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so that P lies on a parabola which has 0 for vertex, OM for
axis, and OB for latus rectum,
(2) the similar relation AO . ON = OM 2, = PN 2 ,
so that P lies on a parabola which has 0 for vertex, ON for
axis, and OA for latus rectum.
In order therefore to find P, we have only to construct the
two parabolas with OM, ON for axes and OB, OA for latera
recta respectively ; the intersection of the two parabolas gives
a point P such that
AO.PN = PN: PM = PM: OB,
and the problem is solved.
(We shall see later on that Menaechmus did not use the
names parabola and hyperbola to describe the curves, those
names being due to Apollonius.)
(e) The solution attributed to Plato.
This is the first in Eutocius’s arrangement of the various
solutions reproduced by him. But there is almost conclusive
reason for thinking that it is wrongly attributed to Plato.
No one but Eutocius mentions it, and there is no reference to
it in Eratosthenes’s epigram, whereas, if a solution by Plato
had then been known, it could hardly fail to have been
mentioned along with those of Archytas, Menaechmus, and
Eudoxus. Again, Plutarch says that Plato told the Delians
that the problem of the two mean proportionals was no easy
one, but that Eudoxus or Helicon of Cyzicus would solve it
for them; he did not apparently propose to attack it himself.
And, lastly, the solution attributed to him is mechanical,
whereas we are twice told that Plato objected to mechanical
solutions as destroying the good of geometry. 1 Attempts
have been made to reconcile the contrary traditions. It is
argued that, while Plato objected to mechanical solutions on
principle, he wished to show how easy it was to discover
such solutions and put forward that attributed to him as an
illustration of the fact. I prefer to treat the silence of
Eratosthenes as conclusive on the point, and to suppose that
the solution was invented in the Academy by some one con
temporary with or later than Menaechmus.
1 Plutarch, Quaest. Conviv. 8. 2. l,p. 718 e,f; Vita Marcelli, c. 14.5.
