Page:The Rhind Mathematical Papyrus, Volume I.pdf/23

tells us which numbers are to be regarded as mere numbers, and which represent things of some kind.^[1]

But sometimes they wished to keep in mind the nature of the quantities in the problem, and then, instead of solving it as a problem in multiplication, they used a process of trial that has been called “false position” and will be explained below.

The Egyptians commenced their descending series with the quantity ²⁄₃, and it is interesting to note that in the papyrus of Akhmîm (Baillet, 1892), written in Greek about 600 A.D., the Egyptian system of unit fractions is still used, and with the same apparent exception of ²⁄₃. Griffith (volume 16, page 168), regarding the short line in the hieroglyphic sign as a half of the longer line, saw in it a symbolic expression for 1 divided by 1+¹⁄₂, but it is known now that at first the two strokes were of the same length, so that this idea applies, if at all, only to later Egyptian times.^[2]

Special Processes Emphasized in the Rhind Papyrus

I will mention three special processes:

1. A method in which a fractional expression is applied to some particular number;

2. The solution of problems by false position;

3. A process of completion used for determining the amount to be added to an approximation to a given number in order to get the number.

It may be well at this point to explain these processes somewhat carefully.

1. If we wish to add two groups of fractions, say ¹⁄₃ ¹⁄₅ and ¹⁄₅ ¹⁄₇, we reduce them to the common denominator 105. Then as many times as each denominator has to be multiplied to produce the common denominator, so many times must the numerator of that fraction be multiplied to produce the new numerator, and these new numerators,

1. In Problem 64 also this is shown by the notation. See page 30.
2. Sethe (1916, pages 91 ff.) finds some traces among the Egyptians, as well as with other ancient peoples, of a system of complementary fractions ²⁄₃, ³⁄₄, ⁴⁄₅, . . . They called these expressions “the 2 parts,” “the 3 parts,” etc., the 2 parts being what is left after taking away ¹⁄₃, and so with the others. Thus in the Hebrew Scriptures we read (Genesis 47, 24) that Joseph said to the Egyptians when they sold themselves and their land for food,“Ye shall give the fifth part unto Pharaoh and four parts shall be your own.” ²⁄₃ is the only fraction of this system to be found in the Rhind papyrus. In the papyrus of Akhmîm the sign for ²⁄₃ is always preceded by the singular article, τὸ, as if it were one part (Baillet, page 19).