the combination of three lines in definite proportions give us? It gives us, simply, the coordinates of a _point_. Here again we see a great difference between the colour and space-manifolds. In colours, the combination of magnitudes gives a new magnitude of the same kind; in space, it defines, not a magnitude at all, but a would-be element of a different kind from the defining magnitudes. In the tone-manifold, we should find still different conditions. Here, no one of the measuring magnitudes can vanish without the tone vanishing too, and all three are so bound up together, in the single resulting sensation, that none can exist without a finite quantity of the others. They are all qualitatively different, both from each other, and from any possible tone, being constituents of it, as mass and velocity are constituents of momentum. All these different conditions require to be examined, before a manifold can be completely defined; and until we have conducted such an examination in detail, we cannot pronounce as to the _à priori_ or empirical nature of the laws of the manifold. As regards space, I have attempted such an examination in the third and fourth chapters of this Essay. =65.= I do not wish to deny, however, the great value of the conception of space as a manifold. On the contrary, this conception seems to have become essential to any treatment of the question. I only wish to urge that the purely algebraical treatment of any manifold, important as it may be in deducing fresh consequences from known premisses, tends rather to conceal than to make clear the basis of the premisses themselves, and is therefore misleading in a philosophical investigation. For mathematics, where quantity reigns supreme, Riemann's conception has proved itself abundantly fruitful; for philosophy, on the contrary, where quantity appears rather as a cloak to conceal the qualities it abstracts from, the conception seems to me more productive of error and confusion than of sound doctrine.