One of the objects of a Congress such as this is to exhibit the connecting links of the different departments of Science, taking that term in its widest acceptation. Hence the organisers of this meeting have taken care that the relations between the different sections shall be exhibited in broad relief. In undertaking a sketch of this kind, which must necessarily be somewhat indefinite in character, we must forget that all is in all; as far as algebra and analysis are concerned, the follower of Pythagoras would be disconcerted by the extent of his task, remembering the celebrated formula of the School: “Things are numbers.” From this point of view my subject would be in-exhaustible. But I shall make no such pretensions, and for the best of reasons. By passing in rapid review the development of our science throughout the ages, and particularly in the last century, I hope to be able to give an adequate indication of the part played by mathematical analysis in its relations to other sciences.

In the following articles the method of Averages, used in the Oct. 1903 issue of the Gazette to effect the expansion of the simple Algebraical functions, is employed for the Trigonometrical Expansions and for Taylor’s Theorem, and the results are applied to the question of convergence. The method was adopted as the simplest I could devise for even the Binomial, subject to the condition that no infinite series should be used unless the limits of error after a finite number of terms were explicitly stated. I would suggest that this condition might well be laid down in a school course, as the uncertainty that comes over our Algebra as soon as we leave Geometrical Progressions ‘to infinity’ delays a clear grasp of Mathematics more than a fuller consideration of the simpler infinite series would. The method is, of course, essentially integral calculus, but I have found—and I hope my subjects have not been specially favourable—that the proofs can be readily followed by students to whom the ideas and notation of the Differential and Integral will not be familiar for a considerable time, and in whom perhaps they will never inspire a sense of absolute security.

To prove that if ƒ represents a function satisfying certain conditions for all values of the variable from a to a + x, then ƒ(a + x) can be expanded in powers of x.

Assume that, as in § I., where ƒ′ is another function which we call the derived function of ƒ.