# Ancient Babylonian tablet discovery rewrites Pythagoras’ claim to famous theorem

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Pythagoras was an influential Greek mathematician and philosopher, best known for the theory which he gave his name. Very little is known about his life, however. He is believed to have been born on the Greek island of Samos around 580 BC, travelling widely during his youth, visiting Egypt and Persia.

Eventually, he settled in the city of Crotone in southern Italy, and is thought to have been killed or died around 500 BC.

It is in Crotone that Pythagoras began teaching and gained a reputation for his lively philosophy based mathematics.

But a fresh discovery has cast doubt on his claim to creating his now world-famous theorem.

A 3,700-year-old tablet discovered in Iraq in 1894 has been revisited by a mathematician who claims that its carvings show the theorem’s formula long before Ancient Greece was flourishing.

The tablet, named Si.427, had been sitting in a museum in Istanbul for over a century.

That is until Dr Daniel Mansfield from the University of New South Wales, Australia, was given access to the clay tablet, meticulously studying and uncovering its meaning.

He told BBC Science Focus Magazine: “Si.427 dates from the Old Babylonian (OB) period – 1900 to 1600 BC.

“It’s the only known example of a cadastral document from the OB period, which is a plan used by surveyors to define land boundaries.

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“In this case, it tells us legal and geometric details about a field that’s split after some of it was sold off.”

The 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.

The fact is extremely useful for making accurate rectangles — constructing a triangle whose sides are a Pythagorean triple gives you a right angle every time.

As a result, it makes Si.427 the earliest-known example of applied geometry.

Dr Mansfield added: “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.

“The discovery and analysis of the tablet have important implications for the history of mathematics.

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“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.”

But this mathematics wasn’t always simple for the Babylonians, as their number system was different from the one we use today.

While ours is in a system called base 10, they used a much more complex number system, called base 60.

It is similar to how we keep time: 60 seconds make up one minute, and 60 minutes make up one hour.

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

In 2017, he studied another tablet later in the same period.

Called Plimpton 322, it contained what he calls “proto-trigonometry”: a table studying different types of triangles.

This, he said, makes it appear as though “the author of Plimpton 322 went through all these Pythagorean shapes to find these useful ones,” but is “completely different to our modern trigonometry involving sin, cos, and tan.”

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