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