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Vergleichende Betrachtungen über neuere geometrische Forschungen.


Leipzig, B.G. Teubner, 1893. 8vo. Orig. printed wrappers (no backstrip) to Heft 1, 43. Bd. of "Mathematische Annalen", the whole issue pp. IV,144. Klein's paper: pp.63-100. Frontwrapper repaired with the same kind of paper and without loss of letters. The sewing somewhat loose. Frontwrapper loose. ¶ This is the second printing of Klein's famous "Erlanger Programm" having Klein's own improvements.

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.

The Erlanger Programm was translated into six languages in the following two decades, and it has had an immense influence on geometry up to and throughout the 20th century.

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