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First edition in French, and the first translation, of Riemann's epoch-making lecture on the foundations of geometry, one of the most important works in the history of mathematics. Although it was presented in 1854, Riemann's lecture did not appear in print "until 1868, two years after the author's death, in part since Riemann made no particular effort to publish it" (Knoebel, Mathematical Masterpieces, 217). "To complete his Habilitation Riemann had to give a lecture. He prepared three lectures, two on electricity and one on geometry. Gauss had to choose one of the three for Riemann to deliver and, against Riemann's expectations, Gauss chose the lecture on geometry. Riemann's lecture "Über die Hypothesen welche der Geometrie zu Grunde liegen", delivered on 10 June 1854, became a classic of mathematics. There were two parts to Riemann's lecture. In the first part he posed the problem of how to define an n-dimensional space and ended up giving a definition of what today we call a Riemannian space. Freudenthal writes in [DSB]: 'It possesses shortest lines, now called geodesics, which resemble ordinary straight lines. In fact, at first approximation in a geodesic coordinate system such a metric is flat Euclidean, in the same way that a curved surface up to higher-order terms looks like its tangent plane. Beings living on the surface may discover the curvature of their world and compute it at any point as a consequence of observed deviations from Pythagoras's theorem.' In fact the main point of this part of Riemann's lecture was the definition of the curvature tensor. The second part of Riemann's lecture posed deep questions about the relationship of geometry to the world we live in. He asked what the dimension of real space was and what geometry described real space. The lecture was too far ahead of its time to be appreciated by most scientists of that time. Monastyrsky writes in [Riemann, Topology, and Physics, 1987]: 'Among Riemann's audience, only Gauss was able to appreciate the depth of Riemann's thoughts . . . The lecture exceeded all his expectations and greatly surprised him. Returning to the faculty meeting, he spoke with the greatest praise and rare enthusiasm to Wilhelm Weber about the depth of the thoughts that Riemann had presented.' It was not fully understood until sixty years later. Freudenthal writes in [DSB]: 'The general theory of relativity splendidly justified his work. In the mathematical apparatus developed from Riemann's address, Einstein found the frame to fit his physical ideas, his cosmology, and cosmogony: and the spirit of Riemann's address was just what physics needed: the metric structure determined by data'" (MacTutor). 4to, pp. [iv], 338. Contemporary half-calf and marbled boards, spine lettered in gilt (rubbed, library stamp on recto and verso of title, bookplates on front paste-down). Codice articolo ABE-1595071730088
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