conic sections

The curves known as conic sections, the ellipse, hyperbola, and parabola, were investigated intensely in Greek mathematics. The most famous work on the subject was the Conics, in eight books by Apollonius of Perga, but conics were also studied earlier by Euclid and Archimedes, among others. Conic sections were important not only for purely mathematical endeavors such as the problem of doubling the cube, but also in other scientific matters such as burning mirrors and sundials. How the ancient theory of conics is to be understood also played a role in the general development of the historiography of Greek mathematics.

Apollonius’ Problem: A Study of Solutions and Their Connections

In Tangencies Apollonius of Perga showed how to construct a circle that is tangent to three given circles. More generally, Apollonius' problem asks to construct the circle which is tangent to any three objects that may be any combination of points, lines, and circles. The case when all three objects are circles is the most complicated case since up to eight solution circles are possible depending on the arrangement of the given circles. Within the last two centuries, solutions have been given by J. D. Gergonne in 1816, by Frederick Soddy in 1936, and most recently by David Eppstein in 2001. In this report, we illustrate the solution using the geometry software Cinderella™, survey some connections among the three solutions, and provide a framework for further study.