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

136 INTRODUCTION TO MATHEMATICS
SP to PN (. e. SP is constant, where PN is
the perpendicular on the directrix from P,
and P is any point on the curve. Here we
have finally found the desired property of the
curves which does not require us to leave
the plane, and is stated uniformly for all
three curves. For ellipses, the ratio N is less
than 1, for parabolas it is equal to 1, and for
hyperbolas it is greater than 1.
When Pappus had finished his investigations, he must have felt that, apart from
minor extensions, the subject was practically
exhausted; and if he could have foreseen the
history of science for more than a thousand
years, it would have confirmed his belief.
Yet in truth the really fruitful ideas in connection with this branch of mathematics had
not yet been even touched on, and no one
had guessed their supremely important applications in nature. No more impressive
warning can be given to those who would
confine knowledge and research to what is
apparently useful, than the reflection that
conic sections were studied for eighteen hundred years merely as an abstract science,
without a thought of any utility other than
to satisfy the craving for knowledge on the
part of mathematicians, and that then at the
end of this long period of abstract study, they