Note 155 — Introductory material

The axiom of Free Mobility involves, if it is to be true, the homogeneity of space, or the complete relativity of position. For if any shape, which is possible in one part of space, be always possible in another, it follows that all parts of space are qualitatively similar, and cannot, therefore, be distinguished by any intrinsic property. Hence positions in space, if our axiom be true, must be wholly defined by external relations, i.e. Position is not an intrinsic, but a purely relative, property of things in space. If there could be such a thing as absolute position, in short, metrical Geometry would be impossible. This relativity of position is the fundamental postulate of all Geometry, to which each of the necessary metrical axioms leads, and from which, conversely, each of these axioms can be deduced.