# Babylonian artefact reveals history of maths

A maths tablet dating back to 1900 BCE depicts 'Pythagorean triples' a millennium before Pythagoras' birth
10 August 2021

## Interview with

Daniel Mansfield, The University of New South Wales Sydney

## Clay maths tablet

But first, we probably all remember learning Pythagoras’s theorem in school, and there was always one kid who asked, “but when are we ever going to use this in real life”? Well it turns out that not only did ancient Babylonians learn about Pythagorean triangles 1000 years before Pythagoras was born, they also used them in real life to measure out land. An old clay tablet inscribed with symbols found lying in a museum was the clue that led researchers to this ancient form of maths, as Sally has been hearing...

Daniel - We always knew the ancient Babylonians were aware of Pythagorean triples, but we had no idea why. We had no idea what they were doing with these objects.

Sally - That’s Daniel Mansfield from the University of New South Wales, Sydney, who discovered that the ancient Babylonians were using Pythagorean triples. So, what is a Pythagorean triple? We all remember right angled triangles from school and calculating the length of the longest side, the hypotenuse. Because of all the square roots involved, even if two of the sides are nice whole numbers, like 5 and 6cm long, the third side is usually a horribly complicated number with lots of decimal places, which in our case as you will have all worked out already from remembering Pythagoras’s theorem, is of course the square root of 25 plus 36, otherwise known as root 61, or the rather unpleasant 7.8102496... However, for a special group of right angle triangles, all the sides are lovely round numbers. For example, 3, 4, and 5. And if you make a triangle with sides that measure 3cm, 4cm and 5cm, it’s impossible for it to be anything other than a perfect right angle. These special triangles are called Pythagorean triples. . Mathematician Daniel was looking at markings written on palm sized flat discs of clay from 4000 years ago…

Daniel - You know, this is about a thousand years before Pythagoras was even born.

Sally - Two tablets in particular caught his eye in the museum. One, the Plimpton 322 had already been deciphered in 1945 which was simply a long list of these Pythagorean triples, much like the trigonometric tables you may have looked up in school. But the ancient Babylonians probably weren’t studying for their Maths GCSEs, so why did they need these triples? That’s when a small, round clay tablet drew Daniel’s attention, with rectangles and measurements that he realised was a map of field boundaries.

Daniel - Now, thanks to this new tablet, we know they were using them to solve problems about land. In particular, they were using them to create perpendicular boundaries.

Sally - Why did Babylonians need to measure out their land so accurately with such precise right angles?

Daniel - Let's slow it down. Because I want to tell you a story about lands and gods and math. So right at the beginning, Babylonian society is based on these small cities built around the local temple and all the land around is owned by the local temple or maybe a king or a palace. And they have surveyors, but the surveyors, unlike our surveyors, they don't measure boundaries. They're just there to tell the local prince or the local temple administrators how much land they've got. They can decide for themselves what they own and they practically own everything. They don't need no surveyor to tell them where their boundaries are. Then at about this time is old Babylonian period, something changes. And we now see land in the hands of private individuals and private individuals can't decide between themselves necessarily where their boundaries are. You need a surveyor to do that for you.

Daniel - Now we see surveyors change what they do instead of just saying, "Oh, prince, your land is this large". They say "Two people, here's your land. Here's where your land stops. And here's where your land stops". And they start making boundaries. Now to maintain good boundaries between neighbours, you need to have some confidence in those boundaries and to a Babylonian that means you get the right angles just so. In modern society, that's easy. You just use trigonometry to make the right angles, but this is a thousand years before trigonometry would have been invented. So instead the Babylonians are using Pythagorean triples.

Sally - Why should we care about old maths when we've got perfectly functioning new maths?

Daniel - It's so different. Their whole approach to mathematics is just unbelievably different. We tend to think of mathematics as this kind of universal language where everyone can agree on what at least what mathematics is. And if aliens came from out of space, then we could, we may not be able to communicate them in words, but we could, you know, we could share math with them.

Sally - Didn't we send maths on that golden disc out into space.

Daniel - Yeah, I'm afraid we may, we may have overestimated how universal that was going to be.

Sally - Has anyone tried using Babylonian maths to solve modern problems?

Daniel - We're still trying to figure out what Babylonian maths is. This is all, this is all happening very quickly. Actually the original computers used a Babylonian method of performing division because it's very, very simple. I like to think there's still a place in the world for them though, for fantastically simple calculations that sure, aren't as accurate as what you can do now with a modern computer. But geez, they're efficient, just so very efficient. So I like to think there's a place in the world for those kinds of things like in computer graphics, where speed is more important than accuracy. Maybe this Babylonian approach might have some kind of advantages.