1 Axiomatic Systems and Finite Geometries.- 2 Non-Euclidean Geometry.- 3 Geometric Transformations of the Euclidean Plane.- 4 Projective Geometry.- Appendixes.- A. Euclid's Definitions, Postulates, and the First 30 Propositions of Book I.- B. Hilbert's Axioms for Plane Geometry.- C. Birkhoff's Postulates for Euclidean Plane Geometry.- D. The S.M.S.G. Postulates for Euclidean Geometry.- E. Some S.M.S.G. Definitions for Euclidean Geometry.- F. The A.S.A. Theorem.- G. References.

A Course in Modern Geometries is designed for a junior-senior level course for mathematics majors, including those who plan to teach in secondary school. Chapter 1 presents several finite geometries in an axiomatic framework. Chapter 2 introduces Euclid's geometry and the basic ideas of non-Euclidean geometry. The synthetic approach of Chapters 1 - 2 is followed by the analytic treatment of transformations of the Euclidean plane in Chapter 3. Chapter 4 presents plane projective geometry both synthetically and analytically. The extensive use of matrix representations of groups of transformations in Chapters 3 - 4 reinforces ideas from linear algebra and serves as excellent preparation for a course in abstract algebra. Each chapter includes a list of suggested sources for applications and/or related topics.