ON INFINITESIMAL CALCULUS; CONTAINING DIFFERENTIAL AND INTEGRAL CALCULUS, CALCULUS OF VARIA- AND ANALYTICAL MECHANICS. BY BARTHOLOMEW PRICE, M.A., F.R.S., F.R.A.S., FELLOW AND TUTOR OF PEMBROKE COLLEGE, AND SEDLEIAN Professor of NATURAL PHILOSOPHY, oxford. VOL. I. DIFFERENTIAL CALCULUS. SECOND EDITION. "Les progrès de la science ne sont vraiment fructueux, que quand ils amènent OXFORD: AT THE UNIVERSITY PRESS. M.DCCC.LVII. PREFACE. TO THE SECOND EDITION. THE character of Infinitesimal Calculus as a progressive science has been so well sustained during the last few years, that in preparing for the press this second edition of my first volume I have found it necessary to make more alterations than might otherwise have been desirable. In Great Britain, no less than in other parts of the world, and especially in Germany and France, have the boundaries of our science been advanced: new methods and new processes have been devised, new theorems have been discovered, and to new forms of subject-matter the principles of the Calculus have been applied. Every one who has watched the progress of the science will hardly need to be told of Sir W. R. Hamilton, Cayley, Boole, Donkin, Spottiswoode, Sylvester, Salmon, Hargreave, any more than of Jacobi, Dirichlet, Joachimsthal, Hesse, Plücker, Steiner, Sturm, Liouville. To all these in their several degrees is due the credit of having brought the subject to a state of perfection never heretofore attained; and a treatise pretending to be didactic and educational would ill deserve those epithets if the discoveries of such eminent mathematicians were not noticed in it. 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 aware 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'Abbé 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 |