UNSW/Andrew Kelly

Greek astronomer Hipparchus is often referred to as the father of trigonometry, but a new analysis argues that the ancient Babylonians may have beat him to it by over a thousand years.

University of New South Wales mathematics professors Daniel Mansfield and Norman Wildberger spent over two years studying Plimpton 322 (P322), a 3,700-year-old clay tablet containing rows of cuneiform, an early form of writing. They concluded that P322 may have been used for trigonometry, which, if true, would make the artifact the oldest known trigonometric table.

For decades, mathematicians have known that P322 was a table representing Pythagorean triples—values for A, B, and C that solve the Pythagorean theorem, A^{2} + B^{2} = C^{2}—but it was unclear what the tablet was used for. One widely accepted theory is that the tablet was a mathematical teaching device. Mansfield and Wildberger recently theorized that the tablet is a trigonometric reference, for use in construction or surveying.

Their conclusions are based on reinterpreting the figures on the tablet as a study of ratios in a triangle. Greek trigonometry focused on angles, resulting in the sines, cosines, and tangents that are still taught in math classes today. Babylonian trigonometry, in contrast, described triangles in terms of the ratios of their three sides, say Mansfield and Wildberger, and the rows of cuneiform on Plimpton 322 depict just such a ratio-focused form of trigonometry.

While Mansfield was familiar with P322, he had never set out to study it, until he noticed that the tablet had parallels to work by Wildberger, his colleague down the hall. “[Wildberger] wrote the book on how to do trigonometry without angles, and this gave us a new approach to understanding the tablet that had not been considered before,” said Mansfield. This new theory by the two professors provides a different way of understanding mathematical history. Perhaps, Manfield says, “the study of trigonometry in high school could be enriched by studying triangles from two different cultural perspectives: the Babylonian and the Greek.” (*Historia Mathematica*)