The one intelligible theory of the universe is that of objective idealism, that matter is effete mind, inveterate habits becoming physical laws. But before this can be accepted it must show itself capable of explaining the tridimensionality of space, the laws of motion, and the general characteristics of the universe, with mathematical clearness and precision; for no less should be demanded of every Philosophy. Modern mathematics is replete with ideas which may be applied to philosophy. I can only notice one or two. The manner in which mathematicians generalise is very instructive. Thus, painters are accustomed to think of a picture as consisting geometrically of the intersections of its plane by rays of light from the natural objects to the eye. But geometers use a generalised perspective. For instance, in the figure let _O_ be the eye, let _A B C D E_ be the edgewise view of any plane, and let _a f e D c_ be the edgewise view of another plane. The geometers draw rays through _O_ cutting both these planes, and treat the points of intersection of each ray with one plane as representing the point of intersection of the same ray with the other plane. Thus, _e_ represents _E_, in the painter's way. _D_ represents itself. _C_ is represented by _c_, which is further from the eye; and _A_ is represented by _a_ which is on the other side of the eye. Such generalisation is not bound down to sensuous images. Further, according to this mode of representation every point on one plane represents a point on the other, and every point on the latter is represented by a point on the former. But how about the point _f_ which is in a direction from _O_ parallel to the represented plane, and how about the point _B_ which is in a direction parallel to the representing plane? Some will say that these are exceptions; but modern mathematics does not allow exceptions which can be annulled by generalisation. As a point moves from _C_ to _D_ and thence to _E_ and off toward infinity, the corresponding point on the other plane moves from _c_ to _D_ and thence to _e_ and toward _f_. But this second point can pass through _f_ to _a_; and when it is there the first point has arrived at _A_. We therefore say that the first point has passed _through infinity_, and that every line joins in to itself somewhat like an oval. Geometers talk of the parts of lines at an infinite distance as points. This is a kind of generalisation very efficient in mathematics. [Illustration]