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## 20/11/2010

### A KAHUN MATHEMATICAL FRAGMENT ## John A.R. Legon

[Based on the author's article in Discussions in Egyptology 24 (1992), p.21-24]

 Among the various papyri discovered by Flinders Petrie in the Middle Kingdom town of El-Lahun (Kahun) were found some fragments dealing with mathematical problems, some of which were understood at the time by F.L. Griffith , while others were explained by Schack-Schackenburg . The problem represented by columns 11 and 12 of Kahun fragment IV.3, however, was not fully understood, and much confusion has resulted from the analysis by R.J. Gillings . In the present article we will show that contrary to Gillings' view, the text contains a straightforward and complete example of the Egyptian method of calculating an arithmetical progression. The significance of the numbers written in hieratic in column 12 of the fragment (see fig. 1) was in fact first recognised by Moritz Cantor , who noticed that these numbers form the ten terms of an arithmetical progression with a common difference between the terms of 2/3 + 1/6 (or "3 '6 to use the notation of fig. 1). Cantor also realised that since the sum of the ten terms is just 100, the hieratic signs for 100 and 10 which stand at the head of column 12, probably denote this sum and the number of terms, and not the number 110 which was transcribed by Griffith. If the scribe had intended to write the number 110, then the hieratic sign for 10 would be expected to stand above the tail of the 100 sign, so that the two signs could be read together as a single value; but the tail of the 100 sign in fact only runs into the side of 10 sign because of the cramped working, viz: and the reading of these signs as two numbers is quite possible. The scribe has thus given a brief statement of the problem, which is to divide a quantity of 100 into 10 shares in arithmetical progression.  