The History of Mathematics: A Brief Course

278 10. EUCLIDEAN GEOMETRY

Axis

S

R Q Ï (Vertex) R

Q

FIGURE 6. Sections of a cone. Top left: through the axis. Top

right: perpendicular to the axis. Bottom: perpendicular to the

generator OR at a point Q lying at distance u from the vertex O.

The fundamental relation is v^2 = h^2 + 2uh — w^2. The length h has

a fixed ratio to w, depending only on the shape of the triangle

We begin by looking at a general conic section, shown in Fig. 6. When a cone
is cut by a plane through its axis, the resulting figure is simply a triangle. The end
that we have left open by indicating with arrows the direction of the axis and two
generators in this plane would probably have been closed off by Menaechmus. If it is
cut by a plane perpendicular to the axis, the result is a circle. The conic section itself
is obtained as the intersection with a plane perpendicular to one of its generators at
a given distance (marked u in the figure) from the vertex. The important relation
needed is that between the length of a horizontal chord (double the length marked
v) in the conic section and its height (marked w) above the generator that has
been cut. Using only similar triangles and the fact that a half chord in a circle is
the mean proportional between the segments of the diameter through its endpoint,
Menaechmus would easily have derived the fundamental relation

Although we have written this relation as an equation with letters in it, Menaechmus
would have been able to describe what it says in terms of the lines v, u, h, and
w, and squares and Tectangles on them. He would have known the value of the
ratio h/w, which is determined by the shape of the triangle ROC. In our terms
h = uitan(y>/2), where ö is the vertex angle of the cone.

OCR.

(1)

v^2 = h^2 + 2uh - w^2.