This period is characterized chiefly by its close relationship to the theory of surfaces. RIEMANN'S Habilitation Lecture on _The Hypotheses Which Constitute the Bases of Geometry_ marks the beginning of this epoch. In this dissertation, RIEMANN not only promulgated the system upon which GAUSS had spent more than thirty years of his life in elaborating, for he was a disciple of GAUSS; but he disclosed his own views with respect to space which he regarded as a particular case of manifold. His work contains two fundamental concepts, namely, the _manifold_ and the _measure of curvature_ of a continuous manifold, possessed of what he called _flatness_ in the smallest parts. The conception of the measure of curvature is extended by RIEMANN from surfaces to spaces and a new kind of space, finite, but unbounded, is shown to be possible. He showed that the dimensions of any space are determined by the number of measurements necessary to establish the position of a point in that space. Conceiving, therefore, that space is a manifold of finite, but unbounded, extension, he established the fact that the passage from one element of a manifold to another may be either discrete or continuous and that the manifold is discrete or continuous according to the manner of passage. Where the manifold is regarded as discrete two portions of it can be compared, as to magnitude, by counting; where continuous, by measurement. If the whole manifold be caused to pass over into another manifold each of its elements passing through a one-dimensional manifold, a two-dimensional manifold is thus generated. In this way, a manifold of _n_-dimensions can be generated. On the other hand, a manifold of _n_-dimensions can be analyzed into one of one dimension and one of (_n_-1) dimensions. To RIEMANN, then, is due the credit for first promulgating the idea that space being a special case of manifold is generable, and therefore, _finite_. He laid the foundation for the establishment of a special kind of geometry known as the "elliptic." Space, as viewed by him, possessed the following properties, viz.: generability, divisibility, measurability, ponderability, finity and flexity. These are the six pillars upon which rests the structure of hyperspace analyses.[5] [5] Vide _Nature_, Vol. VIII, pp. 14-17; 36, 37 (1873); also _Mathematical Papers_, pp. 65-71.