The Columbia Electronic Encyclopedia, 6th ed. Non-Euclidean Geometry and Curved Space.The results of these two types of non-Euclidean geometry are identical with those of Euclidean geometry in every respect except those propositions involving parallel lines, either explicitly or implicitly (as in the theorem for the sum of the angles of a triangle). Although hyperbolic geometry needed only to contradict the Parallel Postulate, spherical geometry doesnt get off so easy. The second alternative, which allows no parallels through any external point, leads to the elliptic geometry developed by the German Bernhard Riemann in 1854. The word geometry is derived from the Greek words ‘geo’ meaning Earth and ‘metrein’ meaning ‘To measure’. A rigorous development of metric and synthetic approaches to Euclidean and non-Euclidean geometries using an axiomatic format.
Euclidean geometry is based on different axioms and theorems. There are two main types of non-Euclidean geometries, spherical (or elliptical) and hyperbolic. Lobachevsky in 1826 and independently by the Hungarian Janos Bolyai in 1832. Euclidean geometry is the study of 2-Dimensional geometrical shapes and figures. This work of Gauss, published after his death in 1855, led many mathematicians to take non-Euclidean geometry. Spherical geometry is called elliptic geometry, but the space of elliptic geometry is really has points antipodal pairs on the sphere. Euclidean geometry in this classication is parabolic geometry, though the name is less-often used. While spherical geometry has no parallels, in saddle geometry many lines can be drawn through the same point, all parallel to the same line. The negatively curved non-Euclidean geometry is called hyperbolic geometry. Allowing two parallels through any external point, the first alternative to Euclid's fifth postulate, leads to the hyperbolic geometry developed by the Russian N. The figure shows a natural rock formation that is much like a triangle in saddle geometry. Non-Euclidean geometry is the study of geometry on surfaces which are not flat. Non-Euclidean geometry, branch of geometry in which the fifth postulate of Euclidean geometry, which allows one and only one line parallel to a given line through a given external point, is replaced by one of two alternative postulates.
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