To account for the first proof of existence of an irrational magnitude, paradigm of a mathematical proof, historians of science as well as commentators of Aristotle refer to the texts on the incommensurability of the diagonal in the Prior Analytics, the most ancient texts on the subject. The usual proofs, using the representation of fractions as ratio of relatively prime integers i.e. the proposition VII.22 of the Elements, are based on a proposition at the end of the Book X of Euclid’s Elements. But their conclusions do not match the Aristotelian texts. In this talk, we propose a new demonstration conform to these texts. It is based on very old results of the odd/even theory, probably already known by the Babylonian and Egyptian mathematicians. As a consequence, we will see the irrationality of √2 was the first result impossible to prove directly. It was the birth of a new kind of proofs, the demonstrations ad absurdum, and, in a way, of ‘modern’ mathematics.