Plato, Theaetetus

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.)