It might be said that, as to the validity of mathematics, Vico is in agreement with Descartes; he differs from him only in his reason for this validity. And, admitting that Vico's reason must be thought the more profound, this would only enhance and strengthen the mathematical ideal which Descartes had set before science. If mathematics is the one perfect form of knowledge attained by the human mind, obviously we must found the others upon it, and either remodel or condemn them according to its pattern. Vico, in short, was hasty in declaring Descartes wrong: he had found a better argument whose existence the latter had not suspected. But, however strongly this may appear at first sight (and so it has appeared to some commentators), on a closer examination it is seen that the high perfection attributed by Vico to mathematics is more apparent than real; that the vaunted conclusiveness of its method is by his own confession gained at the expense of truth: in a word, that the stress of his theory falls less on the truth of mathematics than on its arbitrary nature. The fact is, that man, while occupying himself with the investigation of the nature of things, and ultimately realising his total inability to attain it, not having in himself the elements of which they are composed, which are indeed all external to his nature, is led by degrees to the intention of profiting by this very fault of his mind. By means of abstraction--not, be it remembered, abstraction from material things, for Vico is opposed to the empirical origin of mathematics, but abstraction brought to bear on metaphysical entities--he creates two fictions, _duo sibi confingit_: the point in geometrical figures, and unity in multiplication. Each is a fiction, _utrumque fictum,_ because the point when drawn is no longer a point, and the unit when multiplied is no longer one. Then, from these fictions, by his own arbitrary fiat, _proprio iure,_ he assumes an infinite process, so that lines may be produced or the unit multiplied _ad infinitum._ Thus he constructs for his own purposes a world of forms and numbers, all of which he embraces within himself; and by lengthening, shortening, and combining the lines and adding and subtracting the numbers, he performs infinite operations and learns infinite truths. Since he cannot define things, he defines names; since he cannot reach the elements of reality he satisfies himself with imaginary elements, the ideas arising from which admit of no dispute. Like God, _ad Dei instar,_ from no material substrate and, as it were, out of nothing he creates the point, the line, and the surface; the point, assumed as that which has no parts; the line, as the locus of a point, or as length without breadth or depth; and the surface, as the meeting of two different lines in one point, that is, length and breadth without depth. Thus mathematics overcomes the failing of human knowledge, that its objects are always external to itself, and that the