Can You Really Derive Conic Formulae from a Cone? - Conic Sections of Apollonius - Introduction

Little is known of the life of Apollonius. The information we have is from his own writings and, once again, from Eutocius, in his work Commentary on Apollonius' Conics. Apollonius says of his work Conics,

Of the eight books the first four belong to a course in the elements. The first book contains the generation of the three sections and of the opposite branches, and the principal properties in them worked out more fully and universally than in the writings of others [1, p. 603].

Eutocius illuminates this further by stating

Apollonius of Perga proved generally that all the sections can be obtained in any cone, whether right or scalene, according to different relations of the plane to the cone [15, p. 279].

Indeed, Apollonius redefined the cone:

If from a point a straight line is joined to the circumference of a circle which is not in the same plane with the point, and the line is produced in both directions, and if, with the point remaining fixed, the straight line being rotated about the circumference of the circle returns to the same place from which it began, then the generated surface composed of the two surfaces lying vertically opposite one another, each of which increases indefinitely as the generating straight line is produced indefinitely, I call a conic surface, and I call the fixed point the vertex, and the straight line drawn from the vertex to the center of the circle the axis [1, p. 604].

Above: Drag point B to generate a nappe of the cone

Compare this with Euclid's definition of a cone on page 4.

Unlike earlier geometers who intersected planes at right angles to different shaped cones, Apollonius varied the plane, not the cone. The angle of intersection of the plane to the cone determined the conic. Let's see how this generates the three conic sections:

Parabola (plane slices the cone at a right angle)
Right click and drag to change the viewing angle.

Ellipse (plane slices the cone at an acute angle)
Right click and drag to change the viewing angle.

Hyperbola (plane slices the cone at an obtuse angle)
Right click and drag to change the viewing angle.

The Conics is a tour de force of information about conic sections. It would take us too far afield to discuss all of its wonderful propositions, many of which today require the aid of calculus for proof. The interested reader can read Apollonius's own account of his work from his introduction to Book I. In this work Apollonius gives us the names "parabola," ellipse," and "hyperbola," and we will look at how he derived the symptom of each of these conics in the following sections.