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, 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. * See the Preliminary Definitions and Principles, page 1. 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 than in other sciences, merely because its language is so remarkably concise and comprehensive ; for if this circumstance causes any difference, it must increase, not diminish, the difficulty. COMPARISON BETWEEN THE ALGEBRAICAL AND GEOMETRICAL METHODS. It appears frequently that the investigation of a subject by the geometrical method is exceedingly difficult, and remarkably simple by analytical methods. This, however, generally happens when the subject may be considered as merely of an elementary character in the analytical department; for this branch properly begins at that point, at which common Geometry—from its being a comparatively feeble, and, generally, in the higher problems, an inadequate instrument of investigation-ought to terminate. In applying these two methods to mathematical subjects, the most useful mode of proceeding would certainly be to confine the geometrical method to the more elementary portions of a subject, so far as it is applicable without introducing prolixity, both on account of its greater simplicity, and the peculiar elegance of many of its results; and also because there are many students who cannot devote sufficient time to enable them to acquire that degree of proficiency in the analytical department, which is indispensable to the successful cultivation, to any considerable extent, of the interesting and extensive subjects to which it is applied. The geometrical method, however, even in regard to many subjects, to the investigation of which it is adequate in other respects, is inferior to the analytical, in the generality of its results. A property, for instance, may be proved by the former method to belong to some curves; but it is by the latter method that the more general truth is established, that this property is possessed by no other curves: the former method affords one means of reaching a certain point, the latter shows by what and by how many ways it be attained ; the one is only particular, the other is general, in its conclusions. As to the comparative difficulty of the two methods, it is one of the most certain truths that that difficulty is as unlimited in degree, as the field for investigation is unbounded in extent; with, however, this important distinction, that by means of the superior power of the analytical method, it is possible, with a certain expenditure of intellectual labour, to penetrate to an incomparably greater extent into the unexplored regions of science. A subject must undergo a mathematical analysis before its conditions can be combined and a solution obtained; and if it is a physical subject, it must undergo a physical as well as a mathematical analysis. Since, therefore, the analysis of a subject generally constitutes the chief difficulty in its investigation, and the chief exercise of the mind, and since those subjects to which the analytical method is applied are usually most complex, it is an inevitable consequence that the analytical departments afford in these, the most important points, a superior exercise to the mind. It no doubt sometimes happens that in the application of this method in original researches, a considerable part of the labour consists in the transformation of expressions; but in such cases these operations are not always performed according to rule, for, sometimes, to effect them, it is necessary to investigate new methods of resolution, which alone not unfrequently require much address and inventive power of no ordinary kind to accomplish; and thus a series of difficulties of different kinds must be surmounted before the ultimate object of research can be attained. How mistaken, then, is the opinion that the analytical methods are mere mechanical processes; that difficulties of the most abstruse and recondite character can be solved by means of it by the smallest intellectual effort! Were this opinion true, how great would be our admiration of those minds that invented and fabricated such a miraculous instrument of science, compared with the power of which the supernatural agency of the rod of the magician is impotent! The Higher Analysis is certainly a very efficient instrument, but even its power is frequently inadequate to surmount difficulties that lie on every hand as immoveable obstacles to the progress of the mathematician. What constitutes one of the peculiar difficulties of Mathematical Science, is, that, from the certainty of its principles, it is capable of being indefinitely extended; and the system already formed is of such formidable magnitude, that the student must acquire a great proficiency in a very extended course of study, before he can engage, with the prospect of great success, in original researches. OF THE PRACTICAL UTILITY OF ALGEBRA. By the principles of Algebra, rules are investigated for the solution of questions in Compound Interest, and in Certain and Contingent Annuities, and in various other important branches. The importance of this science, in a practical point of view, may be comprehended in the statement, that the rules in many of the practical branches of Mathematics and Physical Science are expressed by Algebraical Formulas, which are well adapted for this purpose by their conciseness and precision; as these rules are in many instances incapable of intelligible enunciation in common language. HISTORY OF ALGEBRA. The term Algebra is of Arabic origin, being composed of the two words al and gebr ; but the title of the earliest European treatises on this science was Alghebra e Almukabala, which means resolution and composition. The origin of the science, like that of some others, is involved in much uncertainty; whether we are indebted to the Greeks or Hindoos for its first principles, is still an undetermined question. About the middle of the fourth century, the arithmetic of the ancients contained only the four fundamental rules, and the extraction of the square and cube roots, when Diophantus (350 A. D.), one of the most distinguished Grecian mathematicians of the Alexandrian school, invented a new class of arithmetical problems, and the very ingenious methods of solution employed by him have a great resemblance to those of Algebra. But in the first six books of his works, which are all that remain of the thirteen com |