A history of Greek mathematics - Wilbourhall.org

PERSEUS 205
dicular distance of the plane of section from the axis of rotation,
we can distinguish the following cases :
(1)
c + a>d>c. Here the curve is an oval.
(2) d = c: transition from case (1) to the next case.
(3) od>c — a. The curve is now a closed curve narrowest
in the middle.
(4) d = c — a. In this case the curve is the hi r ppopede
(horse-fetter), a curve in the shape of the figure of 8. The
lemniscate of Bernoulli is a particular case of
namely in which c = 2 a.
this curve, that
(5) c — a>d>0. In this case the section consists of two
ovals symmetrical with one another.
The three curves specified
by Proclus are those corresponding
to (1), (3) and (4).
When the tore is ' continuous ' or closed, c = a, and we have
sections corresponding to (1), (2) and (3) only; (4) reduces to
two circles touching one another.
But Tannery finds in the third, the interlaced, form of tore
three new sections corresponding to (1) (2) (3), each with an
oval in the middle. This would make three curves in addition
to the five sections, or eight curves in all. We cannot be
certain that this is the true explanation of the phrase in the
epigram ;
but it seems to' be the best suggestion that has
been made.
According to
Proclus, Perseus worked out the property of
his curves, as Nicomedes did that of the conchoid, Hippias
that of the quadratrix, and Apollonius those of the three
conic sections. That is, Perseus must have given, in some
form, the equivalent of the Cartesian equation by which we
can represent the different curves in question.
If we refer the
tore to three axes of coordinates at right angles to one another
with the centre of the tore as origin, the axis of y being taken
to be the axis of
revolution, and those of 0, x being perpendicular
to it in the plane bisecting the tore (making it a splitring),
the equation of the tore is
(x 2 -f y 2 + z 2 + c 2 — a 2 )
2
= 4 c 2 (z 2 + x 2 ),