The History of Mathematics: A Brief Course

1. THE EARLIEST GREEK GEOMETRY 279

A

Ε

d^2 = 2AE

v^2 =w(w + 2d)

{x + y)^2 = 2(w + d)^2

÷^2 + y^2 = (w + d)^2 + V^2

2xy = d^2

FIGURE 7. One of Menaechmus' solutions to the problem of two

mean proportionals, as reported by Eutocius.

The simplest case is that of the parabola, where the vertex angle is 90° and
h — w. In that case the relation between õ and w is

é;^2 = 2uw.

In the problem of putting two mean proportionals Β and à between two lines A
and E, Menaechmus took this u to be \A, so that v^2 = Aw.
The hyperbola Menaechmus needed for this problem was a rectangular hyper-
2 2*
bola, which results when the triangle ROC is chosen so that RC = 20C , and
therefore OR^2 = 30C. Such a triangle is easily constructed by extending one
side of a square to the same length as the diagonal and joining the endpoint to the
opposite corner of the square. In any triangle of this shape the legs are the side
and diagonal of a square. For that case Menaechmus would have been able to show
that the relation
v^2 = w{w + 2d)
holds, where d is the diagonal of a square whose side is u. To solve the prob-
lem of two mean proportionals, Menaechmus took u = V'AE, that is, the mean
proportional between A and E. Menaechmus' solution is shown in Fig. 7.
The solution fits perfectly within the framework of Pythagorean-Euclidean
geometry, yet people were not satisfied with it. The objection to it was that the
data and the resulting figure all lie within a plane, but the construction requires
the use of cones, which cannot be contained in the plane.
Trisecting the angle. The practicality of trisecting an angle is immediately evident:
It is the vital first step on the way to dividing a circular arc into any number of
equal pieces. If a right angle can be divided into ç equal pieces, a circle also can
be divided into ç equal pieces, and hence the regular n-gon can be constructed.
The success of the Pythagoreans in constructing the regular pentagon must have
encouraged them to pursue this program. It is possible to construct the regular
n-gon for ç = 3, 4, 5, 6, 8, 10, but not 7 or 9. The number 7 is awkward, being the
only prime between 5 and 10, and one could expect to have difficulty constructing

xy = AE x^2 = Ay