This mathematics of ours was bound in due course to reach the point at which not merely the limits of artificial geometrical form but the limits of the visual itself were felt by theory and by the soul alike as limits indeed, as obstacles to the unreserved expression of inward possibilities—in other words, the point at which the ideal of transcendent extension came into fundamental conflict with the limitations of immediate perception. The Classical soul, with the entire abdication of Platonic and Stoic ἀταραξία, submitted to the sensuous and (as the erotic under-meaning of the Pythagorean numbers shows) it rather _felt_ than _emitted_ its great symbols. Of transcending the corporeal here-and-now it was quite incapable. But whereas number, as conceived by a Pythagorean, exhibited the essence of individual and discrete _data_ in “Nature” Descartes and his successors looked upon number as _something to be conquered_, to be _wrung out_, an abstract relation royally indifferent to all phenomenal support and capable of holding its own against “Nature” on all occasions. The will-to-power (to use Nietzsche’s great formula) that from the earliest Gothic of the Eddas, the Cathedrals and Crusades, and even from the old conquering Goths and Vikings, has distinguished the attitude of the Northern soul to its world, appears also in the sense-transcending energy, the _dynamic_ of Western number. In the Apollinian mathematic the intellect is the servant of the eye, in the Faustian its master. Mathematical, “absolute” space, we see then, is utterly un-Classical, and from the first, although mathematicians with their reverence for the Hellenic tradition did not dare to observe the fact, it was something different from the indefinite spaciousness of daily experience and customary painting, the _a priori_ space of Kant which seemed so unambiguous and sure a concept. It is a pure abstract, an ideal and unfulfillable postulate of a soul which is ever less and less satisfied with sensuous means of expression and in the end passionately brushes them aside. _The inner eye has awakened._ And then, for the first time, those who thought deeply were obliged to see that the Euclidean geometry, which is the _true and only_ geometry of the simple of all ages, is when regarded from the higher standpoint nothing but a _hypothesis_, the general validity of which, since Gauss, we know it to be quite impossible to prove in the face of other and perfectly non-perceptual geometries. The critical proposition of this geometry, Euclid’s axiom of parallels, is an _assertion_, for which we are quite at liberty to substitute another assertion. We may assert, in fact, that through a given point, no parallels, or two, or many parallels may be drawn to a given straight line, and all these assumptions lead to completely irreproachable geometries of three dimensions, which can be employed in physics and even in astronomy, and are in some cases preferable to the Euclidean.