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Über die sogenannte Nicht-Euclidische Geometri. (Erster-) Zweiter Aufsatz.


(Leipzig, B.F. Teubner, 1871 a. 1873). Without wrappers, (wrappers blank to Second Part) as published in "Mathematische Annalen. Hrsg. von Felix Klein, Walter Dyck, Adolph Mayer." Vol. IV, pp. 573-625 and vol. VI, pp. 112-145. Kept in a cloth-portfolio. ¶ First edition. In these groundbreaking papers Klein established that if Euclidean geometry is consistent then non-Euclidean geometry is consistent as well and he introduces the adjectives "parabolic", "elliptic", and "hyperbolic" for the respective geometries of Georg Riemann, of Nicolai Lobachevsky, of C.F. Gauss and Janos Bolyai.
"Cayley's idea (that metrical geometry is part of projective geometry) was taken over by Felix Klein (1849-1925) and generalized so as to include the non-Euclidan geometries. Klein, a professor at Göttingen, was one of the lading mathematicians in Germany during the last part of the nineeeeteenth and first part of the twentieth century. During the years 1869-70 he larned the work of Lobatchevsky, Bolyai, von Staudt, and Cayley; however, even in 1871he did not know Laguerre's result. It seemed to him to be posible to subsume the non-Euclidean geometries, hyperbolic, and double elliptic geometry, under projective geometry byexploiting Cayley's idea. He gave a sketch og his thoughts in a paper of 1871, and then developed them in two papers (1871 a. 1873, the ppers offered here). Klein was the first to obtain models of non-Euclidean geometries." (Morris Kline). - Sommerville, Bibliography of Non-Euclidean Geometry p.45 (1871) and p. 49 (1873).

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