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Zur Nicht-Euklidischen Geometrie (+) Ueber die Nullstellen der hypergeometrischen Reihe.


Leipzig, B.G. Teubner, 1890. 8vo. Original printed wrappers, no backstrip and a small nick to front wrapper. In "Mathematische Annalen. Begründet durch Rudolf Friedrich Alfred Clebsch. XXXVII. [37]. Band. 4. Heft." Entire issue offered. Internally very fine and clean. [Klein:] Pp. 544-72; 573-90. [Entire issue: Pp. pp. 465-604]. ¶ First printing of Klein's important contribution to non-Euclidean geometry. Klein saw a fundamental unity in the subject of non-Euclidean geometry. Rather than a heterogeneous collection of abstruse mathematics, non-Euclidean geometry was in Klein's view a "concrete discipline".

For over two millennia geometry had been the study of theorems which could be proved from Euclid's axioms. However, in the beginning of the 19th century it was proved that there exist other geometries than that of Euclid. Motivated by the emergence of the new geometries of Bolyai, Lobachevsky, and Riemann, Klein proposed to define a geometry, not by a set of axioms, but instead in terms of the transformations that leave it invariant; according to Klein, a geometric structure consists of a space together with a particular group of transformations of the space. A valid theorem in that particular geometry is one that holds under this group of transformations. This controversial idea did not only give a more systematic way of classifying the different geometries, but also gave birth to new geometric structures such as manifolds.

Landmark Writtings in Western Mathematics 1640-1940, p.544-52.

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