We now enter into the strange land of modern OB scholarship. I found much confused raveling of Robson's thought.
I shall quote from page 182 that follows directly beneath her Figure 3:
Although the diameter tallum regularly crops
up in OB problems about circles --- and, one could argue, it was necessary in
order to conceive of a circumference or circle as a loop whose opposite points
are all equidistant --- the radius pirkum is never mentioned. That
is not to say, though, that the radius played no part in OB geometry. We find
it, for instance, in problems about semicircles. In those contexts, however, it
functions as the short transversal of the figure, perpendicular to the diameter
or long transversal, tallum. Indeed, this is the function of the
pirkum in the context of all OB geometrical shapes [Robson 1999, 38]; it is
never conceptualized as a rotatable line.
Figure 3, from a "text-book" on finding the areas of geometrical figures, shows a circle inscribed in a square (BM 15285 [Robson 1999, 208-217]; probably from Larsa). It gives clear evidence that circles could be drawn with compasses from a central point. My argument is certainly not that the radius was not known in the Old Babylonian period, but simply that it was not central to the ancient mathematical concept of a circle.
In short, to treat Plimpton 322 as a trigonometric table of any kind does extreme violence to Criteria 1-2. The Old Babylonian circle was a figure --- like all OB geometrical figures --- conceptualized from the outside in. In such a situation, there could be no notion of measurable angle in the Old Babylonian period . . .
She goes on to illustrate where the radius of a circle is specified as part of a problem, TMS 3, and justifies its insignificance by noting that it is listed last. (The designation TMS, BM, YBC, and so on, refer to the Museum that now houses the sample.) She then says:
The illustration (her Figure 3) does show though, that there was a concept of "quasi-perpendicularity," or more-or-less right-angledness. The squares on the tablet are definitely skew: it is not my drawing skills at fault, but the ancient scribe's conception of squareness. We do not know exactly what the Old Babylonian perception of perpendicularity was, but it seems to have been something like the range of angles for which the "Pythagorean Rule" is reasonably accurate [cf. Hoyrup 1999a, 403].
These remarks are at the heart of her demoting Plimpton 322.
there could be no notion of measurable angle in the Old Babylonian period . . .
there was a concept of "quasi-perpendicularity," or more-or-less right-angledness. . .
The squares on the tablet are definitely skew: it is not my drawing skills at fault, but the ancient scribe's conception of squareness . . .
We do not know exactly what the Old Babylonian perception of perpendicularity was . . .
the range of angles for which the "Pythagorean Rule" is reasonably accurate . . .
Hoyrup, from whom Robson borrowed much of her thought, on page 275, with example BM 85196, #9, illustrates the "ladder against the wall" problem. This is an example of many similar problems found in the ancient Near East over the next three millennium. Inherent in the conceptualization of this problem is the notion of verticality. The wall is vertical, and was understood as vertical. The wall does not lean at some angle. (The ladder here is at first vertical, but then slides down to some other position.) The notion of the wall being vertical has nothing to do with modern algebra or Greek geometry, but is a function of the human mind. If the wall is vertical, then it is regarded in relation to the ground on which it rests, and since that is at a right angle, the wall is perpendicular to the ground. Again, the ground was understood as being at right angles to the wall. In this one problem we have a vertical wall, a flat piece of ground, and the notion of a right angle. As Hoyrup notes, the solution requires application of the Pythagorean Rule, which again invokes the necessity of right-angles. But Robson regards this problem as within "the range of angles for which the "Pythagorean Rule" is reasonably accurate . . .," not exactly accurate. She thus betrays her attitude toward Pythagorean right angles.
Hoyrup discusses Perpendicularity and Orientation on pg 228 after he discusses Angles and Similarity. He summarizes the notion of right angles in a pun wherein he says the "Old Babylonian right angle was understood in its opposition to a wrong angle." I personally cannot accept that the OB scribes conceived right angles in such a way. They understood right angles. Hoyrup then talks about the metaphors of OB and modern minds but is unwilling to accept that the OB understood verticality. ". . . it is a horizontal and no vertical dimension . . ." He uses trapezoids to show that "the metaphorical plumb line might point in several directions."
Does Hoyrup truly believe that the plumb line, metaphorical or otherwise, was not understood in OB minds? A plumb line is the ideal way to determine verticality. It always points directly to the center of the earth. (Perception may differ if one is foolish enough to stand on a hillside.) In so doing it forms a right angle to the plane of the earth. Plumb lines do not point in several directions, and they do not denote horizontal directions. He attempts to show that the drawing of the problem is laid out on a horizontal plane: a clay tablet, a dusty shelf where it could be sketched, or some similar horizontal operation. By so doing he violates the principle of plumb lines.
Now consider how Robson treats this (to modern scholars) conceptual puzzle.
As she says: The squares on the tablet are definitely skew: it is not my drawing skills at fault, but the ancient scribe's conception of squareness . . . She definitely believed that the OB scribe did not truly understand the concept of squareness. If they did not understand squareness they would not recognize right angles, perpendicularity, verticality, or any other mathematical property so important to their work.
Now consider this clay tablet from OB mathematical repertoire.
This little piece of clay was discussed by Neugebauer, pg 42, by Hoyrup, pg 261, and many, many other people.
Three properties make it fascinating: a) its dependence on the notion of perpendicularity, b) its invoking of the Pythagorean rule, and c) its approximation to the square root of two.
Note how it is "crudely" drawn. A straight edge was used to draw the lines. But the lines do not intersect exactly, not because the scribe did not know they should, but because he is using it as illustration, and not perfect construction. The same practice is used by the scribe in Robson's Fig 3. The scribe is not revealing his concept of "squareness" but merely illustrating a circle within a square. And from this Robson deduces the primitive state of the Semitic Akkadian mind .
The square of YBC 7289 is inscribed with sexagesimal 30 along one edge. It then has sexagesimal 1 24 51 10 along the diagonal, and 42 25 35 in the lower segment of the square. The first sexagesimal string converts to the √2: 1 + 24/60 + 51/3600 + 10/216,000 = 1.414212963. The actual value of the √2 to the same modern calculator precision is 1.414213562. If you raise the last digit in the OB sexagesimal string by one you get 1.414217592. If you lower it by one digit you get 1.414208333. Clearly the scribe knew the √2 within his stated precision of seven decimal places. If you multiply the OB number at the scribe's precision times 30, you get 42.42638889. The same value using the √2 to modern calculator precision gives 42.42640687. This is the distance across the diagonal of the square, denoted by 42 25 35.
Neugebauer discussed the reiterative process to produce the √2 to the desired precision. But note that Neugebauer's process is a modern mathematical exercise. If the OB mathematicians were primitive, how did they know there was an iterative process to obtain √2 to any precision?
(The OB value for √2 is listed in YBC 7243, a table of numerical constants, and could be consulted by a scribe. However, this does not relieve the question of how the √2 was calculated in the first place.)
Hoyrup goes on to remark:
Since both diagonals are drawn, we may guess that the diagram is also meant to suggest the naive-geometric argument by which the square on the side can be seen to be twice the square on the bisected diagonal (and, equivalently but not directly to be seen, the square on the diagonal to be twice the square on the side). This is only a special case of the Pythagorean rule, but enough to support the assumption that the rule had been discovered by means of such arguments . . .
Note that he uses the term "naive-geometric argument." If we follow Hoyrup's illustration we obtain 302 = 900 while 2 X (1/2 X 42.42639)2 = 900 and 42.426392 = 2 X 900. However, this is nothing more than D2 = 2 S2 and its rearrangement. He suggests that the Pythagorean rule was discovered by such methods of finding squares. He does not believe the Pythagorean rule was discovered on theoretical grounds, contrary to the sophistication of Plimpton 322.
Conceptually, the illustrated square requires that it be exact. We must accept this requirement for the square to be conceptually and mathematically coherent. First, the OB scribe understood the four sides as exactly equal to one another. Second, the OB scribe understood the four sides as exactly perpendicular to one another. One cannot invoke this mathematical example unless the four sides are exactly equal to one another and exactly perpendicular to one another. Are they on the illustration? No, because this is a conceptual study, not a practical execution. Thus Robson's notion of "OB squareness" as indicating a lack of understanding is just plain foolishness. So also is Hoyrup's notion of a "naive-geometric argument."
The Old Semitic Akkadian scribes understood a Perfect Square.
Third, when invoking the Pythagorean rule to determine the diagonal the scribe took the result to six (seven) decimal places. But there was no practical method, either in the third millennium BC, or in the modern world, to measure the diagonal to six (seven) decimal places with the naked eye.
How well could the ancients obtain resolution on the measure of a practical distance? Perhaps to the third decimal place, but not better than that. Let's look at some examples. Consider that the standard unit of measure was the cubit. This cubit was essentially that of the Egyptian Royal cubit, 20 inches, or thereabouts, and again shows cross-culture exchange, or borrowing from a common past. Refer to my work on cubits:
The Egyptian cubit rods were divided into "fingers" and "units" that were about a millimeter, about 0.04 inches per division. We should assume that the Mesopotamian cubit was divided into similar units. Resolution better than that was not easily discernible to the naked eye.
We cannot discern a thousandth of an inch by the naked eye in modern measure. We multiply our ability to discern such small difference by a mechanical instrument, a vernier micrometer. For resolution better than that we must use other methods, such as optical benches. (We now could do it by more modern methods, such as lasers.) Hence we are limited by the naked eye to about 25 parts per thousandth of an inch.
If we were to measure a distance of 1,000 meters we could easily discern one meter (1 part per thousand), and subdivide that into much small units, and so on. Unfortunately, we must take into consideration how well we can hold to 1,000 meters. Stretch ropes won't do it. They could be off by several meters in 1,000. Hence, when we talk about resolution of measure we are limited by our methods.
Looking again at the square with a calculated diagonal of 42.42638889 susi, (42.42640687 susi theoretical) we see that we cannot discriminate with the naked eye better than 0.02 susi. So how did the ancient mathematician resolve his difference to 0.0001 susi?
He didn't. He couldn't.
He would have been unable to measure to the precision stated. So why does the scribe state results with a resolution beyond the range of any practical measuring devices and the unaided human eye? Because he is not presenting a practical result; he is presenting a theoretical result. This clay object is illustrating a theoretical idea. The ancient Semitic Akkadian scribes could think theoretically. When they gave the result to six decimal places they understood that it could not be measured by any practical device; they were demonstrating theoretical mathematics.
But how did they arrive at this knowledge in the first place? Did they borrow it by rote from some unknown past? Did they calculate it for themselves? They had to know the general rule that 12 + 12 = x2 = 2 and hence that x = √2. This is from the Pythagorean rule. They not only had to know √2 theoretically, they had to know the Pythagorean relationship theoretically. The question whether the Semitic scribes of 1800 BC borrowed by rote from some unknown past become moot. Someone in ancient times had to obtain this knowledge from theoretical considerations.
Now where is Robson's belief that the ancient scribes were illustrating their primitive conception of squareness? Where is Hoyrup's belief that the Pythagorean rule was discovered by naive geometric arguments?
If the Akkadian scribes could not measure angle, how did they arrive at an angular relationship that was exactly perpendicular? Did they espouse a concept of "quasi-perpendicularity," or more-or-less right-angledness? Was this a range of angles for which the "Pythagorean Rule" is reasonably accurate.
We can see how ridiculous are the current notions of modern scholars concerning OB mathematics.
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