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 conics, the parabola and the rectangular 
hyperbola, actually appear in Menaechmus’s solutions. The 
question arises, how did Menaechmus come to think of obtain 
ing curves by cutting a cone ? On this wè have no informa 
tion 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 presum 
ably be noticed first, and some attempt would be made to 
investigate the nature and geometrical measure of the elonga 
tion 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