Following up on my “origins of the alphabet” chart, here is one for numerals. The chart was produced using R, and the pictures are purely illustrative – unlike the pictures in the alphabet chart, they do not relate to the origins of the symbols.

# Tag Archives: Babylonia

# 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 C^{2} − B^{2} is always a perfect square (yellow in the diagram). Column A is exactly equal to C^{2} / (C^{2} − B^{2}), 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} + 2^{2} = (*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 = *z*^{2} / (*z*^{2} − *y*^{2}).

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.

# 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: C^{2} − B^{2} is always a perfect square (yellow in the diagram), and C^{2} / (C^{2} − B^{2}), the ratio of blue to yellow, is always equal to A.

C^{2} − B^{2} is a perfect square |
C^{2} / (C^{2} − B^{2}) = A |
D |
---|---|---|

169^{2} − 119^{2} = 14400 = 120^{2} |
169^{2} / 14400 = 1.983402777777778 |
#1 |

4825^{2} − 3367^{2} = 11943936 = 3456^{2} |
4825^{2} / 11943936 = 1.949158552088692 |
#2 |

6649^{2} − 4601^{2} = 23040000 = 4800^{2} |
6649^{2} / 23040000 = 1.918802126736111 |
#3 |

18541^{2} − 12709^{2} = 182250000 = 13500^{2} |
18541^{2} / 182250000 = 1.886247906721536 |
#4 |

97^{2} − 65^{2} = 5184 = 72^{2} |
97^{2} / 5184 = 1.815007716049383 |
#5 |

481^{2} − 319^{2} = 129600 = 360^{2} |
481^{2} / 129600 = 1.785192901234568 |
#6 |

3541^{2} − 2291^{2} = 7290000 = 2700^{2} |
3541^{2} / 7290000 = 1.719983676268861 |
#7 |

1249^{2} − 799^{2} = 921600 = 960^{2} |
1249^{2} / 921600 = 1.692709418402778 |
#8 |

769^{2} − 481^{2} = 360000 = 600^{2} |
769^{2} / 360000 = 1.642669444444444 |
#9 |

8161^{2} − 4961^{2} = 41990400 = 6480^{2} |
8161^{2} / 41990400 = 1.586122566110349 |
#10 |

75^{2} − 45^{2} = 3600 = 60^{2} |
75^{2} / 3600 = 1.5625 |
#11 |

2929^{2} − 1679^{2} = 5760000 = 2400^{2} |
2929^{2} / 5760000 = 1.489416840277778 |
#12 |

289^{2} − 161^{2} = 57600 = 240^{2} |
289^{2} / 57600 = 1.450017361111111 |
#13 |

3229^{2} − 1771^{2} = 7290000 = 2700^{2} |
3229^{2} / 7290000 = 1.430238820301783 |
#14 |

53^{2} − 28^{2} = 2025 = 45^{2} |
53^{2} / 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.