Note 158 — Introductory material

We have seen, in discussing the axiom of Free Mobility, that all position is relative, that is, a position exists only by virtue of relations[164]. It follows that, if positions can be defined at all, they must be uniquely and exhaustively defined by some finite number of such relations. If Geometry is to be possible, it must happen that, after enough relations have been given to determine a point uniquely, its relations to any fresh known point are deducible from the relations already given. Hence we obtain, as an à priori condition of Geometry, logically indispensable to its existence, the axiom that Space must have a finite integral number of Dimensions. For every relation required in the definition of a point constitutes a dimension, and a fraction of a relation is meaningless. The number of relations required must be finite, since an infinite number of dimensions would be practically impossible to determine. If we remember our axiom of Free Mobility, and remember also that space is a continuum, we may state our axiom, for metrical Geometry, in the form given by Helmholtz (v. Chap. I. § 25): "In a space of n dimensions, the position of every point is uniquely determined by the measurement of n continuous independent variables (coordinates).[165]"