A history of Greek mathematics Vol.II from Aristarchus to Diophantus by Heath, Thomas Little, Sir, 1921

DIOCLES. PERSEUS 203
1
Proclus on Eucl. I, 2
See Tannery, Memoires scientifiques, II, pp. 24-8.
204 SUCCESSORS OF THE GREAT GEOMETERS
above shown, be proved to bring about ignition at the point
like renown as discoverers of other curves to be obtained by
indicated.'
cutting well-known solid figures other than the cone and
Heiberg held that the style of this fragment is Byzantine
cylinder. A particular case of one such solid figure, the
and that it is probably by Anthemius. Cantor conjectured
cnreipa, had already been employed by Archytas, and the more
that here we might, after all, have an extract from Diocles's
general form of it would not unnaturally be thought of as
work. Heiberg's supposition seems to me untenable because
likely to give sections worthy of investigation. Since Geminus
of the author's use (1) of the ancient terms section of
' is Proclus's authority, Perseus may have lived at any date from
a right-angled cone ' for parabola and diameter ' ' for axis
Euclid's time to (say) 75 B.C., but the most probable supposition
seems to be that he came before Apollonius and .near to
(to say nothing of the use of the parameter, of which there is
no word in the genuine fragment of Anthemius), and (2) of
Euclid in date.
'
the mixed angles of contact '. Nor does it seem likely that
The spire in one of its forms is what we call a tore, or an
even Diocles, living a century after Apollonius, would have
anchor- ring. It is generated by the revolution of a circle
spoken of the 'section of a right-angled cone' instead of a
about a straight line in its plane in such a way that the plane
parabola, or used the mixed ' ' angle of which there is only the
of the circle always passes through the axis of revolution. It
merest survival in Euclid. The assumption of the equality
takes three forms according as the axis of revolution is
of the two angles made by the curve with the tangent on
(a) altogether outside the circle, when the spire is open
both sides of the point of contact reminds us of Aristotle's
(Sizyjis), (b) a tangent to the circle, when the surface is continuous
(avvexvs)' or
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
( c )
a chord of the circle, when it is interlaced
(efj.7r€7rXeyfiei/r]), or crossing-itself (kiraWdrTova-a) ; an
inclined, therefore, to date the fragment much earlier even
alternative name for the surface was KpiKos, a ring. i Perseus
than Diocles. Zeuthen suggested that the property of the
celebrated his discovery in an epigram to the effect that
paraboloidal mirror may have been discovered by Archimedes,
'
Perseus on his discovery of three lines (curves) upon five
who, according to a Greek tradition, wrote Gatoptrica. This,
sections gave thanks to the gods therefor'. 1 There is some
however, does not receive any confirmation in Ibn al-Haitham
doubt about the meaning of three
or in Anthemius, and we can only say that the fragment lines upon five sections'
at
(Tpet? ypafipLas kirl irevre rouaTs). We gather from Proclus's
least goes back to an original which was probably not later
account of three sections distinguished by Perseus that the
than Apollonius.
plane of section was always parallel to the axis of revolution
Perseus is only known, from allusions to him in Proclus, 1
or perpendicular to the plane which cuts the tore symmetrically
like the division in a split-ring. It is difficult to inter-
as the discoverer and investigator of the spiric sections. They
are classed by Proclus among curves obtained by cutting
pret the phrase if it means three curves made by five different
solids, and in this respect they are associated with the conic
sections. Proclus indeed implies that the three curves were
sections. We may safely infer that they were discovered
sections of the three kinds of tore respectively (the open, the
after the conic sections, and only after the theory of conies
closed, and the interlaced), but this is evidently a slip.
had been considerably developed. This was already the case
Tannery interprets
'
the phrase as meaning three curves in
in Euclid's time, and it is probable, therefore, that Perseus was
addition to five sections '. 2 Of these the five sections belong
not earlier than Euclid. On the other hand, by that time
to the open tore, in which the distance (c) of the centre of the
the investigation of conies had brought the exponents of the
generating circle from the axis of revolution is greater than
subject such fame that it would be natural for mathematicians
the radius (a) of the generating circle. If d be the perpento
see whether there was not an opportunity for winning a
1
Proclus on Eucl. I, p. 112. 2.