Non-Euclidean geometry
Non-Euclidean geometry
Euclid’s postulates for geometry included one, the parallel postulate, which was regarded from earliest times (perhaps even by Euclid himself) as less satisfactory than the others. This may be stated as follows (not Euclid’s original form but an equivalent one) Through a given point P not on a given line l there passes at most one line, in the plane of P and l, which does not intersect l. Here “line” means a straight line extended infinitely in both directions (not a line segment).
Attempts to prove the parallel postulate from the other postulates of Euclidean geometry were unsuccessful. The undertaking of Saccheri (1733) to make a proof by reductio ad absurdum of the parallel postulate by deducing consequences of its negation did, however, lead to his developing many of the theorems of what is now known as hyperbolic geometry. The proposal that this hyperbolic geometry, in which Euclid’s parallel postulate is replaced by its negation, is a system equally valid with the Euclidean originated with Bolyai and Lobachevsky (independently, c 1825). Proof of the self-consistency of hyperbolic geometry, and thus of the impossibility of Saccheri’s undertaking, is contained in results of Cayley (1859) and was made explicit by Klein in 1871; for the two-dimensional case another proof was given by Beltrami in 1868.
The name non-Euclidean geometry is applied to hyperbolic geometry and generally to any system in which one or more postulates of Euclidean geometry are replaced by contrary assumptions. (But geometries of more than three dimensions, if they otherwise follow the postulates of Euclid, are not ordinarily called non-Euclidean.)
Closely related to the hyperbolic geometry is the elliptic geometry, which was introduced by Klein on the basis of ideas of Riemann. In this geometry lines are of finite total length and closed, and every two coplanar lines intersect in a unique point.
Still other non-Euclidean geometries are given an actual application to physical space — or rather, space-time — in the General Theory of Relativity.
Contemporary ideas concerning the abstract nature of mathematics (q. v.) and the status of applied geometry have important historical roots in the discovery of non-Euclidean geometries. — A.C.
G. Saccheri,
Euclides Vindicatus, translated into English by G. B Halsted, Chicago and London, 1920.
H. P. Manning,
Non-Euclidean Geometry, 1901.
J. L. Coolidge,
The Elements of Non-Euclidean Geometry, Oxford. 1909.