Introduction to Non-Euclidean Geometry

By: HAROLD E WOLFE

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Paperback | 30 November 2012

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254 Pages

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One of the first college-level texts for elementary courses in non-Euclidean geometry, this concise, readable volume is geared toward students familiar with calculus. A full treatment of the historical background explores the centuries-long efforts to prove Euclid's parallel postulate and their triumphant conclusion. Numerous original exercises form an integral part of the book.Topics include hyperbolic plane geometry and hyperbolic plane trigonometry, applications of calculus to the solutions of some problems in hyperbolic geometry, elliptic plane geometry and trigonometry, and the consistency of the non-Euclidean geometries. Extensive appendixes offer background information on the foundation of Euclidean geometry, circular and hyperbolic functions, the theory of orthogonal circles and allied topics, and the elements of inversion.

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Non-Fiction Science Science in General History Mathematics Geometry Non-Euclidean Geometry

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

Introduction to Non-Euclidean Geometry

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