Projective Geometry as the Fundamental Geometry

Abstract

Projective geometry became regarded by the mid 19th century as the fundamental geometry. This was very much the view of the English mathematicians Arthur Cayley, James Joseph Sylvester, and Henry Smith, and of George Salmon in Ireland, as it was of the Italian mathematician Luigi Cremona, whose book Elementi di geometria projettiva may be the first to have given the subject its present name. We look briefly at this book, concentrating on Cremona’s treatment of duality. Then we look at the disquiet over the foundations of geometry that were addressed by Moritz Pasch in his Vorlesungen über neuere Geometrie (1882), who attempted to build elementary geometry axiomatically from ideas laid down empirically by Hermann von Helmholtz. Helmholtz, whose acceptance of non-Euclidean geometry was influential, based his approach on an analysis of what is required for a geometry to capture the idea of the free mobility of figures in space.