几何三部曲 第1卷 几何的公理化方法 英文版【2025|PDF|Epub|mobi|kindle电子书版本百度云盘下载】

几何三部曲 第1卷 几何的公理化方法 英文版PDF格式电子书版下载

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1 Pre-Hellenic Antiquity1

1.1 Prehistory1

1.2 Egypt3

1.3 Mesopotamia5

1.4 Problems6

1.5 Exercises7

2 Some Pioneers of Greek Geometry9

2.1 Thales of Miletus10

2.2 Pythagoras and the Golden Ratio13

2.3 Trisecting the Angle16

2.4 Squaring the Circle18

2.5 Duplicating the Cube23

2.6 Incommensurable Magnitudes29

2.7 The Method ofExhaustion34

2.8 On the Continuity of Space38

2.9 Problems40

2.10 Exercises41

3 Euclid's Elements43

3.1 Book 1:Straight Lines44

3.2 Book 2:Geometric Algebra64

3.3 Book 3:Circles68

3.4 Book 4:Polygons74

3.5 Book 5:Ratios77

3.6 Book 6:Similarities78

3.7 Book 7:Divisibility in Arithmetic85

3.8 Book 8:Geometric Progressions90

3.9 Book 9:More on Numbers90

3.10 Book 10:Incommensurable Magnitudes91

3.11 Book 11:Solid Geometry92

3.12 Book 12:The Method of Exhaustion100

3.13 Book 13:Regular Polyhedrons106

3.14 Problems109

3.15 Exercises110

4 Some Masters of Greek Geometry111

4.1 Archimedes on the Circle112

4.2 Archimedes on the Numberπ113

4.3 Archimedes on the Sphere120

4.4 Archimedes on the Parabola124

4.5 Archimedes on the Spiral127

4.6 Apollonius on Conical Sections130

4.7 Apollonius on Conjugate Directions135

4.8 Apollonius on Tangents139

4.9 Apollonius on Poles and Polar Lines143

4.10 Apollonius on Foci146

4.11 Heron on the Triangle149

4.12 Menelaus on Trigonometry151

4.13 Ptolemy on Trigonometry154

4.14 Pappus on Anharmonic Ratios157

4.15 Problems162

4.16 Exercises164

5 Post-Hellenic Euclidean Geometry167

5.1 Still Chasing the Number π168

5.2 The Medians of a Triangle170

5.3 The Altitudes of a Triangle172

5.4 Ceva's Theorem172

5.5 The Trisectrices of a Triangle174

5.6 Another Look at the Foci ofConics177

5.7 Inversions in the Plane180

5.8 Inversions in Solid Space184

5.9 The Stereographic Projection186

5.10 Let us Burn our Rulers!189

5.11 Problems195

5.12 Exercises195

6 Projective Geometry197

6.1 Perspective Representation198

6.2 Projective Versus Euclidean202

6.3 Anharmonic Ratio205

6.4 The Desargues and the Pappus Theorems208

6.5 Axiomatic Projective Geometry210

6.6 Arguesian and Pappian Planes214

6.7 The Projective Plane over a Skew Field219

6.8 The Hilbert Theorems222

6.9 Problems240

6.10 Exercises241

7 Non-Euclidean Geometry243

7.1 Chasing Euclid's Fifth Postulate245

7.2 T he Saccheri Quadrilaterals251

7.3 The Angles of a Triangle259

7.4 The Limit Parallels264

7.5 The Area of a Triangle273

7.6 The Beltrami-K1ein and Poincaré Disks280

7.7 Problems302

7.8 Exercises303

8 Hilbert's Axiomatization of the Plane305

8.1 The Axioms of Incidence306

8.2 The Axioms of Order307

8.3 The Axioms of Congruence319

8.4 The Axiom of Continuity335

8.5 The Axioms of Parallelism351

8.6 Problems353

8.7 Exercises353

Appendix A Constructibility355

A.1 The Minimal Polynomial355

A.2 The Eisenstein Criterion358

A.3 Ruler and Compass Constructibility360

A.4 Constructibility Versus Field Theory363

Appendix B The Classical Problems369

B.1 Duplicating the Cube369

B.2 Trisecting the Angle369

B.3 Squaring the Circle371

Appendix C Regular Polygons379

C.1 What the Greek Geometers Knew379

C.2 The Problem in Algebraic Terms380

C.3 Fermat Primes382

C.4 Elements of Modular Arithmetic384

C.5 A Flavour of Galois Theory387

C.6 The Gauss-Wantzel Theorem390

Referencesand Further Reading395

Index397