A history of Greek mathematics - Wilbourhall.org

204 SUCCESSORS OF THE GREAT GEOMETERS
like renown as discoverers of other curves to be obtained by
cutting well-known solid figures other than the cone and
cylinder. A particular case of one such solid figure, the
cnreipa, had already been employed by Archytas, and the more
general form of it would not unnaturally be thought of as
likely to give sections worthy of investigation. Since Geminus
is Proclus's authority, Perseus may have lived at any date from
Euclid's time to (say)
75 B.C., but the most probable supposition
seems to be that he came before Apollonius and .near to
Euclid in date.
The spire in one of its forms is what we call
a tore, or an
anchor- ring. It is generated by the revolution of a circle
about a straight line in its plane in such a way that the plane
of the circle always passes through the axis of revolution.
takes three forms according as the axis of revolution is
(a) altogether outside the circle, when the spire is open
(Sizyjis), (b) a tangent to the circle, when the surface is continuous
(avvexvs)' or (
c )
a chord of the circle, when it is interlaced
(efj.7r€7rXeyfiei/r]), or crossing-itself (kiraWdrTova-a) ; an
alternative name for the surface was KpiKos, a ring. i
It
Perseus
celebrated his discovery in an epigram to the effect that
'
Perseus on his discovery of three lines (curves) upon five
sections gave thanks to the gods therefor'. 1 There is some
doubt about the meaning of ' three lines upon five sections'
(Tpet? ypafipLas kirl irevre rouaTs). We gather from Proclus's
account of three sections distinguished by Perseus that the
plane of section was always parallel
or perpendicular to the plane which cuts
the tore symmetrically
like the division in a split-ring.
to the axis of revolution
It is difficult to interpret
the phrase if it means three curves made by five different
sections.
Proclus indeed implies that the three curves were
sections of the three kinds of tore
respectively (the open, the
closed, and the interlaced), but this is evidently a slip.
'
Tannery interprets the phrase as meaning three curves in
addition to five sections '. 2 Of these the five sections belong
to the open tore, in which the distance (c)
generating circle from the axis of revolution is
of the centre of the
greater than
the radius (a) of the generating circle. If d be the perpen-
1
Proclus on Eucl. I, p. 112. 2.
2
See Tannery, Memoires scientifiques, II, pp. 24-8.