THE PHILOSOPHY OF MATHEMATICS 289
thing called a circle and known by that name; next there
is (2) its definition as that in which the distances from its
extremities in all directions to the centre are always equal,
for this may be said to be the definition of that to which the
names ‘ round ’ and ‘ circle ’ are applied; again (3) we have
the circle which is drawn or turned: this circle is perishable
and perishes; not so, however, with (4) avrbs 6 kvkXos, the
essential circle, or the idea of circle: it is by reference to
this that the other circles exist, and it is different from each
of them. The same distinction applies to anything else, e. g.
the straight, colour, the good, the beautiful, or any natural
or artificial object, fire, water, &c. Dealing separately with
the four things above distinguished, Plato observes that there
is nothing essential in (1) the name: it is merely conventional;
there is nothing to prevent our assigning the name * straight ’
to what we now call £ round ’ and vice versa; nor is there any
real definiteness about (2) the definition, seeing that it too
is made up of parts of speech, nouns and verbs. The circle
(3), the particular circle drawn or turned, is not free from
admixture of other things: it is even full of what is opposite
to the true nature of a circle, for it will anywhere touch
a straight line ’, the meaning of which is presumably that we
cannot in practice draw a circle and a tangent with only one
point common (although a mathematical circle and a mathe
matical straight line touching it meet in one point only). It
will be observed that in the above classification there is no
place given to the many particular mathematical circles which
correspond to those which we draw, and are intermediate
between these imperfect circles and the idea of circle which
is one only.
(a) The hypotheses of mathemoMcs.
The hypotheses ol mathematics are discussed by Plato in
the Republic.
1 1 think you know that those who occupy themselves with
geometries and calculations and the like take for granted the
odd and the even, figures, three kinds of angles, and other
things cognate to these in each subject; assuming these things
as known, they take them as hypotheses and thenceforward
they do not feel called upon to give any explanation with
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1623