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

CONIC SECTIONS 129
to be made shorter. It was to this request
that Menaechmus replied: "In the country
there are private and even royal roads, but
in geometry there is only one road for all."
This reply no doubt was true enough in
the sense in which it would have been immediately understood by Alexander. But if
Menaechmus thought that his proofs could
not be shortened, he was grievously mistaken; and most modern mathematicians
would be horribly bored, if they were compelled to study the Greek proofs of the properties of conic sections. Nothing illustrates
better the gain in power which is obtained
by the introduction of relevant ideas into a
science than to observe the progressive
shortening of proofs which accompanies the
growth of richness in idea. There is a certain type of mathematician who is always
rather impatient at delaying over the ideas
of a subject: he is anxious at once to get on
to the proofs of "important" problems. The
history of the science is entirely against him.
There are royal roads in science; but those
who first tread them are men of genius and
not kings.
The way in which conic sections first presented themselves to mathematicians was as
follows: think of a cone (cf. fig. 15), whose
vertex (or point) is V, standing on a circular
base STU. For example, a conical shade to