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

CONIC SECTIONS 131
example the plane of the curve D1AiD1' in
the diagram is parallel to the generating line
VS; the curve is still confined to one of the
half-cones, but it is now not a closed oval
curve, it goes on endlessly as long as the
generating lines of the half-cone are produced away from the vertex. Such a conic
section is called a parabola.
(3) The plane may cut both the halfcones, so that the complete curve consists of
two detached portions, or "branches" as
they are called; this case is illustrated by
the two branches G2A2G2' and L2A2'L2'
which together make up the curve. Neither
branch is closed, each of them spreading out
endlessly as the two half-cones are prolonged
away from the vertex. Such a conic section
is called a hyperbola.
There are accordingly three types of conic
sections, namely, ellipses, parabolas, and
hyperbolas. It is easy to see that, in a sense,
parabolas are limiting cases lying between
ellipses and hyperbolas. They form a more
special sort and have to satisfy a more particular condition. These three names are
apparently due to Apollonius of Perga (born
about 260 B.C., and died about 200 B.C.),
who wrote a systematic treatise on conic
sections which remained the standard work
till the sixteenth century.
It must at once be apparent how awkward