Introduction to Non-Euclidean Geometry

An Introduction to Non-Euclidean Geometry covers some introductory topics related to non-Euclidian geometry, including hyperbolic and elliptic geometries. This book is organized into three parts encompassing eight chapters. The first part provides mathematical proofs of Euclid’s fifth postulate concerning the extent of a straight line and the theory of parallels. The second part describes some problems in hyperbolic geometry, such as cases of parallels with and without a common perpendicular. This part also deals with horocycles and triangle relations. The third part examines single and double elliptic geometries. This book will be of great value to mathematics, liberal arts, and philosophy major students.

Preface

Historical Introduction

I Euclid's Fifth Postulate

1. Introduction

2. Euclid's Stated Assumptions

3. The Extent of a Straight Line

4. Euclid's Theory of Parallels

5. Further Consequences of Postulate 5

6. Substitutes for Postulate 5

II Attempts to Prove the Fifth Postulate

1. Introduction

2. Euclid's Choices

3. Posidonius and His Followers

4. Ptolemy and Proclus

5. Saccheri

6. Lambert

7. Legendre

8. The Discovery of Non-Euclidean Geometry

Hyperbolic Geometry

III Parallels With a Common Perpendicular

1. Introduction

2. The Basis E

3. The Initial Theorems of Hyperbolic Geometry

4. The Hyperbolic Parallel Postulate

5. Immediate Consequences of the Postulate

6. Further Properties of Quadrilaterals

7. Parallels With a Common Perpendicular

8. The Angle-Sum of a Triangle

9. The Defect of a Triangle

10. Quadrilaterals Associated with a Triangle

11. The Equivalence of Triangles

12. Area of a Triangle

13. Implications of the Area Formula

14. Circles

IV Parallels Without a Common Perpendicular

1. Introduction

2. Parallels Without a Common Perpendicular

3. Properties of Boundary Parallels

4. Trilateral

5. Angles of Parallelism

6. Distance Between Two Lines

7. The Uniqueness of Parallels Without a Common Perpendicular

8. Perpendicular Bisectors of the Sides of a Triangle

V Horocycles

1. Introduction

2. Corresponding Points

3. Definition of a Horocycle

4. Arcs and Chords of a Horocycle

5. Codirectional Horocycles

6. Arc Length on a Horocycle

7. Formulas Related to k-Arcs

VI Triangle Relations

1. Introduction

2. Associated Right Triangles

3. Improved Angle of Parallelism Formulas

4. Remarks on the Trigonometric Functions

5. Right Triangle Formulas

6. Comparison with Euclidean Formulas

7. Formulas for the General Triangle

8. Hyperbolic Geometry in Small Regions

9. Hyperbolic Geometry and the Physical World

Elliptic Geometry

VII Double Elliptic Geometry

1. Introduction

2. Riemann

3. The Elliptic Geometries

4. Geometry on a Sphere

5. A Description of Double Elliptic Geometry

6. Double Elliptic Geometry and the Physical World

7. An Axiomatic Presentation of Double Elliptic Geometry

VIII Single Elliptic Geometry

1. Intoduction

2. Geometry on a Modified Hemisphere

3. A Description of Single Elliptic Geometry

4. An Axiomatic Presentation of Single Elliptic Geometry

Appendix: A Summary from Euclid's Elements

Bibliography

Answers to Selected Exercises

Index