288
PLATO
business of the cylinders of Archytas or the cutting of the
cone in the three curves of Menaechmus It would therefore
be quite possible for Plato to regard Archytas and Menaechmus
as having given constructions that were ultra-mechanical, since
they were more mechanical than the ordinary constructions by
means of the straight line and circle; and even the latter, which
alone are required for the processes of ‘ squaring ’, ‘ applying
(a rectangle) ’ and ‘ adding ’, are according to Plato no part of
theoretic geometry. This banning even .of simple constructions
from true geometry seems, incidentally, to make it impossible
to accept the conjecture of Hankel that we owe to Plato the
limitation, so important in its effect on the later development
of geometry, of the instruments allowable in constructions to
the ruler and compasses. 1 Indeed, there are signs that the
limitation began before Plato’s time (e. g. this may be the
explanation of the two constructions attributed to Oenopides),
although no doubt Plato’s influence would help to keep the
restriction in force; for other instruments, and the use of
curves of higher order than circles in constructions, were
expressly barred in any case where the ruler and compasses
could be made to serve (cf. Pappus’s animadversion on a solu
tion of a ‘ plane ’ problem by means of conics in Apollonius’s
Conics, Book Y).
Contributions to the philosophy of mathematics.
We find in Plato’s dialogues what appears to be the first
serious attempt at a philosophy of mathematics. Aristotle
says that between sensible objects and the ideas Plato placed
‘things mathematical’ (ra ¡ia6r¡\xaTLKa), which differed from
sensibles in being eternal and unmoved, but differed again
from the ideas in that there can be many mathematical
objects of the same kind, while the idea is one only; e. g. the
idea of triangle is one, but there may be any number of
mathematical triangles as of visible triangles, namely the
perfect triangles of which the visible triangles are imper
fect copies. A passage in one of the Letters (No. 7, to the
friends of Dion) is interesting in this connexion. 2 Speaking
of a circle by way of example, Plato says there is (1) some-
1 Hankel, op. cit., p. 156.
2 Plato, Letters, 842 b, c, 848 A, B.
