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

As is the case with a great deal of interesting mathematics, the conic sections are believed to have been discovered in an attempt to solve a problem, a problem which on the surface seems to have nothing to do with conic sections. The problem is the problem of doubling the cube, one of the three famous problems of antiquity. While we are not certain as to the exact moment of discovery, most authors attribute it to Menaechmus (4th century B.C.) [8, pp. 251-253]. The evidence for this is based on a work of Eutocius (6th century A.D.), Commentary on Archimedes’ Sphere and Cylinder. In this work Eutocius describes solutions to the problem of doubling the cube, that is, constructing a cube with twice the volume of a given cube.

As we shall now see, this problem is reduced to the problem of finding two mean proportionals between two given lengths. To see this, consider a cube of side a. We wish to construct the lengths x and y (the two mean proportionals) such that

a : x = x : y = y : 2a.

Before we see how Menaechmus constructed these mean proportionals, we must review the meaning of the "latus rectum" of a conic. The latus rectum of a conic is the chord through the focus parallel to the directrix. (The focus and the directrix can serve to define a conic section.) In the diagram below, you can see how the latus rectum controls the "opening" of a parabola.

What follows is Menaechmus' construction, according to Eutocius, of the two mean proportionals between two given lengths. (Note that Menaechmus did not use the terms "parabola" and "hyperbola." They were coined later by Apollonius.)