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Re: RMP # 48
Posted:
Oct 6, 2007 10:15 AM
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Milo,
Thank you for your interest and for your detailed response. Here are my thoughts in reply to the points you have raised.
I use the term "diagrammatic geometry" to denote a geometry empirical in nature, that is not "axiomatic" and is not "proof' driven. Results are accepted as "true' when one can tell from empirical measurement and experience that they are true, or nearly enough so for all practical purposes.
This is the case with RMP 48, where it appears that the area of a circle is approximated as a result of a diagrammatic relationship. A major part of this problem's importance to us is that it shows us such a diagrammatic usage in the act. It furthermore shows us the interest in the inscribed circle phenomenon. And, it further shows us (as do a number of other papyri problems) the Egyptian method for squaring the circle in terms of area.
You ask "if 64/81 was so important, why was the approximation not improved until Greek times?" I believe that the Egyptians were also aware of the alternate "formula" for circumference of 11/14 x Diameter x 4. As I mentioned in my preceding post, this is easily discoverable empirically when one but makes the diameter of a circle equal to a royal cubit, and hence equal to 28 fingers in length. The circumference will then be found to be exceeding close to 88 fingers - the difference being all but indistinguishable by direct measurement. It is hard for me to believe that no architect, scribe, or whomever, would not have come across this relationship. (A circle with a circumference of 88 units will have the same perimeter length as a square having 22 of these units to a side - i.e., the circle is squared for circumference. It is from this occurrence that one can then arrive at the 11/14ths "formula" for circumference).
If I am right in my interpretation of MMP 10, then it would seem to be telling us that the everyday scribe would find a circle's circumference essentially by multiplying the diameter by 4 and then multiplying this by (8/9)sqrd. To me, your question then becomes why would they have used this route for finding circumference rather than the 11/14ths route.
RMP 48 may hold a clue to the answer in that it essentially provides a single diagram which can be used to provide both the area, and the circumference, relationships. This duality of derivation, and/or its mnemonic attributes, may have provided the reason for the choice. I cannot now prove any of this, but it is at minimum food for thought.
You are correct, we must await direct evidence to substantiate any deduced connections. I have offered a number of possible ways to test my theory for the design basis of four of the Old Kingdom pyramids, and should these tests ever be performed and find my predictions to be true, it will then go a long way toward substantiating the connections and conjectures of which i speak of here. This will be a proof found not in any surviving written text of the period, but in a surviving geometrical construct. Since I fear the finding any further surviving papyric mathematical texts to be extremely unlikely at this point, it seems to me to be a most reasonable alternative to see what (if any) "mathematics" may be gleaned from the surviving monuments of the period.
Very best wishes,
Lee
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