Between Euclid's time and 1829, the year Lobachevskiĭ's On the Principles of Geometry appeared, most of the critics of the Elements were concerned with the “purification” of Euclid's work from its perceived imperfections. So strong was the conviction that Postulate V depended on Postulates I through IV that some of them did not see in their work the basis for a new geometry. We begin this chapter with one such critic, Girolamo Saccheri (1667–1733). Further in the chapter the viewpoint changes when we consider the work of Gauss, Bolyai, and Lobachevskiĭ, the founders of a “new geometry.”

What distinguishes this chapter from the next is the role of analysis in the discussion, as well as the role of space (three-dimensional). In this chapter we restrict ourselves, for the most part, to synthetic (not analytic) arguments and arguments in the plane.

The work of Saccheri

Girolamo Saccheri was a Jesuit priest and professor at the University of Pavia. His Euclides ab omni naevo vindicatus (“Euclid vindicated of every flaw,” Saccheri (1733)) marks a triumph of logic in the pursuit of a proof of Postulate V. It also contains the beginning of the study of non-Euclidean geometry, disguised by a flaw in Saccheri's work.

The goal is to prove that Postulates I, II, III, and IV imply V. This can be accomplished through a reductio ad absurdum argument; assume Postulates I, II, III, and IV and the negation of Postulate V and then reason to a contradiction.