Projective Geometry

Abstract

In his first book, An Essay on the Foundations of Geometry, Russell, following Felix Klein,¹ explained how the three classical metrical three-dimensional geometries (the Euclidean, the hyperbolic and the elliptic) could be developed in a projective setting. The method was rather complicated (see Chapter 2), but the mathematical reasoning was widely known and used at the time. If Russell did not completely endorse the reduction of all the metrical concepts to the projective ones, he clearly assumed that metrical geometry presupposed the projective framework. For Russell, then, defining the nature of space amounted ultimately to defining the nature of projective space. He had not changed his mind by 1903. PoM VI is indeed composed of three parts: the first one (from chapters 44 to 46), devoted to the definition of space, is a study of projective geometry; the second part (from chapters 47 to 49) is dedicated to metrical geometry; the third one (from chapters 50 to 53) is more philosophical in nature — it is a rebuttal of certain arguments (from Lotze and Kant, aimed at showing that space is a contradictory concept).² There is thus no doubt that Russell, in PoM, resumed Arthur Cayley’s slogan according to which ‘projective geometry is all geometry’.