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

Rh tells us which numbers are to be regarded as mere numbers, and which represent things of some kind.

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.

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 $2/3$ $3/4$ and $4/5$ $1/3$, 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,