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The investigations on the foundation of geometry of RIEMANN, HELMHOLTZ and LIE have as a basal notion the distance between two points of space, and the analytic representation of this distance plays an important r6le. A point in space is determined by means of three independent coordinates; and the distance between two points is a certain function of their six coordinates. Rigid bodies are used and the continuous motions of such, determined by the continuous variation of the co6rdinates of their points. This notion of three independent and continuous co6rdinates of a point is, in reality, not simple, as has been pointed out by LIE and others. In an article entitled Ein Beitrag zur Jllannigfaltigkeitslehre, Borchhardt's Journal, vol. 84 (1877), G. CANTOR describes a space in which the co6rdinates of a point are neither independent nor continuous. It does not seem easy, however, to determine the distance between two points of space if the co6rdinates of a point are neither independent nor continuous. It would not be evident, a priori, that certain loci (as spheres and circles) exist, defined by means of certain equations in the coordinates of their points. Such loci are used in the processes employed by HELMHOLTZ and LIE, and the question arises whether their axioms involving such loci and the motion of points in space, etc., should be modified or replaced by others. One such new axiom, used in the following paper, refers to distance-relations, i. e., it is assumed that certain definite relations connect the mutual distances between any n + 2 different points in space of n dimensions.t One notices that the assumption, used by HELMHOLTZ and LIE and their successors, that the distance between two points in space is a certain function of the

On the determination of the distance between two points in space of 𝑛 dimensions