Euclidean and Non-Euclidean Geometries: Development and History (Hardcover, 4)

Chapter 1 Euclid's Geometry.- Very Brief Survey of the Beginnings of Geometry.- The Pythagoreans.- Plato?.- 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 Geometry.- Elementary Logic?.- Theorems and Proofs.- RAA Proofs?.- Negation?.- Quantifiers?.- Implication?.- Law of Excluded Middle and Proof by Cases?.- Brief Historical Remarks?.- Incidence Geometry?.- Models?.- Consistency?.- Isomorphism of Models.- Projective and Affine Planes?.- Brief History of Real Projective Geometry?.- Conclusion Chapter 3 Hilbert's Axioms.- Flaws in Euclid?.- Axioms of Betweenness?.- Axioms of Congruence.- Axioms of Continuity.- Hilbert'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 Geometry<.- Janos 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 Poincare Models?.- Perpendicularity in the Beltrami-Klein Model?.- A Model of the Hyperbolic Plane from Physics?.- Inversion in Circles, Poincare Congruence?.- The Projective Nature of the Beltrami-Klein Model?.- Conclusion Chapter 8 Philosophical Implications, Fruitful Applications.- What 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 Transformations.- Klein'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 Poincare Model?.- Congruence Described by Motions?.- Symmetry?Chapter 10 Further Results in Real Hyperbolic Geometry.- Area 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 A.- Appendix B.- Axioms.- Bibliography.- Symbols.- Name Index.- Subject Index DIV>.