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; *525* 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. *526* 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 {ciii} 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. *527* 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 soul, and create the mind of philosophy; it should raise up what has fallen down, not to speak of lesser uses in war and military tactics, and in the improvement of the faculties.