Elementary Considerations relating to the Absolute

Non-Euclidean geometry in the narrowest sense is that system of geometry which is usually associated with the names of Lobachevskij and Bolyai, and which arose from the substitution for Euclid's parallel-postulate of a postulate admitting an infinity of lines through a fixed point not intersecting a given line, the two limits between the intersectors and the non-intersectors being called the parallels to the given line through the fixed point. In a wider sense, any system of geometry which denies one or more of the fundamental assumptions upon which Euclid's system is based is a non-euclidean geometry. Of special interest are, however, those which touch only the question of parallel lines ; and there exists, in addition to Lobachevskij's geometry, another, commonly associated with the name of Riemann, in which the parallels to any line through a fixed point are imaginary. The three geometries, Lobachevskij's, Euclid's, and Riemann's, thus form a trio characterised by the existence of real, coincident, or imaginary pairs of parallels through a given point to a given line. With reference to this criterion, a consistent nomenclature was introduced by Klein, who called these three geometries respectively Hyperbolic, Parabolic, and Elliptic.