Kahun Papyrus

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The Kahun Paprus(KP) has been long known as a medical text discussing several important topics. It's many fragments are kept in University College London and were discovered by Flinders Petrie in 1889. Most of the texts are dated to 1825 BC to the reign of Amenemhat III. One of its fragments deals with gynaecological matters. In some ways it is similar to the Edwin Smith papyrus which discusses ailments of women suffering from teeth and gum problems.

However, one fragment began with a 2/n table. It is an abbreviated version compared to the 51 terms found in the Rhind Mathematical Papyrus (RMP). Considering the KP's 10 term arithmetic progression, summing to 100, with a difference of 5/6, the innovative math should allow the KP to be seen as a mathematical text.

To discuss the mathematical contents of an arithmetic progression two columns of data were published by Gillings in 1972. Column 11 multiples 5/12 times 9, a key fact that was needed to find the largest term in the arithmetic progression. The scribe added 10 and wrote out the correct largest term arithmetic progression, and then by subtracting 5/6 nine times to find the remaining terms.

Gillings' 1972 analysis failed to parse the scribal method. He had noticed similar problems in the RMP (RMP 40, 64) yet he muddled three pages of analysis.

In 1987, Egyptologist Gay Robins, and spouse Charles Shute, writing on the Rhind Mathematical Papyrus, and Egyptologist John Legon, 1992, writing on the KP, described that the RMP and KP scribes had used the same method to find the largest term of closely related arithmetic progressions. The method: take 1/2 of the difference, 1/2 of 5/6 (5/12 in the KP) times the number of differences (nine times 5/12 = 15/4 in the KP) plus the sum of the A.P progession (100 in the KP) divided by the number of terms (10 , meaning 100/10 = 10 in the KP). Finally sdd column 11's result, 3 3/4, to 10, and the largest term, 13 3/4.

To repeat, add column 11, 5/12 times 9, or 45/12, or 3 3/4, 3 2/3 1/12 in Egyptian fractions to 10 in column 12 beginning with the largest term 13 2/3 1/12. The scribe subtracted 5/6 nine times, creating the remaining terms of the progression.

Robins-Shute confused the problem by omitting the sum divided by the number of terms parameter in the RMP. An algebraic statement could have been created by Robins-Shute from matched pairs that added to 20, five pairs summing to 100, as potentially related to RMP 40. A modern mathematician Carl Gauss added 1 to 100 noting 50 pairs of 101, finding the sum of an arithmetic progression to be 5050, an aspect of any arithmetic progression, facts that were apparently known to Ahmes and the KP scribe.

The KP method found the largest term, and used other facts that have been reported in RMP 64, and RMP 40, by John Legon in 1992. Scholars, at other times, have attempted to parse Rhind Mathematical Papyrus 40, a problem that asks 100 loaves of bread to be shared between five men by finding the smallest term of an arithmetic progression.

A confirmation of the Kahun methods are reported by RMP 64. Ahmes was asked 10 men to share 10 hekat of barley, with a differential of 1/8, by using an arithmetical progression. Robins and Shute reported, "the scribe knew the rule that, to find the largest term of the arithmetical progession, he must add half the difference to the average number of terms as many times as there are common differences, that is, one less than the number of terms" (and omitted the sum divided by the number of terms), as noted by:

1. number of terms: 10

2. arithmetical progression difference: 1/8

3. arithmetic progression sum: 10

The scribe used the following facts to find the largest term.

1. one-half of differences, 1/16, times number of terms minus one, 9,

  1/16 times 9 = 9/16

2. The computed parameter(1), was found by 10, the sum, divided by 10, the number of terms. It was inserted by Robins-Shute, but had not been high-lighted, citing 1 + 1/2 + 1/16, or 1 9/16, the largest term. The remaining nine terms were found by subtracting 1/8 nine times to obtain the remaining barley shares.

That is, the KP scribe used formula 1.0:

(1/2)d(n-1) + S/n = Xn (formula 1.0)

with,

d = differential, n = number of terms in the series, S = sum of the series, Xn = largest term in the series.

allowing 2 or 3 parameters: d, n, S and Xn, to be cited to find the missing parameter(s).

In summary, the KP and RMP scribes used an identical, easy to parse, method to calculate largest terms in arithmetic progressions. Despite this agreement, several basic questions remain open for scholars to resolve. For example, what were the scribal intermediate arithmetic steps, particularly the scribal subtraction and division operations? Asking the question in other terms, why were vulgar fractions used in all intermediate calculations in the KP, as was the case in the RMP? That is, did the KP scribe use false position division, the standard method of division suggested by Peet in 1923 that required all MK scribes to employ at all times? Is it possible, or likely, that the KP scribe only created Egyptian fractions as final statements, since Egyptian fractions were only a convenient notation to uniquely write vulgar fractions?