=154.= It is to be observed that the axiom of Free Mobility, as I have enunciated it, includes also the axiom to which Helmholtz gives the name of Monodromy. This asserts that a body does not alter its dimensions in consequence of a complete revolution through four right angles, but occupies at the end the same position as at the beginning. The supposed mathematical necessity of making a separate axiom of this property of space has been disproved by Sophus Lie (v. Chap. I. § 45); philosophically, it is plainly a particular case of Free Mobility[161], and indeed a particularly obvious case, for a translation really does make some change in a body, namely, a change in position, but a rotation through four right angles may be supposed to have been performed any number of times without appearing in the result, and the absurdity of ascribing to space the power of making bodies grow in the process is palpable; everything that was said above on congruence in general applies with even greater evidence to this special case. =155.= 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.