Babylonian tablets show maths 1,000 years before Ancient Greeks started doing sums

The world’s oldest example of applied geometry has been discovered on a 3,700 year old Babylonian clay tablet.

The Greek mathematician Pythagoras was thought to have invented many of the first geometrical formulas around 500 BC – but the newly discovered tablet contains similar calculations more than 1,000 years earlier.

Named Si.427, and dated to between 1900 and 1600 BC, the tablet appears to calculate the size and shape of land boundaries for a field.

Though the tablet was first discovered at the end of the 19th century, in modern-day Iraq, it was sat gathering dust in a Turkish archaeological museum until two Australian researchers tracked it down.

Dr. Daniel Mansfield and Norman Wildberger, researchers at the University of New South Wales, had previously analysed another Babylonian tablet, Plimpton 322, that appeared to show the world’s oldest geometrical tables.

The tablet displayed rudimentary trigonometry, a technique that uses triangles to calculate unknown length. Specifically, it contained Pythagorean triples, the same ratios that were used in Ancient Greece 1,000 years later.

‘It is generally accepted that trigonometry – the branch of maths that is concerned with the study of triangles – was developed by the ancient Greeks studying the night sky in the second century BCE,’ said Dr. Mansfield.

‘But the Babylonians developed their own alternative ‘proto-trigonometry’ to solve problems related to measuring the ground, not the sky.’

The tablet that Mansfield and his colleague found next, Si.427, is thought to be even older than Plimpton 322.

‘There is a whole zoo of right triangles with different shapes. But only a very small handful can be used by Babylonian surveyors. Plimpton 322 is a systematic study of this zoo to discover the useful shapes,’ said Mansfield.

Mansfield and his colleague believe that both tablets were essential for practical tasks in ancient Babylonian society, to calculate the value of a piece of land that’s being sold.

In the cuneiform script, an ancient alphabet used by Babylonians, the tablet appears to show a field containing marshy areas, alongside a threshing floor and tower.

‘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,’ said Mansfield.

‘And this is what this tablet immediately says. It’s a field being split, and new boundaries are made.’

The rectangles used in the calculation have opposite sides of equal length, which requires land surveyors to draw accurate perpendicular lines – this would have required a certain level of technical sophistication for the society at the time.

Published in the journal Foundations of Science, the researchers’ analysis shows how the tablet uses a base 60 number system (similar to today’s clocks), and marks the beginning of private land rights.

One mystery that’s still to solve for Mansfield is a strange number engraved on the back of the tablet – 25:29. The number, which appears to be in the sexagesimal script, based in the number 6, doesn’t appear to have an easy explanation.

‘I can’t figure out what these numbers mean – it’s an absolute enigma,’ said Mansfield.

‘I’m keen to discuss any leads with historians or mathematicians who might have a hunch as to what these numbers trying to tell us!’

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