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Historical
The origins of geometry	p. 1
Euclid's Elements; axioms and postulates	p. 2
The Parallel-postulate; attempts to prove it	p. 3
Playfair's Axiom	p. 4
Thibaut's rotation proof; the direction fallacy	p. 6
Bertrand's proof by infinite areas	p. 7
Equidistant lines	p. 9
First glimpses of non-euclidean geometry; Saccheri	p. 11
Lambert	p. 13
Gauss	p. 14
Schweikart, Taurinus and Wachter	p. 14
Legendre	p. 16
The theory of parallels in Britain	p. 19
The discovery of non-euclidean geometry; Lobachevsky	p. 20
Bolyai	p. 21
The later development	p. 24
Examples I	p. 25
Elementary Hyperbolic Geometry
Fundamental assumptions or axioms	p. 27
Parallel lines; the three geometries	p. 29
Definition of parallel lines	p. 30
Plane Geometry. Properties of parallelism	p. 31
Theorem on the transversal	p. 33
Theorem of the exterior angle	p. 34
The parallel-angle	p. 35
Theorems on the quadrilateral	p. 36
Intersecting lines are divergent	p. 37
Parallels are asymptotic in one direction and divergent in the other	p. 38
Parallel lines meet at infinity at a zero angle	p. 39
Non-intersectors; common perpendicular	p. 40
Classification of pairs of lines	p. 41
Solid Geometry. Planes, dihedral angles, etc.	p. 42
Systems of parallel lines, and lines perpendicular to a plane	p. 43
Pencils and bundles of lines	p. 45
Points at infinity; the absolute	p. 46
Ideal points	p. 47
Parallel lines and planes	p. 49
Principle of duality	p. 50
The circle; horocycle; equidistant-curve	p. 51
The sphere; horosphere; equidistant-surface	p. 52
Circles determined by three points or three tangents	p. 53
Geometry of a bundle of lines and planes	p. 55
Trigonometry. The circular functions	p. 56
Ratio of arcs of concentric horocycles	p. 57
The parallel-angle	p. 58
Two formulae for the horocycle	p. 61
The right-angled triangle; notation; complementary angles and segments	p. 63
Correspondence between rectilinear and spherical triangles	p. 63
Associated triangles	p. 65
Trigonometrical formulae for a right-angled triangle	p. 66
Engel-Napier rules	p. 67
Spherical trigonometry the same as in euclidean space	p. 69
Correspondence between a right-angled triangle and a tri-rectangular quadrilateral	p. 70
Napier's rules for a tri-rectangular quadrilateral	p. 74
Formulae for any triangle	p. 74
Euclidean geometry holds in the infinitesimal domain	p. 75
Circumference of a circle	p. 76
Sum of the angles of a triangle; defect and area	p. 77
Relation between the units of length and area; area of a sector and of a triangle by integration	p. 79
The maximum triangle	p. 81
Gauss' proof of the defect-area theorem	p. 82
Area of a polygon	p. 83
Another proof that the geometry on the horosphere is euclidean	p. 84
Examples II	p. 84
Elliptic Geometry
Absolute pole and polar	p. 88
Spherical and elliptic geometries	p. 89
The elliptic plane is a one-sided surface	p. 91
Absolute polar system	p. 92
Projective geometry; summary of theorems	p. 93
The absolute	p. 98
Principle of duality	p. 100
Relation between distance and angle	p. 101
Area of a triangle	p. 103
The circle; duality	p. 104
Cylinder; rectilinear generators	p. 105
Common perpendiculars to two lines in space	p. 106
Paratactic lines or Clifford's parallels	p. 108
Construction for common perpendiculars	p. 109
Paratactic lines cut the same two generators of the absolute	p. 110
Comparison between parataxy and euclidean parallelism	p. 111
Clifford's Surface	p. 112
Trigonometrical formulae; circumference of a circle	p. 114
The right-angled triangle	p. 116
Associated triangles	p. 117
Napier's Rules	p. 119
Spherical trigonometry the same as in euclidean space	p. 120
The trirectangular quadrilateral	p. 121
Examples III	p. 122
Analytical Geometry
Coordinates of a point; Weierstrass' coordinates	p. 125
The absolute	p. 127
Equation of a straight line; Weierstrass' line-coordinates	p. 128
Distance between two points	p. 129
The absolute in elliptic geometry	p. 130
Angle between two lines	p. 131
Distance of a point from a line	p. 132
Point of intersection of two lines; imaginary points	p. 132
Line joining two points	p. 134
Minimal lines	p. 134
Concurrency and collinearity	p. 135
The circle	p. 136
Coordinates of point dividing the join of two points into given parts	p. 137
Middle point of a segment	p. 138
Properties of triangles; centroid, in- and circum-centres	p. 139
Explanation of apparent exception in euclidean geometry	p. 139
Polar triangles; orthocentre and orthaxis	p. 141
Desargues' Theorem; configurations	p. 142
Desmic system	p. 144
Concurrency and collinearity	p. 145
Position-ratio; cross-ratio; projection	p. 147
Examples IV	p. 149
Representations of Non-Euclidean Geometry in Euclidean Space
The problem	p. 153
Projective Representation	p. 154
The absolute	p. 154
Euclidean geometry	p. 155
The circular points	p. 156
Expression of angle by logarithm of cross-ratio	p. 156
Projective expression for distance and angle in non-euclidean geometry	p. 157
Metrical geometry reduced to projective; Cayley-Klein	p. 158
Example: construction of middle points of a segment	p. 159
Classification of geometries with projective metric	p. 160
Distance in euclidean geometry	p. 161
Geometry in which the perimeter of a triangle is constant	p. 162
Extension to three dimensions	p. 163
Application to proof that geometry on the horosphere is euclidean	p. 164
Geodesic Representation	p. 165
Geometry upon a curved surface	p. 166
Measure of curvature	p. 166
Surfaces of constant curvature; Gauss' theorem	p. 168
The Pseudosphere	p. 168
The Cayley-Klein representation as a projection	p. 170
Meaning of Weierstrass' coordinates	p. 171
Conformal Representation. Stereographic projection	p. 172
The orthogonal circle or absolute	p. 174
Conformal representation	p. 175
Point-pairs	p. 175
Pencils of lines; concentric circles	p. 176
Distance between two points	p. 179
Motions	p. 179
Reflexions	p. 180
Complex numbers	p. 181
Circular transformation; conformal and homographic	p. 182
Inversion	p. 183
Types of motions	p. 185
The distance-function	p. 186
The line-element	p. 187
Simplification by taking fixed circle as a straight line	p. 188
Angle at which an equidistant-curve meets its axis	p. 189
Extension to three dimensions	p. 191
"Space-Curvature" and the Philosophical Bearing of Non-Euclidean Geometry
Four periods in the history of non-euclidean geometry	p. 192
"Curved space"	p. 193
Differential geometry; Riemann	p. 194
Free mobility of rigid bodies; Helmholtz	p. 195
Continuous groups of transformations; Lie	p. 197
Assumption of coordinates	p. 199
Space-curvature and the fourth dimension	p. 199
Proof of the consistency of non-euclidean geometry	p. 202
Which is the true geometry?	p. 203
Attempts to determine the space-constant by astronomical measurements	p. 203
Philosophy of space	p. 207
The inextricable entanglement of space and matter	p. 209
Radical Axes, Homothetic Centres, and Systems of Circles
Common points and tangents to two circles	p. 211
Power of a point with respect to a circle	p. 212
Power of a point with respect to an equidistant-curve	p. 213
Reciprocal property	p. 216
Angles of intersection of two circles	p. 218
Radical axes	p. 219
Homothetic centres	p. 221
Radical centres and homothetic axes	p. 221
Coaxal circles in elliptic geometry	p. 222
Homocentric circles	p. 224
Comparison with euclidean geometry	p. 226
Linear equation of a circle	p. 227
Systems of circles	p. 228
Correspondence between circles and planes in hyperbolic geometry; marginal images	p. 229
The marginal images of two planes intersect in the marginal image of the line of intersection of the planes	p. 231
Two planes intersect at the same angle as their marginal images	p. 231
Systems of circles	p. 232
Types of pencils of circles	p. 233
Examples VII	p. 235
Inversion and Allied Transformations
Conformal and circular transformations	p. 236
A circular transformation is conformal	p. 236
Every congruent transformation of space gives a circular transformation of the plane	p. 237
Converse	p. 238
The general circular transformation is compounded of a congruent transformation and a circular transformation which leaves unaltered all the straight lines through a fixed point	p. 239
Inversions and radiations	p. 240
Formulae for inversion	p. 241
Comparison with euclidean inversion	p. 244
Congruent transformations; transformation of coordinates	p. 245
Equations of transformation	p. 247
Position of a point in terms of a complex parameter	p. 248
Expression for congruent transformation by means of homographic transformation of the complex parameter	p. 248
Groups of motions	p. 249
Connection with quaternions	p. 250
Equation of a circle in terms of complex parameters	p. 251
Equation of inversion	p. 252
Examples VIII	p. 253
The Conic
Equation of the second degree	p. 256
Classification of conics	p. 257
Centres and axes; foci and directrices	p. 258
Focal distance property	p. 260
Focus-tangent properties	p. 261
Focus-directrix property	p. 262
Geometrical proof of the focal distance property	p. 262
Examples IX	p. 265
Index	p. 269
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The Elements of Non-Euclidean Geometry

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