Now, in Geometry, the result of two successive motions or collineations of a figure can always be obtained by a single motion or collineation, and any motion or collineation can be built up of a series of infinitesimal motions or collineations. Moreover the analytical expression of either is a certain transformation of the coordinates of all the points of the figure[61]. Hence the transformations determining a motion or a collineation are such as to form a continuous group. But the question of the projective equivalence of two figures, to which all projective Geometry is reducible, must always be dealt with by a collineation; and the question of the equality of two figures, to which all metrical Geometry is reducible, must always be decided by a motion such as to cause superposition; hence the whole subject of Geometry may be regarded as a theory of the continuous groups which define all possible collineations and motions. Now Sophus Lie has developed, at great length, the purely analytical theory of groups; he has therefore, by this method of formulating the problem, a very powerful weapon ready for the attack. In two papers "On the foundations of Geometry[62]," undertaken at Klein's urgent request, he takes premisses which roughly correspond to those of Helmholtz, omitting Monodromy, and applies the theory of groups to the deduction of their consequences[63]. Helmholtz's work, he says, can hardly be looked upon as _proving_ its conclusions, and indeed the more searching analysis of the group-theory reveals several possibilities unknown to Helmholtz. Nevertheless, as a pioneer, devoid of Lie's machinery, Helmholtz deserves, I think, more praise than Lie is willing to give him[64]. Lie's method is perfectly exhaustive; omitting the premiss of Monodromy, the others show that a body has six degrees of freedom, _i.e._ that the group giving all possible motions of a body will have six independent members; if we keep one point fixed, the number of independent members is reduced to three. He then, from his general theory, enumerates all the groups which satisfy this condition. In order that such a group should give possible motions, it is necessary, by Helmholtz's second axiom, that it should leave invariant some function of the coordinates of any two points. This eliminates several of the groups previously enumerated, each of which he discusses in turn. He is thus led to the following results: I. _In two dimensions_, if free mobility is to hold _universally_, there are no groups satisfying Helmholtz's first three axioms, except those which give the ordinary Euclidean and non-Euclidean motions; but if it is to hold only _within a certain region_, there is also a possible group in which the curve described by any point in a rotation is not closed, but an equiangular spiral. To exclude this possibility, Helmholtz's axiom of Monodromy is required.