His idea that the _a priori_-ness of space was proved by and through the unconditional validity of simple geometrical facts rests, as we have already remarked, on the all-too-popular notion that mathematics are either geometry or arithmetic. Now, even in Kant’s time the mathematic of the West had got far beyond this naïve scheme, which was a mere imitation of the Classical. Modern geometry bases itself not on space but on multiply-infinite number-manifolds—amongst which the three-dimensional is simply the undistinguished special case—and within these groups investigates functional formations with reference to their structure; that is, there is no longer any contact or even possibility of contact between any possible kind of sense-perception and mathematical facts in the domain of such extensions as these, and yet the demonstrability of the latter is in no wise impaired thereby. Mathematics, then, are independent of the perceived, and the question now is, how much of this famous demonstrability of the forms of perception is left when the artificiality of juxtaposing both in a supposedly single process of experience has been recognized. Footnote 183: It is true that a geometrical theorem may be proved, or rather demonstrated, by means of a drawing. But the theorem is differently constituted in every kind of geometry, and that being so, the drawing ceases to be a _proof_ of anything whatever. Footnote 184: So much so that Gauss said nothing about his discovery until almost the end of his life for tear of “the clamour of the Bœotians.” Footnote 185: The distinction of right and left (see p. 169) is only conceivable as the outcome of this directedness in the dispositions of the body. “In front” has no meaning whatever for the body of a plant. Footnote 186: It may not be out of place here to refer to the enormous importance attached in savage society to initiation-rites at adolescence.—_Tr._ Footnote 187: Either in Greek or in Latin, τόπος (= _locus_) means spot, locality, and also social position; χώρα (= _spatium_) means space-between, distance, rank, and also ground and soil (e.g., τὰ ἐκ τῆς χώρας, produce); τὸ κένον (_vacuum_) means quite unequivocally a hollow body, and the stress is emphatically on the envelope. The literature of the Roman Imperial Age, which attempted to render the _Magian_ world- feeling through Classical words, was reduced to such clumsy versions as ὁρατὸς τόπος (sensible world) or _spatium inane_ (“endless space,” but also “wide surface”—the root of the word “spatium” means to swell or grow fat). In the true Classical literature, the idea not being there, there was no necessity for a word to describe it. Footnote 188: It has not hitherto been seen that this fact is implicit in Euclid’s famous parallel axiom (“through a point only one parallel to a straight line is possible”).