Page:The Rhind Mathematical Papyrus, Volume I.pdf/52

area of a circle and its diameter. He does not explain how he discovered this relation, but from the fact that this problem in volume precedes 48, which states the relation for areas, it may be supposed that he determines the volume of a cylinder by some such process as the following:

He would quite possibly start with a cylinder of 9 units diameter because 9 with the Egyptians was a very important number and represented a group of the principal divinities.^[1] He would then construct square prisms of the same height but of different bases, and he would find that the cylinder filled with water would almost exactly fill the prism whose side was 8 units. He was then able to judge that the base of the prism would be determined by subtracting from the diameter its ¹⁄₉, and this gives the value of π as 3.1605. The fact that in this case a whole number for the side of the prism gives such a close approximation was probably a happy accident. In the next problem, having determined this relation. he takes for diameter 10, which does not give him a whole number for the side of the prism. This method, which we have already noted (page 8), of taking a simple numerical example and generalizing from that. is not from the mathematical point of view strictly legitimate and is liable to error unless the generalization is afterwards proved.

Determination of Areas

In Problem 48 is indicated the relation between a circle and its circumscribing square.^[2]

In Problems 51-53 the Egyptian determines the area of a triangle by multiplying ¹⁄₂ of its base, and the area of a trapezoid by multiplying ¹⁄₂ of the sum of its bases, by the length of a line (meret) which, so far as our present knowledge goes, might be either the side or a line representing the altitude. In the latter case he would be correct. In case the triangle is isosceles with a narrow base as compared with its height, he would be nearly correct, even if meret means side. Personally I am inclined to think that this word does mean side in geometry, and that the author intended to consider only isosceles triangles with narrow bases. In Problem 51 the base is comparatively narrow, 4, with meret equal to 10.^[3] In 53 it seems to be 4+¹⁄₂ with meret 14, while in 52 the

1. Sethe (1916) page 38. There were a greater and a lesser “ennead,” the personnel of which varied in different ages and localities.
2. Such a figure occurs on a Babylonian tablet of 2000 B.C.; compare Gadd, 1922.
3. The error is but little over 2%. In the other cases it is still smaller.