license. |
| Colebrooke's translation of Brahmagupta and Bhaskara, with Bhaskara II's rebuttal at Lilavati 167 to 175 | Algebra, with Arithmetic and Mensuration, from the Sanscrit | Internet Archive | unknown no license field is recorded | No license field recorded. The 1817 printing is old, but the repository says nothing, so the cell stays unknown. |
| Cotes, Harmonia mensurarum, 1722, page 28, the complex logarithm passage | Harmonia mensurarum | Bayerische Staatsbibliothek, Munich, via Internet Archive | public_domain the Internet Archive item records public domain | Public domain per the Internet Archive item. Whether the same passage is in the 1714 Logometria printing is an open question in chapter 7, so caption the edition exactly. |
| First page of Datta and Singh's Hindu Trigonometry, the source of the jya, koti-jya and utkrama-jya definitions | Hindu trigonometry, Indian Journal of History of Science 18(1) | Indian National Science Academy | unknown the PDF is free to read but no license is stated | BLOCKER for reproduction. Free to read on the INSA site with no explicit license, and a 1983 article is squarely in copyright. Quote with citation instead. The local page images were made for OCR, not for print. |
| The 70 page images of Datta and Singh's Hindu Trigonometry captured for local OCR, pages dt-01 to dt-70 | Hindu trigonometry, Indian Journal of History of Science 18(1) | Indian National Science Academy | unknown no license stated | BLOCKER for reproduction, working copies only. Same status as the first page. The PDF has no text layer, which is why the pages were rasterized; any quotation should be checked against the page image before print. |
| Copernicus and Rheticus, De lateribus et angulis triangulorum, Wittenberg 1542, the first printed appearance of the trigonometric chapters of De revolutionibus | De lateribus et angulis triangulorum | Biblioteka Slaska, Katowice, via the Silesian Digital Library | public_domain the catalog record states Domena publiczna | The record states Domena publiczna, public domain. Usable. The title page is image 5 of 68 in the PDF, which is the copy already on disk. |
| Euler, Introductio in analysin infinitorum, volume 1, 1748, section 126 and section 127, where sin. z becomes a function of a bare number | Introductio in analysin infinitorum, vol. 1 (E101) | Euler Archive, University of the Pacific Scholarly Commons | unknown nothing stated on the record page | BLOCKER until asked. The Euler Archive is a scholarly open repository but states no license on the record page. The 1748 imprint is old; the scan is somebody's work, and this book does not assume public domain from age. |
| Fincke, Geometriae rotundi libri XIIII, Basel 1583, definitions 21, 22 and 27, where tangens and secans are coined, and the six contractions in Liber XIV | Geometriae rotundi libri XIIII | Det Kongelige Bibliotek, Copenhagen, via Internet Archive | restricted the courtesy line and the ProQuest notice on the scan were read | BLOCKER. Images are reproduced by courtesy of the Royal Library, Copenhagen, and the scan carries Early European Books, Copyright (c) 2009 ProQuest LLC notices on interleaved pages. A courtesy line is not a license, and the ProQuest claim covers the scan. Ask the Royal Library, or find another copy. |
| Fourier, Theorie analytique de la chaleur, 1822, the statement that any function can be written as a sum of sines | Theorie analytique de la chaleur | University of Toronto Library, via Internet Archive | public_domain the Internet Archive item records public domain | Public domain per the Internet Archive item, 1822 imprint. A second scan of the same book from Gallica is mirrored at archive.org/details/bnf-bpt6k1045508v. |
| Gunter, Canon triangulorum, London 1620, the introduction where co.sinus and cotangens first appear | Canon triangulorum | Internet Archive, from Early English Books microfilm | public_domain the Internet Archive item records public domain | Public domain per the Internet Archive item. The words are in the introduction, not in the table headings, so pick the page with care. |
| Heath's Works of Archimedes, the hexagon to 96-gon chain that is a chord computation | The Works of Archimedes | Internet Archive | public_domain the item metadata records public domain | Public domain as recorded by Internet Archive. |
| Heath's Greek text and translation of Aristarchus, On Sizes and Distances, the small-angle bounds | Aristarchus of Samos, the ancient Copernicus | Internet Archive | public_domain the item metadata records public domain | Public domain as recorded by Internet Archive. |
| Heiberg's Greek critical edition of the Syntaxis, Pars I, the printed Greek of the chord table | Claudii Ptolemaei opera quae exstant omnia, Syntaxis mathematica Pars I | Internet Archive, Google digitized | public_domain the item metadata records public domain | Public domain as recorded by Internet Archive. Watch the Google watermark page, which some libraries ask not be cropped out. |
| Helmholtz, On the Sensations of Tone, third English edition, the law behind beats | On the Sensations of Tone | Internet Archive | public_domain the Internet Archive item records public domain | Public domain per the Internet Archive item. |
| Manitius's Greek and German edition of Hipparchus, Commentary on the Phaenomena | Hipparchi in Arati et Eudoxi Phaenomena commentariorum libri tres | Internet Archive, University of Illinois copy | public_domain the item metadata records public domain | Public domain as recorded by Internet Archive. |
| Ibn Yunus's Hakimi Zij, chapter 2, folios 44r to 44v, the account of al-Mamun's degree measurement | Hakimi Zij | Bibliotheque nationale de France | unknown no scan was located | WANTED, not located. Shelfmark from the secondary source only. |
| The page of Inman's Navigation and Nautical Astronomy carrying log. haversines, the earliest printing of the word this book verified | Navigation and Nautical Astronomy for the Use of British Seamen | Internet Archive | public_domain the Internet Archive item records public domain | Public domain per the Internet Archive item. Caption honestly: this is the 1858 edition, not the 1821 or 1835 edition on which the coinage claim turns, and neither of those could be located. |
| Ricci and Xu Guangqi's Chinese Euclid, Jihe yuanben, books 1 to 6, four volumes | Jihe yuanben | Library of Congress, World Digital Library collection | no_known_restrictions the Library of Congress rights advisory was read | The Library of Congress states these World Digital Library materials are free to use and reuse with no known copyright restrictions. The safest Chinese image in the set. |
| Al-Khalili's prayer tables for Damascus, folios 10v to 11r, reproduced as plate 4.17 in King's chapter | Prayer tables for Damascus | Bibliotheque nationale de France | unknown no scan was located and no statement read | WANTED, not located. The shelfmark comes from King's chapter; a Gallica scan could not be found. Rights unknown because there is nothing to read yet. |
| Sengupta's translation of the Khandakhadyaka, with Brahmagupta's second-order interpolation rule | The Khandakhadyaka | Digital Library of India via Internet Archive | unknown no license field is recorded | BLOCKER. No license field recorded and a 1934 imprint. Quote rather than reproduce until the status is settled. |
| Maskelyne's letters to Henry Andrews, the paper trail of the Nautical Almanac computers | Royal Greenwich Observatory Archives, papers of Nevil Maskelyne | Cambridge University Library | unknown nothing is declared on the page | BLOCKER. Nothing declared. The page offers a Request Rights link and a IIIF manifest. Do not reproduce without checking with Cambridge Imaging Services. |
| Menelaus, Spherics, in al-Harawi's Arabic recension, copied at Damascus in 1153, with 118 spherical figures | Kitab Manalawus fi al-ashkal al-kurriyah | British Library, via the Qatar Digital Library | public_domain the QDL item page states Public Domain | Stated as Public Domain under Creative Commons on the QDL item page. Usable. Note that other QDL item pages returned HTTP 403 to automated requests, so check each one by hand. |
| Egyptian merkhet, a bronze sighting instrument for aligning to a star | Merkhet | Science Museum Group, London | cc-by-nc-sa-4.0 the Science Museum Group terms page was read | Images are licensed CC BY-NC-SA 4.0, catalog data CC0 and descriptions CC BY 4.0. Non-commercial suits a free student book, but the share-alike term binds any derivative, so do not composite it into a figure the book licenses differently. |
| MUL.APIN tablet 1, the Babylonian astronomical compendium that fixes 30 US to a beru | MUL.APIN | British Museum | unknown the object page was read but no license statement was | BLOCKER until checked. Object page read: 86 by 60 by 16 mm, 1000 to 500 BCE, excavated in southern Iraq, acquired 1899. |
| Nallino's edition of al-Battani's Opus astronomicum, Arabic text with Latin translation, the source for the shadow tables | Al-Battani sive Albatenii Opus astronomicum | Internet Archive | public_domain the item metadata gives a Public Domain Mark license URL | The combined upload records licenseurl http://creativecommons.org/publicdomain/mark/1.0/. Other uploads of the same work record only the pre-1929 date, which is an inference rather than a statement; prefer this one. |
| Napier, Mirifici logarithmorum canonis descriptio, Edinburgh 1614, the logarithms of sines | Mirifici logarithmorum canonis descriptio | Smithsonian Libraries, via Internet Archive | public_domain the Internet Archive item records public domain | Public domain per the Internet Archive item. |
| The four Nature letters of 1910 in which Muir and Thomson each explain how the word radian was coined | Nature, vol. 83 | Internet Archive scans of Nature volume 83 | unknown the article bodies are behind a paywall at the publisher and the scans carry no statement that was read | BLOCKER for reproduction. Full text of all four letters is on disk from the block 9 acquisition, so the content is settled; the page images are a separate question and their rights were not read. |
| Rheticus and Otho, Opus palatinum de triangulis, Neustadt 1596, a page of the six-function tables | Opus palatinum de triangulis | ETH-Bibliothek Zurich, via e-rara | unknown the IIIF manifest was read and contains no license field | BLOCKER. The IIIF manifest carries no license field, so e-rara's default terms apply and were not read. This is the page that shows Rheticus naming no function at all, which is the evidence for the cosecant correction, so it is worth clearing. |
| The manuscript of the main table of the Opus palatinum | Opus palatinum main table manuscript | British Library | restricted the rights line is quoted at second hand from Roegel | BLOCKER. Roegel reproduces it by permission of the British Library, which is a permission granted to him and not to this book. Apply separately. |
| A second copy of the Opus palatinum, cited by Roegel | Opus palatinum de triangulis | Staats- und Universitatsbibliothek Dresden | unknown the site is behind a bot check and was not opened | Not verified: the page could not be opened. Recorded so the alternative copy is not lost. |
| Thibaut and Dvivedi's Pancasiddhantika, chapter IV with Varahamihira's sine table | The Panchasiddhantika | Digital Library of India via Internet Archive | unknown no license field is recorded | No license field recorded. Do not assume public domain from the 1889 imprint: record the absence and ask, or use the alternate Google scan whose terms are stated. |
| Pitiscus, Trigonometriae sive de dimensione triangulorum libri quinque, Augsburg 1600 | Trigonometriae libri quinque | ETH-Bibliothek Zurich, via e-rara | public_domain the IIIF manifest states CC Public Domain Mark 1.0 | CC Public Domain Mark 1.0 stated in the IIIF manifest. Usable, and the obvious substitute if the 1595 section title page cannot be cleared. |
| The section title page of Pitiscus's Trigonometria, the first printing of the word trigonometry | Trigonometria, appended to Scultetus, Sphaericorum libri tres | e-rara, holding institution not displayed | unknown the catalog page sits behind a browser check and could not be read | BLOCKER, and this is the single most quotable image in chapter 6. e-rara items are generally CC Public Domain Mark 1.0 but this record could not be opened, so the license, the holding institution and the shelfmark are all unread. Worth one manual visit. |
| Composite photograph of Plimpton 322: obverse, all four edges and reverse. The reverse carries the ink stamp PLIMPTON LIBRARY and a penciled 322 | Plimpton 322 | Cuneiform Digital Library Initiative, photographed for CDLI; tablet held by the Rare Book and Manuscript Library, Columbia University | unknown the About page was read and carries no rights statement to verify | BLOCKER. No license declaration of any kind appears on the CDLI About page. Nothing states the terms for reuse, so a free student book cannot rely on it. Written permission from CDLI and from Columbia is needed before print. |
| Plimpton 322, obverse only, the four ruled columns of numbers | Plimpton 322 | Rare Book and Manuscript Library, Columbia University Libraries | unknown nothing stated on the item page | BLOCKER. The exhibition item page states no rights at all. Ask Columbia RBML directly, since this is the owning institution and its answer settles the CDLI photograph too. |
| Plimpton 322 obverse, exhibition image | Plimpton 322 | Institute for the Study of the Ancient World, New York University, lent by Columbia University | unknown nothing stated | BLOCKER. No statement found. Third image of the same object with no rights declaration, which makes the Columbia permission the single thing worth chasing. |
| Plimpton 322 reverse, exhibition image | Plimpton 322 | Institute for the Study of the Ancient World, New York University, lent by Columbia University | unknown nothing stated | BLOCKER. No statement found. |
| Qatar Digital Library holdings of Islamic astronomy manuscripts, including al-Tusi and Ulugh Beg under IO Islamic 1148 and al-Biruni's Kitab al-tafhim under Or 8349 | Various Islamic astronomical manuscripts | British Library and Qatar Foundation Partnership | unknown the finding aid was read but the item pages returned HTTP 403 | QDL content is generally openly licensed and the Menelaus item states Public Domain, but the individual item pages here could not be opened, so the license for each is unread. Check per item. |
| Regiomontanus, De triangulis omnimodis, Nuremberg 1533, title page and the Book I definitions of the sinus rectus | De triangulis omnimodis libri quinque | Internet Archive, from the Biblioteca de la Universidad de Sevilla | public_domain the Internet Archive item records public domain | Public domain per the Internet Archive item. Note the title page OCR is ambiguous between 1533 and 1534, so caption the imprint from the DSB rather than from the page image. |
| Regiomontanus, Tabulae directionum, Augsburg 1490, the Tabella sinus recti | Tabulae directionum et profectionum | Keio University Libraries | all_rights_reserved the notice on the collection page was read | BLOCKER. Quoted from the repository: COPYRIGHT (C) KEIO UNIVERSITY ALL RIGHTS RESERVED. Do not use without written permission. |
| Rhind Mathematical Papyrus, first section, which carries the pyramid problems 56 to 60 | Rhind Mathematical Papyrus | British Museum | unknown the object page was read but the image license itself was not | BLOCKER until checked. The object page was fetched and read; the museum's collection online terms govern the image and were not read at the level of a license. No image file URL was captured. |
| Rhind Mathematical Papyrus, second section | Rhind Mathematical Papyrus | British Museum | unknown the object page was read but the image license itself was not | BLOCKER until checked, as for EA10057. |
| The printed general title page of Scultetus, Sphaericorum libri tres, the volume Pitiscus's Trigonometria is bound into | Sphaericorum libri tres | Google Books, holding library not confirmed | unknown only the full view flag was seen, no license statement | BLOCKER. Google Books full view is an access setting, not a license, and the holding library is not identified. This is the only route seen to the printed general title page. |
| Sedillot's translation of the prolegomena to Ulugh Beg's tables | Prolegomenes des tables astronomiques d'Oloug-Beg | Internet Archive, Google digitized from the Biblioteca Universitaria di Bologna | public_domain the item metadata gives a Public Domain Mark license URL | licenseurl http://creativecommons.org/publicdomain/mark/1.0/ recorded in the item metadata. |
| Egyptian altitude sundial, or shadow clock, in pine | Egyptian shadow clock | Science Museum Group, London | cc-by-nc-sa-4.0 the Science Museum Group terms page was read | CC BY-NC-SA 4.0 for images. Same share-alike caution as the merkhet. |
| Daniel Mansfield holding Si.427, a press photograph rather than a study image | Si.427 | UNSW Sydney press office; tablet held by Istanbul Arkeoloji Muzeleri | unknown press page carries no license | BLOCKER for reuse. A university press image with no stated license. Useful only if UNSW grants permission, and it is a context photograph, not evidence. |
| Si.427 obverse with the museum scale bar, the Old Babylonian field plan with perpendicular sides | Si.427 | Istanbul Arkeoloji Muzeleri; this reproduction published in Mansfield 2021, Foundations of Science | cc-by-4.0 the article carries an open access CC BY 4.0 statement, read 14 August 2026 | The article is copyright The Author(s) 2021, open access under CC BY 4.0, and the photograph is credited to the museum. The museum's own terms for the photograph were not read, so credit both. Caption must not date the tablet: Si.427 is undated, and Sin-bel-apli is the landowner, not the surveyor. |
| Smith, History of Mathematics volume 2, 1925, the page that turns Ball's I think into a flat statement about the cosecant | History of Mathematics, vol. 2 | Internet Archive | public_domain the item is recorded as public domain in the United States | Recorded as public domain in the United States. This page is the evidence for the cosecant myth-check, so reproducing it is worth more than describing it. |
| SMSG, Elementary Functions, Student's Text, Unit 21, 1961, the moment the American curriculum moves to circular functions on the unit circle | Elementary Functions, Student's Text, Unit 21 | Internet Archive, from ERIC microfiche ED135629 | no_known_restrictions the ERIC and National Science Foundation sponsorship statement was read | Sponsored by the National Science Foundation and distributed as an ERIC document, which is freely distributable. The closest thing to a dated origin the book has for the memorized unit circle chart, and the chart itself is still undated. |
| Steinmetz's 1893 paper at page 33 of the Proceedings of the International Electrical Congress, where a sinusoid becomes a constant complex number | Proceedings of the International Electrical Congress, Chicago 1893 | Internet Archive, Google digitized | public_domain the Internet Archive item records public domain | Public domain per the Internet Archive item. |
| Burgess's translation of the Surya Siddhanta, the page carrying the 24 tabular sines and the versed sines | Translation of the Surya-Siddhanta | Internet Archive, uploader supplied copy | cc-by-nc-nd-3.0 the license field in the item metadata was read | CC BY-NC-ND 3.0 as recorded in the item metadata. The 1860 text underneath is old enough to be out of copyright in most places, but the tag is the uploader's own and ND forbids derivatives, so do not crop or recolor. Prefer a different scan if the book needs to edit the image. |
| Page 31 of the 1867 first edition, the page where Miller's source claims the word radian appears. The full text search shows it does not | A Treatise on Natural Philosophy, vol. 1, first edition | Internet Archive | unknown no image of the page was obtained, only the full text | WANTED. A photograph of this one page settles a priority claim that this book corrects, so it is worth having even though the text search already answers the question. No item page URL was recorded and no rights statement read. |
| The page of the 1879 Thomson and Tait carrying section 41, For brevity we shall call this angle a radian | Treatise on Natural Philosophy, vol. 1 part 1, new edition | Internet Archive, Google digitized from the University of Michigan copy | public_domain the Internet Archive item records public domain | Public domain per the Internet Archive item. This is the earliest verified printing of the sentence in a book, and the full text on disk shows nine occurrences of radian here against zero in 1867. |
| Al-Tusi's recension of the Almagest, a fifteenth century copy | Tahrir al-Majisti | Bibliotheque nationale de France, Departement des Manuscrits | public_domain the Gallica OAI record states domaine public | The Gallica OAI record carries domaine public. Gallica asks for a credit line naming the BnF and the shelfmark; include it. |
| Ulugh Beg's Zij-i jadid-i Sultani in Persian, the sine table computed to great precision | Zij-i jadid-i Sultani | Bodleian Libraries, University of Oxford | unknown nothing stated on the record fetched | BLOCKER until checked. No rights statement on the record fetched. Digital Bodleian publishes terms elsewhere on its site; read them before using an image. |
| The oldest fully digitized Greek witness of the Almagest, 582 openings, including the chord table of Book I.11 | Ptolemy, Almagest, Greek manuscript | Biblioteca Apostolica Vaticana | restricted the Vatican Library statement on the viewer was read | Quoted from the repository: Free use of this image is only for personal use or study purposes. Publication rights on request. BLOCKER for a published book, even a free one, because publication is exactly what is excluded. Apply for publication rights or use a different witness. |
| A second copy of Viete's Canon mathematicus | Canon mathematicus seu ad triangula | Bibliotheque nationale de France | unknown the record was not opened | Not verified. Gallica usually states domaine public, which would make this the better copy of the two; one visit would settle it. |
| Viete, Canon mathematicus seu ad triangula, Paris 1579 | Canon mathematicus seu ad triangula | ETH-Bibliothek Zurich, via e-rara | unknown the IIIF manifest was read and contains no license field | BLOCKER. No license field in the IIIF manifest. |
| The gougu, or hypotenuse diagram, pages of the 1603 Ming printing of the Zhoubi suanjing | Zhoubi suanjing, Ming printing | Smith and Plimpton Collections, Columbia University, presented through MAA Convergence | unknown no explicit rights statement on the page | BLOCKER. No explicit statement on the MAA page. Confirm with Columbia RBML, which is the same institution that holds Plimpton 322, so one letter can cover both. |
| Zhoubi suanjing, juan xia page 2, the page carrying the gnomon rule | Zhoubi suanjing | Chinese Text Project digital library, from the Sibu congkan photo-reprint of a Ming edition | unknown the site terms were read and they state no license for the images | BLOCKER. Quoted from the site: content should be cited with a link back to the site, and bulk automated download is prohibited. No license is granted for republication, so ask before printing a page image. The scanned copy carries the collection stamps of the Lu Muzhai collection and Tsinghua University Library. |
H.2 Portraits: what each one is, and its rights
47 portraits ship in this book, every one with a license read at the repository rather than assumed. 9 of them are marked Not from life: no likeness made while the person was alive is known to survive, so what you are looking at is a later artist’s invention. The caption says so at every place the portrait appears, not only here.
The other 138 people in the portrait register have no picture here, because no freely licensed likeness was found or none is known to exist. Every one of them has an empty frame in Appendix A saying which of those two it is, and an invitation to send one if you know of it.
| ID | Person | What it is | Creator | Year | License | Credit and file page | Where it appears |
|---|---|---|---|---|---|---|---|
| PTR-002 | Abraham de Moivre | Abraham de Moivre, 1667 to 1754. Jacques-Antoine Dassier, 1741. CC0, via Wikimedia Commons. | Jacques-Antoine Dassier | 1741 | CC0 | Jacques-Antoine Dassier, 1741. CC0, via Wikimedia Commons. https://commons.wikimedia.org/wiki/File:Abraham_de_Moivre_MET_DP-1424-031.jpg File page | Appendix A, chapter cast |
| PTR-016 | Alice Everett | Alice Everett, 1865 to 1949. Morgan and Kidd, 1893. Public domain, via Wikimedia Commons. | Morgan and Kidd | 1893 | Public domain | Morgan and Kidd, 1893. Public domain, via Wikimedia Commons. https://commons.wikimedia.org/wiki/File:Alice_Everett.jpg File page | Appendix A, chapter cast |
| PTR-017 | Amenemhat III | Amenemhat III, 1849 BCE to 1801 BCE. ArchaiOptix, 2014. CC BY-SA 4.0, via Wikimedia Commons. | ArchaiOptix | 2014 | CC BY-SA 4.0 | ArchaiOptix, 2014. CC BY-SA 4.0, via Wikimedia Commons. https://commons.wikimedia.org/wiki/File:Head_of_portrait_statue_of_pharaoh_Amenemhat_III_wearing_the_crown_of_Upper_Egypt_01.jpg File page | Appendix A, chapter cast |
| PTR-027 | Al-Biruni Not from life | Al-Biruni, 973 to 1048. Michel Bakni, 2020. CC BY-SA 4.0, via Wikimedia Commons. No portrait made from life survives. This is a later image, not a record of the face. | Michel Bakni | 2020 | CC BY-SA 4.0 | Michel Bakni, 2020. CC BY-SA 4.0, via Wikimedia Commons. https://commons.wikimedia.org/wiki/File:Al-Biruni_Portrait.jpg File page | Appendix A, chapter cast |
| PTR-029 | Brook Taylor | Brook Taylor, 1685 to 1731. An unidentified artist. CC BY 4.0, via Wikimedia Commons. | Unidentified artist | not stated | CC BY 4.0 | An unidentified artist. CC BY 4.0, via Wikimedia Commons. https://commons.wikimedia.org/wiki/File:Brook_Taylor._Line_engraving_after_R._Earlom._Wellcome_V0005740.jpg File page | Appendix A, chapter cast |
| PTR-031 | Carl Friedrich Gauss | Carl Friedrich Gauss, 1777 to 1855. Christian Albrecht Jensen, 1840. Public domain, via Wikimedia Commons. | Christian Albrecht Jensen | 1840 | Public domain | Christian Albrecht Jensen, 1840. Public domain, via Wikimedia Commons. https://commons.wikimedia.org/wiki/File:Carl_Friedrich_Gauss_1840_by_Jensen.jpg File page | Appendix A, chapter cast |
| PTR-033 | Charles Proteus Steinmetz | Charles Proteus Steinmetz, 1865 to 1923. An unidentified artist, c. 1900. Public domain, via Wikimedia Commons. | Unidentified artist | 1900 | Public domain | An unidentified artist, c. 1900. Public domain, via Wikimedia Commons. https://commons.wikimedia.org/wiki/File:ETH-BIB-Steinmetz_,_Charles_Proteus_(1865-1923)-Portrait-Portr_03022.tif File page | Appendix A, chapter cast |
| PTR-036 | Christopher Clavius | Christopher Clavius, 1538 to 1612. Francesco Villamena, 1606. CC0, via Wikimedia Commons. | Francesco Villamena | 1606 | CC0 | Francesco Villamena, 1606. CC0, via Wikimedia Commons. https://commons.wikimedia.org/wiki/File:Portrait_of_Cardinal_Christopher_Clavius_MET_DP218205.jpg File page | Appendix A, chapter cast |
| PTR-037 | Claudius Ptolemy Not from life | Claudius Ptolemy, 100 to 175. Miscellaneous Items in High Demand, PPOC, Library of Congress, 1886. Public domain, via Wikimedia Commons. No portrait made from life survives. This is a later image, not a record of the face. | Miscellaneous Items in High Demand, PPOC, Library of Congress | 1886 | Public domain | Miscellaneous Items in High Demand, PPOC, Library of Congress, 1886. Public domain, via Wikimedia Commons. https://commons.wikimedia.org/wiki/File:Claudius_Ptolemy,_half-length_portrait,_facing_right_LCCN93515230.jpg File page | Appendix A, chapter cast |
| PTR-038 | Colin Maclaurin | Colin Maclaurin, 1698 to 1746. S. Freeman, 1801. Public domain, via Wikimedia Commons. | S. Freeman | 1801 | Public domain | S. Freeman, 1801. Public domain, via Wikimedia Commons. https://commons.wikimedia.org/wiki/File:Maclaurin_Colin_engraving.jpg File page | Appendix A, chapter cast |
| PTR-040 | Daniel Bernoulli | Daniel Bernoulli, 1700 to 1782. An unidentified artist, c. 1750. Public domain, via Wikimedia Commons. | Unidentified artist | 1750 | Public domain | An unidentified artist, c. 1750. Public domain, via Wikimedia Commons. https://commons.wikimedia.org/wiki/File:ETH-BIB-Bernoulli,_Daniel_(1700-1782)-Portrait-Portr_10971.jpg File page | Appendix A, chapter cast |
| PTR-055 | François Thureau-Dangin | François Thureau-Dangin, 1872 to 1944. Unidentified photographer, 20th century. Public domain, via Wikimedia Commons. | Unidentified photographer | not stated | Public domain | Unidentified photographer, 20th century. Public domain, via Wikimedia Commons. https://commons.wikimedia.org/wiki/File:Thureau-dangin.jpg File page | Appendix A, chapter cast |
| PTR-056 | François Viète | François Viète, 1540 to 1603. Charles Méryon. Public domain, via Wikimedia Commons. | Charles Méryon | not stated | Public domain | Charles Méryon. Public domain, via Wikimedia Commons. https://commons.wikimedia.org/wiki/File:Meryon_-_Portrait_of_Fran%C3%A7ois_Vi%C3%A8te,_1861,_1938.1666.jpg File page | Appendix A, chapter cast |
| PTR-058 | Gaspard de Prony | Gaspard de Prony, 1755 to 1839. David d'Angers, Pierre-Jean (Angers, 12-03-1788 - Paris, 05-01-1856), sculpteur, 1833. CC0, via Wikimedia Commons. | David d'Angers, Pierre-Jean (Angers, 12-03-1788 - Paris, 05-01-1856), sculpteur | 1833 | CC0 | David d'Angers, Pierre-Jean (Angers, 12-03-1788 - Paris, 05-01-1856), sculpteur, 1833. CC0, via Wikimedia Commons. https://commons.wikimedia.org/wiki/File:Portrait_de_Gaspard-Clair-Fran%C3%A7ois-Marie_Riche,_baron_de_Prony_(1755-1839),_ing%C3%A9nieur,_S446.jpg File page | Appendix A, chapter cast |
| PTR-066 | Gerardus Mercator | Gerardus Mercator, 1512 to 1594. Hendrik Goltzius / Frans Hogenberg. CC0, via Wikimedia Commons. | Hendrik Goltzius / Frans Hogenberg | not stated | CC0 | Hendrik Goltzius / Frans Hogenberg. CC0, via Wikimedia Commons. https://commons.wikimedia.org/wiki/File:Musea_Brugge,_HUB,_2014_GRO1150_III.jpg File page | Appendix A, chapter cast |
| PTR-070 | Guo Shoujing Not from life | Guo Shoujing, 1231 to 1316. Wcr1993, 2019. CC BY-SA 4.0, via Wikimedia Commons. No portrait made from life survives. This is a later image, not a record of the face. | Wcr1993 | 2019 | CC BY-SA 4.0 | Wcr1993, 2019. CC BY-SA 4.0, via Wikimedia Commons. https://commons.wikimedia.org/wiki/File:Xingtai_Guo_Shoujing%27s_Statue.jpg File page | Appendix A, chapter cast |
| PTR-074 | Henry Briggs | Henry Briggs, 1561 to 1630. An unidentified artist, 1738. Public domain, via Wikimedia Commons. | Unidentified artist | 1738 | Public domain | An unidentified artist, 1738. Public domain, via Wikimedia Commons. https://commons.wikimedia.org/wiki/File:Henry-Briggs.jpg File page | Appendix A, chapter cast |
| PTR-075 | Hermann von Helmholtz | Hermann von Helmholtz, 1821 to 1894. Ludwig Knaus, 1881. Public domain, via Wikimedia Commons. | Ludwig Knaus | 1881 | Public domain | Ludwig Knaus, 1881. Public domain, via Wikimedia Commons. https://commons.wikimedia.org/wiki/File:Hermann_von_Helmholtz_by_Ludwig_Knaus.jpg File page | Appendix A, chapter cast |
| PTR-079 | Hypatia Not from life | Hypatia, 370 to 415. Jules Maurice Gaspard, 1908. Public domain, via Wikimedia Commons. No portrait made from life survives. This is a later image, not a record of the face. | Jules Maurice Gaspard | 1908 | Public domain | Jules Maurice Gaspard, 1908. Public domain, via Wikimedia Commons. https://commons.wikimedia.org/wiki/File:Hypatia_portrait.png File page | Appendix A, chapter cast |
| PTR-082 | Isaac Barrow | Isaac Barrow, 1630 to 1677. An unidentified artist. CC BY 4.0, via Wikimedia Commons. | Unidentified artist | not stated | CC BY 4.0 | An unidentified artist. CC BY 4.0, via Wikimedia Commons. https://commons.wikimedia.org/wiki/File:Isaac_Barrow._Engraving_by_W._Roffe_after_M._Noble._Wellcome_V0000372.jpg File page | Appendix A, chapter cast |
| PTR-083 | Isaac Newton | Isaac Newton, 1642 to 1727. John Vanderbank / Formerly attributed to Godfrey Kneller, first quarter of 18th century. Public domain, via Wikimedia Commons. | John Vanderbank / Formerly attributed to Godfrey Kneller | not stated | Public domain | John Vanderbank / Formerly attributed to Godfrey Kneller, first quarter of 18th century. Public domain, via Wikimedia Commons. https://commons.wikimedia.org/wiki/File:Kneller_Isaac_Newton.jpg File page | Appendix A, chapter cast |
| PTR-085 | James Gregory | James Gregory, 1638 to 1675. George Dawe, 1805. Public domain, via Wikimedia Commons. | George Dawe | 1805 | Public domain | George Dawe, 1805. Public domain, via Wikimedia Commons. https://commons.wikimedia.org/wiki/File:Portrait_of_James_Gregory,_M.D_(4672280).jpg File page | Appendix A, chapter cast |
| PTR-093 | Johann Bernoulli | Johann Bernoulli, 1667 to 1748. Johann Jakob Haid / After Johann Rudolf Huber, 1742. Public domain, via Wikimedia Commons. | Johann Jakob Haid / After Johann Rudolf Huber | 1742 | Public domain | Johann Jakob Haid / After Johann Rudolf Huber, 1742. Public domain, via Wikimedia Commons. https://commons.wikimedia.org/wiki/File:Johann_Bernoulli.jpg File page | Appendix A, chapter cast |
| PTR-094 | Johann Heinrich Lambert | Johann Heinrich Lambert, 1728 to 1777. An unidentified artist, c. 1770. Public domain, via Wikimedia Commons. | Unidentified artist | 1770 | Public domain | An unidentified artist, c. 1770. Public domain, via Wikimedia Commons. https://commons.wikimedia.org/wiki/File:Johann-Heinrich_Lambert.png File page | Appendix A, chapter cast |
| PTR-097 | Johann Radon | Johann Radon, 1887 to 1956. An unidentified artist, c. 1920. Public domain, via Wikimedia Commons. | Unidentified artist | 1920 | Public domain | An unidentified artist, c. 1920. Public domain, via Wikimedia Commons. https://commons.wikimedia.org/wiki/File:Johann_Radon.png File page | Appendix A, chapter cast |
| PTR-101 | John Herschel | John Herschel, 1792 to 1871. Maull and co, c. 1800. Public domain, via Wikimedia Commons. | Maull and co | 1800 | Public domain | Maull and co, c. 1800. Public domain, via Wikimedia Commons. https://commons.wikimedia.org/wiki/File:John_Herschel_portrait.jpg File page | Appendix A, chapter cast |
| PTR-104 | John Napier | John Napier, 1550 to 1617. An unidentified artist. CC BY 4.0, via Wikimedia Commons. | Unidentified artist | not stated | CC BY 4.0 | An unidentified artist. CC BY 4.0, via Wikimedia Commons. https://commons.wikimedia.org/wiki/File:John_Napier._Stipple_engraving._Wellcome_V0004221.jpg File page | Appendix A, chapter cast |
| PTR-107 | John Wallis | John Wallis, 1616 to 1703. An unidentified artist. CC BY 4.0, via Wikimedia Commons. | Unidentified artist | not stated | CC BY 4.0 | An unidentified artist. CC BY 4.0, via Wikimedia Commons. https://commons.wikimedia.org/wiki/File:John_Wallis._Line_engraving_by_D._Loggan,_1678,_after_himsel_Wellcome_V0006130.jpg File page | Appendix A, chapter cast |
| PTR-110 | Joseph Fourier | Joseph Fourier, 1768 to 1830. Julien-Léopold Boilly, early 19th century. Public domain, via Wikimedia Commons. | Julien-Léopold Boilly | not stated | Public domain | Julien-Léopold Boilly, early 19th century. Public domain, via Wikimedia Commons. https://commons.wikimedia.org/wiki/File:Fourier2_-_restoration1.jpg File page | Appendix A, chapter cast |
| PTR-111 | Joseph-Louis Lagrange | Joseph-Louis Lagrange, 1736 to 1813. Miscellaneous Items in High Demand, PPOC, Library of Congress, 1833. Public domain, via Wikimedia Commons. | Miscellaneous Items in High Demand, PPOC, Library of Congress | 1833 | Public domain | Miscellaneous Items in High Demand, PPOC, Library of Congress, 1833. Public domain, via Wikimedia Commons. https://commons.wikimedia.org/wiki/File:Joseph_Louis_Lagrange,_1736-1813,_head-and-shoulders_portrait_LCCN2005691518.jpg File page | Appendix A, chapter cast |
| PTR-112 | Jost Bürgi | Jost Bürgi, 1552 to 1632. An unidentified artist, c. 1600. Public domain, via Wikimedia Commons. | Unidentified artist | 1600 | Public domain | An unidentified artist, c. 1600. Public domain, via Wikimedia Commons. https://commons.wikimedia.org/wiki/File:ETH-BIB-B%C3%BCrgi,_Jost_(1552-1632)-Portrait-Portr_10998.tif File page | Appendix A, chapter cast |
| PTR-115 | Kangxi Emperor | Kangxi Emperor. Anonymous Unknown author Qing Dynasty Court Painter, late Kangxi period. Public domain, via Wikimedia Commons. | Anonymous Unknown author Qing Dynasty Court Painter | not stated | Public domain | Anonymous Unknown author Qing Dynasty Court Painter, late Kangxi period. Public domain, via Wikimedia Commons. https://commons.wikimedia.org/wiki/File:Emperor_Kangxi.PNG File page | Appendix A |
| PTR-117 | Karlheinz Brandenburg | Karlheinz Brandenburg, 1954. An unidentified artist, 2013. CC BY 2.0, via Wikimedia Commons. | Unidentified artist | 2013 | CC BY 2.0 | An unidentified artist, 2013. CC BY 2.0, via Wikimedia Commons. https://commons.wikimedia.org/wiki/File:Karlheinz_Brandenburg_cropped.jpg File page | Appendix A, chapter cast |
| PTR-118 | Al-Khwarizmi Not from life | Al-Khwarizmi, 780 to 850. Michel Bakni, 2020. Public domain, via Wikimedia Commons. No portrait made from life survives. This is a later image, not a record of the face. | Michel Bakni | 2020 | Public domain | Michel Bakni, 2020. Public domain, via Wikimedia Commons. https://commons.wikimedia.org/wiki/File:Al-Khwarizmi_portrait.jpg File page | Appendix A, chapter cast |
| PTR-121 | Leonhard Euler | Leonhard Euler, 1707 to 1783. Jakob Emanuel Handmann, 1753. Public domain, via Wikimedia Commons. | Jakob Emanuel Handmann | 1753 | Public domain | Jakob Emanuel Handmann, 1753. Public domain, via Wikimedia Commons. https://commons.wikimedia.org/wiki/File:Leonhard_Euler.jpg File page | Appendix A, chapter cast |
| PTR-129 | Matteo Ricci | Matteo Ricci, 1552 to 1610. An unidentified artist, c. 1610. Public domain, via Wikimedia Commons. | Unidentified artist | 1610 | Public domain | An unidentified artist, c. 1610. Public domain, via Wikimedia Commons. https://commons.wikimedia.org/wiki/File:Matteo_Ricci_2.jpg File page | Appendix A, chapter cast |
| PTR-136 | Nasir al-Din al-Tusi Not from life | Nasir al-Din al-Tusi, 1201 to 1274. Michel Bakni, 2021. Public domain, via Wikimedia Commons. No portrait made from life survives. This is a later image, not a record of the face. | Michel Bakni | 2021 | Public domain | Michel Bakni, 2021. Public domain, via Wikimedia Commons. https://commons.wikimedia.org/wiki/File:Nasir_al-Din_al-Tusi_portrait.jpg File page | Appendix A, chapter cast |
| PTR-139 | Nicolaus Copernicus | Nicolaus Copernicus, 1473 to 1543. J. Falck, 17th century (Ref.: Nicolaus Copernicus. Reproduction of line engraving after J. Falck. on Wellcome Library Catalogue). Public domain, via Wikimedia Commons. | J. Falck | not stated | Public domain | J. Falck, 17th century (Ref.: Nicolaus Copernicus. Reproduction of line engraving after J. Falck. on Wellcome Library Catalogue). Public domain, via Wikimedia Commons. https://commons.wikimedia.org/wiki/File:Nicolaus_Copernicus._Reproduction_of_line_engraving.jpg File page | Appendix A, chapter cast |
| PTR-146 | Pierre-Simon Laplace | Pierre-Simon Laplace, 1749 to 1827. Jean-François Villain, c. 1818. CC0, via Wikimedia Commons. | Jean-François Villain | 1818 | CC0 | Jean-François Villain, c. 1818. CC0, via Wikimedia Commons. https://commons.wikimedia.org/wiki/File:Musea_Brugge,_HUB,_2014_GRO2049_III.jpg File page | Appendix A, chapter cast |
| PTR-150 | Richard of Wallingford Not from life | Richard of Wallingford, 1292 to 1336. user:Leinad-Z. Public domain, via Wikimedia Commons. No portrait made from life survives. This is a later image, not a record of the face. | user:Leinad-Z | not stated | Public domain | user:Leinad-Z. Public domain, via Wikimedia Commons. https://commons.wikimedia.org/wiki/File:Abbot_Richard_Wallingford.jpg File page | Appendix A, chapter cast |
| PTR-152 | Roger Cotes | Roger Cotes, 1682 to 1716. An unidentified artist, Unknown date. Public domain, via Wikimedia Commons. | Unidentified artist | not stated | Public domain | An unidentified artist, Unknown date. Public domain, via Wikimedia Commons. https://commons.wikimedia.org/wiki/File:Roger_Cotes.png File page | Appendix A, chapter cast |
| PTR-156 | Seki Takakazu Not from life | Seki Takakazu, 1640 to 1708. Masahiko Fujiwara, c. 1708. Public domain, via Wikimedia Commons. No portrait made from life survives. This is a later image, not a record of the face. | Masahiko Fujiwara | 1708 | Public domain | Masahiko Fujiwara, c. 1708. Public domain, via Wikimedia Commons. https://commons.wikimedia.org/wiki/File:Seki_Takakazu.jpg File page | Appendix A, chapter cast |
| PTR-171 | Tycho Brahe | Tycho Brahe, 1546 to 1601. nicht anwendbar, c. 1791. Public domain, via Wikimedia Commons. | nicht anwendbar | 1791 | Public domain | nicht anwendbar, c. 1791. Public domain, via Wikimedia Commons. https://commons.wikimedia.org/wiki/File:ETH-BIB-Brahe,_Tycho_(1546-1601)-Portrait-Portr_10895.tif File page | Appendix A, chapter cast |
| PTR-172 | Ulugh Beg Not from life | Ulugh Beg, 1394 to 1449. 1425-1450 artist, c. 1425. Public domain, via Wikimedia Commons. No portrait made from life survives. This is a later image, not a record of the face. | 1425-1450 artist | 1425 | Public domain | 1425-1450 artist, c. 1425. Public domain, via Wikimedia Commons. https://commons.wikimedia.org/wiki/File:Ulugh_Beg_portrait.jpg File page | Appendix A, chapter cast |
| PTR-176 | Vincenzo Riccati | Vincenzo Riccati, 1707 to 1775. An unidentified artist, 18th century. Public domain, via Wikimedia Commons. | Unidentified artist | not stated | Public domain | An unidentified artist, 18th century. Public domain, via Wikimedia Commons. https://commons.wikimedia.org/wiki/File:Vincenzo_Riccati.jpeg File page | Appendix A, chapter cast |
| PTR-178 | William Jones | William Jones, 1675 to 1749. Joshua Reynolds, 1811. Public domain, via Wikimedia Commons. | Joshua Reynolds | 1811 | Public domain | Joshua Reynolds, 1811. Public domain, via Wikimedia Commons. https://commons.wikimedia.org/wiki/File:Portrait_of_Sir_William_Jones_(4671550).jpg File page | Appendix A, chapter cast |
| PTR-179 | Willibald Pirckheimer | Willibald Pirckheimer, 1470 to 1530. After Albrecht Dürer, c. 1524. Public domain, via Wikimedia Commons. | After Albrecht Dürer | 1524 | Public domain | After Albrecht Dürer, c. 1524. Public domain, via Wikimedia Commons. https://commons.wikimedia.org/wiki/File:After_Albrecht_D%C3%BCrer_001.jpg File page | Appendix A, chapter cast |
Appendix I
Where each story goes in your course
50 placements across 46 sections of Megan's Trigonometry course. Every section link opens the course book at the exact section it belongs to.
The kinds: 17 worked example, 11 origin, 7 myth check, 5 open question, 4 why it is called that, 4 person, 2 guess line.
How to use this. You do not have to read this book in order, or at all, to get something from it. Arrive from a link in your course book, read one box, and go back. The placements are chosen so that each one answers a question the course has just made you ask, rather than a question the history happens to be able to answer.
| Course section | Title | Kind | What goes there | Chapters | Sources |
|---|---|---|---|---|---|
| 1.0 | Unit 1 Opener and Objectives | origin | The oldest mathematics in this course was written by people counting sheep and measuring fields, and nobody in it ever measured an angle. | Ch. 1 | S001, S002 |
| 1.1 | The Pythagorean Theorem and Its Converse | myth-check | The theorem with Pythagoras's name on it was in use long before he was born, and nobody can show that he proved it. | Ch. 8;Ch. 1 | S001, S494 |
| 1.2 | SOH-CAH-TOA | open-question | Nobody knows who invented SOH-CAH-TOA: it is in none of the standard reference works, and the one date in circulation, 1944, comes with no book, no author and no page. | Ch. 8 | S446, S447, S449 |
| 1.3 | The Law of Sines and the Law of Cosines | origin | Euclid wrote the law of cosines twice, once for obtuse triangles and once for acute, because he had no negative numbers to let him write it once. | Ch. 8 | S428, S076, S077, S440 |
| 1.P | Mixed Practice | worked-example | Slip one 3,500 year old problem into today's mixed set with no commentary: Rhind papyrus problem 58, a pyramid of base 140 and height 93 and a third, find the seked. | Ch. 1 | S005 |
| 1.R | Recap: Check Yourself | myth-check | Here are the famous claims about Egyptian and Babylonian triangles that do not survive contact with the sources. | Ch. 1 | S001, S002, S003, S004 |
| 2.0 | Unit 2 Opener and Objectives | origin | The first people to build a sine table set the radius at 3438, and once you know why, radian measure stops looking arbitrary. | Ch. 3 | S122, S128 |
| 2.1 | The Law of Sines: The Ambiguous Case | open-question | The ambiguous case is in every trigonometry book on Earth and no historian has worked out where it came from. | Ch. 8 | S421, S423, S427, S450 |
| 2.2 | Angle Terms and Radian Measurement | why-it-is-called-that | A degree is a Greek word for a portion, a minute is a small part, a second is the second small part, and the radian got its name on an examination paper in Belfast on 5 June 1873. | Ch. 8;Ch. 7 | S485, S486, S487, S488, S489, S490, S381 |
| 2.3 | Sine, Cosine, and Tangent in Quadrants II, III, and IV | why-it-is-called-that | For a thousand years tables listed the versed sine instead of the cosine, for one reason: 1 minus cosine never goes negative, and nobody had negative numbers. | Ch. 8;Ch. 3 | S121, S421, S427 |
| 2.4 | Special Right Triangles and Cofunctions | worked-example | A 3-4-5 triangle is sitting inside a pyramid problem written around 1550 BCE, and the scribe never mentions it. | Ch. 8;Ch. 1 | S005, S061 |
| 2.5 | The Reciprocal Trig Functions | myth-check | Somebody once wrote I think in a history book, the hedge got dropped, and now textbooks credit the word cosecant to a man who never wrote it. | Ch. 8;Ch. 6 | S350, S427, S481 |
| 2.6 | The Quotient Identities | origin | Tangent starts life as the length of a shadow cast by a stick, which is why it took centuries to become sine over cosine. | Ch. 8;Ch. 4 | S181, S207, S481 |
| 2.P | Mixed Practice | worked-example | Slip in Sea Island Manual problem 1 with no commentary: measure an island you cannot reach, using two poles and one subtraction. | Ch. 5 | S247, S248, S275 |
| 2.R | Recap: Check Yourself | myth-check | Eleven popular claims about Islamic trigonometry, checked one at a time against what the specialists say. | Ch. 4 | S181, S182 |
| 3.0 | Unit 3 Opener and Objectives | person | Every repeating thing you can name is a sum of sines, and the man who claimed that took it to a hostile committee in Paris in 1807 and got nowhere. | Ch. 7 | S371, S372 |
| 3.1 | Graphing Sine and Cosine | origin | The first sine curve ever drawn was a byproduct: Roberval was computing the area under a cycloid around 1634 and this fell out of it. | Ch. 8 | S426, S427 |
| 3.1 | Graphing Sine and Cosine | open-question | If somebody asks where the word midline comes from, the honest answer is that nobody knows and nobody appears to have looked. | Ch. 8 | |
| 3.2 | Modeling with Sinusoidal Functions | person | Fourier said you could draw any curve you liked, freehand, with corners, and he would write it as a sum of sines, and Lagrange refused to accept it. | Ch. 7 | S371, S372 |
| 3.3 | Graphing Tangent and Cotangent | open-question | James Gregory drew the tangent curve and stopped at the first quadrant, so he never drew an asymptote, and nobody has found who first described one. | Ch. 8 | S426, S427 |
| 3.4 | Graphing Secant and Cosecant | worked-example | Edward Wright built the Mercator map by adding up 3,600 secants by hand, which is a Riemann sum decades before anyone had the calculus to name it. | Ch. 6 | S324 |
| 3.4 | Graphing Secant and Cosecant | open-question | The exsecant has a name, a symbol and a documented job on the railways, and no known coiner: Cajori prints exsec A and says nothing about who invented it. | Ch. 8 | S423 |
| 3.5 | Solving Linear Trig Equations | why-it-is-called-that | The arc in arcsin means a literal arc, a length measured along a circle, which is the only reason the name makes sense. | Ch. 8;Ch. 7 | S363, S423 |
| 3.P | Mixed Practice | worked-example | Slip one row of Ptolemy's chord table into the set with no commentary: chord 120 degrees is 103;55,23 in base 60, so what is it in decimals, and what angle is that? | Ch. 2 | S061 |
| 3.R | Recap: Check Yourself | myth-check | De Moivre never stated de Moivre's formula, and four more things everyone says about this era that the documents do not support. | Ch. 7 | S368, S369, S395 |
| 4.0 | Unit 4 Opener and Objectives | origin | Five practical problems, most of them about where to stand and when to pray, funded six hundred years of trigonometry. | Ch. 4 | S181, S182 |
| 4.1 | Simplifying and Verifying Trig Identities | worked-example | Abu al-Wafa wrote the angle addition formula with square roots where the cosines should be, and your job today is to show it is the same formula. | Ch. 8;Ch. 4 | S181, S481 |
| 4.2 | Solving Linear Trig Equations with Basic Identities | worked-example | Ptolemy could not get the chord of one degree exactly, so he proved it was larger than one number and smaller than another and read the answer off the gap. | Ch. 2 | S061, S481 |
| 4.3 | Solving Quadratic Trig Equations | worked-example | Finding sin 1 degree from sin 3 degrees means solving a cubic in the sine, and al-Kashi's move was to refuse to solve it and rearrange it instead. | Ch. 4 | S184, S484, S481 |
| 4.4 | The Pythagorean Identity | myth-check | The claim that Varahamihira first stated the Pythagorean identity comes from misreading one carefully hedged sentence, and today we read the sentence. | Ch. 8;Ch. 3 | S124, S481 |
| 4.5 | Solving Quadratic Trig Equations with Identities | person | Adriaan van Roomen challenged the mathematicians of the world with an equation of degree 45 in 1593, and Viete cracked it because he noticed that 45 is 3 times 3 times 5. | Ch. 6 | S316, S329 |
| 4.6 | Solving All Types of Trig Equations | worked-example | When no formula exists you guess, substitute and repeat, and al-Kashi did that by hand in the 1420s and got 16 correct significant digits. | Ch. 4 | S184, S484 |
| 4.7 | Additional Identities | origin | Before logarithms, astronomers multiplied by turning products into sums with a cosine identity, which turned a ten minute multiplication into a thirty second subtraction. | Ch. 8;Ch. 6 | S321, S212, S425 |
| 4.P | Mixed Practice | worked-example | Slip in one prosthaphaeresis multiplication with no commentary and let the class race a calculator to the answer. | Ch. 6 | S321 |
| 4.R | Recap: Check Yourself | myth-check | Five widely repeated claims about Indian trigonometry, including a famous sine formula that returns 16 when you feed it 90 degrees. | Ch. 3 | S122, S128 |
| A.0 | Algebra Opener | origin | Base 60 is not a decision anybody made; it is the fossil of a set of accounting units, and it is why your clock and your protractor look the way they do. | Ch. 1 | S026, S006, S020 |
| A.1 | Unit Conversions | worked-example | An Egyptian scribe gave the slope of a pyramid in palms and fingers, 7 palms to the cubit and 4 fingers to the palm, which is a unit conversion and a cotangent at the same time. | Ch. 1 | S005, S027 |
| A.2 | Simplifying Radicals | worked-example | Before Ptolemy could write one row of his chord table he needed the square root of 2 and the square root of 3 in base 60: chord 90 degrees is 84;51,10 and chord 120 degrees is 103;55,23. | Ch. 2 | S061 |
| A.3 | Operations with Radicals | worked-example | Viete's formula for 2 over pi is an endless product of nested square roots, each one the square root of 2 plus the one before it. | Ch. 6 | S316, S329 |
| A.4 | Operations with Fractions | worked-example | Egyptian arithmetic had no 0.72: the Rhind papyrus writes it as one half plus one fifth plus one fiftieth, because every fraction had to be a sum of distinct unit fractions. | Ch. 1 | S005, S027, S028 |
| A.5 | Operations with Fractions and Radicals | worked-example | Al-Kashi's sine of one degree is a sexagesimal fraction, and every place after the semicolon is a fraction over a power of 60. | Ch. 4 | S184, S481, S484 |
| A.6 | Factoring Strategies | worked-example | Viete solved an equation of degree 45 by factoring the exponent: 45 is 3 times 3 times 5, so the monster is three angle-division steps stacked. | Ch. 6 | S316 |
| A.7 | Solving Quadratic Equations | origin | Completing the square is Old Babylonian: it is the method Robson reconstructs behind the numbers on Plimpton 322, more than a thousand years before anyone wrote an equation. | Ch. 1 | S001, S002 |
| B.1 | Notation and Symbols | origin | Every symbol on your worksheet was once somebody's proposal, several of the proposals lost, and one of them was designed to pick a fight. | Ch. 7;Ch. 8 | S363, S423 |
| B.2 | Glossary | why-it-is-called-that | The word sine is Latin for a bay, standing in for an Arabic word for a fold, standing in for a Sanskrit word for a bowstring, and the person who made the swap is unknown. | Ch. 8 | S421, S425, S427, S481 |
| B.3 | Methods | worked-example | Liu Hui measured an island he could not reach, using two poles and one subtraction, and no angle at all. | Ch. 5 | S247, S248, S275 |
| B.4 | Formulas | person | A table of sines cost Rheticus 4,400 Gulden and years of paid labor, and when the finished thing was printed in 1596 it was wrong. | Ch. 6 | S307, S327, S352, S481 |
| B.5 | Properties and Identities | origin | Most of your formula sheet falls out of one theorem about a quadrilateral inscribed in a circle. | Ch. 8;Ch. 2 | S061, S421 |
| 2.2 | Angle Terms and Radian Measurement | guess-line | Ask the class who decided the radius of the unit circle should be 1, and when. The answer everybody reaches for is Euler in 1748, and it is in print in respectable places, including a 2010 article in Mathematics Teacher. It is roughly eight centuries early: Abu al-Wafa was using a unit radius in the tenth century, and al-Biruni built a sine table on one. Then the better question, which is the actual lesson: if they had it first, why does Euler get the credit? Because they used it as a scaling choice inside a table and mostly dropped it, and he stopped treating sine as a length in a circle at all. | Ch. 4;Ch. 7 | S481, S181, S425 |
| A.5 | Operations with Fractions and Radicals | guess-line | Put al-Kashi 1;2,49,43,11,14,44,16,26,17 on the board as his sine of one degree and ask whether they can check it. They can: it is a sexagesimal fraction, and every place after the semicolon is a sixtieth of the one before. Work it out and it is wrong in the ninth place. The true string ends 26,18. The wrong one has been copied for a century because a number that good does not invite rechecking, which is the point worth making. | Ch. 4 | S481, S184, S484 |
Appendix J
Bibliography
332 sources in APA 7, grouped by how much weight each can carry. The access column is the honesty control: a source marked abstract only or not obtained supports no claim anywhere in this book, and is listed so you know it exists and know it is not load-bearing.
Every source pointer in the text, the small boxed S numbers, links here.
Tier 1: primary sources and critical scholarly editions of them
39 sources.
| ID | Reference (APA 7) | Language | Access |
|---|---|---|---|
| S184 | Aaboe, A. (1951). Al-Kashi's iteration method for the determination of sin 1 degree. Scripta Mathematica, 17, 355 to 359 https://www.jphogendijk.nl/samarkand/Kashi-Aaboe.pdfCited in Chapter 4 13 times and 1500 to 1620. | en | read in full |
| S075 | Bernard, A. (2019). Review of R. Rashed and A. Papadopoulos, Menelaus' Spherics. Bryn Mawr Classical Review, 2019.01.30 https://bmcr.brynmawr.edu/2019/2019.01.30/Cited in Chapter 2 4 times and 1950 to the present. | en | read in full |
| S362 | Bruce, I. (n.d.). Euler's Introductio in analysin infinitorum, vol. 1, Chapter 8: On transcending quantities arising from the circle [English translation with facing Latin original]. 17centurymaths.com. http://www.17centurymaths.com/contents/euler/introductiontoanalysisvolone/ch8vol1.pdfCited in Chapter 7 22 times and 1620 to 1750. | en;la | read in full |
| S482 | Cullen, C. (1982). An eighth century Chinese table of tangents. Chinese Science, 5, 1 to 33Not cited in the story. It backs a register entry. | en | read in full |
| S368 | de Moivre, A. (1707). Aequationum quarundam potestatis tertiae, quintae, septimae, nonae, et superiorum ... resolutio analytica. Philosophical Transactions, 25(309), 2368 to 2371, R. J. Pulskamp (Trans.), Xavier University. https://probabilityandfinance.com/pulskamp/Moivre/aequationum%20quarundam.pdfCited in Chapter 7 3 times and 1620 to 1750. | en | read in full |
| S369 | de Moivre, A. (1722). De sectione anguli. Philosophical Transactions, 32(374), 228 to 230, R. J. Pulskamp (Trans.), Xavier University. https://probabilityandfinance.com/pulskamp/Moivre/de_%20sectione_anguli.pdfCited in Chapter 7 4 times and 1620 to 1750 twice. | en | read in full |
| S181 | Debarnot, M.-T. (1996). Trigonometry. Encyclopedia of the History of Arabic Science, 2, 160 to 203, Routledge (R. Rashed, Ed.). https://archive.org/stream/RoshdiRasheded.EncyclopediaOfTheHistoryOfArabicScienceVol.3Routledge1996/Rashed%20R.-Encyclopedia%20of%20the%20History%20of%20Arabic%20Science.%202-Routledge%20(1996)_djvu.txtCited in Chapter 4 96 times, 750 to 1200 5 times and 1200 to 1500 twice. | en | read in full |
| S064 | Duke, D. W. (n.d.). The very early history of trigonometry. Florida State University (author's PDF; no journal, volume or year printed on the PDF). https://people.sc.fsu.edu/~dduke/earlytrig12.pdfCited in Chapter 2 8 times and 300 BCE to 400 CE. | en | read in full |
| S495 | Duke, D. W. (n.d.). Hipparchus' eclipse trios and early trigonometry. Florida State University (author's PDF).Not cited in the story. It backs a register entry. | en | read in full |
| S496 | Duke, D. W. (n.d.). Associations between the ancient star catalogues. Florida State University (author's PDF).Not cited in the story. It backs a register entry. | en | read in full |
| S497 | Duke, D. W. (n.d.). An early use of the chain rule. Florida State University (author's PDF).Not cited in the story. It backs a register entry. | en | read in full |
| S498 | Duke, D. W. (n.d.). Hipparchus' coordinate system. Florida State University (author's PDF).Not cited in the story. It backs a register entry. | en | read in full |
| S499 | Duke, D. W. (n.d.). The very early history of trigonometry. Florida State University (author's PDF).Not cited in the story. It backs a register entry. | en | read in full |
| S500 | Duke, D. W. (n.d.). Dating the Almagest star catalogue using proper motions: A reconsideration. Florida State University (author's PDF).Not cited in the story. It backs a register entry. | en | read in full |
| S501 | Duke, D. W. (n.d.). Greek angles from Babylonian numbers. Florida State University (author's PDF).Not cited in the story. It backs a register entry. | en | read in full |
| S063 | Duke, D. W. (2005). Hipparchus' eclipse trios and early trigonometry. Centaurus, 47(3), 163 to 177, Florida State University (author's PDF). https://people.sc.fsu.edu/~dduke/alm411-3.pdfCited in Chapter 2 11 times and 1950 to the present twice. | en | read in full |
| S428 | Euclid (n.d.). Elements, Propositions I.47, I.48, II.12 and II.13. D. E. Joyce (Ed.), after T. L. Heath's translation. Clark University. https://mathcs.clarku.edu/~djoyce/java/elements/bookII/propII12.htmlCited in Chapter 8 10 times, 300 BCE to 400 CE and 1620 to 1750. | en | read in full |
| S429 | Euler, L. (1748). Introductio in analysin infinitorum, Vol. 1, Chapter 8: On transcending quantities arising from the circle. 1, I. Bruce (Trans.), 2013. https://www.17centurymaths.com/ (local copy: `sources/euler_intro_ch8.txt`)Cited in Chapter 8 9 times and 1620 to 1750. | en | read in full |
| S185 | Gomez Gomez, A. (2023). Biruni's measurement of the Earth. International Journal of Science and Research, 12(3), 1535 to 1542 https://www.ijsr.net/archive/v12i3/SR23326005322.pdfCited in Chapter 4 15 times and 750 to 1200 twice. | en | read in full |
| S492 | Heideman, M. T. (1985). Gauss and the history of the fast Fourier transform. Archive for History of Exact Sciences, 34(3), 265 to 277Not cited in the story. It backs a register entry. | en | read in full |
| S494 | Hoyrup, J. (n.d.). Pythagorean "rule" and "theorem": Mirror of the relation between Babylonian and Greek mathematics. 393 to 407, Roskilde University.Not cited in the story. It backs a register entry. | en | read in full |
| S253 | Kotyk, J. (2022). The astronomical innovations of monk Yixing 一行 (673 to 727). Religions, 13(6), 543 https://www.mdpi.com/2077-1444/13/6/543 (PDF read at https://www.wisdomlib.org/uploads/journals/mdpi-relig/2022-volume-13-issue-6--2077-1444-13-6-543-.pdf)Cited in Chapter 5 23 times and 400 to 750 5 times. | en | read in full |
| S003 | Mansfield, D. F. (2017). Plimpton 322 is Babylonian exact sexagesimal trigonometry. Historia Mathematica, 44(4), 395 to 419 https://jmhammond.github.io/history/Mansfield2017.pdf (publisher OA: https://www.sciencedirect.com/science/article/pii/S0315086017300691)Cited in Chapter 1 42 times, c. 3000 to 300 BCE, 1850 to 1950 twice and 1950 to the present. | en | read in full |
| S483 | Mansfield, D. F. (2020). Perpendicular lines and diagonal triples in Old Babylonian surveying. Journal of Cuneiform Studies, 72, 87 to 99 https://doi.org/10.1086/709309Cited in Chapter 1 7 times. | en | read in full |
| S004 | Mansfield, D. F. (2021). Plimpton 322: A study of rectangles. Foundations of Science, 26(4), 977 to 1005 https://link.springer.com/content/pdf/10.1007/s10699-021-09806-0.pdfCited in Chapter 1 16 times, c. 3000 to 300 BCE, 1850 to 1950 and 1950 to the present. | en | read in full |
| S485 | Muir, T. (1910). The term "radian" in trigonometry [letter, 7 April 1910]. Nature, 83(2110), 156Cited in Chapter 7 and Chapter 8. | en | read in full |
| S487 | Muir, T. (1910). The term "radian" in trigonometry [letter, 16 June 1910]. Nature, 83(2120), 459 to 460Cited in Chapter 7 and Chapter 8. | en | read in full |
| S484 | Riahi, F. (1995). An early iterative method for the determination of sin 1 degree. The College Mathematics Journal, 26(1), 16 to 21, Taylor & Francis on behalf of the Mathematical Association of America.Cited in Chapter 4. | en | read in full |
| S002 | Robson, E. (2001). Neither Sherlock Holmes nor Babylon: A reassessment of Plimpton 322. Historia Mathematica, 28(3), 167 to 206 https://uruk-warka.dk/mathematics/ER13%20neither-sherlock.pdfCited in Chapter 1 37 times, c. 3000 to 300 BCE, 1850 to 1950 3 times and 1950 to the present 3 times. | en | read in full |
| S001 | Robson, E. (2002). Words and pictures: New light on Plimpton 322. The American Mathematical Monthly, 109(2), 105 to 120 https://uruk-warka.dk/mathematics/ER12%20words-pictures.pdf (mirror: https://jmhammond.github.io/history/Robson2002.pdf)Cited in Chapter 1 33 times, c. 3000 to 300 BCE twice, 1850 to 1950 3 times and 1950 to the present. | en | read in full |
| S491 | Steele, J. M. (2013). Shadow-length schemes in Babylonian astronomy. SCIAMVS, 14, 3 to 39Not cited in the story. It backs a register entry. | en | read in full |
| S489 | Thomson, W. (1867). A treatise on natural philosophy, Vol. I (first edition). 1, Oxford: at the Clarendon Press. With P. G. Tait.Cited in Chapter 7 4 times and Chapter 8 3 times. | en | read in full |
| S490 | Thomson, W. (1879). Treatise on natural philosophy, Vol. I Part I (new edition). 1, Cambridge: at the University Press. With P. G. Tait.Cited in Chapter 7 3 times and Chapter 8 twice. | en | read in full |
| S486 | Thomson, J. (1910). The term "radian" in trigonometry [letter, 21 April 1910]. Nature, 83(2112), 217Cited in Chapter 7 3 times and Chapter 8 3 times. | en | read in full |
| S488 | Thomson, J. (1910). The term "radian" in trigonometry [letter, 16 June 1910]. Nature, 83(2120), 460Cited in Chapter 7 3 times and Chapter 8 twice. | en | read in full |
| S257 | Wagner, D. B. (2012). Shen Gua and an ignorant editor on the length of an arc [research note]. https://donwagner.dk/Shen-Gua-arc.htmCited in Chapter 5 11 times, 750 to 1200 and 1200 to 1500. | en | read in full |
| S493 | Wilkinson, L. (1861). Translation of the Surya Siddhanta, and of the Siddhanta Siromani. Calcutta: printed by C. B. Lewis at the Baptist Mission Press. Translated by Bapu Deva Sastri and L. Wilkinson, revised by Bapu Deva Sastri.Not cited in the story. It backs a register entry. | en | read in full |
| S033 | (n.d.). Photograph of Plimpton 322 [JPEG, 2159 x 2886 px]. Cuneiform Digital Library Initiative. https://cdli.mpiwg-berlin.mpg.de/dl/photo/P254790.jpgCited in Chapter 1 twice. | en | read in full |
| S034 | (2021). Si.427 obverse [photograph, Fig. 2 in S004, 1062 x 1164 px]. İstanbul Arkeoloji Müzeleri. https://media.springernature.com/lw1200/springer-static/image/art%3A10.1007%2Fs10699-021-09806-0/MediaObjects/10699_2021_9806_Fig2_HTML.pngCited in Chapter 1 3 times. | en | read in full |
Tier 2: peer reviewed scholarship
117 sources.
| ID | Reference (APA 7) | Language | Access |
|---|---|---|---|
| S379 | Abbott, B. P. (2016). Observation of gravitational waves from a binary black hole merger. Physical Review Letters, 116(6), 061102 https://arxiv.org/pdf/1602.03837 (author accepted version of the PRL article)Cited in Chapter 7 5 times and 1950 to the present. | en | read in full |
| S396 | Ahmed, N. (1974). Discrete cosine transform. IEEE Transactions on Computers, C-23(1), 90 to 93Cited in Chapter 7 3 times and 1950 to the present. | en | not obtained |
| S220 | al-Battani (1899). Al-Battani sive Albatenii Opus astronomicum. 1 to 3, C. A. Nallino (Ed. and Trans.). U. Hoepli, Milan. https://archive.org/details/albattanisivealb00battCited in Chapter 4 twice. | ar;la | read in full |
| S186 | al-Nadim, Ibn (1970). The Fihrist of al-Nadim: A tenth-century survey of Muslim culture. 1 to 2, B. Dodge (Trans.). Columbia University Press. https://archive.org/details/fihrist-of-al-nadim.-a-tenth-century-survey-of-muslim-culture.-ed.-and-transl.-bCited in Chapter 4 5 times and 750 to 1200. | en | read in full |
| S222 | al-Tusi, Nasir al-Din (n.d.). Tahrir al-Majisti (Recension of the Almagest), 15th-century copy. BnF, Departement des Manuscrits, Arabe 2485. https://gallica.bnf.fr/ark:/12148/btv1b110017717Cited in Chapter 4. | ar | read in full |
| S438 | Allen, F. B. (1961). Elementary functions, student's text, Unit 21. School Mathematics Study Group, Stanford University. ERIC ED135629. https://archive.org/download/micro_IA41153328_0765/micro_IA41153328_0765_djvu.txtCited in Chapter 8 6 times and 1950 to the present. | en | read in full |
| S399 | Blanton, J. D. (1988). Leonhard Euler: Introduction to analysis of the infinite, Book I. New York: Springer-Verlag. https://archive.org/details/introductiontoan0000eule (controlled digital lending)Not cited in the story. It backs a register entry. | en | not obtained |
| S378 | Brandenburg, K. (1999). MP3 and AAC explained. Proceedings of the AES 17th International Conference on High Quality Audio Coding, Audio Engineering Society. https://www.iis.fraunhofer.de/content/dam/iis/de/doc/ame/conference/AES-17-Conference_mp3-and-AAC-explained_AES17.pdfCited in Chapter 7 6 times. | en | read in full |
| S030 | Britton, J. P. (2011). Plimpton 322: A review and a different perspective. Archive for History of Exact Sciences, 65, 519 to 566 https://link.springer.com/article/10.1007/s00407-011-0083-4Cited in Chapter 1 twice and 1950 to the present. | en | not obtained |
| S121 | Burgess, E. (1860). Translation of the Surya-Siddhanta, a text-book of Hindu astronomy. P. C. Sengupta (Ed.). Calcutta University reprint. Internet Archive. https://archive.org/details/SuryaSiddhantaTranslation (full text: https://archive.org/download/SuryaSiddhantaTranslation/surya_siddhanta_english_djvu.txt)Cited in Chapter 3 20 times and 400 to 750. | en | read in full |
| S443 | Burgess, E. (1860). Translation of the Surya-Siddhanta, a text-book of Hindu astronomy. Journal of the American Oriental SocietyCited in Chapter 8 3 times and 400 to 750. | en | read in full |
| S364 | Cajori, F. (1919). A history of mathematics (2nd ed.). New York: Macmillan. https://archive.org/details/historyofmathema00cajouoftCited in Chapter 7 6 times, Chapter 8 twice, 1750 to 1850 and 1850 to 1950 twice. | en | read in full |
| S424 | Cajori, F. (1928). A history of mathematical notations: Vol. 1. Notations in elementary mathematics. 1, Open Court. https://archive.org/download/historyofmathema031756mbp/historyofmathema031756mbp_djvu.txtCited in Chapter 8 5 times. | en | read in full |
| S363 | Cajori, F. (1929). A history of mathematical notations, Volume II: Notations mainly in higher mathematics. 2, Chicago: Open Court. https://archive.org/details/historyofmathema027671mbpCited in Chapter 7 39 times, Chapter 8, 1500 to 1620 twice, 1620 to 1750 twice and 1750 to 1850 4 times. | en | read in full |
| S423 | Cajori, F. (1929). A history of mathematical notations: Vol. 2. Notations mainly in higher mathematics. 2, Open Court. https://archive.org/download/historyofmathema027671mbp/historyofmathema027671mbp_djvu.txtCited in Chapter 6 twice, Chapter 7 11 times, Chapter 8 37 times, 1500 to 1620, 1620 to 1750 3 times, 1750 to 1850 3 times and 1850 to 1950. | en | read in full |
| S073 | Cameron, A. (1990). Isidore of Miletus and Hypatia: On the editing of mathematical texts. Greek, Roman and Byzantine Studies, 31(1), 103 to 127 https://grbs.library.duke.edu/index.php/grbs/article/download/4171/5587/0Cited in Chapter 2 11 times and 400 to 750. | en | read in full |
| S273 | Chen, J.-P. J. (2015). Trigonometric tables: explicating their construction principles in China. Archive for History of Exact Sciences, 69(5) https://link.springer.com/article/10.1007/s00407-015-0162-zNot cited in the story. It backs a register entry. | en | abstract only |
| S122 | Clark, W. E. (1930). The Aryabhatiya of Aryabhata: An ancient Indian work on mathematics and astronomy. University of Chicago Press. Internet Archive. https://archive.org/details/The_Aryabhatiya_of_Aryabhata_Clark_1930 (full text: https://archive.org/download/The_Aryabhatiya_of_Aryabhata_Clark_1930/The_Aryabhatiya_of_Aryabhata_Clark_1930_djvu.txt)Cited in Chapter 3 23 times and 400 to 750 twice. | en | read in full |
| S444 | Clark, W. E. (1930). The Aryabhatiya of Aryabhata. University of Chicago Press.Cited in Chapter 8 twice and 400 to 750. | en | read in full |
| S126 | Colebrooke, H. T. (1817). Algebra, with arithmetic and mensuration, from the Sanscrit of Brahmegupta and Bhascara. John Murray. Internet Archive. https://archive.org/details/1817-henry-colebrooke-algebra-with-arithmetic-and-mensuration-from-the-sanskrit-of-brCited in Chapter 3 5 times. | en | read in full |
| S266 | Cooke, R. (2013). Traditional Japanese mathematics. The history of mathematics: a brief course (chapter 24), Reproduced at schoolbag.info. https://schoolbag.info/mathematics/history/25.htmlCited in Chapter 5 19 times and 1620 to 1750. | en | read in full |
| S376 | Cooley, J. W. (1965). An algorithm for the machine calculation of complex Fourier series. Mathematics of Computation, 19(90), 297 to 301 https://www.ams.org/journals/mcom/1965-19-090/S0025-5718-1965-0178586-1/Cited in Chapter 7 and 1950 to the present. | en | read in full |
| S302 | Copernicus, N. (1542). De lateribus et angulis triangulorum, tum planorum rectilineorum, tum sphaericorum, libellus eruditissimus & utilissimus ... Additus est Canon semissium subtensarum rectarum linearum in circulo. Excusum Vitrembergae per Iohannem Lufft (G. J. Rheticus, Ed.). Copy: Biblioteka Śląska, shelfmark 223856 II. https://www.sbc.org.pl/dlibra/publication/440324/edition/412962/ (PDF: http://sbc.org.pl/Content/412962/PDF/ii223856-0000-00-0001.pdf)Cited in Chapter 6 9 times and 1500 to 1620. | la | read in full |
| S366 | Cotes, R. (1722). Harmonia mensurarum, sive analysis et synthesis per rationum et angulorum mensuras promotae. R. Smith (Ed.). Cambridge. Bayerische Staatsbibliothek copy. https://archive.org/details/10525453bsbCited in Chapter 7 5 times and 1620 to 1750. | la | read in full |
| S256 | Cullen, C. (1982). An eighth century Chinese table of tangents. Chinese Science, 5, Now hosted as East Asian Science, Technology, and Medicine 5(1). https://brill.com/view/journals/east/5/1/article-p1_2.xmlCited in Chapter 5 5 times. | en | read in full |
| S274 | Cullen, C. (2002). Revisiting an eighth-century Chinese table of tangents [book chapter]. Springer. https://link.springer.com/chapter/10.1007/978-94-015-9862-0_17Not cited in the story. It backs a register entry. | en | abstract only |
| S128 | Datta, B. (1983). Hindu trigonometry. Indian Journal of History of Science, 18(1), 39 to 108, K. S. Shukla (Rev.). https://insa.nic.in/writereaddata/UpLoadedFiles/IJHS/Vol18_1_5_BDatta.pdfCited in Chapter 3 59 times, 400 to 750 twice, 750 to 1200 3 times, 1500 to 1620 and 1620 to 1750 twice. | en | read in full |
| S146 | Datta, B. (2019). Hindu trigonometry. Studies in the history of Indian mathematics, Springer reprint. https://link.springer.com/content/pdf/10.1007/978-981-13-7326-8_16Not cited in the story. It backs a register entry. | en | abstract only |
| S361 | Euler, L. (1748). Introductio in analysin infinitorum, volume 1 (E101). 1, Lausanne: Marcum-Michaelem Bousquet. Euler Archive record and PDF scan. https://scholarlycommons.pacific.edu/euler-works/101/ (scan: https://scholarlycommons.pacific.edu/cgi/viewcontent.cgi?article=1100&context=euler-works )Cited in Chapter 7 3 times and 1620 to 1750. | la | read in full |
| S304 | Fincke, T. (1583). Thomae Finkii Flenspurgensis Geometriae rotundi libri XIIII. Basileae: per Sebastianum Henricpetri. Copy: Det Kongelige Bibliotek, Copenhagen, LN 599 4to copy 1. https://archive.org/details/den-kbd-pil-130018099382-001Cited in Chapter 6 8 times and 1500 to 1620. | la | read in full |
| S371 | Fourier, J. (1822). Theorie analytique de la chaleur. Paris: Firmin Didot. University of Toronto copy. https://archive.org/details/thorieanalytiq00fourCited in Chapter 7 5 times and 1750 to 1850. | fr | read in full |
| S066 | Grasshoff, G. (2024). An astronomical analysis of the data in the pseudo-Hipparchus palimpsest in the Codex Climaci Rescriptus. Journal for the History of Astronomy, 55(3), 332 to 349 https://edoc.hu-berlin.de/items/00a2b3a1-636e-41ed-b948-4c1fca173920Cited in Chapter 2 5 times, 300 BCE to 400 CE and 1950 to the present. | en | read in full |
| S309 | Gunter, E. (1620). Canon triangulorum, sive tabulae sinuum et tangentium artificialium ad radium 10000,0000 & ad scrupula prima quadrantis. London: William Jones. https://archive.org/details/bim_early-english-books-1475-1640_canon-triangulorum-sive_gunter-edmund_1620Cited in Chapter 6 3 times, 1500 to 1620 and 1620 to 1750. | la | read in full |
| S065 | Gysembergh, V. (2022). New evidence for Hipparchus' Star Catalogue revealed by multispectral imaging. Journal for the History of Astronomy, 53(4), 383 to 393 https://journals.sagepub.com/doi/10.1177/00218286221128289Cited in Chapter 2 10 times, 300 BCE to 400 CE, 400 to 750, 750 to 1200 and 1950 to the present. | en | read in full |
| S067 | Gysembergh, V. (2025). A note on the new evidence for Hipparchus' star catalogue. Journal for the History of Astronomy, 56(3), 287 to 290 https://journals.sagepub.com/doi/10.1177/00218286251335640Cited in Chapter 2 3 times and 1950 to the present. | en | abstract only |
| S504 | Hannah, J. (2010). Review of Mathematics in India by Kim Plofker. Aestimatio, 7, 45 to 53Not cited in the story. It backs a register entry. | en | read in full |
| S503 | Hannah, J. (2022). Review of The Doctrine of Triangles: A History of Modern Trigonometry by Glen Van Brummelen. Aestimatio, ns 3.1, 134 to 143Not cited in the story. It backs a register entry. | en | read in full |
| S070 | Heath, T. L. (1897). The works of Archimedes. Cambridge University Press. https://archive.org/details/worksofarchimede00archCited in Chapter 2 3 times and 300 BCE to 400 CE. | en | read in full |
| S069 | Heath, T. L. (1913). Aristarchus of Samos, the ancient Copernicus... together with Aristarchus's treatise On the sizes and distances of the sun and moon. Clarendon Press. https://archive.org/details/aristarchusofsam00heatCited in Chapter 2 6 times and 300 BCE to 400 CE. | en | read in full |
| S068 | Heath, T. L. (1921). A history of Greek mathematics, Vol. II: From Aristarchus to Diophantus. 2, Clarendon Press. https://archive.org/details/historyofgreekma029268mbpCited in Chapter 2 32 times, 300 BCE to 400 CE 3 times, 750 to 1200, 1200 to 1500 and 1500 to 1620. | en | read in full |
| S062 | Heiberg, J. L. (1898). Claudii Ptolemaei opera quae exstant omnia, Vol. I.1: Syntaxis mathematica (Greek text). 1.1, Teubner. https://archive.org/details/claudiiptolemae00ptolgoogCited in Chapter 2 10 times. | grc | read in full |
| S400 | Heideman, M. T. (1984). Gauss and the history of the fast Fourier transform. IEEE ASSP Magazine, 1(4), 14 to 21 https://doi.org/10.1109/MASSP.1984.1162257Not cited in the story. It backs a register entry. | en | not obtained |
| S375 | Heideman, M. T. (1985). Gauss and the history of the fast Fourier transform. Archive for History of Exact Sciences, 34(3), 265 to 277Cited in Chapter 7 4 times, 1750 to 1850, 1850 to 1950 and 1950 to the present. | en | read in full |
| S380 | Helmholtz, H. (1895). On the sensations of tone as a physiological basis for the theory of music (3rd English ed.). A. J. Ellis (Trans.). London: Longmans, Green. https://archive.org/details/onsensationsofto00helmrichCited in Chapter 7 5 times and 1850 to 1950. | en | read in full |
| S248 | Hong, T. C. (n.d.). The Sea Island Mathematical Manual of Liu Hui [PDF lecture text]. Malaysian Institute of Sinology. https://mysinology.org.my/wp-content/uploads/2020/01/%E6%B4%AA%E5%A4%A9%E8%B3%9C-The-Sea-Island-Mathematical-Manual.pdfCited in Chapter 5 5 times. | en | read in full |
| S007 | Hunger, H. (2019). The Babylonian astronomical compendium MUL.APIN. Routledge. https://peachv.org/images/MuslimGeo/BabyAstroMulApinHunger.pdfCited in Chapter 1 12 times and c. 3000 to 300 BCE twice. | en | read in full |
| S435 | Inman, J. (1858). Navigation and nautical astronomy for the use of British seamen. https://archive.org/download/navigationnautic00inma/navigationnautic00inma_djvu.txtCited in Chapter 8 3 times and 1750 to 1850. | en | read in full |
| S182 | King, D. A. (1996). Astronomy and Islamic society: Qibla, gnomonics and timekeeping. Encyclopedia of the History of Arabic Science, 1, 128 to 184, Routledge (R. Rashed, Ed.). https://archive.org/stream/encyclopedia-of-the-history-of-arabic-science-volume-1/Encyclopedia%20of%20the%20History%20of%20Arabic%20Science%2C%20Volume%201_djvu.txtCited in Chapter 4 23 times and 1200 to 1500 twice. | en | read in full |
| S255 | Kotyk, J. (2018). The sinicization of Indo-Iranian astrology in medieval China. Sino-Platonic Papers, 282 https://sino-platonic.org/complete/spp282_Indo-Iranian_Astrology_China.pdfCited in Chapter 5 3 times and 400 to 750. | en | read in full |
| S212 | Kuehn, K. (2009). Prosthaphaeresis and Johannes Werner (1468 to 1522). Journal of the Oughtred Society https://www.oughtred.org/jos/articles/PROSTHAPHAERESISandWERNERfinal.jmccLR8.8.pdfCited in Chapter 4 5 times. | en | read in full |
| S244 | Li, Y. (2009). Gnomon shadow lengths recorded in the Zhoubi Suanjing: the earliest meridian observations in China?. Research in Astronomy and Astrophysics, 9(12), 1377 to 1386 https://www.raa-journal.org/issues/all/2009/v9n12/202203/P020220325531194532388.pdfCited in Chapter 5 15 times, c. 3000 to 300 BCE, 300 BCE to 400 CE, 400 to 750 and 1200 to 1500. | en | read in full |
| S247 | Liu Hui 劉徽 (n.d.). Haidao suanjing 海島算經. Chinese Text Project digital edition. https://ctext.org/hai-dao-suan-jing/zhsCited in Chapter 5 5 times and 300 BCE to 400 CE. | zh | read in full |
| S078 | Manitius, K. (1894). Hipparchi in Arati et Eudoxi Phaenomena commentariorum libri tres (Greek text with German translation). Teubner. https://archive.org/details/ipparchoutonarat00hippCited in Chapter 2 twice. | de | read in full |
| S032 | Mansfield, D. F. (2020). Perpendicular lines and diagonal triples in Old Babylonian surveying. Journal of Cuneiform Studies, 72, 87 to 99Cited in c. 3000 to 300 BCE. | en | read in full |
| S029 | Miatello, L. (2012). A debated but little examined mathematical text: Papyrus Berlin 6619. Zeitschrift für Ägyptische Sprache und Altertumskunde, 139, 158 to 170 https://www.academia.edu/1915436/Not cited in the story. It backs a register entry. | en | abstract only |
| S502 | Montelle, C. (2010). Review of The Mathematics of the Heavens and the Earth: The Early History of Trigonometry by Glen van Brummelen. Aestimatio, 7, 1 to 7, With Kathleen M. Clark.Not cited in the story. It backs a register entry. | en | read in full |
| S436 | Muir, T. (1910). The term "radian" in trigonometry. Nature, 83(2120), 459 to 460 https://www.nature.com/articles/083459d0Cited in 1850 to 1950 twice. | en | read in full |
| S305 | Napier, J. (1614). Mirifici logarithmorum canonis descriptio, ejusque usus, in utraque trigonometria. Edinburgi: ex officina A. Hart. Copy: Smithsonian Libraries. https://archive.org/details/mirificilogarit00napiCited in Chapter 6 9 times and 1500 to 1620. | la | read in full |
| S031 | Neugebauer, O. (1945). Mathematical cuneiform texts (American Oriental Series 29). American Oriental Society.Cited in Chapter 1 3 times. | en | not obtained |
| S006 | Neugebauer, O. (1957). The exact sciences in antiquity (2nd ed.). Brown University Press. https://archive.org/stream/TheExactSciencesInAntiquity/The%20Exact%20Sciences%20in%20Antiquity_djvu.txtCited in Chapter 1 7 times and 300 BCE to 400 CE. | en | read in full |
| S365 | Newton, I. (1745). Sir Isaac Newton's two treatises of the quadrature of curves, and analysis by equations of an infinite number of terms, explained. J. Stewart (Trans. and Comm.). London. https://archive.org/details/sirisaacnewtons00stewgoogCited in Chapter 7 6 times and 1620 to 1750. | en | read in full |
| S265 | Osada, N. (2011). The early history of convergence acceleration methods [author preprint; published version: Numerical Algorithms, 60 (2012), 205 to 221]. Tokyo Woman's Christian University. https://www.lab.twcu.ac.jp/osada/sc2011_rev.pdfCited in Chapter 5 9 times and 1620 to 1750 3 times. | en | read in full |
| S017 | Ossendrijver, M. (2016). Ancient Babylonian astronomers calculated Jupiter's position from the area under a time-velocity graph. Science, 351(6272), 482 to 484 https://fermatslibrary.com/s/ancient-babylonian-astronomers-calculated-jupiter-s-position-from-the-area-under-a-time-velocity-graphCited in Chapter 1 5 times, 300 BCE to 400 CE and 1950 to the present. | en | read in full |
| S018 | Ossendrijver, M. (2018). Bisecting the trapezoid: Tracing the origins of a Babylonian computation of Jupiter's motion. Archive for History of Exact Sciences, 72, 145 to 189 https://link.springer.com/article/10.1007/s00407-018-0204-4Cited in Chapter 1 5 times, c. 3000 to 300 BCE and 300 BCE to 400 CE. | en | read in full |
| S005 | Peet, T. E. (1923). The Rhind mathematical papyrus, British Museum 10057 and 10058. University Press of Liverpool / Hodder & Stoughton. https://archive.org/details/Peet_1923Cited in Chapter 1 28 times and c. 3000 to 300 BCE 3 times. | en | read in full |
| S303 | Pitiscus, B. (1595). Trigonometria: sive de solutione triangulorum tractatus brevis & perspicuus. A. Scultetus, Sphaericorum libri tres methodice conscripti et utilibus scholiis expositi, Heidelberg. Copy digitised by e-rara, page image ID 265972. https://www.e-rara.ch/zut/content/zoom/265972 (image: https://www.e-rara.ch/download/webcache/2000/265972)Cited in Chapter 6 8 times and 1500 to 1620. | la | read in full |
| S306 | Pitiscus, B. (1600). Bartholomaei Pitisci ... Trigonometriae sive de dimensione triangulorum libri quinque. [Augsburg]: [Manger]. Copy: ETH-Bibliothek Zürich, Rar 5201. https://doi.org/10.3931/e-rara-4035Cited in Chapter 6 5 times and 1500 to 1620. | la | read in full |
| S085 | Ptolemy (1515). Almagestum Cl. Ptolemei Pheludiensis Alexandrini. Gerard of Cremona (Trans.). Venice: Petrus Liechtenstein. Digitised copy, Deutsches Museum, via the Internet Archive. https://archive.org/details/almagestumcl.ptolemeideutschesmuseumCited in Chapter 2 and 1500 to 1620. | la | read in full |
| S086 | Ptolemy (1515). Almagestum. Ptolemaeus Arabus et Latinus, print no. 1, Venice: Petrus Liechtenstein. https://ptolemaeus.badw.de/print/1/70/1rCited in Chapter 2 twice and 1500 to 1620. | la | read in full |
| S061 | Ptolemy (1984). Ptolemy's Almagest. G. J. Toomer (Trans. and Ed.). Princeton University Press. Scanned copy of the Duckworth/Princeton text, 693 pp.. https://classicalliberalarts.com/resources/PTOLEMY_ALMAGEST_ENGLISH.pdfCited in Chapter 2 68 times, 300 BCE to 400 CE 4 times, 750 to 1200, 1500 to 1620 twice and 1950 to the present. | en | read in full |
| S074 | Rashed, R. (2017). Menelaus' Spherics: Early translation and al-Māhānī / al-Harawī's version (Scientia Graeco-Arabica 21). De Gruyter. ISBN 978-3-11-056823-3. https://api.pageplace.de/preview/DT0400.9783110571424_A31167978/preview-9783110571424_A31167978.pdfCited in Chapter 2 13 times, 300 BCE to 400 CE, 750 to 1200, 1200 to 1500 and 1950 to the present. | en | read in full |
| S301 | Regiomontanus, J. (1533). Doctissimi viri et mathematicarum disciplinarum eximij professoris Ioannis de Regio Monte De triangulis omnimodis libri quinque. Norimbergae: Io. Petreius (J. Schöner, Ed.). Copy: Biblioteca de la Universidad de Sevilla. https://archive.org/details/ARes352131Cited in Chapter 6 18 times and 1500 to 1620. | la | read in full |
| S445 | Regiomontanus, J. (1533). De triangulis omnimodis libri quinque. Nuremberg.Cited in Chapter 8 5 times, 1200 to 1500 and 1500 to 1620. | la | read in full |
| S307 | Rheticus, G. J. (1596). Opus palatinum de triangulis. [Neostadii in Palatinatu]: excudebat Mathaeus Harnisius. Copy: ETH-Bibliothek Zürich, Rar 9660. https://doi.org/10.3931/e-rara-9112Cited in Chapter 6 4 times, 1200 to 1500 and 1500 to 1620 5 times. | la | read in full |
| S020 | Robson, E. (2000). Mesopotamian mathematics: Some historical background. Using history to teach mathematics: An international perspective, 149 to 158, MAA (V. Katz, Ed.). https://uruk-warka.dk/mathematics/ER10%20background.pdfCited in Chapter 1 4 times and c. 3000 to 300 BCE 3 times. | en | read in full |
| S331 | Roegel, D. (2010). A reconstruction of Gunter's Canon triangulorum (1620). LOCOMAT project technical report. Nancy: LORIA. https://locomat.loria.fr/gunter1620/gunter1620doc.pdfCited in Chapter 6 7 times and 1620 to 1750. | en | read in full |
| S373 | Roegel, D. (2011). The great logarithmic and trigonometric tables of the French Cadastre: a preliminary investigation. LOCOMAT project research report. LORIA, Nancy. https://locomat.loria.fr/cadastre/analysis.pdfCited in Chapter 7 22 times, 1620 to 1750 and 1750 to 1850 3 times. | en | read in full |
| S327 | Roegel, D. (2021). A reconstruction of the tables of Rheticus' Opus Palatinum (1596). LOCOMAT project technical report, orig. 2011. Nancy: LORIA. https://locomat.loria.fr/rheticus1596/rheticus1596doc.pdfCited in Chapter 6 19 times and 1500 to 1620 4 times. | en | read in full |
| S328 | Roegel, D. (2021). A reconstruction of the tables of Pitiscus' Thesaurus mathematicus (1613). LOCOMAT project technical report, orig. 2011. Nancy: LORIA. https://locomat.loria.fr/pitiscus1613/pitiscus1613doc.pdfCited in Chapter 6 16 times and 1500 to 1620 4 times. | en | read in full |
| S329 | Roegel, D. (2021). A reconstruction of Viète's Canon mathematicus (1579). LOCOMAT project technical report. Nancy: LORIA. https://locomat.loria.fr/viete1579/viete1579doc1.pdfCited in Chapter 6 8 times and 1500 to 1620 twice. | en | read in full |
| S330 | Roegel, D. (2021). A reconstruction of the tables of Rheticus' Canon doctrinae triangulorum (1551). LOCOMAT project technical report, orig. 2010. Nancy: LORIA. https://locomat.loria.fr/rheticus1551/rheticus1551doc.pdfCited in Chapter 6 4 times and 1500 to 1620. | en | read in full |
| S183 | Rosenfeld, B. A. (1996). Geometry. Encyclopedia of the History of Arabic Science, 2, 115 to 159, Routledge (R. Rashed, Ed.). https://archive.org/stream/RoshdiRasheded.EncyclopediaOfTheHistoryOfArabicScienceVol.3Routledge1996/Rashed%20R.-Encyclopedia%20of%20the%20History%20of%20Arabic%20Science.%202-Routledge%20(1996)_djvu.txtCited in Chapter 4 5 times. | en | read in full |
| S127 | Sarma, K. V. (2008). Ganita-Yukti-Bhasa (Rationales in mathematical astronomy) of Jyesthadeva. Volume I: Mathematics. 1, Hindustan Book Agency / Springer. https://www.ms.uky.edu/~sohum/ma330/files/chennai_talks/Ganitayuktibhasa%20Vol%20I%20Sarma_Ramasubramanian_Srinivas_Sriram%20(2008).pdfCited in Chapter 3 43 times, 1500 to 1620 4 times, 1750 to 1850 twice and 1950 to the present twice. | en | read in full |
| S354 | Scultetus, A. (1595). Abrahami Sculteti Grünbergensis Silesii Sphaericorum libri tres methodice conscripti & utilibus scholiis expositi. Accessit De solutione triangulorum tractatus brevis & perspicuus Bartholomaei Pitisci Grünbergensis. Google Books full-view digitisation. https://books.google.com/books?id=CZ8jx2-edGYC&printsec=frontcoverCited in Chapter 6 3 times and 1500 to 1620. | la | read in full |
| S221 | Sedillot, L. P. E. A. (1853). Prolegomenes des tables astronomiques d'Oloug-Beg: traduction et commentaire. Firmin Didot, Paris. https://archive.org/details/bub_gb_3J5SAAAAcAAJCited in Chapter 4. | fr | read in full |
| S125 | Sengupta, P. C. (1934). The Khandakhadyaka: An astronomical treatise of Brahmagupta. University of Calcutta. Internet Archive. https://archive.org/details/dli.calcutta.10407Cited in Chapter 3 6 times and 400 to 750. | en | read in full |
| S123 | Shukla, K. S. (1976). Aryabhatiya of Aryabhata (Vol. 1, English translation). 1, Indian National Science Academy. Internet Archive. https://archive.org/details/aryabhatiyaofaryabhattaenglishtranslationkripashankarshuklasarmak.v.vol1_954_lCited in Chapter 3 twice. | en;sa | read in full |
| S072 | Sidoli, N. (2006). The sector theorem attributed to Menelaus. SCIAMVS, 7, 43 to 79 http://individual.utoronto.ca/acephalous/Sidoli_2006.pdfCited in Chapter 2 8 times. | en | read in full |
| S071 | Sidoli, N. (2007). The Arabic version of Ptolemy's Planisphere, or Flattening the surface of the sphere: Text, translation, commentary. SCIAMVS, 8, 37 to 139 http://individual.utoronto.ca/acephalous/Sidoli_Berggren_2007.pdfCited in Chapter 2 4 times and 750 to 1200. | en | read in full |
| S422 | Smith, D. E. (1923). History of mathematics: Vol. 1. General survey of the history of elementary mathematics. 1, Ginn and Company. https://archive.org/download/historyofmathema01smit/historyofmathema01smit_djvu.txtCited in Chapter 3 twice, Chapter 6 3 times, Chapter 8 10 times and 750 to 1200 twice. | en | read in full |
| S421 | Smith, D. E. (1925). History of mathematics: Vol. 2. Special topics of elementary mathematics. 2, Ginn and Company. https://archive.org/download/historyofmathema02smit/historyofmathema02smit_djvu.txtCited in Chapter 3 4 times, Chapter 6 9 times, Chapter 8 82 times, c. 3000 to 300 BCE twice, 300 BCE to 400 CE twice, 400 to 750 twice, 750 to 1200 3 times, 1200 to 1500 4 times, 1500 to 1620 5 times, 1620 to 1750 7 times and 1750 to 1850 twice. | en | read in full |
| S015 | Steele, J. M. (2016). Geminos and Babylonian astronomy. The frontiers of ancient science, ISAW open repository copy. https://archive.nyu.edu/bitstream/2451/61288/57/12.%20Steele.pdfCited in Chapter 1. | en | read in full |
| S016 | Steele, J. M. (2018). The development of the Babylonian zodiac: Some preliminary observations. Mediterranean Archaeology and Archaeometry, 18(4) https://www.academia.edu/39946417/THE_DEVELOPMENT_OF_THE_BABYLONIAN_ZODIAC_SOME_PRELIMINARY_OBSERVATIONSCited in Chapter 1 3 times and c. 3000 to 300 BCE. | en | read in full |
| S377 | Steinmetz, C. P. (1894). Complex quantities and their use in electrical engineering. Proceedings of the International Electrical Congress held in the city of Chicago, August 21st to 25th, 1893, 33 to 74, New York: American Institute of Electrical Engineers. https://archive.org/details/proceedingsinte00chicgoogCited in Chapter 7 5 times and 1850 to 1950. | en | read in full |
| S079 | Strabo (n.d.). Geography 12.4.9. Greek: A. Meineke (Ed.), Geographica, Teubner, 1877, via Perseus Digital Library. English: H. L. Jones (Trans.), Loeb Classical Library, 1928, via LacusCurtius. https://www.perseus.tufts.edu/hopper/dltext?doc=Perseus%3Atext%3A1999.01.0197%3Abook%3D12%3Achapter%3D4%3Asection%3D9 and https://penelope.uchicago.edu/Thayer/E/Roman/Texts/Strabo/12D*.htmlCited in Chapter 2. | grc;en | read in full |
| S275 | Swetz, F. J. (1992). The Sea Island mathematical manual: surveying and mathematics in ancient China. Pennsylvania State University Press. Internet Archive lending copy. https://archive.org/details/isbn_2083776009956Cited in Chapter 5. | en | not obtained |
| S124 | Thibaut, G. (1889). The Panchasiddhantika: The astronomical work of Varaha Mihira. E. J. Lazarus, Benares. Internet Archive (Digital Library of India scan). https://archive.org/details/in.ernet.dli.2015.110157Cited in Chapter 3 11 times. | en | read in full |
| S434 | Thomson, W. (1879). Treatise on natural philosophy (Vol. 1, Part 1, 2nd ed.). 1, Cambridge University Press. https://archive.org/download/treatiseonnatur01darwgoog/treatiseonnatur01darwgoog_djvu.txtCited in Chapter 7 twice, Chapter 8 and 1850 to 1950. | en | read in full |
| S437 | Thomson, J. (1910). [Letter to the editor]. Nature, 83(2120), 460 https://www.nature.com/articles/083460a0Cited in 1850 to 1950 3 times. | en | read in full |
| S087 | Toomer, G. J. (1974). The chord table of Hipparchus and the early history of Greek trigonometry. Centaurus, 18(1), 6 to 28 https://onlinelibrary.wiley.com/doi/10.1111/j.1600-0498.1974.tb00205.xCited in Chapter 2 twice and 1950 to the present. | en | not obtained |
| S326 | Tracey, K. (2021). "Disturbed" by Euclid: Thomas Fincke and the reading of Ramist mathematics in sixteenth-century Germany. Historia Mathematica, Author accepted manuscript, Maynooth University repository. https://mural.maynoothuniversity.ie/id/eprint/18601/1/KevinTraceyEuclid2021.pdfCited in Chapter 6 6 times, 1500 to 1620 and 1750 to 1850. | en | read in full |
| S225 | Ulugh Beg (n.d.). Zij-i jadid-i Sultani, MS. Greaves 5 (English catalogue record read; the Persian manuscript images were not). Bodleian Library, University of Oxford. https://digital.bodleian.ox.ac.uk/objects/8772a1fe-ab37-45d6-80ff-f1430f0e6585/Cited in Chapter 4. | en | read in full |
| S481 | Van Brummelen, G. (2009). The mathematics of the heavens and the earth: The early history of trigonometry. Princeton University Press. ISBN 978-0-691-12973-0.Cited in Chapter 4 6 times, Chapter 6 3 times and Chapter 8 3 times. | en | read in full |
| S308 | Viète, F. (1579). Canon mathematicus seu ad triangula. Cum adpendicibus. Lutetiae: apud Ioannem Mettayer. Copy: ETH-Bibliothek Zürich, Rar 3769. https://doi.org/10.3931/e-rara-18548Cited in Chapter 6 3 times and 1500 to 1620. | la | read in full |
| S024 | Vodolazhskaya, L. N. (2014). Reconstruction of ancient Egyptian sundials. Archaeoastronomy and Ancient Technologies, 2(2), 1 to 18 https://arxiv.org/pdf/1408.0987Cited in Chapter 1 3 times, c. 3000 to 300 BCE twice and 1950 to the present. | en | read in full |
| S425 | von Braunmühl, A. (1900). Vorlesungen über Geschichte der Trigonometrie: Erster Teil. 1, B. G. Teubner. https://archive.org/download/vorlesungenberg00braugoog/vorlesungenberg00braugoog_djvu.txtCited in Chapter 3 3 times, Chapter 4, Chapter 6 4 times, Chapter 8 41 times, 300 BCE to 400 CE, 750 to 1200 3 times, 1500 to 1620 and 1850 to 1950. | de | read in full |
| S426 | von Braunmühl, A. (1903). Vorlesungen über Geschichte der Trigonometrie: Zweiter Teil. 2, B. G. Teubner. https://archive.org/download/vorlesungenberg02braugoog/vorlesungenberg02braugoog_djvu.txtCited in Chapter 8 20 times, 1500 to 1620, 1620 to 1750 4 times and 1850 to 1950. | de | read in full |
| S258 | Wagner, D. B. (n.d.). Guo Shoujing's conversion of ecliptic to equatorial coordinates [research note]. http://donwagner.dk/gsj/gsj.htmlCited in Chapter 5 7 times and 1200 to 1500. | en | read in full |
| S260 | Wang, W. (2022). Science, religion and Sino-Western exchanges: literati-Jesuit translation of Euclidean geometry and its reception from late Ming to mid-Qing. Journal of Chinese History, 7, 115 to 148, Open access, CC BY 4.0. https://www.cambridge.org/core/services/aop-cambridge-core/content/view/710CD29EDADCD7FF7F362721A5A37920/S205916322200038Xa.pdfCited in Chapter 5 27 times, 1500 to 1620 3 times and 1620 to 1750 3 times. | en | read in full |
| S254 | Wu, C.-Y. (2023). The search for the Tang royal domain (Wangji 王畿): how Yixing (683 to 727) used the Zhou-era "Nine Domains" (Jiufu 九服) to map the Tang dynasty's new terrestrial realm. Nuncius, 38(2), 311 to 339 https://pure.mpg.de/rest/items/item_3515480_2/component/file_3515481/contentCited in Chapter 5 11 times, 300 BCE to 400 CE and 400 to 750 twice. | en | read in full |
| S241 | (n.d.). Zhoubi suanjing 周髀算經, juan shang. Chinese Text Project digital edition, with machine-assisted English rendering. https://ctext.org/zhou-bi-suan-jing/juan-shang/zhsCited in Chapter 5 8 times and 300 BCE to 400 CE. | zh | read in full |
| S242 | (n.d.). Zhoubi suanjing 周髀算經, juan xia. Chinese Text Project digital edition. https://ctext.org/zhou-bi-suan-jing/juan-xia/zhsCited in Chapter 5 twice. | zh | read in full |
| S243 | (n.d.). Zhoubi suanjing 周髀算經, juan xia, page 2. Facsimile of the Sibu congkan chubian 四部叢刊初編 reprint (vols. 388 to 389) of a Ming printed edition from the Xu family Jixuezhai 積學齋 of Nanling; Tsinghua University Library. https://ctext.org/library.pl?if=gb&file=77746&page=2Cited in Chapter 5. | zh | read in full |
| S246 | (n.d.). Jiuzhang suanshu 九章算術, chapter 9 Gougu 勾股. Chinese Text Project digital edition (base texts: Sibu congkan chubian and Qinding Siku quanshu). https://ctext.org/nine-chapters/gou-gu/zhsCited in Chapter 5 12 times and 300 BCE to 400 CE. | zh | read in full |
| S356 | (1542). Catalogue record for Rheticus and Copernicus, De lateribus et angulis triangulorum (1542). Library of Congress. https://www.loc.gov/item/35022703/Not cited in the story. It backs a register entry. | en | not obtained |
| S449 | (1990). Sharing teaching ideas: The legend of Soh Cah Toa. Mathematics Teacher, 83(4), 286 https://pubs.nctm.org/view/journals/mt/83/4/article-p286.xmlCited in Chapter 8 twice and 1950 to the present. | en | not obtained |
| S145 | (2023). Article on the use of derivatives in Karanottama and Drkkarana. Indian Journal of History of Science https://link.springer.com/article/10.1007/s43539-023-00090-4Not cited in the story. It backs a register entry. | en | abstract only |
Tier 3: reference works, encyclopedias and press
176 sources.
| ID | Reference (APA 7) | Language | Access |
|---|---|---|---|
| S135 | Almeida, D. F. (2007). Kerala mathematics and its possible transmission to Europe. Philosophy of Mathematics Education Journal, 20, Republished by MuslimHeritage.com. https://muslimheritage.com/kerala-math-transmission-europe/Cited in Chapter 3 3 times. | en | read in full |
| S233 | Azarian, M. K. (2019). An overview of mathematical contributions of Ghiyath al-Din Jamshid al-Kashi. Mathematics Interdisciplinary Research, 4(1), 11 to 19 https://mir.kashanu.ac.ir/article_88765.htmlNot cited in the story. It backs a register entry. | en | abstract only |
| S383 | Bal, D. (n.d.). Leibniz, Bernoulli and the logarithms of negative numbers. [Expository paper]. Montclair State University. https://msuweb.montclair.edu/~bald/files/LeibBernLogs.pdfCited in Chapter 7 9 times, 1620 to 1750 3 times and 1750 to 1850. | en | read in full |
| S448 | Ballew, P. (2024). Sine: Etymology and history of math terms; Versine, haversine: History and etymology of math terms. Pat'sBlog. https://pballew.blogspot.com/2024/10/20-sine-etymology-and-history-of-math.htmlNot cited in the story. It backs a register entry. | en | read in full |
| S193 | Brentjes, S. (2007). Khwarizmi: Muhammad ibn Musa al-Khwarizmi. Biographical Encyclopedia of Astronomers, Springer. https://mathshistory.st-andrews.ac.uk/BEA/al_khwarizmi_bea.pdfCited in Chapter 4 twice. | en | read in full |
| S019 | Bulmer-Thomas, I. (1970). Hypsicles of Alexandria. Dictionary of Scientific Biography, Scribner (MacTutor scanned copy). https://mathshistory.st-andrews.ac.uk/DSB/Hypsicles.pdfCited in Chapter 1 3 times and 300 BCE to 400 CE. | en | read in full |
| S355 | Christie, T. (2021). Renaissance science VIII. The Renaissance Mathematicus [blog]. https://thonyc.wordpress.com/2021/04/21/renaissance-science-viii/Not cited in the story. It backs a register entry. | en | read in full |
| S398 | Croarken, M. (2002). Providing longitude for all: the eighteenth century computers of the Nautical Almanac. Journal for Maritime Research, 4(1), 107 to 118 https://www.tandfonline.com/doi/abs/10.1080/21533369.2002.9668324Not cited in the story. It backs a register entry. | en | not obtained |
| S397 | Croarken, M. (2003). Mary Edwards: computing for a living in 18th-century England. IEEE Annals of the History of Computing, 25(4), 9 to 15 https://doi.org/10.1109/MAHC.2003.1253887Cited in Chapter 7 and 1750 to 1850. | en | not obtained |
| S187 | Debarnot, M.-T. (2007). Habash al-Hasib. Biographical Encyclopedia of Astronomers, Springer (T. Hockey et al., Eds.). https://islamsci.mcgill.ca/RASI/BEA/Habash_al-Hasib_BEA.htmCited in Chapter 4 and 750 to 1200. | en | read in full |
| S384 | Euler, L. (1751). De la controverse entre Mrs. Leibnitz et Bernoulli sur les logarithmes des nombres negatifs et imaginaires (E168, written 1747). Memoires de l'academie des sciences de Berlin, 5, 139 to 179, Euler Archive record. https://scholarlycommons.pacific.edu/euler-works/168/Cited in Chapter 7 twice and 1750 to 1850. | en | read in full |
| S385 | Euler, L. (1751). Recherches sur les racines imaginaires des equations (E170, written 1746). Memoires de l'academie des sciences de Berlin, 5, 222 to 288, Euler Archive record. https://scholarlycommons.pacific.edu/euler-works/170/Cited in Chapter 7 twice and 1750 to 1850. | en | read in full |
| S144 | Fleron, J. F. (n.d.). Bhaskara's approximation to and Madhava's series for sine: A mini-primary source project for Calculus 2 students. MAA Convergence. https://old.maa.org/press/periodicals/convergence/bh-skara-s-approximation-to-and-m-dhava-s-series-for-sine-a-mini-primary-source-project-for-calculusCited in Chapter 3 3 times and 400 to 750. | en | read in full |
| S028 | Greenberg, M. (n.d.). The Rhind Mathematical Papyrus [course page]. University of Washington Department of Mathematics. https://sites.math.washington.edu//~greenber/Rhind.htmlCited in Chapter 1 twice. | en | read in full |
| S133 | Harper, D. (n.d.). Sine. Online Etymology Dictionary. https://www.etymonline.com/word/sineCited in Chapter 3 twice. | en | read in full |
| S442 | Harper, D. (n.d.). Online etymology dictionary, sixteen entries: asymptote, amplitude, phase, frequency, period, sine, sinus, sinusoid, tangent, secant, trigonometry, versed, harmonic, radian, degree, minute. https://www.etymonline.com/Cited in Chapter 8 14 times and 1750 to 1850. | en | read in full |
| S188 | Hartner, W. (2007). Battani. Biographical Encyclopedia of Astronomers, Springer. https://islamsci.mcgill.ca/RASI/BEA/Battani_BEA.htmCited in Chapter 4 twice and 750 to 1200. | en | read in full |
| S076 | Joyce, D. E. (1996). Euclid's Elements, Book II, Proposition 12. Clark University. https://mathcs.clarku.edu/~djoyce/java/elements/bookII/propII12.htmlCited in Chapter 2 3 times. | en | read in full |
| S077 | Joyce, D. E. (1996). Euclid's Elements, Book II, Proposition 13. Clark University. https://mathcs.clarku.edu/~djoyce/java/elements/bookII/propII13.htmlCited in Chapter 2 twice. | en | read in full |
| S208 | Kari-Niazov, T. N. (1970). Ulugh Beg. Dictionary of Scientific Biography, Scribner (PDF hosted by MacTutor). https://mathshistory.st-andrews.ac.uk/DSB/Ulugh_Beg.pdfCited in Chapter 4 10 times. | en | read in full |
| S196 | King, D. A. (2007). Ibn Yunus. Biographical Encyclopedia of Astronomers, Springer. https://islamsci.mcgill.ca/RASI/BEA/Ibn_Yunus_BEA.htmCited in Chapter 4 3 times. | en | read in full |
| S200 | King, D. A. (2007). Ibn al-Shatir. Biographical Encyclopedia of Astronomers, Springer. https://islamsci.mcgill.ca/RASI/BEA/Ibn_al-Shatir_BEA.htmCited in Chapter 4 and 1200 to 1500. | en | read in full |
| S231 | King, D. A. (2019). Islamic sacred geography and the qibla (survey PDF). Goethe University, Frankfurt, hosted by Muslim Heritage. https://muslimheritage.com/wp-content/uploads/1799/02/davidking-sacredgeography.pdfCited in Chapter 4 3 times. | en | read in full |
| S014 | Lamb, E. (2017). Don't fall for Babylonian trigonometry hype. Scientific American, Roots of Unity blog. https://blogs.scientificamerican.com/roots-of-unity/dont-fall-for-babylonian-trigonometry-hype/Cited in Chapter 1 twice and 1950 to the present. | en | read in full |
| S249 | Madden, J. (2006). The Sea Island Problem [course page]. Louisiana State University, Department of Mathematics. https://www.math.lsu.edu/~madden/M4005s2006/SeaIsland/Not cited in the story. It backs a register entry. | en | read in full |
| S209 | Maor, E. (n.d.). Trigonometry: India and the Islamic world. Encyclopaedia Britannica, Encyclopaedia Britannica editors. https://www.britannica.com/science/trigonometry/India-and-the-Islamic-worldCited in Chapter 4 7 times and 750 to 1200. | en | read in full |
| S332 | Miller, J. (n.d.). Earliest known uses of some of the words of mathematics (S). MacTutor History of Mathematics Archive. https://mathshistory.st-andrews.ac.uk/Miller/mathword/s/Cited in Chapter 6 14 times. | en | read in full |
| S333 | Miller, J. (n.d.). Earliest known uses of some of the words of mathematics (T). https://jeff560.tripod.com/t.htmlCited in Chapter 6 9 times and 1500 to 1620. | en | read in full |
| S381 | Miller, J. (n.d.). Earliest known uses of some of the words of mathematics (R): RADIAN. Hosted by MacTutor, University of St Andrews. https://mathshistory.st-andrews.ac.uk/Miller/mathword/r/Cited in Chapter 7 twice and 1850 to 1950 3 times. | en | read in full |
| S427 | Miller, J. (n.d.). Earliest known uses of some of the words of mathematics. MacTutor History of Mathematics Archive mirror. https://mathshistory.st-andrews.ac.uk/Miller/mathword/Cited in Chapter 3 3 times, Chapter 6 5 times, Chapter 8 55 times, 750 to 1200 twice, 1500 to 1620 twice, 1620 to 1750 4 times, 1750 to 1850 twice and 1850 to 1950 3 times. | en | read in full |
| S259 | Nakayama, S. (2007). Guo Shoujing. Biographical Encyclopedia of Astronomers, Springer (PDF hosted by MacTutor). https://mathshistory.st-andrews.ac.uk/BEA/guo_shoujing_bea.pdfCited in Chapter 5 16 times, 750 to 1200 and 1200 to 1500 3 times. | en | read in full |
| S319 | North, J. D. (2008). Richard of Wallingford. Complete dictionary of scientific biography, Charles Scribner's Sons. https://www.encyclopedia.com/science/dictionaries-thesauruses-pictures-and-press-releases/richard-wallingfordCited in Chapter 6 13 times, 1200 to 1500 3 times and 1950 to the present. | en | read in full |
| S201 | O'Connor, J. J. (n.d.). Al-Kashi. MacTutor History of Mathematics Archive, University of St Andrews. https://mathshistory.st-andrews.ac.uk/Biographies/Al-Kashi/Cited in Chapter 4 5 times and 1200 to 1500 3 times. | en | read in full |
| S202 | O'Connor, J. J. (n.d.). Al-Battani. MacTutor History of Mathematics Archive https://mathshistory.st-andrews.ac.uk/Biographies/Al-Battani/Cited in Chapter 4 3 times and 750 to 1200 twice. | en | read in full |
| S203 | O'Connor, J. J. (n.d.). Nasir al-Din al-Tusi. MacTutor History of Mathematics Archive https://mathshistory.st-andrews.ac.uk/Biographies/Al-Tusi_Nasir/Cited in Chapter 4 6 times and 1200 to 1500. | en | read in full |
| S204 | O'Connor, J. J. (n.d.). Al-Biruni. MacTutor History of Mathematics Archive https://mathshistory.st-andrews.ac.uk/Biographies/Al-Biruni/Cited in Chapter 4 twice. | en | read in full |
| S205 | O'Connor, J. J. (n.d.). Abu Nasr Mansur ibn Ali ibn Iraq. MacTutor History of Mathematics Archive https://mathshistory.st-andrews.ac.uk/Biographies/Mansur/Cited in Chapter 4 twice. | en | read in full |
| S206 | O'Connor, J. J. (n.d.). Al-Khwarizmi. MacTutor History of Mathematics Archive https://mathshistory.st-andrews.ac.uk/Biographies/Al-Khwarizmi/Cited in Chapter 4 twice. | en | read in full |
| S207 | O'Connor, J. J. (n.d.). The trigonometric functions. MacTutor History of Mathematics Archive https://mathshistory.st-andrews.ac.uk/HistTopics/Trigonometric_functions/Cited in Chapter 4 10 times. | en | read in full |
| S251 | O'Connor, J. J. (n.d.). Liu Hui. MacTutor History of Mathematics Archive, University of St Andrews. https://mathshistory.st-andrews.ac.uk/Biographies/Liu_Hui/Cited in Chapter 5 4 times and 300 BCE to 400 CE. | en | read in full |
| S252 | O'Connor, J. J. (n.d.). Zu Chongzhi. MacTutor History of Mathematics Archive https://mathshistory.st-andrews.ac.uk/Biographies/Zu_Chongzhi/Cited in Chapter 5 7 times and 400 to 750. | en | read in full |
| S262 | O'Connor, J. J. (n.d.). Mei Wending. MacTutor History of Mathematics Archive https://mathshistory.st-andrews.ac.uk/Biographies/Mei_Wending/Cited in Chapter 5 twice and 1620 to 1750. | en | read in full |
| S263 | O'Connor, J. J. (n.d.). Takakazu Seki. MacTutor History of Mathematics Archive https://mathshistory.st-andrews.ac.uk/Biographies/Seki/Cited in Chapter 5 twice and 1620 to 1750. | en | read in full |
| S264 | O'Connor, J. J. (n.d.). Takebe Katahiro. MacTutor History of Mathematics Archive https://mathshistory.st-andrews.ac.uk/Biographies/Takebe/Cited in Chapter 5 twice and 1620 to 1750. | en | read in full |
| S334 | O'Connor, J. J. (n.d.). Bartholomeo Pitiscus. MacTutor History of Mathematics Archive https://mathshistory.st-andrews.ac.uk/Biographies/Pitiscus/Cited in Chapter 6 3 times and 1500 to 1620 4 times. | en | read in full |
| S335 | O'Connor, J. J. (n.d.). The trigonometric functions. MacTutor History of Mathematics Archive https://mathshistory.st-andrews.ac.uk/HistTopics/Trigonometric_functions/Cited in Chapter 6 twice. | en | read in full |
| S336 | O'Connor, J. J. (n.d.). Levi ben Gerson. MacTutor History of Mathematics Archive https://mathshistory.st-andrews.ac.uk/Biographies/Levi/Cited in Chapter 6 3 times and 1200 to 1500. | en | read in full |
| S337 | O'Connor, J. J. (n.d.). Albert Girard. MacTutor History of Mathematics Archive https://mathshistory.st-andrews.ac.uk/Biographies/Girard_Albert/Cited in Chapter 6 4 times and 1620 to 1750. | en | read in full |
| S338 | O'Connor, J. J. (n.d.). Nasir al-Din al-Tusi. MacTutor History of Mathematics Archive https://mathshistory.st-andrews.ac.uk/Biographies/Al-Tusi_Nasir/Cited in Chapter 6 4 times. | en | read in full |
| S353 | O'Connor, J. J. (n.d.). Edmund Gunter and his measuring devices. MacTutor History of Mathematics Archive https://mathshistory.st-andrews.ac.uk/SH/gunter_sh.pdfNot cited in the story. It backs a register entry. | en | read in full |
| S367 | O'Connor, J. J. (n.d.). Roger Cotes. MacTutor History of Mathematics Archive, University of St Andrews. https://mathshistory.st-andrews.ac.uk/Biographies/Cotes/Cited in Chapter 7 6 times and 1620 to 1750. | en | read in full |
| S370 | O'Connor, J. J. (n.d.). Abraham de Moivre. MacTutor History of Mathematics Archive https://mathshistory.st-andrews.ac.uk/Biographies/De_Moivre/Cited in Chapter 7 3 times. | en | read in full |
| S372 | O'Connor, J. J. (n.d.). Jean Baptiste Joseph Fourier. MacTutor History of Mathematics Archive https://mathshistory.st-andrews.ac.uk/Biographies/Fourier/Cited in Chapter 7 3 times and 1750 to 1850 twice. | en | read in full |
| S374 | O'Connor, J. J. (n.d.). Gaspard Clair Francois Marie Riche de Prony. MacTutor History of Mathematics Archive https://mathshistory.st-andrews.ac.uk/Biographies/De_Prony/Cited in Chapter 7 4 times and 1750 to 1850. | en | read in full |
| S386 | O'Connor, J. J. (n.d.). Vincenzo Riccati. MacTutor History of Mathematics Archive https://mathshistory.st-andrews.ac.uk/Biographies/Riccati_Vincenzo/Cited in Chapter 7 3 times and 1750 to 1850 twice. | en | read in full |
| S387 | O'Connor, J. J. (n.d.). Colin Maclaurin. MacTutor History of Mathematics Archive https://mathshistory.st-andrews.ac.uk/Biographies/Maclaurin/Cited in Chapter 7 and 1620 to 1750. | en | read in full |
| S388 | O'Connor, J. J. (n.d.). Brook Taylor. MacTutor History of Mathematics Archive https://mathshistory.st-andrews.ac.uk/Biographies/Taylor/Cited in Chapter 7 twice and 1620 to 1750. | en | read in full |
| S394 | O'Connor, J. J. (n.d.). Leonhard Euler. MacTutor History of Mathematics Archive https://mathshistory.st-andrews.ac.uk/Biographies/Euler/Cited in Chapter 7 twice. | en | read in full |
| S430 | O'Connor, J. J. (n.d.). Al-Jayyani. MacTutor History of Mathematics Archive https://mathshistory.st-andrews.ac.uk/Biographies/Al-Jayyani/Cited in Chapter 8 4 times and 750 to 1200. | en | read in full |
| S431 | O'Connor, J. J. (n.d.). Abu Nasr Mansur. MacTutor History of Mathematics Archive https://mathshistory.st-andrews.ac.uk/Biographies/Mansur/Cited in Chapter 8 twice. | en | read in full |
| S432 | O'Connor, J. J. (n.d.). Al-Khujandi. MacTutor History of Mathematics Archive https://mathshistory.st-andrews.ac.uk/Biographies/Al-Khujandi/Cited in Chapter 8 3 times. | en | read in full |
| S433 | O'Connor, J. J. (n.d.). Nasir al-Din al-Tusi. MacTutor History of Mathematics Archive https://mathshistory.st-andrews.ac.uk/Biographies/Al-Tusi_Nasir/Cited in Chapter 8 and 1200 to 1500. | en | read in full |
| S084 | O'Connor, J. J. (1999). MacTutor History of Mathematics Archive biographies: Hipparchus, Aristarchus, Eratosthenes, Hypsicles, Menelaus, Hypatia. University of St Andrews. https://mathshistory.st-andrews.ac.uk/Biographies/Hipparchus/ (and parallel URLs)Cited in Chapter 2 11 times, 300 BCE to 400 CE and 400 to 750. | en | read in full |
| S142 | O'Connor, J. J. (2000). Madhava of Sangamagramma. MacTutor History of Mathematics Archive. https://mathshistory.st-andrews.ac.uk/Biographies/Madhava/Cited in Chapter 3 twice and 1200 to 1500. | en | read in full |
| S143 | O'Connor, J. J. (2000). Bhaskara II. MacTutor History of Mathematics Archive. https://mathshistory.st-andrews.ac.uk/Biographies/Bhaskara_II/Cited in Chapter 3 twice. | en | read in full |
| S224 | Parviz, F. (n.d.). Practical astronomy in the Islamicate world: The significance of Ulugh Beg's Zij-i Sultani; and List of different editions of Zij-i Sultani. Stanford University. https://web.stanford.edu/~fparviz/introduction.htmlCited in Chapter 4 8 times and 1200 to 1500 twice. | en | read in full |
| S131 | Pearce, I. G. (2002). Keralese mathematics IV: Possible transmission of Keralese mathematics to Europe. Indian mathematics: Redressing the balance, ch. 19, MacTutor History of Mathematics Archive. https://mathshistory.st-andrews.ac.uk/Projects/Pearce/chapter-19/Cited in Chapter 3 3 times. | en | read in full |
| S228 | Peterson, D. (n.d.). Trig terminology: What do those words mean?. The Math Doctors. https://www.themathdoctors.org/trig-terminology-what-do-those-words-mean/Cited in Chapter 4 9 times. | en | read in full |
| S132 | Peterson, D. (2021). Trig terminology: What do those words mean?. The Math Doctors. https://www.themathdoctors.org/trig-terminology-what-do-those-words-mean/Cited in Chapter 3. | en | read in full |
| S199 | Ragep, F. J. (2007). Tusi: Nasir al-Din al-Tusi. Biographical Encyclopedia of Astronomers, Springer. https://islamsci.mcgill.ca/RASI/BEA/Tusi_BEA.htmCited in Chapter 4 twice and 1200 to 1500. | en | read in full |
| S439 | Raimi, R. A. (2005). The CEEB Commission on Mathematics, 1959. https://nonpartisaneducation.org/Review/Resources/RalphRaimiWebpages/ceeb_59.htmlCited in Chapter 8 5 times and 1950 to the present. | en | read in full |
| S352 | Roegel, D. (n.d.). LOCOMAT: reconstructed tables (index). LORIA. https://locomat.loria.fr/locomat/reconstructed.htmlNot cited in the story. It backs a register entry. | en | read in full |
| S392 | Roegel, D. (n.d.). The irrationality of pi. LOCOMAT project. https://locomat.loria.fr/roegel/pi.htmlCited in Chapter 7 twice and 1750 to 1850. | en | read in full |
| S310 | Rosen, E. (2008). Regiomontanus, Johannes. Complete dictionary of scientific biography, Charles Scribner's Sons. https://mathshistory.st-andrews.ac.uk/DSB/Regiomontanus.pdfCited in Chapter 6 25 times, 1200 to 1500 7 times and 1500 to 1620 twice. | en | read in full |
| S312 | Rosen, E. (2008). Rheticus, George Joachim. Complete dictionary of scientific biography, Charles Scribner's Sons. https://mathshistory.st-andrews.ac.uk/DSB/Rheticus.pdfCited in Chapter 6 25 times and 1500 to 1620 7 times. | en | read in full |
| S245 | Theobald, U. (2025). Zhoubi suanjing 周髀算經. ChinaKnowledge.de. http://www.chinaknowledge.de/Literature/Science/zhoubisuanjing.htmlCited in Chapter 5 12 times. | en | read in full |
| S318 | Vernet, J. (2008). Levi ben Gerson. Complete dictionary of scientific biography, Charles Scribner's Sons. https://mathshistory.st-andrews.ac.uk/DSB/Levi.pdfCited in Chapter 6 9 times and 1200 to 1500 twice. | en | read in full |
| S026 | Wang, Y. S. (2024). Theories on the origins of the sexagesimal system [HOM SIGMAA student paper]. Mathematical Association of America. https://old.maa.org/sites/default/files/images/upload_library/46/HOMSIGMAA/2024-Y_Shane_Wang.pdfCited in Chapter 1. | en | read in full |
| S446 | Weisstein, E. W. (2004). SOHCAHTOA. Wolfram MathWorld. https://mathworld.wolfram.com/SOHCAHTOA.htmlCited in Chapter 8 5 times and 1950 to the present. | en | read in full |
| S008 | (n.d.). The Rhind Mathematical Papyrus, museum number EA10057. British Museum. https://www.britishmuseum.org/collection/object/Y_EA10057Cited in Chapter 1 5 times, c. 3000 to 300 BCE and 1850 to 1950 twice. | en | read in full |
| S009 | (n.d.). The Rhind Mathematical Papyrus, museum number EA10058. British Museum. https://www.britishmuseum.org/collection/object/Y_EA10058Cited in Chapter 1 3 times, c. 3000 to 300 BCE and 1850 to 1950. | en | read in full |
| S010 | (n.d.). Tablet (MUL.APIN tablet 1), museum number 86378, registration 1899,0610.108. British Museum. https://www.britishmuseum.org/collection/object/W_1899-0610-108Cited in Chapter 1 3 times. | en | read in full |
| S011 | (n.d.). MCT 038, Plimpton 322 (P254790). Cuneiform Digital Library Initiative. https://cdli.mpiwg-berlin.mpg.de/artifacts/254790Cited in Chapter 1, c. 3000 to 300 BCE and 1850 to 1950. | en | read in full |
| S012 | (n.d.). Plimpton 322. Jewels in her crown: Treasures of Columbia University Libraries special collections, Columbia University Libraries. https://exhibitions.library.columbia.edu/exhibits/show/jewels/themes/science/158Cited in Chapter 1 twice and 1850 to 1950. | en | read in full |
| S021 | (n.d.). Egyptian merkhet, object number 1929-585. Science Museum Group. https://collection.sciencemuseumgroup.org.uk/objects/co500/egyptian-merkhetCited in Chapter 1 3 times and c. 3000 to 300 BCE. | en | read in full |
| S022 | (n.d.). Copy of a merkhet and bay, object number 1913-573. Science Museum Group. https://collection.sciencemuseumgroup.org.uk/objects/co1219/copy-of-a-merkhet-and-bay-instrumentsCited in Chapter 1 twice. | en | read in full |
| S023 | (n.d.). Ancient Egyptian altitude sundial, or shadow clock, in pine, object number 1926-992. Science Museum Group. https://collection.sciencemuseumgroup.org.uk/objects/co473/ancient-egyptian-altitude-sundial-or-shadow-clock-in-pine-woodCited in Chapter 1 3 times and c. 3000 to 300 BCE. | en | read in full |
| S027 | (n.d.). Seked. Wikipedia, Wikipedia. https://en.wikipedia.org/wiki/SekedCited in Chapter 1 5 times. | en | read in full |
| S080 | (n.d.). Entries: Ptolemy, Analemma (Greek), work 150; and Ptolemy, Planispherium (Greek), work 152. Ptolemaeus Arabus et Latinus (PAL), Bayerische Akademie der Wissenschaften. https://ptolemaeus.badw.de/work/150 and https://ptolemaeus.badw.de/work/152Cited in Chapter 2 twice and 750 to 1200. | en | read in full |
| S081 | (n.d.). Claudii Ptolemaei Opera, MS Vat.gr.1594, saec. IX, 582 imaged openings. Biblioteca Apostolica Vaticana, DigiVatLib (Polonsky Foundation Digitization Project). https://digi.vatlib.it/view/MSS_Vat.gr.1594 (IIIF manifest: https://digi.vatlib.it/iiif/MSS_Vat.gr.1594/manifest.json)Cited in Chapter 2. | en | read in full |
| S082 | (n.d.). Kitab Manalawus fi al-ashkal al-kurriyah (al-Harawi's recension of Menelaus' Spherics), MS Or. 13127, copied by Isma'il at Damascus, 4 Rabi' II 548 AH / 29 June 1153 CE, 55 fols.. British Library, via the Qatar Digital Library. https://www.qdl.qa/en/archive/81055/vdc_100023511683.0x000065Cited in Chapter 2 4 times and 750 to 1200. | en | read in full |
| S083 | (n.d.). Codex Climaci Rescriptus, MS.000149.1-.86, 137 folios; multispectral images of fols. 47r to 64v released under CC BY-SA 4.0. Museum of the Bible. https://collections.museumofthebible.org/artifacts/32858-codex-climaci-rescriptus-uncial-0250 and https://www.museumofthebible.org/ccr-creative-commons-licensed-imagesCited in Chapter 2 3 times and 750 to 1200. | en | read in full |
| S088 | (n.d.). Crossref REST API record for Toomer, The chord table of Hipparchus. Crossref. https://api.crossref.org/works?query.bibliographic=Chord+Table+of+Hipparchus+Early+History+Greek+TrigonometryCited in Chapter 2 twice and 1950 to the present. | en | read in full |
| S089 | (n.d.). LSJ entry for χορδή (chorde). A Greek-English Lexicon (9th ed.), Liddell, Scott and Jones, with Pape, Bailly, Dvoretsky and Liddell-Scott entries, via lsj.gr. https://lsj.gr/wiki/%CF%87%CE%BF%CF%81%CE%B4%CE%AECited in Chapter 2 5 times. | en | read in full |
| S129 | (n.d.). Madhava's sine table. Wikipedia, Wikipedia contributors. https://en.wikipedia.org/wiki/Madhava%27s_sine_tableNot cited in the story. It backs a register entry. | en | read in full |
| S130 | (n.d.). Jya, koti-jya and utkrama-jya. Wikipedia, Wikipedia contributors. https://en.wikipedia.org/wiki/Jya,_koti-jya_and_utkrama-jyaCited in Chapter 3 5 times. | en | read in full |
| S134 | (n.d.). Kerala school of astronomy and mathematics. Wikipedia, Wikipedia contributors. https://en.wikipedia.org/wiki/Kerala_school_of_astronomy_and_mathematicsCited in Chapter 3 3 times. | en | read in full |
| S136 | (n.d.). Aryabhata. Wikipedia, Wikipedia contributors. https://en.wikipedia.org/wiki/AryabhataCited in Chapter 3 4 times. | en | read in full |
| S137 | (n.d.). Brahmagupta. Wikipedia, Wikipedia contributors. https://en.wikipedia.org/wiki/BrahmaguptaCited in Chapter 3 3 times and 400 to 750 twice. | en | read in full |
| S138 | (n.d.). Varahamihira. Wikipedia, Wikipedia contributors. https://en.wikipedia.org/wiki/Var%C4%81hamihiraCited in Chapter 3 4 times and 400 to 750. | en | read in full |
| S139 | (n.d.). Bhaskara II. Wikipedia, Wikipedia contributors. https://en.wikipedia.org/wiki/Bh%C4%81skara_IICited in Chapter 3 4 times and 750 to 1200. | en | read in full |
| S140 | (n.d.). Madhava of Sangamagrama. Wikipedia, Wikipedia contributors. https://en.wikipedia.org/wiki/Madhava_of_SangamagramaCited in Chapter 3 6 times and 1200 to 1500 twice. | en | read in full |
| S141 | (n.d.). Nilakantha Somayaji. Wikipedia, Wikipedia contributors. https://en.wikipedia.org/wiki/Nilakantha_SomayajiCited in Chapter 3 3 times and 1500 to 1620. | en | read in full |
| S189 | (n.d.). Buzjani (Abu al-Wafa). Biographical Encyclopedia of Astronomers, Springer. https://islamsci.mcgill.ca/RASI/BEA/Buzjani_BEA.htmCited in Chapter 4 3 times and 750 to 1200. | en | read in full |
| S190 | (n.d.). Biruni. Biographical Encyclopedia of Astronomers, Springer. https://islamsci.mcgill.ca/RASI/BEA/Biruni_BEA.htmCited in Chapter 4 and 750 to 1200. | en | read in full |
| S191 | (n.d.). Kashi. Biographical Encyclopedia of Astronomers, Springer. https://islamsci.mcgill.ca/RASI/BEA/Kashi_BEA.htmCited in Chapter 4 3 times and 1200 to 1500. | en | read in full |
| S192 | (n.d.). Ulugh Beg. Biographical Encyclopedia of Astronomers, Springer. https://islamsci.mcgill.ca/RASI/BEA/Ulugh_Beg_BEA.htmCited in Chapter 4 9 times. | en | read in full |
| S194 | (n.d.). Khujandi. Biographical Encyclopedia of Astronomers, Springer. https://islamsci.mcgill.ca/RASI/BEA/Khujandi_BEA.htmCited in Chapter 4 3 times and 750 to 1200. | en | read in full |
| S195 | (n.d.). Ibn Iraq (Abu Nasr Mansur). Biographical Encyclopedia of Astronomers, Springer. https://islamsci.mcgill.ca/RASI/BEA/Ibn_Iraq_BEA.htmCited in Chapter 4 twice. | en | read in full |
| S197 | (n.d.). Sijzi. Biographical Encyclopedia of Astronomers, Springer. https://islamsci.mcgill.ca/RASI/BEA/Sijzi_BEA.htmCited in Chapter 4 twice. | en | read in full |
| S198 | (n.d.). Thabit ibn Qurra. Biographical Encyclopedia of Astronomers, Springer. https://islamsci.mcgill.ca/RASI/BEA/Thabit_ibn_Qurra_BEA.htmCited in Chapter 4 twice. | en | read in full |
| S210 | (n.d.). Abu al-Wafa. Encyclopaedia Britannica, Encyclopaedia Britannica editors. https://www.britannica.com/biography/Abu-al-WafaCited in Chapter 4. | en | read in full |
| S211 | (n.d.). Risalah al-watar wa'l-jaib. Encyclopaedia Britannica, Encyclopaedia Britannica editors. https://www.britannica.com/topic/Risalah-al-watar-wal-jaibCited in Chapter 4 twice and 1200 to 1500. | en | read in full |
| S226 | (n.d.). Al-Battani explained. Everything Explained Today (Wikipedia-derived mirror). https://everything.explained.today/Al-Battani/Cited in Chapter 4 4 times. | en | read in full |
| S227 | (n.d.). Algorithm. Online Etymology Dictionary, Etymonline. https://www.etymonline.com/word/algorithmCited in Chapter 4 twice. | en | read in full |
| S229 | (n.d.). The Starry Messenger: Ibn Yunus and mathematical techniques. Cambridge HPS. http://www.sites.hps.cam.ac.uk/starry/ibnyunusmaths.htmlNot cited in the story. It backs a register entry. | en | read in full |
| S232 | (n.d.). Medieval Islamic enumeration (excerpt used in MA330, University of Kentucky, 2023). Unattributed course reading, University of Kentucky. https://www.ms.uky.edu/~dhje223/MA330Fall2023/Medieval%20Islamic%20Enumeration.pdfCited in Chapter 4 twice and 1200 to 1500. | en | read in full |
| S234 | (n.d.). Biography: Jamshid al-Kashi. HandWiki. https://handwiki.org/wiki/Biography:Jamsh%C4%ABd_al-K%C4%81sh%C4%ABNot cited in the story. It backs a register entry. | en | read in full |
| S250 | (n.d.). Haidao Suanjing. Wikipedia, Wikipedia. https://en.wikipedia.org/wiki/Haidao_SuanjingCited in Chapter 5 3 times and 300 BCE to 400 CE. | en | read in full |
| S261 | (n.d.). Euclid, Ricci, Xu and Clavius, Ji he yuan ben: liu juan 幾何原本 六卷 [1606], LCCN 2021666487. Library of Congress, World Digital Library collection. https://www.loc.gov/item/2021666487/Cited in Chapter 5 3 times and 1500 to 1620. | en | read in full |
| S267 | (n.d.). Takebe Kenko. Wikipedia, Wikipedia. https://en.wikipedia.org/wiki/Takebe_Kenk%C5%8DCited in Chapter 5 4 times. | en | read in full |
| S268 | (n.d.). Mathematical treasures: Zhoubi suanjing. MAA Convergence (Smith and Plimpton Collections, Columbia University). https://old.maa.org/press/periodicals/convergence/mathematical-treasures-zhoubi-suanjingCited in Chapter 5. | en | read in full |
| S269 | (n.d.). Gautama Siddha. Wikipedia, Wikipedia. https://en.wikipedia.org/wiki/Gautama_SiddhaCited in Chapter 5. | en | read in full |
| S270 | (n.d.). Geyuan milu jiefa. Encyclopaedia Britannica. https://www.britannica.com/topic/Geyuan-milu-jiefaCited in Chapter 5 twice and 1750 to 1850. | en | read in part |
| S271 | (n.d.). 劉徽 (Liu Hui), 祖冲之 (Zu Chongzhi), 関孝和 (Seki Takakazu), article titles in Chinese and Japanese. Wikipedia, Wikipedia, consulted only to confirm character forms. https://zh.wikipedia.org/wiki/%E5%88%98%E5%BE%BD ; https://zh.wikipedia.org/wiki/%E7%A5%96%E5%86%B2%E4%B9%8B ; https://ja.wikipedia.org/wiki/%E9%96%A2%E5%AD%9D%E5%92%8CCited in 400 to 750. | zh;ja | read in full |
| S272 | (n.d.). Chongzhen calendar. Wikipedia, Wikipedia. https://en.wikipedia.org/wiki/Chongzhen_calendarCited in Chapter 5. | en | read in full |
| S325 | (n.d.). Robert of Chester. Science and its times, Encyclopedia.com. https://www.encyclopedia.com/science/encyclopedias-almanacs-transcripts-and-maps/robert-chesterCited in Chapter 6 3 times and 750 to 1200 twice. | en | read in full |
| S339 | (n.d.). Bartholomaeus Pitiscus. Wikipedia, Wikipedia contributors. https://en.wikipedia.org/wiki/Bartholomaeus_PitiscusCited in Chapter 6 twice and 1500 to 1620. | en | read in full |
| S340 | (n.d.). Bartholomäus Pitiscus. Wikipedia, Wikipedia-Autoren (German edition). https://de.wikipedia.org/wiki/Bartholom%C3%A4us_PitiscusCited in Chapter 6 twice and 1500 to 1620. | de | read in full |
| S341 | (n.d.). Edward Wright (mathematician). Wikipedia, Wikipedia contributors. https://en.wikipedia.org/wiki/Edward_Wright_(mathematician)Cited in Chapter 6 4 times and 1500 to 1620 twice. | en | read in full |
| S342 | (n.d.). Mercator 1569 world map. Wikipedia, Wikipedia contributors. https://en.wikipedia.org/wiki/Mercator_1569_world_mapCited in Chapter 6 twice and 1500 to 1620. | en | read in full |
| S343 | (n.d.). Albert Girard. Wikipedia, Wikipedia contributors. https://en.wikipedia.org/wiki/Albert_GirardCited in Chapter 6 4 times and 1620 to 1750. | en | read in full |
| S344 | (n.d.). Plato Tiburtinus. Wikipedia, Wikipedia contributors. https://en.wikipedia.org/wiki/Plato_TiburtinusCited in Chapter 6. | en | read in full |
| S345 | (n.d.). Regiomontanus. Wikipedia, Wikipedia contributors. https://en.wikipedia.org/wiki/RegiomontanusCited in Chapter 6 twice. | en | read in full |
| S346 | (n.d.). Richard of Wallingford. Wikipedia, Wikipedia contributors. https://en.wikipedia.org/wiki/Richard_of_WallingfordCited in Chapter 6 4 times and 1200 to 1500. | en | read in full |
| S347 | (n.d.). De revolutionibus orbium coelestium. Wikipedia, Wikipedia contributors. https://en.wikipedia.org/wiki/De_revolutionibus_orbium_coelestiumCited in Chapter 6 3 times and 1500 to 1620. | en | read in full |
| S349 | (n.d.). Catalogue record for the 1542 De lateribus et angulis triangulorum (S302). Śląska Biblioteka Cyfrowa. https://www.sbc.org.pl/dlibra/publication/440324/edition/412962/Cited in Chapter 6 4 times and 1500 to 1620. | en | read in full |
| S351 | (n.d.). Richard of Wallingford. Biographical encyclopedia of astronomers, Springer. https://link.springer.com/referenceworkentry/10.1007/978-1-4419-9917-7_1167Cited in Chapter 6 twice. | en | read in part |
| S357 | (n.d.). Prosthaphaeresis. Wikipedia, Wikipedia contributors. https://en.wikipedia.org/wiki/ProsthaphaeresisCited in Chapter 6 twice. | en | read in full |
| S358 | (n.d.). New results in the research on some mathematical works of Nasir al-Din al-Tusi. Muslim Heritage. https://muslimheritage.com/research-math-works-al-tusi/Cited in Chapter 6 3 times. | en | read in full |
| S382 | (n.d.). James Thomson (engineer). Wikipedia, Wikipedia. https://en.wikipedia.org/wiki/James_Thomson_(engineer)Cited in Chapter 7 twice and 1850 to 1950 twice. | en | read in full |
| S389 | (n.d.). Women at Royal Observatory Greenwich: Alice Everett [Blog article]. Royal Museums Greenwich. https://rmg.co.uk/stories/blog/women-rog-alice-everettCited in Chapter 7 9 times and 1850 to 1950. | en | read in full |
| S390 | (n.d.). Papers of Nevil Maskelyne: Letters from Maskelyne to Henry Andrews, MS-RGO-00004-00149. Cambridge University Library, Cambridge Digital Library. https://cudl.lib.cam.ac.uk/view/MS-RGO-00004-00149Cited in Chapter 7 5 times and 1750 to 1850. | en | read in full |
| S391 | (n.d.). Mary Edwards (human computer). Wikipedia, Wikipedia. https://en.wikipedia.org/wiki/Mary_Edwards_(human_computer)Cited in Chapter 7 5 times and 1750 to 1850. | en | read in full |
| S395 | (n.d.). De Moivre's formula. Wikipedia, Wikipedia. https://en.wikipedia.org/wiki/De_Moivre%27s_formulaCited in Chapter 7 twice. | en | read in full |
| S440 | (n.d.). Loi des cosinus. Wikipedia, Wikipedia, French edition. https://fr.wikipedia.org/wiki/Loi_des_cosinusCited in Chapter 4 twice, Chapter 8 3 times, 1200 to 1500 and 1950 to the present. | fr | read in full |
| S441 | (n.d.). Versine; and Haversine formula. Wikipedia, Wikipedia. https://en.wikipedia.org/wiki/Versine ; https://en.wikipedia.org/wiki/Haversine_formulaCited in Chapter 8 4 times and 1750 to 1850 twice. | en | read in full |
| S447 | (n.d.). SOHCAHTOA. Dictionary.com acronym entry. https://www.dictionary.com/culture/acronyms/sohcahtoaCited in Chapter 8 3 times. | en | read in full |
| S450 | (n.d.). Solution of triangles. Wikipedia, Wikipedia. https://en.wikipedia.org/wiki/Solution_of_trianglesCited in Chapter 8 twice. | en | read in full |
| S451 | (n.d.). Learn to find all trigonometric ratios in 2 minutes. Maths Vidya Institute. https://www.mathsvidya.com/blog/Cited in Chapter 8 twice. | en | read in full |
| S348 | (1490). Regiomontanus, Tabulae directionum et profectionum. Tabella sinus recti, shelfmark 120X@753@1. Augsburg: Erhard Ratdolt, 2 Jan. 1490. Keio University Libraries, Digital Collections. https://dcollections.lib.keio.ac.jp/en/incunabula/024Cited in Chapter 6 4 times and 1200 to 1500. | en | read in full |
| S311 | (2008). Peurbach (or Peuerbach), Georg. Complete dictionary of scientific biography, Charles Scribner's Sons. https://mathshistory.st-andrews.ac.uk/DSB/Peurbach.pdfCited in Chapter 6 5 times, 1200 to 1500 and 1500 to 1620. | en | read in full |
| S313 | (2008). Pitiscus, Bartholomeo. Complete dictionary of scientific biography, Charles Scribner's Sons. https://www.encyclopedia.com/science/dictionaries-thesauruses-pictures-and-press-releases/pitiscus-bartholomeoCited in Chapter 6 10 times and 1500 to 1620 6 times. | en | read in full |
| S314 | (2008). Fink (Fincke), Thomas. Complete dictionary of scientific biography, Charles Scribner's Sons. https://mathshistory.st-andrews.ac.uk/DSB/Fincke.pdfCited in Chapter 6 5 times and 1500 to 1620. | en | read in full |
| S315 | (2008). Edmund Gunter. Complete dictionary of scientific biography, Charles Scribner's Sons. https://mathshistory.st-andrews.ac.uk/DSB/Gunter.pdfCited in Chapter 6 6 times and 1620 to 1750. | en | read in full |
| S316 | (2008). Francois Viete. Complete dictionary of scientific biography, Charles Scribner's Sons. https://mathshistory.st-andrews.ac.uk/DSB/Viete.pdfCited in Chapter 6 6 times and 1500 to 1620 3 times. | en | read in full |
| S317 | (2008). John Napier. Complete dictionary of scientific biography, Charles Scribner's Sons. https://mathshistory.st-andrews.ac.uk/DSB/Napier.pdfCited in Chapter 6 11 times, 1500 to 1620 twice and 1620 to 1750. | en | read in full |
| S320 | (2008). Adelard of Bath. Complete dictionary of scientific biography, Charles Scribner's Sons. https://mathshistory.st-andrews.ac.uk/DSB/Adelard.pdfCited in Chapter 6 5 times and 750 to 1200. | en | read in full |
| S321 | (2008). Brahe, Tycho. Complete dictionary of scientific biography, Charles Scribner's Sons. https://mathshistory.st-andrews.ac.uk/DSB/Brahe.pdfCited in Chapter 6 4 times and 1500 to 1620 twice. | en | read in full |
| S322 | (2008). Bürgi, Joost. Complete dictionary of scientific biography, Charles Scribner's Sons. https://mathshistory.st-andrews.ac.uk/DSB/Burgi.pdfCited in Chapter 6 twice, 1500 to 1620 and 1620 to 1750. | en | read in full |
| S323 | (2008). Henry Briggs. Complete dictionary of scientific biography, Charles Scribner's Sons. https://mathshistory.st-andrews.ac.uk/DSB/Briggs.pdfCited in Chapter 6 twice. | en | read in full |
| S324 | (2008). Wright, Edward. Complete dictionary of scientific biography, Charles Scribner's Sons. https://www.encyclopedia.com/science/dictionaries-thesauruses-pictures-and-press-releases/wright-edwardCited in Chapter 6 5 times and 1500 to 1620. | en | read in full |
| S013 | (2010). Plimpton 322. Before Pythagoras: The culture of Old Babylonian mathematics, Institute for the Study of the Ancient World, NYU. https://isaw.nyu.edu/exhibitions/before-pythagoras/items/plimpton-322/Cited in Chapter 1 3 times and 1850 to 1950 twice. | en | read in full |
| S223 | (2015). Finding aid: Astronomy and astrology in the Islamic world. British Library and Qatar Foundation, Qatar Digital Library. https://www.qdl.qa/en/finding-aid-astronomy-and-astrology-islamic-worldCited in Chapter 4. | en | read in full |
| S350 | (2018). The Library of Erwin Tomash, lot 566: Rheticus, Opus palatinum de triangulis, Neustadt 1596. Sotheby's. https://www.sothebys.com/en/auctions/ecatalogue/2018/the-library-of-erwin-tomash-l18409/lot.566.htmlCited in Chapter 6 3 times and 1500 to 1620. | en | read in full |
| S025 | (2021). Daniel Mansfield uncovers world's oldest known example of applied geometry [News release]. UNSW Sydney. https://www.unsw.edu.au/news/2021/08/daniel-mansfield-uncovers-world-s-oldest-known-example-of-applieCited in Chapter 1 twice and c. 3000 to 300 BCE. | en | read in full |
| S393 | (2022). NAVSTAR GPS space segment / navigation user segment interfaces, IS-GPS-200N, 01-AUG-2022. Global Positioning Systems Directorate. https://www.gps.gov/technical/icwg/IS-GPS-200N.pdfCited in Chapter 7 3 times and 1950 to the present. | en | read in full |
| S213 | (2026). Al-Battani. Wikipedia, Wikipedia contributors. https://en.wikipedia.org/wiki/Al-BattaniCited in Chapter 4 7 times. | en | read in full |
| S214 | (2026). Abu al-Wafa' al-Buzjani. Wikipedia, Wikipedia contributors. https://en.wikipedia.org/wiki/Abu_al-Wafa%27_al-BuzjaniCited in Chapter 4 3 times. | en | read in full |
| S215 | (2026). Al-'Ijliyyah. Wikipedia, Wikipedia contributors. https://en.wikipedia.org/wiki/Al-%CA%BBIjliyyahCited in Chapter 4 4 times and 750 to 1200. | en | read in full |
| S216 | (2026). Muwaqqit. Wikipedia, Wikipedia contributors. https://en.wikipedia.org/wiki/MuwaqqitCited in Chapter 4 5 times and 1200 to 1500. | en | read in full |
| S217 | (2026). Law of cosines. Wikipedia, Wikipedia contributors. https://en.wikipedia.org/wiki/Law_of_cosinesCited in Chapter 4 4 times and 1200 to 1500. | en | read in full |
| S218 | (2026). Zij-i Sultani. Wikipedia, Wikipedia contributors. https://en.wikipedia.org/wiki/Zij-i_SultaniCited in Chapter 4 3 times, 1200 to 1500 twice and 1850 to 1950. | en | read in full |
| S219 | (2026). Habash al-Hasib. Wikipedia, Wikipedia contributors. https://en.wikipedia.org/wiki/Habash_al-HasibCited in Chapter 4 and 750 to 1200. | en | read in full |
| S230 | (2026). Al-Khwarizmi; Al-Biruni; Nasir al-Din al-Tusi; Ulugh Beg; Ibn Yunus; Thabit ibn Qurra; Abu Nasr Mansur; Abu-Mahmud Khujandi. Wikipedia, Wikipedia contributors (used only for Arabic-script name forms and basic dates). https://en.wikipedia.org/wiki/Al-KhwarizmiCited in Chapter 4 10 times and 1200 to 1500 twice. | en | read in full |
| S235 | (2026). Spherical law of cosines. Wikipedia, Wikipedia contributors. https://en.wikipedia.org/wiki/Spherical_law_of_cosinesNot cited in the story. It backs a register entry. | en | read in part |
Appendix K
What could not be read, and what to request
18 sources this book could not read in full: 11 not obtained at all, 7 available only as an abstract.
Nothing in this book rests on any of them. They are listed for two reasons. The first is honesty: a bibliography that quietly omits what the author could not get makes the research look more complete than it was. The second is practical. If you have a university library card, several of these are one request away, and one of them, Jensen's 1972 article in Centaurus, is the single document that would settle whether Abu Nasr Mansur belongs beside Abu al-Wafa in Chapter 4.
| ID | Reference (APA 7) | Access | Language |
|---|---|---|---|
| S396 | Ahmed, N. (1974). Discrete cosine transform. IEEE Transactions on Computers, C-23(1), 90 to 93 | not obtained | en |
| S233 | Azarian, M. K. (2019). An overview of mathematical contributions of Ghiyath al-Din Jamshid al-Kashi. Mathematics Interdisciplinary Research, 4(1), 11 to 19 https://mir.kashanu.ac.ir/article_88765.html | abstract only | en |
| S399 | Blanton, J. D. (1988). Leonhard Euler: Introduction to analysis of the infinite, Book I. New York: Springer-Verlag. https://archive.org/details/introductiontoan0000eule (controlled digital lending) | not obtained | en |
| S030 | Britton, J. P. (2011). Plimpton 322: A review and a different perspective. Archive for History of Exact Sciences, 65, 519 to 566 https://link.springer.com/article/10.1007/s00407-011-0083-4 | not obtained | en |
| S273 | Chen, J.-P. J. (2015). Trigonometric tables: explicating their construction principles in China. Archive for History of Exact Sciences, 69(5) https://link.springer.com/article/10.1007/s00407-015-0162-z | abstract only | en |
| S398 | Croarken, M. (2002). Providing longitude for all: the eighteenth century computers of the Nautical Almanac. Journal for Maritime Research, 4(1), 107 to 118 https://www.tandfonline.com/doi/abs/10.1080/21533369.2002.9668324 | not obtained | en |
| S397 | Croarken, M. (2003). Mary Edwards: computing for a living in 18th-century England. IEEE Annals of the History of Computing, 25(4), 9 to 15 https://doi.org/10.1109/MAHC.2003.1253887 | not obtained | en |
| S274 | Cullen, C. (2002). Revisiting an eighth-century Chinese table of tangents [book chapter]. Springer. https://link.springer.com/chapter/10.1007/978-94-015-9862-0_17 | abstract only | en |
| S146 | Datta, B. (2019). Hindu trigonometry. Studies in the history of Indian mathematics, Springer reprint. https://link.springer.com/content/pdf/10.1007/978-981-13-7326-8_16 | abstract only | en |
| S067 | Gysembergh, V. (2025). A note on the new evidence for Hipparchus' star catalogue. Journal for the History of Astronomy, 56(3), 287 to 290 https://journals.sagepub.com/doi/10.1177/00218286251335640 | abstract only | en |
| S400 | Heideman, M. T. (1984). Gauss and the history of the fast Fourier transform. IEEE ASSP Magazine, 1(4), 14 to 21 https://doi.org/10.1109/MASSP.1984.1162257 | not obtained | en |
| S029 | Miatello, L. (2012). A debated but little examined mathematical text: Papyrus Berlin 6619. Zeitschrift für Ägyptische Sprache und Altertumskunde, 139, 158 to 170 https://www.academia.edu/1915436/ | abstract only | en |
| S031 | Neugebauer, O. (1945). Mathematical cuneiform texts (American Oriental Series 29). American Oriental Society. | not obtained | en |
| S275 | Swetz, F. J. (1992). The Sea Island mathematical manual: surveying and mathematics in ancient China. Pennsylvania State University Press. Internet Archive lending copy. https://archive.org/details/isbn_2083776009956 | not obtained | en |
| S087 | Toomer, G. J. (1974). The chord table of Hipparchus and the early history of Greek trigonometry. Centaurus, 18(1), 6 to 28 https://onlinelibrary.wiley.com/doi/10.1111/j.1600-0498.1974.tb00205.x | not obtained | en |
| S356 | (1542). Catalogue record for Rheticus and Copernicus, De lateribus et angulis triangulorum (1542). Library of Congress. https://www.loc.gov/item/35022703/ | not obtained | en |
| S449 | (1990). Sharing teaching ideas: The legend of Soh Cah Toa. Mathematics Teacher, 83(4), 286 https://pubs.nctm.org/view/journals/mt/83/4/article-p286.xml | not obtained | en |
| S145 | (2023). Article on the use of derivatives in Karanottama and Drkkarana. Indian Journal of History of Science https://link.springer.com/article/10.1007/s43539-023-00090-4 | abstract only | en |
Appendix L
Glossary
94 terms a 14-to-18-year-old reader, or a teacher from another subject, may not know. Mathematical terms are defined as this book uses them, not in full generality.
(Arabic al-shakl al-mughni)
the name for the theorem that replaced the Menelaus configuration
IdeaChapter 4
(Arabic al-Khwarizmi, a byname meaning the man from Khwarazm,)
a finite step by step procedure for computing something. The name is al-Khwarizmi's byname, 'the man from Khwarazm', worn smooth: Latinised as algorismus it meant the Hindu-Arabic reckoning method, then any method, and nobody put his name on it on purpose. English algorism is attested in the early 13th century and algorithm from the 1690s
IdeaChapter 3
(Latin amplus)
half the distance between the maximum and minimum of a sinusoid. The general English sense is 1540s and Legendre uses the word mathematically in 1786; the first use for the height of a trigonometric graph was not dated by this book
IdeaChapter 7
(Ancient Greek asymptotos)
a line that a curve approaches without ever meeting, as the tangent curve approaches a quarter turn
CurvesChapter 8
(Akkadian beru, written with the Sumerian sign DANNA)
the Babylonian time unit of 30 US, attested in MUL.APIN before about 750 BCE
IdeaChapter 8
(Chinese bi)
the gnomon of the Zhoubi suanjing. Theobald's rendering of the old title as trigonometry of circles is a modern gloss, not an ancient one
Idea
(Chinese chong cha)
the double sighting survey method that measures an unreachable height and distance with no angle at all
IdeaChapter 5
(Ancient Greek chorde)
the straight line joining two points of a circle. When the geometric sense first attached to the word was not established by this book, and Ptolemy never uses chorde for the object
IdeaChapter 1
(Latin complementi secans)
the reciprocal of the sine. The object is much older than the word: Abu al-Wafa uses a shadow diameter for it, and Bianchini tabulates a cosecant at R = 10,000 in 1463
IdeaChapter 4
(Sanskrit koti-jya by analogy, and Latin complementi sinus)
the sine of the complementary angle. Datta and Singh argue that kojya became co-sinus by descent; the Indian track records against this that early medieval texts say complementi sinus, so the resemblance may be coincidence. That link is unresolved
IdeaChapter 1
(Latin complementi tangens)
the reciprocal of the tangent, and the ratio the Islamic shadow tables call the straight shadow
IdeaChapter 1
(Latin co- from complementi plus versus plus sinus)
1 minus the sine of the angle, a navigation table function now obsolete
IdeaChapter 8
(Chinese cun qian li)
the shadow rule of the Zhoubi suanjing, which the survey of 724 to 725 measured and disproved
IdeaChapter 5
Named after de Moivre, who never stated it in his works. A closely related formula appears in a paper of 1707 and again in a 1722 publication, but he was eliminating a variable between two polynomials, not thinking about points on a circle. He had the machinery before Euler and got a formula named after him that he never wrote.
PeopleChapter 7
(possibly Greek omicron from moira, or a florescent form of Latin gradus. Cajori says the Greek line of descent has not been established)
the small raised circle that marks degrees of arc
IdeaChapter 7
(Ancient Greek euthys, straight,)
Ptolemy's own name for what we call chords
IdeaChapter 8
(Latin ex-, out of, plus secans)
the amount by which the secant of an angle exceeds the radius, used by railway surveyors to lay out curves
IdeaChapter 8
(Arabic al-suds al-Fakhri)
A 60-degree arc about 43 meters across, built by al-Khujandi and named for the patron who funded it rather than the man who built it.
IdeaChapter 4
(Latin frequentia from frequentem)
the number of cycles per unit time. English 1550s in the general sense, 1831 in the physics sense
IdeaChapter 7
(Ancient Greek gonia plus metron)
the measurement of angles, an older name for the analytic side of trigonometry
IdeaChapter 8
(Chinese gougu, also written with the variant character for gou)
the Chinese name for the right triangle relation. Modern Chinese for the Pythagorean theorem is gougu dingli
IdeaChapter 5
Named after Gregory, who published it in 1671. The same series is derived, geometrically and constructively, in the Kerala text this book quotes in Chapter 3, two centuries earlier.
PeopleChapter 3
(Arabic)
Ibn Yunus compiled the tables; they carry the name of the caliph who paid for them.
IdeaChapter 4
(Ancient Greek harmos)
a sinusoidal component whose frequency is a whole multiple of a fundamental. English 1560s of music; harmonic analysis is credited to William Thomson and is in print by December 1863
IdeaChapter 7
(English half plus versed plus sine, matching Latin semiversus)
(1 minus cos theta) divided by 2, the function in the haversine formula for great circle distance
IdeaChapter 8
(Chinese huiyuan)
Shen Kuo's technique for the arc of a circular segment, the start of the Chinese circle-division tradition
IdeaChapter 5
(Arabic ilm al-miqat)
the Islamic discipline that paid for six centuries of applied trigonometry. It names a genre from at least the 13th century
IdeaChapter 4
(Akkadian indanum)
a Babylonian slope quantity, a property of the whole shape rather than of one side
IdeaChapter 1
(Latin baculus Jacobi, from a Hebrew poem)
The cross-staff European navigators used to find latitude for three hundred years. The name comes from an allusion to Genesis 32:10 in a Hebrew poem, which Latin readers took as an attribution to a man called Jacob. The instrument is described in Levi ben Gerson's Sefer Tekunah, and Peter of Alexandria put that into Latin in 1342.
IdeaChapter 6
(Sanskrit jya plus utpatti plus ganita)
the indigenous Indian name for trigonometry. Nothing cognate with the Greek trigonometria was used
IdeaChapter 3
(Sumerian KI)
the word standing before the row number in column IV of Plimpton 322
Idea
(Akkadian kippatum, from kapapum,)
the Old Babylonian word for a circle, meaning both the disc and the circumference that defines it
IdeaChapter 1
The arctangent series at t = 1. It falls out of the same Kerala derivation as the series named after Gregory, one substitution later.
PeopleChapter 3
(Ancient Greek mekos and platos)
the east-west and north-south extension of a constellation. Grasshoff and Hoffmann dispute that this usage is diagnostic of Hipparchan authorship
IdeaChapter 8
The theorem Ptolemy proves and leans on throughout Almagest Book I. Ptolemy never credits Menelaus for it: Toomer notes that the Almagest mentions Menelaus only as an observer, for two nights of star-watching. No Greek manuscript of Menelaus's own Sphaerica survives, so what he proved cannot be read directly.
ManuscriptsChapter 4
(English mid plus line)
the horizontal center line of a sinusoid, the line y = D about which the graph oscillates
IdeaChapter 8
(Akkadian mithartum, the reflexive stem of maharum)
the Old Babylonian word for a square, meaning both the square and its side
IdeaChapter 8
(Sumerian written form for Akkadian sumsu)
the standard heading of the final, line-numbering column of a Babylonian table, used in Larsa-area administrative tables from 1822 BCE and in Plimpton 322 column IV
IdeaChapter 1
(Akkadian mutarrittum)
the perpendicular side of a shape, named by metaphor from the plumb line
IdeaChapter 1
(Arabic muwaqqit)
a professional astronomer salaried by a mosque or madrasa. The office appears in Egypt in the 13th century; Wikipedia, citing King, names ibn Simun as the first known holder, which is less firm than the century
IdeaChapter 4
(Ancient Greek periodos)
the length of one full cycle of anything that repeats: a graph, a sequence, or a motion. English from the early 15th century, and the sense time of one revolution by 1727
IdeaChapter 1
(Ancient Greek phasis)
the horizontal shift of a sinusoid. English 1705 of the moon, and 1861 for a stage in a recurring movement
IdeaChapter 2
(English phase plus vector)
a constant complex number standing in for a sinusoid of time
IdeaChapter 7
(Greek letter pi, the initial of periphereia)
the ratio of a circle's circumference to its diameter
Idea
(modern museum designation)
the museum name of the Old Babylonian tablet at the center of chapter 1
IdeaChapter 1
(Greek astronomical practice through medieval Latin)
the marks for minutes and seconds of arc, against m and s for minutes and seconds of time
IdeaChapter 7
(Latin coinages by Viete as replacements for tangent and secant)
Viete's rejected names for the tangent and the secant, offered because he judged tangent ambiguous
IdeaChapter 6
(Ancient Greek prosthesis plus aphairesis)
the pre-logarithm trick that turns a multiplication into an addition using a product to sum identity. Braunmuhl finds its ancestor in Ibn Yunus's use of the formula for cos a cos b
IdeaChapter 4
Plimpton 322 holds fifteen rows of Pythagorean triples, written roughly 1,200 years before Pythagoras was born.
IdeaChapter 3
Named for the Scottish lawyer and antiquarian who bought it at Thebes around 1858, not for the scribe who wrote and signed it.
ManuscriptsChapter 1
(Arabic sahm)
the Latin name for the versed sine, the arrow between the bow and the bowstring
IdeaChapter 5
(Japanese sangaku)
a votive wooden tablet carrying a geometry problem, hung at a temple or shrine
IdeaChapter 5
(Arabic shadow diameter as the object, and Latin secans, present participle of secare,)
the reciprocal of the cosine
IdeaChapter 4
(Egyptian skd)
the horizontal run per one cubit of rise of a pyramid face, a cotangent in mixed units. The attestation is secure and the etymology is Peet's own hedged suggestion
LanguageChapter 1
(Akkadian siliptum, from salapum,)
the diagonal of a rectangle, and by the Mesopotamian identity of figure and defining line the rectangle itself. It heads column III of Plimpton 322
IdeaChapter 8
(English, by analogy with d)
the superscript minus one that marks an inverse trigonometric function, and the deliberate collision with the reciprocal that students still trip over
IdeaChapter 8
(Latin abbreviation in Euler's usage)
the modern reading of sine as a function of a real number
PeopleChapter 7
(Latin abbreviations of sinus, tangens and secans, with complementi for the co-functions)
the standard function abbreviations on every modern calculator
Idea
(Sanskrit ardha-jya and jya-ardha, shortened)
the ratio of the opposite side to the hypotenuse, and the y coordinate on the unit circle
IdeaChapter 1
(Latin abbreviations plus English hyperbolic)
the hyperbolic functions, written with the h that has kept its place ever since
IdeaChapter 7
(Latin sinus rectus, translating Sanskrit krama-jya)
the medieval Latin name for the sine proper, as against the versed sine
IdeaChapter 6
(Latin sinus plus Ancient Greek -oeides, form or shape,)
the curve of sines, or any graph of the form y = A sin(Bx + C) + D. The Latin phrase linea sinuum is earlier, in Honore Fabri, 1659
CurvesChapter 7
(Chinese suifang yan fa, calquing Sanskrit sva-desa-aksa)
terrestrial latitude, built out of native Chinese words to carry an imported Sanskrit idea
IdeaChapter 5
(Akkadian takiltum, from the verb kullum)
the disputed heading word of Plimpton 322 column I, on which the reading of the whole tablet partly turns
IdeaChapter 1
(Arabic zill, shadow, for the object, and Latin tangens, present participle of tangere,)
the ratio of the opposite side to the adjacent side, and the length of the shadow that started it
IdeaChapter 1
Printed in 1715, ignored for fifty-seven years, promoted by Lagrange in 1772 to the status of founding principle of the calculus, and only called the Taylor series in 1785: seventy years after Taylor wrote it, and fifty-four years after he died.
PeopleChapter 7
(Ancient Greek trigonon plus metron)
the subject of this book. The 1595 print appearance is firm; none of the fetched sources spells out the Greek roots, so treat the root gloss as background
IdeaChapter 1
(Sanskrit trikona plus miti, a modern calque on the Greek)
the modern Sanskrit and Hindi name for trigonometry
IdeaChapter 3
(Ancient Greek tripleuron, from tri- three and pleura side)
Menelaus's word for a spherical triangle
IdeaChapter 2
(Egyptian wha-tbt)
the side of the square base of a pyramid
IdeaChapter 1
(Akkadian ukullu)
the reciprocal slope, run over rise, usually given as a certain length per cubit
IdeaChapter 8
(Arabic straight shadow)
the shadow of a gnomon standing on a horizontal plane, which behaves as the cotangent
IdeaChapter 4
(Arabic turned shadow)
the shadow of a gnomon set into a vertical wall, which behaves as the tangent
IdeaChapter 4
(English unit plus circle)
the circle of radius 1 centered at the origin, on which the coordinates are the cosine and the sine
IdeaChapter 2
(Sumerian sign US used as a unit)
the Babylonian unit that becomes the degree of arc in Late Babylonian astronomy from about 400 BCE
Idea
(Sanskrit utkrama-jya)
R times (1 minus cos theta). Tables gave it for a thousand years because it never goes negative and nobody had negative numbers
IdeaChapter 3
(Chinese xian)
the hypotenuse of a right triangle, and separately the chord of a circular arc
IdeaChapter 5
(Japanese enri)
the Japanese family of infinitesimal and series methods for arcs, areas and volumes
IdeaChapter 5
(Arabic zill)
the Arabic ancestor of the tangent and cotangent. High confidence on the Arabic side, medium on the Latin dating
IdeaChapter 4
(Akkadian ziqpu)
the ziqpu stars, named for culminating at the meridian, in MUL.APIN I iii 49 to I iv 9
IdeaChapter 1
Back matter
Provenance and reproducibility
Where every number on these pages comes from, and how to check it.
This book is generated, not typed. Eight registers hold the structured data; eight tagged narrative files hold the prose; a build script turns them into the page you are reading. No table in this book is hand written. If a count appears in a sentence, it was computed from the register at build time, which is why the sentence cannot go stale when the register changes.
| Register | Rows |
|---|---|
| conflicts | 84 |
| dates | 260 |
| images | 67 |
| people | 185 |
| placements | 50 |
| portraits | 185 |
| sources | 332 |
| terms | 94 |
The gates. Nothing is published until 46 checks pass. Among them:
- The register gate reads all eight registers, 332 sources and 185 people among them, checking structure, duplicates, dangling references and style.
- A self-test proves each of those checks fires, by breaking a register on purpose and confirming the check goes red.
- The narrative lint requires every prose block to be tagged and every source pointer to resolve.
- The numerical verification recomputes 93 historical calculations at sixty significant digits, and reports 81 more values without asserting anything about them.
- The anchor gate checks every link into the course book against that book's own live table of contents.
The check that matters most is the one on the checks. A gate that passes because it is looking in the wrong place is worse than no gate, because it produces confidence instead of doubt.
So the gates are mutation tested: the build breaks the book on purpose and confirms the gate goes red. 70 mutations across 38 of the 46 checks, plus a twenty-two check self-test on the register gate, nine probes on the prose gate, and thirteen on the portrait name matcher. The rest are not mutation tested, and GATE-APPLICABILITY.md in the repository names every one of them, along with two checks that currently measure nothing at all because this book has no content of the kind they look for. A gate that stays green while its own subject is broken is reported as blind and does not count as passing.
That is not a boast, it is a repair. Two gates in this book once could not fail at all. One printed its findings and then exited zero regardless, so every green it had ever reported was uninformative. The other measured the accent colors of a different book, printed sixteen lines of "not checked", and then said the contrast was fine. Both were found by a fresh-eyes review that was told to break things rather than to confirm them, and both are fixed and mutation tested now.
Known limits, stated plainly. 73 disagreements between sources are unresolved and Appendix F lists them. 18 sources could not be read in full and Appendix K lists them. One of the 185 people has no established pronunciation. Where a portrait could not be found or could not be cleared, the page shows an empty frame and says why, and carries one line inviting you to send a likeness if you know of a free one: 138 people have that frame, and 47 have a face.
Back matter
About this book
Written as the free companion reader for Megan Warren's Trigonometry course, 2026-2027. It exists because the one-sentence version of this history is not merely thin, it is misleading in a way that costs students understanding. A reader who thinks trigonometry began as a subject about triangles will never work out why the unit circle turns up, or why there are six functions, or why the sine is named after a bay.
The course itself is Megan Warren's design. This book was written to that design and keyed to it section by section: Appendix I holds 50 placements linking each story to the exact place in her textbook where it does the most good.
Every claim traces to a source. 332 sources went in, and where a source could not be obtained the book says so in the reader's own language rather than in a note to the author: Appendix K lists every one. 84 disagreements between sources are logged and 73 are still open. Every calculation in Appendix E was re-derived at sixty significant digits and checked in code before it shipped, and that appendix is the script's own output rather than a transcription of it.
Design and voice: hers as well, navy with a teal accent. Math is typeset with KaTeX and fully embedded, so the book renders with no internet connection.
Like what you've seen here? Megan takes on a small number of commissions at a time. She builds custom interactive textbooks like this one, and custom software for classrooms and small teams, on commission from Boston.