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.

Contenu

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
7. Introduction
8. Euclid's Choices
9. Posidonius and His Followers
10. Ptolemy and Proclus
11. Saccheri
12. Lambert
13. Legendre
14. The Discovery of Non-Euclidean Geometry
    Hyperbolic Geometry
    III Parallels With a Common Perpendicular
15. Introduction
16. The Basis E
17. The Initial Theorems of Hyperbolic Geometry
18. The Hyperbolic Parallel Postulate
19. Immediate Consequences of the Postulate
20. Further Properties of Quadrilaterals
21. Parallels With a Common Perpendicular
22. The Angle-Sum of a Triangle
23. The Defect of a Triangle
24. Quadrilaterals Associated with a Triangle
25. The Equivalence of Triangles
26. Area of a Triangle
27. Implications of the Area Formula
28. Circles
    IV Parallels Without a Common Perpendicular
29. Introduction
30. Parallels Without a Common Perpendicular
31. Properties of Boundary Parallels
32. Trilateral
33. Angles of Parallelism
34. Distance Between Two Lines
35. The Uniqueness of Parallels Without a Common Perpendicular
36. Perpendicular Bisectors of the Sides of a Triangle
    V Horocycles
37. Introduction
38. Corresponding Points
39. Definition of a Horocycle
40. Arcs and Chords of a Horocycle
41. Codirectional Horocycles
42. Arc Length on a Horocycle
43. Formulas Related to k-Arcs
    VI Triangle Relations
44. Introduction
45. Associated Right Triangles
46. Improved Angle of Parallelism Formulas
47. Remarks on the Trigonometric Functions
48. Right Triangle Formulas
49. Comparison with Euclidean Formulas
50. Formulas for the General Triangle
51. Hyperbolic Geometry in Small Regions
52. Hyperbolic Geometry and the Physical World
    Elliptic Geometry
    VII Double Elliptic Geometry
53. Introduction
54. Riemann
55. The Elliptic Geometries
56. Geometry on a Sphere
57. A Description of Double Elliptic Geometry
58. Double Elliptic Geometry and the Physical World
59. An Axiomatic Presentation of Double Elliptic Geometry
    VIII Single Elliptic Geometry
60. Intoduction
61. Geometry on a Modified Hemisphere
62. A Description of Single Elliptic Geometry
63. An Axiomatic Presentation of Single Elliptic Geometry
    Appendix: A Summary from Euclid's Elements
    Bibliography
    Answers to Selected Exercises
    Index