On Plimpton 322. Pythagorean numbers in Babylonian mathematics

Pythagorean Numbers in Babylonian Mathematics by 5 Table t. I il Ill IV 1,59, 0,15 1,59 2,49 ! I, 56, 56, 58, 14, 50, 6, 15 1,55, 7,41, 15, 33, 45 1, 53, 10, 29, 32, 52, 16 56, 7 1, 16, 41 3,31, 49 3, 12,1 1, 50, 49 5, 9,1 2 3 4 28 53 15 OLAF SCHMIDT* 1, 23, 13, 46, 40 Since the publication of Plimpton 322 (a text containing Pythagorcan numbers) by O. Neugebauer and A. Sachs this particular text has been mentioned in several papers and many attempts to explain the construction of the Pythagorean numbers have been given. ‘Thus it is with some hesitation that the writing of another paper dealing with this Plimpton text is undertaken. ‘The Plimpton text in its present form (see table 1) contains four columns, I, li, HI, and IV, and each column contains 15 numbers. A break at the left hand side of the text indicates that the original tablet was larger. Column IV is the least interesting of the four columns: it enumerates the lines. The integer numbers in column Il and [II are denoted by b andd respectively, and Neugebauer showed that to each pair of numbers (6, d) on the same line corresponds an integer number and II] respectively are relatively prime, and they vary rather irregularly. Itis a remarkable fact, however, that the numbers (4) in column I are monotonely decreasing. In the original discussion by Neugebauer and Sachs two explanations of the construction of the text are given. The first explanation (and the one Neugebauer and Sachs consider most likely) takes as starting point the Pythagorean equation and assumes that the text somehow uses that the number triple (x, y, Z) given by P+ b? = a. SR. I such that = 2pq =p? —q? The number triple (£, b, d) is in other words a Pythagorean number triple. If we for each number triple compute the number where p and gq are integers, is a solution to (1). The second explanation takes as starting point the equation 4 \2 4 2’-y?=1. (2) we obtain the numbers in column L The numbers6 and d in column II A rational solution to (2) gives an integral solution to (1), and rational *Kubenhavns Universitet, Matematish Institut, Universitetsparken 5, DK-2100 Kobenhuvn 0), solutions to (2) can be obtained from a table of reciprocals by noticing Denmark that if (z, y) is a solution of (2) then the pair of numbers Cemanrus 1980. val 24

Table 2 where « and b are rational numbers. A priori this system has nothing 2; 8 0: 28, 2; 5 first: + then (4 ) > de 7, 30 0; 30 1; SS, 12 0; 31, 15 1; 52, 30 0; 32 1,51, 1,48 0; 32, 24 0; 33, 20 and rae . ral 2 ras 0; 29, 37, 46, di | 0; 28, 48 2; 1,30 a, (5) = 0,27 0, 27, 46, 40 hoi 0; 26, 40 2, 13, 20 2; 9,36 | 2,15 role 0; 25, 18, 45 0; 25, 36 0; 25, $$, 12 i T 2; 22, 13, 20 2, 20, 37, 30 2; 18, 53, 20 and y be rational for given a and b we arrive at an identity (12) which may lead to Pythagorean numbers. In detail this is seen as follows. In the text material a solution (x, y) to (3) and (4) is found by computing I 0; 25 Le ni 2, 24 6,40 q to do with Pythagorean numbers. But by giving a detailed description of how such a system is solved, and by requiring that the solutions x x D: x + 6 The solution (x, y) is given by the last two numbers. To this procedure for finding the solutions to a quadratic equation given on a normal form we may add the following comments. Nowhere does the Babylo- (zy, 2+y) nian text material reveal how these solutions were obtained. By the (“On Plimpton 322”, p. 630) has computed the whole table 2 and argued strongly in favour of this second explanation. The present explanation leads to a construction of Plimpton 322 which is based upon a table of reciprocals, and in this respect it coincides with the construction given by E. M. Bruins and with the construction rejected by Neugebauer and Sachs. Our explanation deviates however from the explanations by either Neugebauer, Sachs or Bruins in that it does not take either equation (1) or equation (2) as starting point. li takes as starting point an identity (12) (sec infra) which can be derived from a normal form of a quadratic equation, i.e. a system of equations x+yFa (3) x-y=b, (4) following simple reasoning we may however justify the procedure that leads to the solution. First we assume that x and y are equal. Then each of them is equal to a I} and thus equation (3) is satisfied. In order to decide whether equation (4) is satisfied we compute. $°4=(3) ì and if this is equal to b the problem is solved. If not we assume that x li is a pair of reciprocals. Neugebauer and Sachs actually constructed the first four pairs of such a table of reciprocals (see table 2) from which the numbers in Plimpton 322 can be computed, and E. M. Bruins =35 + “something” y=73 - “something”. This rather inconventional way of writing x and y has been used in

Order to emphasize that an apparently sophi stica ted mathematical table like Plimpton 322 can be worked out by very eleme ntary reasonunknowns a ings and without the use of any mathematical notation. The critical reader may object that in writing “x” and “y” for the modern notation has already been used. It is, however, a simple matand On Plumpton 322 a=xty b=x'y. gy is (9) It is now evident that if a and 6 are chosen as in (9), then (8) satisfied. In other words it is evident that for any rational number sx and y 4 [v2 (x+y) - xy = O. (10) ion by replacing x ter to make the explanation free from modern notat and y by eg. “length” and “width”, terms sometimes used in this connection by the texts. The problem now is to adjust “some thing” so Ir ty)? - xy = [2 (fe (11) Thus the Babylonian computer who was familiar with a norma l form of a quadratic equation most likely would know that DU = {2 (a-y)]?. ıy=b that the equation (7) We may even assume that it also was known that the Q in (10) is the same as (“something”)? in (7), and therefore (“something”)? or DI can be found from (6) by a subtraction so that is satisfied. To this end we compute xy and find (4) — (“something”)? This most be equal to b, and hence (“something”)? = (4)? -b and from this we find “something” When we say that (11) was known we mean that it was known in the form that half the sum of two numbers raised to the second power by extracting the square root, and thus we have explained the procedure used by the text, minus the product of the numbers is equal to half the difference of the two numbers raised to the second power. Next we consid er a special case Of a normal form of a quadratic equation, namely a system in X+ty=a xy=l La (ati)? I = 4 1 = Pe (Gp. x (12) and denoting the reciprocal of a number x by x this system may be written in the form which b = 1. Such a system actually occurs in the text material (Neugebauer, Mathematische Keilskrift-Texte, vol. 1, p. 106. The system is (8) We have given a detailed exposition of this procedure in order to point out that itis almost certain that the attentive user of this procedure noticed that the computations may get to a stand still at (5) or (7) unless 4 and b are chosen properly, ie. unles s a and b are chosen such that 6} -b=0 the ques- Il lati and the identity (11) now becomes > = where LI denotes the square of a rational numbe r. And now tion: how was such a choice made, natur ally sugge sts itself. We shall of course never know for certain how the Babylonians managed to choose the numbers a and b so that (8) is satisfied. An obvious guess is that they did not start the system of equations (3) and (4) by choosing 4 and b. They rather started by choosing the solutions x and y, and then the numbers a and b were computed by putting

On Plimpton 322 a 3 4 10 We now maintain that (12) combined with table 2 (i.e. a table of Yo (atx), Wa (xx). reciprocals of altogether 15 pairs of numbers of the form x and x) is the basis of Plimpton 322 in the following sense. From table 2 we compute 15 pairs of numbers From (12) it follows that (2 +2) - [2 (kP = N. (15) Each pair in (13) is multiplied by a suitable power of 60, say 60%, so that a pair of integers Ha (x + à) 60%, 12 (x — x) 604 is oblained. The greatest common divisor of the integers (15) divides the sum of the integers, ie. x 60%, and thus the greatest common and 6) which satisfy a relation like (14), namely divisor is regular. The integers (15) divided by their greatest common divisor is a pair of relatively prime numbers (let us call the numbers d d?—b? =D. It now turns out that the numbers d and b computed in this manner 0 1,50,49 5,9,1 1:23,6,45 1:22,24.16 oa 2 55.1 are precisely the numbers in columns II and Il of Plimpton 322, and 56,7 1,16,41 3,31,49 1;59,0,15 1;56,56,58,14,50,6,15 1:55,7,41.15.33,45 1;53,10,29,32,52.16 1:24.30 0:59,0,15 0:56.56,58.14,50.6,15 0:55,7,41,15,33,4 5 0;53,10.29,32,52.16 1:2 3.46.2.30 0:39,30 258,27 .17.30 0 157,304 G:56,29.4 LER further, the numbers in column I are the numbers IV 025.536 [Va (x+x)}?. IN 0;25,18,45 We have thus given an exceedingly simple direction tor constructing the numbers in Plimpton 322. We notice that in the explanations given by Neugebauer and Sachs, and E. M. Bruins the number/ determined P=d-b? nl 10] [Me (+4)}? [iz (xt)? N mM Nz (x4à) Ha (x) 11) [IV] by plays a rather significant role. In our explanation the number/ does not Occur explicitly. Nor does/ occur explicitly in the text proper, and Table 3.

in this respect our explanation agrees better with the text than the two previous explanations. Implicitly the number / does of course enter into our explanation; it is readily seen that/ is the greatest common divisor of the integers obtained from (13) by a multiplication of a suitable power of 60. A remark by Neugebauer that / most likely was computed in a column prior to column I is not confirmed by our explanation. Our explanation rather leads to a text which in its complete form contains the following columns: LV]. (4), [HI], [4 {O}, 1, EL HI, IV where columns 1, Columns I and Hi contain numbers proportional to the numbers in column [I] and [1], and reduced in the sense explained above. The computation of the numbers in columns II and HE may be illustrated by the following example. The numbers in columns [-H] and [—I], line 3 multiplied by 60° are equal to 57,30,45 and 1,23,6,45. The greatest common divisor of these numbers is regular (see supra). By dividing the numbers by 3-3:5 we obtain the numbers 1,16,41 and 1,50,49. ..., IV are the actually preserved columns, and columns |-IV], ..., [U] are reconstructed columns, (See table 3.) Columns |-IV] and [IH] contain 15 pairs of numbers 13 ‘These numbers do not have any regular common divisor and hence they are relatively prime, and they are the numbers in columns H and . x and x Hl, line 3. Thus it is possible to construct a text of altogether 9 columns (or 8 given in table 2. Columns [-I1] and [--1] contain 15 pairs of numbers Yo (wat) and '2 (xt) computed from the two preceding columns. Columns [0] and I contain 15 pairs of numbers columns if column [0] which deviates from column I by one is left out) of which the last four are the columns appearing in the fragment of Plimpton 322. We notice that this reconstructed text does not have any obvious relation to a right triangle. In the heading of columns I and [Il we do find, however, the words “width” and “diagonal”. In order to explain this fact we refer to the identity (12) which shows that the difference between the squares of 1/2 (v+x) and Y2 (rx) is a [2 Kk)? and [42 +)? square, and this remains true if the numbers are reduced, i.e. multicomputed trom the two preceding columns. ‘The identity (12) shows that the difference between two numbers from columns I and [0] is 1. Thus if the number in column I, line 2 is this way (i.e. the numbers in columns II and III) deserve the names plicd by a proper constant, and so the pairs of numbers obtained in “diagonal” and “width” (of a rectangle). 1; 56,56,58,14,50,6,15 then the number in column [0], line 2 is 0; 56,56,58,14,50,6,15 and having noticed this very simple rule, there is hardly any need of both columns [0] and 1. One of them will do. BIBLIOGRAPHY E. M. Bruins, “On Plimpton 322. Pythagorean numbers in Babylonian Mathemalies”, Ronmklyke Nederlandsche Akadenne van Weienschappen. Proceedings, Vol 191 -194. 52 (1949), E. M. Bruins, “Reciprocals and Pythagorean Triads”, Physis, Vol. 9 (1967), 371-392. O. Neugebauer and A. Sachs, Mathematical Cuneiform Texts, New Haven, Connecticut, 1945. The text Plimpton 322 ts published and discussed on p. 38-41. O. Neugebauer, Mathematische Keilschrift- Texte, Vol, 1-3. Reprint, Berlin, 1973. Centaurus XXIV