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Topic: Scribal division reported within a volume unit
Replies: 13   Last Post: Mar 2, 2011 12:21 PM

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Milo Gardner

Posts: 1,105
Registered: 12/3/04
Re: Scribal division reported within a volume unit
Posted: Nov 3, 2010 12:08 PM
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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 400-hekat and 100-hekat units. RMP 41, 42, 43, 44, 45, 46, and 47 mixed hieratic symbols for 400-hekat and 100-hekat. Scribal two-part 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 400-hekat multiplications by 1/10 and 1/20 reporting correct answers with 100-hekat (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 4-hekat quotients were added to 1-ro remainders. Correct RMP 47 quotient and remainder answer are reported by two sets of balanced statements:

A. Table A with 4-hekat quotient and 4-ro remainder answers:

following 4 x (6400/64)/n = (Q/64) 4-hekat + (5R/n)4-ro

1/10 gives (10) 4-hekat
1/20 gives (5) 4-hekat
1/30 gives (3 1/4 1/16 1/64) 4-hekat + (1 2/3)4-ro
1/40 gives (2 1/2) 4-hekat
1/50 gives (2) 4-hekat
1/60 gives (1 1/2 1/8 1/32) 4-hekat + (3 1/3)4-ro
1/70 gives (1 1/4 1/8 1/32 1/64) 4-hekat + (2 1/14 1/21 1/42) 4-ro
1/80 gives (1 1/4) 4-hekat
1/90 gives (1 1/16 1/32 1/64) 4-hekat (1/2 1/18)4-ro
1/100 gives (1) 4-hekat

B. And Table B with 1-hekat quotient and 1-ro remainder answers:

following 1 x (6400/64)/n = (Q/64)1-hekat + (5R/n)1-ro

1/10 gives (10) 1-hekat
1/20 gives (5) 1-hekat
1/30 gives (3 1/4 1/16 1/64) 1-hekat + (1 2/3)1-ro
1/40 gives (2 1/2) 1-hekat
1/50 gives (2) 1-hekat
1/60 gives (1 1/2 1/8 1/32) 1-hekat + (3 1/3)1-ro
1/70 gives (1 1/4 1/8 1/32 1/64) 1-hekat + (2 1/14 1/21 1/42)1-ro
1/80 gives (1 1/4) 1-hekat
1/90 gives (1 1/16 1/32 1/64) 1-hekat (1/2 1/18)1-ro
1/100 gives (1) 1- hekat

C. The quadruple (400) hekat case is also made by 4-sack and 4-hekat 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) 4-hekat (wheat), 2-hekat (flour) scaled to (15/16) 4-hekat (flour), 383 deben (meal) and 31 loaves of 10 3/4 deben/loaf summed to 33 1/4 deben. Balanced 4-hekat, 4-ro or 1-hekat, 1-ro 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 100-hekat initial divisions byt rational numbers have been translated into one hekat into 4800 ccm, 1/10 of a hekat (hin) into 480 ccm, 4-ro into 60 ccm, and 1-ro = 15 ccm by scholars, often muddling scribal 4-hekat and 4-ro intermediate details.



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