Page:The Rhind Mathematical Papyrus, Volume I.pdf/46

In Problem 64, 10 hekat of barley are supposed to be divided among 10 men in such a way that the amounts received by the different men fonn an arithmetical progression with a common difference of ¹⁄₈^[1] of a hekat. Thus the terms, their sum, and the common difference are all concrete quantities, namely, certain amounts of barley, and the author is careful to use the special notation required for these quantities (the special forms of numbers of hekat and the “Horus eye” forms; see page 31), but the ordinary notation for numbers which represent the number of men, or differences, and for numbers used as multipliers. Thus there are given the number of terms, 10; the sum of the terms, 10 hekat of barley; and the common difference, ¹⁄₈ of a hekat. First he gets the mean or average, which is 1 hekat. Then to get the largest term he would add to this average the common difference one-half as many times as there are differences in all, but as the number of differences is an odd number he cleverly adds instead one-half of the common difference as many times as there are differences, that is, 9 times. Having obtained the amount of the largest portion, he gets the others by subtracting the common difference a sufficient number of times, and writes down the entire progression in descending order, each term as a certain amount of barley.

Problem 79 is a problem in which is calculated in two ways the sum of a geometrical progression. There are two columns. The first column indicates, it seems to me, a numerical method for determining the sum when the first term is equal to the ratio. This method may be stated in the following rule: In any geometrical progression whose first term is equal to the ratio, the sum of any number of terms is equal to the sum of one less number of terms plus 1 multiplied by the ratio. This rule the author follows in the first column, and in the second he performs the ordinary process of multiplication by the ratio and adds the terms together, getting the same result and thinking perhaps that he was proving the rule.^[2]

1. I am using black-faced type to represent the “Horus eye” forms. See page 31.
2. Neugebauer (1926) constructs a table in which these two columns appear as the last column and row, and the calculations are all reduced to doublings and additions. See Bibliography.