And next, shall we enquire whether the kindred science also concerns us?
You mean geometry?
Exactly so.
Clearly, he said, we are concerned with that part of geometry which relates to war; for in pitching a camp, or taking up a position, or closing or extending the lines of an army, or any other military manoeuvre, whether in actual battle or on a march, it will make all the difference whether a general is or is not a geometrician.
Yes, I said, but for that purpose a very little of either geometry or calculation will be enough; the question relates rather to the greater and more advanced part of geometry--whether that tends in any degree to make more easy the vision of the idea of good; and thither, as I was saying, all things tend which compel the soul to turn her gaze towards that place, where is the full perfection of being, which she ought, by all means, to behold.
True, he said.
Then if geometry compels us to view being, it concerns us; if becoming only, it does not concern us?
Yes, that is what we assert.
Yet anybody who has the least acquaintance with geometry will not deny that such a conception of the science is in flat contradiction to the ordinary language of geometricians.
How so?
They have in view practice only, and are always speaking?
in a narrow and ridiculous manner, of squaring and extending and applying and the like--they confuse the necessities of geometry with those of daily life; whereas knowledge is the real object of the whole science.
Certainly, he said.
Then must not a further admission be made?
What admission?
That the knowledge at which geometry aims is knowledge of the eternal, and not of aught perishing and transient.
That, he replied, may be readily allowed, and is true.
Then, my noble friend, geometry will draw the soul towards truth, and create the spirit of philosophy, and raise up that which is now unhappily allowed to fall down.
Nothing will be more likely to have such an effect.
Then nothing should be more sternly laid down than that the inhabitants of your fair city should by all means learn geometry.

Moreover the science has indirect effects, which are not small.
Of what kind?
he said.
There are the military advantages of which you spoke, I said; and in all departments of knowledge, as experience proves, any one who has studied geometry is infinitely quicker of apprehension than one who has not.
Yes indeed, he said, there is an infinite difference between them.
Then shall we propose this as a second branch of knowledge which our youth will study?
Let us do so, he replied.
And suppose we make astronomy the third--what do you say?
I am strongly inclined to it, he said; the observation of the seasons and of months and years is as essential to the general as it is to the farmer or sailor.
I am amused, I said, at your fear of the world, which makes you guard against the appearance of insisting upon useless studies; and I quite admit the difficulty of believing that in every man there is an eye of the soul which, when by other pursuits lost and dimmed, is by these purified and re-illumined; and is more precious far than ten thousand bodily eyes, for by it alone is truth seen.

Now there are two classes of persons: one class of those who will agree with you and will take your words as a revelation; another class to whom they will be utterly unmeaning, and who will naturally deem them to be idle tales, for they see no sort of profit which is to be obtained from them.

And therefore you had better decide at once with which of the two you are proposing to argue.

You will very likely say with neither, and that your chief aim in carrying on the argument is your own improvement; at the same time you do not grudge to others any benefit which they may receive.
I think that I should prefer to carry on the argument mainly on my own behalf.
Then take a step backward, for we have gone wrong in the order of the sciences.
What was the mistake?
he said.
After plane geometry, I said, we proceeded at once to solids in revolution, instead of taking solids in themselves; whereas after the second dimension the third, which is concerned with cubes and dimensions of depth, ought to have followed.
That is true, Socrates; but so little seems to be known as yet about these subjects.
Why, yes, I said, and for two reasons:--in the first place, no government patronises them; this leads to a want of energy in the pursuit of them, and they are difficult; in the second place, students cannot learn them unless they have a director.