A history of Greek mathematics - Wilbourhall.org

;
200 SUCCESSORS OF THE GREAT GEOMETERS
The second name is that of Diocles. We have already
(vol. i,
pp. 264-6) seen him as the discoverer of the curve
known as the cissoid, which he used to solve the problem
of the two mean proportionals, and also (pp. 47-9 above)
as the author of a method of solving the equivalent of
a certain cubic equation by means of the intersection
of an ellipse and a hyperbola. We are indebted for our
information on both these subjects to Eutocius, 1 who tells
us that the fragments which he quotes came from Diocles's
work rrepl nuptiais, On burning-mirrors. The connexion of
the two things with the subject of this treatise is not obvious,
and we may perhaps infer that it was a work of considerable
scope. What exactly were the forms of the burning-mirrors
discussed in the treatise it is not possible to say, but it is
probably safe to assume that among them were concave
mirrors in the forms (1) of a sphere, (2) of a paraboloid, and
(3) of the surface described by the revolution of an ellipse
about its major axis. The author of the Fragmentum mathematicum
Bobiense says that Apollonius in his book On the
burning-mirror discussed the case of the concave spherical
mirror, showing about what point ignition would take place
and it is certain that Apollonius was aware that an ellipse has
the property of reflecting all rays through one focus to the
other focus. Nor is it likely that the corresponding property
of a parabola with reference to rays parallel to the axis was
unknown to Apollonius. Diocles therefore, writing a century
or more later than Apollonius, could hardly have failed to
deal with all three cases. True, Anthemius (died about
A. D. 534) in his fragment on burning-mirrors says that the
ancients, while mentioning the usual burning-mirrors and
saying that such figures are conic sections, omitted to specify
which conic sections, and how produced, and gave no geometrical
proofs of their properties. But if the properties
were commonly known and quoted, it is obvious that they
must have been proved by the ancients, and the explanation
of Anthemius's remark is presumably that the original works
in which they were proved (e.g. those of Apollonius and
Diocles) were already lost when he wrote.
There appears to
be no trace of Diocles's work left either in Greek or Arabic,
1
Eutocius, loc. cit., p. 66. 8 sq., p. 160. 3 sq.