The study of the non-Euclidean geometries resulting from changes in the Euclidean parallel postulate often strengthens an individual's awareness and understanding of the deductive systems of mathematics in general. Students at both the high school and college levels are frequently fascinated by the study of deductive systems such as the two non-Euclidean systems considered in this book. In chapter 1 several simple finite deductive systems are considered. The mathematical system considered in chapter 2 is usually referred to as Lobachevskian geometry, and that discussed in chapter 3 is Riemannian geometry. The book can be used as supplemental material for mathematics courses at both the high school and the college levels or as a primary text for a mini-course. In addition, it can be valuable for secondary school students doing individualized or independent study. (Author/MP)