In short, mathematics is exact and universally credible just because it is so empty and so logically formal, being always "in the abstract," in that ideal, wholly blessed region, where there is no disputing, where all men readily admit anything that can be [p.180] suggested; and its being exact for this cause made it the only credible knowledge for Descartes' time, a time at once of scepticism and mysticism. With Vanini, then, and Charron, who were separately engaged, as was remarked, in a single activity, we may associate the mathematicians of the day, among whom none were more distinguished than Descartes himself and the members of the Cartesian school. To Descartes we are largely indebted for the Analytic Geometry, and Pascal did important work in the Theory of Equations. In rough outline we now have the times of Descartes before us, and with deepened meaning I may say again that Descartes came into European life at a crucial moment. Materialism was rife, not merely theoretically among a few scientists and philosophers, nor practically in some isolated class of dissipated human beings, but really and more or less openly everywhere in the whole life and feeling of society. Even the devout played into the hands of the worldly by their very purism. And an accompanying doubt, cropping out significantly, now in positive irreverence, now in mysticism, now in intellectual formalism, appears to have thoroughly possessed the minds of men. There was, too, in Descartes' day a growing sensitiveness to the paradoxes of man's experience which have occupied so much of our attention. Nothing was what it seemed. One writer boldly declared--not much later--that France, nay, the whole world, could not be happy until all should turn atheists. The boast of Louis XIV, "I am the State," whether literally made or not, was hardly less startling. The sensualism of [p.181] the Catholic Church or the Pharisaism of the Protestant was flagrantly paradoxical, and was keenly felt to be so on all sides. Men turned doubters perforce, and in the fact that with their scepticism rose also a movement at once of individualism and cosmopolitanism, we cannot fail to see how the course of history illustrates the conclusions of a previous chapter. The time was one in which through its humanism, or its cosmopolitan individualism, civilization was to reap the harvest from the medieval organization of society.