A new analysis of a 3,700-year-old Babylonian cuneiform tablet suggests that the ancient Babylonians were using an advanced form of trigonometry roughly a millennium before ancient Greek mathematicians recorded what is known as the Pytharoean theorem. In addition to the tablet’s antiquity, the tables inscribed on it also suggest that the Mesopotamians’ approach to this form of mathematics may be superior to the function we use today.

Currently stored at Columbia University, tablet Plimpton 322 (P322) is believed to have been written around 1,800 BCE. It was obtained from antiquities dealer Edgar James Banks in 1922 by New York publisher George Arthur Plimpton, and subsequently given to Columbia University in the 1930s. While the tablet was originally suspected to depict a trigonometric table, this idea was abandoned at some point, with other administrative functions attributed to it.

University of New South Wales (UNSW) mathematician Daniel Mansfield encountered tablet P322 when he was developing a new course for high school math teachers, and was intrigued by its inscriptions: "It took me 2 years of looking at this [tablet] and saying ‘I’m sure it’s trig, I’m sure it’s trig, but how?" The tablet was otherwise missing the sine, cosine and tangent entries requisite for the implementation of modern trigonometry. Rather, it recorded the relationships between short and long sides of the triangle, and of its short side to its diagonal.

Mansfield’s collaboration with fellow UNSW mathematician Norman Wildberger yielded the conclusion that the ancient Babylonians used the exact ratios of the lengths of the right sides of a triangle to express their form of trigonometry, rather than using the angles, as prescribed by Pythagoras in the fourth century BCE.

If this theory is correct, tablet P322 represents the earliest known recording of trigonometric theory, and also represents a more accurate approach to solving trig problems. The Babylonian method uses exact values for the sides of the right triangles represented, while the Greek method relies on extremely close approximations provided by sine and cosine values.

Regardless, the find may well underscore how different the thought processes of an ancient culture might be to what we’re accustomed to today. "This is a whole different way of looking at trigonometry," explains Mansfield. "We prefer sines and cosines … but we have to really get outside our own culture to see from their perspective to be able to understand it."

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