The ancient Babylonians understood key concepts in geometry, including how to make precise right-angled triangles. They used this mathematical know-how to divide up farmland – more than 1000 years before the Greek philosopher Pythagoras, with whom these ideas are associated.

“They’re using a theoretical understanding of objects to do practical things,” says Daniel Mansfield at the University of New South Wales in Sydney, Australia. “It’s very strange to see these objects almost 4000 years ago.”

Babylonia was one of several overlapping ancient societies in Mesopotamia, a region of southwest Asia that was situated between the Tigris and Euphrates rivers. Babylonia existed in the period between 2500 and 500 BC, and the First Babylonian Empire controlled a large area between about 1900 and 1600 BC.

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Mansfield has been studying a broken clay tablet from this period, known as Plimpton 322. It is covered with cuneiform markings that make up a mathematical table listing “Pythagorean triples”. Each triple is the lengths of the three sides of a right-angled triangle, where each side is a whole number. The simplest example is (3, 4, 5); others include (5, 12, 13) and (8, 15, 17).

The triangles’ sides are these lengths because they obey Pythagoras’s theorem: the square of the longest side is equal to the sum of the squares of the other two sides. This classic bit of mathematics is named for the Greek philosopher Pythagoras, who lived between about 570 and 495 BC – long after the Plimpton 322 tablet was made.

“They [the early Babylonians] knew Pythagoras’ theorem,” says Mansfield. “The question is why?”

Mansfield thinks he has found the answer. The key clue was a second clay tablet, dubbed Si.427, excavated in Iraq in 1894. Mansfield tracked it down to the Istanbul Archaeology Museums.

Si.427 was a surveyor’s tablet, used to make the calculations necessary to fairly share out a plot of land by dividing it into rectangles. “The rectangles are always a bit wonky because they’re just approximate,” says Mansfield. But Si.427 is different. “The rectangles are perfect,” he says. The surveyor achieved this by using Pythagorean triples.

“Even the shapes of these tablets tell a story,” says Mansfield. “Si.427 is a hand tablet… Someone’s picked up a piece of clay, stuck it in their hand and wrote on it while surveying a field.” In contrast, Plimpton 322 seems to be more of an academic text: a systematic investigation of Pythagorean triples, perhaps inspired by the difficulties surveyors had. “Someone’s got a huge slab of clay… [and] squashed it flat” while sitting at a desk, he says.

Journal reference: *Foundations of Science*, DOI: 10.1007/s10699-021-09806-0

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