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

A source book in mathematics

David Eugene Smith

New York : McGraw-Hill Book Co. | 1929

online: archive.org

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Vol. 1

Dover Publications Inc. | 1993 | 324 páginas | djvu | 10,87 Mb

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The writings of Newton, Liebniz, Pascal, Riemann, Bernoulli, and others in a comprehensive selection of 125 treatises, articles from the Renaissance to end of the 19th century—most unavailable elsewhere. Grouped in five sections: Number; Algebra; Geometry; Probability; and Calculus, Functions, and Quaternions. Index. 83 illustrations.

Vol. 2

Dover Publications Inc. | 1993 | 418 páginas | djvu | 12,53Mb

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A Long Way from Euclid

Constance Reid

 Thomas Y. Crowell Company | 1834

online: archive.org

Dover Publications | 2004 | 304 páginas

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This lively guide by a prominent historian focuses on the role of Euclid's Elements in mathematical developments of the last 2,000 years. No mathematical background beyond elementary algebra and plane geometry is necessary to appreciate the clear and simple explanations, which are augmented by more than 80 drawings. 1963 edition. 

The Canterbury Puzzles with Solutions

Henry Ernest Dudeney

W. Heinemann | 1907 |195 páginas  |

online: archive.org
gutenberg.org

PDF | 14 MB

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This book includes 110 puzzles, not as individual problems but as incidents in connected stories. The first 31 are amusingly posed by pilgrims in Chaucer's Canterbury Tales. Additional puzzles are presented using different characters. Many require only the ability to exercise logical or visual skills; others offer a stimulating challenge to the mathematically advanced.

The foundations of mathematics; a contribution to the philosophy of geometry

Paul Carus

Chicago, The Open Court Publishing Co. | 1908

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Magic squares and cubes

William Symes Andrews

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In the introduction to Magic Squares and Cubes, W.S. Andrews wrote writes, "The study of magic squares probably dates back to prehistoric times. Examples have been found in Chinese literature written about A. D. 1125 which were evidently copied from still older documents. It is recorded that as early as the ninth century magic squares were used by Arabian astrologers in their calculations of horoscopes, etc. Hence, the probable origin of the term magic, which has survived to the present day." He added that "a magic square consists of a series of numbers so arranged in a square that the sum of each row and column and of both the corner diagonals shall be the same amount which may be termed the summation.

Mathematics and the Imagination

Edward Kasner & James Newman

G. Bell & Sons Ltd.| 1949 | 393 páginas |  pdf | 52,1 Mb

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Anyone who gambles, plays cards, loves puzzles, or simply seeks an intellectual challenge will love this amusing and thought-provoking book. With wit and clarity, the authors deftly progress from simple arithmetic to calculus and non-Euclidean geometry. "Charming and exciting." — Saturday Review of Literature. Includes 169 figures.