a definite process which they generally used. The remainder, being a small number, would consist of one or more reciprocal numbers. For one of these numbers the third step may be expressed by the rule: To get the multiplier that will produce the reciprocal of a given whole number as a product multiply the multiplicand by the number itself and take the reciprocal of the result of this multiplication.[1] If, for example, we wish to multiply 17 so as to get ⅓, we take 3 times 17, which is 51, and then we can say that 1⁄51 of 17 equals ⅓.[2] See page 17. In both kinds of multiplication the product is formed by adding certain multiples of the multiplicand, and so the multiplicand and product, instead of being mere numbers, may be things of some kind, while the multiplier must always be a mere number.
On the other hand, the Egyptians could not solve directly the problem of finding the multiplicand when the multiplier and product were given. At the present time we scarcely think of this as a new problem; in fact, we never notice the distinction between multiplier and multiplicand, often saying that we multiply two numbers together. Even when some of our numbers represent things, we always think of them as mere numbers while we are multiplying or dividing. The Egyptian method emphasized this distinction, even when all the quantities were mere numbers. However, it was known that the product then would be the same if the multiplier and the multiplicand were interchanged, and sometimes, even in the first kind of multiplication, they were interchanged, simply because this would make the multiplication easier.[3] When the multiplier and product were given the Egyptians usually made this interchange, thinking of all their quantities as mere numbers;[4] that is, they changed their problem into one in which the given multiplier became the multiplicand and the required multiplicand was to be obtained as a multiplier, only at the end to be interpreted as the problem required. Problems 30-34 are particularly good illustrations of this method; also some of the pefsu problems (69-78), where the notation sometimes
- ↑ To use Gunn’s expression (pages 129 and 134) we might say, “Multiply by the number itself and dot the result.”
- ↑ This rule is illustrated in Problems 34 and 70, and all through the table for the division of 2 by odd numbers. The formation of this table is for each number a multiplication problem of the second kind, and, except for the cases of 3, 35, and 91, the three steps of the process are always present.
- ↑ This is done in the third step in the solutions of Problems 24-27.
- ↑ In Problem 55, instead of regarding the number of the given product as a mere number, the author regards the given multiplier, now taken as a multiplicand, as representing things of the same kind as the product. See notes to that problem.
