Before BOOK VIII.
Project Gutenberg EPUB edition
Plato and Benjamin Jowett
That is very true.
526 Now, suppose a person were to say to them: O my friends, what are these wonderful numbers about which you are reasoning, in which, as you say, there is a unity such as you demand, and each unit is equal, invariable, indivisible,—what would they answer?
They would answer, as I should conceive, that they were speaking of those numbers which can only be realized in thought.
Then you see that this knowledge may be truly called Bnecessary, necessitating as it clearly does the use of the pure intelligence in the attainment of pure truth?
Yes; that is a marked characteristic of it.
The arithmetician is naturally quick, and the study of arithmetic gives him still greater quickness. And have you further observed, that those who have a natural talent for calculation are generally quick at every other kind of knowledge; and even the dull, if they have had an arithmetical training, although they may derive no other advantage from it, always become much quicker than they would otherwise have been.
Very true, he said.
C And indeed, you will not easily find a more difficult study, and not many as difficult.
You will not.
And, for all these reasons, arithmetic is a kind of knowledge in which the best natures should be trained, and which must not be given up.
I agree.
Let this then be made one of our subjects of education. And next, shall we enquire whether the kindred science also concerns us?
You mean geometry?
Exactly so.
DGeometry has practical applications; 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.
these however are trifling in comparison with that greater part of the science which tends towards the good, Yes, I said, but for that purpose a very little of either geometry or calculation will be enough; the question relates 229 rather to the greater and more advanced part of geometry—Ewhether 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?
527 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 Breal object of the whole science.
Certainly, he said.
Then must not a further admission be made?
What admission?
and is concerned with the eternal. 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.
C 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. 230
Then shall we propose this as a second branch of knowledge which our youth will study?
Let us do so, he replied.
D And suppose we make astronomy the third—what do you say?
Astronomy, like the previous sciences, is at first praised by Glaucon for its practical uses. 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 Eother 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 528to 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.
Source notes
4 Meaning either (1) that they integrate the number because they deny the possibility of fractions; or (2) that division is regarded by them as a process of multiplication, for the fractions of one continue to be units.