The Plimpton 322 tablet re-examined

Several years back I blogged about the Plimpton 322 tablet – a Babylonian clay tablet from around 1,800 BC. It contains four columns of numbers, written in base 60 (with a small number of errors, as well as some numbers missing through damage – these are corrected below). For example, 1.59:00:15 = 1 + 59/60 + 0/3600 + 15/216000 = 1.983402777777778.

Column B of the tablet (with a label on the tablet containing the word “width”) is one of the sides of a Pythagorean triangle, and column C (with a label on the tablet containing the word “diagonal”) is the hypotenuse, such that C2 − B2 is always a perfect square (yellow in the diagram). Column A is exactly equal to C2 / (C2 − B2), the ratio of blue to yellow.

A B (“width”) C (“diagonal”) D
1.59:00:15 = 1.983402777777778 1:59 = 119 2:49 = 169 #1
1.56:56:58:14:50:06:15 = 1.949158552088692 56:07 = 3367 1:20:25 = 4825 #2
1.55:07:41:15:33:45 = 1.918802126736111 1:16:41 = 4601 1:50:49 = 6649 #3
1.53:10:29:32:52:16 = 1.886247906721536 3:31:49 = 12709 5:09:01 = 18541 #4
1.48:54:01:40 = 1.815007716049383 1:05 = 65 1:37 = 97 #5
1.47:06:41:40 = 1.785192901234568 5:19 = 319 8:01 = 481 #6
1.43:11:56:28:26:40 = 1.719983676268861 38:11 = 2291 59:01 = 3541 #7
1.41:33:45:14:03:45 = 1.692709418402778 13:19 = 799 20:49 = 1249 #8
1.38:33:36:36 = 1.642669444444444 8:01 = 481 12:49 = 769 #9
1.35:10:02:28:27:24:26:40 = 1.586122566110349 1:22:41 = 4961 2:16:01 = 8161 #10
1.33:45 = 1.5625 45 1:15 = 75 #11
1.29:21:54:02:15 = 1.489416840277778 27:59 = 1679 48:49 = 2929 #12
1.27:00:03:45 = 1.450017361111111 2:41 = 161 4:49 = 289 #13
1.25:48:51:35:06:40 = 1.430238820301783 29:31 = 1771 53:49 = 3229 #14
1.23:13:46:40 = 1.38716049382716 28 53 #15

What is this table all about? A good discussion is by Eleanor Robson [Words and pictures: new light on Plimpton 322,” American Mathematical Monthly, 109 (2): 105–120]. Robson sees Plimpton 322 as fitting into standard Babylonian mathematics, and interprets it as a teacher’s effort to produce a list of class problems.

Specifically, Robson believes that the table was generated by taking values of x (in descending order of x) from standard Babylonian reciprocal tables (specifically the values 2:24, 2:22:13:20, 2:20:37:30, 2:18:53:20, 2:15, 2:13:20, 2:09:36, 2:08, 2:05, 2:01:30, 2, 1:55:12, 1:52:30, 1:51:06:40, and 1:48) and then using the relationship (x − 1 / x)2 + 22 = (x + 1 / x)2 to generate Pythagorean triples. If we let y = (x − 1 / x) / 2 and z = (x + 1 / x) / 2, then B and C are multiples of y and z, and A = z2 / (z2 − y2).

Just recently, Daniel F. Mansfield and N. J. Wildberger [Plimpton 322 is Babylonian exact sexagesimal trigonometry,” Historia Mathematica, online 24 August 2017] interpret the table as proto-trigonometry. I find their explanation of the first column (“a related squared ratio which can be used as an index”) unconvincing, though. Why such a complex index? Robson calls such trigonometric interpretations “conceptually anachronistic,” and points out that there is no other evidence of the Babylonians doing trigonometry.

Mansfield and Wildberger also suggest that “the numbers on P322 are just too big to allow students to reasonably obtain the square roots of the quantities required.” However, I don’t think that’s true. The Babylonians loved to calculate. Using the standard square-root algorithm, even simplistic starting guesses for the square roots of the numbers in column A give convergence in 2 or 3 steps every time. For example, to get the square root of 1.59:00:15 (1.983402777777778), I start with 1.30:00:00 (1.5) as a guess. That gives 1.24:40:05 as the next iteration, then 1.24:30:01, and then 1.24:30:00 (1.408333333333333), which is the exact answer. That said, however, calculating those square roots was not actually necessary for the class problems envisaged by Robson.

Sadly, I do not think that Mansfield and Wildberger have made their case. Robson is, I believe, still correct on the meaning of this tablet.


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Plimpton 322: Mathematics 3,800 years ago

The Plimpton 322 tablet is a Babylonian clay tablet, written in cuneiform, from around 1,800 BC (now held at Columbia University). The tablet contains four columns of numbers, written in base 60 (a system that survives in our hours, minutes, and seconds):

A B (“width”) C (“diagonal”) D
1.59:00:15 = 1 + 59/60 + 15/216000 = 1.983402777777778 1:59 = 60 + 59 = 119 2:49 = 2×60 + 49 = 169 #1
1.56:56:58:14:50:06:15 = 1.949158552088692 56:07 = 3367 1:20:25 = 4825 #2
1.55:07:41:15:33:45 = 1.918802126736111 1:16:41 = 4601 1:50:49 = 6649 #3
1.53:10:29:32:52:16 = 1.886247906721536 3:31:49 = 12709 5:09:01 = 18541 #4
1.48:54:01:40 = 1.815007716049383 1:05 = 65 1:37 = 97 #5
1.47:06:41:40 = 1.785192901234568 5:19 = 319 8:01 = 481 #6
1.43:11:56:28:26:40 = 1.719983676268861 38:11 = 2291 59:01 = 3541 #7
1.41:33:45:14:03:45 = 1.692709418402778 13:19 = 799 20:49 = 1249 #8
1.38:33:36:36 = 1.642669444444444 8:01 = 481 12:49 = 769 #9
1.35:10:02:28:27:24:26:40 = 1.586122566110349 1:22:41 = 4961 2:16:01 = 8161 #10
1.33:45 = 1.5625 45 1:15 = 75 #11
1.29:21:54:02:15 = 1.489416840277778 27:59 = 1679 48:49 = 2929 #12
1.27:00:03:45 = 1.450017361111111 2:41 = 161 4:49 = 289 #13
1.25:48:51:35:06:40 = 1.430238820301783 29:31 = 1771 53:49 = 3229 #14
1.23:13:46:40 = 1.38716049382716 28 53 #15

Column B (with a label on the tablet containing the word “width”) is one of the sides of a Pythagorean triangle, and C (with a label on the tablet containing the word “diagonal”) is the hypotenuse. Column A is the ratio of the square on the hypotenuse to the square on the third side. The tablet does contain a small number of errors, as well as some numbers missing through damage, but correcting them gives the very accurate results below: C2 − B2 is always a perfect square (yellow in the diagram), and C2 / (C2 − B2), the ratio of blue to yellow, is always equal to A.

C2 − B2 is a perfect square C2 / (C2 − B2) = A D
1692 − 1192 = 14400 = 1202 1692 / 14400 = 1.983402777777778 #1
48252 − 33672 = 11943936 = 34562 48252 / 11943936 = 1.949158552088692 #2
66492 − 46012 = 23040000 = 48002 66492 / 23040000 = 1.918802126736111 #3
185412 − 127092 = 182250000 = 135002 185412 / 182250000 = 1.886247906721536 #4
972 − 652 = 5184 = 722 972 / 5184 = 1.815007716049383 #5
4812 − 3192 = 129600 = 3602 4812 / 129600 = 1.785192901234568 #6
35412 − 22912 = 7290000 = 27002 35412 / 7290000 = 1.719983676268861 #7
12492 − 7992 = 921600 = 9602 12492 / 921600 = 1.692709418402778 #8
7692 − 4812 = 360000 = 6002 7692 / 360000 = 1.642669444444444 #9
81612 − 49612 = 41990400 = 64802 81612 / 41990400 = 1.586122566110349 #10
752 − 452 = 3600 = 602 752 / 3600 = 1.5625 #11
29292 − 16792 = 5760000 = 24002 29292 / 5760000 = 1.489416840277778 #12
2892 − 1612 = 57600 = 2402 2892 / 57600 = 1.450017361111111 #13
32292 − 17712 = 7290000 = 27002 32292 / 7290000 = 1.430238820301783 #14
532 − 282 = 2025 = 452 532 / 2025 = 1.38716049382716 #15

The obvious question here is: why on earth were they doing this? There have been quite a few theories. A good recent discussion is by Eleanor Robson [Words and pictures: new light on Plimpton 322,” American Mathematical Monthly 109 (2): 105–120]. We don’t really know enough about how Babylonian mathematicians thought to ever be totally certain why they were doing this, but they were clearly very good at this kind of geometric calculation. Which makes this 3,800-year-old clay tablet really cool.

An Assyrian star map

The badly damaged Assyrian clay disc below (K 8538) is an interesting relic of ancient astronomy. It was held in the Library of Ashurbanipal, 2600 years ago. Today it is in the British Museum, where I was fortunate enough to see it once. The disc maps the sky as eight 45° sections and has annotations in cuneiform (both names and numbers). In the view shown here, the constellation Orion is superimposed on the disc’s own crude figure (which could also represent the constellation Gemini):