Irrationality of the square root of 2: the early Pythagorean proof, Theodorus's and Theaetetus's generalizations
This is a lucid presentation of the existing explanations for the possible proof methods by which the early Pythagoreans, Theodorus and later Theaetetus, arrived at the realization that square roots of non-square integers cannot be ratios of integers. One finds the explanation based on a pure theory of the even and the odd, going back to the early Pythagoreans (one reads that it is ``more probable that Pythagoras, and not Hippasus, could have been the author of the the doctrine of irrational and incommensurable quantities and of the [\(\ldots\)] proof'' (p.\ 30) of the irrationality of \(\sqrt{2}\)), which works for all non-squares that are not of the form \(8k+1\), and which helps explain why Theodorus, according to Plato's \textit{Theaetetus}, stopped proving the irrationality of square roots when he reached \(\sqrt{17}\), the significance of the geometric passage in Plato's \textit{Meno}, and an explanation of the way in which Euclid deals with the irrationality of square roots in his \textit{Elements}, primarily in VIII.14.
scientific article; zbMATH DE number 6536979