PLIMPTON 322

Part 1: Introduction & bibliography

The clay tablet with the catalog number 322 in the G. A. Plimpton Collection at Columbia University may be the most well known mathematical tablet, certainly the most photographed one, but it deserves even greater renown. It was scribed in the Old Babylonian period between -1900 and -1600 and shows the most advanced mathematics before the development of Greek mathematics.

References:

O. Neugebauer and A. Sachs. Mathematical Cuneiform Texts. Amer. Oriental Series 29. American Oriental Society, New Haven, 1945

E. M. Bruins. On Plimpton 322, Pythagorean numbers in Babylonian mathematics. Afdeling Naturkunde, Proc. 52 (149), 629-632.

Akad. v. Wetenshoppen, Amsterdam.

Kurt Vogel. Vorgriechische Mathematik. Two volumes. Hermann Schroedel Verlag KG, Hannover, 1959.

O. Neugebauer. The Exact Sciences in Antiquity. Second edition, 1957, Brown Univ. Press. Dover reprint, 1969.

R. Creighton Buck. Sherlock Holmes in Babylon. Amer. Math. Monthly 87 (1980), 335-345.

Jöran Fribert. Methods and traditions of Babylonian Mathematics I: Plimpton 322, Pythagorean triples, and the Babylonian triangle parameter equations. Historia Math. 8 (1981), 277-318.

Part 2: Transcriptions of the tablet

Here's a transcription of the Plimpton 322 tablet using modern digits. The numbers on the cuneiform tablet are written sexigesimally (in base 60) with combinations of two symbols, one for tens and one for units. Several unit symbols are placed in particular patterns to form the digits 1 through 9, and several ten symbols are placed in particular patterns to form the five digits 10, 20, 30, 40, and 50. Spaces are left to indicate zeros in the unit's place and zeros in the ten's place. In this transcription a colon is used to separate different sexagesimal digits for ease of reading. This is the same notation we use for time. For instance, 3:10:15 means 10 and 1/4 minutes after 3 o'clock.

	                           width     diagonal
	1:59:00:15                 1:59      2:49      1
	1:56:56:58:14:50:06:15    56:07      1:20:25   2
	1:55:07:41:15:33:45        1:16:41   1:50:49   3
	1:53:10:29:32:52:16        3:31:49   5:09:01   4
	1:48:54:01:40              1:05      1:37      5
	1:47:06:41:40              5:19      8:01      6
	1:43:11:56:28:26:40       38:11     59:01      7
	1:41:33:45:14:03:45       13:19     20:49      8
	1:38:33:36:36              8:01     12:49      9
	1:35:10:02:28:27:24:26     1:22:41   2:16:01  10
	1:33:45                   45         1:15     11
	1:29:21:54:02:15          27:59     48:49     12
	1:27:00:03:45              2:41      4:49     13
	1:25:48:51:35:06:40       29:31     53:49     14
	1:23:13:46:40             56         1:46     15

There are fifteen rows and four columns on the tablet. It isn't clear whether the first column has all its numbers beginning with 1 or not; unfortunately the tablet is broken off at that point. There may have been more columns in the broken off portion of the tablet. Also, there are a couple of chips off the tablet, but it's clear how to fill in the missing digits. Within each column the numbers aren't lined up quite as nicely as they are here, but the numbers in each column are left justified.

The last column is clearly just the row number. The second and third columns are headed by words that can be interpreted as "width" and "diagonal."

The transcription above already includes four corrections. In the second row, third column, the original number was 3:12:01 rather than 1:20:25. In the ninth row, second column, the original number was 9:01 instead of 8:01. In the 13th row, second column, the original number was 7:12:01 instead of 2:41. And in the 15th row, third column, the original number was 53 rather than 1:46.

Often, this tablet is converted to decimal form, but it isn't clear whether the numbers are fractions or integers. The last column clearly contains integers, the first almost certainly contains fractions, and the second and third might contain fractions but probably contain integers. Here's a translation of the second, third, and fourth columns into decimal form under the assumption that all the numbers are integers

width diagonal

119 169 1 3367 4825 2 4601 6649 3 12709 18541 4 65 97 5 319 481 6 2291 3541 7 799 1249 8 481 769 9 4961 8161 10 45 75 11 1679 2929 12 161 289 13 1771 3229 14 56 106 15

Part 3: Interpretation and possible construction

The numbers in the first column are interesting since each is a perfect square and subtracting one from each leaves a perfect square. Consider, for instance, line 11. The number 1:33:45 represents 1 + 33/60 + 45/3600 = 1 + 9/16 = 25/16 which is the square of 5/4. One less than 25/16 is 9/16, the square of 3/4. The second and third entries in this row represent these fractions: 45 represents 45/60 = 3/4 and 1:15 represents 1 + 15/60 = 5/4. (Or perhaps 45 represents 45 and 1:15 represents 75 in which case these two entries are proportional to the fractions.)

For another example, consider row 5. 1:48:54:01:40 represents 1 + 48/60 + 54/3600 + 1/216000 + 40/12960000 = 1 + 4225/5184 = 9409/5184, which is the square of 97/72. And 4225/5184 is the square of 65/72. The second and third entries in row 5 are 1:05 representing 65, and 1:37 representing 97. Nearly always, the second and third entries aren't equal to the square roots, but just proportional to them.

This table is usually considered in relation to Pythagorean triples. In that interpretation, the second column is a, one side of a right triangle or the width of a rectangle, and the third column is c, the hypotenuse of the right triangle or the diagonal of the rectangle, and the other side of the triangle or rectangle, b, doesn't appear on the table. In this interpretation the first column is then (c/b)² = 1 + (a/b)².

The Old Babylonians knew the Pythagorean theorem (better called the rule of the right triangle for them since there's no evidence that they had a proof; Gillings calls the term "the Pythagorean theorem" a true mumpsimus), since there are examples of its use in various problems of the period. Along with the headings of the second and third columns, that justifies believing that this table relates to Pythagorean triples and right triangles.

Some historians have noticed that (1) each first column entry is the square of the cosecant of an angle of a right triangle, and (2) the associated angles are roughly one degree apart. So they have suggested that this is a trigonometric table of squares of cosecants for 45° down to 30°. This would be the earliest instance of cosecants by millennia, and the earliest instance of a trigonometric function by over a thousand years (the first trig function being the chord of an angle). It would also be the earliest instance of degree measurement by over a thousand years. In other words, that's a bold claim. A weaker claim is that the table was constructed to make the first column uniformly decreasing.

Anyway, here are the actual associated angles in a table. The angles are given in degrees, but for perversity, their fractions are given decimally. A column for the remaining side is also displayed as well as columns for parameters p and q generating Pythagorean triples where a = p² - q², b = 2pq, and c = p² + q².

angle A    (c/b)²                   a         c        b      p     q     n
 44.76    1:59:00:15                1:59     2:49     2      12      5    1
 44.25    1:56:56:58:14:50:06:15   56:07     1:20:25 57:36    1:04  27    2
 43.79    1:55:07:41:15:33:45       1:16:41  1:50:49  1:20    1:15  32    3
 43.27    1:53:10:29:32:52:16       3:31:49  5:09:01  3:45    2:05  54    4
 42.08    1:48:54:01:40             1:05     1:37     1:12    9      4    5
 41.54    1:47:06:41:40             5:19     8:01     6      20      9    6
 40.32    1:43:11:56:28:26:40      38:11    59:01    45      54     25    7
 39.77    1:41:33:45:14:03:45      13:19    20:49    16      32     15    8
 38.72    1:38:33:36:36             8:01    12:49    10      25     12    9
 37.44    1:35:10:02:28:27:24:26    1:22:41  2:16:01  1:48    1:21  40   10
 36.87    1:33:45                   3        5        4       2      1   11
 34.98    1:29:21:54:02:15         27:59    48:49    40      48     25   12
 33.86    1:27:00:03:45             2:41     4:49     4      15      8   13
 33.26    1:25:48:51:35:06:40      29:31    53:49    45      50     27   14
 31.89    1:23:13:46:40            56        1:46     1:30    9      5   15

Note that the every value of p and q is a regular sexagesimal integer, that is, their only prime divisors are 2, 3, and 5. That is necessary to make b regular so that the fraction c/b terminates. The Babylonians didn't like to divide by irregular numbers.

It may be that the Babylonians actually used the parameters p and q to generate this table. There is no direct evidence that they did, but there is some indirect evidence. When you consider all possible regular values of p and q with p less than or equal to 125 (and q < p, of course), tabulate the associated triangles, and sort them by p/q, or what is the same order, by (c/b)², then there are exactly 16 triangles in the range of the table. That is, only one extra triangle.

angle A    (c/b)²                   a         c        b      p     q     n
 44.76    1:59:00:15                1:59     2:49     2      12      5    1
 44.25    1:56:56:58:14:50:06:15   56:07     1:20:25 57:36    1:04  27    2
 43.79    1:55:07:41:15:33:45       1:16:41  1:50:49  1:20    1:15  32    3
 43.27    1:53:10:29:32:52:16       3:31:49  5:09:01  3:45    2:05  54    4
 42.08    1:48:54:01:40             1:05     1:37     1:12    9      4    5
 41.54    1:47:06:41:40             5:19     8:01     6      20      9    6
 40.32    1:43:11:56:28:26:40      38:11    59:01    45      54     25    7
 39.77    1:41:33:45:14:03:45      13:19    20:49    16      32     15    8
 38.72    1:38:33:36:36             8:01    12:49    10      25     12    9
 37.44    1:35:10:02:28:27:24:26    1:22:41  2:16:01  1:48    1:21  40   10
 36.87    1:33:45                   3        5        4       2      1   11
 35.78    1:31:09:09:25:42:02:15    3:12:09  5:28:41  4:26:40 2:05  1:04 **
 34.98    1:29:21:54:02:15         27:59    48:49    40      48     25   12
 33.86    1:27:00:03:45             2:41     4:49     4      15      8   13
 33.26    1:25:48:51:35:06:40      29:31    53:49    45      50     27   14
 31.89    1:23:13:46:40            56        1:46     1:30    9      5   15

Only one extra line. Perhaps it wasn't used because of the large values of q and b. Perhaps it wasn't copied into the table; there were, after all, four other errors. Under two other hypotheses, it ought to be in the table, too, since it fills in a missing angle and a gap in the (c/b)² column.

David E. Joyce, 1995.
Department of Mathematics and Computer Science
Clark University