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Introduction 7

Chapter 1．Euclid's Geometry 7

1．A First Look at Euclid's Elements 8

2．Ruler and Compass Constructions 18

3．Euclid's Axiomatic Method 27

4．Construction of the Regular Pentagon 45

5．Some Newer Results 51

Chapter 2．Hilbert's Axioms 65

6．Axioms of Incidence 66

7．Axioms of Betweenness 73

8．Axioms of Congruence for Line Segments 81

9．Axioms of Congruence for Angles 90

10．Hilbert Planes 96

11．Intersection of Lines and Circles 104

12．Euclidean Planes 112

Chapter 3．Geometry over Fields 117

13．The Real Cartesian Plane 118

14．Abstract Fields and Incidence 128

15．Ordered Fields and Betweenness 135

16．Congruence of Segments and Angles 140

17．Rigid Motions and SAS 148

18．Non-Archimedean Geometry 158

Chapter 4．Segment Arithmetic 165

19．Addition and Multiplication of Line Segments 165

20．Similar Triangles 175

21．Introduction of Coordinates 186

Chapter 5．Area 195

22．Area in Euclid's Geometry 196

23．Measure of Area Functions 205

24．Dissection 212

25．Quadratura Circuli 221

26．Euclid's Theory of Volume 226

27．Hilbert's Third Problem 231

Chapter 6．Construction Problems and Field Extensions 241

28．Three Famous Problems 242

29．The Regular 17-Sided Polygon 250

30．Constructions with Compass and Marked Ruler 259

31．Cubic and Quartic Equations 270

32．Appendix:Finite Field Extensions 280

Chapter 7．Non-Euclidean Geometry 295

33．History of the Parallel Postulate 296

34．Neutral Geometry 304

35．Archimedean Neutral Geometry 319

36．Non-Euclidean Area 326

37．Circular Inversion 334

38．Digression:Circles Determined by Three Conditions 346

39．The Poincaré Model 355

40．Hyperbolic Geometry 373

41．Hilbert's Arithmetic of Ends 388

42．Hyperbolic Trigonometry 403

43．Characterization of Hilbert Planes 415

Chapter 8．Polyhedra 435

44．The Five Regular Solids 436

45．Euler's and Cauchy's Theorems 448

46．Semiregular and Face-Regular Polyhedra 459

47．Symmetry Groups of Polyhedra 469

Appendix:Brief Euclid 481

Notes 487

References 495

List of Axioms 503

Index of Euclid's Propositions 505

Index 507