PREFACE. THE present treatise of Algebra contains all the subjects in theory and practice usually comprehended in an elementary work. In the composition of the treatise, it has been a special object to explain, as clearly and concisely as possible, the principles of the science, to illustrate them fully by appropriate examples, and to prescribe a sufficient number of exercises for solution by the student, in order to impress the principles on his memory, and enable him to acquire a competent skill in Algebraical Computation. And as to the general character of the work, it may be added, that the two extremes of a merely practical and superficial treatise on the one hand, and of a purely theoretical and abstract work on the other, have been equally avoided. One new feature of the present work consists in the manner of treating Insulated Negative Quantities. The view taken of this subject, however, is not altogether new, as it is supported by several eminent mathematicians; and all the merit claimed on this point consists in the new demonstrations which are advanced. There are also several new demonstrations regarding the identity of the products of any number of Factors,hether numerical or algebraical, when taken in any order; and the chain of Theorems on this subject is made more complete, by proceeding in a systematic order from first principles. The subject of the Least Common Multiple has also been more fully treated of tiran usual, on account of its importance, especially in the calculation of fractions. A new proposition is added in the last article of the method of Undetermined Coefficients, which is of importance in some of the higher branches of the science, but which is assumed without demonstration. The summation of some of those species of Infinite Series that are usually given in similar treatises to the present, is here accomplished by a new and simple method, which is both concise and general, and requires for determining the sum, merely the general term of the Series. In Indeterminate Equations, a new method is given for the solution of an equation containing two unknown quantities, which is simple, concise, and direct; and a new theorem is added to the case of two equations with three unknown quan tities, explaining a rather singular fact not hitherto observed, which generally leads to a more simple solution than could otherwise be anticipated. After the student has made himself familiar with the subjects of this treatise, he will be prepared for entering on the applications of this science in the subjects of Analytical Trigonometry and Analytical Geometry, and for the study of the higher branches of Algebra, including the important subject of the General Theory of Equations. Although the mode of arrangement of the subjects of this, as well as every other treatise, is to some extent arbitrary, yet it is not permitted to distribute all the subjects exactly in the order of difficulty, as this would produce a confused and unsystematic disposition of parts. The order, however, in which the matter of each subject, especially in the more elementary portion of the work, has been stated, nearly effects the object that would be attained by such an arrangement. The method generally adopted is as follows:-The rules are first stated as clearly and concisely as possible; then they are illustrated by examples; and the student being thus made familiar with the subject, the demonstrations of the rules are then given; and, lastly, a collection of exercises is prescribed. By this means the transition is very gradual from the more simple to the more difficult parts of the subjects. There are some articles near the beginning of the treatise, which it may be proper to omit in studying the subject for the first time, till the pupil has made some progress in the more advanced parts. The discussions regarding Insulated Negative Quantities in Addition and Subtraction, are of this description; and also the examples with Literal Coefficients in Addition, Subtraction, and Multiplication, which ought to be postponed till the student has advanced through the third case of Division. After finishing Division, the student will be qualified to solve some of the more simple exercises in Simple Equations; and by doing so he will perceive more readily the utility of the science, and will afterwards proceed with greater interest in his study. It would be superfluous to add more on this subject, as the judicious teacher will readily perceive the proper means to be adopted, in order to smooth the difficulties that may occur to the student, to create in him an interest in the subject, and to form a proper taste for the science, without which no great progress can be made. September 24, 1839. INTRODUCTION. NATURE AND OBJECT OF ALGEBRA. IN Algebra, quantities are represented by symbols.* The notation or language of this science is remarkable for its concise and comprehensive character, by means of which many useful and remarkable relations of abstract quantity have been discovered, which could never have been elicited by the use of a less perfect language; and many of these properties are so complex, that, although they had been known, it would have been impossible to express them intelligibly in common language. Instead of using this concise notation to express the reasoning in algebraical investigations, common language might in some simple instances be employed; but, generally, the expressions, from their prolixity, would be altogether unintelligible. Although algebraical symbols represent abstract numbers, which generally denote the numerical values of quantities, yet, when investigating their properties, if particular values were assigned to them, the results obtained would generally be useless, for these results could be asserted to belong only to the given combination of the particular numbers assumed-hence the necessity and the origin of algebraical language. By reasoning in regard to particular numbers, general properties may sometimes be discovered, and their truth satisfactorily established; but, * See the Preliminary Definitions and Principles, page 1. unless in simple cases, these properties can be investigated only by means of the general symbols of this science, which are necessary in order to generalise the results obtained, which show how the various quantities are combined together, and hence also the relation existing among them. THE STUDY OF ALGEBRA AS A DISCIPLINE FOR THE MIND. The study of Theoretical Algebra, which, like the other theoretical branches of Mathematical Science, consists of a continuous process of reasoning, affords an excellent discipline for the highest faculty of the mind. The study of this subject would certainly be of a very superficial nature, were the opinion true, that the symbolical processes of investigation are of so mechanical a character as scarcely to require any intellectual effort. In Practical Algebra, as in the practical application of most sciences and principles, the operations, though certainly not mechanical in a literal sense, may be considered to be so to some extent in a figurative sense; as they consist chiefly in the execution of various transformations effected according to prescribed rules on the algebraical expressions, in order to reduce them, when practicable, to the required form of solution; and any person at all conversant with the sciences, is aware that a species of mechanical process is not peculiar to Practical Algebra, but, on the contrary, that it is common to the practical department of most sciences. Adequate judges who have impartially reflected on the subject, can never be chargeable with such a palpable misapprehension as to confound the practical application of the science with its theoretical investigations-mere operations executed according to established rules, with processes of reasoning frequently of a very subtle nature, and requiring more than an ordinary power of abstraction. Theoretical Algebra is as much a system of abstract science as Theoretical Geometry; and, therefore, the propositions of which it is composed must be as rigorously demonstrated, and the unity of the system as carefully preserved; and the reasoning in this, as in the other theoretical branches of analysis, is certainly not more simple |