That ancient Babylonian “trigonometry” tablet isn’t really trigonometry

The new craze on the Internet seems to be that ancient “Plimpton 322” (P322) Babylonian tablet in which trigonometry is invented millennia before it was first thought to have been. The authors of the paper themselves, University of New South Wales mathematicians Daniel Mansfield and Norman Wildberger, are responsible for much of the hyperbole, when they say things like, “It’s a trigonometric table that’s so unfamiliar and advanced that in some respects it’s superior even to modern trigonometry.” Or this gem: “It’s actually trigonometry, but a different kind of trigonometry…this is a ratio-based trigonometry.” They even call it “Babylonian Exact Sexagesimal Trigonometry.”

None of this is really true.

Trigonometry at its most basic level is about the relationships between angles in a triangle and the ratios of the side lengths. They tout the newness of “ratio-based trigonometry” though trig has always been ratio-based. For example, in a 30-60-90 right triangle, the ratio of the leg opposite the 30° angle to the hypotenuse is 1/2. (To see why, start with an equilateral triangle and cut it in half. Notice that each half is a 30-60-90 triangle. Now think some more!) In modern trigonometry, that 1/2 ratio is called the sine of the 30° angle or the cosine of the 60° angle. How do we determine this using P322 “trigonometry”? We don’t, because P322 contains no idea of angle. Triangle ratios is trigonometry like Dick Smothers is the Smothers Brothers.

What P322 actually contains is a series of “Pythagorean triples,” along with a squared ratio of either the short leg to the long leg or of the hypotenuse to the long leg (we’re not sure since the left side has broken off and the only difference would be whether they start with a one or a zero). These fractions are written as sexagesimals, which are like decimals but in base 60 rather than base 10. A Pythagorean triple is a sequence of integers like 3,4,5, in which the squares of the first two add up to the square of the third, making them legs and hypotenuse of a right triangle. In this case, 3²+4²=5². Assuming the left column contains squared ratios of the short to long legs, we would get (3/4)² or 9/16. We could leave it at that, but let’s convert it to decimal: .5625. That’s the exact squared ratio as a decimal. So far so good. The next Pythagorean triple is 5,12,13. If we compute the same squared ratio, we get (5/12)², or 25/144. That’s still exact, but if we try to convert it to decimal we get a repeating decimal, .1736111…, which we can’t write exactly if we limit ourselves to a finite number of digits. That’s where the sexagesimal (base 60) numeral system of the Babylonians wins out, because many more fractions are terminating sexagesimals than are terminating decimals. (I call it a numeral system rather than the more natural number system to emphasize that these aren’t new numbers, they are just new, to most of us, ways of writing numbers, i.e. numerals.) So many more fractions are terminating sexagesimals than are terminating decimals. That’s cool, but far from Earth-shattering. Maybe that was very useful to the Babylonians, but it’s not trigonometry. And it certainly won’t help us at all today.

Depending on which ratios were in the original table, these would either be the squared tangents or squared cosecants of the angle opposite the short side, which sounds a lot like trigonometry except that P322 is completely silent about angles! So all we really have is a table containing Pythagorean triples and one of their (squared) ratios.

What of the claim that the Babylonians knew the Pythagorean Theorem  a millennium before Pythagoras? That’s not news either. For example the Encyclopedia Brittanica already knew this. The origin (or origins) of the theorem and its proof are lost to history, and it may have been the Babylonians, but being aware of it, as the authors of P322 surely were, is not the same as having proved it.

What was P322 good for? If you know two sides of a right triangle, including the long leg, you can find the squared ratio of the side you know to the long leg, and find the closest match in the table. I’m not even sure what that’s good for (since you already know the other ratio since the squared ratio of the hypotenuse to the long leg is just the squared ratio of the short leg to the long leg plus one). There are some theories, but it’s not trigonometry.

Is the “exact” nature of it new? No, just look up any list of Pythagorean triples and compute any ratios you want. Or you can just use the Pythagorean Theorem directly, e.g. if you know the two legs of a right triangle are 6 and 8, just compute the square root of 6²+8² and get 10 for the hypotenuse. Nothing to see here, move along!

As final “proof” that this is not trigonometry, I give you an example of a typical problem given to beginning trig students: If a 10′ ladder (teachers of trig are obsessed with ladders) is placed at a 70° angle to the ground against a wall, how high is the top of the ladder from the ground? P322 is completely useless here, but using modern trigonometry, the sine of 70°, about .94, is the ratio of the side opposite that angle to the hypotenuse. Multiply that by the hypotenuse of 10′ and you get the length of the opposite side, which in this case is the height, about 9.4′. Of course, if you also knew the distance from the bottom of the ladder to the wall along the ground, you could also figure this out with P322 or the Pythagorean Theorem, but calling it trigonometry is still a big stretch.

Incidentally, I’m not alone. Evelyn Lamb in Scientific American is also critical of the hype, with much more detail, though she doesn’t seem as concerned as I am about the complete absence of angles in this “trigonometry.”

Leave a Reply