Fundamental Concepts of Geometry

Description

Demonstrates in a clear and lucid manner the relationships between several types of geometry. This highly regarded work is a superior teaching text, especially valuable in teacher preparation, as well as providing an excellent overview of the foundations and historical evolution of geometrical concepts. Preface. Index. Bibliography. Exercises (no solutions). Includes 98 illustrations. Reprint of the Addison-Wesley Publishing Company, Reading, MA, 1955 edition. Table of Contents CHAPTER 1. FOUNDATIONS OF GEOMETRY 1-1 Logical systems 1-2 Logical notations 1-3 Inductive and deductive reasoning 1-4 Postulates 1-5 Independent postulates 1-6 Categorical sets of postulates 1-7 A geometry of number triples 1-8 Geometric invariants CHAPTER 2. SYNTHETIC PROJECTIVE GEOMETRY 2-1 Postulates of incidence and existence 2-2 Properties of a projective plane 2-3 Figures 2-4 Duality 2-5 Perspective figures 2-6 Projective transformations 2-7 Postulate of Projectivity 2-8 Quadrangles 2-9 Complete and simple n-points 2-10 Theorem of Desargues 2-11 Theorem of Pappus 2-12 Conics 2-13 Theorem of Pascal 2-14 Survey CHAPTER 3. COORDINATE SYSTEMS 3-1 Quadrangular sets 3-2 Properties of quadrangular sets 3-3 Harmonic sets 3-4 Postulates of Separation 3-5 Nets of rationality 3-6 Real projective geometry 3-7 Nonhomogeneous coordinates 3-8 Homogeneous coordinates 3-9 Survey CHAPTER 4. ANALYTIC PROJECTIVE GEOMETRY 4-1 Representations in space 4-2 Representations on a plane 4-3 Representations on a line 4-4 Matrices 4-5 Cross ratio 4-6 Analytic and synthetic geometries 4-7 Groups 4-8 Classification of projective transformations 4-9 Polarities and conics 4-10 Conics 4-11 Involutions on a line 4-12 Survey CHAPTER 5. AFFINE GEOMETRY 5-1 Ideal points 5-2 Parallels 5-3 Mid-point 5-4 Classification of conics 5-5 Affine transformations 5-6 Homothetic transformations 5-7 Translations 5-8 Dilations 5-9 Line reflections 5-10 Equiaffine and equiareal transformations 5-11 Survey CHAPTER 6. EUCLIDEAN PLANE GEOMETRY 6-1 Perpendicluar lines 6-2 Similarity transformations 6-3 Orthogonal line reflections 6-4 Euclidean transformations 6-5 Distances 6-6 Directed angles 6-7 Angles 6-8 Common figures 6-9 Survey CHAPTER 7. THE EVOLUTION OF GEOMETRY 7-1 Early measurements 7-2 Early Greek influence 7-3 Euclid 7-4 Early euclidean geometry 7-5 The awakening in Europe 7-6 Constructions 7-7 Descriptive geometry 7-8 Seventeenth ce 7-9 Eighteenth century 7-10 Euclid's fifth postulate 7-11 Nineteenth and twentieth centuries 7-12 Survey CHAPTER 8. NONEUCLIDEAN GEOMETRY 8-1 The absolute polarity 8-2 Points and lines 8-3 Hyperbolic geometry 8-4 Elliptic and spherical geometries 8-5 Comparisons CHAPTER 9. TOPOLOGY 9-1 Topology 9-2 Homeomorphic figures 9-3 Jordan Curve Theorem 9-4 Surfaces 9-5 Euler's Formula 9-6 Tranversable networks 9-7 Four-color problem 9-8 Fixed-point theorems 9-9 Moebius strip 9-10 Survey BIBLIOGRAPHY INDEX