Babylonian mathematics
Babylonian mathematics

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Babylonian mathematics

1.5 Plimpton 332

1.5.1 Uncertain origins

The tablet is called Plimpton 322, and is described by Neugebauer (The Exact Sciences in Antiquity (Dover, 1969) p. 40) as ‘one of the most remarkable documents of Old-Babylonian mathematics’. The name arises simply from the fact that the tablet has catalogue number 322 in the George A. Plimpton collection at Columbia University, New York. Plimpton bought it in about 1923 from a Mr Banks who lived in Florida; it is not certain where he obtained it, but it may have been dug up at Larsa in Mesopotamia. The left hand side of the original tablet appears to have been broken off, and traces of modern glue suggest that this has happened since its excavation. (All of this is a fairly typical example of the random, not to say slapdash, way in which things have emerged from under the sand into the eventual light of public knowledge.)

Figure 4
Figure 4 Plimpton 322

Look at the photograph and notice the main features of what remains: four columns of numbers, with words at the head of each column. Now look at the transcription (we have labelled the columns A, B, C for ease of reference), and see if any pattern is evident to you

[1;59,0,]15 1,59 2,49 1
[1;56,56,]58,14,5O,6,15 56,7 3,12,1 2
[1;55,7,]41,15,33,45 1,16,41 1,50,49
[1;]5[3,1]0,29,32,52,16 3,31,49 5.9,1 4
[1;]48,54,1,40 1,5 1,37 5
[1;]47,6,41,40 5,19 8,1 6
[1 ;]43,11,56,28,26.4O 38,11 59,1 7
[1;]41,33,45,14,3,45 13,19 20,49 8
[1;]38,33,36,36 9,1 12,49 9
1;35,10,2,28,27,24,26,40 1,22,41 2,16,1 10
1 ;33,45 45 1,15 11
1;29,21,54,2,15 27,59 48,49 12
[1;]27,0,3,45 7,12,1 4,49 13
1;25,48,51,35,6,40 29,31 53,49 14
[1;]23.13,46,40 56 53 15

At first sight, this is not very promising! Something seems to be being listed, as the lines are numbered (final column); and the numbers in column A do diminish fairly regularly, from just under 2, down to just over 1⅓. (But if you did manage to notice that it was partly the effect of the editorially informed semi-coIons, and Neugebauer's reconstructions in square brackets, of course.) Otherwise, the numbers look fairly random, and there is little to tell that this is not a Babylonian supermarket till receipt. (Until Neugebauer studied the tablet, it was, in fact, catalogued as a ‘commercial account’.) But Neugebauer discovered—presumably after a considerable amount of conjecture and refutation—that the numbers in each line can be related as:

and this was the basis for his reconstruction of the illegible entries. Let us check this out on the simplest looking complete case, line 11.

So it is possible to find a relation, albeit a somewhat devious one, between the columns of the tablet. (In fact, for this to hold consistently throughout, the underlined numbers have to be considered as mistakes by the scribe, a point to which we shall return.) Before trying to decide what this is all about, let us investigate the numbers a little more. If we calculate C2B2 for each entry, and then take its square root, something rather surprising emerges. Look at the next table, in which we have calculated the values (and the scribe's ‘errors’ have been corrected’). Ignore for the time being the final two columns which we have labeled p and q.

B C p q
(decimal) (sexagesimal)
119 169 120 2,0 12 5
3367 4825 3456 57,36 64 27
4601 6649 4800 1,20,0 75 32
12709 18541 13500 3,45,0 125 54
65 97 72 1,12 9 4
319 481 360 6,0 20 9
2291 3541 2700 45,0 54 25
799 1249 960 16,0 32 15
481 769 600 10,0 25 12
4961 8161 6480 1,48,0 81 40
45 75 60 1,0
1679 2929 2400 40,0 48 25
161 289 240 4,0 15 8
1771 3229 2700 45,0 50 27
56 106 90 1,30 9 5

Notice that, compared wilh columns B and C, the values computed in D are remarkably simple looking numbers (this is particularly noticeable in their sexagesimal representation). You may indeed have recognised some as old friends, from the reciprocal table. In fact, all of them are regular numbers (whereas B and C are all, except for line 1L, non-regular), which explains why their squares could divide into C2 and yield an exact, finite sexagesimal expression.


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