# Foundation of Euclidean and Non-Euclidean Geometries according to F. Klein, Volume 97

## 1st Edition

**Authors:**L. Redei

**Editors:**I. N. Sneddon M. Stark

**eBook ISBN:**9781483282701

**Imprint:**Pergamon

**Published Date:**1st January 1968

**Page Count:**410

**View all volumes in this series:**International Series in Pure and Applied Mathematics

## Table of Contents

Preface

Chapter I Axioms

§ 1. Axioms of Incidence

§ 2. Axioms of Betweenness

§ 3. Axiom of Continuity

§ 4. Axioms of Motion

Chapter II Consequences of the System of Axioms I

§ 5. Simple Properties of Straight Lines and Planes

§ 6. Desargues Configurations

§ 7. Linear Subspaces

§ 8. The Lattice of Linear Subspaces

§ 9. Basic Projective Configurations

§ 10. Projection and Intersection

Chapter III Simple Consequences of the Systems of Axioms I—II

§ 11. Segments. Triangles

§ 12. Properties of Segments

§ 13. Linear Ordering

§ 14. Properties of Triangles

§ 15. The Tetrahedron

§ 16. Neighbourhoods

§ 17. Validity of the Systems of Axioms I—II for the Basic Domain ℛ

§ 18. Generalization of the Notion of Space

§ 19. The Extension and Restriction of Spaces

Chapter IV Projective Closure

§ 20. Half-Subspaces

§ 21. Half-Pencils. Angles

§ 22. Some Properties of Pencils and Bundles

§ 23. Coplanar Desargues Configurations

§ 24. Improper Pencils of Lines

§ 25. Improper Bundles of Lines

§ 26. The Projective Closure ℛ of ℛ

§ 27. The Projective Axioms

§ 28. The General Case

Chapter V Investigation of the Projective Space

§ 29. Preliminaries

§ 30. Theorem of Duality in Projective Space

§ 31. Collineations

§ 32. The Erlangen Programme

§ 33. Theorem of Duality of the Plane

§ 34. Perspectivities and Projectivities

§ 35. Central Collineations of the Plane

§ 36. Separation

§ 37. Cyclic Ordering

§ 38. Projective Segments and Angles

§ 39. Complete Quadrangles. Harmonic Points

§ 40. Preliminaries About Coordinate Systems

§ 41. Coordinates in Projective Scales

§ 42. Halving a Projective Scale

§ 43. Coordinates for Dyadic Sets of Points on a Line

Chapter VI Consequences of the Systems of Axioms I, II, III

§ 44. Preliminaries

§ 45. Theorem Concerning the Infinite Point

§ 46. Coordinates in an Affine Line

§ 47. Coordinates on the Basic Projective Configurations of the First Degree

§ 48. Point-Coordinates in an Affine Plane

§ 49. The Fundamental Theorem of Projective Geometry

§ 50. Point-Coordinates in an Affine Space

§ 51. Vectors

§ 52. Homogeneous Point- and Plane-Coordinates in Space. Point- and Line-Coordinates in a Plane

§ 53. Determination of All Collineations of the Space

§ 54. Determination of the Coordinate Transformations of Space

§ 55. Transformation of Projective Coordinates

§ 56. Cross Ratio

§ 57. Imaginary Points

§ 58. Fixed Elements of Projectivities

§ 59. Involutions

§ 60. Involutory Collineations of a Plane

Chapter VII Consequences of the Systems of Axioms I, II, III, IV

§ 61. Extended Motions

§ 62. The Comparability of Segments

§ 63. Reflections and Rotations. Absolute Polar Plane

§ 64. Metric Scales. Infinite and Ultra-Infinite Points. Elliptic, Parabolic and Hyperbolic Geometries

§ 65. Absolute Involution of Points on a Proper Line

§ 66. Midpoint and Bisector

§ 67. The Lines Perpendicular to a Proper Plane

§ 68. Motions as Products of Reflections

§ 69. Polarities with Respect to Surfaces and Curves of the Second Order

§ 70. The Absolute Configuration in the Elliptic Case

§ 71. The Absolute Configuration in the Hyperbolic Case

§ 72. Characterization of Motions in the Non-Parabolic Case

§ 73. The Absolute Configuration and Characterization of Motions in the Parabolic Case

§ 74. Formulae of Motion of the Three Geometries

§ 75. The Consistency of the Three Geometries

§ 76. Measuring of Segments

§ 77. Measuring of Angles

§ 78. Applications to Trigonometry

Bibliography

Index

## Description

Foundation of Euclidean and Non-Euclidean Geometries according to F. Klein aims to remedy the deficiency in geometry so that the ideas of F. Klein obtain the place they merit in the literature of mathematics.

This book discusses the axioms of betweenness, lattice of linear subspaces, generalization of the notion of space, and coplanar Desargues configurations. The central collineations of the plane, fundamental theorem of projective geometry, and lines perpendicular to a proper plane are also elaborated. This text likewise covers the axioms of motion, basic projective configurations, properties of triangles, and theorem of duality in projective space. Other topics include the point-coordinates in an affine space and consistency of the three geometries.

This publication is beneficial to mathematicians and students learning geometry.

## Details

- No. of pages:
- 410

- Language:
- English

- Copyright:
- © Pergamon 1968

- Published:
- 1st January 1968

- Imprint:
- Pergamon

- eBook ISBN:
- 9781483282701