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.


30 thoughts on “The Plimpton 322 tablet re-examined

  1. Please could you enlighten me. Are current disputes about the content of the tablet or its purpose? In other words is it generally accepted that it can function as a trig table but disputed whether it was designed as such.

    Incidentally you miss a method for square roots which is intermediate between the trial and error method and the Babylonian method. Take the integer above and below the desired sq root, add numerators and add denominators to form a new fraction which will fall between the two, test to see if it is an upper or lower approximation. You now have a new pair of limits. Repeat as much as you want. The sequence for sq rt 2 begins 1/1. 2/1. 3/2. 4/3. 7/5
    Theon of Smyrna’s ‘side and diameter’ numbers are simply the alternate values of this sequence.

    • As to content, everyone is agreed that the tablet has the numbers and relationships I describe. The debate is indeed about purpose, as you suggest. The idea that it is a trig table (of an unusual kind) doesn’t really fit well with what we know of Babylonian mathematics. I would recommend reading the article by Robson (which I’ve linked to in the post).

      • Tthank you very much. I tried to Robson article and it’s quite dense, so wanted to clarify the issue in dispute and you have confirmed what I thought. Incidentally, as a middle-aged mathematical novice, I find Wildenberger has some interesting ways of relating key concepts regardless of whether his and Mansfield’s views on the function of Plimpton 322 are correct. I think his views/speculations of the history of maths deserve consideration.

      • One of the key points Robson makes is that one shouldn’t simply try to understand Plimpton 322 on its own, but in the context of how Babylonians did mathematics.

        Mansfield and Wildberger suggest the tablet teaches (an unusual style of) trigonometry, but there is no evidence of such trigonometry among the thousands of other tablets that the Babylonians left us. On the other hand, we do know that the Babylonians were interested in Pythagorean triangles and reciprocal pairs, and the tablet fits very well with that.

        Robson suggests (and I agree) that the tablet was produced by a mathematics teacher as a list of class problems, and there are indeed other tablets like that in the historical record.

      • Yes, but that begs the question as to why the values in column A decrease steadily. What relevance does that have to classroom exercises in triples and reciprocals only.

        Anyway if it is a classroom text howcdo we know it was not copied from some trig tables with more information than needed for classroom purposes. That would bevlike someone todaybphotocopying a page of Euclid for a class and skipping sayva lemma or corollary.

      • Presumably, the values in column A decrease steadily because the triples were generated in sequence (probably using additional information on the broken-off left part of the tablet).

        And, like I said, there is no evidence at all that the Babylonians did trigonometry or had trig tables. If trig was part of their mathematics, you’d expect evidence on some of the thousands of other tablets.

      • Yes, you can find lists of triples on wikipedia and elsewhere, but they are in order of size of sides not size of angles. Whyever would they be in size of angles? Indeed given other tablets this may be improbable but it is virtually impossible that they would be in order of angle size unless someone had calculated it. Sometimes we have to rely on a single source, as with the Anikythera mechanism, and I note that Alexander Jones who is prominent in that research, thinks M and W’s intepretation possible. (I should admit that my own view is influence by my belief that the data on various Mesopotamian royal inscriptions is actually scrambled astronomical data.)

    • You could be right, but like I said, I think Robson has the right interpretation.

      As to Jones, I believe that what he said was “I think the interpretation is possible, but we don’t have much in the way of contexts of use from any Babylonian tablets that would confirm such an intention, so it remains rather speculative.

  2. Jones is a satisfactory place to leave it. I’m glad to have found someone who supports Robson I could discuss it with. Robson herself doesn’t want to comment.

    • Thanks for dropping by and commenting!

      Robson’s paper is from 15 years ago, and she’s apparently been working on other stuff since. But anything she said would probably just repeat what her paper says: she critiques the trigonometric interpretation (which Mansfield and Wildberger did not invent) in that paper.

      • Okay I’ll give it another go. However she has to give an explanation of why the triples are in order of angle size becaus the probability of 15 triples being in such order is 2/15! which I make 1 divided by 2 billion. (million million)

      • Robson says that the table is 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.

        (I have added that to the post).

      • Tony,

        Thank you for explaining things to me which I could get out of Robson’s paper. Robson would also need to account for the fairly even spacing of the values.

        Maybe someone somewhere will provide a simple layperson’s guide to the controversy, one which generates light rather than heat, identifying common ground and areas of disagreement and differences in methodology. Unfortunately we are not likely to see that on wikipedia soon, and Robson made a couple of What I thought were rude and ad hominem tweets which appeared in the New York Times and which she seems to have deleted and then declines any comment.

        I can’t see how this can be definitively settled. If Robson’s is the best explanation so far there is always the possibility that something will emerge to support the trig explanation. And because I have prior reason to find Mansfield and Wildenberger’s approach plausible I am more open to their reading than many historians of maths, but I am probably going to stay agnostic. The various papers on this may make for some good and instructive autumn bedside reading.

      • I looked at Robson’s longer article and whilst I could appreciate the ordering of angles couldn’t find anything to explain the fairly even spacing of angles.

        I also took a look at Mansfield and Wildenberger and they do claim to be saying something new. It’s not new to claim it is a trig table, but they say it is a particular type of trigonometry. And so you have to enquire whether it is. What I am sure about is that is a simple whole number arithmetic which has been rather lost, as evinced by the square root method I sent you, so I find that approach plausible. But you may need to study it a bit to appreciate it. What I do hear is a chorus of attacks on M and W by people who give the impression of not knowing anything about rational trig. But maybe these will diminish when people read and critique the article.

        And all a lay person can really think is that it got past the referees of Historia Mathematica. And we are supposed to believe experts because they are experts, and if experts disagree amongst themselves we get confused.

    • The spacing of the numbers comes from the spacing in standard Babylonian tables of reciprocals.

      As Robson correctly points out, the tablet has to be understood in the context of the rest of Babylonian mathematics.

      • On p.199 of her Historia Mathematica article Robson describes the scribe’s choice as ‘shockingly ad hoc’.

  3. Regarding square roots, you say “…even simplistic starting guesses for the square roots of the numbers in column A give convergence in 2 or 3 steps every time”.

    For us, this is certainly true. We can make an initial guess, apply the square root algorithm, and then use the result as an improved guess for another application of the square root algorithm.

    But for a Babylonian there is an extra condition: the initial guess must be regular (i.e. a number with no prime factors other than 2, 3 or 5).

    The output of the square root algorithm will not, in general, be regular. So you cannot expect a Babylonian to be able to reuse the (irregular) result of the square root algorithm as the (regular) initial guess for a subsequent application of the square root algorithm. They are numerically restricted to a single application only, as is apparent in texts such as IM 54 472.

    I might also mention that Robson’s interpretation does not involve diagonals, which are quite prominently mentioned in the headings of the table.

    • You are suggesting that Babylonians couldn’t actually calculate square roots. That seems wrong. However, even if true, that doesn’t affect Robson’s interpretation (which does involve diagonals).

      • Yes, I would say that the Babylonians couldn’t calculate a square root in every case. For example the number in row 4 column 1 can be simplified to = (5.20.53)^2, but calculating sq.rt.( = 5.20.53 is problematic using only Babylonian techniques.

        If this is an table of exercises for “completing the square” style problems such as YBC 6967 then you do need to take a square root at some point. In this problem, and every single known similar example, the square root is always easy to calculate using only simplification rules and standard table of square roots – both of which are well attested to at the time. So in addition to the theoretical limits of the square root procedure, taking square roots of integers such as (5.20.53)^2 is outside their pedagogical tradition by two orders of magnitude.

        Personally, I would regard the interpretation of Britton, Proust and Shnider (2011) as an improved version of Robson’s interpretation based on new evidence that was not available to Robson in 2001/2002. There is also the “factor reduced core” interpretation from Friberg (2007), which at least identifies the square root issue.

      • and you do need to calculate a square root using according to Robson’s hypothesis, where this is a teachers aide for problems of the type found in YBC 6967. Here you start by knowing only the difference between a regular number x and its reciprocal 1/x. Your task is to find x and 1/x. First you calculate the half difference (x-1/x)/2. From this you calculate the square of the half sum as [(x-1/x)/2]^2 + 1 = [(x+1/x)/2]^2. Then take the square root to find the half sum (x+1/x).

        Once you have the half sum and half difference you can find
        ‘the igi’ x = (x+1/x)/2 + (x-1/x)/2 and
        ‘its igibi’ 1/x = (x+1/x)/2 – (1-1/x)2.

        You can interpret the numbers (x + 1/x)/2 as a diagonal, and I agree that you should. This is certainly consistent with later evidence such as MS 3971#3. But here you still need to compute a square root.

      • I’m afraid you’ve misread Robson’s work. Calculation of square roots are not actually required. See the paragraph in my post beginning “Specifically, Robson believes…”

      • I agree that the square root is not used in the generation procedure. I was taking about the purpose of the tablet.

        The discussion about square roots goes to the heart of the “class problems envisaged by Robson”. These problems (YBC 6769, but a better example would be MS 3971) involve a square root.

      • Well, on Robson’s analysis (which is still,, I think, definitive) it isn’t 100% obvious what problems the students would be set from this list. But the use of well-known reciprocal pairs argues for an attempt to smooth the students’ way.

      • I think that Britton (2011) would have agreed with you that it is not obvious what what problems the students would be set from this list. He said

        “Thus, despite their sometimes spirited differences Friberg and Robson arrive by
        different paths at a largely similar view that Plimpton 322 was intended to serve as a teachers’ aide linking reciprocal relationships to the Diagonal Rule, without ever quite pinpointing what problems it would have uniquely helped construct, or what mathematical purpose it might have served.”

        I think that if the squared quantity is an intermediate step in some student problem, then a square root would have to be taken at some point. Regardless of what the student’s problem might be.

        Regarding the use of well-known reciprocal pairs, Britton has quite a lot to say on this point. I will not attempt to recite his entire argument, you can read it for yourself on pages 531-2. I will quote one paragraph

        “Robson (2001) champions this interpretation but also acknowledges its shortcomings, namely that among OB mathematical texts no such tables of extended reciprocals are known”

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