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Index
Cover
Title Page
Copyright Page
Preface
Contents
Chapter I: Historical Development of the Modern View
1. The Rise of Postulational Geometry. Euclid’s Elements
2. Some Comments on Euclid’s System
3. The Fifth Postulate
4. Saccheri’s Contribution
5. Substitutes for the Fifth Postulate
6. Non-euclidean Geometry—a Nineteenth Century Revolution
7. Later Developments
8. The Role of Non-euclidean Geometry in the Development of Mathematics
Chapter II: Sets and Propositions
1. Abstract Sets
2. The Russell Paradox
3. Operations on Sets
4. One-to-one Correspondence. Cardinal Number
5. Finite and Infinite Sets. The Trichotomy Theorem and the Axiom of Choice
6. Propositions
7. Truth Tables
8. Forms of Argumentation
9. Deductive Theory
Chapter III: Postulational Systems
1. Undefined Terms and Unproved Propositions
2. Consistency, Independence, and Completeness of a Postulational System
3. The Postulational System 73
4. A Finite Affine Geometry
5. Hilbert’s Postulates for Three-dimensional Euclidean Geometry
Chapter IV: Coordinates in an Affine Plane
Foreword
1. The Affine Plane
2. Parallel Classes
3. Coordinatizing the Plane π
4. Slope and Equation of a Line
5. The Ternary Operation T
6. The Planar Ternary Ring [Γ, T]
7. The Affine Plane Defined by a Ternary Ring
8. Introduction of Addition
9. Introduction of Multiplication
10. Vectors
11. A Remarkable Affine Plane
Chapter V: Coordinates in an Affine Plane with Desargues and Pappus Properties
Foreword (The First Desargues Property)
1. Completion of the Equivalence Definition for Vectors
2. Addition of Vectors
3. Linearity of the Ternary Operator
4. Right Distributivity of Multiplication Over Addition
5. Introduction of the Second Desargues Property
6. Introduction of the Third Desargues Property
7. Introduction of the Pappus Property
8. The Desargues Properties as Consequences of the Pappus Property
9. Analytic Affine Geometry Over a Field
Chapter VI: Coordinatizing Projective Planes
Foreword
1. The Postulates for a Projective Plane, and the Principle of Duality
2. Homogeneity of Projective Planes. Incidence Matrices of Finite Projective Planes
3. Introduction of Coordinates
4. The Ternary Operation in Γ. Addition and Multiplication
5. Configurations
6. Configurations of Desargues and Pappus
7. Veblen-Wedderburn Planes
8. Alternative Planes
9. Desarguesian and Pappian Planes
10. Concluding Remarks
Chapter VII: Metric Postulates for the Euclidean Plane
Foreword
1. Metric Space. Some Metric Properties of the Euclidean Plane
2. A Set of Metric Postulates. The Space m
3. An Important Property of Space m
4. Straight Lines of Space m
5. Oriented Lines, Angles, and Triangles of Space m
6. Metric Postulates for Euclidean Plane Geometry
7. Concluding Remarks
Chapter VIII: Postulates for the Non-euclidean Planes
Foreword
1. Poincaré’s Model of the Hyperbolic Plane
2. Some Metric Properties of the Hyperbolic Plane
3. Postulates for the Hyperbolic Plane
4. Two-dimensional Spherical Geometry
5. Postulates for Two-dimensional Spherical Space
6. The Elliptic Plane Ɠ2, r
7. Postulates for the Elliptic Plane
Index
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