Every product of the waking consciousness of the Classical world, then, is elevated to the rank of actuality by way of sculptural definition. That which cannot be drawn is not “number.” Archytas and Eudoxus use the terms surface- and volume-numbers to mean what we call second and third powers, and it is easy to understand that the notion of higher integral powers did not exist for them, for a fourth power would predicate at once, for the mind based on the plastic feeling, an extension in four dimensions, and four _material_ dimensions into the bargain, “which is absurd.” Expressions like ε^{ix} which we constantly use, or even the fractional index (e.g., 5^½) which is employed in the Western mathematics as early as Oresme (14th Century), would have been to them utter nonsense. Euclid calls the factors of a product its sides πλευραί and fractions (finite of course) were treated as whole-number relationships between two lines. Clearly, out of this no conception of zero as a number could possibly come, for from the point of view of a draughtsman it is meaningless. We, having minds differently constituted, must not argue from our habits to theirs and treat their mathematic as a “first stage” in the development of “Mathematics.” Within and for the purposes of the world that Classical man evolved for himself, the Classical mathematic was a complete thing—it is merely not so _for us_. Babylonian and Indian mathematics had long contained, as essential elements of _their_ number-worlds, things which the Classical number- feeling regarded as nonsense—and not from ignorance either, since many a Greek thinker was acquainted with them. It must be repeated, “Mathematics” is an illusion. A mathematical, and, generally, a scientific way of thinking is right, convincing, a “necessity of thought,” when it completely expresses the life-feeling proper to it. Otherwise it is either impossible, futile and senseless, or else, as we in the arrogance of our historical soul like to say, “primitive.” The modern mathematic, though “true” only for the Western spirit, is undeniably a master-work of that spirit; and yet to Plato it would have seemed a ridiculous and painful aberration from the path leading to the “true”—to wit, the Classical—mathematic. And so with ourselves. Plainly, we have almost no notion of the multitude of great ideas belonging to other Cultures that we have suffered to lapse because _our_ thought with its limitations has not permitted us to assimilate them, or (which comes to the same thing) has led us to reject them as false, superfluous, and nonsensical. VI