Before coordinates and calculations, there was pure geometric thinking. This 1917 course in synthetic projective geometry invites readers to rediscover space through relationships rather than measurements. Lehmer builds from the ground up: what happens when we consider parallel lines as meeting at infinity, how points and lines create elegant correspondences, why the ancient Greeks saw geometry as the purest form of reasoning. The book moves through one-to-one correspondence between geometric forms, the interplay of point-rows and pencils of rays, and the theorems that reveal how shapes transform under projection. Lehmer's pedagogical aim was revolutionary for its time: not to drill formulas but to develop genuine geometric intuition. He insisted students need only elementary geometry as a foundation, though those with analytical geometry and calculus will find additional purchase. The result is a book that treats mathematics as an art of seeing, where the mind's eye traces relationships that persist regardless of how we measure or where we stand.
