a non-Euclidean geometry that rejects the validity of Euclid's fifth, the “parallel,” postulate. Simply stated, this Euclidean postulate is: through a point not on a given line there is exactly one line parallel to the given line. In hyperbolic geometry, through a point not on a given line there are at least two lines parallel to the given line. The tenets of hyperbolic geometry, however, admit the other four Euclidean postulates.

Although many of the theorems of hyperbolic geometry are identical to those of Euclidean, others differ. In Euclidean geometry, for example, two parallel lines are taken to be everywhere equidistant. In hyperbolic geometry, two parallel lines are taken to converge in one direction and diverge in the other. In Euclidean, the sum of the angles in a triangle is equal to two right angles; in hyperbolic, the sum is less than two right angles. In Euclidean, polygons of differing areas can be similar; and in hyperbolic, similar polygons of differing areas do not exist.

The first published works expounding the existence of hyperbolic and other non-Euclidean geometries are those of a Russian mathematician, Nikolay Ivanovich Lobachevsky, who wrote on the subject in 1829, and, independently, the Hungarian mathematicians Farkas and János Bolyai, father and son, in 1831.