Of the relation of metaphysics to pure mathematics it would be impertinent for any but a trained mathematician to say very much. I must therefore be content to point out that the same difficulty in drawing boundary lines meets us here as in the case of logic. Not so long ago this difficulty might have been ignored, as it still is by too many writers on the philosophy of science. Until recently mathematics would have been thought to be adequately defined as the science of numerical and quantitative relations, and adequately distinguished from metaphysics by the non-quantitative and non-numerical character of the latter, though it would probably have been admitted that the problem of the definition of quantity and number themselves is a metaphysical one. But in the present state of our knowledge such an account seems doubly unsatisfactory. On the one hand, we have to recognize the existence of branches of mathematics, such as the so-called descriptive geometry, which are neither quantitative nor numerical, and, on the other, quantity as distinct from number appears to play no part in mathematical science, while number itself, thanks to the labors of such men as Cantor and Dedekind, seems, as I have said before, to be known now to be only a special type of order in a series. Thus there appears to be ground for regarding serial order as the fundamental category of mathematics, and we are thrown back once more upon the difficult task of deciding how many ultimately irreducible types of order there may be before we can undertake any precise discrimination between mathematical and metaphysical science. However we may regard the problem, it is at least certain that the recent researches of mathematicians into the meaning of such concepts as continuity and infinity have, besides opening up new metaphysical problems, done much to transfigure the familiar ones, as all readers of Professor Royce must be aware. For instance I imagine all of us here present, even the youngest, were brought up on the Aristotelian doctrine that there is and can be no such thing as an actually existing infinite collection, but which of us would care to defend that time-honored position to-day? Similarly with continuity all of us were probably once on a time instructed that whereas "quantity" is continuous, number is essentially "discrete," and is indeed the typical instance of what we mean by the non-continuous. To-day we know that it is in the number series that we have our one certain and familiar instance of a perfect continuum. Still a third illustration of the transforming light which is thrown upon old standing metaphysical puzzles by the increasing formal development of mathematics may be found in the difficulties attendant upon the conception of the "infinitely little," once regarded as the logical foundation of the so-called Differential Calculus. With the demonstration, which maybe found in Mr. Russell's important work, that "infinitesimal," unlike "infinite," is a