Marvin J. Greenberg
(University of California, Santa Cruz)
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Chapter 1 Euclid’s GeometryVery Brief Survey of the Beginnings of GeometryThe PythagoreansPlato Euclid of Alexandria The Axiomatic Method Undefined Terms Euclid’s First Four Postulates The Parallel Postulate Attempts to Prove the Parallel Postulate The Danger in Diagrams The Power of Diagrams Straightedge-and-Compass Constructions, Briefly Descartes’ Analytic Geometry and Broader Idea of Constructions Briefly on the Number ð Conclusion
Chapter 2 Logic and Incidence GeometryElementary Logic Theorems and ProofsRAA Proofs Negation Quantifiers Implication Law of Excluded Middle and Proof by Cases Brief Historical Remarks Incidence Geometry Models Consistency Isomorphism of ModelsProjective and Affine Planes Brief History of Real Projective Geometry Conclusion
Chapter 3 Hilbert’s AxiomsFlaws in Euclid Axioms of Betweenness Axioms of CongruenceAxioms of ContinuityHilbert’s Euclidean Axiom of Parallelism Conclusion
Chapter 4 Neutral Geometry Geometry without a Parallel Axiom Alternate Interior Angle Theorem Exterior Angle Theorem Measure of Angles and Segments Equivalence of Euclidean Parallel Postulates Saccheri and Lambert Quadrilaterals Angle Sum of a Triangle Conclusion
Chapter 5 History of the Parallel Postulate Review Proclus Equidistance Wallis Saccheri Clairaut’s Axiom and Proclus’ Theorem Legendre Lambert and Taurinus Farkas Bolyai
Chapter 6 The Discovery of Non-Euclidean GeometryJános Bolyai Gauss Lobachevsky Subsequent Developments Non-Euclidean Hilbert Planes The Defect Similar Triangles Parallels Which Admit a Common Perpendicular Limiting Parallel Rays, Hyperbolic Planes Classification of Parallels Strange New Universe?
Chapter 7 Independence of the Parallel Postulate Consistency of Hyperbolic Geometry Beltrami’s Interpretation The Beltrami–Klein Model The Poincaré Models Perpendicularity in the Beltrami–Klein Model A Model of the Hyperbolic Plane from Physics Inversion in Circles, Poincaré Congruence The Projective Nature of the Beltrami–Klein Model Conclusion
Chapter 8 Philosophical Implications, Fruitful ApplicationsWhat Is the Geometry of Physical Space? What Is Mathematics About? The Controversy about the Foundations of Mathematics The Meaning The Fruitfulness of Hyperbolic Geometry for Other Branches of Mathematics, Cosmology, and Art
Chapter 9 Geometric TransformationsKlein’s Erlanger Programme Groups Applications to Geometric Problems Motions and Similarities Reflections Rotations Translations Half-Turns
Ideal Points in the Hyperbolic Plane Parallel Displacements Glides Classification of Motions Automorphisms of the Cartesian Model Motions in the Poincaré Model Congruence Described by Motions Symmetry
Chapter 10 Further Results in Real Hyperbolic GeometryArea and Defect The Angle of Parallelism Cycles The Curvature of the Hyperbolic Plane Hyperbolic Trigonometry Circumference and Area of a Circle Saccheri and Lambert Quadrilaterals Coordinates in the Real Hyperbolic Plane The Circumscribed Cycle of a Triangle Bolyai’s Constructions in the Hyperbolic Plane
Appendix AAppendix BAxiomsBibliographySymbolsName IndexSubject Index
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