Note 18 — Introductory material

(1) Conception of a manifold[17]. The general purpose of Riemann's dissertation is, to exhibit the axioms as successive steps in the classification of the species space. The axioms of Geometry, like the marks of a scholastic definition, appear as successive determinations of class-conceptions, ending with Euclidean space. We have thus, from the analytical point of view, about as logical and precise a formulation as can be desired—a formulation in which, from its classificatory character, we seem certain of having nothing superfluous or redundant, and obtain the axioms explicitly in the most desirable form, namely as adjectives of the conception of space. At the same time, it is a pity that Riemann, in accordance with the metrical bias of his time, regarded space as primarily a magnitude[18], or assemblage of magnitudes, in which the main problem consists in assigning quantities to the different elements or points, without regard to the qualitative nature of the quantities assigned. Considerable obscurity thus arises as to the whole nature of magnitude[19]. This view of Geometry underlies the definition of the manifold, as the general conception of which space forms a special case. This definition, which is not very clear, may be rendered as follows.