Note on the Sphero-Conic

Extract

(i.) A quadric cone is a cone having any conic for base, and a point O on the line perpendicular to the plane of the conic through the focus for vertex.

(ii.) A sphero-conic is the curve in which any quadric cone cuts a sphere whose centre is the vertex of the cone.

In an introductory course on space-geometry, where only a passing mention of sphero-conics seems possible, I have found it useful to show directly the fundamental property that the sum or difference of the focal distances is constant, instead of following the classical method, which consists in deducing the property by the use of infinitesimals, from the fact that these focal distances make equal angles with the curve; this latter fact being in effect a direct consequence of Desargue’s theorem.