Note on Legendre's and Bertrand's Proofs of the Parallel-Postulate by Infinite Areas

Abstract

One of the most plausible of the host of “proofs” that have ever been offered for Euclid's parallel-postulate is that known as Bertrand's, which is based upon a consideration of infinite areas. The area of the whole plane being regarded as an infinity of the second order, the area of a strip of plane surface bounded by a linear segment AB and the rays AA′, BB perpendicular to AB is an infinity of the first order, since a single infinity of such strips is required to cover the plane. On the other hand, the area contained between two intersecting straight lines is an infinity of the same order as the plane, since the plane can be covered by a finite number of such sectors. Hence if AP is drawn making any angle, however small, with AA′, the area A′AP, an infinity of the second order, cannot be contained within the area A′ABB′, an infinity of the first order, and therefore AP must cut BB′. And this is just Euclid's postulate.