A history of Greek mathematics - Wilbourhall.org

XIV
CONIC SECTIONS. APOLLONIUS OF PERGA
A. HISTORY OF CONICS UP TO APOLLONIUS
Discovery of the conic sections by Menaechmus.
We have seen that Menaechmus solved the problem of the
two mean proportionals (and therefore the duplication of
the cube) by means of conic sections, and that he is credited
with the discovery of the three curves ; for the epigram of
Eratosthenes speaks of ' the triads of Menaechmus ', whereas
of course only two conies, the parabola and the rectangular
hyperbola, actually appear in Menaechmus's solutions. The
question arises, how did Menaechmus come to think of obtaining
curves by cutting a cone 1 On this we have no information
whatever. Democritus had indeed spoken of a section of
a cone parallel and very near to the base, which of course
would be a circle, since the cone would certainly be the right
circular cone. But it is probable enough that the attention
of the Greeks, whose observation nothing escaped, would be
attracted to the shape of a section of a cone or a cylinder by
a plane obliquely inclined to the axis when it occurred, as it
often would, in real life ; the case where the solid was cut
right through, which would show an ellipse, would presumably
be noticed first, and some attempt would be made to
investigate the nature and geometrical measure of the elongation
of the figure in relation to the circular sections of the
same solid ; these would in the first instance be most easily
ascertained when the solid was a right cylinder ; it would
then be a natural question to investigate whether the curve
arrived at by cutting the cone had the same property as that
obtained by cutting the cylinder. As we have seen, the