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Non-Euclidean Geometry: A Critical and Historical Study of its Development

Roberto Bonola
Dover Publications
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The Basic Library List Committee recommends this book for acquisition by undergraduate mathematics libraries.

[Reviewed by
P. N. Ruane
, on

If this wasn’t the first book to provide an historical overview of non-Euclidean geometry, it certainly continues to be one of the most influential. Included in the bibliographies of many subsequent historical works, it has been described in many quarters as being a classic. Written in Italian in 1906, the English version was published in 1911, and then again by Dover in 1955. The copy under review is a recent re-print of that edition.

The narrative begins by describing how Greeks, Arabs, and many mathematicians of the renaissance, failed to convert Euclid’s parallel postulate to the status of a theorem by deducing it from his first four postulates. The work of a much later range of mathematicians then comes under scrutiny, and this group is described as being the ‘forerunners of non-Euclidean geometry’, whose principle members were Saccheri, Lambert, Legendre and Thibaut.

The founders of non-Euclidean geometry are said to be Schweikart, Taurinus and, unsurprisingly, Gauss, Bolyai and Lobachevsky. But it is the work of this last pair that receives most attention in this book. And, apart from the overview provided on their innovative geometrical thinking, the complete texts of their founding monographs are appended to Bonola’s study.

In discussion of later developments, Bonola treats the work of Riemann, Helmholz and Lie, whereby the impossibility of proving Euclid’s parallel postulate is established. In addition, there are fascinating appendices non-Euclidean Statics, Clifford’s parallels, and the independence of projective geometry from Euclid’s postulate.

In truth, there is an abundance material in this book that is not covered in any other single book of my acquaintance. And yet, because of the amount of analytic and historical detail supplied by Bonola, readers with no prior background knowledge of non-Euclidean geometry may benefit from reading it in conjunction with the books by Adler, Gray or Greenberg. Bonola’s book is one to which I shall return time and time again.


A New Look at Geometry, by Irving Adler

Ideas of Space, by J. J. Gray

Euclidean and Non-Euclidean Geometry, by Marvin J Greenberg.

Peter Ruane’s career was centred upon primary and secondary mathematics education.

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. (bis) BERNHARD FRIEDRICH THIBAUT (1776-1832)
Chapter III. The Founders of Non-Euclidean Geometry.
31-34. KARL FRIEDRICH GAUSS (1777-1855)
36-38. FRANZ ADOLF TAURINUS (1794-1874)
Chapter IV. The Founders of Non-Euclidean Geometry (Cont.).
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___________________follow