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Non-Euclidean geometries are specific examples of modern geometries with different set of axioms from those of Euclidean Geometry. They differ from the geometry of Euclid because they substitute alternatives to his fifth postulate on parallels.
Non-Euclidean geometry provided Einstein with a suitable model for his work on relativity. While it also occurs in differential geometry and elsewhere, it is often studied to develop a real understanding of postulate, by its use of postulates that do not seem self-evident or practical.
From the time Euclid states his postulates, about 300 B.C., controversies arose concerning the fifth postulate on parallels. Many substitutes (including Playfair’s version) were suggested for it, but more significantly, mathematicians felt it was not a postulate at all. For almost 2000 years, many people attempted to show that the statement was actually a theorem that could be proved from the rest of the postulates. The work of Saccheri and Lambert continued the attempt at its proof. These two people used the indirect method of proposing an alternate parallel postulate and attempting the reach a contradiction. The fifth postulate was never proved, nor was a contradiction reached.
Shortly after 1800, Carl Gauss (1777-1855) began to realize that the fifth postulate could never be proved from the others, because it was indeed an independent postulate in the set of Euclidean postulates, not a theorem. Attempts to prove the postulate by denying it had already produced strange theorems that had to be accepted as valid if some other substitute postulate was to be used.
More than one person shares the credit for the discovery of non-Euclidean geometry. Though Gauss was aware of the significance of the subject, he did not publish any material. The first account of non-Euclidean geometry to be published was based on the assumption that, through a point not on a given line, more than one line can be drawn parallel to a given line in the plane. This type of geometry, called hyperbolic geometry, was discovered independently by a Russian, Nikolia Lobachevsky (1793-1856), and, at about the same time, by a Hungarian, Johann Bolyia (1802-1860). The results were published in 1830. Not long after the development of hyperbolic geometry, the German mathematician G.F.B. Riemann (1826-1866) suggested a geometry, now called elliptic geometry, based on the alternative to the fifth postulate, which states that there are no parallel to a line through a point not on the line.
Source:
Smart, James R. Modern Geometries. 3rd Edition. Belmont, CA: Brooks/Cole, 1988