The ignoraunte multitude doeth, but as it was euer wonte, enuie that knoweledge, whiche thei can not attaine, and wishe all men ignoraunt, like unto themself. . .

Yea, the pointe in Geometrie, and the unitie in Arithmetike, though bothe be undiuisible, doe make greater woorkes, & increase greater multitudes, then the brutishe bande of ignoraunce is hable to withstande. . .

But yet one commoditie moare. . . I can not omitte. That is the filying, sharpenyng, and quickenyng of the witte, that by practice of Arithmetike doeth insue. It teacheth menne and accustometh them, so certainly to remember thynges paste: So circumspectly to consider thynges presente: And so prouidently to forsee thynges that followe: that it maie truelie bee called the File of witte.

"Suppose now, Glaucon, someone were to ask them, 'My good friends, what numbers are these you are talking about, in which the one is such as you postulate, each unity equal to every other without the slightest difference and admitting no division into parts?' What do you think would be their answer?"

"This, I think--that they are speaking of units which can only be conceived by thought, and which it is not possible to deal with in any other way."

"You see, then, my friend," said I, "that this branch of study really seems to be indispensable for us, since it plainly compels the soul to employ pure thought with a view to truth itself."

"It most emphatically does."

"Again, have you ever noticed this, that natural reckoners are by nature quick in virtually all their studies? And the slow, if they are trained and drilled in this, even if no other benefit results, all improve and become quicker than they were1?"

"It is so," he said.

"And, further, as I believe, studies that demand more toil in the learning and practice than this we shall not discover easily nor find many of them."

"You will not, in fact."

"Then, for all these reasons, we must not neglect this study, but must use it in the education of the best endowed natures..."

"I agree", he said.

"Assuming this one point to be established," I said, "let us in the second place consider whether the study that comes next2 is suited to our purpose."

"What is that? Do you mean geometry," he said.

"Precisely that," said I.

"So much of it," he said, "as applies to the conduct of war is obviously suitable. For in dealing with encampments and the occupation of strong places and the bringing of troops into column and line and all the other formations of an army in actual battle and on the march, an officer who had studied geometry would be a very different person from what he would be if he had not."

"But still," I said, "for such purposes a slight modicum of geometry and calculation would suffice. What we have to consider is whether the greater and more advanced part of it tends to facilitate the apprehension of the idea of good. That tendency, we affirm, is to be found in all studies that force the soul to turn its vision round to the region where dwells the most blessed part of reality, which it is imperative that it should behold."

"You are right," he said.

"Then if it compels the soul to contemplate essence, it is suitable; if genesis, it is not."

"So we affirm."