An introduction to mathematics, by A. N. Whitehead.

130 INTRODUCTION TO MATHEMATICS
an electric light is often an example of such
a surface. Now let the "generating" lines
which pass through V and lie on the surface
be all produced backwards; the result is a
double cone, and PQR is another circular
cross section on the opposite side of V to the
cross section STU. The axis of the cone
CVC' passes through all the centres of these
circles and is perpendicular to their planes,
which are parallel to each other. In the
diagram the parts of the curves which are
supposed to lie behind the plane of the paper
are dotted lines, and the parts on the plane
or in front of it are continuous lines. Now
suppose this double cone is cut by a plane
not perpendicular to the axis CVC', or at
least not necessarily perpendicular to it.
Then three cases can arise:(1) The plane may cut the cone in a
closed oval curve, such as ABA'B' which
lies entirely on one of the two half-cones.
In this case the plane will not meet the other
half-cone at all. Such a curve is called an
ellipse; it is an oval curve. A particular
case of such a section of the cone is when
the plane is perpendicular to the axis CVC',
then the section, such as STU or PQR, is a
circle. Hence a circle is a particular case of
the ellipse.
(2) The plane may be parallel to one of
the "generating" lines of the cone, as for