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PREFACE TO THE SECOND EDITION.
In the interval which has elapsed since the publication of the first edition of this work, considerable progress has been made in the Integral Calculus; and much matter of the science which was at that time beyond the range and scope of a didactic treatise has been brought within its limits. I thought fit in that edition to omit the whole theory of Definite Integrals except the barest outline of some of its elementary theorems which were required for subsequent applications ; and the Gamma-function and its allied transcendants did not seem to me to be within the scope of an elementary work. Now, on the contrary, no treatise of that character would be complete unless these subjects were discussed at considerable length, and the useful theorems arising out of them, and due to Dirichlet and Liouville, were explained and applied. At that time also it was unnecessary to give any general process for evaluating Definite Integrals; now Cauchy's theory has been
received by mathematicians, and it has become necessary to insert it as the nearest approach to a general method of evaluation of these transcendants. The process of evaluating Definite Integrals has been treated much more fully than heretofore; and the application of the Integral Calculus to the Theory of Series, to the peculiarities of Periodic Series, and to the Calculus of Probabilities has been discussed at considerable length, although the higher parts of these subjects are omitted because they are not suited to a treatise intended mainly for educational use. The several parts of the treatise have been enlarged; of the more difficult portions fuller explanations have been given, and many illustrative examples have been added. The symbols of Determinants have been employed freely in many places; and light is cast by a symmetrical notation on theorems which would be made obscure by a cumbrous symbolism. I have not ventured on the theory of doubly Periodic Functions; for they are too important for cursory and superficial treatment; and a full discussion of their properties would require more space than could be given to it, in a volume which has already exceeded the usual limits of similar works. The latest treatises in which these important transcendants are discussed are mentioned in page 204 ; and the student will in them find much of the information he is in quest of, although for complete information he must have recourse to the original Memoirs of Abel and Jacobi, and other writers of their School.
I have even more consistently than in the first edition constructed the Calculus on its own basis. It has been hitherto for the most part established on an inversion of the rules of the Differential Calculus; it has had scarcely any principles of its own, and of these none independent of those of the Differential Calculus; the student has been obliged to burden his memory with certain rules which he mechanically applies; he has not been taught to deduce them from first principles, because he has had no principles pregnant with such rules; and of them, at least in the early stages of his knowledge, he can give neither intelligible account nor interpretation; and it is only when he arrives at the first geometrical application that he gets an insight into the meaning of the processes; and his view is even then obscured by an expansion into a series, which he no sooner obtains than he omits all terms, save one, of it.
Now in a science replete with applications so large and so important as those of the Integral Calculus, such a method is unsatisfactory, not to say unphilosophical; and it is neither desirable nor necessary to leave it in this state. Most foreign mathematicians have been alive to the defects, and have succeeded in remedying them: why then should Englishmen be behind ? Professor De Morgan is, as far as I know, the first English author who constructed the science on a more philosophical basis; and in his large Treatise on the Differential and Integral Calculus, the Integral Calculus is established on sound principles,
and placed early in the course. For purely scientific reasons such an arrangement may be the best, but it may fairly be questioned whether it is convenient for didactic purposes: I have chosen to place it after the Differential Calculus.
In the following treatise the Integral Calculus is considered a part of Infinitesimal Calculus, and as such, is founded on an intelligible conception of Infinitesimals; it is thus a branch of the science of continuous number; its principles are involved in, and effluent from, that fundamental idea; it assumes the existence of an infinitesimal element-function, formed according to an assigned law, the law being involved in the symbolical form of the infinitesimal; and the primary problem is, to determine the finite number or function of number of which the given infinitesimal is the constituent elemental part; that is, Given the infinitesimal element, to find the finite quantity of which it is the infinitesimal element. The required result can evidently only be definite, when the sum of the infinitesimal elements is to be taken between certain fixed limits, which are at a finite distance apart. Thus the primary problem is one of summation of a series, of which the law is given, (for the symbolical form of the element-function, or type-term, determines that,) and the first and the last terms are given, and the suni of these infinitesimal elementfunctions or differentials is called the Definite Integral. The notion of a Definite Integral is therefore the fundamental one of the Integral Calculus; and the work of the Calculus is, to discover rules for the formation of these, to construct the code of laws which they are subject to, and to investigate the conditions necessary for their application to other subject-matter. Hence it is that the Definite Integrals of simple element-functions are investigated in the early part of the treatise from first principles, and it is only when I have rigorously proved in the most general case that the Definite Integral may be found by an inversion of the process of Differentiation that I have considered myself free to make use of the knowledge of the Differential Calculus, which has been (usually) previously acquired. By these means our labour is diminished, and nothing of principle is lost, because the rules thus found might have been discovered directly from the peculiar principles of the Integral Calculus.
In support of the view of the subject here taken, I allege that on this conception of Infinitesimal elements, and on this conception only, is the Integral Calculus applied to the problems of Rectification, Quadrature and Cubature, and in proof of this allegation I appeal to the processes of Chaps. VI, IX and X; in them the infinitesimal element-function exists previous to the finite function, and the latter is found by the summation of an infinite number of the former. And this is undoubtedly the process, and the only intelligible process, of determining the finite results of an ever-varying law: it is, I assert, on the notion of Infinitesimals only, that the problems of varying velocity can be intelligibly treated by the Integral Calculus.