Egypt
Origins
Origins
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.
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.
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.