This Ancient Tablet Secretly Held The Oldest Evidence Of Applied Geometry In The World

Si.427 is a hand tablet from 1900-1600 BCE, created by an Old Babylonian surveyor. It’s made out of clay and the surveyor wrote on it with a stylus. Image credit: UNSW Sydney

Once upon a time, in a kingdom far, far away, a wealthy landowner decided to sell part of their field. Specifically, it was about 4,000 years ago, in what is now central Iraq – and that land transaction would eventually lead to a complete upending of how we understand the history of mathematics.

New research published this week in the journal Foundations of Science suggests that Si.427 – a small, unassuming clay tablet that has been sitting in a museum in Istanbul for the past 100 years – is in fact the oldest known example of applied geometry in the world. What’s more, this tablet reveals something else extraordinary: the inspiration, and methodology, that allowed its ancient Babylonian authors to beat Pythagoras to his famous Theorem by a good couple of millennia.

“Any history book will tell you that trigonometry goes back to ancient Greek astronomers,” author Daniel Mansfield told IFLScience. “I like to think of the Babylonian understanding as an unexpected prequel, which really is an independent story because the Babylonians weren't using it to measure the stars, they were using it to measure the ground.”

Four years ago, Mansfield made the headlines when he and his colleague Norman Wildberger became the first to decode the ancient clay tablet known as Plimpton 322. This nearly 4,000-year-old artifact was covered in meticulously organized groups of numbers known to modern mathematicians as Pythagorean triples – that is, whole numbers which satisfy Pythagoras’s theorem, like (3, 4, 5) or (5, 12, 13).

Mansfield and Wildberger worked out that Plimpton 322 was a trigonometric table – a kind of ancient “cheat sheet” for working out geometric problems. But why, mathematical novelty notwithstanding, would an ancient Babylonian need something like that?

That’s where Mansfield’s new research comes in. Si.427 isn’t just some abstract math puzzle – it’s a 4,000-year-old legal document, he says.

“It’s the only known example of a cadastral document from the OB [Old Bablylon] period, which is a plan used by surveyors to define land boundaries,” Mansfield explained. “In this case, it tells us legal and geometric details about a field that’s split after some of it was sold off.”

On the front, we see a diagram of a field. The field is being split, and some of it is being sold.  The lines demarcate the boundaries of the different fields. Image: UNSW Sydney

Tantalizingly, Si.427 is thought to predate Plimpton 322 – if that’s the case, then this discovery is not only the oldest known example of geometry in the world, but also an important clue to the context behind Babylonian math.

“Babylonian [math] … was very sophisticated for the time,” Mansfield told IFLScience, “and we now know that they used this understanding to solve contemporary problems about land ownership and boundaries … Now that we know what problems they faced, other tablets start to make more sense.”

This is important because the standard story of trigonometry is that it was invented by ancient Greek astronomers to study the night sky. Si.427 turns that theory on its head. Trigonometry – or as Mansfield puts it, “proto-trigonometry” – was developed independently by the Babylonians thousands of years earlier, and its inspiration was unmistakably terrestrial.

““With this new tablet, we can actually see for the first time why they were interested in geometry: to lay down precise land boundaries,” explained Mansfield. “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 neighborly relationships. And this is what this tablet immediately says. It's a field being split, and new boundaries are made.”

On the back of the tablet is text written in cuneiform, one of the earliest systems of writing. The text corresponds to the diagram on the front – describing the size of the field. Image credit: UNSW Sydney

But the Babylonians didn’t simply “beat” the Greeks to the same result, Mansfield explained – this “proto-trigonometry” comes from a completely different perspective. There are no degrees or functions like sin or cos. In fact, from a modern perspective, the problems tackled by tablets like Si.427 and Plimpton 322 seem almost completely backwards: while today’s math students are used to questions that ask for the lengths of the sides of a right triangle, Babylonian mathematicians and surveyors were instead focused on which set of side lengths would result in a nice pair of perpendicular lines.

“It's fun because this approach to geometry is completely unexpected,” Mansfield told IFLScience. “It has come to us from far outside our mathematical culture. So it seems new and fresh to us, even though it's almost 4,000 years old.”

The tablet may be nearly four millennia old, but that doesn’t mean it can’t teach us anything – and Mansfield believes this ancient way of doing math could have some important modern-day applications.

“Ancient mathematics is not as sophisticated as modern mathematics. But sometimes you want simple answers instead of sophisticated ones,” Mansfield told IFLScience. “And I'm not just talking about how mathematics students want their exams to be. The advantage of a simple approach is that it’s fast … [it] might be of benefit in computer graphics or any application where speed is more important than precision.”


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