Babylonians were using Pythagoras’ Theorem over 1,000 years before he was born
An ancient clay tablet shows that the Babylonians used Pythagorean triples to measure accurate right angles for surveying land
Students may complain that Pythagoras’ Theorem has no real-world uses, but a 3,700-year-old tablet demonstrates that the formula was being used long before Pythagoras even wrote it down. The artefact, named Si.427, shows how ancient land surveyors used geometry to draw boundaries accurately.
First discovered in central Iraq in 1894, Si.427 sat in a museum in Istanbul for over a century. It was only when mathematician Dr Daniel Mansfield from the University of
New South Wales, Australia, began to study the clay tablet that its meaning was uncovered.
“Si.427 dates from the Old Babylonian (OB) period – 1900 to 1600 BC',” said Mansfield. “It’s the only known example of a
cadastral document from the OB period, which is a plan used by surveyors define land boundaries.
“This is from a period where land is starting to become private – people started thinking about land in terms of ‘my
land and your land’, wanting to establish a proper boundary to have positive neighbourly relationships. And this is what this tablet immediately says. It’s a field being split, and new
boundaries are made.”
As many will remember from their school days, Pythagoras’ Theorem states that the sides of a right-angled triangle obey the formula a2 + b2 = c2, where a and b are the lengths of the short sides, and c is the length of the longest side.
A Pythagorean triple is a set of numbers – usually whole numbers – that fits this relation, such as 3, 4 and 5, or 5, 12
and 13. Any triangle with sides of these lengths must be a right-angled triangle – a fact that is useful for marking out accurate rectangles. The surveyor who created Si.427 used Pythagorean triples to make accurate right angles, making it the earliest known example of applied geometry.
However, the number system used by the Babylonians was different from the one we use now. Ours is a system called base 10: numbers are written by breaking them down into hundreds, tens, units, and so on. The Babylonians used the more complex base 60, similar to how we keep time: 60 seconds make up one minute, and 60 minutes make up one hour meaning that only certain Pythagorean shapes can be used. “Nobody expected that the Babylonians were using Pythagorean triples in this way, it is more akin to pure mathematics, inspired by the practical problems of the time,” explained Mansfield. “The discovery and analysis of the tablet
have important implications for the history of mathematics. For instance, this is over a thousand years before Pythagoras was born.”