The missing columns in Plimpton 322

Anthony Phillips
Mathematics Department, Stony Brook University
April 2004

Plimpton 322 is probably the best-known Babylonian mathematical text. A recent publication by Eleanor Robson [Amer Math Monthly Feb 2002 105-120] has given a linguistically, culturally and archaeologically satisfying analysis of the content and purpose of the tablet. This note corroborates Robson's interpretation by providing a mathematical characterization of the hypothetical list of numbers used to generate the entries in the table.

image from Casselman's webpage http://www.math.ubc.ca/~cass/courses/m446-03/pl322/pl322.html

Here is a transcription of the numerical part of the tablet, from Robson's article.

[(1) 59] 00 15                 1 59        2 49     KI.1 
[(1) 56 56] 58 14 50 06 15    56 07        1 20 25  KI.2 
[(1) 55 07] 41 15 33 45        1 16 41     1 50 49  KI.3 
(1) 53 10 29 32 52 16          3 31 49     5 09 01  KI.4 
(1) 48 54 01 40                1 05        1 37     KI.[5] 
(1) 47 06 41 40                5 19        8 01    [KI.6] 
(1) 43 11 56 28 26 40         38 11       59 01     KI.7 
(1) 41 33 45 14 3 45          13 19       20 49     KI.8 
(1) 38 33 36 36                8 01       12 49     KI.9 
(1) 35 10 02 28 27 24 26 40    1 22 41     2 16 01  KI.10 
(1) 33 45                     45           1 15     KI.11 
(1) 29 21 54 2 15             27 59       48 49     KI.12 
(1) 27 00 03 45                2 41        4 49     KI.13 
(1) 25 48 51 35 6 40          29 31       53 49     KI.14 
(1) 23 13 46 40               28          53        KI.15

Text in [brackets] is a reconstitution of missing material. The (1)s are implied by the mathematical structure of the data and may have been written ahead of or along the documented break in the tablet. Comparison of this tablet with others of the period suggests strongly that the calculation is carried over from a now missing left-hand companion tablet. The cleanness of the break suggests that a special, long tablet was fashioned by pushing two standard tablets end to end. Although this joint would have been fragile, the reported traces of modern glue on the join surface suggest that the two halves survived together until modern times.

This transcription includes four unadvertised corrections of the cuneiform entries. According to Joyce: "In the second row, third column, the original number was 3:12:01 rather than 1:20:25. In the ninth row, second column, the original number was 9:01 instead of 8:01. In the 13th row, second column, the original number was 7:12:01 instead of 2:41. And in the 15th row, third column, the original number was 53 rather than 1:46." Robson's transcription differs from Joyce's in that, in row 15, she corrects the 56 to a 28 and leaves the 53. Our analysis will show that this was indeed the correct correction.

It should be noted that although Babylonian numbers were very well developed as a place-value system, with base 60, the system lacked a "decimal point." So that 1:59 could mean 1 x 60 + 59 = 119 or just as well 1 + 59/60, etc. (The story goes that in any particular practical context the order of magnitude would have been obvious.)

The uncontested mathematical properties of the numbers on the tablet are as follows (the first three columns are labeled A and S, D following Robson, for Short and Diagonal; the fourth column needs no explanation).

The numbers in column A decrease from top to bottom.
Each number in column A is a perfect square AND one more than a perfect square. This works as stated when the leftmost place in A is taken as units; if that place represents an even power of 60, it works with "one" replaced by that power.
In each row, the numbers in column S and column D are relatively prime. (One exception: row 11).
In each row, the numbers S and D are two parts of a Pythagorean triple (i.e. D² - S² is a perfect square).

This last fact about S and D is quite remarkable; here it is worked out in base-10 notation, with L² = D² - S²:

row      S       D       L
1       119     169     120
2      3367    4825    3456
3      4601    6649    4800
4     12709   18541   13500
5        65      97      72
6       319     481     360
7      2291    3541    2700
8       799    1249     960
9       481     769     600
10     4961    8161    6480
11       45      75      60
11     1679    2929    2400
13      161     289     240
14     1771    3229    2700
15       28      53      45

It is obvious but still important to mention that there is no column L on the tablet. So even though Plimpton 322 has information equivalent to a collection of Pythagorean triples, it is inaccurate to say that it contains such a list, or even that, to its writer, it necessarily implied such a list.

There is no obvious relation between the elements in column A and those in columns S and D. But according to the "spreadsheet" mentality of the Babylonians, they must be part of the same sequence of calculations, a sequence that presumably began on the missing half of the tablet. The riddle is, what was in the missing columns?

Robson comes down on the side of an interpretation going back to work of Bruins in 1949-1955. In her words, with slightly changed notation, it reads:

"The entries in the table are derived from reciprocal pairs x and 1/x, running in descending numerical order from 2 24 ~ 0 25 to 1 48 ~ 0 33 20 (where ~ marks sexagesimal reciprocity). From these pairs the following reduced triples can be derived: s = (x - 1/x)/2, d = (x + 1/x)/2 , l = 1. The values given on the tablet, according to this theory, are all scaled up or down by common factors 2, 3, and 5 until the co-prime values S and D are reached."

There are many possible choices of reciprocal pairs between 2 24 and 1 48. The explanation can be made more convincing by exhibiting a rule which explains why these particular numbers x are used.

Rule: the entries were generated from a list of especially nice numbers. These are the numbers x which are "round and reciprocally round" in the following precise sense:
x and y=1/x are 4-digit base-60 numbers, with last digit a multiple of 10.

Remember that we are talking about sexagesimal reciprocity, which means that the product xy must equal a power of 60. Here is an observation which I believe has not been made before.

There are exactly 15 numbers, lying between 2 24 and 1 48, which satisfy this criterion.

(It is possible that Plimpton 322 is one of a series of tablets, with entries built from higher or lower round and reciprocally round values of x).

To determine these numbers I proceeded as follows. The largest is 2 24 00 00. To use a decimal spreadsheet I moved everything into the lowest place, giving 0 + 0x60 + 24x3600 + 2x216000 = 518400. The smallest number is 1 48 00 00 = 388800. The spreadsheet listed all 12961 multiples of 10 in this range. If such a number x is to have a 4-place base-60 reciprocal y, then xy (written as decimals) must be a multiple of 60⁷. For each x in the list the spreadsheet calculated y = 60⁷/x, and checked if it was an integer. Seventeen x qualified, but for two of them (468750 and 409600) the corresponding y, although an integer, was not a multiple of 10. This left the fifteen following (x,y) pairs, written again in base-60 notation.

	          			
row    x              y 	 	 	 	 	 	 	 	 
1    2 24           25	 	 	 
2    2 22 13 20     25 18 45	 
3    2 20 37 30     25 36	 	 
4    2 18 53 20     25 55 12	 
5    2 15           26 40	 	 
6    2 13 20        27	 	 	 
7    2 09 36        27 46 40	 
8    2 08           28 07 30	 
9    2 05           28 48	 	 
10   2 01 30        29 37 46 40
11   2 00           30	 	 	 
12   1 55 12        31 15	 	 
13   1 52 30        32	 	 	 
14   1 51 06 40     32 24	 	 
15   1 48           33 20

Now, following Robson, we calculate the two numbers s=(x-y)/2 and d=(x+y)/2. They come out as in the following table. This calculation is not place-independent, so we take the first place of x as units; correspondingly the first place of y will be multiples of 1/60.

So in row 1 we have
s = (2 24 - 0 25)/2 = (1 59)/2 = 0 59 30
d = (2 24 + 0 25)/2 = (2 59)/2 = 1 24 30.


row        s                   d				
1       59 30             1 24 30	 	 
2       58 27 17 30       1 23 46 02 30
3       57 30 45          1 23 06 45	 
4       56 29 04          1 22 24 16	 
5       54 10             1 20 50	 	 
6       53 10             1 20 10	 	 
7       50 54 40          1 18 41 20	 
8       49 56 15          1 18 03 45	 
9       48 06             1 16 54	 	 
10      45 56 06 40       1 15 33 53 20
11      45                1 15	 	 	 
12      41 58 30          1 13 13 30	 
13      40 15             1 12 15	 	 
14      39 21 20          1 11 45 20	 
15      37 20             1 10 40

The squares of the elements in the d column are the elements A in the first column of Plimpton 322. E.g. (1 24 30)² = 1 59 00 15. The S and D columns are derived from s and d by eliminating common factors so as to have a relatively prime pair in each row (exception: row 11). In order to work with integers in this calculation, I multiplied each s and d by 60⁴ = 12960000.

    s       d     (s,d)   S=s/(s,d)  D=d/(s,d)
12852000 18252000 108000   119         169
12626250 18093750 3750     3367        4825
12422700 17952300 2700     4601        6649
12200640 17799360 960      12709       18541
11700000 17460000 180000   65          97  
11484000 17316000 36000    319         481
10996800 16996800 4800     2291        3541
10786500 16861500 13500	   799         1249
10389600 16610400 21600    481         769
9922000  16322000 2000     4961        8161
9720000	 16200000 3240000  3           5
9066600	 15816600 5400     1679        2929
8694000	 15606000 54000    161         289
8500800	 15499200 4800     1771        3229
8064000	 15264000 288000   28          53

Now write the relatively prime pairs in base-60

row     S           D		
1      1 59        2 49
2     56 07        1 20 25
3      1 16 41     1 50 49
4      3 31 49     5 09 01
5      1 05        1 37
6      5 19        8 01
7     38 11       59 01
8     13 19       20 49
9      8 01       12 49
10     1 22 41     2 16 01
11     3           5
12    27 59       48 49
13     2 41        4 49
14    29 31       53 49
15    28          53

This generates exactly the S and D columns of the tablet (except for row 11, which was left in the unreduced form 45, 1 15).

We are now in a position to reconstruct the missing part of Plimpton 322. There must have been 4 additional columns, corresponding to our numbers x, y, s, d, with perhaps an auxiliary column for the greatest common divisor (s,d). The relation between the columns, to summarize, would have been

y = 1/x

s = (x-y)/2

d = (x+y)/2

A = d²

S = s/(s,d)

D = d/(s,d)

The properties of A, S and D follow from their construction, as has been observed elsewhere. Namely,

Let y = 1/x. Then A = (x+y)²/4 is a perfect square and so is A-1 = (x+y)²/4 - xy = (x-y)²/4. In fact the s column gives the square root of A-1.
Similarly d² - s² = (x+y)²/4 - (x-y)²/4 = xy = 1, so s and d are always two thirds of a pythagorean triple, and this property is not changed when they are reduced together to be co-prime.

Finally, this construction explains the unusual roundness of the third sides, the elements of our column L. Since x and y are round and reciprocally round, xy = 60⁷ and has only 2, 3 and 5 as prime factors, as does its quotient by the common denominator of s and d.