guished from one another. Again, the sight beholds
great and small, but only in a confused chaos, and not until they are
distinguished does the question arise of their respective natures; we
are thus led on to the distinction between the visible and intelligible.
That was what I meant when I spoke of stimulants to the intellect; I was
thinking of the contradictions which arise in perception. The idea
of unity, for example, like that of a finger, does not arouse thought
unless involving some conception of plurality; but when the one is also
the opposite of one, the contradiction gives rise to reflection; an
example of this is afforded by any object of sight. All number has also
an elevating effect; it raises the mind out of the foam and flux of
generation to the contemplation of being, having lesser military and
retail uses also. The retail use is not required by us; but as our
guardian is to be a soldier as well as a philosopher, the military one
may be retained. And to our higher purpose no science can be better
adapted; but it must be pursued in the spirit of a philosopher, not of a
shopkeeper. It is concerned, not with visible objects, but with abstract
truth; for numbers are pure abstractions--the true arithmetician
indignantly denies that his unit is capable of division. When you
divide, he insists that you are only multiplying; his 'one' is not
material or resolvable into fractions, but an unvarying and absolute
equality; and this proves the purely intellectual character of his
study. Note also the great power which arithmetic has of sharpening the
wits; no other discipline is equally severe, or an equal test of general
ability, or equally improving to a stupid person.

Let our second branch of education be geometry. 'I can easily see,'
replied Glaucon, 'that the skill of the general will be doubled by his
knowledge of geometry.' That is a small matter; the use of geometry, to
which I refer, is the assistance given by it in the contemplation of the
idea of good, and the compelling the mind to look at true being, and not
at generation only. Yet the present mode of pursuing these studies,
as any one who is the least of a mathematician is aware, is mean and
ridiculous; they are made to look downwards to the arts, and not upwards
to eternal existence. The geometer is always talking of squaring,
subtending, apposing, as if he had in view action; whereas knowledge is
the real object of the study. It should elevate the so