Fundamental Concepts of Geometry

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 century 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