The Babylonian civilization was at its peak roughly 4,000 years ago, with architecturally advanced cities throughout the region known today as Iraq. Babylonians were especially brilliant with math, and they invented the idea of zero as well as the base 60 number system we still use today to describe time (where there are 60 minutes in an hour). Now it appears that the Babylonians invented trigonometry, almost 1,000 years before Pythagoras was born.

University of New South Wales mathematicians Daniel Mansfield and Norman Wildberger discovered this after a breakthrough analysis of an ancient cuneiform tablet, written between 1822-1762 BCE in the Babylonian city of Larsa. Long a mystery, the tablet shows three columns of numbers. Describing their work in Historica Mathematica, the researchers call the tablet "a trigonometric table of a completely unfamiliar kind and... ahead of its time by thousands of years."

What made it hard for scholars to figure this out before was the complete unfamiliarity of the Babylonians' trigonometric system. Today we use the Greek system, which describes triangles using angles that are derived from putting the triangle inside a circle. The Babylonians, however, used ratios of the line lengths of the triangle to figure out its shape. They did it by putting the triangle inside a rectangle and completely circumvented the ideas of sin, cos, and tan, which are key to trigonometry today.

“This is a whole different way of looking at trigonometry,” Mansfield told Science News. “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.”

In the Conversation, Mansfield and Wildberger explain the Babylonian system:

Fundamentally a trigonometric table must describe three ratios of a right triangle. So we throw away sin and cos and instead start with the ratios b/l and d/l. The ratio which replaces tan would then be b/d or d/b, but neither can be expressed exactly in sexagesimal [base 60]. Instead, information about this ratio is split into three columns of exact numbers. A squared index and simplified values of b and d to help the scribe make their own approximation to b/d or d/b.

There are a lot of advantages to the Babylonian trig system, according to Wildberger. He and Mansfield say that the base 60, or sexagesimal, number system is far more accurate than the decimal system we're used to. That's because there are no approximations in Babylonian trig. Base 60 allows mathematicians to do more with whole numbers. Of course there are plenty of disadvantages to a system without imaginary numbers and decimals. But it's perfect for what the Babylonians were doing, namely constructing large buildings, calculating the steepness of grades, and measuring land areas for agricultural use.

No one is certain why the Babylonian trig system died out, even though we retained knowledge of zero and the base 60 system. It's a reminder that intellectual breakthroughs can be forgotten for centuries, only to reappear in a new form.

Historica Mathematica, 2017. DOI: 10.1016/j.hm.2017.08.001