About the book

This book gives a systematic presentation of the fundamentals of Euclidean geometry, non-Euclidean geometry of Lobachevsky and Riemann, projective geometry and the geometrical aspects of spe­cial relativity theory. It also gives a general idea about constant curvature geometries. The subject matter has been shaped by the needs of the students and by my own experience.

A few words on the use of the book as a textbook in a course of the foundations of geometry are in order. The subject matter is distribut­ed over the first two parts. It is presented systematically throughout the book and practically does not omit any of the details of the ar­guments (except for the proofs of certain theorems of elementary geometry). It is evident that in lectures such a detailed presentation would be unprofitable (even if many hours were assigned to this course). The most difficult portion is Chapter 2 in the first part of the book; I feel that in lectures one should discuss from this chapter the formulations of the axioms and present examples of rigorous proofs of certain theorems; moreover, the most important aspects, for exam­ple, the measurement of length, the equivalence of Archimedes’ and Cantor’s axioms to Dedekind’s axiom and the significance of these axioms for the substantiation of analytic geometry, should be discussed in detail. The proof of the majority of the initial theorems of elementary geometry should be left for self-study.

Contents

Preface to the English Edition 9

Part 1. The Foundations of Geometry

Chapter 1.A Short Review of Investigations into the Foundations of Geometry

1.1 Euclid’s definitions, postulates, and axioms

1.2 The fifth postulate 16

1.3 Lobachevsky and his geometry 33

1.4 The formation of the notion of geometrical space 36

Chapter 2.The Axioms of Elementary Geometry

2.1 Geometrical elements 42

2.2 Group I: axioms of incidence 42

2.3 Group II: axioms of betweenness 45

2.4 Corollaries of the axioms of incidence and betweenness 45

2.5 Group III: axioms of congruence 53

2.6 Corollaries of Axioms I-III 57

2.7 Group IV: axioms of continuity 69

2.8 Group V: the axiom of parallelism. Absolute and divergent lines 96

Chapter 3. The Non-Euclidean Theory of Parallels

3.1 Lobachevsky’s definition of parallels 82

3.2 Peculiarities in the location of parallel lines 85

3.3 Lobachevsky’s function II(x) 100

3.4 Lines and planes in Lobachevskian space 104

3.5 Equidistants and oricycles 111

3.6 Equidistant surfaces and horospheres 121

3.7 Elementary geometry on surfaces in Lobachevskian space 126

3.8 Area of a triangle 136

3.9 Proof of logical consistency of Lobachevskian geometry 146

3.10 The basic metric relationships in Lobachevskian geometry 165

3.11 Riemannian geometry: a brief survey 178

Chapter 4. Investigation of the Axioms of Elementary Geometry

4.1 Three basic problems of axiomatic theory 188

4.2 Consistency of the axioms of Euclidean geometry 191

4.3 Proof of independence of some axioms of Euclidean geometry 205

4.4 Axiom of completeness 216

4.5 The completeness of axioms of Euclidean geometry 220

4.6 The axiomatic method in mathematics

Part 2. Projective Geometry

Chapter 5. The Foundations of Projective Geometry

5.1 The subject matter of projective geometry 223

5.2 Desargues’ theorem. Harmonic sets of elements 226

5.3 Order of points on the projective line 231

5.4 Division of harmonic pairs. Continuity of the harmonic correspondence 244

5.5 The axiom of continuity. Projective system of coordinates on a line 252

5.6 Projective coordinate systems on a plane and in space 258

5.7 Projective correspondence between elements of one-dimensional manifolds 270

5.8 Projective correspondences between two- and three-dimensional manifolds 283

5.9 Analytic representation for projectivities. Involution 292

5.10 Transformation formulas for projective coordinates. Cross ratio of four elements 300

5.11 The principle of duality 316

5.12 Algebraic curves and pencils. Algebraic surfaces and bundles. Complex projective plane and complex projective space 325

5.13 Images of second degree. The theory of polars 337

5.14 Constructive theorems and problems of projective geometry 346

Chapter 6. Group-Theoretic Principles in Geometry. Groups of Transformations

6.1 Geometry and the theory of groups 362

6.2 The projective group and its basic subgroups 391

6.3 Geometries of Lobachevsky, Riemann, and Euclid in the projective setting 408

Chapter 7. Minkowski Space

7.1 Multidimensional affine space 425

7.2 Euclidean spaces and Minkowski space 440

7.3 The space of events of the special relativity theory 455

Chapter 8. Differential Properties of Non-Euclidean Metric

8.1 Metric form of the Euclidean plane 474

8.2 The distance between two points on the Lobachevskian plane 478

8.3 Metric form of the Lobachevskian plane 489

8.4 Intrinsic geometry of a surface and the Beltrami problem 505

8.5 Geometry on a surface of constant curvature 511

8.6 Derivation of basic metric relations in Lobachevskian geometry 523

Part 3. Constant Curvature Geometry

Chapter 9. Spatial Forms in Constant Curvature Geometry

9.1 Two-dimensional manifolds with differential-geometric metric 529

9.2 Parabolic spatial forms 537

9.3 Elliptic spatial forms 544

9.4 Hyperbolic spatial forms 547

Name Index 553