Vergleichende Betrachtungen üBer Neuere Geometrische Forschungen Klein, Christian Felix Geometry, Number Theory,Mathematics, Analysis, Algebra

THE FAMOUS 'ERLANGEN PROGRAM': THE UNIFICATION OF GEOMETRY. First edition, rare in the original printed wrappers, of Klein's 'Erlangen Program,' his most famous and influential work. 'Klein's most important achievements in geometry, however, were the projective foundation of the non-Euclidean geometries and the creation of the 'Erlangen Progamm'" (DSB). After lecturing at Göttingen for a year, Klein joined the faculty at the University of Erlangen in 1872. "As was the custom, Klein had to present an Inaugural Address . It is commonly confused with the Erlangen Program, but that was not the Address. Rather, the Erlangen Program was a pamphlet printed by Deichert in Erlangen and distributed to those who came to the Inauguration. A few copies were doubtless distributed to friends and colleagues abroad, and to some libraries, because that was customary at the time; but the informal nature of the publication partially accounts for the negligible response to the Erlangen Program in 1872" (Gray in Landmark Writings, p. 546). It "comprised a proposal for the unification of Euclidean geometry with the geometries that had been devised during the nineteenth century by mathematicians such as Karl Gauss, Nicolai Lobachevsky, Janos Bolyai and Bernhard Riemann. He showed that the different geometries are each associated with a separate 'collection' or 'group' of transformations. Seen in this way, the geometries could all be treated as individual members of one overall family, and from this very connection conclusions and inferences could be drawn. Klein . demonstrated that every individual geometry could be constructed purely projectively; he produced projective models for Euclidean, elliptic, and hyperbolic geometries. Much later in his life, Klein returned to the Erlangen progamme to apply it to problems in theoretical physics, with special relevance to the theory of relativity" (Hutchinson DSB, p. 393). "For Klein, his Erlangen Program was an attempt to create an underlying unity for what had become the fragmented discipline of geometry. He did this through his innovative use of the group concept, which was not then widely known . Mathematicians were accustomed to using transformations of figures, say to replace a figure with an equivalent but simpler one, or to choose more convenient coordinate axes. Klein shifted attention from the figures to the transformations, and argued that henceforth geometry should be about groups as well as the properties of shapes. So a geometric property was one that was invariant under all the operations of the group associated to that geometry. He specifically employed the idea of one group being a subgroup of another. This enabled him to fix a space but vary the group, either to introduce a new geometry or to recognise a known one in an unexpected setting . Klein recognised that by selecting a figure in a space and considering the subgroup that maps that figure to itself was a fundamental way to inter-relate geometries and so to find a unifying principle that would encompass all of geometry . Klein ended his Erlangen Program with a series of seven notes of varying length and significance. One is worth picking out. Note 5 referred to what Klein cautiously continued to call the 'so-called non-Euclidean geometry' in order to avoid debates with non-mathematicians. Non-Euclidean geometry was the subject of two important memoirs by Klein written on either side of the Erlangen Program which probably did more to convey the message of the Program than did his obscurely published pamphlet" (Gray, Worlds Out of Nothing (2011), pp. 235-6). RBH lists two copies, neither in the original printed wrappers. "Klein chose the title to his essay carefully: he intended first to review, and then to compare, a number of recent researches in different areas of geometry. He claimed no novelty for the way he treated specific topics; what was original was the unified viewpoint he offered and its suggestions for the direction of futu