Babylonian mathematics
Babylonian mathematics

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Babylonian mathematics

1.3 The historical study of cuneiform

Now, how did historical study reach the stage where Neugebauer and Sachs could pick up a tablet in a library and translate it so as to provide a fair degree of understanding? As with Egyptian hieroglyphs, cuneiform studies date from the last century. Their equivalent of the Rosetta Stone—a trilingual inscription for which one of the languages could be partially understood—was a sheer rock-face at Behistun in south-western Iran into which a text was carved in three languages, Old Persian, Elamite and Babylonian, proclaiming the victories of Darius the Great (520 BC). It was the British Consul in Baghdad, Henry Rawlinson, who rediscovered this inscription and between 1835 and 1851 copied it (at the risk to his life that any amateur mountaineer faces 300 feet up a precipice) and began to decipher both the script and the languages. Shortly thereafter, the burgeoning science of archaeology resulted in excavations of cuneiform tablets from ancient sites in Mesopotamia. These have sometimes been unearthed in vast quantities, with the result that there are now many more tablets available, in museums and universities throughout the world, than have been translated or even catalogued. It is only a small proportion of these that have been shown to have mathematical content, perhaps five hundred or so, compared with the several hundred thousand extant tablets. The results of studying these emerged in the 1920s and 1930s, and led to a considerable re-evaluation of the Babylonians, who within a decade changed from being a bare footnote to biblical studies (as in the Tower of Babel), to being a culture whose mathematical attainments put those of the Greeks of 1200 years later into a fresh perspective.

The earliest understanding to emerge was that of the Babylonians' remarkable numeration system. This discovery was due, once again, to Henry Rawlinson, who in 1855 was studying a tablet from the ancient city of Larsa. Look at the illustration and see if you can identify some of its main features, then come back to the description here.

Figure 3
Figure 3 Tablet from Larsa

It seems to consist of four columns, of which the second and fourth do not change, but the first and third do. The third, especially, changes in so regular a way that it is fair to infer that this is a column of successive numbers, constructed on a principle like that of the Egyptian hieroglyphic numbers. If represents 1, and is 10, then the third column would be the numbers 49, 50,51,…, 58, 59, and then 1, for a reason that is not yet clear. The first column, though, is not so regular and has the curious feature that like symbols (if that is what they are) are not all collected together. The third line, for instance, has four 0s, then three 1s, then two 10s, then one 1. Rawlinson realised that all could be consistently explained if the assumption were dropped that a number sign could represent only one number value. So he suggested that the 1-symbol at the foot of the third column was to be understood as 60, and that the third line's first-column number was forty-three 60s and twenty-one 1s, This is, then, a place-value system (see Box 1), in which the value of each component number symbol depends on its place in the numeral as a whole.

Box 1: A note on numeration systems

  1. We write numerals in what is called the decimal place-value system: in ‘88’, for example, the first 8 has a value which is ten times that of the 8 in the units place.

  2. We also have distinct symbols, 1,2,…, 9, 0, to put in each place without involving repetition; we have enough distinct symbols to avoid having to repeat ‘1’ eight times to signify 8, for instance.

The Babylonians had a numeration system as in A, except that it was sexagesimal—each place has value sixty times the next, compared with our ten times.

For constructing numbers within each place, the Babylonians used a repetitive system as with the Egyptian hieroglyphs. If there were no value in some place (which is what our zero symbol signifies) a space was sometimes left, but otherwise meant 1 or 60 or 3600 (or, indeed, or etc.) according to context. In much later sources, mainly astronomical texts dating from c. 300 BC onwards, a zero symbol is found to mark empty places within numerals; but not at the end of a numeral, so the absolute value of the whole is still left ‘floating’.

Question 2

Try to transcribe the Larsa tablet. Can you suggest what the cuneiform words (columns two and four) might mean? (Hint: You may find that it helps to form some initial hypothesis about relationships among the numbers, and see if this is borne out elsewhere. So try first to work out the relationship of numbers in the second line (what relation does ‘forty-one sixties and forty’ have to fifty?), then see if the third line confirms this, and so to the whole tablet.)

Discussion

If you followed the hint, you should have found that

(forty-one sixties and forty) comes to 2500 in our numerals, which is the square of

(fifty). So we should try to see if this also works for other numbers. The next line we might hope would be the square of fifty-one, (that is, 2601), which indeed is forty-three sixties and twenty-one. It looks as though our hypothesis is on the right lines. Now we can go back to the first line, (which the hint steered you away from as it is slightly trickier), and note that if the first number is to be 49 squared (2401), then the cuneiform symbols cannot be forty-one anything—even though that is what they look like—but must be forty sixties and one unit.

It seems safe, then, to infer that column two carries the meaning ‘is’ or ‘equals’, and column four means ‘squared’, so the first line would read

2401 equals 49 squared

and so on, down to

3600 equals 60 squared.

There is one further thing to notice about our interpretation of the tablet. Suppose that all the numbers but one, say, had fitted our conjectured pattern: how should we respond to the inconsistent entry? It is just possible, number patterns being indefinitely many, that some other much more complicated interpretation could be found to cover every number without exception. Historians generally adopt the simpler view that the scribe must have made a mistake. Primary sources are not necessarily ‘correct’ merely by virtue of being old! Note that our confidence in sometimes changing the content of a tablet is only possible because of the mathematical structuring we presume it to have. Indeed, if a tablet is quite badly damaged it may be only that presumption that enables it to be reconstructed at all. (Perhaps this is a distinction between the history of mathematics and that of more empirical subjects.)

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