Euclid’s parallel postulate, in its modern reformulation, holds that, on a plane, given a line and a point not on the line, only one line can be drawn through the point parallel to the line. Gerolamo Saccheri (1667-1733) brilliantly attempted to prove this through a reductio ad absurdum argument. There were two ways to contradict the postulate: space could have 1) no parallel lines (straight lines in a plane will always meet if extended far enough), or 2) multiple straight lines through a given point parallel to a given line in the plane. These become non-Euclidean axioms. Saccheri convincingly achieved his reductio for the first possibility with the innocent assumption that straight lines are infinite [cf. Jeremy Gray, Ideas of Space Euclidean, Non-Euclidean, and Relativistic, Oxford, 1989; p. 64]. Later David Hilbert (1862-1953) would point out that the same reductio proof could be achieved by assuming that given three points on a line only one can be between the other two [David Hilbert and S. Cohn-Vossen Geometry and the Imagination (Anschauliche Geometrie--better translated Intuitive Geometry), Chelsea Publishing Company, 1952; p. 240]. For the second possibility, however, Saccheri did not achieve a good proof. And it was using just such an axiom that the first complete non-Euclidean geometries were achieved by Bolyai (1802-1860) and Lobachevskii (1792-1856).

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