The idea that the originator of Plimpton 322 probably pulled from a large data base those Pythagoreans that would give him the quantities he wanted has not been probed to the depth that the Tablet deserves.
A reason existed why we see holes in the array. Robson did not recognize this reason and thought that the OB scribe merely skipped some numbers, or that they were lost in the evolution of the Tablet.
A far better rationale existed behind the "missing" numbers. The Pythagorean triplets were selected to produce a profound relationship. This would not be evident to us with Robson's number inserted into the list.
Neugebauer recognized that the ratios of Pythagorean triangles, with their right angles, provided angular measure. They are shown on his Fig 2, pg 39. I now plot the D/L Ratio Squared for the original 15 numbers of the Plimpton list. Remember, if these are Pythagorean right angles D/L means what we in the modern world denote as the sec θ, where θ is the angle formed by D/L. This trigonometric relationship absorbed Neugebauer's attention, so that he failed to grasp the deeper significance of Plimpton 322.
This is a most remarkable data plot:
It appears to be linear with Line Number. This is shown by Regression Coefficient, R, which has a calculated value of 0.998. That in itself is remarkable: the data points hold exceptionally well to a straight line.
The variability of the data points has some scatter. They are displaced around the regression line with a calculated 3-sigma distribution of +/- 0.038 ratio/Line Number. If this number is divided by the median or average data point of 1.65 D/L, the 3-sigma error would be about 0.038/1.65 or about 2.0 %. (If the calculated error is considered from around the spread of the data points from 1.35 to 2.00 the 0.65 the error would be about 6%.) This is exceedingly good control by the scribe who selected the Pythagorean values.
The slope of the line is about 1/22.5 (D/L)2 versus Line Number. This slope does not, in itself, carry a recognizable meaning. However, on his page 14 Neugebauer shows a recognized reciprocal of 22,30 as 2,40. For information purposes I have indicated by arrows on the plot the location of slopes of 1/20 and 1/25. You can see that 1/22.5 at Line 15 falls midway between them. These would carry sexagesimal values of 0;3 and 0;2 24.
Far more importantly, the intercept on Line 1 is 2.00. This is a calculated number, not a guess. My calculated intercept at Line Number 0 is 2.0433. The calculated slope is -0.0444. One Line Number difference is 2.0433 - 0.0444 = 1.9989. Obviously one cannot distinguish the difference from a pure 2.00 on a practical plot. The designer wanted the regression line to hit 2.00 at Line 1, not a hypothetical Line 0. I found Line 0 only because my statistical calculation was designed to start at Line 0 Note that the individual data point at Line 1 is somewhat below this. For practical reasons he could not find a Pythagorean that would have a value of 2.00 but his choice of all Pythagoreans led to a regression line that pointed to that value. This aspect is important because it shows his grasp of data plots with regression lines. He was expert in his mathematical tools.
Before getting into further implications of this data plot I shall now present the simple D/L ratio with respect to Line Number. This shows even more remarkable evidence of the mathematical skill of the Old Scribe
If the intercept on the (D/L)2 plot at Line Number 1 is 2.00, since it is a square value, it must show on the simple D/L plot as 1.414, the √2. Remember Tablet YBC 7289? The Old Babylonian scribes must have held this number in great respect.
My calculated value at Line Number 0 intercept is 1.4339. The slope is calculated at -0.017168 per Line Number. The difference between these two numbers at Line Number 1 is 1.4167. Again, clearly, this difference from 1.414 would be difficult to distinguish on a graphical plot, three parts out of 1400.
The calculated regression line is again exceptionally linear, with a regression coefficient of 0.998.
The calculated 3-sigma limits of error is +/- 0.015 (in decimal) around the plot. Since the line runs from about 1.17 to about 1.41 D/L the error is about +/- 0.015/0.24 or slightly more than +/- 6%. (The error considered around the average data point of 1.26 would make the error about 1.0 per cent.)
We can readily see the remarkably good control exercised by the scribe on selection of D/L ratio for the Pythagorean triangles. This good linearity and correlation coefficient suggests the scribe chose the D/L ratio as his criteria for selection, rather than D/L Squared, the S/L, or some other relationship. Other reasons exist why he chose D/L.
The surprising property of this plot is that the slope is very nearly 1/60, or sexagesimal 0;1! (The calculated slope of the actual regression line is 1/58.2.) I show the theoretical slope with the broken line in Figure 2. I place the 1/60 theoretical line to begin at 1.414 because it is apparent that this is what the OB scribe desired. The OB scribe clearly chose the regression line to go through that point when the Line Number was 1.
What an interesting sexagesimal relationship! A slope that fits with sexagesimal mathematics. The originator of this plot was thinking in sexagesimal terms. He did not present his material decimally.
I do not translate these data plots into sexagesimal mathematics simply because we, who are accustomed to decimal mathematics, would not recognize them. The Line Numbers would remain the same. The first part of the Pythagorean ratio as an integer would remain the same; the decimal points would merely break down into component sexagesimal units. 1.6 would become 1;36, and so on.
An important property now emerges concerning Plimpton 322. What was regarded as mere line counting prior to this study, and had no apparent other purpose, now becomes important. The Line Numbers become part of the intrinsic purpose of the tablet. The above graphical plot is against the Line Numbers, as numbers, and not just an ordered array. The scribe included them on the tablet because of their important contribution to his purpose. They become a variable, the independent variable. How fortunate we are that they survived the millennia.
Note that I could have defined these plots from the remaining extant evidence, with a calculation of D/L from the Pythagorean relationship. The loss of possible data from the broken part of the tablet does not limit me.
I now understand why the (D/L)2 was included on the Tablet. It served as a pointer to the D/L ratio, from 2.00 to the √2. We know from YBC 7289 that the Semitic Akkadian scribes understood the theoretical meaning of √2. Using that as his initial value for Line 1 was within his mathematical repertoire. His use of it here underlines his theoretical understanding.
I also understand why the Long Side was not included in the list. The clay tablet became a cumbersome medium. A Pythagorean triplet has three sides. But any two sides determine the third, if we know our mathematics. Hence, in order to conserve space on the Tablet he did not need to include the Long Side. He lost no information by neglecting to include that Pythagorean side. Haven't we all calculated that missing side?
But the mathematical knowledge of the scribe was not limited to theoretical understanding of mere numbers. He understood and was a master at graphical plots, with independent and dependent variables. Just consider how he was able to select Pythagoreans, with their scatter, that would give him a desired regression. This fact alone raises the mathematics of ancient times to a level vaster than anything previously recognized by man, for more than 3,000 years.
We know from the errors incorporated into Plimpton 322 that the scribes who preserved it did not recognize those errors. They were borrowing from previous copies. We do not know at what stage the several errors became incorporated. Second copy? Third? But at some point back in time the originators of the material did know their mathematics. This is exactly the type of evidence I found in the Great Egyptian Pyramids. Not only did they preserve knowledge of their mathematics; they were able to marshal social forces to build vast monuments for the preservation that knowledge. I believe the originators of the Egyptian evidence knew that a loss of knowledge would ensue, and built a memorial against that loss. They knew and understood that they came from a race of men who were superior, and recognized that their superiority would be lost within a few generations through interbreeding with inferior genetic stocks. We have exactly the same phenomenon in ancient Babylon. At some point men with superior intellect were able to create Plimpton 322. They may have left it as evidence of themselves against the day when they would be gone. The mathematics that remains in ancient Babylon is only a remnant of that more ancient knowledge, a carry over from times of much greater knowledge and understanding. What we witness today are remaining clay shards.
If Robson were to maintain her thesis that this plot came from a list of reciprocals, with display of a "naive geometry," she would have to overcome this data. This plot resulted from far more than a list of reciprocals. It is far more than "naive geometry." Something else is going on here that Robson, with her apparently elementary knowledge of mathematics, does not appreciate. Since she did not recognize that her Plimpton 322 reciprocals were equivalent to Neugebauer's "p" and "q" solutions we can only regard her approach as mathematically naive. (Or she may have hid her knowledge, not wishing to upset her thesis.) Her explanation of "naive geometry" is due to her naive mathematical understanding. She did not do her homework. She obviously does not understand the heavy implications to have a list of Pythagorean triples with such great regularity and order, and then array them against Line Numbers that make the data come alive. The difficulty for Robson is that her attitude hampers her ability to properly research those ancient people. She has preconditioned them. She has forced them into a mold of her own creation. Her modern Old Babylonian compatriots, those scholars who began to redefine those old people, beginning in the 1970's, contributed to the creation of these new attitudes. They are redefining the old language, (note the new translations), and they are redefining the evidence to suit their ideas of the evolution of man. They shuttle aside, probably without recognition, details of their studies that would force acknowledgement of the true depth of the phenomenon of OB mathematics.
One cannot objectively understand ancient people if one conditions them by modern notions of social and biological evolution.
How unfortunate that someone with Robson's abilities should squander them on such modern godless notions.
Aren't Robson's talents testimony to a genetic inheritance that came from far more than primitive savagery?
March 7, 2005
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