Janos Bolyai Appendix (eBook)

Front Cover 1
Appendix: The Theory of Space 4
Copyright Page 5
CONTENTS 6
Preface to the English edition 8
Part I: Evolution of the space concept up to the discovery of non-Euclidean geometry 10
Chapter 1. From the empirical study of space to deductive geometry 12
Chapter 2. Attempts to prove Postulate 5 15
Chapter 3. Reviving investigations at the beginning of the 19th century 20
Chapter 4. The meditations of Gauss and their results 28
Chapter 5. The geometric investigations of Lobachevsky 30
Chapter 6. The mathematical studies of János Bolyai 32
Chapter 7. The discovery of absolute geometry 33
Part II: The absolute geometry of János Bolyai The Appendix 40
Appendix(facsimile) 42
Appendix (translation) 72
Explanation of signs 74
Chapter I. Parallelism (§§ 1—10) 76
Chapter II. The paracycle and the parasphere (§§ 11—24) 84
Chapter III. Trigonometry (§§ 25—31) 93
Chapter IV. Application of the methods of analysis, relation between geometry and reality (§§32—33) 103
Chapter V. Constructions (§§ 34—43) 110
Part III: Remarks 122
Chapter 1. The Hilbertian system of axioms for Euclidean geometry 124
Chapter 2. Remarks to §§ 1—43 128
Part IV: The work of Bolyai as reflected by subsequent investigations 178
Chapter I. The construction of geometry by elementary methods 182
1. Further investigations of János Bolyai in the field of absolute geometry 182
2. Elliptic geometry 183
3. The commentary literature 184
4. Foundation of hyperbolic plane geometry without using the axioms of continuity 186
Chapter II. The consistency of non-Euclidean geometries 193
5. On the proof of the consistency 193
6. Beltrami's model 195
7. The Cayley–Klein model 198
8. Poincaré's model 208
Chapter III. The effect of the discovery of Non-Euclidean geometry on recent evolution of mathematics 215
9. The formation and development of the concept of mathematical space 215
10. Axiomatic method and modern mathematics 219
Supplement 221
Literature 239
Supplementary Literature 240