普林普顿322Plimpton 322

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17° Plimpton 322. Archeologists working in Mesopotamia have systematically unearthed, since before the middle of the nineteenth century, some half-million inscribed clay tablets. Of these half-million tablets, about three hundred have so far been identified as strictly mathematical tablets containing mathematical tables and lists of mathematical problems. We owe most of our knowledge of ancient Babylonian mathematics to the scholarly deciphering and interpretation of many of these mathematical tablets.

Perhaps the most remarkable of the Babylonian mathematical tablets yet analyzed is that known as Plimpton 322, meaning that it is the item with catalogue number 322 in the G. A. Plimpton collection at Columbia University. The tablet is written in Old Babylonian script, which dates it somewhere from 1900 to 1600 B.C., and it was first described by Otto Neugebauer and A. J. Sachs in 1945.

Figure 1 gives an idea of the shape of the tablet. Unfortunately a missing piece has been broken from the entire left edge and the tablet is further marred by a deep chip near the middle of the right edge and a flaked area in the top left corner. Upon examination, crystals of modern glue were found along the left broken edge of the tablet. This suggests that the tablet was probably complete when excavated, that it subsequently broke, that an attempt was made to glue the pieces back together, and that later the pieces again separated. Thus the missing piece of the tablet may still be in existence but, like a needle in a haystack, lost somewhere among the collections of these ancient tablets. As we shall shortly see, it would be very interesting if this missing piece should be found.

The tablet contains three essentially complete columns of figures which, for convenience, are reproduced on Figure 1 in our own decimal notation. There is a fourth and partly incomplete column of figures along the broken edge. We shall later reconstruct this column.

It is clear that the column on the extreme right merely serves to number the lines. The next two columns seem, at first glance, to be rather haphazard. With study, however, one discovers that corresponding numbers in these columns, with four unfortunate exceptions, constitute the hypotenuse and a leg of integral-sided right triangles. The four exceptions are noted in Figure 1 by placing the original readings in parentheses to the right of the corrected readings. The exception in the second line has received an involved explanation, but the other three exceptions can be easily accounted for. Thus, in the ninth line, 481 and 541 appear as (8, 1) and (9, 1) in the sexagesimal system. Clearly the occurrence of 9 instead of 8 could be a mere slip of the stylus when writing these numbers in cuneiform script. The number in line 13 is the square of the corrected value, and that in the last line is half of the corrected value, showing that the squares and the halves of certain numbers in the table probably played a role in the construction of the table.

Now a set of three positive integers, like (3, 4, 5), which can be the sides of a right triangle, is known as a Pythagorean triple. Again, if the triple contains no common factor other than unity, it is known as a primitive Pythagorean triple. Thus (3, 4, 5) is a primitive triple, whereas (6, 8, 10) is not. One of the achievements of the Arabians, two thousand years after the date of the Plimpton tablet, was to show that all primitive Pythagorean triples (a, b, c) are given parametrically by

a = 2uv, b = u² − v², c = u² + v²,

where u and v are relatively prime, of different parity, and u > v. Thus if u = 2 and v = 1, we obtain the primitive triple a = 4, b = 3, c = 5.

Suppose we compute the other leg a of the integral-sided right triangles determined by the given hypotenuse c and leg b on the Plimpton tablet. We find the following Pythagorean triples:

| lines | a | b | c | u | v | |---:|---:|---:|---:|---:|---:| | 1 | 120 | 119 | 169 | 12 | 5 | | 2 | 3456 | 3367 | 4825 | 64 | 27 | | 3 | 4800 | 4601 | 6649 | 75 | 32 | | 4 | 13500 | 12709 | 18541 | 125 | 54 | | 5 | 72 | 65 | 97 | 9 | 4 | | 6 | 360 | 319 | 481 | 20 | 9 | | 7 | 2700 | 2291 | 3541 | 54 | 25 | | 8 | 960 | 799 | 1249 | 32 | 15 | | 9 | 600 | 481 | 769 | 25 | 12 | | 10 | 6480 | 4961 | 8161 | 81 | 40 | | 11 | 60 | 45 | 75 | 2 | 1 | | 12 | 2400 | 1679 | 2929 | 48 | 25 | | 13 | 240 | 161 | 289 | 15 | 8 | | 14 | 2700 | 1771 | 3229 | 50 | 27 | | 15 | 90 | 56 | 106 | 9 | 5 |

It will be noticed that all of these triples, except the ones in lines 11 and 15, are primitive triples. For discussion we have also listed the values of the parameters u and v leading to these Pythagorean triples. The evidence seems good that the Babylonians of this remote period were acquainted with the general parametric representation of primitive Pythagorean triples as given above. This evidence is strengthened when we notice that u and v, and hence also a (since a = 2uv), are regular sexagesimal numbers—that is, are numbers of the form 2^p 3^q 5^r and thus have their reciprocals expressible as terminating sexagesimal fractions. It appears that the table on the tablet was constructed by deliberately choosing small regular numbers for the parameters u and v.

This choice of u and v must have been motivated by some subsequent process involving division, for regular numbers appear in tables of reciprocals and are useful in reducing division to multiplication. An examination of the fourth, and partially destroyed, column gives the answer. For this column is found to contain the values of (c/a)² for the different triangles. To carry out the division, the side a, and hence the numbers u and v, had to be regular.

It is worth examining the column of values for (c/a)² a little more deeply. This column, of course, is a table giving the square of the secant of the angle B opposite side b of the right triangle. Because side a is regular, sec B has a finite sexagesimal expansion. Moreover it turns out, with the particular choice of triangles as given, that the values of sec B form a surprisingly regular sequence which decreases by almost exactly 1/60 as we pass from one line of the table to the next, and the corresponding angle decreases from 45° to 31°. We thus have a secant table for angles from 45° to 31°, formed by means of integral-sided right triangles, in which there is a uniform jump in the function rather than in the corresponding angle. All this is truly remarkable. It seems highly probable that there were companion tables giving similar information for angles ranging from 30° to 16° and from 15° to 1°.

The analysis of Plimpton 322 shows the careful examination to which some of the Babylonian mathematical tablets must be subjected. Formerly such a tablet might have been summarily dismissed as merely a business list or record.