
Re: RMP # 48
Posted:
Oct 5, 2007 3:58 PM


Lee,
Let me comment on your latest points, per ** comment**:
The distinction which you note between the exactitude sought and realized in the hekat calculations as opposed to the fudging apparently allowed in the geometrical solutions is an interesting point.
**to me, this point means that modular arithmetic, quotients and its exact Egyptian fraction remainders, was more important to Ahmes and the other scribes than geometry when approximations were involved. Of course, geometry was important when its data could be presented exactly, such as the MMP details. **
I am also not at ease with the word "sacred", but it would seem that they desired to utilize circle and square relationships in order to arrive at various understandings.
** I can not determine Ahmes' desires in this respect. Others have inferred proportions and other relationships, views that are not directly documented in any text that I have seen. That is, modern geometry's use of Greek models and theorems makes many of the points that you cite. Egyptian texts do not. The problem is, therefore, how can your views be confirmed in the RMP or any other Egyptian text? **
I argue elsewhere that a paramount concern for them was in regard to squaring the circle for both circumference and area. I believe that the design of Old Kingdom pyramid interiors is derivable from this point of beginning.
** squaring the circle and one approximation 256/81 was used by Egyptians for a very a long period of time (maybe 1,000 years). I have no proven idea why a more accurate value for pi was not selected. Jumping back to pyramid interior dimensions is another matter. This subject is not well documented by numbers in scribal texts, hence it is much harder to draw inferences and connect to formal scribal methods than the texts that Ahmes and others left. **
RMP 48, I believe, allows for at least the following inferences:
1) that there was a more than casual interest in what can be termed diagrammatic geometry;
** why do you use the term diagrammatic geometry? One interpretation is that a visual or implied context was at work, however, the scribal numerical data does not disclose such details. **
2) that they clearly understood that there was a constant relationship (ie, 64/81) between the area of a circle and the area of the square whose sides were the same length as the diameter of that circle;
** if 64/81 was so important, why was the approximation not improved until Greek times? The problem was very difficult to solve since Egyptians only worked within the realm of rational numbers. Greeks, starting with Pyathagoras newly accepted irrational numbers, beginning a debate that continues today (in certain respects, as Gauss discovered when he formally introduced complex numbers to solve the fundamental theorem of algebra, that an nth degree equation contains exactly n roots).**
3) that they would have extrapolated upon the concept and have drawn the square inscribed within the circle, and the circle circumscribed around the square  forcing a formalizing of the square root of 2 understanding;
** Visually, or by diagram, you are correct. However, using only rational numbers there are no solutions to many classes of geometry problems, as Greeks discovered with their straight edge and rule construction problems. **
4) that it is more than likely that they would have asked the question, "if the area of the inscribed circle is in a constant relationship (ie, 64/81) with the area of the circumscribed square, would not the circumference of the circle be in this same constant relationship (64/81) with the perimeter of the square?"
** Asking a question, and not allowing the appropriate class of number (any number beyond the rationals) to be used to find a solution was the limitation reached by Egyptians. It is sad that Egyptian geometry did not theoretically proceed in documented ways (as visually, or diagrammatically, to use your term) that a few scribes may have been understood, yet a mathematical dead end was reached by all scribes **
Since each side of the square is the same length as the circle's diameter, this latter inference of course leads to the circle's circumference being equated with 64/81 times 4 times this diameter  the understanding which i believe is being implemented in MMP 10.
That the equality was more or less correct could easily have been confirmed through empirical measurement. Had they already reached a similar understanding via a C = 11/14 times 4 times this diameter "formula" as a result of empirical implementation using their 28 finger royal cubit system, then they could have confirmed the 64/81 result by comparison of the respective unit fraction chains.
I believe that Problem 48 tells us all of this and more. I continue to look for others who have considered this problem in this way as I am sure that someone must have.
**belief is a wonderful thing. In the case of Egyptian geometry, we'll all have to wait until direct evidence is presented that documents the use of irrational numbers or better approximations of pi, two subjects that exploded under the Greeks. Yet, Egyptian arithmetic formed the basis of Greek arithmetic, and was not improved upon in the manner that Greeks clearly improved Egyptian geometry by adding irrational numbers**
Best Regards,
Milo

