# 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.