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Topic: Egyptian Math and Sylvia Couchoud
Replies: 2   Last Post: Sep 1, 2010 11:37 AM

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Milo Gardner

Posts: 1,105
Registered: 12/3/04
Egyptian Math and Sylvia Couchoud
Posted: Aug 31, 2010 9:28 AM
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Sylvia Couchod's "views of Egyptian math", translated from the original French from Wikipedia, will be discussed in two sections.

1. "The requirement of scribe Ahmose returned to replace the area of a round field of diameter 9 khet by the area of a square field of side 8 khet. The equivalent in our language today, is used for pi approximation fractional follows:

4 x (8 / 9) x (8 / 9), 3.16, which overlooks ft accuracy of 0.6%.

If the text is not a conceptual definition[2] of pi, we can reduce the requirement set out in a strictly empirical to solve a practical problem. Indeed, only with 9 and 8, the numbers just chosen by the scribe, that the approximation is accurate, then we do not often find a round field of diameter 9 khet in practice! The teacher has in mind something like the formula S = (64/81) x (DXD), but he can not afford to write it."

is incomplete and misleading in several respects. Sylvia considered a valuation of pi = 256/81 rather than looking for a formula in which pi and a semi-diameter D/2 replaced radius R, were computed by Ahmes, and MMP and KP scribes. Gillings made a similar error in judgment by guessing that a circumferences of a circle formula C = (pi)(D) was used in some way.

Ahmes' logic simply input pi = 256/81 and r = (D/2) into A = (pi)(r^2) such that:

1. A = (256/81)(D/2)(D/2) = (64/81)(D)^2, and placing sqrt (64/81) = 8/9 as a coefficient of D

2. A = [(8/9)D]^2

a. MMP 10 set D = 9 in a semi-circle (asking for the area of an egg) computed: (9 - 9/9) = (8 x 8)/2 = 32 (no unit attached)

b. RMP 41 set D =9 in a circular granary computed: (9 - 9/9) = (8 x 8) = 64 cubits^2

c. RMP 42 set D = 9, height(H) = 10 in a circular granary computed (10 - 10/9) = (80/9) x (80/9) = 6400/81 cubits^2, with volume computed by 6400/81 x 10 - 64000/81 = 790 10/81 cubits^3

2. "Sylvia Couchoud has also prepared a glossary of 124 mathematical terms that are found regularly in the papyri (pp. 194-204). This glossary, which includes the hieroglyphics in Egyptian transcription and translation in French, is a convenient tool to tackle any mathematical text in hieroglyphics.

Couchoud wrote (p. 1-2 and p. I):

"There is one important difference between Greek mathematics and those of Pharaonic Egypt. The Greeks were interested in the formulas and theories and they brought to their proofs of evidence. The Egyptians, for their part, were concerned primarily with the results of their calculation and they were also looking for evidence for the latter, it was only to demonstrate their numerical accuracy.

I dedicate this work with my admiration and gratitude to Ahmose, the scribe who copied it four thousand years ago, this mathematical papyrus now known as the Rhind."

Couchoud's views contain serious oversights that relate to the EMLR, the RMP and all the hieratic math texts. Her undocumented conclusion that Ahmes (Ahmose) only demonstrated numerical accuracy was incomplete and misleading.

A counter example will be provided by Ahmes' written information that demonstrated numerical facts that later scribes (one was Fibonacci, "Liber Abaci"), using the same demonstration approach, ended with a set of rules (seven distinctions by L.E. Sigler's count).

Several counter examples validate that Ahmes' text included rules (and formulas), even though it is true that Ahmes never concluded in his hand writing explicit rules and formulas (i.e. the area of a circle, or rules that converted rational numbers to concise unit fraction series). But implicit rules and formulas abound in Ahmes' shorthand notes, meta facts that scribes reported in all the Middle Kingdom mathematical texts.

For example in reading (decoding) Ahmes 87 problems, it is clear that a small number of rational numbers in Ahmes' time, and today, can not be converted by one scaling factor (ie. 28/97 in RMP 31, and 30/53 in RMP 35). In these cases Ahnes replaced n/p by (n-2)/p + 2/p, a general rule, and solved two separate problems, THEREBY establishing a core reason for spending 1/3 of the RMP on the 2/n table. I'd be pleased to show the details that Ahmes used to scale 26/97 by LCM 4, 2/97 by LCM 56; 28/53 by LCM 2, and 2/53 by LCM 30, much as Fibonacci converted rational numbers in a related subtraction context.

Conclusions: Reading (decoding) Ahmes' implicit rules, Egyptian scribes looked for and found one or two scaling factors that converted every rational number to concise unit fraction series, and used other formulas as needed, two being the area of a circle by

A = [(8/9)(D)]^2 cubits^2

and two volume formulas for a circular granary

V = (H)[(8/9)(D)]^2 and

V = (2/3)(H)[(4/3)(D)]^2

all based on pi = 256/81 and r = D/2 substituted into an area of a circle formula A = pi(r)^2 and H = height.

The EMLR was only one critical document that validates scribal student learning in which alternative scaling factors, three for 1/8, were calculated by brute force. Concise unit fraction series represented in 2/n tables, and in Ahmes' 87 problems, the MMP's 25 problems, and other texts, in total validate that a student became a scribe, double checked answers and thereby confirmed theoretical and practical aspects of scribal math methods.

Comments would be appreciated.

Best Regards,

Milo Gardner

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