mind which endeavours to know them has not created them. Mathematics creates what it knows; it contains in itself its own elements, and thus forms a perfect copy of the divine knowledge (_scientiae divinae similes evadunt_). The reader of these and other similar descriptions and praises by Vico of the processes of mathematics seems to observe in them something like a tinge of irony; which, if not actually intentional, certainly results from the facts of the case. The brilliant truth of mathematics arises, it appears, from despair of attaining truth; its tremendous power from the knowledge of impotence. The similarity of the mathematician to God is not altogether unlike that of the imitator of an object to its creator. What God is in the universe of reality, man is in the universe of quantity and number,--a universe indeed, but one peopled by abstractions and fictions. The divinity which has been conferred upon man is only, so to speak, a Twelfth-night Godhead. The different origin assigned by Vico to mathematics results in a correspondingly profound change in the validity of its truth. Mathematics no longer, as with Descartes, stands at the summit of human knowledge, an aristocratic science, destined to reclaim and to rule over the inferior sciences. It occupies a field as strictly limited as it is unique, beyond which if it ever attempts to pass it loses in a moment its magical virtue. The power of mathematics is met by obstacles both _a parte ante_ and _a parte post,_ in its foundations and in the superstructure which in its turn it is to support. In its foundations, because if it creates its own elements, that is to say, the initial fictions, it does not create the matter of which they are formed, which is given to it no less than to the other human sciences by metaphysics, which while it cannot supply it with its true subject matter, supplies it with definite images of it. From metaphysics, geometry takes the point by drawing it, that is by annihilating it as a point, and arithmetic the unit by multiplying it, that is by destroying it _qua_ unit. But since metaphysical truth, however certain it may seem to consciousness, is indemonstrable, mathematics itself rests in the last resort upon authority and probability. This is enough to expose the fallaciousness of any mathematical treatise which makes use of metaphysics. Vico seems to be involved in a kind of circle between geometry and metaphysics, of which the former, according to him, owes its truth to the latter, and after receiving it gives it back again to metaphysics, thus in turn supporting the human science by the divine. But this conception, the truth of which is more than doubtful, indeed we may frankly call it inconsistent and contradictory, recalls, whatever its value, the metaphysical or rather poetical or symbolic use made of mathematics by Pythagoras and other philosophers of antiquity and the Renaissance, and has no resemblance to a mathematically-treated philosophy like the