SOC. 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?
THEAET. Perhaps.
SOC. 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?
THEAET. By no means.
SOC. Then he does not understand knowledge of shoes if he does not know knowledge.
THEAET. No.
SOC. Then he who is ignorant of knowledge does not understand cobblery or any other art.
THEAET. That is true.
SOC. 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.
THEAET. So it seems.
SOC. 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.
THEAET. 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.
SOC. What kind of thing was that, Theaetetus?
THEAET. 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. Footnote 1A 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.)
SOC. And did you find such a name?
THEAET. I think we did. But see if you agree.
SOC. Speak on.
THEAET. 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.
SOC. Well done!