Trigonometry: Secant-ing out New Life in Ancient Civilizations

Plimpton 322 from

It’s irresistible. The siren song of Wikipedia calls to me. All I was trying to do was find out which Greek invented trigonometry. Was it Pythagoras and his bean-renouncing cult or someone else? And I come across this enticing little tidbit, a curious little reference which, to a history buff is like the smell of fresh cookies…

Based on one interpretation of the Plimpton 322 cuneiform tablet (c. 1900 BC), some have even asserted that the ancient Babylonians had a table of secants.[8] There is, however, much debate as to whether it is a table of Pythagorean triples, a solution of quadratic equations, or a trigonometric table.

Wikipedia: History of Trigonometry

Much debate? Some have asserted? This sounds like historical mystery to me. I was instantly overjoyed at the thought of poking around to see if anyone denounced anyone else in the public square or started fistfights or wrote long letters to the editor of scientific journals about how their enemies were cretins who didn’t know a hypotenuse from a hippopotamus. And I wasn’t disappointed.

Don’t Be Afeared, it’s Just Math

First, a few definitions. Even if you’ve never taken trigonometry or if the very word causes you to put a blanket over your head, don’t worry. Imagine that it’s a warm sunny day in Greece (or Babylonia or Sumeria or Egypt) and you notice that the pillar of the nearby temple, next to where you are sunning yourself, throws a shadow. Since you like to measure things, you get out your handy measuring stick and you measure the length of the shadow. You know the length of the pillar. You start doing calculations.

Eventually, you develop a table of relationships of that triangle, and all others like it, that shows the length of the pillar (or table leg if you like), the shadow, and the distance between the two. Pythagoras is credited with the “rule,” or theorem, that the pillar-squared and the shadow-squared would equal the square of the distance between the two. That a2 + b2 = c2 is the “Pythagorean triple.”

The triple has theme and variations. Somebody–lots of somebodies actually–figured out that if you could measure the angle as well as one of the distances, then you could know the other two distances. Very handy if you don’t actually know the height of the tree or pillar and you don’t have a ladder.

That’s trigonometry.

Trigonometry: if you can measure the angle and one side of a right triangle, you can know the length of the other two sides. Or if you know two sides, you can calculate the third.

Hammurabi and his Calculating Friends

You might remember the Babylonians if you’ve ever heard about Hammurabi’s Code, which was one of the first known legal codes written down. It had the earliest example of justice through retribution, codified “eye for an eye” stuff. The Babylonians, and their earlier cousins from modern day Iraq, the Sumerians, also loved to keep track of things, as a lot of the clay tablets dug up by archaeologists include long inventory lists of grain, oil, and food stores. If they counted things, they probably added, which means they cross-footed and probably used primitive Excel, so Accountants! My people!

You’re an accountant! You’re in a noble profession! The word “count” is part of your title!

Max Bialystock from The Producers

The other things that the Babylonians did was create numbering systems that led to how we think about time and circles. They used Base 60. Yep. Why they used 60 itself is a source of much curiosity, explained in an interesting analysis from J.F. O’Connor and E. F. Robertson here. Plausible explanations include either that A) 60 is the first number divisible evenly by 1, 2, 3, 4, and 5; B), there are roughly 12 months in a solar year times 5 visible planets of 1500 B.C.; or C) the number of 12 non-thumb knuckles on one hand times the 5 fingers on your hand could let you count to 60. These might seem far-fetched, but if you are an ancient accountant without even an abacus, much less a calculator button on your smart phone, then you probably created all sorts of counting methods using your body.

What seems harder to understand is that the Babylonians–and pre-Greek civilizations in general–didn’t use angles. That is, they didn’t have notations or measurements that could be interpreted as angles. They kept extensive records on “the rising and setting of stars, the motion of the planets, and the solar and lunar eclipses, all of which required familiarity with angular distances measured on the celestial sphere. [Wikipedia/Eli Maor]” They subdivided circles into 360 degrees and created hours and minutes by on 60 unit intervals.

It’s a little hard to believe or understand how Babylonians could do all this sophisticated math-y stuff and not use angles, but as scholars later showed, not everyone came to mathematical truths in the same manner that the Greeks did. What they did do is create tables of numbers–possibly the relationships between the length of the sides of triangles?–which were very exact. We know this from a Babylonian tablet called Plimpton 322, and that’s where some juicy scholarly fighting comes to the fore.

Stealing from the Non-Europeans

First, remember that there’s an automatic assumption by scholars for hundreds of years that Greeks invented everything. Typical was the assertion by mathematics professor Morris Kline in 1964, who wrote that compared to the Greeks, “the mathematics of Egyptians and Babylonians is the scrawling of children just learning to write, as opposed to great literature.”

In response to such a Euro-biased point of view, others began to put together analyses of Indian, Arabic, and Babylonian contributions to mathematics, such as George Joseph’s Crest of the Peacock. Unfortunately, Joseph and collaborator Dennis Almeida were accused of plagiarism of the work of Indian mathematician C. K. Raju which ended with the University of Exeter dismissing Almeida and the University of Manchester disassociating itself with Joseph’s work.

Seems a bit of classic irony, doesn’t it? British professors Joseph and Almeida write a best-seller about how non-European mathematicians predated Europeans in calculus and geometry by stealing borrowing extensively (without acknowledgement) from scholars in India.

Forget Scholarship, All You Need is YouTube

Aside from that little side scandal, what Crest of the Peacock actually includes is the debate about tablet 322. Over dozens of years, researchers had determined that the table includes four columns, one (far right) being the row number, and the other three sets of numbers which have some unknown relationship. (See? Ancient Excel spreadsheet!)

There was agreement on what the numbers were and agreement on the pronunciation of the words that headed the columns. But whether the words meant “square of the diagonal” or “square root” was debatable. Then, there was a ragged line on the left side of the tablet, which meant perhaps that numbers were missing. Some experts suggested numbers were missing, others theorized based only on what was there. All of it, remember, base 60.

Plimpton 322 translated, photo from

As I dug further into this research about the Babylonians and trigonometry, one of the next things that kept popping up was a reference to the Austin Powers-lookalike in the picture below, smiling and holding the infamous Plimpton 322 tablet. Multiple articles from August 2017 described how he and his research pal at the University of New South Wales had discovered “new information” about this tablet. In Science News, a dubious publication rife with ads for new bras and suggestions about how to make my money off of my utility’s impending bankruptcy, Professor Daniel Mansfield wrote about how he made this startling discovery. It revealed, he said, that these were trigonometric combinations. The article even attached a simulation showing how the tablet created right triangle combinations, a YouTube video which quickly morphs into a suggestion that you next watch a Ted Talk or an Interview with a Four-Month Old.

Daniel Mansfield of USNW, who could give P.T. Barnum a run for his money

The issue with Dan and Norm’s new theory, their Theory Number One and it’s their theory, all theirs, and they are going now to relate it, this theory of theirs which belongs to them copyright DMUSNW (see Ann Elk)… is that it’s not original and not particularly theirs. It appears to be primarily a publicity stunt. Scholars had been puzzling over the exact purpose and structure of Plimpton 322 for the entire century since the tablet was bought by collector George Plimpton in 1922. Mansfield’s claims of discovery just repeated what others had said, which even prompted a response in Scientific American entitled “Don’t Fall for Babylonian Trigonometry Hype.

Earlier researchers from the 1980s like Creighton Buck had long before put forth the idea that the table was a list of secants, i.e. relationships between the hypotenuse and the height of my hypothetical temple pillar. Buck had written a well-known paper called “Sherlock Holmes in Babylon” (1980). Linked to the idea was the fact that using base 60 would allow the Babylonians to be more precise, which might explain how they could create trigonometric ratios without using angles (sines/cosines) at all.

Eleanor Smacks Down Sherlock and Ends My Babylonian Dreams

On the other hand, Oxford scholar Eleanor Robson, in her counterargument “Neither Sherlock Holmes Nor Babylon,” argued convincingly that the table was neither related to trigonometry or triangles at all, but was rather a list of reciprocal pairs. While I prefer to imagine brandishing Plimpton 322 — Voila! Trigonometry!, Robson makes a pretty good case that because the Babylonians didn’t think about angles (or even circles) in the same manner as the Greeks (and us), then we shouldn’t superimpose our ideas on their Excel spreadsheet cuneiform table. She points out that if you look at the column headings, which are there for a reason, you can extract what they mean based on the language rather than impose meaning that makes sense to a mathematician. In other words, the columns probably describe a common mathematical thing, like 1/x, rather than a never-before-seen-in-Babylon idea, like a secant.

Robson makes her case, not just with math ideas but also based on a linguistic expertise of the words and a knowledge from ancient Mesopotamian culture of what kind of person probably wrote the table. Based on archaeological evidence, rather than mathematical theorizing, she suggests the writer was a scribe or a teacher who was creating a series of exercises involving x and 1/x. There have been plenty of other cuneiform tables found that do something similar. She was awarded the Ford Prize for her work, which means her take-down of earlier theories was considered pretty definitive. So much for Mansfield, so much for Friberg, Buck, Neugebauer & Sachs. Easy come, easy go-sine.

I guess this means the Babylonians didn’t invent trigonometry after all. On the other hand, the Egyptians…

3 Replies to “Trigonometry: Secant-ing out New Life in Ancient Civilizations”

    1. But the Egyptians… used it for the pyramids. Plus “perfected” might just be a Greek Platonic concept, so circular? But lots of fingers in lots of pis, for sure!

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