and could have replaced 1/40 with 1/32, as the square root of 164 recommended showed
2^1/2 approx = (1 + 1/3 + 1/24 + 1/32)^2 with E2 = not enough
2^1/2 approx = (1 + 1/3 + 1/24 + 1/30)^2 with E2 = again, not enough
2^1/2 was likely approx by (1 + 1/3 + 1/24 + 1/28)^2 with E2 = about right
E. CONCLUSION: Egyptian and Greek square root exactly solved squared rational numbers 4, 9, 16, 25, .... by an inverse proportion and other steps. The square root method accurately estimated irrational numbers 2, 3, 5, 6, ... and found a quotient (Q) by the inverse method and estimated a remainder (R), in unit fractions, by a subtractive method. The scribe kept in mind the initial rational number error amount (E1), while searching for a smaller irrational error amount (E2). The method's final step squared (Q + R) and determined rational unit fraction roots, plus irrational errors(E2)in proofs.
That is scribes focused upon were remainders(R). Initial remainders were often estimated in n/5 parts such that n/5 was scaled by 24/24 = 24n/120 as discussed by:
1/5(24/24) = 24/120 = (20 + 3 + 1)/120
2/5(24/24).= 48/120 = (40 + 5 + 3)/120
3/5(24/24) = 72/120 = (60 + 8 + 3 + 1)/120
4/4(24/24) = 96/120 = (80 + 8 + 5 + 3)/120
and other partitions.
as Ahmes and earlier Middle Kingdom scribes scaled a volume unit hekat to (64/64) in terms of binary quotients and remainders scaled by (5/5) to 1/320 (ro) of a hekat.
Initial estimated square root of N statements used quotient (Q) in the form
(Q + n/5)^2
The raw square root data was processed in shorthand formats that considered a rational error E1 that scribes reduced to a lower acceptable error E2.
A broadly defined historical method (found by Occam's Razor, and historical data) will be posted here. A finalized, but not final, methodology will describe selections of (irrational) errors E2 associated with adding or subtracting from the last term (1/40, most often, and as I suspect, at other times 1/120).
Math Forum discussions; Why Study Egyptian Fractions
Planetmath Remainder Arithmetic entry;
Planetmath Kahun Papyrus entry. Bibliography
1 A.B. Chace, Bull, L, Manning, H.P., Archibald, R.C., The Rhind Mathematical Papyrus, Mathematical 2 Marshall Clagett Ancient Egyptian Science, Volume III, American Philosophical Society, Philadelphia, 1999. 3 Richard Gillings, Mathematics in the Time of the Pharaohs, Dover Books, 1992. 4 H. Schack-Schackenburg, "Der Berliner Papyreys 6619", Zeitscrift fur Agypyische Sprache , Vol 38 (1900), pp. 135-140 and Vol. 40 (1902), p. 65f.