THE CALCULUS FOR ENGINEERS.

CHAPTER I.

INTRODUCTORY.

1. Integration more useful than Differentiation.-In physical and engineering investigations the Integral Calculus lends more frequent aid than does the Differential Calculus, and the problems involving the Integral are more often of a practically important type than those requiring the Differential Calculus alone in their solution. But the ordinary student of mathematics never reaches even an elementary knowledge of Integration until he has mastered all but the most recondite portions of the science of Differentiation.

It seems a priori irrational, and contrary to a liberal conception of educational policy, to teach the higher mathematics in a manner so contrary to almost self-evident utility. Adherence to this the orthodox method of teaching in the Schools and Universities is, no doubt, responsible for the persistent unpopularity of this branch of knowledge and intellectual training among the classes devoted to practical work.

2. Method of the Schools. It must be admitted that no great progress can be made in Integration without help from the results obtained by Differentiation. Therefore, so long as the two are taught as distinct subjects, by the aid of separate text-books, it is a distinct convenience to the teachers to finish off one before entering upon the other. If they be thus separated into two successive periods of study, it becomes a practical necessity to give Differentiation the priority in point of time.

3. Rational Method. Still, it by no means follows that the whole of the science of Differentiation must be known before any of that of Integration can be explained, thoroughly mastered, and

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utilised. The ordinary system of teaching the subject forces the practical student to spend on Differentiation an amount of time altogether needless for his professional objects before he enters upon Integration. Much of the former he will never use. The latter, from the very beginning, will supply him with abundant problems of immediate interest and importance in his own special work, and will, moreover, furnish him with a powerful engine that will enormously lighten the difficulties of his own professional subjects and make his progress in these tenfold more rapid.

Let it be noted, also, that very frequently the reasoning used to find an integral is essentially the same as that used to find the inverse differential. It is thus pure waste of time to go through this reasoning twice over. Once understood, it leads to the simultaneous recognition of the two inverse results, both of them, it may be, eminently useful. Therefore, as far as practicable, the study of Differentiation and Integration ought to be pursued pari passu.

4. Active Interest in the Study.-In modern education, in which such large demands are made upon the intellectual energies of the pupil, the necessity of the stimulus of a real active interest, opening out easily recognised prospects of broadening and deepening knowledge and of utilitarian advantage, ought to be conceded in the freest and most liberal fashion. Moreover, it is right to lead the pupil along the easiest road, provided it be a legitimate one. The thoroughness of the training he receives in habits of sound, trustworthy scientific thought depends more upon the length of time he is guided within the limits of correct method, and less upon whether he travels a short distance on a rugged and difficult path or a long distance upon a plainer and smoother route.

5. Object of Present Treatise. The object of the present treatise is to introduce the student at once to the fundamental and important uses of the Integral Calculus, and incidentally to those of much of the Differential Calculus. This we desire to do in such a way as to stimulate a growing desire to progress always further in a branch of science which soon shows itself capable of supplying the key to so many practical investigations.

6. Clumsiness of Common Modes of Engineering Analysis.At the present time our technical text-books are loaded with tedious and clumsy demonstrations of results that can be obtained "in the twinkling of an eye" by one who has grasped even only the elements of the Calculus. These demonstrations are supposed to be "elementary." They are not really so; each of them really contains, hidden with more or less skill, identically the same reasoning as that employed in establishing the Calculus formulas applicable to the case in hand. They are, in fact, simply laboured

methods of cheating the student into using the Calculus without his knowing that he is so doing. There is no good reason for this. The elements of the Calculus may be made as easy as those of Algebra or of Trigonometry. More good, useful scientific result can be obtained with less labour by the study of the Calculus than by that of any other branch of mathematics.

7. Graphic Method.-Much of the Calculus can be rigorously proved by the graphic method; that is, by diagram. This method is here used wherever it offers the simplest and plainest proof; but where other methods seem easier and shorter they are preferred. The present book is strictly confined to its own subject; and, wherever it is necessary, the results proved in books on Geometry, Algebra, Trigonometry, etc., are freely made use of; employing always, however, the most elementary and most generally known of these results as may be sufficient for the purpose.

8. Illustrations. Everywhere the meaning and the utility of the results obtained are illustrated by applications to mechanics, thermodynamics, electrodynamics, problems in engineering design, etc., etc.

9. Classified List of Integrals.-The part of the book which is looked upon by its authors as the most important and the most novel is the last, namely, the Classified Reference List of Integrals. This is really a development of a Classified List of Integrals which one of the authors made twenty years ago to assist him in his theoretical investigations, and which he has found to be continuously of very great service. He has never believed in the policy of a practical man's burdening his memory with a load of theoretical formulas. Let him make sure of the correctness of these results, and of the methods by which they were reached. Let him very thoroughly understand their general meaning, and especially the limits of their range of applicability; let him recognise clearly the sort of problem towards the solution of which they are suited to help; let him practise their application to this sort of problem to an extent sufficient to make him feel sure of himself in using them in the future in the proper way. Then let him keep notes of these results in such a manner as will enable him to find them when wanted without loss of time; and let him particularly avoid wasting his brain-power by preserving them in his memory. The more brain-power is spent in memorising, the less is there left for active service in vigorous and wary application in new fields to attain new results. Formulas have a lamentable characteristic in the facility they offer for wrong application. A formula fixed perfectly in the memory, and the exact meaning and correct mode and limits of whose application are imperfectly understood, is a