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.