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The character of my treatise is, as heretofore, elementary. I have endeavoured to explain as clearly as possible the language, the symbols, the first processes of Infinitesimal Calculus; and with the object of presenting the principles in a form less repulsive than is usual with English writers many illustrations are introduced which may at first sight appear foreign to the subject; yet they are not so; for the thoughtful reader will detect in these applications abstract conceptions which are fundamental in the Calculus. And he will also see that though he may have been ignorant of the nomenclature of the science, yet that neither the idea nor the apparatus of it is entirely new to him. Throughout the Treatise, and especially in the early part of it, examples are inserted to give the student an aptness in applying general rules to particular cases, as well as to aid him in the detection of universal principles lying under particular examples. The Treatise also includes the higher parts of the science, because it is intended for the use of the most advanced students in our universities; and thus in many parts theorems are introduced which are close upon the boundaries of our knowledge.
The matter was first delivered orally in lecture, and subsequently written; whence has arisen the colloquial style; and which it has been thought expedient to retain, under the conviction that it invests a book with a personal and living character more akin to the explanations of a speaking teacher, and thereby infuses life into what might be otherwise dry text; and in some few passages wherein disputed questions are discussed, objections are stated as if urged by an opponent.
I do not care to claim any originality as to the matter of the Treatise; it is enough for me if I have been able to state the principles of our science with perspicuity, and to arrange its several parts in such order as to remove difficulties which I venture to think are not necessarily connected with the subject. The sources whence the matter is derived are various, and for the most part foreign; all cannot be specified, and probably I am indebted to others for more than I am aAvare of; every book, and perhaps every sentence and every conversation, may leave an impress which, unconsciously to himself, modifies an author's opinions. Almost all which has been taken from other sources, and is not specially acknowledged, has been so long known and commented on, as to have become publici juris; Euler, Lagrange, Lacroix, Peacock, Monge, Dupin, it is quite unnecessary to mention; a few names occur in the foot- and other notes. I am however under especial obligation to M. Cauchy in his various treatises and memoirs, to his Redacteur M. l'Abbe" Moigno; to M. Navier; to the late Mr. D. F. Gregory; to Professor De Morgan, the author of the treatise published by the Society for the Diffusion of Useful Knowledge; to Professor Donkin; to Mr. W. Spottiswoode, M. A.; both of the University of Oxford, and from whom I have received valuable assistance and advice in many parts of the Treatise: and generally too to the Journals of Liouville and Crelle, mines of precious materials to the mathematical student.
The process of assimilating a body of matter so large, and of such diverse origin, as that of Infinitesimal Calculus, is necessarily long; and in the course of it the question arises, under what principle is all to be harmonized? the parts are seemingly dissonant, what renders them consistent ?" whence is the string obtained on which the pearls are to be hung?"
An inquiry of this kind is far too large to be answered within the limits of a preface, but some few remarks are necessary for a due understanding of our method.
Infinitesimal Calculus, both in its pure and applied forms, whether of Geometry or of Mechanics, is a branch of the science of Number; its symbols are of the same kind, and are operated on according to the same laws; they are applied subject to the same conditions, are interpreted on the same principle, and lead to analogous results. What then is its specific characteristic? In the parts of the science of Number which are supposed to have been previously studied, viz. in Arithmetic and Algebra (as it is called), numbers are finite and discontinuous; but Number also admits of being continuous; that is, is capable of gradual growth and of infinitesimal increase Number under this aspect is what Infinitesimal Calculus contemplates: and investigates the new properties of it, the new symbols required to express them, and the new laws to which they are subject; it has thus to create its own materials, and these materials are infinitesimals.
The method therefore has been at first to unfold those properties of Number which are necessary to the construction of the science of infinitesimals; and then so to describe continuous Number and the infinitesimal elements of its growth, in their essential qualities, as to be able to enunciate certain axiomatic properties of them, from which the Calculus may be evolved. These are stated in seven Theorems in Art. 9; of which perhaps the most important is Theorem VI, presenting the character of infinitesimals in their broadest view, viz. that they are such that a finite number of them has no value at all when added to a finite quantity. These Theorems are the ultimate propositions of the science: from them the other truths are inferred, and on them does the correctness of the processes depend. In order to the subsequent use of the advantage in the way of inference possessed by algebraical symbols, we express them in mathematical language, and then proceed to deduce the consequences with which they are pregnant; and this deductive process is continued through the Treatise. In the course of it however, and especially in the applied parts, the subject-matter becomes enlarged, new relations are introduced, and new names are required; thus definitions become necessary, and these are so constructed as to contain in germ the substance of the Articles dependent on them.
A due understanding of these axioms and definitions is plainly of the utmost importance; because on it does it depend whether we work with mere symbols, or whether the symbols are eweta of philosophical ideas which we comprehend. With the object of guarding against such superficial knowledge, which is useful neither in its results nor as an intellectual exercise, geometrical interpretation of infinitesimals has been often introduced, and magnified diagrams exhibit lines, areas, angles, &c. which are represented by symbols of infinitesimals: every process, nay every step in every process, admits of such geometrical translation; and it is most desirable that the student should exercise himself in it: by so doing he will have a most certain test whether his operations are according to the laws of correct inference, or whether he is merely applying mnemonic rules and groping his way in the dark by some obscure road, and drawing his conclusion as it were only by riddles.
The Calculus then is that of infinitesimals; but as the most convenient mode of forming them shews them in the light of differences between two states at a "very small" distance apart, they have obtained the name of Differentials; and hence the name of "Differential Calculus" has arisen; the latter term is retained, although it refers to the mode of generating the materials, and not to any pregnant property of them when generated. The notation also and language of Lagrange's Calculus of derivedfunctions has been employed, because it expresses in a most convenient form some of the earliest results of operations with infinitesimals, and their relations to finite quantities: but I would warn the reader especially against supposing that our Calculus is founded on the notion of derived-functions as the coefficients of the terms of a series; they are not considered in that relation at their first admission, and it is only by a course of reasoning that they afterwards become so. Expansion in a series has been admitted fundamentally but once, viz. in Art. 21, and whatever may be the final issue of the question as to Convergent and Divergent Series, it cannot affect that which I have employed, inasmuch as it is proved to be Convergent.