264
THE DUPLICATION ^OF THE CUBE
a
problem of the two mean proportionals by means of the points
of intersection of any two of these conics.
But, if we add the first two equations, we have
x 2 + y 2 — bx —ay = 0,
which is a circle passing through the points common to the
two parabolas x 2 = ay, y 2 = hx.
Therefore we can equally obtain a solution by means of
the intersections of the circle x 2 + y 2 — bx — ay = 0 and the
rectangular hyperbola xy = ab.
This is in effect what Pinion does, for, if AF, AG are the
coordinate axes, the circle x 2 + y 2 — hx — ay= 0 is the circle
BDHC, and xy = ah is the rectangular hyperbola with
AF, AG as asymptotes and passing through D, which
hyperbola intersects the circle again in H, a point such
that FD = HG.
(i) Diodes and the cissoid.
We gather from allusions to the cissoid in Proclus’s com
mentary on Eucl. I that the curve which Geminus called by
that name was none other than the curve invented by Diodes
and used by him for doubling the cube or finding two mean
proportionals. Hence Diodes must have preceded Geminus
(fl. 70 B.c.). Again, we conclude from the two fragments
preserved by Eutocius of a work by him, nepl nvpeicor, On
burning-mirrors, that he was later than Archimedes and
Apollonius. He may therefore have flourished towards the
end of the second century or at the beginning of the first
century b.c. Of the two fragments given by Eutocius one
contains a solution by means of conics of the problem of
dividing a sphere by a plane in such a way that the volumes
of the resulting segments shall be in a given ratio—a problem
equivalent to the solution of a certain cubic, equation—while
the other gives the solution of the problem of the two mean
proportionals by means of the cissoid.
Suppose that AB, DC are diameters of a circle at right
angles to one another. Let E, F be points on the quadrants
BD, BG respectively such that the arcs BE, BF are equal.
Draw EG, FH perpendicular to DC. Join GE, and let P be
the point in which GE, FH intersect.
