The Babylonian tablet Plimpton 322

This mathematical tablet was recovered from an unknown place in the Iraq desert. It can be determined, apparently from its style, that it was written originally sometime around 1800 BCE. It is now located at Columbia University.

[ An image at higher resolution ]

How to read it

The Babylonian number system. The Babylonians `wrote' on wet clay by pressing a stylus or wedge into it in one of a small number of ways. The Babylonian number system, already from a very early era, was remarkably sophisticated. The number 1 was `written' with a single `stroke'

and the numbers 2 through 9 were written by combining multiples of a single stroke:

The number 10 was written in a single character and the numbers 20 to 50 were written with multiples of this character:

The numbers up to 59 were obtained by combining these much as we do. For example, 11 was written as . But the system was not actually based on 10, but rather on 60 - and it was a true floating point system so that 1 and 60 (as well as, say, 3600) were indistiguishable. All numbers could be obtained by combining these in order, writing from left to right as we do. Thus seventy was written as . Sometime before 300 BCE, but after Plimpton 322 was written, a special symbol was devised as a zero, but in Plimpton 322 there is potential confusion because of this problem. The conventional way to write floating point sexagesimal numbers is by using comma separators, so that 1,29 is 60+29 = 89 in decimal notation, and 1,1,1 is 3661.

The last two columns. At any rate, we now know almost enough to read the right hand columns of the tablet. The last column (with a few natural interpolations to take into account missing symbols for 5, 6, and 15, ) simply numbers the line of numerical data. In the fourth column is written the word pronounced ki which can be translated loosely here as number, so that the fourth and fifth column together (number 1, number 2, etc.) just enumerate the different lines of data. Interpolations are in green.

For the rest of the columns, we show in order the original column, the sexagesimal numbers written in conventional notation, and then the decimal equivalents. Error corrections are in red. Both interpolations and corrections are explained later on. All the zeroes are interpolated, also, since the Babylonians at this time did not have them.

Column two. The heading of the second column includes the word for width.

Column three. The heading of the third column includes the word for diagonal.

Column one. The heading of the first column has not been translated. We shall see in a moment the reasons for placing the `decimal point'.

What it means

Following perhaps the hints given by the words diagonal and width, Neugebauer and Sachs discovered that if w is the entry in the second column and d is the entry in the third column, then in all but a few cases d²-w² turned out to be a perfect integer square l². In other words, assuming the exceptions to be caused by error, this table contains part of a list of Pythagorean triples, that is to say integers w, l, d with

w² + l² = d²

which form the sides of right triangles. Here is the resulting table of calculations, in modern notation (with discrepancies in square brackets):

A bit later, we shall confirm this theory by explaining the errors.

As for the first column, it contains the values of d² / l² . For example, in the first row d = 169, l = 120, and d²/l² = 1.9834 ... = 1,59,0,15.

How the table was produced

How are the Pythagorean triples in this table distinguished? If (A, B, C) is Pythagorean triple, then we can write it as (ma, mb, mc) where (a, b, c) is a primitive Pythagorean triple - one in which the numbers are relatively prime in pairs. Primitive Pythagorean triples are parametrized by pairs of intgers (p, q) satisfying these conditions:

p and q are both positive;
p is greater than q;
One of them is odd, the other even;
p and q are relatively prime.

The pair (p, q) gives rise to the triple (p²-q², 2pq, p²+q²). The pair (p, q) can be easily recovered from (a, b, c) by the formulas p² = (a + c)/2, q = b/2p.

Here are the values of p, q, and m for the triples in the table:

The ratio c/b is equal to (p² + q²)/2pq = (1/2)(p/q + q/p). Therefore this ratio, the square of which appears in the first column of the tablet, will have a finite expression in base 60 if 1/p and 1/q do. The Babylonians almost certainly understood the difference between finite sexagesimal expansions and repeating ones, and in particular we have found tables of reciprocals 1/p for many values of p where the expansion is finite. Such numbers p are called regular by Neugebauer. It is not likely to be a coincidence that the values of both p and q associated to the rows of the tablet are regular, and in fact that in all but one case the expansions of 1/p and 1/q appear in the tables of reciprocals that have been found. It seems plausible, therefore, that the Babylonians knew how to generate primitive Pythagorean triples.

We know something, therefore, about how the tablet was constructed, but we do not know exactly why it was constructed. The ordering of the rows according to the size of the first column suggests that it might have been used in an early form of trigonometry. Perhaps it was constructed from Pythagorean triples just to make arithmetic easier.

However incomplete our present knowledge of Babylonian mathematics may be, so much is establshed beyond any doubt: we are dealing with a level of mathematical development which can in many aspects be compared with the mathematics, say, of the early Renaissance. (O. Neugebauer in The exact sciences in antiquity)

Accounting for the errors

As confirmation of both the interpretation of the table and this conjecture regarding p and q, the four apparent errors can be reasonably explained:

The number [9, 1] in row 9 should be [8, 1] - a simple copying error.
[7,12,1] in row 13 is the square of [2,41], which would be the correct value - a mistake particularly easy to make since the squares also appear in the conjectured calculation.
The correct value to replace [53] in row 15 is [1,46], which is twice the erroneous value.
As for the fourth error in row 2, where [3,12,1] occurs instead of [1,20,25], there have been a couple of solutions proposed. None are entirely convincing. The possibility proposed by Gillings suggests strongly that those who made up the table had values of p and q at hand.

References

O. Neugebauer and A. Sachs, Mathematical cuneiform texts, American Oriental Society, 1945. This, as far as I know, is where the tablet was first analyzed.
O. Neugebauer, The exact sciences in antiquity, Dover, 1969. This is a reprint of the 1957 second edition. Chapter 2 spends a great deal of effort on the tablet.
R. J. Gillings, Australian Journal of Science 16 (1953), pages 54-56. Mentiuoned in Neugebauer's book.
R. Creighton Buck, Sherlock Holmes in Babylon, the Mathematical Monthly, 1964.
The Akkadian home page. The Internet has a large collection of material on cuneiform and Bablyonian tablets, although not much concerned with mathematics, and no recent photographs of the tablet Plimpton 322.

Written by Bill Casselman.