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Ueber die sogenannte Nicht-Euklidische Geometrie (+) Ueber die sogenannte Nicht-Euklidische Geometrie. (Zweiter Aufsatz).


Leipzig, B.G.Teubner, 1871 u. 1873. Bound in 2 later full cloth. Small stamp on foot of titlepages.In. "Mathematische Annalen. In Verbindung mit C. Neumann begründet durch Rudolf Friedrich Alfred Clebsch", IV. und VI. Band. (4),637 pp. a. (4),642 pp., 6 plates. Klein's papers: pp. 573-625 a. pp. 112-145. Both volumes offered. ¶ First edition of these 2 papers which unifies the Euclidean and Non-Euclidean geometries, by reducing the differences to expressions of the "distance function", and introducing the concepts "parabolic", "elliptic" and "hyperbolic" for the geometries of Euclid, Riemann and of Lobatschewski, Gauss and Bolyai. He further eliminates Euclid's parallel-axiom from projective geometry, as he shows that the quality of being parallel, is not invariant under projections.
Klein build his work on Cayley's "distant measure" saying, that "Metrical properties are not properties of the figure per se but of the figure in relation to the absolute." This is Cayley's idea of the general projective determination of metrics. The place of the metric concept in projective geometry and the greater generality of the latter were described by Cayley as "Metrical geometry is part of projective geometry." Cayley's idea was taken over by Felix Klein....It seemed to him to be possible to subsume the non-Euclidean geometries, hyperbolic and double elliptic geometry, under projective geometry by exploring Cayley's idea. He gave a sketch of his thoughts in a paper of 1871, and then developed them in two papers (the papers offered here).Klein was the first to recognize that we do not need surfaces to obtain models of non-Euclidean geometries....The import which gradually emerged from Klein's contributions was that projective geometry is really logically independent of Euclidean geometry....By making apparent the basic role of projective geometry Klein paved the way for an axiomatic development which could start with projective geometry and derive the several metric geometries from it."(Morris Kline).
The offred volumes cntains other importen mathematical papers by f.i. by Klebsch, Lipschitz, Neumann, Noether, Thomae, Gordan, Lie, Du Bois-Raymond, Cantor (Über trigonometrische Reihen),etc.
(Sommerville: Bibliography of Non-Euclidean Geometry p. 45 a. 49.)

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