238 9. MEASUREMENTIt has been imagined that the Egyptians regarded one triangle
above all others, likening it to the nature of the universe. And in his
Republic Plato seems to have used it in arranging marriages. This
triangle has 3 on the vertical side, 4 on the base, and a hypotenuse
of 5, equal in square to the other two sides. It is to be imagined
then that it was constituted of the masculine on the vertical side,
and the feminine on the base; also, Osiris as the progenitor, Isis as
the receptacle, and Horus as the offspring. For 3 is the first odd
number and is a perfect number; the 4 is a square formed from an
even number of dyads; and the 5 is regarded as derived in one way
from the father and another way from the mother, being made up
of the triad and the dyad.Still further, Berlin Papyrus 6619 contains a problem in which one square equals
the sum of two others. It is hard to imagine anyone being interested in such
conditions without knowing the Pythagorean theorem. Against the conjecture,
we could note that the earliest Egyptian text that mentions a right triangle and
finds the length of all its sides using the Pythagorean theorem dates from about
300 BCE, and by that time the presence of Greek mathematics in Alexandria was
already established. None of the older papyri mention or use by implication the
Pythagorean theorem.
On balance, one would guess that the Egyptians did know the Pythagorean
theorem. However, there is no evidence that they used it to construct right angles,
as Cantor conjectured. There are much simpler ways of doing that (even involving
the stretching of ropes), which the Egyptians must have known. Given that the
evidence for this conjecture is so meager, why is it so often reported as fact? Simply
because it has been repeated frequently since it was originally made. We know
precisely the source of the conjecture, but that knowledge does not seem to reach
the many people who report it as fact.^5
Spheres or cylinders? Problem 10 of the Moscow Papyrus has been subject to
various interpretations. It asks for the area of a curved surface that is either half of
a cylinder or a hemisphere. In either case it is worth noting that the area is obtained
by multiplying the length of a semicircle by another length in order to obtain the
area. Finding the area of a hemisphere is an extremely difficult problem. Intuitive
techniques that work on flat or ruled surfaces break down, as shown in Problem 9.20.
If the Egyptians did compute this area, no one has given any reasonable conjecture
as to how they did so. The difficulty of this problem was given as one reason for
interpreting the figure as half of a cylinder. Yet the plain language of the problem
implies that the surface is a hemisphere. The problem was translated into German
by the Russian scholar V. V. Struve (1889-1965); the following is a translation from
the German:
The way of calculating a basket, if you are given a basket with an
opening of 4 2. 0, tell me its surface!(^5) This point was made very forcefully by van der Waerden (1963, p. 6). In his later book,
Geometry and Algebra in Ancient Civilizations, van der Waerden claimed that integer-sided right
triangles, which seem to imply knowledge of the Pythagorean theorem, are ubiquitous in the oldest
megalithic structures. Thus, he seems to imply that the Egyptians knew the theorem, but didn't
use it as Cantor suggested.