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Euclidean and Non-Euclidean Geometries Development and History,9780716799481
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Euclidean and Non-Euclidean Geometries Development and History



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W. H. Freeman
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  • Euclidean and Non-Euclidean Geometries: Development and History
    Euclidean and Non-Euclidean Geometries: Development and History


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

Prefacep. xiii
Introductionp. xxv
Euclid's Geometryp. 1
Very Brief Survey of the Beginnings of Geometryp. 1
The Pythagoreansp. 3
Platop. 5
Euclid of Alexandriap. 7
The Axiomatic Methodp. 9
Undefined Termsp. 11
Euclid's First Four Postulatesp. 15
The Parallel Postulatep. 20
Attempts to Prove the Parallel Postulatep. 23
The Danger in Diagramsp. 25
The Power of Diagramsp. 27
Straightedge-and-Compass Constructions, Brieflyp. 29
Descartes' Analytic Geometry and Broader Idea of Constructionsp. 34
Briefly on the Number [pi]p. 38
Conclusionp. 40
Logic and Incidence Geometryp. 53
Elementary Logicp. 53
Theorems and Proofsp. 55
RAA Proofsp. 58
Negationp. 60
Quantifiersp. 61
Implicationp. 64
Law of Excluded Middle and Proof by Casesp. 65
Brief Historical Remarksp. 66
Incidence Geometryp. 69
Modelsp. 72
Consistencyp. 76
Isomorphism of Modelsp. 79
Projective and Affine Planesp. 81
Brief History of Real Projective Geometryp. 89
Conclusionp. 90
Hilbert's Axiomsp. 103
Flaws in Euclidp. 103
Axioms of Betweennessp. 105
Axioms of Congruencep. 119
Axioms of Continuityp. 129
Hilbert's Euclidean Axiom of Parallelismp. 138
Conclusionp. 142
Neutral Geometryp. 161
Geometry Without a Parallel Axiomp. 161
Alternate Interior Angle Theoremp. 162
Exterior Angle Theoremp. 164
Measure of Angles and Segmentsp. 169
Equivalence of Euclidean Parallel Postulatesp. 173
Saccheri and Lambert Quadrilateralsp. 176
Angle Sum of a Trianglep. 183
Conclusionp. 190
History of the Parallel Postulatep. 209
Reviewp. 209
Proclusp. 210
Equidistancep. 213
Wallisp. 214
Saccherip. 218
Clairaut's Axiom and Proclus' Theoremp. 219
Legendrep. 221
Lambert and Taurinusp. 223
Farkas Bolyaip. 225
The Discovery of Non-Euclidean Geometryp. 239
Janos Bolyaip. 239
Gaussp. 242
Lobachevskyp. 245
Subsequent Developmentsp. 248
Non-Euclidean Hilbert Planesp. 249
The Defectp. 252
Similar Trianglesp. 253
Parallels Which Admit a Common Perpendicularp. 254
Limiting Parallel Rays, Hyperbolic Planesp. 257
Classification of Parallelsp. 262
Strange New Universe?p. 264
Independence of the Parallel Postulatep. 289
Consistency of Hyperbolic Geometryp. 289
Beltrami's Interpretationp. 293
The Beltrami-Klein Modelp. 297
The Poincare Modelsp. 302
Perpendicularity in the Beltrami-Klein Modelp. 308
A Model of the Hyperbolic Plane from Physicsp. 311
Inversion in Circles, Poincare Congruencep. 313
The Projective Nature of the Beltrami-Klein Modelp. 333
Conclusionp. 346
Philosophical Implications, Fruitful Applicationsp. 371
What Is the Geometry of Physical Space?p. 371
What Is Mathematics About?p. 374
The Controversy about the Foundations of Mathematicsp. 376
The Meaningp. 380
The Fruitfulness of Hyperbolic Geometry for Other Branches of Mathematics, Cosmology, and Artp. 382
Geometric Transformationsp. 397
Klein's Erlanger Programmep. 397
Groupsp. 399
Applications to Geometric Problemsp. 403
Motions and Similaritiesp. 408
Reflectionsp. 411
Rotationsp. 414
Translationsp. 417
Half-Turnsp. 420
Ideal Points in the Hyperbolic Planep. 422
Parallel Displacementsp. 424
Glidesp. 426
Classification of Motionsp. 427
Automorphisms of the Cartesian Modelp. 431
Motions in the Poincare Modelp. 436
Congruence Described by Motionsp. 444
Symmetryp. 448
Further Results in Real Hyperbolic Geometryp. 475
Area and Defectp. 476
The Angle of Parallelismp. 480
Cyclesp. 481
The Curvature of the Hyperbolic Planep. 483
Hyperbolic Trigonometryp. 487
Circumference and Area of a Circlep. 496
Saccheri and Lambert Quadrilateralsp. 500
Coordinates in the Real Hyperbolic Planep. 507
The Circumscribed Cycle of a Trianglep. 515
Bolyai's Constructions in the Hyperbolic Planep. 520
Elliptic and Other Riemannian Geometriesp. 541
Hilbert's Geometry Without Real Numbersp. 571
Axiomsp. 597
Bibliographyp. 603
Symbolsp. 611
Name Indexp. 613
Subject Indexp. 617
Table of Contents provided by Ingram. All Rights Reserved.

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