Ronald wrote;

"The "squaring the circle" arguments are a product of modern minds. One has to try to replace her/himself into the mind of the Ancient Egyptians. These people certainly did not think within the concepts of maths we have today. There also was simply no need to do so. I retain my view that you do not need much 'math' (what's in a name ?) to build tens of quadrangles of diminishing size on top of each other (limestone-block courses) to achieve a pyramidal shape."


I agree with you that the ancient Egyptian "concepts of maths" were different from that of today, and that it is important to make the attempt to enter into their ways of thinking and world view. I absolutely and totally disagree, however, that the "squaring of the circle arguments are a product of modern minds". I find this statement somewhat stunning in light of the Egyptian method for finding the area of a circle. Given problems 48 and 50 of the RMP (not to mention MMP 10), it is not clear to me what level of proof one would want in order to accept that the Egyptians squared the circle in terms of area. (Gillings refers to the scribe of the RMP as being "the first authentic circle-squarer in recorded history").

Note that I am not saying here that the Egyptians thought of things in terms of Pi, as we might do today. I believe their conceptual approach in this situation was a diagrammatic - or graphic - one (Peet uses the word "graphically" to describe the Egyptian method used in MMP 6). Friberg, in his Unexpected Links Between Egyptian and Babylonian Mathematics, exposes much of the geometric underpinnings of Old Babylonian mathematics (his term for their methodology is "naive geometry"), and after extensive comparisons between OB and Egyptian mathematical texts comes to the conclusion that "Middle Egyptian and OB mathematics must have influenced each other in decisive ways" (p. 103). He implies that the Egyptians likely also thought "graphically" when dealing with certain mathematical situations. (As one instance, he points to P. Berlin 6619, showing that in computing square roots, the method involved the computation of "squaresides, that is sides of squares". p. 82)

I have gotten the sense from some of the discussions here that there is a view that the ancient Egyptians were, in certain ways, a bunch of simpletons, incapable of seeing - for instance - that a circle which used a royal cubit for its diameter would then have a perimeter length of almost exactly 88 fingers. No Pi here, just simple empirical measurement. The surviving record is incomplete, and so there is no surviving explicit statement of this fact. However, is it credible to believe that no ancient Egyptian scribe, advanced scribe, or architect ever discovered this relationship? I think not.

We would be best served, I believe, by familiarizing ourselves with the methods AND intelligence of the ancient Egyptians. They were extremely clever, they were extremely capable, and they could be as precise as desired. Friberg states that "second millennium BCE Egyptian mathematics was just as advanced as Babylonian mathematics!" (exclamation point his). Gillings notes that within their own methodological structures the Egyptians "reached a relatively high level of mathematical sophistication".

At this point in time, I submit that we should be focusing our attention on the ways in which Egyptian design appears to have been preoccupied with the "squaring the circle" theme, not in attempting to ignore, deny or discredit all of the "coincidences" which point directly to this fact.


Lee Cooper