Century of Geometry, 1830-1930: Epistemology, History, and by L. Boi, D. Flament, Jean-Michel Salanskis

By L. Boi, D. Flament, Jean-Michel Salanskis

Within the first 1/2 the nineteenth century geometry replaced greatly, and withina century it helped to revolutionizeboth arithmetic and physics. It additionally placed the epistemologyand the philosophy of technological know-how on a brand new footing. In thisvolume a legitimate assessment of this improvement is given byleading mathematicians, physicists, philosophers, andhistorians of technological know-how. This interdisciplinary procedure givesthis assortment a different personality. it may be used byscientists and scholars, however it additionally addresses a generalreadership.

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Surprisingly there are no indications whatsoever that Riemann knew more than superficially of Bolyai's and Lobachevskii's work and maybe even not at all Consequently he did not bother about the intimate potential connection between his and their considerations. I shall try to give the main arguments for this thesis which I have discussed more in detail elsewhere [Scholz 1982b]. It may even be surprising that in Riemann's inaugural lecture the only "modern reformer of geometry" cited by name was Legendre.

Vol. 2. Boston: Publish or Perish. J. Gray Faculty of Mathematics, Open University, Milton Keynes, MK7 6AA, England The theme of this conference which I wish to address is the status of geometry, in particular the introduction of group-theoretic ideas into geometry; I hope that what I shall say will provide an introduction to some of the papers to be presented later. I shall go on to describe how Poincard came to develop his ideas about non-Euclidean geometry at the very start of his career. More generally, I shall claim that non-Euclidean geometry was a decisive arena for the recognition of the importance that ideas of group theory play in geometry of any kind.

His paper on this topic had been entered for the prize offered by the Acadgmie des Sciences. He had obtained a spider's web (in his phrase) of triangles in the unit disc and needed to know if they ever overlapped s. Since their sides were generally curved, in fact arcs of circles perpendicular to the unit circle, he had decided to study them by straightening them out, thus obtaining a different figure, still bounded by the unit circle, but in which all the sides were straight. His realisation on boarding the bus was that this second picture was the Kleinian (or projective) one of non-Euclidean geometry and so, necessarily, his original picture must be.

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