non-Euclidean geometry those axiomatized versions of geometry in which the parallel axiom of Euclidean geometry is rejected, after so many unsuccessful attempts to prove it. As in so many branches of mathematics, C. F. Gauss had thought out much of the matter first, but he kept most of his ideas to himself. As a result, credit is given to J. Bolyai and N. Lobachevsky, who worked independently from the late 1820s. Instead of assuming that just one line passes through a point in a plane parallel to a non-coincident coplanar line, they offered a geometry in which a line admits more than one parallel, and the sum of the ‘angles’ between the ‘sides’ of a ‘triangle’ lies below 180°. Then in mid-century G. F. B. Riemann conceived of a geometry in which lines always meet (so no parallels), and the sum of the ‘angles’ exceeds 180°. In this connection he distinguished between the unboundedness of space as a property of its extent, and the special case of the infinite measure over which distance might be taken (which is dependent upon the curvature of that space). Pursuing the (published) insight of Gauss, that the curvature of a surface could be defined in terms only of properties dependent solely on the surface itself (and later called ‘intrinsic’), Riemann also defined the metric on a surface in a very general and intrinsic way, in terms of the differential arc length. Thereby he clarified the ideas of ‘distance’ that his non-Euclidean precursors had introduced (drawing on trigonometric and hyperbolic functions); arc length was now understood geodesically as the shortest ‘distance’ between two ‘points’ on a surface, and was specified independent of any assumptions of a geometry within which the surface was embedded. Further properties, such as that pertaining to the ‘volume’ of a three-‘dimensional’ solid, were also studied. The two main types of non-Euclidean geometry, and its Euclidean parent, may be summarized as follows: OF Reaction to these geometries was slow to develop, but their impact gradually emerged. As mathematics, their legitimacy was doubted; but in 1868 E. Beltrami produced a model of a Bolyai-type two-dimensional space inside a planar circle. The importance of this model was to show that the consistency of this geometry depended upon that of the Euclidean version, thereby dispelling the fear that it was an inconsistent flash of the imagination. During the last thirty years of the nineteenth century a variety of variant geometries were proposed, and the relationships between them were studied, together with consequences for projective geometry. On the empirical side, these geometries, and especially Riemann’s approach, affected the understanding of the relationship between geometry and space; in particular, it posed the question whether space is curved or not (the latter being the Euclidean answer). The geometries thus played a role in the emergence and articulation of relativity theory, especially the differential geometry and tensorial calculus within which its mathematical properties could be expressed.
Philosophically the new geometries stressed the hypothetical nature of axiomatizing, in contrast to the customary view of mathematical theories as true in some (usually) unclear sense. This feature led to the name ‘metageometry’ for them; it was intended (as an ironical proposal of opponents) to be in line with the hypothetical character of metaphysics in philosophy. They also helped to encourage conventionalist philosophy of science (with Poincaré, e.g.), and put fresh light on the age-old question of the (im)possibility of a priori knowledge.
See also EUCLIDEAN GEOMETRY, PHILOSO- PHY OF MATHEMATIC. I.G.-G.