Preface to the second edition .................................. ix
Introduction ................................................... xi
Part A
Prelude and themes: Synthetic methods and results
1 Spherical geometry ........................................... 3
2 Euclid ...................................................... 12
Euclid's theory of parallels ................................ 19
Appendix: The Elements: Book I .............................. 24
3 The theory of parallels ..................................... 27
Uniqueness of parallels ..................................... 28
Equidistance and boundedness of parallels ................... 29
On the angle sum of a triangle .............................. 31
Similarity of triangles ..................................... 34
The work of Saccheri ........................................ 37
4 Non-Euclidean geometry ...................................... 43
The work of Gauss ........................................... 43
The hyperbolic plane ........................................ 47
Digression: Neutral space ................................... 55
Hyperbolic space ............................................ 62
Appendix: The Elements: Selections from Book XI ............. 72
Part B
Development: Differential geometry
5 Curves in the plane ......................................... 77
Early work on plane curves (Huygens, Leibniz, Newton, and
Euler) ...................................................... 81
The tractrix ................................................ 84
Oriented curvature .......................................... 86
Involutes and evolutes ...................................... 89
6 Curves in space ............................................. 99
Appendix: On Euclidean rigid motions ....................... 110
7 Surfaces ................................................... 116
The tangent plane .......................................... 124
The first fundamental form ................................. 128
Lengths, angles, and areas ................................. 130
jbu Map projections ........................................ 138
Stereographic projection ................................... 143
Central (gnomonic) projection .............................. 147
Cylindrical projections .................................... 148
Sinusoidal projection ...................................... 152
Azimuthal projection ....................................... 153
8 Curvature for surfaces ..................................... 156
Euler's work on surfaces ................................... 156
The Gauss map .............................................. 159
9 Metric equivalence of surfaces ............................. 171
Special coordinates ........................................ 179
10 Geodesies .................................................. 185
Euclid revisited I: The Hopf-Rinow Theorem ................. 195
11 The Gauss-Bonnet Theorem ................................... 201
Euclid revisited II: Uniqueness of lines ................... 205
Compact surfaces ........................................... 207
A digression on curves ..................................... 211
12 Constant-curvature surfaces ................................ 218
Euclid revisited III: Congruences .......................... 223
The work of Minding ........................................ 224
Hilbert's Theorem .......................................... 231
Part C
Recapitulation and coda
13 Abstract surfaces .......................................... 237
14 Modeling the non-Euclidean plane ........................... 251
The Beltrami disk .......................................... 255
The Poincare disk .......................................... 262
The Poincare half-plane .................................... 265
15 Epilogue: Where from here? ................................. 282
Manifolds (differential topology) .......................... 283
Vector and tensor fields ................................... 287
Metrical relations (Riemannian manifolds) .................. 289
Curvature .................................................. 294
Covariant differentiation .................................. 303
Riemann's Habilitationsvortrag: On the hypotheses which
lie at the foundations of geometry ......................... 313
Solutions to selected exercises ............................ 325
Bibliography .................................................. 341
Symbol index .................................................. 351
Name index .................................................... 352
Subject index ................................................. 354
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