KLEIN, Felix Christian. Vergleichende Betrachtungen über neuere geometrische Forschungen.
Erlangen: Deichert, 1872. First edition, first printing.

A fine copy, unopened in original wrappers, of his famous unification of geometry through group theory - the ‘Erlanger Programm - which “was an outstanding landmark in the mathematics of the nineteenth century.” (Bell). Klein’s unification of projective and non-Euclidean geometry “brought full recognition at last to the neglected theories of Bolyai and Lobachevsky. Their logical consistency was now established.” (Struik).

Landmark Writings in Western Mathematics, no. 54.

“The conceptions grouped together under the name ‘Erlanger Programm’ were presented in 1872 in ‘Vergleichende Betrachtungen über neuere geometrische Forschungen’. This work reveals the early familiarity with the concept of group that Klein acquired chiefly through his contact with Lie and from C. Jordan. The essence of the ‘Erlanger Programm’ is that every geometry known so far is based on a certain group, and the task of the geometry in question consists in setting up the invariants of this group. The geometry with the most general group, which was already known, was topology; it is the geometry of the invariants of the group of all continuous transformations—for example, of the plane. Klein then successively distinguished the projective, the affine, and the equiaffine or principal group of the particular dimension; in certain cases the succeeding group is a subgroup of the previous one. To these groups belong the projective, affine, and equiaffine geometries with their invariants, whereby the equiaffine geometry is the same as the Euclidean elementary geometry.

“The non-Euclidean geometries accounted for with the aid of the Cayley-Klein models, as well as the various types of circular and spherical geometries devised by Moebius, Laguerre, and Lie, could likewise be viewed as the invariant theories of certain subgroups of the projective groups. In his later years Klein returned to the ‘Erlanger Programm’ and, in a series of works, showed how theoretical physics, and especially the theory of relativity, which had emerged in the meantime, can be understood on the basis of the ideas presented there. The ‘Programm’ was translated into six languages and guided much work undertaken in the following years: for example, the analytic geometry of Lothar Heffter, school instruction, and the lifelong efforts of W. Blaschke in differential geometry. Only later in the twentieth century was it superseded.” (DSB VII: 397).

Rosenfeld, A History of Non-Euclidean Geometry, pp. 338-44; Parkinson, Breakthroughs, p.381; Struik, A Concide History of Mathematics, pp.272; Boyer, A History of Mathematics, pp.592-94; Bell, The Development of Mathematics, pp.444-47; Gray, Worlds out of Nothing, pp.227-29; Klein, Mathematical Thought from Ancient to Modern Times, pp.917-20.

8vo: 234 x 153 mm. Original yellow printed wrappers, lower corners slightly chipped, otherwise fine. Uncut and unopened. Rare in such fine condition. 48 pp.