A history of Greek mathematics - Wilbourhall.org

DIOCLES. PERSEUS 203
above shown, be proved to bring about ignition at
indicated.'
the point
Heiberg held that the style of this fragment is Byzantine
and that it is probably by Anthemius. Cantor conjectured
that here we might, after all, have an extract from Diocles's
work. Heiberg's supposition seems to me untenable because
of the author's use (1) of the ancient terms ' section of
a right-angled cone ' for parabola and ' diameter ' for axis
(to say nothing of the use of the parameter, of which there is
no word in the genuine fragment of Anthemius), and (2) of
'
the mixed
angles of contact '. Nor does it seem likely that
even Diocles, living a century after Apollonius, would have
spoken of the 'section of a right-angled cone' instead of a
parabola, or used the ' mixed ' angle of which there is only the
merest survival in Euclid. The assumption of the equality
of the two angles made by the curve with the tangent on
both sides of the point of contact reminds us of Aristotle's
assumption of the equality of the angles ' of a segment ' of
a circle as prior to the truth proved in Eucl. I. 5. I am
inclined, therefore, to date the fragment much earlier even
than Diocles. Zeuthen suggested that the property of the
paraboloidal mirror may have been discovered by Archimedes,
who, according to a Greek tradition, wrote Gatoptrica. This,
however, does not receive any confirmation in Ibn al-Haitham
or in Anthemius, and we can only say that the fragment at
least
goes back to an original which was probably not later
than Apollonius.
Perseus is only known, from allusions to him in Proclus, 1
as the discoverer and investigator of the spiric sections.
They
are classed by Proclus among curves obtained by cutting
solids, and in this respect they are associated with the conic
sections. We may safely infer that they were discovered
after the conic sections, and only after the theory of conies
had been considerably developed. This was already the case
in Euclid's time, and it is probable, therefore, that Perseus was
not earlier than Euclid. On the other hand, by that time
the investigation of conies had brought the exponents of the
subject such fame that it would be natural for mathematicians
to see whether there was not an opportunity for winning a
1
Proclus on Eucl. I, pp. 111. 23-112. 8, 356. 12. Cf. vol. i, p. 226,