## The brilliance of Babylonian mathematics

- March 7, 2024
- Mark Ronan

The Babylonians did things differently. Unlike the Greeks, they treated numbers in the more abstract way we do today.

Many years ago I was in Berlin for a performance of Wagner’s *Ring* cycle, and met a university colleague who worked in Hebrew and Jewish Studies. Since I was a mathematician who had learned to read cuneiform texts in Akkadian and Sumerian, he wanted to introduce me to an expert on cuneiform mathematics. My purpose with cuneiform had been to read the *Epic of Gilgamesh*, rather than ancient mathematics, but I accepted the invitation.

At the end of our meeting the German scholar handed me a long paper entitled ‘Kannten die Babylonier den Satz des Pythagoras?’ (Did the Babylonians know the theorem of Pythagoras?) from which I gathered there must be some doubt about it. I was aware of a tablet called Plimpton 322, and had once read a brief piece by Otto Neugebauer – the doyen of cuneiform mathematics – showing its numbers were related to Pythagoras’s theorem.

Neugebauer was an Austrian mathematician working in Germany, who kept one step ahead of the Nazis, first moving to Denmark then New York. There he read the Plimpton tablet, donated to Columbia University by George Plimpton, an American collector of ancient texts. When Neugebauer read its table of numbers, some having many digits, he saw that they were related to Pythagoras’ theorem. That was the limit of my knowledge, until one day my colleague presented me with a large and recently published book on Babylonian mathematical cuneiform texts and asked me to review it. I reluctantly agreed – it was a huge compendium – and looked for a place to start.

I quickly found a section on rectangles and their diagonals and was hooked. Here were ancient exercises on cuneiform tablets telling the reader how to compute the diagonal of a rectangle: square the lengths of both sides, add together and take the square root. This is what we call Pythagoras’s theorem, so they obviously knew the result. There were several related exercises, but one was very striking indeed: given the area of a rectangle and the length of its diagonal, find the lengths of the two sides. The solution it gave conjured up a picture that commonly appears in proofs of Pythagoras’s theorem.

What could this mean? The main purpose of all ancient cuneiform mathematical exercises was to give the student, or indeed the teacher, a clear view of what is going on, and the exercise computing the two sides of a rectangle, given its area and diagonal, provides a clear view of why Pythagoras’s theorem is true. The Babylonians did not write proofs, unlike the later Greeks, but they certainly had some highly competent teachers who could explain why the methods worked.

For the Greeks, numbers could be lengths, areas or volumes, and the proof of Pythagoras’s theorem in Book One of the *Elements* shows three squares attached to three sides of a right-angled triangle. It shows that the sum of the square areas on the rectangular sides, equals the square area on the diagonal side.

The Babylonians did things differently. They treated numbers in the more abstract way we do today. They could state and solve, for example, quadratic equations (which involve adding a square area to a length), and their method of solution was to apply a standard recipe that we call the quadratic formula. They were good at formulas, and clearly had one to derive the numbers on Plimpton 322. Unlike the Greeks, the Babylonians were unconcerned with whole numbers, so to find three numbers for the dimensions of a rectangle and its diagonal they took the long side of the rectangle to have length number one. They could then scale up or down as necessary.

This we know from the work of a Swedish mathematician, Jöran Friberg, who has produced extensive research on Babylonian mathematics. He shows how the Babylonians started with a single number *s*, which for technical reasons had to lie between 1 + √2 and 1, and used a simple formula to create two further numbers, one for the short side of the rectangle, and one for the diagonal. When *s* is close to 1 + √2 the rectangle is almost square, and as *s* decreases it becomes increasingly elongated. In modern terms we would call *s* a parameter, and by choosing it to be a fraction, whose numerator and denominator have a suitable form in the base-60 system they used, both the short side and diagonal of the rectangle can be computed precisely.

This is remarkable. It yields right-angled triangles whose three sides are given with no need for approximation, like the examples having sides of lengths (3,4,5) or (8,15,17). The Greeks could not have done this because in modern terms the method was algebraic not geometric.

Moreover the listing of data on the tablet, which on the face of it appears somewhat random, is very methodical. In Friberg’s reconstruction of this broken tablet, where the left half is missing completely, it starts with the parameter *s* being 12/5 (which is close to 1+√2) and gradually decreasing until in line 15 it is 9/5. The Babylonians often used both sides of their tablets, and the reverse is already scored with columns for further data.

By contrast Euclid produces a geometric method of constructing two squares whose areas add up to that of a third square, but there is no control on the shape of the right-angled triangle.

The Babylonians remained ahead of the game, yet the brilliance of their mathematics is still poorly recognised.