The Greek mathematic, as a science of perceivable magnitudes, deliberately confines itself to facts of the comprehensibly present, and limits its researches and their validity to the near and the small. As compared with this impeccable consistency, the position of the Western mathematic is seen to be, practically, somewhat illogical, though it is only since the discovery of Non-Euclidean Geometry that the fact has been really recognized. Numbers are images of the perfectly desensualized understanding, of pure thought, and contain their abstract validity within themselves.[52] Their exact application to the actuality of conscious experience is therefore a problem in itself—a problem which is always being posed anew and never solved—and the congruence of mathematical system with empirical observation is at present anything but self-evident. Although the lay idea—as found in Schopenhauer—is that mathematics rest upon the direct evidences of the senses, Euclidean geometry, superficially identical though it is with the popular geometry of all ages, is only in agreement with the phenomenal world approximately and within very narrow limits—in fact, the limits of a drawing-board. Extend these limits, and what becomes, for instance, of Euclidean parallels? They meet at the line of the horizon—a simple fact upon which all our art-perspective is grounded. Now, it is unpardonable that Kant, a Western thinker, should have evaded the mathematic of distance, and appealed to a set of figure-examples that their mere pettiness excludes from treatment by the specifically Western infinitesimal methods. But Euclid, as a thinker of the Classical age, was entirely consistent with its spirit when he refrained from proving the phenomenal truth of his axioms by referring to, say, the triangle formed by an observer and two infinitely distant fixed stars. For these can neither be drawn nor “intuitively apprehended” and his feeling was precisely the feeling which shrank from the irrationals, which did not dare to give nothingness a value as zero (i.e., a number) and even in the contemplation of cosmic relations shut its eyes to the Infinite and held to its symbol of Proportion.