The Elements Of Non Euclidean Geometry

Julian Lowell Coolidge

OXFORD AT THE CLARENDON PRESS | 1909

online: archive.org

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Chapters Include: Foundation For Metrical Geometry In A Limited Region; Congruent Transformations; The Three Hypotheses; The Introduction Of Trigonometric Formulae; Analytic Formulae; Consistency And Significance Of The Axioms; The Geometric And Analytic Extension Of Space; The Groups Of Congruent Transformations; Point, Line, And Plane Treated Analytically; The Higher Line Geometry; The Circle And The Sphere; Conic Sections; Quadric Surfaces; Areas And Volumes; Introduction To Differential Geometry; etc.

Introduction To Non Euclidean Geometry

Harold E. Wolfe

The Dryden Press | 1945

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Introduction to NON-EUCLIDEAN GEOMETRY by HAROLD E. WOLFE . PREFACE This book has been written in an attempt to provide a satisfactory textbook to be used as a basis for elementary courses in Non-Euclid ean Geometry. The need for such a volume, definitely intended for classroom use and containing substantial lists of exercises, has been evident for some time. It is hoped that this one will meet the re quirements of those instructors who have been teaching the subject tegularly, and also that its appearance will encourage others to institute such courses. x The benefits and amenities of a formal study of Non-Euclidean Geometry are generally recognized. Not only is the subject matter itself valuable and intensely fascinating, well worth the time of any student of mathematics, but there is probably no elementary course which exhibits so clearly the nature and significance of geometry and, indeed, of mathematics in general. However, a mere cursory acquaintance with the subject will not do. One must follow its development at least a little way to see how things come out, and try his hand at demonstrating propositions under circumstances such that intuition no longer serves as a guide. For teachers and prospective teachers of geometry in the secondary schools the study of Non-Euclidean Geometry is invaluable. With out it there is strong likelihood that they will not understand the real nature of the subject they are teaching and the import of its applications to the interpretation of physical space. Among the first books on Non-Euclidean Geometry to appear in English was one, scarcely more than a pamphlet, written in 1880 by G. Chrystal. Even at that early date the value of this study for those preparing to teach was recognized. In the preface to this little brochure, Chrystal expressed his desire to bring pangeometrical speculations under the notice of those engaged in the teaching of geometry He wrote It will not be supposed that I advocate the introduction of pan geometry as a school subject it is for the teacher that I advocate vi PREFACE such a study. It is a great mistake to suppose that it is sufficient for the teacher of an elementary subject to be just ahead of his pupils. No one can be a good elementary teacher who cannot handle his subject with the grasp of a master. Geometrical insight and wealth of geometrical ideas, either natural or acquired, are essential to a good teacher of geometry and I know of no better way of cultivat ing them than by studying pan geometry. Within recent years the number of American colleges and uni versities which offer courses in advanced Euclidean Geometry has increased rapidly. There is evidence that the quality of the teaching of geometry in our secondary schools has, accordingly, greatly improved. But advanced study in Euclidean Geometry is not the only requisite for the good teaching of Euclid. The study of Non-Euclidean Geometry takes its place beside it as an indispensable part of the training of a well-prepared teacher of high school geometry. This book has been prepared primarily for students who have completed a course in calculus. However, although some mathe matical maturity will be found helpful, much of it can be read profitably and with understanding by one who has completed a secondary school course in Euclidean Geometry. He need only omit Chapters V and VI, which make use of trigonometry and calcu lus, and the latter part of Chapter VII. In Chapters II and III, the historical background of the subject has been treated quite fully. It has been said that no subject, when separated from its history, loses more than mathematics. This is particularly true of Non-Euclidean Geometry...

Non-Euclidean Geometry: A Critical And Historical Study Of Its Development

Roberto Bonola

Chicago Open Court Pub. Co | 1912

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Examines various attempts to prove Euclid's parallel postulate — by the Greeks, Arabs and Renaissance mathematicians. Ranging through the 17th, 18th, and 19th centuries, it considers forerunners and founders such as Saccheri, Lambert, Legendre, W. Bolyai, Gauss, Schweikart, Taurinus, J. Bolyai and Lobachewsky. Includes 181 diagrams.

Table of Contents

Chapter I. The Attempts to prove Euclid's Parallel Postulate.

1-5. The Greek Geometers and the Parallel Postulate

6. The Arabs and the Parallel Postulate

7-10. The Parallel Postulate during the Renaissance and the 17th Century

Chapter II. The Forerunners on Non-Euclidean Geometry.

11-17. GEROLAMO SACCHERI (1667-1733)

18-22. JOHANN HEINRICH LAMBERT (1728-1777)

23-26. The French Geometers towards the End of the 18th Century

27-28. ADRIEN MARIE LEGENDRE (1752-1833)

29. WOLFGANG BOLYAI (1775-1856)

30. FRIEDRICH LUDWIG WACHTER (1792-1817)

30. (bis) BERNHARD FRIEDRICH THIBAUT (1776-1832)

Chapter III. The Founders of Non-Euclidean Geometry.

31-34. KARL FRIEDRICH GAUSS (1777-1855)

35. FERDINAND KARL SCHWEIKART (1780-1859)

36-38. FRANZ ADOLF TAURINUS (1794-1874)

Chapter IV. The Founders of Non-Euclidean Geometry (Cont.).

39-45. NICOLAI IVANOVITSCH LOBATSCHEWSKY (1793-1856)

46-55. JOHANN BOLYAI (1802-1860)

56-58. The Absolute Trigonometry

59. Hypotheses equivalent to Euclid's Postulate

60-65. The Spread of Non-Euclidean Geometry

Chapter V. The Later Development of Non-Euclidean Geometry.

66. Introduction

Differential Geometry and Non-Euclidean Geometry

67-69. Geometry upon a Surface

70-76. Principles of Plane Geometry on the Ideas of RIEMANN

77. Principles of RIEMANN'S Solid Geometry

78. The Work of HELMHOLTZ and the Investigations of LIE

Projective Geometry and Non-Euclidean Geometry

79-83. Subordination of Metrical Geometry to Projective Geometry

84-91. Representation of the Geometry of LOBATSCHEWSKY-BOLYAI on the Euclidean Plane

92. Representation of RIEMANN'S Elliptic Geometry in Euclidean Space

93. Foundation of Geometry upon Descriptive Properties

94. The Impossibility of proving Euclid's Postulate

Appendix I. The Fundamental Principles of Statistics and Euclid's Postulate.

1-3. On the Principle of the Lever

4-8. On the Composition of Forces acting at a Point

9-10. Non-Euclidean Statics

11-12. Deduction of Plane Trigonometry from Statics

Appendix II. CLIFFORD'S Parallels and Surface. Sketch of CLIFFFORD-KLEIN'S Problems.

1-4. CLIFFORD'S Parallels

5-8. CLIFFORD'S Surface

9-11. Sketch of CLIFFORD-KLEIN'S Problem

Appendix III. The Non-Euclidean Parallel Construction and other Allied Constructions.

1-3. The Non-Euclidean Parallel Construction

4. Construction of the Common Perpendicular to two non-intersecting Straight Lines

5. Construction of the Common Parallel to the Straight Lines which bound an Angle

6. Construction of the Straight Line which is perpendicular to one of the lines bounding an acute Angle and Parallel to the other

7. The Absolute and the Parallel Construction

Appendix IV. The Independence of Projective Geometry from Euclid's Postu

1. Statement of the Problem

2. Improper Points and the Complete Projective Plane

3. The Complete Projective Line

4. Combination of Elements

5. Improper Lines

6. Complete Projective Space

7. Indirect Proof of the Independence of Projective Geometry from the Fifth Postulate

8. BELTRAMI'S Direct Proof of this Independence

Appendix V. The Impossibility of proving Euclid's Postulate. An Elementary Demonstration of this Impossibility founded upon the Properties of the System of Circles orthogonal to a Fixed Circle.

1. Introduction

2-7. The System of Circles passing through a Fixed Point

8-12. The System of Circles orthogonal to a Fixed Circle

Index of Authors

The Science of Absolute Space and the Theory of Parallels