Non-Euclidean Geometry

Abstract

MR. FRANKLAND (NATURE, September 7) has raised the old problem of Bertrand's proof of the parallel-axiom by a consideration of infinite areas. This is perhaps the most subtle and the most specious of all the attempted proofs, and this character it owes to the fact that a process of reasoning which is sound for finite magnitudes is extended to a field which is beyond our powers of comprehension—the field of infinity. The fallacy which underlies Bertrand's proof becomes more apparent in Legendre's simpler device (“Elements de Geometrie, ” 12 ed., Note ii.). A straight line divides a plane in which it lies into two congruent parts—;this, of course, has no real meaning, since we are dealing with infinite areas, but such is the argument—;and two rays from a point enclose an (infinite) area which is less than half the whole plane. Hence, if two intersecting lines are both parallel to the same straight line, the area of half the plane can be enclosed within an area which is less than half the plane.