Recently, I got hooked when I stumbled over a historical remark mentioning the title of an article named “Sherlock Holmes in Babylon“. I have now spent a little while on this article and the stuff surrounding it: it deals with a cuneiform writing from the 2nd millenium BC.

The writing is labelled “Plimpton 322“, as such clay tablets usually are indexed with consecutive numbers within the collection in which they are kept: in this case the Plimpton Collection of Columbia University. The tablet itself contains a set of numbers, organised quite similar to a modern-day tabular matrix: one column with rather large looking numbers, two colums with smaller numbers and one column that contains the indexing 1 – 15 (there are 15 lines of such numbers). Above, there is a headline in text.

Myself, I am not familiar with cuneiform writing at all, so I can only guess about the difficulty in translating those ancient texts. As far as the numbers are concerned, it is rather standard knowledge that the Babylonians used a hexagesimal number system, based on the number 60 instead of our present-day 10. The numbers themselves are written in a decimal fashion, having special signs for the numbers 1 to 9 and for the numbers 10, 20, … , 50. A little speciality is that the concept of a positional number system is not completely identical to ours: there is no “hexagesimal point” to distinguish numbers smaller than 1, and you cannot tell if the symbol 30 actually means $30 = 30 . 60 ^{0}$ or rather $30 . 60 ^{-1} = \frac12$ or even $30 . 60 ^{1} = 1800$.

Now, why the interest in this particular clay tablet? Plimpton 322 is famous because one can guess about its contents: there is some mathematics going on, but you can’t be really sure what was actually meant by this. So people have guessed that the Babylonians knew about Pythagorean numbers ($l^2+b^2 = d^2$ for integers $l, b, d$) or about trigonometric functions (namely, the tangent) a lot earlier than we thought. Hence the title for the survey paper that got me hooked: it’s a mystery – we have many hints, we have a lot of clues, but we don’t know for sure.

The first scientific look at the mathematics of Plimpton 322 was taken, to my knowledge, by Neugebauer in the 1940s. He states that the tablet deals with pythageoren numbers in the colums 2 and 3, every line containing one of the smaller sides of the rectangular triangle and the hypothenuse. Strangely, the other of the smaller sides is missing. The first colum then contains the ration $\frac{d^2}{l^2}$, that means the square of tangent of one of the angles in the triangle (which is computed using a number that doesn’t come up on the tablet). Even more: the values in the first column are sorted in descending order, starting at an angle of 45° and ending about 31°, with more or less constant steps.

Re-discovering the structure of the numbers presented here was a nice little exciting exercise for me. The ratios in column 1 actually fit perfectly the numbers to the right, down to four or five hexagesimal places. There is no coincidence in this tablet, this was manufactured for some good reason.

Once the system has been found, some writing errors can be corrected. Some of them are trivial copy errors, other follow from the computation that was used to construct the columns. The “Sherlock Holmes”-paper does a nice job about showing how these errors may have come to pass.

However, Neugebauer could not, as he openly states, bring a meaning to numbers on the tablet or a sensible translation to the headline of column 1: “the translation causes serious difficulties“. The question stays: how were the numbers on the tablet chosen to fit this scheme so precisely? What was the use of the tablet, why would one pick just these computations to write down?

From Neugebauer’s statements, people have concluded that the Babylonians not only knew about rectangular triangles with integer side lengths, they also knew about the generating-parameters to find new such pythagorean numbers, and about the computation of the tangent function. There has been a vast activity about this particular clay tablet, dealing with the implications that this discovery has for our understanding of how advanced the ancient Babylonians have been. Many of the articles on this topic use modern methods of making sense from the numbers of Plimpton 322 – methods that can’t possibly have been known 3000 years ago.

The two concurrent theories are

• pythagorean numbers: we have lost the generating numbers that once were on a now broken part of the tablet, and the remaining colunns contain integer-sized rectangular triangles.
• reciprocal pairs: the tablet serves as a list of exercises for students, where they must solve the equation x + 1/x = d; the tablet then holds the solutions and some intermediate results, when the Babylonians solved the equation basically by a geometric argument that was memorized via concrete algorithmic instructions. This kind of exercise is not unusual in other clay tablets from that period.

A direct follow-up on these assertions is an article named “Neither Sherlock Holmes nor Babylon“, which sounds like an attack to the “Sherlock Holmes”-paper, but it actually isn’t. It’s a quite thorough investigation about the logical surroundings of this tablet. In particular, there are six criteria that a theory explaining the tablet needs to satisfy before it can become credible – among these criteria is „historical sensitivity“ (consider the historical context of the other clay tablets that we know), „calculational plausibility“ (can the theory explain how the numbers were found and why the errors have been made like they were) and „tabular order“ (why is the supposedly complicated column 1 the first with fewer errors, instead of the obviously easier numbers in columns 2 and 3).

Besides, the author Eleanor Robson believes strongly, that Plimpton 322 is not as important to the history of mathematics as many others think it is. It might not be so very special. In a way, I as a reader could see this even from the deep arguments about the language of the headlines that are presented in the article: a lot is known about the writing and the languages in the cuneiform tablets. This is a tiny hint at how many texts are known and how much analysing has been done in this field. The experts are well beyond educated guesses when it comes to interpretations of the clay tablets. But it is still extremely hard to give a profound analysis of what has actually been going on in the ancient times. Apparently, there were no mathematicians doing their science “for fun”. The scribes were trained experts in writing and computing, but they did it for economical reasons: they were accountants or architects or the like, not scientists. From what we know of the historical context, there was no “l’art pour l’art” in ancient mathematics. And in particular, the applications for a theory explaining Plimpton 322 would have to be found in some sort of “real life” application and therefore in some other clay tablet. It cannot be considered in its own right, as much of modern-day mathematics can. This has to be kept in mind when the interpretations for Plimpton 322 run wild.

The theory on pythagorean numbers is dismissed on these grounds. In particular, the Babylonians didn’t have a concept of angles in a triangle (or, at most, of the right angles), hence there can be no dealing with the tangent function – there was no use to measure the slope inside that triangle. At least no use that would find any parallel in the other Babylonian texts. In the “Neither Sherlock”-article, there is a very readable account about what the Babylonians did and did not consider in geometrical shapes – if only limited to circles.

The theory on reciprocals fits better in the historical context and it can explain much of the shape in which we can find the numbers presented on Plimpton 322. It is suggested that, in that tablet, we have a good copy of a series of exercises for students, all yielding “nice” results, possibly missing some of the columns that once were present to the left of what we see today.

But in the end, “The Mystery of the Cuneiform Tablet has not yet been fully solved“. There is a good chance that the mystery will never be solved at all – but maybe new ideas or new findings will prove me wrong here. It’s still a mystery and that is what makes it exciting.