This volume, as originally planned, was intended to conclude the whole work with a review, chiefly in differential equations, of such standard theory of the calculus as could be exhibited without a detailed study of analysis. I soon found, however, that analytical requirements kept penetrating and could not be kept out without loss of intellectual honesty. The volume is therefore much longer than I intended, and includes, substantially, a whole freshman's course of analysis, and more in addition. Nevertheless, my aim remained to keep the exposition as simple as possible within clearly stated limitations.

The theme of the volume is the differential equation and its solution; and it is hoped that the treatment shows how the processes of solution demand extended definitions of functions (for example, series and integrals) together with a technique (analysis) for studying and controlling their behaviour. The aim is not so much to elaborate the detailed properties of such fresh functions, as to instil methods which the student can apply or, better, adapt himself when faced later with the need for extending his mathematical vocabulary.

The work is, in essence, familiar, but it ought perhaps to be remarked that there are a number of points where the details vary from standard practice.

The functions which we have already met may be summarized briefly. They are the powers of x and combinations of them such as polynomials and rational functions, the trigonometric functions, the logarithmic and exponential functions, and the hyperbolic functions.

The list is substantial, but it forms only a beginning, and many other functions remain for our attention. The work of this section is directed towards the setting-up of certain basic techniques which enable necessary extensions to be made. The question that we pose is less ‘What particular functions are there at my disposal?’ than ‘How can I set about to find such functions when I need them?’ The results are all well established, and it is hoped that the presentation will enable the student to see some ways of extending his mathematical vocabulary while at the same time absorbing standard information.

The plan of these four volumes has been to present Calculus in the spirit of Analysis, but without detailed examination of the properties which belong essentially to the latter. It is natural that analytical ideas should become increasingly pressing as we approach the later stages, and it seems wise to insert a section now on convergence and similar topics lest the processes which form our main theme later should be treated on a purely mechanical basis. As in the book as a whole, however, so here also we shall try to clarify the guiding principles rather than to establish the wealth of detail which the serious student of Analysis must always require.

The aim of these volumes is that they shall together form a complete course in Calculus from its beginnings up to the point where it joins with the subject usually known as analysis. The whole conception is based on considerable dissatisfaction with much that seems rough-and-ready in the basic ideas with which pupils reach the universities, so that almost anything seems acceptable for ‘proof’ which is superficially plausible. Of course the early work cannot be treated with the rigour appropriate to more mature judgement; but I have tried here, however unsuccessfully, to present the subject in such a way that the more exact treatment, when it comes, can follow by natural development, without being forced to return to a fresh beginning which is often felt to be both unnecessary and even pointless. (How many students lose the thread of analysis just because they do not see any reason for the first few lectures and therefore do not give them serious attention?)

The first volume deals with the basic ideas of differentiation and integration. Graphical methods are used freely, but, it is hoped, in such a way that the essential logical development is never far away. The examples at this stage are mainly very simple, and beginners should have no difficulty in acquiring a fluent technique. Integration appears from the start as area and summation, the method of calculation by inverse differentiation being deduced.