Hints of Trigonometry on a 3,700-Year-Old Babylonian Tablet

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One of the columns on Plimpton 322 is just a numbering of the rows from 1 to 15.

The other three columns are much more intriguing. In the 1940s, Otto E. Neugebauer and Abraham J. Sachs, mathematics historians, pointed out that the other three columns were essentially Pythagorean triples — sets of integers, or whole numbers, that satisfy the equation a2 + b2 = c2.

This equation also represents a fundamental property of right triangles — that the square of the longest side, or hypotenuse, is the sum of the squares of the other two shorter sides.

That by itself was remarkable given that the Greek mathematician Pythagoras, for whom the triples were named, would not be born for another thousand years.

Why the Babylonians compiled the triples and wrote them down has remained a matter of debate. One interpretation was that it helped teachers generate and check problems for students.

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Dr. Mansfield, who was searching for examples of ancient mathematics to intrigue his students, came across Plimpton 322 and found the previous explanations unsatisfying. “None of them really seemed to nail it,” he said.

Other researchers have postulated that the tablet originally had additional columns listing ratios of the sides. (There’s a break along the left side of the tablet.)

But what is conspicuously missing is the notion of angle, the central concept impressed upon students learning trigonometry today. Dr. Wildberger, down the hall from Dr. Mansfield, had a decade earlier proposed teaching trigonometry in terms of ratios rather than angles, and the two wondered that Babylonians took a similar angle-less approach to trigonometry.

“I think the interpretation is possible,” said Alexander R. Jones, director of the Institute for the Study of the Ancient World at New York University, who was not involved with the research, “but we don’t have much in the way of contexts of use from any Babylonian tablets that would confirm such an intention, so it remains rather speculative.”

Eleanor Robson, a Mesopotamia expert now at University College London who proposed the idea of the tablet as a teacher’s guide, is not convinced. Although she declined interviews, she wrote on Twitter that the trigonometry interpretation ignores the historical context.

Perhaps the strongest argument in favor of the hypothesis of Dr. Mansfield and Dr. Wildberger is that the table works for trigonometric calculations, that someone had put in the effort to generate Pythagorean triples to describe right triangles at roughly one-degree increments.

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“You don’t make a trigonometric table by accident,” Dr. Mansfield said. “Just having a list of Pythagorean triples doesn’t help you much. That’s just a list of numbers. But when you arrange it in such a way so that you can use any known ratio of a triangle to find the other sides of a triangle, then it becomes trigonometry. That’s what we can use this fragment for.”

A Babylonian faced with the ziggurat word problem may have found it easy to set up: a right triangle with the long side, or hypotenuse, 56 cubits long, and one of the shorter sides 45 cubits. Next, the problem solver could have calculated the ratio 56/45, or about 1.244 and then looked up the closest entry on the table, which is line 11, which lists the ratio 1.25.

From that line, it is then a straightforward calculation to produce an answer of 33.6 cubits. In their paper, Dr. Mansfield and Dr. Wildberger show that this is better than what would be calculated using a trigonometric table from the Indian mathematician Madhava 3,000 years later.

These days, someone with a calculator can quickly come up with a bit more accurate answer: 33.3317.

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