The History of Mathematics: A Brief Course

400 13. PROBLEMS LEADING TO ALGEBRA

notion that multiplication is distributive over addition (another way of saying that

proportions are preserved). But of course, since multiplication was thought of in a

peculiar way in Egyptian culture, the algebraic reasoning was very likely as follows:

Such-and-such operations applied to 8 will yield 19. If I first add the seventh part

of 7 to 7, I will get 8 as a result. If I then perform those operations on 8, I will

get 19. Therefore, if I first perform those operations on 7, and then add the seventh

part of the result to itself, I will also get 19.

The computation is carried out by the standard Egyptian method. First find

the operations that must be performed on 8 in order to yield 19:

1 8

2 16*

2 4

4 2*

8 1 *

2 4 8 19 Result.

Next, perform these operations on 7:

1 7

2 14*

2 3 2

4 12 4*

8 2 4 8*

2 4 8 16 2 8 Result.

This is the answer. The scribe seems quite confident of the answer and does
not carry out the computation needed to verify that it works.
The Egyptian scribes were capable of performing operations more complicated
than mere proportion. They could take the square root of a number, which they
called a corner. The Berlin Papyrus 6619, contains the following problem (Gillings,
1972, p. 161):

The area of a square of 100 is equal to that of two smaller squares.

The side of one is 2 4 the side of the other. Let me know the sides

of the two unknown squares.

Here we are asking for two quantities given their ratio (|) and the sum of their
squares (100). The scribe assumes that one of the squares has side 1 and the other
has side 2 4. Since the resulting total area is 1 2 16, the square root of this quantity
is taken (14), yielding the side of a square equal to the sum of these two given
squares. This side is then multiplied by the correct proportionality factor so as to
yield 10 (the square root of 100). That is, the number 10 is divided by 1 4, giving 8
as the side of the larger square and hence 6 as the side of the smaller square. This
example, incidentally, was cited by van der Waerden as evidence of early knowledge
of the Pythagorean theorem in Egypt.