Page:The Rhind Mathematical Papyrus, Volume I.pdf/41

Rh 61B, which gives a rule for ﬁnding $2/3$ of a fraction. The rule says, “To find $2/3$ of $1/undefined$ take thou its double and its six-fold, and do thou likewise for any fraction that may occur.” In the table at the beginning of the papyrus for the division of 2 by odd numbers we notice that the quotient for 3 is simply $2/3$, but if the author had used the method employed for all multiples of 3 after 15 he would have obtained $1/undefined$ $1/undefined$, and he sometimes finds it convenient as here to use this expression in place of $2/3$. This rule is used a number of times in the arithmetical portions of the papyrus. In Problem 33 it is employed with the fraction $2/3$.

It is to be noticed that in dealing with numbers which the modern mathematician calls fractions, the Egyptian regards the denominator as the important element, and therefore, although he wants to multiply the 5 by 2 and 6, he says, “Multiply $1/undefined$.”

Problems 24-38 are all essentially problems in division by a fractional expression. Problem 67 may also be included in this list. In other words, these are multiplication problems in which the product is one of the given numbers. But in all of them it is the multiplier and not the multiplicand that is the other given number. And therefore, as they stand, they cannot be solved directly by the Egyptian process of multiplication. We have seen (pages 6 and 7) that there are two methods of solving such problems. One method was to regard all of the numbers as mere numbers and change the problem so as to make the given multiplier a multiplicand and obtain the answer first as a multiplier. The other method was that of false position. The former method is used in Problems 30-34. The latter seems to be the method in all except these.

The first eleven of these problems (24-34) are sometimes called ꜥahaꜥ or quantity problems because they nearly all begin with the word ꜥahaꜥ and use this word to denote the result of their calculations. In Problems 35, 37, and 38 the quantity is a measure of grain. In 36 the word for the measure is omitted and the problem by itself is a purely