optical_ analysis of things-become on the basis of sculptural-Classical bounds. They used deeply-thought-out (and for us hardly understandable) methods of integration, but these possess only a superficial resemblance even to Leibniz’s definite-integral method. They employed geometrical loci and co-ordinates, but these are always specified lengths and units of measurement and never, as in Fermat and above all in Descartes, unspecified spatial relations, values of points in terms of their positions in space. With these methods also should be classed the exhaustion-method of Archimedes,[57] given by him in his recently discovered letter to Eratosthenes on such subjects as the quadrature of the parabola section by means of inscribed rectangles (instead of through similar polygons). But the very subtlety and extreme complication of his methods, which are grounded in certain of Plato’s geometrical ideas, make us realize, in spite of superficial analogies, what an enormous difference separates him from Pascal. Apart altogether from the idea of Riemann’s integral, what sharper contrast could there be to these ideas than the so-called quadratures of to-day? The name itself is now no more than an unfortunate survival, the “surface” is indicated by a bounding function, and the _drawing_ as such, has vanished. Nowhere else did the two mathematical minds approach each other more closely than in this instance, and nowhere is it more evident that the gulf between the two souls thus expressing themselves is impassable. In the cubic style of their early architecture the Egyptians, so to say, concealed pure numbers, fearful of stumbling upon their secret, and for the Hellenes too they were the key to the meaning of the become, the stiffened, the mortal. The stone statue and the scientific system deny life. Mathematical number, the formal principle of an extension-world of which the phenomenal existence is only the derivative and servant of waking human consciousness, bears the hall-mark of causal necessity and so is linked with _death_ as chronological number is with becoming, with _life_, with the necessity of destiny. This connexion of strict mathematical form with the _end_ of organic being, with the phenomenon of its organic remainder the corpse, we shall see more and more clearly to be the origin of all great art. We have already noticed the development of early ornament on funerary equipments and receptacles. _Numbers are symbols of the mortal._ Stiff forms are the negation of life, formulas and laws spread rigidity over the face of nature, numbers make dead—and the “Mothers” of Faust II sit enthroned, majestic and withdrawn, in “The realms of Image unconfined. ... Formation, transformation, Eternal play of the eternal mind With semblances of all things in creation For ever and for ever sweeping round.”[58]