Diocles

c. 240-c. 180 b.c.

Greek Mathematician

The details of his life are virtually unknown, and Diocles is remembered almost entirely for a fragmentary manuscript entitled On Burning Mirrors. In it he discussed not only the physical problem referred to in the title, but such subjects as cutting sphere with a plane, as well as the famous Delian problem of doubling the cube.

On Burning Mirrors may actually have been a collection of three separate short works, combined under a single title that does not reflect the whole. In any case, the book consisted of 16 geometric propositions, most of which involved conics.

In the first proposition, Diocles put forth the focal property of the parabola, and in propositions 2 and 3 presented the properties of spherical mirrors. The next two propositions show the focus directrix parabola construction, which as with the statement on the focal property of that shape are innovations credited to Diocles. In propositions 7 and 8, Diocles examined a problem first put forth by Archimedes (c. 287-212 b.c.), on how to cut a sphere with a plane.

Propositions 10 through 12 offered Diocles's solution to the Delian problem. The latter had long perplexed Greek mathematicians who, using only a straightedge and compass, attempted to double the volume of a cube. Today this problem could be expressed in simple terms: find length x for a cube such that x³ = 2a³, where a is the length of the known cube. The Greeks, however, possessed no such sophisticated algebraic notation, and attempted to work the problem geometrically. Diocles's solution to the problem involved a special curve called a cissoid, which he used to find two mean proportionals. The use of the cissoid, however, involved more than a straightedge and compass, and in fact by the nineteenth century a.d. mathematicians recognized that it was impossible to solve the problem using only those tools.

In a 1976 article for a journal on the history of mathematics and the sciences, G. J. Toomer made a case for Diocles, and not his contemporary Apollonius of Perga (c. 262-c. 190 b.c.), as the originator of such familiar terms as hyperbola, parabola, and ellipse. In any case, On Burning Mirrors discussed a number of interesting subjects, not least of which was the one contained in the title. In this regard, Diocles put forth the problem of finding a mirror such that the reflected light forms certain curves, questions that would have implications for the creation of sundials.

JUDSON KNIGHT