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 Trepi TTvpeioov, 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 mathe 
maticum 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 geo 
metrical 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.