Table of Contents
Chapter I The Foundation of Euclidean Geometry
1 Introduction 1
2 The Definitions 2
3 The Common Notions 4
4 The Postulates 4
5 Tacit Assumptions Made by Euclid. Superposition 5
6 The Infinitude of the Line 6
7 Pasch's Axiom 9
8 The Principle of Continuity 10
9 The Postulate System of Hilbert 12
Chapter II The Fifth Postulate
10 Introduction 17
11 Substitutes for the Fifth Postulate 20
12 Playfair's Axiom 20
13 The Angle-Sum of a Triangle 21
14 The Existence of Similar Figures 23
15 Equidistant Straight Lines 25
16 Other Substitutes 25
17 Attempts to Prove the Fifth Postulate 26
18 Ptolemy 26
19 Proclus 27
20 Nasiraddin 28
21 Wallis 29
22 Saccheri 30
23 Lambert 33
24 Legendre 34
25 Some Fallacies in Attempts to Prove the Postulate 39
26 The Rotation Proof 40
27 Comparison of Infinite Areas 41
Chapter III The Discovery of Non-Euclidean Geometry
28 Introduction 44
29 Gauss 45
30 Bolyai 48
31 Lobachewsky 53
32 Wachter, Schweikart and Taurinus 56
33 Riemann 60
34 Further Developments 63
35 Conclusion 63
Chapter IV Hyperbolic Plane Geometry
36 Introduction 65
37 The Characteristic Postulate of Hyperbolic Geometry 66
38 Elementary Properties of Parallels 68
39 Ideal Points 71
40 Some Properties of an Important Figure 72
41 The Angle of Parallelism 76
42 The Saccheri Quadrilateral 77
43 The Lambert Quadrilateral 79
44 The Sum of the Angles of a Triangle 81
45 The Common Perpendicular of Two Non-Intersecting Lines 84
46 Ultra-Ideal Points 85
47 The Variation in the Distance between Two Lines 86
48 The Perpendicular Bisectors of the Sides of a Triangle 90
49 The Construction of the Parallels to a Line through a Point 93
50 The Construction of a Common Parallel to Two Intersecting Lines 97
51 The Construction of a Line Perpendicular to One of Two Intersecting Lines and Parallel to the Other 99
52 Units of Length and Angle 100
53 Associated Right Triangles 101
54 The Construction of a Triangle when Its Angles Are Given 105
55 The Absolute 107
56 Circles 109
57 Corresponding Points 110
58 Limiting Curves and Their Properties 113
59 Equidistant Curves and Their Properties 117
60 The Limiting Curve as Related to Circles and Equidistant Curves 119
61 Area 120
62 Equivalence of Polygons and Triangles 122
63 Measure of Area 127
64 The Triangle with Maximum Area 128
Chapter V Hyperbolic Plane Trigonometry
65 Introduction 131
66 The Ratio of Corresponding Arcs of Concentric Limiting Curves 131
67 Relations between the Parts of an Important Figure 136
68 A Coördinate System and another Important Figure 138
69 The Relations between Complementary Segments 140
70 Relations among the Parts of a Right Triangle 142
71 Relations among the Parts of the General Triangle 145
72 The Relation between a Segment and Its Angle of Parallelism 148
73 Simplified Formulas for the Right Triangle and the General Triangle 152
74 The Parameter 153
Chapter VI Applications of Calculus to the Solutions of Some Problems in Hyperbolic Geometry
75 Introduction 157
76 The Differential of Arc in Cartesian Coördinates 158
77 The Differential of Arc in Polar Coördinates 160
78 The Circumference of a Circle and the Lengths of Arcs of Limiting Curve and Equidistant Curve 161
79 The Area of a Fundamental Figure 163
80 Limiting Curve Coördinates 165
81 The Element of Area 166
82 The Area of a Circle 169
83 The Area of a Lambert Quadrilateral 169
84 The Area of a Triangle 171
Chapter VII Elliptic Plane Geometry and Trigonometry
85 Introduction 173
86 The Characteristic Postulate of Elliptic Geometry and Its Immediate Consequences 174
87 The Relation between Geometry on a Sphere and Elliptic Geometry 177
88 The Two Elliptic Geometries 179
89 Properties of Certain Quadrilaterals 180
90 The Sum of the Angles of a Triangle 182
91 The Trigonometry of the Elliptic Plane 185
92 The Trigonometric Functions of an Angle 185
93 Properties of a Variable Lambert Quadrilateral 190
94 The Continuity of the Function φ(x) 192
95 An Important Functional Equation 193
96 The Function φ(x) 194
97 The Relation among the Parts of a Right Triangle 196
Chapter VIII The Consistency of the Non-Euclidean Geometries
98 Introduction 201
99 The Geometry of the Circles Orthogonal to a Fixed Circle 204
100 The Nominal Length of a Segment of Nominal Line 206
101 Displacement by Reflection 208
102 Displacement in the Geometry of the Nominal Lines 209
103 The Counterparts of Circles, Limiting Curves and Equidistant Curves 213
104 The Relation between a Nominal Distance and Its Angle of Parallelism 214
105 Conclusion 217
Appendix
I The Foundation of Euclidean Geometry
1 The Definitions of Book I 218
2 The Postulates 219
3 The Common Notions 220
4 The Forty-Eight Propositions of Book I 220
II Circular and Hyperbolic Functions
5 The Trigonometric Functions 223
6 The Hyperbolic Functions 225
7 The Inverse Hyperbolic Functions 228
8 Geometric Interpretation of Circular and Hyperbolic Functions 229
III The Theory of Orthogonal Circles and Allied Topics
9 The Power of a Point with Regard to a Circle 232
10 The Radical Axis of Two Circles 232
11 Orthogonal Circles 234
12 Systems of Coaxal Circles 235
IV The Elements of Inversion
13 Inversion 237
14 The Inverse of a Circle and the Inverse of a Line 238
15 The Effect of Inversion on Angles 240
16 The Peaucellier Inversor 240
Index 245
