AN ELEMENTARY TREATISE ON ALGEBRA, THEORETICAL AND PRACTICAL. BY JAMES THOMSON, LL.D. PROFESSOR OF MATHEMATICS IN THE UNIVERSITY OF GLASGOW. LONDON: PRINTED FOR LONGMAN, BROWN, GREEN, AND LONGMANS, PATERNOSTER-ROW. 1844. PREFACE. In the following Treatise, it has been the aim of the Author to render the various investigations and processes as simple, natural, and easy as possible, and to establish and illustrate the principles in a plain and familiar manner. He trusts, in particular, that the modes in which he has established the rule for subtraction (not new, he admits, in principle), and the rule for the signs in multiplication and division, and also the methods which he has employed in explaining the theory of fractions and radicals, will divest those subjects of much of the mystery and difficulty which have unnecessarily been thrown around them. In a similar manner, using freely, but not servilely, what has been done by previous writers, he has endeavoured to simplify the elementary principles and processes in the resolution of equations; and he hopes that the views and explanations which he has given regarding infinite quantities and the sums of infinite series, will remove much of the difficulty which is very generally and very naturally felt in reference to those subjects. Of the binomial theorem, he has given a proof which he considers simple and easy; and the experienced algebraist will recognise various other improvements in the course of the work, which it is unnecessary here to particularise. Much absolute novelty, unless in the mode of exposition, cannot now be expected in a work on algebra. Some things, however, the Author believes are new altogether, while others are now for the first time fully developed, and extensively and advantageously applied. Of this kind is the method of detached coefficients, which is explained and exemplified in pages 26. 35. 84. &c.; a method which is at once greatly shorter and easier than the common one, and which, throughout the operation, relieves the mind from the trouble and fatigue of considering what powers of the quantities are to be written in the several terms. It prepares the student also for many of the investigations connected with the theory and resolution of equations, and it is employed with advantage on many other occasions. He believes also, that the methods of resolving quadratic equations by means of the rules in pages 135. and 139. will be found to be much superior, in point of facility and simplicity, to any other method that has yet appeared; as those rules preclude the necessity of "completing the square," extracting the square root, and transposing; and they prevent all trouble in the management of fractions; while, in the common method, the fractional operations often constitute the principal part of the labour.* In page 63. there is given a new method of finding the greatest common divisor, by means of which the required result is often obtained much more easily than by the common method, according to the rule in page 59. In pages 49. and 73., the advantage of employing negative indices in certain cases, so as to prevent the necessity of performing fractional operations, is pointed out and exemplified; and a new and easy method of "proving" operations in multiplication and division is given in the notes in pages 24. and 33. In Section X. a brief outline of some of the principal subjects comprehended in the theory of equations is given ; and, along with it, the beautiful method of resolving numerical equations, discovered by the late Mr. Horner, is exhibited at considerable length, and in a mode which it is hoped will be found to be very simple and easy, both in its theory, and in practice. That the brief introduction to the theory of equations here given must be very * The first of these rules was published by the Author in a Belfast periodical in 1825, and the second occurred to him at a later period. A particular case of the second rule is given by Bonnycastle in his larger Algebra. The Bija Ganita, a Hindoo work on algebra of the twelfth or thirteenth century, contains a method of solution which is much superior, in many instances, to the one which is commonly employed in Europe, as it prevents fractions from arising in the course of the operation. It requires, however, the formal completing of the square the extraction of the square root, &c. The rules here given combine the advantages of both these methods, and are free from their disadvantages, giving in every case the values of the unknown quantity without any intermediate work, and always in the simplest and best form. It is strange, that, while the modern analysts have so zealously and successfully exerted themselves in deriving easy rules for practical purposes in numberless cases, they should not have thought of establishing similar rules for the solution of quadratic equations, but should have gone on in the old way, always dividing by the coefficient of the highest power, completing the square, extracting the square root, and transposing. imperfect will be readily anticipated, if it be considered that Professor Young of Belfast has published a work in five hundred octavo pages on the Theory and Solution of Algebraical Equations of the higher Orders. What is here given on this curious and interesting branch of analysis will perhaps be sufficient for the greater number of mathematical students; but those who may wish to obtain a more extensive knowledge of the subject, and to devote the time and labour necessary for accomplishing that object, may have recourse to Mr. Young's work above referred to. The examples and exercises are, in the great majority of instances, new. Several, however, have been taken from foreign, and particularly German, works; and some are from English writers of past times, but none from living English authors, except in a few instances in which improved solutions are given or indicated. Several instances of this kind will be found in the Section on the Diophantine Analysis. As to the mode of employing the work as a text-book, the teacher must be guided mainly by his own judgment. It may be remarked, however, that in a first course, unless the pupil pos-. sess more than ordinary ability, the more difficult parts may be omitted, and may be taken up afterwards, when the learner has acquired more power in the management of algebraic quantities, and in the performance of operations. Thus, there are portions of Section III., which, however important and valuable, will at first be found by most learners to be somewhat abstract, and not so easily followed out as many subsequent parts of the work, and which may therefore be postponed. For the same reason, several things regarding fractions, radicals, involution, evolution, and some other subjects, may at first be omitted with advantage; and the same may be done, with great propriety, regarding several of the more difficult exercises in the various Sections. What has now been said will be applicable, in a still greater degree, in reference to those who may study the work without the aid of a teacher. The learner, whether he has a teacher or not, should by no means be dispirited by feeling at first some difficulty in the study of algebra. Few studies are free at their commencement from something that is unattractive, or even forbidding: and though in algebra, from its peculiar symbolical language, and its abstract and general character, some difficulty will often be felt for a short time, this will soon be removed by steady application, and by continued practice; and the student, on returning to investiga |