Theaetetus and the History of the Theory of Numbers

Abstract

This famous passage has given rise to much discussion and some perplexity. Theodoras the mathematician is represented by Theaetetus as proving the irrationality of the square roots of the (non-square) numbers from 3 to 17:‘He took the separate cases up to the root of 17 square feet; and there, for some reason, he stopped.’ (Transl. Cornford.)The passage is of great importance in the history of Greek mathematics for more than one reason. Theaetetus is said to have generalized the proof of the irrationality of square roots of non-square integers; and thus his connexion with this passage is important because Plato here obviously implies that Theodorus was not giving a generalized proof—otherwise, why should he go up to 17 ? If Theodorus did not know the generalized proof, he clearly had to proceed by enumeration and proof of particular case