ABOUT THE BOOK: This book is based on courses of lectures on Elementary Analysis given at Queen’s College, Galway, during each of the sessions 19021907. But additions have naturally been made in preparing the manuscript. In Particular the whole of chapter XI and greater part of the appendices have been added. This historic book may have numerous typos and missing text. In Chapter 1., a preliminary account is given of the notions of a limit and of convergence. I have not in this chapter attempted to supply arithmetic proofs of the fundamental theorems concerning the existence of limits, but have allowed their truth to rest on an appeal to the readers intuition, in the hope that the discussion may thus be made more attractive to beginners. An arithmetic treatment will be found in AppendixI., where Dedekind’s definition of irrational numbers is adopted as fundamental, this method leads at once the monotonic principle of convergence, from which the existence limits is deduced.