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