- THE EARLIEST GREEK GEOMETRY^277
FIGURE 5. The three conic sections, according to Menaechmus.The three surfaces intersect in a point from which the two mean proportionals
can easily be determined. A later solution by Menaechmus may have arisen as a
simplification of Archytas' rather complicated construction. It requires intersecting
two cones, each having a generator parallel to a generator of the other, with a plane
perpendicular to both generators. These intersections form two conic sections, a
parabola and a rectangular hyperbola; where they intersect, they produce the two
mean proportionals.
If Eutocius is correct, the conic sections first appeared, but not with the names
they now bear, in the late fourth century BCE. Menaechmus created these sections
by cutting a cone with a plane perpendicular to one of its generators. When that
is done, the shape of the section depends on the vertex angle of the cone. If that
angle is acute, the section will be an ellipse; if it is a right angle, the section will be
a parabola; if it is obtuse, the section will be a hyperbola. In his commentary on
Archimedes' treatise on the sphere and cylinder, Eutocius tells how he happened to
find a work written in the Doric dialect which seemed to be a work of Archimedes.
He mentions in particular that instead of the word parabola, used since the time of
Apollonius, the author used the phrase section of a right-angled cone, and instead
of hyperbola, the phrase section of an obtuse-angled cone. Since Proclus refers to
"the conic section triads of Menaechmus," it is inferred that the original names of
the conic sections were oxytome (sharp cut), orthotome (right cut), and amblytome
(blunt cut), as shown in Fig. 5. However, Menaechmus undoubtedly thought of the
cone as the portion of the figure from the vertex to some particular circular base.
In particular, he wouldn't have thought of the hyperbola as having two nappes, as
we now do.
How Apollonius came to give them their modern names a century later is
described below. Right now we shall look at the consequences of Menaechmus'
approach and see how it enabled him to solve the problem of two mean proportion-
als. It is very difficult for a modern mathematician to describe this work without
breaking into modern algebraic notation, essentially using analytic geometry. It is
very natural to do so; for Menaechmus, if Eutocius reports correctly, 7 comes very
close to stating his theorem in algebraic language.
7 That is a big "if." Eutocius clearly had read Apollonius; Menaechmus, just as clearly, could not
have done so.