purely relative term, and that there are no infinitesimal real numbers, the supposed logical significance of the concept seems simply to disappear. Instances of this kind could easily be multiplied almost indefinitely, but those already cited should be sufficient to show how important are the metaphysical results which may be anticipated from contemporary mathematical research, and how grave a mistake it would be to regard existing metaphysical construction, _e. g._, that of the Hegelian system, as adequate in principle to the present state of our organized knowledge. In fact, all the materials for a new _Kategorienlehre_, which may be to the knowledge of our day what Hegel's _Logic_ was to that of eighty years ago, appear to lie ready to hand when it may please Providence to send us the metaphysician who knows how to avail himself of them. The proof, given since this address was delivered, by E. Zermelo, that every assemblage can be well ordered, is an even more startling illustration of the remarks in the text. It remains to say something of the relation of metaphysical speculation to the various sciences which make use of empirical premises. On this topic I maybe allowed to be all the more brief, as I have quite recently expressed my views at fair length in an extended treatise (_Elements of Metaphysics_, Bks. 3 and 4), and have nothing of consequence to add to what has been there said. The empirical sciences, as previously defined, appear to fall into two main classes, distinguished by a difference which corresponds to that often taken in the past as the criterion by which science is to be separated from philosophy. We may study the facts of temporal sequence either with a view to the actual control of future sequences or with a view to detecting under the sequence some coherent purpose. It is in the former way that we deal with facts in mechanics, for instance, or in chemistry, in the latter that we treat them when we study history for the purpose of gaining insight into national aims and character. We may, if we please, with Professor Royce, distinguish the two attitudes toward fact as the attitude respectively of description and of appreciation or evaluation. Now as regards the descriptive sciences, the position to which, as I believe, metaphysicians are more and more tending is that here metaphysics has, strictly speaking, no right at all to interfere. Just because of the absence from metaphysics itself of all empirical premises, it can be no business of the metaphysician to determine what the course of events will be or to prescribe to the sciences what methods and hypotheses they shall employ in the work of such determination. Within these sciences any and every hypothesis is sufficiently justified, whatever its nature, so long as it enables us more efficiently than any other to perform the actual task of calculation and prediction. And it was owing to neglect of this caution