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Published online by Cambridge University Press: 05 November 2012
Basically the actual content of the entire argument belongs to a higher area of the general abstract study of quantity, independent of the spatial, whose object is the combinations of quantities that are connected through continuity, an area that is presently still little developed, and in which we also cannot move without a language borrowed from spatial pictures.
C.-F. GAUSS (1849)The concept “two-dimensional manifold” or “surface” will not be associated with points in three-dimensional space; rather it will be a much more general abstract idea.
HERMANN WEYL (1913)The development of differential geometry led us to a set of requirements for a model of non-Euclidean geometry-a geodesically complete surface of constant negative curvature. Hilbert's Theorem (Theorem 12.11) proves that there is no example of such a surface that is a regular surface in ℝ3. To widen the search we turn to a more abstract notion of a surface. Other objects of geometric investigations in the 19th century suggested the need for a broader definition of surface. In 1808, Etienne Louis Malus (1775–1812) had proven that if a system of light rays emanating from a point are reflected in an arbitrary surface, then the system of rays will remain orthogonal to the surface of points given by the wave front, the set of points reached at a fixed time from the origin. Such a surface is determined implicitly and its properties derived from the optics, not from an explicit description. William Rowan Hamilton (1805–65) generalized Malus's Theorem in a context of higher dimensional spaces on which he based his analytical mechanics (Lützen 1995).
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