(3) _As regards free mobility._ Every point can pass freely and continuously from one position to another. From (2) and (3) it follows, that if two systems _A_ and _B_ can be brought into congruence in any one position, this is also possible in every other position. (4) _As regards independence of rotation in rigid bodies_ (Monodromy). If (_n_ - 1) points of a body remain fixed, so that every other point can only describe a certain curve, then that curve is closed. These axioms, says Helmholtz, suffice to give, with the axiom of three dimensions, the Euclidean and non-Euclidean systems as the only alternatives. That they _suffice_, mathematically, cannot be denied, but they seem, in some respects, to go too far. In the first place, there is no necessity to make the axiom of Congruence apply to actual rigid bodies--on this subject I have enlarged in Chapter II.[29] Again, Free Mobility, as distinct from Congruence, hardly needs to be specially formulated: what barrier could empty space offer to a point's progress? The axiom is involved in the homogeneity of space, which is the same thing as the axiom of Congruence. Monodromy, also, has been severely criticized; not only is it evident that it might have been included in Congruence, but even from the purely analytical point of view, Sophus Lie has proved it to be superfluous[30]. Thus the axiom of Congruence, rightly formulated, includes Helmholtz's third and fourth axioms and part of his second axiom. All the four, or rather, as much of them as is relevant to Geometry, are consequences, as we shall see hereafter, of the one fundamental principle of the relativity of position.