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Eleanor Robson and Plimpton 322

Part VII - Fetching Pythagoreans

We now come to an important question. How did Robson fetch the Pythagorean triples she inserted into the array? They appear to be obtained by trial and error, guessing here and there for values that would fit the rest of the tablet. As we shall see, they are very large numbers, far beyond the average level of the original Plimpton tablet.

The more important question is how did the Semitic Akkadian scribe fetch the numbers? Was there some consistent technique by which he structured the tablet? (Keep in mind that the original of the Plimpton Tablet may go back in time, to a point where there were no errors. When I speak of the Semitic Akkadian scribe I refer to this original work.) Clearly, the tablet is structured, and it is this underlying structure which requires a systematic approach and understanding. I do not accept that the work was done by trial and error.

Neugebauer gave the reciprocals for the first four lines of Plimpton 322. He concluded that Plimpton 322 could not have been generated by using reciprocals but rather was done through the "p" and "q" values. Since Robson knew this, and the "p" and "q" values, it was simple for her to calculate the reciprocals for the entire table. In proposing the use of reciprocals for generating Plimpton 322 she went contrary to Neugebauer, and she went contrary to the underlying mathematics. This idea of reciprocals was not new with Robson, as she admits, her pg 185. She includes a discussion of the history.

We have no means to penetrate mathematically to the starting level of Plimpton 322 from the remaining evidence. Anything that Robson does in suggesting reciprocals as the starting point is strictly speculative. She bases this on her notion that those ancient people were not mathematically sophisticated to the point of understanding "p" and "q" relationships. This involved number theory, as Neugebauer pointed out, and she was not about to go there.

If we take the "p" and "q" numbers, and then examine Robson's reciprocals we can understand why Neugebauer made that choice. I show all value in sexagesimal for the benefit of the reader.

Line
Number
p q x = p/q 1/x = q/p
1 12 5 2;24 0;25
2 1;04 27 2;22 13 20 0;25 18 45
3 1;15 32 2;20 37 20 0;25 36
4 2;05 54 2;18 53 20 0;25 55 12
5 9 4 2;15 0;26 40
6 20 9 2;13 20 0;27
7 54 25 2;09 36 0;27 46 40
8 32 15 2;08 0;28 07 30
9 25 12 2;05 0;28 48
10 1;21 40 2;01 30 0;29 37 46 40
11 10 5 2 0;30
12 48 25 1;55 12 0;31 15
13 15 8 1;52 30 0;32
14 50 27 1;51 06 04 0;32 24
15 9 5 1;48 0;33 20

Quite clearly the original "p" and "q" numbers are much more simple than the reciprocals. Neugebauer reasoned that the inventors of Plimpton 322 would not start with the more complex numbers. (If you examine Robson's reciprocals you can see that they are equal to p/q and q/p.)

Something else happens in the choice of reciprocals: we lose the meaning of the respective columns. We do not recognize how the values of "x" and "1/x" are related to the ratio of S/L and D/L. With the use of "p" and "q" we retain that recognition since (p2 - q2) = S and (p2 + q2) = D while 2pq = L.

The derived ratios of S/L and D/L are exactly the same, regardless of which starting point we choose. We have no way mathematically to distinguish between them. Robson's proposed columns after the first two reciprocal columns have the identical mathematical result. If we were to start with "p" and "q" we would substitute them for Robson's reciprocals, and have the same size original tablet in the restoration.

The extant first column of Plimpton 322 suggested to Robson that the OB scribes used the "squaring" of the Diagonal numbers to somehow obtain their results. We saw this in her previous discussions about YBC 6967 where she added 1 to obtain a square. Why include the squares if they were of no use in generating the final result? But this same riddle exists if we use the "p" and "q" numbers rather than the reciprocals. We do not need that column to find the Pythagorean triplets.

Furthermore, why neglect the "long side" in his final accounting of the triplets? Why not include it? We are unable to explain these riddles with our present understanding of the work of the OB scribes. Perhaps there was a utilitarian purpose to the tablet that escapes us. I shall show otherwise.

I shall now examine Robson's compilation of her proposed restoration of Plimpton 322. I have converted to decimal for convenience in further analysis. I also show Robson's proposed hypothetical insertions for "missing" lines.

Plimpton 322 According to Robson
Calculated Values of Ratios

Robson Numerical Line Insertions Shown in Bold Face

Line
No.
    x 1/x (x - 1/x)/2
S/L
(x + 1/x)/2
D/L
[(x + 1/x)/2]2 Angle
  p q p/q q/p (p2 - q2)/2pq (p2 + q2) /2pq  [(p2 +q2) /2pq]2  
1 12 5 2.400000000 0.416666667 0.991666667 1.40833333 1.98340278 44.7603
2 64 27 2.370370370 0.421875000 0.974247685 1.39612269 1.94915855 44.2527
3 75 32 2.343749990 0.426666668 0.958541661 1.38520833 1.91880212 43.7873
4 125 54 2.314814810 0.432000001 0.941407405 1.37340741 1.88624790 43.2713
4a 288 125 2.304000000 0.434027778 0.934986111 1.36901389 1.87419903 43.0756
5 9 4 2.250000000 0.444444444 0.902777778 1.34722222 1.81500772 42.0750
6 20 9 2.222222220 0.450000000 0.886111110 1.33611111 1.78519290 41.5445
6a 625 288 2.170138888 0.460800000 0.854669444 1.31546944 1.73045986 40.5195
7 54 25 2.160000000 0.462962963 0.848518519 1.31148148 1.71998368 40.3152
8 32 15 2.133333330 0.468750001 0.832291665 1.30104167 1.69270942 39.7703
8a 135 64 2.109375000 0.474074074 0.817650463 1.29172454 1.66855228 39.2712
9 25 12 2.083333330 0.480000001 0.801666665 1.28166667 1.64266944 38.7180
9a 256 125 2.048000000 0.488281250 0.779859375 1.26814063 1.60818064 37.9492
10 81 40 2.025000000 0.493827160 0.765586420 1.25941358 1.58612257 37.4372
11 10 5 2.000000000 0.500000000 0.750000000 1.25000000 1.56250000 36.8699
11a 125 64 1.953125000 0.512000000 0.720562500 1.23256250 1.51921032 35.7751
12 48 25 1.920000000 0.520833333 0.699583333 1.22041667 1.48941684 34.9760
12a 256 135 1.896296290 0.527343752 0.684476269 1.21182002 1.46850776 34.3907
13 15 8 1.875000000 0.533333333 0.670833333 1.20416667 1.45001736 33.8550
14 50 27 1.851851850 0.540000001 0.655925925 1.19592593 1.43023882 33.2619
15 9 5 1.800000000 0.555555556 0.622222222 1.17777778 1.38716049 31.8908

 

Some idea of the missing places in the Plimpton list that were filled in by Robson may be seen by viewing the Figure to the left. I have taken the ratio of S/L, that is, the tangents of the Pythagorean triangles, and plotted them according to their individual values, from Line 1 to Line 15. A value of 1.0 would indicate the slope of a 45 degree angle; a value of 0.6 would indicate the slope of an angle close to 31 degrees. In this illustration I have not included Robson's insertions.

 

One can easily see that the spread of the data points has gaps, or wider separation from line to line. These come between lines 4-5, 6-7, 8-9, 9-10, and 11-12, 12-13, and 14-15. The gap between 11-12 is extra wide, but Robson did not attempt to fill that in except with one line. She also made no attempt to insert a line between 14-15.

The importance of the scribes creation of this structured list may be better understood by examining Robson's insertions. I took her insertions into the body of the table and plotted the values, with the following results.

Figure 2

We can see that we get a linear regression against Line Number. I have extended the meaning of the Line Numbers to create the linear regression. The error spread of the data is about +/- 0.020 as an estimated 3-sigma. Her 4a and 6a lines are the culprits in widening the spread. Her other interpolations are very close to the center of the best-fit line.

Interestingly, her Pythagorean numbers are much larger than the original Semitic Akkadian scribe values. She is consistently higher in all numbers. Following is a table that lists all Pythagorean values.

Plimpton 322 Calculated Values

Robson Inserted Lines in Bold Face

Line
No.
Mult. 60X
[(x - 1/x)/2]
Short
Side
60X
[(x + 1/x)/2]
Diag.
Side
[(x + 1/x)/2]2 Long
Side
Angle
1 2.00 59.5000000 119 84.5000000 169 1.9834027778 120 44.7603
2 57.60 58.4548611 3367 83.7673611 4825 1.9491585517 3456 44.2527
3 80.00 57.5124996 4601 83.1124998 6649 1.9188021154 4800 43.7873
4 225.00 56.4844443 12709 82.4044443 18541 1.8862479013 13500 43.2713
4a 1200.00 56.0991667 67319 82.1408333 98569 1.8741990280 72000 43.0756
5 1.20 54.1666667 65 80.8333333 97 1.8150077160 72 42.0750
6 60.00 53.1666666 3190 80.1666666 4810 1.7851928989 3600 41.5445
6a 6000.00 51.2801666 307681 78.9281666 473569 1.7304598583 360000 40.5195
7 45.00 50.9111111 2291 78.6888889 3541 1.7199836763 2700 40.3152
8 16.00 49.9374999 799 78.0624999 1249 1.6927094150 960 39.7703
8a 288.00 49.0590278 14129 77.5034722 22321 1.6685522796 17280 39.2712
9 10.00 48.0999999 481 76.8999999 769 1.6426694412 600 38.7180
9a 1066.67 46.7915625 49911 76.0884375 81161 1.6081806448 64000 37.9492
10 108.00 45.9351852 4961 75.5648148 8161 1.5861225661 6480 37.4372
11 1.00 45.0000000 45 75.0000000 75 1.5625000000 60 36.8699
11a 266.67 43.2337500 11529 73.9537500 19721 1.5192103164 16000 35.7751
12 40.00 41.9750000 1679 73.2250000 2929 1.4894168403 2400 34.9760
12a 1152.00 41.0685761 47311 72.7092013 83761 1.4685077630 69120 34.3907
13 40.00 40.2500000 1610 72.2500000 2890 1.4500173611 2400 33.8550
14 45.00 39.3555555 1771 71.7555555 3229 1.4302388187 2700 33.2619
15 0.75 37.3333333 28 70.6666667 53 1.3871604938 45 31.8908

I show the multipliers that were used to obtain the Plimpton Pythagorean results. Lines 5 and 15 are not integer multipliers. So also are not two of Robson's lines, 9a and 11a.

Neugebauer and Robson both recognized that the choice of sexagesimal numbers for Plimpton 322 involve "regular" numbers, numbers that one could find in sexagesimal mathematics with a limited number of steps and without repeated fractionalization. In the original data of the D/L squared column this is limited to 8 sexagesimal places in line 10, with one 7 in line 2, and the others less than that. In her insertions lines 4a, 8a, and 11a she limits it to 7 sexagesimal places, while lines 6a, 9a, and 12a are limited to 8 places. But this does not determine the size of the value; it merely determines the form of the value, with more or less sexagesimal places. The initial sexagesimal values lead to her very large numbers. We can see that the multipliers are much beyond what the OB scribe selected. The only one that comes close to her is in Line 4. If we were to calculate the spread of the Pythagoreans from the OB scribe, and from Robson, we would find two distinctly different populations.

Understand that this chart is on a logarithmic scale. This tends to smother the difference between the two populations. I could have chosen other properties, such as the multipliers to show this difference.

We can calculate it differently. The average of the original Plimpton values for the diagonals is 3860 while those of Robson is 129,850. This is a major difference in magnitude for the two different populations.

Clearly, the original Semitic Akkadian scribe had a repertoire of Pythagorean knowledge that was profound. By the necessity of the choices of "low" Pythagorean triangles, with regular sexagesimal numbers,  he must have had available to him a vast array of Pythagoreans. Such an array did not come about through reciprocals; they came about through more sophisticated mathematics, probably the "p" and "q" numbers or their equivalents. He had to have some insight in "number theory."  We can see from Robson's choices where the use of reciprocals leads; very large Pythagoreans.  She must have had considerable difficulty in finding Pythagoreans that would fit with those on the clay tablet.

Furthermore, this sample base could not have been developed on the "primitive" media of clay tablets. Clay is much too crude to produce such a large array. I would suggest that Plimpton 322 originated in an intellectual milieu and culture that was conversant with Pythagoreans at a higher sophisticated level.

Other reasons exist for working with the "p" and "q" numbers.

If you examine the "p" and "q"  values you will see that they range from 5,12 in the first line to 5,9 in the last. Line 11 is 5,10. What happened to 5,11? 5,11 yields a continuing fraction; that is why it was not used. It would fit very nicely in line 6a of Robson's insertions, and with a greatly reduced S = 96, D = 146 and L = 110, but for the fact that it is not a regular number.

If you examine the "p" and "q" numbers in the tabulation above you will instantly recognize why Robson worked under a handicap in fetching her Pythagoreans. She did not believe the ancient scribes used such numbers; she was forced to work with the raw reciprocals. Compare the scribal "p" and "q" values with her reciprocal values. Then note how large her unrecognized "p" and "q" numbers compare with those of the scribe. You can see how simple some of them are: 4:9, 9:20, 12:25, 5:10, and 8:15 compared to her 125:288, 288:625, and 135;256. (S here refers to Semitic Akkadian Scribe and R refer to Robson.)

S S S S R S S R S S R S R S S R S R S S S
5 27 32 54 125 4 9 288 25 15 64 12 125 40 5 64 25 135 8 27 5
12 64 75 125 288 9 20 625 54 32 135 25 256 81 10 125 48 256 15 50 9

One cannot recognize simple ratios from the reciprocals. When she went fetching Pythagoreans Robson ended with raw reciprocal numbers, 2:18 14 24, 2;10 12 30, and so on. But the scribe ended with 5:12, 27:1;04, and so on. This more dramatically shows how the two techniques compare. Remember, we have here the finished product, not the many guesses she had to engage in to obtain the reciprocals.

I can only exclaim over the disdain Robson must have for those ancient minds, minds that were better than hers.

What led to the great regularity and order of the Plimpton list? Surely the ancient Semitic scribe understood his work far better than Robson, indicated by the choice of the Pythagorean triplets.

We can see from the two examples of "p" and "q" tables I earlier offered for generating Pythagorean triplets that a vast array of such numbers exists. The ancient scribe must have been acquainted with such arrays in order to make his choices.

This takes us back to the idea that Plimpton 322 was borrowed from a much earlier work, and that it is a rote copy. Whatever the process to obtain the spectrum of Pythagoreans it must have been based on a much broader knowledge or catalog that has not come down to us. If you examine the ratio of the "p" and "q" numbers you can see how it changes monotonically from the lowest to the highest. In order to maintain his order the ancient scribe had to find numbers that would hold those ratios.

Such order can come about only through an idea of "numbers" that does not appear in the use of reciprocals.

 

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