Indiana Jones and the Mystery of the Maths Tablet

24 August 2017

Interview with

Daniel Mansfield, University of New South Wales

Da da DA DAAA, da da daaa! Indiana Jones may be everyone's favourite fictional archeologist, but he might have been based on a real person. An ancient mathematical mystery has been solved this week by two Australian researchers who have been studying a 3,700 year-old clay tablet that was made originally by the Babylonians. It was first uncovered about a Century ago in what is now southern Iraq and then it was ultimately left, by the owner, to Columbia University in America. Historians knew that the writings on the tablet were numerical, but they didn’t realise their significance. Now mathematicians Norman Wildberger and Daniel Mansfield, from the University of New South Wales in Sydney, have deciphered what they think is the true meaning of the tablet. It describes, they reckon, a previously unknown form of trigonometry for finding the sizes of right-angle triangles, and, get this, it pre-dates Pythagoras by 1000 years… Izzie Clarke caught up with Daniel Mansfield to hear what the tablet looks like, and what it shows…

Daniel - It’s rectangular, and it’s surprising heavy, and it’s written in a very neat hand. Someone went to a great deal of trouble to rule this thing out neatly and carefully write down a whole sequence of numbers. The tablet was obtained by George Arthur Plimpton who was an American publisher, and it was sold to him by a fellow called Edgar Banks who’s the real person upon which the say Indiana Jones was based. He was an archeologist, an academic, an adventurer, and an obtainer of rare antiquities. He’s even got the same hat. He sold it to Plimpton who was a publisher and collector of ancient mathematical texts, but, of course, Plimpton just knew it had maths on it. No-one really understood exactly what this was about. But it really wasn’t known that it was anything special at all until 1945 when Neugebauer and Sachs discovered that it actually contained pythagorean triples.

Izzie - And what exactly are they?

Daniel - They’re three numbers which describe a right triangle. Now, numbers which describe a right triangle are very special because they obey what we call Pythagoras's theorem. So, if you have a triangle with sides A, B, and C where C is the hypotenuse, then the numbers are related - A2, plus B2 is equal to C2. And what Neugebauer and Sachs showed was that Plimpton 322 actually contained pythagorean triples, which is amazing because it shows that the Babylonians knew about Pythagoras's theorem a thousand years before Pythagoras was even born.

Izzie - That is incredible. What do those numbers mean?

Daniel - Well, that what people have been really scratching their heads about. For 70 years we’ve known that this is a very special, highly sophisticated tablet, but why does it have pythagorean triples on it? What is the meaning of it? It’s not just a random sequence, it’s an ordered series of pythagorean triples, and the ordering is very special. It starts with something that’s almost a right isosceles triangle, or almost half square, and then it goes down row by row to flatter kind of triangles. I don’t want to say words like inclinational angle because they don’t exist at this time. The triangles get flatter, if you like.

Izzie - Now Pythagoras’s theorem, which we all learn throughout high school, uses angles. If they weren't using angles, how did Babylonians use trigonometry?

Daniel - They used ratios. This is a thousand years before angles were even thought of but you don’t need an ocean of angle to study trigonometry, you can use ratios alone. In fact, that’s what we propose Plimpton 322 was all about, just a table of ratios of the sides of a right triangle, they used ratios to describe steepness. So you see things like descriptions of a canal or they talk about how much width per unit depth the triangle requires. A long ramp might require a lot of width per unit depth, a really steep one - we call it steep, the Babylonians would say that it’s a very short amount per unit depth.

Izzie - How did these Babylonians use trigonometry?

Daniel - It’s very difficult to say what they used it for; we really don’t know. All we have to explore this world is this tiny window for which we have evidence. Now we have something which appears to be a trigonometric table, but how they used it is really open to speculation. I’m happy to speculate on that. Personally, I think it was used for surveying but, I have to say, all that we have is what’s written on the tablet.

Izzie - This is all mathematics from literally over 3,000 years ago. Can we use this new - well not new, ancient - Babylonian approach for our use?

Daniel - Well I’d love to see this find a way to use this trigonometry. It’s really out there for people to start using it again now that we know what it looks like. I personally think that we should use this to teach trigonometry in school. This is a great way for people to understand triangles and it doesn’t require them to understand what the square root of two is, or trigonometric functions like sin cos and tan, or an angle. You can study triangles without any of these things. It’s very early days; we’ve only just rediscovered this. Certainly, I think there’s a place for it in the world.

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