Theory of Functions can now be built up without a single contradiction or breach of logic. The puzzles about the infinitely great and infinitely small, which used to throw a cloud of mystery over the 'higher' branches of Mathematics, have been finally dissipated by the discovery that the 'infinite' is readily definable in purely ordinal terms and that the 'infinitesimal' does not really enter into the misnamed 'Infinitesimal Calculus' at all. Arithmetic and the theory of serial order have been shown to be the sufficient basis of the whole science which, as Plato long ago remarked, is 'very inappropriately called geometry'. A resume of the work which has been thus done may be found in the stately volumes of the _Principia Mathematica_ of Whitehead and Russell, or--to a large extent--in the _Formulario Matematico_ of Professor Peano. Of other works dealing with the subject, the finest from the strictly philosophical point of view is probably that of Professor G. Frege on _The Fundamental Laws of Arithmetic_. The general result of the whole development is that we are now at last definitely freed from the haunting fear that there is some hidden contradiction in the principles of the exact sciences which would vitiate all our knowledge of universal truths. This removes the chief, if not the only ground for the view that all the truths of Science are only 'partial'. At the same time, the proof that pure mathematics is a strictly logical development and that all its conclusions are of the hypothetical form, 'if _a b c_ ..., then _x_' definitely disproves the popular Kantian doctrine that _sense_-data are a necessary constituent of scientific knowledge. And with this dogma falls the _main_ ground for the denial that knowledge about the soul and God is attainable. The recovery of a sounder philosophical method has, as Mr. Russell himself says, disposed of what was yesterday the accepted view that the function of Philosophy is to narrow down the range of possible interpretations of facts until only one is left. Philosophy rather opens doors than shuts them. It multiplies the number of logically possible sets of premisses from which consequences agreeing with empirical facts may be inferred. Mr. Russell's unreasoned anti-theism seems to me to make him curiously blind to an obvious application of this principle. On the other side, the revived attention to the logical methods of the sciences is killing the crude sensationalism of the days which saw the first publication of Mach's _Science of Mechanics_ and Pearson's _Grammar of Science_. The claims of 'induction' to be a method of establishing truths may be fairly said to have been completely exposed. It is clearer now than it was when Kant made the observation that each of the 'sciences' contains just so much science as it contains mathematics, and that the Critical Philosophy was fully justified in insisting that all science implies universal _a priori_ postulates, though it went wrong in thinking that