Note 44 — Introductory material

We have now, I think, discussed most of the questions concerning the scope and validity of the projective method. We have seen that it is independent of all metrical presuppositions, and that its use of coordinates does not involve the assumption that spatial magnitudes are measured or expressed by them. We have seen that it is able to deal, by its own methods alone, with the question of the qualitative likeness of geometrical figures, which is logically prior to any comparison as to quantity, since quantity presupposes qualitative likeness. We have seen also that, so far as its legitimate use extends, it applies equally to all homogeneous spaces, and that its criterion of an independently possible space—the determination of a straight line by two points[60]—is not subject to the qualifications and limitations which belong, as we have seen in the case of the cylinder, to the metrical criterion of constant curvature. But we have also seen that, when projective Geometry endeavours to grapple with spatial magnitude, and bring distance and the measurement of angles beneath its sway, its success, though technically valid and important, is philosophically an apparent success only. Metrical Geometry, therefore, if quantity is to be applied to space at all, remains a separate, though logically subsequent branch of Mathematics.