By Peter Pesic
Contents comprise "On the Hypotheses which Lie on the Foundations of Geometry" through Georg Friedrich Riemann; "On the evidence which Lie on the Foundations of Geometry" and "On the foundation and importance of Geometrical Axioms" via Hermann von Helmholtz; "A Comparative evaluate of contemporary Researches in Geometry" through Felix Klein; "On the distance thought of subject" by way of William Kingdon Clifford; "On the rules of Geometry" via Henri Poincaré; "Euclidean Geometry and Riemannian Geometry" through Elie Cartan; and "The challenge of house, Ether, and the sector in Physics" by Albert Einstein.
These remarkably available papers will attract scholars of contemporary physics and arithmetic, in addition to a person drawn to the origins and resources of Einstein's so much profound paintings. Peter Pesic of St. John's collage in Santa Fe, New Mexico, offers an advent, in addition to notes that supply insights into each one paper.
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Additional info for Beyond Geometry: Classic Papers from Riemann to Einstein
Is therefore absolutely continuous”; see Rosenfeld 1988, 198-199. Riemann’s last thoughts are his most provocative, even today. By directing our attention to the possibility that solid bodies and rays of light “lose their validity in the infinitely small,” he indicates the way in which the presumed continuity of space might finally be grounded on something not continuous. It is purely an assumption, he notes, that bodies exist independently of position and hence can be of any size (as is not true in Lobachevsky’s geometry, for instance).
At least in the nearby region, the distance between two points should not vary observably if both points are shifted the same amount. The qualification that it is unchanged “to first order” means that the distance cannot have any term like dx in it, but might have in it higher-order terms like (dx)2 or (dx)3, etc. Note that if dx is itself very small, then the higher-order terms are extremely small. To require that a quantity is unchanged in this way is the same as the requirement that a function be a maximum or minimum: one is requiring that one is “at the bottom” of a trough or “at the top” of a peak, where the first derivatives vanish (and hence there is no change to order dx, as stated).
He may have been aware of Thomas Young’s three-receptor theory of color vision (1802), perhaps also of the early work of Hermann von Helmholtz (1852) and Hermann Grassmann (1853); for a helpful collection, see MacAdam 1970, 51-60. In the same year as Riemann’s lecture (1854), James Clerk Maxwell took up this color theory and produced the first color photograph (1861); see Everitt 1975, 63-72, and MacAdam 1970,62-83. Helmholtz later argued that any perceived color can be specified by three quantities (hue, saturation, and luminosity); see his 1866 magnum opus on physiological optics Helmholtz 1962, 2:120-146, excerpted in MacAdam 1970, 84-100; Erwin Schrodinger pointed out that this manifold has a non-Euclidean geometry; his paper is included in MacAdam 1970, 134-193.