By L. Boi, D. Flament, Jean-Michel Salanskis
Within the first 1/2 the nineteenth century geometry replaced appreciably, and withina century it helped to revolutionizeboth arithmetic and physics. It additionally positioned the epistemologyand the philosophy of technology on a brand new footing. In thisvolume a valid assessment of this improvement is given byleading mathematicians, physicists, philosophers, andhistorians of technological know-how. This interdisciplinary method givesthis assortment a distinct personality. it may be used byscientists and scholars, however it additionally addresses a generalreadership.
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Extra info for Century of Geometry, 1830-1930: Epistemology, History, and Mathematics
In his "Mdmoire sur les groupes de mouvements" his theme was the application of group theory to the results of Bravais and others on crystal lattices. idean geometry 2. However, there are others much more intimately involved with geometry as usually understood. Curiously, they all have almost the same date, the date of the publication of Riemann's "Ueber die Hypothesen, welche der Geometrie zu Grunde liegen": 1867=t=1. T h e y are three famous papers by Helmholtz: "Uber die thats~chlichen Grundlagen der Geometrie" (1866) , "Ueber die Thatsachen, die der Geometrie zu Grunde liegen" (1868), with its deliberate echo of Riemann's paper, and " T h e origin and meaning of geometric axioms" (1876, but originally given as a published lecture in 1870); Beltrami's "Saggio di interpetrazione della geometria non-euclidea" (1868), and Hou~l's Essai crilique sur ies principes fondamentauz de la g~omg~rie ~l~mentaire, etc, (1867).
J. Gray As you know, Lobachevskii even conducted an empirical investigation to see if space was Euclidean or non-Euclidean; the results were inconclusive, but the idea that such an investigation was necessary was revolutionary. It threatened one's interest in geometry because it raised the question of what the basic objects of one's spatial intuition are. If, after all, they are not to be described as Euclid had originally done, what, one might ask, was the value of teaching everyone Euclidean geometry.
The next point to mention in this context is Riemann's insight that the meromorphic functions on a curve can in fact be expressed as rational functions in two of them, say z and t, which, read as inhomogeneous coordinates in in P(2, C), let the curve be represented algebraically as F ( z , t ) = 0, F e C[z,t]. This makes it possible, as Riemann stated clearly, to study any compact complex curve from a purely algebraic birational point of view. In particular, the change of representing coordinates (z,t) to (z', t') is given by rational transformations in both directions.
Century of Geometry, 1830-1930: Epistemology, History, and Mathematics by L. Boi, D. Flament, Jean-Michel Salanskis