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Euclid's fifth postulate is often stated in simpler yet equivalent forms, e.g. Playfair's axiom: Through a point not on a line there is exactly one line parallel to the given line.
For mathematicians, a noneuclidean geometry is a geometry based on the negation of Euclid's fifth. (Some disciplines consider any geometry that is not euclidean to be noneuclidean.)
Janos Bolyai , an Hungarian mathematician, and Nikolai Lobachevsky, a Russian mathematician, both independently published an account of a noneuclidean geometry. In this geometry, Euclid's fifth postulate is negated and it is assumed that there is an infinite number of lines that can be drawn parallel to a given line through a point. (Karl F. Gauss claims to have discovered this 20 years previously, but never published.) This geometry has come to be called hyperbolic geometry -- you can get a sense of this geometry by drawing triangles on the surface of a saddle; the sides of the triangle are arcs of circles.
Bernhard Riemann also created a noneuclidean geometry -- indeed an infinite number of them which are now called elliptic geometries. In these there are no parallel lines. (Consider geometry on the curved surface of a sphere.)
Felix Klein continued the work by classifying the geometries (and giving them their current names). He was able to fold all of these geometries into a system called projective geometry, where each geometry is described by varying certain parameters, and where all geometries share a basic set of axioms.
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