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 
u 
1623