few who have reached them, do such imaginings as systems of hypercomplex numbers (e.g., the quaternions of the calculus of vectors) and apparently quite meaningless symbols like ∞^{_n_} acquire the character of something actual. And here if anywhere it must be understood that actuality is not only sensual actuality. The spiritual is in no wise limited to perception-forms for the actualizing of its idea. ----- Footnote 76: From the standpoint of the theory of “aggregates” (or “sets of points”), a well-ordered set of points, irrespective of the dimension figure, is called a corpus; and thus an aggregate of _n_ - 1 dimensions is considered, _relatively_ to one of _n_ dimensions, as a surface. Thus the limit (wall, edge) of an “aggregate” represents an aggregate of lower “potentiality.” ----- XVIII From this grand intuition of symbolic space-worlds came the last and conclusive creation of Western mathematic—the expansion and subtilizing of the function theory in that of _groups_. Groups are aggregates or sets of homogeneous mathematical images—e.g., the totality of all differential equations of a certain type—which in structure and ordering are analogous to the Dedekind number-bodies. Here are worlds, we feel, of perfectly new numbers, which are nevertheless not utterly sense- transcendent for the _inner_ eye of the adept; and the problem now is to discover in those vast abstract form-systems certain elements which, relatively to a particular group of operations (viz., of transformations of the system), remain unaffected thereby, that is, possess invariance. In mathematical language, the problem, as stated generally by Klein, is— given an _n_-dimensional manifold (“space”) and a group of transformations, it is required to examine the forms belonging to the manifold in respect of such properties as are not altered by transformation of the group.