János Bolyai Appendix

Evolution of the Space Concept up to the Discovery of Non-Euclidean Geometry. From the Empirical Study of Space to Deductive Geometry. Attempts to Prove Postulate 5. Reviving Investigations at the Beginning of the 19th Century. The Meditations of Gauss, and Their Results. The Geometric Investigations of Lobachevsky. The Mathematical Studies of János Bolyai. The Discovery of Absolute Geometry. II. The Absolute Geometry of János Bolyai: The Appendix. Facsimile. Translation. Explanation of Signs. Parallelism. The Paracycle and the Parasphere. Trigonometry. Application of the Methods of Analysis, Relation between Geometry and Reality. Constructions. III. Remarks. The Hilbertian System of Axioms for Euclidean Geometry. Remarks. IV. The Work of Bolyai as Reflected by Subsequent Investigations. The Construction of Geometry by Elementary Methods: Further Investigations of János Bolyai in the Field of Absolute Geometry. Elliptic Geometry. The Commentary Literature. Foundation of Hyperbolic Plane Geometry without Using the Axioms of Continuity. The Consistency of Non-Euclidean Geometries: On the Proof of the Consistency. Beltrami's Model. The Cayley-Klein Model. Poincaré's Model. The Effect of the Discovery of Non-Euclidean Geometry on Recent Evolution of Mathematics: The Formation and Development of the Concept of Mathematical Space. Axiomatic Method and Modern Mathematics. Supplement (by B. Szénássy). Literature. Supplementary Literature.