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Plimpton 322: More than just an “accounting tablet” (©UNSW/Andrew Kelly)

It is not often mathematics enters the news, still less from the Old Babylonian period (around 2000-1600BC), yet a paper in Historia Mathematica hit the headlines recently. Two Australian mathematicians, Daniel Mansfield and Norman Wildberger, interpreted an ancient numerical tablet from southern Mesopotamia as showing that the Babylonians had a form of trigonometry more than a thousand years before the Greeks had even invented the concept of an angle.

The tablet itself had been something of a mystery ever since Otto Neugebauer published a concise explanation of its contents in 1945, showing a relation to the right-angled triangles of what we now call Pythagoras’s theorem. Previously classed as “an accounting tablet” in the George A. Plimpton collection at Columbia University Library, it was nothing of the sort. All the long numbers in the first column are squares, though with random numbers of the same size this is a one in a million million million million chance. Indeed the longest number alone has a less than one in ten million chance of being a square, so this is certainly no accounting tablet.

What is it, and where did it come from? The tablet was originally purchased in Iraq by Edgar J. Banks, a real-life model for Indiana Jones, and the first American to do serious archaeological work there. He was told it came from the ancient city of Larsa, and experts on cuneiform suggest it was written about 1800 BC during the Victorian-length reign of  King Rim-Sin. This was a golden age for the city, which only lost its pre-eminence when Hammurabi expanded the power of Babylon to become the new great power in Mesopotamia.

Banks later sold the tablet to George Plimpton, a retired publisher anxious to expand his collection of texts on which our modern civilisation is based. The three columns of figures would surely have piqued his curiosity, but it was Neugebauer’s seminal work that aroused the interest of mathematicians in item 322 of the collection, now widely known as Plimpton 322. Neugebauer’s observations about a connection with right-angled triangles, along with the three columns of non-trivial numbers, has led unwary commentators to assume it gives the three sides of a right-angled triangle. It does not, but to understand it one first needs to understand the number system they used.

Recent publicity has credited ancient India with the invention of zero, but this is false. The Mesopotamians invented the concept before 2000 BC because they needed it for their “place-value” system of numbers. This is like the system we use today where the order of digits in a number is crucial to its value — each step you move from left to right reduces the value of a digit by one-tenth. In the Mesopotamian case they worked to base-60 rather than base-10, but the principle is the same. The system demands a role for zero, which they represented as a space, before coming up with a special symbol for it, and line 13 on the Plimpton tablet shows a four-digit number with a zero in second position.

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