Axiomatic Geometry

John M. Lee, University of Washington, Seattle, WA, USA

In the preface, the author announces a textbook for undergraduate students who plan to teach geometry in a North American high-school; this intention is fulfilled perfectly. The author offers, among others, a comprehensive description of the historical development of axiomatic geometry, a careful approach to all arising problems, well-motivated definitions, an analysis of the procedure of proof writing, and plenty of very aesthetical, helpful diagrams. For the reader's convenience, many theorems are named by well-chosen catchwords, thus a very clearly arranged text is reached." - Zentralblatt Math"Lee's “Axiomatic Geometry” gives a detailed, rigorous development of plane Euclidean geometry using a set of axioms based on the real numbers. It is suitable for an undergraduate college geometry course, and since it covers most of the topics normally taught in American high school geometry, it would be excellent preparation for future high school teachers. There is a brief treatment of the non-Euclidean hyperbolic plane at the end." - Robin Hartshorne, University of California, Berkeley"The goal of Lee's well-written book is to explain the axiomatic method and its role in modern mathematics, and especially in geometry. Beginning with a discussion (and a critique) of Euclid's elements, the author gradually introduces and explains a set of axioms sufficient to provide a rigorous foundation for Euclidean plane geometry... Because they assume properties of the real numbers, Lee's axioms are fairly intuitive, and this results in a presentation that should be accessible to upper level undergraduate mathematics students. Although the pace is leisurely at first, this book contains a surprising amount of material, some of which can be found among the many exercises. Included are discussions of basic trigonometry, hyperbolic geometry and an extensive treatment of compass and straightedge constructions." - I. Martin Isaacs, University of Wisconsin-Madison"Jack Lee's book will be extremely valuable for future high school math teachers. It is perfectly designed for students just learning to write proofs; complete beginners can use the appendices to get started, while more experienced students can jump right in. The axioms, definitions, and theorems are developed meticulously, and the book culminates in several chapters on hyperbolic geometry—a lot of fun, and a nice capstone to a two-quarter course on axiomatic geometry." - John H. Palmieri, University of Washington

• PrefaceEuclidIncidence geometryAxioms for plane geometryAnglesTrianglesModels of neutral geometryPerpendicular and parallel linesPolygonsQuadrilateralsThe Euclidean parallel postulateAreaSimilarityRight trianglesCirclesCircumference and circular areaCompass and straightedge constructionsThe parallel postulate revisitedIntroduction to hyperbolic geometryParallel lines in hyperbolic geometryEpilogue: Where do we go from here?Hilbert’s axiomsBirkhoff’s postulatesThe SMSG postulatesThe postulates used in this bookThe language of mathematicsProofsSets and functionsProperties of the real numbersRigid motions: Another approachReferencesIndex