- MESOPOTAMIA 241
Q R s<FIGURE 2. Dissection of a cube into two square pyramids and two tetrahdra.three pieces that the scribe added to get the volume of the frustum in a way that
is not terribly implausible.
It goes without saying that the last few paragraphs and Figs. 2 and 3 are
conjectures, not facts of history. We do not know how the Egyptians discovered
that the volume of a pyramid is one-third the volume of a prism of the same base and
height or how they found the volume of a frustum. The little story just presented
is merely one possible scenario.
2. Mesopotamia
Mesopotamian geometry, like its Egyptian counterpart, was regarded more as an
application of mathematics than as mathematics proper. The primary emphasis
was on areas and volumes. However, the Mesopotamian tablets suggest a very
strong algebraic component. Many of the problems that are posed in geometric
garb have no apparent practical application but are very good exercises in algebra.
For example, British Museum tablet 13901 contains the following problem: Given
two squares such that the side of one is two-thirds that of the other plus 5 GAR
and whose total area is 25,25 square GAR, what are the sides of the squares?
Where in real life would one encounter such a problem? The tablet itself gives
no practical context, and we conclude that this apparently geometric problem is
really a problem in algebraic manipulation of expressions. As Neugebauer states
(1952, p. 41), "It is easy to show that geometrical concepts play a very secondary
part in Babylonian algebra, however extensively a geometrical terminology may be
used." Both Neugebauer and van der Waerden (1963, p. 72) point out that the
cuneiform tablets contain operations that are geometrically absurd, such as adding