(2) A consequence of this distinction is that only the pure or formal sciences can be matter of rigid logical demonstration. Since the empirical or applied sciences one and all contain empirical premises, _i. e._, premises which we admit as true only because they have always appeared to be confirmed by the appeal to "intuition," and not because the denial of them can be shown to lead to falsehood, the conclusions to which they conduct us must one and all depend, in part at least, upon induction from actual observation of particular temporal sequences. This is as much as to say that all propositions in the applied sciences involve somewhere in the course of the reasoning by which they are established the appeal to the calculus of Probabilities, which is our one method of eliciting general results from the statistics supplied by observation or experiment. That this is the case with the more concrete among such applied sciences has long been universally acknowledged. That it is no less true of sciences of such wide range as mechanics may be said, I think, to have been definitely established in our own day by the work of such eminent physicists as Kirchhoff and Mach. In fact, the recent developments of the science of pure number, to which reference has been made in a preceding paragraph, combined with the creation of the "descriptive" theory of mechanics, may fairly be said to have finally vindicated the distinction drawn by Leibniz long ago between the truths of reason and the truths of empirical fact, a distinction which the Kantian trend of philosophical speculation tended during the greater part of the nineteenth century to obscure, while it was absolutely ignored by the empiricist opponents of metaphysics both in England and in Germany. The philosophical consequences of a revival of the distinction are, I conceive, of far-reaching importance. On the one side, recognition of the empirical and contingent character of all general propositions established by induction appears absolutely fatal to the current mechanistic conception of the universe as a realm of purposeless sequences unequivocally determined by unalterable "laws of nature," a result which has in recent years been admirably illustrated for the English-speaking world by Professor Ward's well-known Gifford lectures on "Naturalism and Agnosticism." Laws of physical nature, on the empiristic view of applied science, can mean no more than observed regularities, obtained by the application of the doctrine of chances,--regularities which we are indeed justified in accepting with confidence as the basis for calculation of the future course of temporal sequence, but which we have no logical warrant for treating as ultimate truths about the final constitution of things. Thus, for example, take the common assumption that our physical environment is composed of a multitude of particles each in every respect the exact counterpart of every other.