An ancient Mesopotamian tablet shows Babylonian trigonometrists had long anticipated the Greeks
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.
The reason for base-60 is that it allows division by 2, 3, 4, 5, 6 and many other useful numbers. Such divisions were done using reciprocals and multiplication tables, which takes a bit of getting used to, but once you have managed it and dropped the training wheels, calculations go very smoothly and quickly. It was a fantastic system, lost to posterity after cuneiform writing gave way to the alphabet (by which time they were speaking Aramaic). Yet the Greeks learned a great deal from Babylonia and adopted base-60 for mathematical and astronomical work, which is why we have 60 minutes in an hour, and 60 seconds in a minute.
To understand the mysterious tablet, working to base-60 is essential (conversion to base-10 only creates confusion), and a first step is to find a method of producing precisely the numbers appearing in each of its 15 rows. The Norwegian mathematician Jöran Friberg did this in the early 1980s, concluding that each row could be obtained from a single parameter. As the parameter decreased in a very natural way it gave 38 results, the first 15 of which appear as the 15 rows of the tablet — the further 23 rows may yet turn up on another tablet. Meanwhile, several things are clear. The Babylonians certainly knew Pythagoras’s theorem, which modern cuneiform scholars call the Babylonian Diagonal Rule because it was used on problems phrased in terms of rectangles and their diagonals. The tablet adds to this knowledge by showing that the Babylonians knew how to obtain precise numbers for the sides and diagonals of rectangles (or in other words the sides of right-angled triangles), but the unanswered question is what they did with this information.
When Neugebauer first studied the tablet he observed traces of glue on the left-hand edge, suggesting it was the right-hand half of a larger tablet with further columns on the left. Friberg reconstructed three such columns, one being the parameter that produced everything else, the other two being the diagonal and the short side of the rectangle (the long side has length 1 in all cases). What Mansfield and Wildberger have done is to explain how these columns and the three columns on the remaining right-hand part of the tablet could be used to calculate practical data on right-angled triangles, such as using two sides to find the third side without calculating square roots. This is just how we might have used a trigonometry table before the existence of electronic calculators.
Indeed the word trigonometry, which conjures up schoolwork using sines and cosines, means triangle measurement, and does not presuppose a need for angles. Angles are difficult to measure accurately and while the angle in a trig table serves as a useful parameter, in this tablet the Babylonians used a different parameter. The great advantage of their system is that it produced exact numbers rather than the approximations that you find in any set of trigonometry tables. Moreover, in its complete form it has 38 entries for angles greater than 0˚ and less than 45˚, and the examples of calculations that Mansfield and Wildberger give in their paper yield more accurate results than an Indian table of sines constructed 3,000 years later. Little wonder the media were so interested.
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