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A branch of mathematics concerned with relationships in space, defined in terms of points, lines, angles, planes, surfaces, and solids. The original application of the term was to measurements on the Earth’s surface—a practical art developed by the Egyptians that was theorised when Thales imported it into the Greek world, its foundations clarified by Euclid’s Elements (fourth centuryb.c.). Euclid used deductive logic to construct a series of theorems—further augmented by such writers as Apollonius and Archimedes— based on five axioms that seemed intuitively indubitable (although the fifth axiom, which relates to parallel lines, seemed less indubitable than the rest). Significant augmentations of geometric technique followed the reintroduction of Euclid’s work to Europe in theRenaissance, the most important being Rene´ Descartes’ development of an ‘‘analytical geometry’’ that permitted the graphical representation of algebraic relationships. Euclidean geometry continued to be considered a definitive description of spatial properties even though notions of ‘‘absolute space’’ were repeatedly challenged by idealistic notions of space as an artefact of perception.Immanuel Kant’s proposal that space was a necessary a priori construct deflected attention away from the possibility that actual space might be significantly different from perceived space—or, at least, from the possibility of ever finding out if that were the case.

When mathematicians began developing ‘‘non-Euclidean’’ geometries based on the variance of Euclid’s fifth postulate in the earlynineteenth century, suspicion was aroused that real space might be non-Euclidean. Carl Friedrich Gauss and Nikolai Ivanovich Lobatchevsky both made measurements of actual triangular relationships in the hope of finding a discrepancy in the sum of their angles that would offer evidence of a curvature in actual space. The formulation of an abstract philosophy of geometry by David Hilbert, which refused tomake assumptions about the properties of real space, did not long precede the development of the general theory of relativity by Albert *Einstein, which extrapolated Hermann Minkowski’s notion of a non-Euclidean ‘‘space-time continuum’’.

Coordinate geometry began to produce literary spinoff when the representation of time as an axis encouraged its mathematical consideration as a fourthdimension equivalent to distance in any of the three spatial dimensions, thus assisting the Notion of time travel. The relevant apologetic jargon was not developed, however, until the mathematical consideration of hypothetical two-dimensional and four-dimensional objects and spaces began to make extensive use of fiction as an imaginative aid, as in Edwin Abbott’s description of Flatland (1884, as by ‘‘ASquare’’). Abbott’s hope was that if readers could be persuaded to identify with the predicament of a two-dimensional being attempting to imagine the third dimension, they would then find it easier to appreciate the limitations of their own perception. He used satirical humour as a means to make his account of Flatland more interesting and engaging, and to reflect the essential oddness of the idea.Abbott’s contemporaries included C. H. Hinton, who made more elaborate attempts to popularise similar ideas in ‘‘A Plane World’’ (1886) and An Episode of Flatland (1907), and Alfred Taylor Schofield, whose Another World; or, The Fourth Dimension (1888) tackled the problem more directly. Claude Bragdon’s A Primer of Higher Space: The Fourth Dimension (1913) is a hectic combination of mathematicsand occultism, concluding with ‘‘Man the Square: A Higher Space Parable’’. This kind of speculative fiction became more than an intellectual game once the refinement of relativity theory required physicists to think in terms of a four-dimensional space-time—a problem sharpened considerably when theoretical physicists in search of a unifying theory began to hypothesise further spatial dimensions....
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