几何三部曲 第1卷 几何的公理化方法 英文版PDF电子书下载

1 Pre-Hellenic Antiquity 1

1.1 Prehistory 1

1.2 Egypt 3

1.3 Mesopotamia 5

1.4 Problems 6

1.5 Exercises 7

2 Some Pioneers of Greek Geometry 9

2.1 Thales of Miletus 10

2.2 Pythagoras and the Golden Ratio 13

2.3 Trisecting the Angle 16

2.4 Squaring the Circle 18

2.5 Duplicating the Cube 23

2.6 Incommensurable Magnitudes 29

2.7 The Method ofExhaustion 34

2.8 On the Continuity of Space 38

2.9 Problems 40

2.10 Exercises 41

3 Euclid's Elements 43

3.1 Book 1:Straight Lines 44

3.2 Book 2:Geometric Algebra 64

3.3 Book 3:Circles 68

3.4 Book 4:Polygons 74

3.5 Book 5:Ratios 77

3.6 Book 6:Similarities 78

3.7 Book 7:Divisibility in Arithmetic 85

3.8 Book 8:Geometric Progressions 90

3.9 Book 9:More on Numbers 90

3.10 Book 10:Incommensurable Magnitudes 91

3.11 Book 11:Solid Geometry 92

3.12 Book 12:The Method of Exhaustion 100

3.13 Book 13:Regular Polyhedrons 106

3.14 Problems 109

3.15 Exercises 110

4 Some Masters of Greek Geometry 111

4.1 Archimedes on the Circle 112

4.2 Archimedes on the Numberπ 113

4.3 Archimedes on the Sphere 120

4.4 Archimedes on the Parabola 124

4.5 Archimedes on the Spiral 127

4.6 Apollonius on Conical Sections 130

4.7 Apollonius on Conjugate Directions 135

4.8 Apollonius on Tangents 139

4.9 Apollonius on Poles and Polar Lines 143

4.10 Apollonius on Foci 146

4.11 Heron on the Triangle 149

4.12 Menelaus on Trigonometry 151

4.13 Ptolemy on Trigonometry 154

4.14 Pappus on Anharmonic Ratios 157

4.15 Problems 162

4.16 Exercises 164

5 Post-Hellenic Euclidean Geometry 167

5.1 Still Chasing the Number π 168

5.2 The Medians of a Triangle 170

5.3 The Altitudes of a Triangle 172

5.4 Ceva's Theorem 172

5.5 The Trisectrices of a Triangle 174

5.6 Another Look at the Foci ofConics 177

5.7 Inversions in the Plane 180

5.8 Inversions in Solid Space 184

5.9 The Stereographic Projection 186

5.10 Let us Burn our Rulers! 189

5.11 Problems 195

5.12 Exercises 195

6 Projective Geometry 197

6.1 Perspective Representation 198

6.2 Projective Versus Euclidean 202

6.3 Anharmonic Ratio 205

6.4 The Desargues and the Pappus Theorems 208

6.5 Axiomatic Projective Geometry 210

6.6 Arguesian and Pappian Planes 214

6.7 The Projective Plane over a Skew Field 219

6.8 The Hilbert Theorems 222

6.9 Problems 240

6.10 Exercises 241

7 Non-Euclidean Geometry 243

7.1 Chasing Euclid's Fifth Postulate 245

7.2 T he Saccheri Quadrilaterals 251

7.3 The Angles of a Triangle 259

7.4 The Limit Parallels 264

7.5 The Area of a Triangle 273

7.6 The Beltrami-K1ein and Poincaré Disks 280

7.7 Problems 302

7.8 Exercises 303

8 Hilbert's Axiomatization of the Plane 305

8.1 The Axioms of Incidence 306

8.2 The Axioms of Order 307

8.3 The Axioms of Congruence 319

8.4 The Axiom of Continuity 335

8.5 The Axioms of Parallelism 351

8.6 Problems 353

8.7 Exercises 353

Appendix A Constructibility 355

A.1 The Minimal Polynomial 355

A.2 The Eisenstein Criterion 358

A.3 Ruler and Compass Constructibility 360

A.4 Constructibility Versus Field Theory 363

Appendix B The Classical Problems 369

B.1 Duplicating the Cube 369

B.2 Trisecting the Angle 369

B.3 Squaring the Circle 371

Appendix C Regular Polygons 379

C.1 What the Greek Geometers Knew 379

C.2 The Problem in Algebraic Terms 380

C.3 Fermat Primes 382

C.4 Elements of Modular Arithmetic 384

C.5 A Flavour of Galois Theory 387

C.6 The Gauss-Wantzel Theorem 390

Referencesand Further Reading 395

Index 397