Page:Chance, love, and logic - philosophical essays (IA chancelovelogicp00peir 0).pdf/213

 *wise 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 generalization 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 generalization. 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 of2. After it was proved that no ratio of two integers could possibly equal [sqrt]2 the idea of number was generalized to include the latter. Fractions and the so-called imaginary numbers illustrate the same process of generalization for the sake of making certain operations (i.e. division and finding the root) continuously applicable. ]]