IN WHICH THE METHOD OF LIMITS IS
EXCLUSIVELY MADE USE OF.
REV. M. O'BRIEN, M.A.,
LATE FELLOW OF CAIUS COLLEGE, CAMBRIDGE.
PRINTED AT THE UNIVERSITY PRESS;
PUBLISHED BY J. & J. J. DEIGHTON, CAMBRIDGE;
JOHN W. PARKER, LONDON.
The method of Limits is generally allowed to be the best and most natural basis upon which to found the principles of the Differential Calculus; in the following pages this method is exclusively adopted, no use whatever being made of series in the demonstration of fundamental propositions. The following is an outline of the work, which is by no means offered to the reader as a complete treatise on the subject, but merely as an exposition of its more prominent and useful principles.
In Chap. I, certain terms, afterwards to be used, are defined and explained. In Chap. II the nature of a Limiting Value is fully set forth, and the important distinction (which ought never to be overlooked) between an actual and a limiting value is pointed out and illustrated by examples. Chap. III contains a set of Lemmas, which are necessary in order to render the use made of limiting values in the Differential Calculus perfectly legitimate; and here I have endeavoured to confine myself to what seems really essential. In Chap. IV certain important limiting values are obtained. Chap. V contains the Rules for Differentiation, in the demonstration of which Lagrange's functional notation is employed, as being the simplest to begin with. In Chap. VI the Differential notation of Leibniz is explained, dy/dx is defined as the quote of the differentials dy and dx, which however are not supposed to be infinitesimals, but simply two arbitrary quantities in a certain ratio. In the case of partial differential coefficients, some modification of the common differential notation du/dx, du/dy, is clearly necessary: I have employed the suffix notation dxu, dyu, as being frequently employed, though not exactly in this manner. I should have much preferred the notation dxu, dyu to denote partial differentials, and dxu/dx, dyu/dy to denote partial differential coefficients. Chap. VII relates to successive differentiation, and the change of the independent variable. Chap. VIII contains certain very important Lemmas upon which the use and application of the Differential Calculus in a great measure depends. Chap. IX contains the theory of Series, based upon one of the preceding Lemmas, without assuming that f (x + h) can be developed in the form
A + Bha + Chb + &c. ...
and here I have endeavoured to show what the real nature of a series is, and to prove rigorously the principle of Indeterminate Coefficients. Chapters X and XI relate to Vanishing Fractions, and Maxima and Minima, and contain some useful simplifications of the common methods. The very insufficient and troublesome criterion usually employed in distinguishing the maxima and minima of functions of two variables is not introduced. Chap. XII relates to Tangents, Normals, &c., the Curvature of Curves, and the properties of the Evolute; and here the arrangement usually adopted is somewhat departed from, and what seems a more natural course pursued, in order to avoid certain difficulties, which I have observed very often impede the student on his first reading of the subject. In Chap. XIII the useful Polar formulae and the differentials of Areas, Volumes, &c., are deduced. Chap. XIV relates to Asymptotes. Chap. XV contains a very simple method of tracing curves. Chap. XVI relates to singular points. Chap. XVII contains the general Theory of Contacts and Ultimate Intersections: no use is made of series in explaining the different orders of contact. The remaining chapters are occupied with Elimination by differentiation, Lagrange's Theorem, the properties of the Cycloid, &c., &c. The Appendix contains Examples worked out.
It was my intention to have added a few more chapters, and among the rest, one on the origin and progress of the Differential Calculus, and another on the Infinitesimal method; but from various circumstances I found it impossible to send the work to the press at the time originally promised to my bookseller, without omitting these concluding chapters. I mention this to account for the absence of allusions to the History of the Differential Calculus, which were all reserved for the final chapter, and the small number of Examples in the Appendix.
Professor Peacock's excellent collection of Examples, which have been of such service to the Mathematical Student, is now out of print; but Mr Gregory's work lately published will supply its place, which contains, not only a great number of well-selected and valuable examples, but also many important explanations and theorems not to be met with in any elementary treatise. In a subject of so much importance as the present, the student ought not to confine his attention to one book or system: for a very valuable treatise on this subject he is referred to that published by the Society for the Diffusion of Useful Knowledge.
In the general plan of this work, and in several particulars, I have deviated from some of the methods often made use of, partly in attempting to put the subject in a simpler and clearer point of view, and partly in avoiding certain steps of reasoning which appear to be defective. One of these is the fallacy of establishing premises on a certain implied condition, and drawing a conclusion from them by a direct violation of that condition. An example of this is to be found in a proof often given of the principle of indeterminate coefficients, in which the factor x is divided out of the equation
Bx + Cx2 + Dx3 + &c. ... = 0,
which of course tacitly assumes the condition that x is not zero; in this manner is obtained the equation
B + Cx + Dx2 + &c. ... = 0;
and then by putting x = 0, contrary to the implied condition, the conclusion B = 0 is arrived at. Another example of this kind of reasoning is given in the first note, page 6.
The assumption, that f(x + h) can be expanded in a series of the form A + Bha + Chb + &c. ... seems to me to be a serious defect in the common method of establishing Taylor's Series, and thereupon the principles of the Differential Calculus. This assumption is usually justified by arguing, that if we find definite values for A, B, C, &c. it shows that the assumption is correct. Now this argument may be stated thus:
"If the assumption that f (x + h) = A + Bha + Chb + &c. be true, then A, B, C, &c., must have definite values. But we can in general obtain definite values for A, B, C, &c.... (e.g. by the method of indeterminate coefficients.) Therefore the assumption is true."
This is clearly a fallacious argument, for to warrant the conclusion the first premise should have been this:
"If the assumption be not true, definite values cannot be obtained for A, B, C, &c."
These defective steps of reasoning and others which might be mentioned, are objectionable, not because they lead to erroneous conclusions, but because they ought not to be found in a subject like the present, in which every thing should be conformable to the strictest rules of logical deduction. M Cauchy has done much towards the improvement and perfection of the Differential Calculus, and his writings on this, like those on the more abstruse branches of mathematics, are most valuable. In one or two places the methods I have employed in the following pages are apparently similar to those of M Cauchy, but in reality they are essentially different: so far as I am aware I am indebted to him only for article 48.