
Re: Scribal division reported within a volume unit
Posted:
Nov 3, 2010 12:08 PM


Footnote:
Aspects of hekat calculations are reported in RMP 43 and the Kahun Papyrus where the volume formula
V = (2/3)(H)[(4/3)(4/3)(D)(D)] (khar)
divided khar data by 20 to reach 400hekat and 100hekat units. RMP 41, 42, 43, 44, 45, 46, and 47 mixed hieratic symbols for 400hekat and 100hekat. Scribal twopart quotient and remainder answers are incorrectly scaled and therefore muddled on Wikipedia:
http://en.wikipedia.org/wiki/Rhind_Mathematical_Papyrus
by:
"Problem 47 gives a table with equivalent fractions for fractions of 100 quadruple hekat of grain. The quotients are expressed in terms of Horus eye fractions...
1/10 gives 10 quadruple hekat 1/20 gives 5 quadruple hekat 1/30 gives 3 1/4 1/16 1/64 (quadruple) hekat and (1 2/3 ro) (error) 1/40 gives 2 1/2 (quadruple) hekat 1/50 gives 2 (quadruple) hekat 1/60 gives 1 1/2 1/8 1/32 (quadruple) hekat (3 1/3) ro (error) 1/70 gives 1 1/4 1/8 1/32 1/64 (quadruple) hekat (2 1/14 1/21) ro (error)* 1/80 gives 1 1/4 (quadruple) hekat 1/90 gives 1 1/16 1/32 1/64 (quadruple) hekat 1/2 1/18 ro (error) 1/100 gives 1 (quadruple) hekat (error)
*Ahmes actually reported one (5R/n)ro remainder as 2 1/14 1/21 1/42 since (150/70)ro = (2 + 1/7)ro with 1/7 = 1/7(6/6) = 6/42 = (3 + 2 + 1)/42 = 1/14 + 1/21 + 1/42 (a 2/n table rule per 2/101, and an EMLR rule per 1/101).
Ahmes mixed initial 400hekat multiplications by 1/10 and 1/20 reporting correct answers with 100hekat (scaled to 6400/64) multiplications by 1/30, 1/40, 1/50, 1/60, 1/70, 1/80, 1/90 and 1/100 into Q/64 quotient and (5R/n) remainders reporting incorrect answers whenever 4hekat quotients were added to 1ro remainders. Correct RMP 47 quotient and remainder answer are reported by two sets of balanced statements:
A. Table A with 4hekat quotient and 4ro remainder answers:
following 4 x (6400/64)/n = (Q/64) 4hekat + (5R/n)4ro
1/10 gives (10) 4hekat 1/20 gives (5) 4hekat 1/30 gives (3 1/4 1/16 1/64) 4hekat + (1 2/3)4ro 1/40 gives (2 1/2) 4hekat 1/50 gives (2) 4hekat 1/60 gives (1 1/2 1/8 1/32) 4hekat + (3 1/3)4ro 1/70 gives (1 1/4 1/8 1/32 1/64) 4hekat + (2 1/14 1/21 1/42) 4ro 1/80 gives (1 1/4) 4hekat 1/90 gives (1 1/16 1/32 1/64) 4hekat (1/2 1/18)4ro 1/100 gives (1) 4hekat
B. And Table B with 1hekat quotient and 1ro remainder answers:
following 1 x (6400/64)/n = (Q/64)1hekat + (5R/n)1ro
1/10 gives (10) 1hekat 1/20 gives (5) 1hekat 1/30 gives (3 1/4 1/16 1/64) 1hekat + (1 2/3)1ro 1/40 gives (2 1/2) 1hekat 1/50 gives (2) 1hekat 1/60 gives (1 1/2 1/8 1/32) 1hekat + (3 1/3)1ro 1/70 gives (1 1/4 1/8 1/32 1/64) 1hekat + (2 1/14 1/21 1/42)1ro 1/80 gives (1 1/4) 1hekat 1/90 gives (1 1/16 1/32 1/64) 1hekat (1/2 1/18)1ro 1/100 gives (1) 1 hekat
C. The quadruple (400) hekat case is also made by 4sack and 4hekat economic shipping units recorded in Northumberland Papyri 1, 2 and 3 published by Barns and Gunn, JEA, 1948. Quadruple sack and hekat initial scaled values (47/60) 4hekat (wheat), 2hekat (flour) scaled to (15/16) 4hekat (flour), 383 deben (meal) and 31 loaves of 10 3/4 deben/loaf summed to 33 1/4 deben. Balanced 4hekat, 4ro or 1hekat, 1ro quotients and remainders was practiced by Ahmes. Ahmes reported khar divided by 20 into 400 hekat units by two volume formulas. The 400 hekat and 100hekat initial divisions byt rational numbers have been translated into one hekat into 4800 ccm, 1/10 of a hekat (hin) into 480 ccm, 4ro into 60 ccm, and 1ro = 15 ccm by scholars, often muddling scribal 4hekat and 4ro intermediate details.

