Euclidean and Non-Euclidean Geometries: Development and HistoryThis classic text provides overview of both classic and hyperbolic geometries, placing the work of key mathematicians/ philosophers in historical context. Coverage includes geometric transformations, models of the hyperbolic planes, and pseudospheres. |
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... Coordinates in the hyperbolic plane 417 The circumscribed cycle of a triangle 423 Review exercise 428 Exercises 429 Appendix A Elliptic and Other Riemannian Geometries 428 Elliptic geometry 438 Riemannian geometry 443 Appendix B ...
... Coordinates in the hyperbolic plane 417 The circumscribed cycle of a triangle 423 Review exercise 428 Exercises 429 Appendix A Elliptic and Other Riemannian Geometries 428 Elliptic geometry 438 Riemannian geometry 443 Appendix B ...
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Contents
INTRODUCTION | 1 |
The axiomatic method | 10 |
The parallel postulate | 18 |
The power of diagrams | 25 |
LOGIC AND INCIDENCE GEOMETRY | 38 |
NEUTRAL GEOMETRY | 115 |
HISTORY OF THE PARALLEL POSTULATE | 148 |
Saccheri | 154 |
PHILOSOPHICAL IMPLICATIONS | 290 |
The mess | 299 |
GEOMETRIC TRANSFORMATIONS | 309 |
Cycles | 392 |
Circumference and area of a circle | 407 |
The circumscribed cycle of a triangle | 423 |
Elliptic geometry | 438 |
Appendix B Geometry Without Continuity | 454 |
Common terms and phrases
AABC ABCD angle sum assume asymptotically automorphism axis Bolyai called Chapter chord circle collinear common perpendicular Congruence Axiom congruent construction continuity principle contradiction coordinates corollary cosh cross-ratio curvature Dedekind's axiom defined definition diameter distinct points elliptic geometry equation Euclid's Euclidean geometry Euclidean plane exists Figure formula Gauss given Hence Hint horocycles hyperbolic geometry hyperbolic plane hypothesis ideal point Incidence Axiom incidence geometry interior angles interpretation intersects invariant inverse isomorphism isosceles Klein model Lambert quadrilateral Lemma length lies limiting parallel ray logic Major Exercise maps mathematician mathematics meet midpoint motions neutral geometry non-Euclidean geometry opposite sides orthogonal parallel lines parallel postulate perpendicular bisector Poincaré model projective plane proof Proposition prove radius real numbers rectangle reflection right angles right triangle rotation Saccheri quadrilateral segment sinh sphere statement symmetry tangent tanh theorem transformation translation ultra-ideal unique line unique point