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[147e] by which we could henceforth call all the roots.1

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.

Well done!

The numbers between these, such as three

1 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.)

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