George Gibson: Calculus
Preface to the 1901 First Edition
The rapid growth in recent years of all branches of applied science and the consequent increasing claims on the time of students have given rise in various quarters to the demand for a change in the character of mathematical text-books. To meet this demand several works have been published, addressed to particular classes of students and designed to supply them with the special kind and quantity of mathematics they are supposed to need.
With many of the arguments urged in favour of the change I am in hearty sympathy, but it is as true now as it was of old that there is no royal road to mathematics, and that no really useful knowledge can be gained except by strenuous effort.
It is sometimes alleged that a thorough knowledge of the derivatives and integrals of the simpler powers, of the exponential and the logarithmic functions, and perhaps of the sine and the cosine, is quite sufficient preparation in the Calculus for the engineer. This contention has a solid substratum of truth; but a knowledge that goes beyond the mere ability to quote results is not to be obtained by the few lessons that are too often considered sufficient to expound these elementary rules. It may be possible to state and illustrate in a few lessons a sufficient amount of the special results of the Calculus to enable a student to follow with some intelligence. the more elementary treatment of mechanical and physical problems; but, though such a meagre course in the Calculus may not be without value, it is quite inadequate, both in kind and in quantity, m a preparation for the serious study of such practical subjects as Alternate Current Theory, Thermodynamics, Hydrodynamics, and the theory of Elasticity, and to a student so prepared much of the recent literature in Physics and Chemistry would be a sealed book. Besides, it should surely be the aim of every well-devised scheme of education to place the student in a position to undertake independent research in his own particular line of work, and the very complexity of the problems presented to modem science, with the vast accumulation of detail so characteristic of it, enhances in no small degree the value of a liberal training in mathematics. Subsequent specialisation makes it the more, not the less, necessary that the mathematical training in the earlier stages should be the same whether the student afterwards devotes himself to pure mathematics or to the more practical branches of science, especially as the processes of thought involved in any serious study of mechanical, physical, or chemical phenomena have much in common with those developed in the study of the Calculus.
The early text-books on the Calculus, such as Maclaurin's or Simpson's, were not written for pure mathematicians alone, but drew their illustrations largely from Natural Philosophy; the later text-books, probably in consequence of the ever-widening range of Physics, gradually dropped physical applications, and even tended to become treatises on Higher Geometry. In the present position of mathematical science, however, it is just as much out of place to make an elementary work on the Calculus a text-book of Higher Geometry as. it would be to make it a textbook of Physics or of Engineering or of Chemistry. What may be reasonably required of an elementary work on the Calculus is that it should prepare the student for immediately applying its principles and processes in any department of his studies in which the Calculus is generally used. With this end in view, the subject should be illustrated from Geometry, Mechanics, and Physics while the peculiar difficulties of these branches are relegated for detailed treatment to special text-books, so that the illustrations may really serve their purpose of throwing light on general principles, and may not introduce rather than remove intellectual obscurity. As regards Chemistry, a sound knowledge of the Calculus is of special importance, since it is the properties of functions -of more than one variable that are predominant in chemical investigations; the lately published book of Van Laar, Lehrbuch der Mathematischen Chemie, is a sign of the times that cannot be mistaken.
In this text-book an effort has been made to realise the aims just indicated. With respect to mathematical attainments, the reader is supposed to be familiar with Geometry, as represented by the parts of Euclid's Elements that are usually read., with Algebra up to the Binomial Theorem for positive integral indices, and with Plane Trigonometry as far as the Addition Theorem; but no use is made of Complex (imaginary) number, nor is a knowledge of Infinite Series presupposed. The excessive refinements of modern mathematics have been deliberately avoided, as being neither profitable nor even intelligible to the young student; constant appeal has been made to geometrical intuitions, while at the same time considerable attention has been paid to the logical development of the subject.
The early chapters may seem to contain a great deal of matter that is foreign to the book: but the theory of graphs and of units is of such importance, and is as yet so imperfectly treated in elementary teaching, that some account of it appeared to be- a necessity. After considerable hesitation I have included in my plan the elements of Coordinate Geometry, so far as these were likely to be of real service in elucidating fundamental principles or important applications; but for many applications of the Calculus an extensive acquaintance with Coordinate Geometry is not necessary, and I hope that a sufficiently clear account of its principles has been given to meet the practical needs of many students. I have, however, excluded the discussion of the theory of Higher Plane Curves and of Surfaces as unsuitable for an elementary treatise.
Another innovation is the chapter on the Theory of Equations; the innovation seems to be justified, not merely as an arithmetical illustration of the Calculus, but also by the practical importance of the subject, and by the absence of elementary works that treat of transcendental equations.
The general development is that which I have followed in class-teaching for several years. The somewhat lengthy discussion of the conceptions of a rate and a limit I have found in practice to be the simplest method of enabling a student to grapple with the special difficulties of the Calculus in its applications to mechanical or physical problems; when these notions have been thoroughly grasped, subsequent progress is more certain and rapid. No rigid line is drawn between differentiation and integration, and several important results requiring integration are obtained before that branch is taken up for detailed treatment. The discussion in Chapter X of areas and of derived and integral curves is designed, not only to furnish a fairly satisfactory basis for the geometrical definition of the definite integral, but also to illustrate a method of graphical integration that is of some importance to engineers, and that may be of some value even in purely theoretical discussions.
As in some of the more recent text-books, the discussion of Taylor's Theorem has been postponed; the Mean Value Theorem is sufficient in the earlier stages, and the somewhat abstract theorems on Convergence and Continuity of Series are most profitably treated towards the end of the course. The treatment, however, is such that teachers who prefer the usual order may at once pass from the Mean Value Theorem to Chapters XVII. and XVIII.
Functions of more than one variable are treated in less detail than functions of one variable; but I have tried to select such portions of the theory as are of most importance in physical applications. The book closes with a short chapter on Ordinary Differential Equations, designed to illustrate the types of equations most frequently met with in dynamics, physics, and mechanical and electrical engineering.
Simple exercises are attached to many of the sections; in the formal sets will be found several theorems and results for which room could not be made in the text, and which are yet of sufficient importance to be explicitly stated. I have tried to exclude all examples that have nothing but their difficulty. to recommend. them; and with the object of encouraging the student to put himself through the drill that is absolutely necessary for the acquisition of facility and confidence in applying the Calculus, I have freely given hints towards the solution of the more important examples.
In the preparation of the book, I have consulted many treatises, and where I am conscious of having adopted a method of exposition that is peculiar to any writer, I have been careful to make due acknowledgment. It is difficult, however, when one has been teaching a subject for years to recognise the sources of his knowledge, and it may well be that I have borrowed more largely than I am aware.
I am greatly indebted to my friends Professor Andrew Gray, F.R.S.; Mr John S Mackay, LL.D.; Mr Peter Bennett; Mr John Dougall, M.A.; and Mr Peter Pinkerton, M.A., for help in the tedious task of the revision of proof-sheets and for useful criticism. In all matters bearing on Physics, Professor Gray's advice has been of the greatest service. To Mr Dougall my obligations are specially great; he has taken a lively interest in the work from its inception, and has read the whole of, it in manuscript, placing at my disposal, in the most generous way, his great knowledge of the subject and the fruits of his experience as a teacher; to him, too, I owe the verification of the examples.
I desire to thank Sir Richard Gregory for his constant and kindly advice on matters relating to the passage of the book through the press. I am also grateful to the printers for the excellence of their share of the work.
GEORGE A GIBSON.
Glasgow, September, 1901.
Preface to the 1906 second Edition
In this edition I have not ventured to make any changes on the first edition; I have however added two chapters with the object of making the book more useful to students of mathematical physics. In the discussion of operations under the sign of Integration I have adopted the method developed by M Charles J de la Vallée Poussin in his Memoir Étude des intégrales à limites infinies; that method seems to me to combine simplicity and rigour in a very remarkable degree. The chapter on the Fourier Series will, I hope, be sufficient as an introduction to the subject; but the student can not be too earnestly recommended to read and to master the fascinating pages in which Fourier himself develops the process of representing an arbitrary function by means of a harmonic series.
I am indebted to my friends Mr John Dougall, M.A., and Mr John Miller, M.A., for their generous help in the revision of the proof-sheets; Mr Dougall has also verified all the examples and supplied answers where these seemed to be necessary.
GEORGE A GIBSON.
Glasgow, November, 1905.
JOC/EFR August 2007
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