published in the Leipziger Magazin für reine und angewandte Mathematik, 1786, in which: (1) The failure of the parallel-axiom in surface-spherics gives a geometry with angle-sum right angles; (2) In order to make intuitive a geometry with angle-sum right angles we need the aid of an "imaginary sphere" (pseudo-sphere); (3) In a space with the angle-sum differing from 2 right angles, there is an absolute measure (Bolyai's natural unit for length).
In 1854, nearly twenty years later, Gauss heard from his
pupil, Riemann, a marvellous dissertation carrying the discussion
one step further by developing the notion of -ply extended magnitude, and the measure-relations of which a manifoldness of dimensions is capable, on the assumption that every line may be measured by every other. Riemann applied his ideas to space. He taught us to distinguish between "unboundedness" and "infinite extent." According
to him we have in our mind a more general notion of space, i.e. a notion of non-Euclidean space; but we learn by experience that our physical space is, if not exactly, at least to high degree of approximation, Euclidean space. Riemann's profound dissertation was not published until 1867, when it appeared in the Göttingen Abhandlungen. Before this the idea of -dimensions had suggested itself under various
aspects to Lagrange, Plücker, and H. Grassmann. About the
same time with Riemann's paper, others were published from the pens of Helmholtz and Beltrami. These contributed powerfully
to the victory of logic over