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.
The epoch-making work of János Bolyai is presented here, together with a supplement outlining Hungarian political and science history to help the reader to get acquainted with the miserable fate of János Bolyai and with his intellectual world.
A facsimile of a copy of Bolyai's original 1831 Scientia Spatii (also known as the Appendix) is included, together with a translation. Comments and notes, and a survey of the effects of his work, complete the volume.
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- © North Holland 1987
- 1st January 1987
- North Holland
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