Kahun Papyrus
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The Kahun Papyrus (KP) is as an ancient Egyptian text discussing mathematical and medical topics. Its many fragments were discovered by Flinders Petrie in 1889 and are kept at the University College London. Most of the texts are dated to ca 1825 BC, to the reign of Amenemhat III. One of its fragments, referred to as the Kahun Gynaecological Papyrus deals with gynaecological illnesses and conditions.
[edit] The Egyptian Fraction Text
The Kahun Papyrus (KP) is also notable for the insights it gives into the use of Egyptian fractions through its exact solution to one arithmetic progression problem. Ancient Egyptian fractions was a distinctive form of arithmetic used to represent positive rational numbers, fractions (2/n, or m/n), as concise sums of unit fractions (typically 5-terms 1/a + 1/b + 1/c + 1/d + 1/e or less) in optimal ways. The Egyptian fraction method of writing fractions was continuously used for over 3,000 years, with the convention ending around 1600 AD with the raise and dominance of our modern algorithmic base 10 decimal arithmetic.
One fragment of the KP begins with a traditional RMP 2/n table, a Middle Kingdom scribe's method of defining arithmetic skills related to writing rational numbers in exact unit fraction series. The KP 2/n table offers an abbreviated version of a 2/n table compared to the longer set of 51 rational number terms written as unit fraction series found in the Rhind Mathematical Papyrus (RMP). Considering the KP's central arithmetic topic, arithmetic progressions, most likely the highest form of Egyptian arithmetic, the scribe defined a 10-term arithmetic progression, summed to 100, with a difference of 5/6. The KP method detailed an arithmetic progression, a form of mathematics that was also discussed in two RMP problems.
To imply a generalized form of arithmetic progressions, the RMP scribe Ahmes listed two columns of data (published by Gillings in 1972). Ahmes's thinking is shown in Gillings' column 11 by multiplying 5/12 times 9, a fact that was needed to find the largest term of the RMP progression. Ahmes then added 10 and wrote out the correct largest term of the arithmetic progression, and subtracted 5/6, nine times. Gillings found the remaining terms in the progression by using a method that was directly related to the KP's method. To fully understand the KP method, readers are requested to attempt to make the exact arithmetic calculations that the Middle Kingdom scribes wrote down in their three problems, thereby double and triple checking your work.
Gillings' 1972 analysis of both RMP versions of Middle Kingdom arithmetic progression failed to parse the method in a manner that was comparable, in every respect, to the KP method. For example, Gillings had noticed similar problems in the RMP (RMP 40, 64) yet Gillings muddled three pages of his analysis, thereby reaching no definitive conclusions on this topic.
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 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 progression (100 in the KP) divided by the number of terms (10 , meaning 100/10 = 10 in the KP). Finally add 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 an aspect of 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 method is reported by RMP 64. Ahmes asked 10 men to share 10 hekats 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 progression, 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" (note that Robins-Shutre 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, KP and RMP scribes used identical methods to calculate the largest term in arithmetic progressions. Despite this agreement several questions remain open for scholars to discuss and resolve. For example, what were the beginning, and intermediate arithmetic steps, particularly scribal subtraction and division operations? Asking the question in other terms, why were vulgar fractions present in beginning and intermediate calculations in the KP, and the RMP? Noting that KP scribe did not use false position division, a method proposed by 'guessing' division steps, as suggested by Peet, and others, what was used? Another direct calculation apparenttly was used in both the KP and the RMP. It is highly likely that the KP scribe created Egyptian fractions as final statements since Egyptian fraction arithmetic employed an exact system that first wrote beginning, and intermediate vulgar fractions. Ahmes and other MK scribes also used direct calculations within a wide range of rational number problems, all of which ended up with exact Egyptian fraction answers. One exception was the calculation of hekat (volume) units, a topic that touches on higher order numbers.
[edit] References
1. Gillings, Richard, "Mathematics in the Time of the Pharaohs", pages 176-180, MIT, Cambridge, Mass, 1972, ISBN 0-486-24315-x
2. Legon, John, A.R., "A KAHUN MATHEMATICAL FRAGMENT", In Discussions in Egyptology 24 (1992), p.21-24.
3. Robins, Gay, and Shute, Charles. " The Rhind Mathematical Papyrus", pages 41-43, British Museum, Dover reprint, 1987, ISBN 0-486-26407-6.
