Students may not believe that Pythagoras’ Theorem has real-world uses, but a 3,700-year-old tablet proves that their maths teachers are right. The artifact, named Si.427, shows how ancient land surveyors used geometry to draw boundaries accurately.

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Discovered in central Iraq in 1894, Si.427 sat in a museum in Istanbul for over a century. Now, mathematician Dr Daniel Mansfield from the University of New South Wales, Australia, has studied the clay tablet and uncovered its meaning.

“Si.427 dates from the Old Babylonian (OB) period – 1900 to 1600 BCE,” 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. In this case, it tells us legal and geometric details about a field that’s split after some of it was sold off.”

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 fit 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.

This fact is useful for marking out accurate rectangles: constructing a triangle whose sides are a Pythagorean triple gives you a right angle every time. This makes Si.427 the earliest-known example of applied geometry.

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“Nobody expected that the Babylonians were using Pythagorean triples in this way,” Mansfield said. “It is more akin to pure mathematics, inspired by the practical problems of the time.

“The discovery and analysis of the tablet have important implications for the history of mathematics,” he said. “For instance, this is over a thousand years before Pythagoras was born.

“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.”

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However, this mathematics wasn’t always simple for the Babylonians. Their number system was different from the one we use now. Ours is in a system called base 10: numbers are written by breaking them down into hundreds, tens, units, and so on. The Babylonian number system, however, used the much more complex base 60, similar to how we keep time: 60 seconds make up one minute, and 60 minutes make up one hour.

“This raises a very particular issue – their unique base 60 number system means that only some Pythagorean shapes can be used,” said Mansfield.

In 2017, Mansfield studied another tablet from later in the same time period. This one, called Plimpton 322, contained what he calls ‘proto-trigonometry’: a table studying different types of triangle.

“It seems that the author of Plimpton 322 went through all these Pythagorean shapes to find these useful ones,” he said. “This deep and highly numerical understanding of the practical use of rectangles earns the name ‘proto-trigonometry’ but it is completely different to our modern trigonometry involving sin, cos, and tan.”

The issue of geometry and land ownership came up over and over for the ancient Babylonians, highlighting just how important this mathematics was.

“Another tablet refers to a dispute between Sin-bel-apli – a prominent individual mentioned on many tablets including Si.427 – and a wealthy female landowner,” Mansfield said.

“The dispute is over valuable date palms on the border between their two properties. The local administrator agrees to send out a surveyor to resolve the dispute. It is easy to see how accuracy was important in resolving disputes between such powerful individuals.”

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## Authors

Sara RigbyOnline staff writer, BBC Science Focus

Sara is the online staff writer at BBC Science Focus. She has an MPhys in mathematical physics and loves all things space, dinosaurs and dogs.

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