=46.= I have now come to the end of my history of Metageometry. It has not been my aim to give an exhaustive account of even the important works on the subject--in the third period, especially, the names of Poincaré, Pasch, Cremona, Veronese, and others who might be mentioned, would have cried shame upon me, had I had any such object. But I have tried to set forth, as clearly as I could, the principles at work in the various periods, the motives and results of successive theories. We have seen how the philosophical motive, at first predominant, has been gradually extruded by the purely mathematical and technical spirit of most recent Geometers. At first, to discredit the Transcendental Aesthetic seemed, to Metageometers, as important as to advance their science; but from the works of Cayley, Klein or Lie, no reader could gather that Kant had ever lived. We have also seen, however, that as the interest _in_ philosophy waned, the interest _for_ philosophy increased: as the mathematical results shook themselves free from philosophical controversies, they assumed gradually a stable form, from which further development, we may reasonably hope, will take the form of growth, rather than transformation. The same gradual development out of philosophy might, I believe, be traced in the infancy of most branches of mathematics; when philosophical motives cease to operate, this is, in general, a sign that the stage of uncertainty as to premisses is past, so that the future belongs entirely to mathematical technique. When this stable stage has been attained, it is time for Philosophy to borrow of Science, accepting its final premisses as those imposed by a real necessity of fact or logic. =47.= Now in discussing the systems of Metageometry, we have found two kinds, radically distinct and subject to different axioms. The historically prior kind, which deals with metrical ideas, discusses, to begin with, the conditions of Free Mobility, which is essential to all measurement of space. It finds the analytical expression of these conditions in the existence of a space-constant, or constant measure of curvature, which is equivalent to the homogeneity of space. This is its first axiom. Its second axiom states that space has a finite integral number of dimensions, _i.e._ in metrical terms, that the position of a point, relative to any other figure in space, is uniquely determined by a finite number of spatial magnitudes, called coordinates. The third axiom of metrical Geometry may be called, to distinguish it from the corresponding projective axiom, the axiom of distance. There exists one relation, it says, between any two points, which can be preserved unaltered in a combined motion of both points, and which, in any motion of a system as one rigid body, is always unaltered. This relation we call distance.