Gothic cathedrals and Doric temples are _mathematics in stone_. Doubtless Pythagoras was the first in the Classical Culture to conceive number scientifically as the principle of a world-order of comprehensible things—as _standard_ and as _magnitude_—but even before him it had found expression, as a noble arraying of sensuous-material units, in the strict canon of the statue and the Doric order of columns. The great arts are, one and all, modes of interpretation by means of limits based on number (consider, for example, the problem of space- representation in oil painting). A high mathematical endowment may, without any mathematical science whatsoever, come to fruition and full self-knowledge in _technical_ spheres. In the presence of so powerful a number-sense as that evidenced, even in the Old Kingdom,[47] in the dimensioning of pyramid temples and in the technique of building, water-control and public administration (not to mention the calendar), no one surely would maintain that the valueless arithmetic of Ahmes belonging to the New Empire represents the level of Egyptian mathematics. The Australian natives, who rank intellectually as thorough primitives, possess a mathematical instinct (or, what comes to the same thing, a power of thinking in numbers which is not yet communicable by signs or words) that as regards the interpretation of pure space is far superior to that of the Greeks. Their discovery of the boomerang can only be attributed to their having a sure feeling for numbers of a class that we should refer to the higher geometry. _Accordingly_—we shall justify the adverb later—they possess an extraordinarily complicated ceremonial and, for expressing degrees of affinity, such fine shades of language as not even the higher Cultures themselves can show. There is analogy, again, between the Euclidean mathematic and the absence, in the Greek of the mature Periclean age, of any feeling either for ceremonial public life or for loneliness, while the Baroque, differing sharply from the Classical, presents us with a mathematic of spatial analysis, a court of Versailles and a state system resting on dynastic relations. It is the style of a Soul that comes out in the world of numbers, and the world of numbers includes something more than the science thereof. ----- Footnote 45: See Vol. II, pp. 327 et seq. Footnote 46: Also “thinking in money.” See Vol. II, pp. 603 et seq. Footnote 47: Dynasties I-VIII, or, effectively, I-VI. The Pyramid period coincides with Dynasties IV-VI. Cheops, Chephren and Mycerinus belong to the IV dynasty, under which also great water-control works were carried out between Abydos and the Fayum.—_Tr._ ----- III From this there follows a fact of decisive importance which has hitherto been hidden from the mathematicians themselves.