On the Hypotheses Which Lie at the Bases of Geometry

III.—Application to Space. § 1.—By means of these inquiries into the determination of the measure relations of an n-fold extent the conditions may be declared which are necessary and sufficient to determine the metric properties of space, if we assume the independence of line-length from position and expressibility of the line-element as the square rout of a quadric differential, that is to say, flatness in the smallest parts.