242 9. MEASUREMENTFIGURE 3. Dissection of a frustum of a pyramid.a length to an area or multiplying two areas. For that reason, our discussion of
such problems is postponed to Chapter 13.2.1. The Pythagorean theorem. In contrast to the case of Egypt, there is clear
proof that the Mesopotamians knew the Pythagorean theorem in full generality at
least 1000 years before Pythagoras. They were thus already on the road to finding
more abstract properties of geometric figures than mere size. Of course, this theo-
rem was known at an early date in India and China, so that one cannot say certainly
where the earliest discovery was and whether the appearance of this theorem in dif-
ferent localities was the result of independent discovery or transmission. But as far
as present knowledge goes, the earliest examples of the use of the "Pythagorean"
principle that the square on the hypotenuse of a right triangle equals the sum of
the squares on the other two legs occur in the cuneiform tablets. Specifically, the
old Babylonian text known as BM 85 196 contains a problem that has appeared in
algebra books for centuries. We give it below as Problem 9.4. In this problem we
are dealing with a right triangle of hypotenuse 30 with one leg equal to 30 —6 = 24.
Obviously, this is the famous 3-4-5 right triangle with all sides multiplied by 6.
Obviously also, the interest in this theorem was more numerical than geometric.
How often, after all, are we called upon to solve problems of this type in everyday
life?
How might the Pythagorean theorem have been discovered? The following
hypothesis was presented by Allman (1889, pp. 35-37), who cited a work (1870)
by Carl Anton Bretschneider (1808-1878). Allman thought this dissection was due
to the Egyptians, since, he said, it was done in their style. If he was right, the
Egyptians did indeed discover the theorem.
Suppose that you find it necessary to construct a square twice as large as a
given square. How would you go about doing so? (This is a problem the Platonic
Socrates poses in the dialogue Meno.) You might double the side of the square, but
you would soon realize that doing so actually quadruples the size of the square. If
you drew out the quadrupled square and contemplated it for a while, you might be