But now, if we are concerned about the likeness of our faces,
we must consider whether he who speaks is a painter, or not.
I think we must.
Well, is Theodorus a painter?
Not so far as I know.
Nor a geometrician, either?
Oh yes, decidedly, Socrates.
And an astronomer, and an arithmetician, and a musician, and in general an educated man?
I think so.
Well then, if he says, either in praise or blame, that we have some physical resemblance, it is not especially worth while to pay attention to him.
Perhaps not.
But what if he should praise the soul of one of us for virtue and wisdom? Is it not worth while for the one who hears to examine eagerly the one who is praised, and for that one to exhibit his qualities with eagerness?
Certainly, Socrates.
Then, my dear Theaetetus, this is just the time for you to exhibit your qualities and for me to examine them; for I assure you that Theodorus, though he has praised many foreigners and citizens to me, never praised anyone as he praised you just now.
A good idea, Socrates; but make sure
that he was not speaking in jest.
That is not Theodorus’s way. But do not seek to draw back from your agreement on the pretext that he is jesting, or he will be forced to testify under oath; for certainly no one will accuse him of perjury. Come, be courageous and hold to the agreement.
I suppose I must, if you say so.
Now tell me; I suppose you learn some geometry from Theodorus?
Yes.
And astronomy and harmony and arithmetic?
I try hard to do so.
And so do I, my boy, from him and from any others who I think know anything about these things. But nevertheless, although in other respects I get on fairly well in them, yet I am in doubt about one little matter, which should be investigated with your help and that of these others. Tell me, is not learning growing wiser about that which one learns?
Of course.
And the wise, I suppose, are wise by wisdom.
Yes.
And does this differ at all from knowledge?
Does what differ?
Wisdom. Or are not people wise in that of which they have knowledge?
Of course.
Then knowledge and wisdom are the same thing?
Yes.
Well, it is just this that I am in doubt about and cannot fully grasp by my own efforts—what knowledge really is.
Can we tell that? What do you say? Who of us will speak first? And he who fails, and whoever fails in turn, shall go and sit down and be donkey, as the children say when they play ball; and whoever gets through without failing shall be our king and shall order us to answer any questions he pleases. Why are you silent? I hope, Theodorus, I am not rude, through my love of discussion and my eagerness to make us converse and show ourselves friends and ready to talk to one another.
That sort of thing would not be at all rude, Socrates; but tell one of the youths to answer your questions; for I am unused to such conversation and, moreover, I am not of an age to accustom myself to it. But that would be fitting for these young men, and they would improve much more than I; for the fact is, youth admits of improvement in every way. Come, question Theaetetus as you began to do, and do not let him off.
Well, Theaetetus, you hear what Theodorus says,
and I think you will not wish to disobey him, nor is it right for a young person to disobey a wise man when he gives instructions about such matters. Come, speak up well and nobly. What do you think knowledge is?
Well, Socrates, I must, since you bid me. For, if I make a mistake, you are sure to set me right.
Certainly, if we can.
Well then, I think the things one might learn from Theodorus are knowledge—geometry and all the things you spoke of just now—and also cobblery and
the other craftsmen’s arts; each and all of these are nothing else but knowledge.
You are noble and generous, my friend, for when you are asked for one thing you give many, and a variety of things instead of a simple answer.
What do you mean by that, Socrates?
Nothing, perhaps; but I will tell you what I think I mean. When you say cobblery you speak of nothing else than the art of making shoes, do you?
Nothing else.
And when you say carpentry? Do you mean anything else than the art of making wooden furnishings?
Nothing else by that, either.
Then in both cases you define that to which each form of knowledge belongs?
Yes.
But the question, Theaetetus, was not to what knowledge belongs, nor how many the forms of knowledge are; for we did not wish to number them, but to find out what knowledge itself really is. Or is there nothing in what I say?
Nay, you are quite right.
Take this example. If anyone should ask us about some common everyday thing, for instance, what clay is, and we should reply that it is the potters’ clay and the oven makers’ clay and the brickmakers’ clay, should we not be ridiculous?
Perhaps.
Yes in the first place for assuming that the questioner can understand from our answer what clay is, when we say clay, no matter whether we add the image-makers’
or any other craftsmen’s. Or does anyone, do you think, understand the name of anything when he does not know what the thing is?
By no means.
Then he does not understand knowledge of shoes if he does not know knowledge.
No.
Then he who is ignorant of knowledge does not understand cobblery or any other art.
That is true.
Then it is a ridiculous answer to the question what is knowledge? when we give the name of some art;
for we give in our answer something that knowledge belongs to, when that was not what we were asked.
So it seems.
Secondly, when we might have given a short, everyday answer, we go an interminable distance round; for instance, in the question about clay, the everyday, simple thing would be to say clay is earth mixed with moisture without regard to whose clay it is.
It seems easy just now, Socrates, as you put it; but you are probably asking the kind of thing that came up among us lately when
your namesake, Socrates here, and I were talking together.
What kind of thing was that, Theaetetus?
Theodorus here was drawing some figures for us in illustration of roots, showing that squares containing three square feet and five square feet are not commensurable in length with the unit of the foot, and so, selecting each one in its turn up to the square containing seventeen square feet and at that he stopped. Now it occurred to us, since the number of roots appeared to be infinite, to try to collect them under one name,
by which we could henceforth call all the roots. A simple form of the first statement would be: the square roots of 3, 5, etc., are irrational numbers or surds. The word δύναμις has not the meaning which we give in English to power, namely the result of multiplication of a number by itself, but that which we give to root, i.e. the number which, when multiplied by itself, produces a given result. Here Theaetetus is speaking of square roots only; and when he speaks of numbers and of equal factors he evidently thinks of rational whole numbers only, not of irrational numbers or fractions. He is not giving an exhaustive presentation of his investigation, but merely a brief sketch of it to illustrate his understanding of the purpose of Socrates. Toward the end of this sketch the word δύναμις is limited to the square roots of oblong numbers, i.e. to surds. The modern reader may be somewhat confused because Theaetetus seems to speak of arithmetical facts in geometrical terms. (Cf. Gow, Short History of Greek Mathematics, p. 85.)
And did you find such a name?
I think we did. But see if you agree.
Speak on.
We divided all number into two classes. The one, the numbers which can be formed by multiplying equal factors, we represented by the shape of the square and called square or equilateral numbers.