Euclidean and Non-Euclidean Geometries Development and History
by Greenberg, Marvin JayEdition: 4th
Format: Hardcover
Pub. Date: 2007-09-28
Publisher(s): W. H. Freeman
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Summary
This is the definitive presentation of the history, development and philosophical significance of non-Euclidean geometry as well as of the rigorous foundations for it and for elementary Euclidean geometry, essentially according to Hilbert. Appropriate for liberal arts students, prospective high school teachers, math. majors, and even bright high school students. The first eight chapters are mostly accessible to any educated reader; the last two chapters and the two appendices contain more advanced material, such as the classification of motions, hyperbolic trigonometry, hyperbolic constructions, classification of Hilbert planes and an introduction to Riemannian geometry.
Table of Contents
| Preface | p. xiii |
| Introduction | p. xxv |
| Euclid's Geometry | p. 1 |
| Very Brief Survey of the Beginnings of Geometry | p. 1 |
| The Pythagoreans | p. 3 |
| Plato | p. 5 |
| Euclid of Alexandria | p. 7 |
| The Axiomatic Method | p. 9 |
| Undefined Terms | p. 11 |
| Euclid's First Four Postulates | p. 15 |
| The Parallel Postulate | p. 20 |
| Attempts to Prove the Parallel Postulate | p. 23 |
| The Danger in Diagrams | p. 25 |
| The Power of Diagrams | p. 27 |
| Straightedge-and-Compass Constructions, Briefly | p. 29 |
| Descartes' Analytic Geometry and Broader Idea of Constructions | p. 34 |
| Briefly on the Number [pi] | p. 38 |
| Conclusion | p. 40 |
| Logic and Incidence Geometry | p. 53 |
| Elementary Logic | p. 53 |
| Theorems and Proofs | p. 55 |
| RAA Proofs | p. 58 |
| Negation | p. 60 |
| Quantifiers | p. 61 |
| Implication | p. 64 |
| Law of Excluded Middle and Proof by Cases | p. 65 |
| Brief Historical Remarks | p. 66 |
| Incidence Geometry | p. 69 |
| Models | p. 72 |
| Consistency | p. 76 |
| Isomorphism of Models | p. 79 |
| Projective and Affine Planes | p. 81 |
| Brief History of Real Projective Geometry | p. 89 |
| Conclusion | p. 90 |
| Hilbert's Axioms | p. 103 |
| Flaws in Euclid | p. 103 |
| Axioms of Betweenness | p. 105 |
| Axioms of Congruence | p. 119 |
| Axioms of Continuity | p. 129 |
| Hilbert's Euclidean Axiom of Parallelism | p. 138 |
| Conclusion | p. 142 |
| Neutral Geometry | p. 161 |
| Geometry Without a Parallel Axiom | p. 161 |
| Alternate Interior Angle Theorem | p. 162 |
| Exterior Angle Theorem | p. 164 |
| Measure of Angles and Segments | p. 169 |
| Equivalence of Euclidean Parallel Postulates | p. 173 |
| Saccheri and Lambert Quadrilaterals | p. 176 |
| Angle Sum of a Triangle | p. 183 |
| Conclusion | p. 190 |
| History of the Parallel Postulate | p. 209 |
| Review | p. 209 |
| Proclus | p. 210 |
| Equidistance | p. 213 |
| Wallis | p. 214 |
| Saccheri | p. 218 |
| Clairaut's Axiom and Proclus' Theorem | p. 219 |
| Legendre | p. 221 |
| Lambert and Taurinus | p. 223 |
| Farkas Bolyai | p. 225 |
| The Discovery of Non-Euclidean Geometry | p. 239 |
| Janos Bolyai | p. 239 |
| Gauss | p. 242 |
| Lobachevsky | p. 245 |
| Subsequent Developments | p. 248 |
| Non-Euclidean Hilbert Planes | p. 249 |
| The Defect | p. 252 |
| Similar Triangles | p. 253 |
| Parallels Which Admit a Common Perpendicular | p. 254 |
| Limiting Parallel Rays, Hyperbolic Planes | p. 257 |
| Classification of Parallels | p. 262 |
| Strange New Universe? | p. 264 |
| Independence of the Parallel Postulate | p. 289 |
| Consistency of Hyperbolic Geometry | p. 289 |
| Beltrami's Interpretation | p. 293 |
| The Beltrami-Klein Model | p. 297 |
| The Poincare Models | p. 302 |
| Perpendicularity in the Beltrami-Klein Model | p. 308 |
| A Model of the Hyperbolic Plane from Physics | p. 311 |
| Inversion in Circles, Poincare Congruence | p. 313 |
| The Projective Nature of the Beltrami-Klein Model | p. 333 |
| Conclusion | p. 346 |
| Philosophical Implications, Fruitful Applications | p. 371 |
| What Is the Geometry of Physical Space? | p. 371 |
| What Is Mathematics About? | p. 374 |
| The Controversy about the Foundations of Mathematics | p. 376 |
| The Meaning | p. 380 |
| The Fruitfulness of Hyperbolic Geometry for Other Branches of Mathematics, Cosmology, and Art | p. 382 |
| Geometric Transformations | p. 397 |
| Klein's Erlanger Programme | p. 397 |
| Groups | p. 399 |
| Applications to Geometric Problems | p. 403 |
| Motions and Similarities | p. 408 |
| Reflections | p. 411 |
| Rotations | p. 414 |
| Translations | p. 417 |
| Half-Turns | p. 420 |
| Ideal Points in the Hyperbolic Plane | p. 422 |
| Parallel Displacements | p. 424 |
| Glides | p. 426 |
| Classification of Motions | p. 427 |
| Automorphisms of the Cartesian Model | p. 431 |
| Motions in the Poincare Model | p. 436 |
| Congruence Described by Motions | p. 444 |
| Symmetry | p. 448 |
| Further Results in Real Hyperbolic Geometry | p. 475 |
| Area and Defect | p. 476 |
| The Angle of Parallelism | p. 480 |
| Cycles | p. 481 |
| The Curvature of the Hyperbolic Plane | p. 483 |
| Hyperbolic Trigonometry | p. 487 |
| Circumference and Area of a Circle | p. 496 |
| Saccheri and Lambert Quadrilaterals | p. 500 |
| Coordinates in the Real Hyperbolic Plane | p. 507 |
| The Circumscribed Cycle of a Triangle | p. 515 |
| Bolyai's Constructions in the Hyperbolic Plane | p. 520 |
| Elliptic and Other Riemannian Geometries | p. 541 |
| Hilbert's Geometry Without Real Numbers | p. 571 |
| Axioms | p. 597 |
| Bibliography | p. 603 |
| Symbols | p. 611 |
| Name Index | p. 613 |
| Subject Index | p. 617 |
| Table of Contents provided by Ingram. All Rights Reserved. |
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