the repres result In likewise of symbolical algebra, where they are general in value as well as in form: thus the product of a" and a", which is am+n when m and n are whole numbers, and therefore general in form though particular in value, will be their product likewise when m and n are general in value as well as in form: the series for (a + b)", determined by the principles of arithmetical algebra, when ʼn is any whole number, if it be exhibited in a general form, Li without reference to a final term, may be shewn, upon the same principle, to be the equivalent series for (a + b)", when n is general both in form and value. for th numb for, i that numb there and letter 2 the e tables orde For, from orde subt form as in This principle, in my former Treatise on Algebra, I denominated the "principle of the permanence of equivalent forms," and it may be considered as merely expressing the general law of transition from the results of arithmetical to those of symbolical algebra: it is this law which secures the complete identity of the two sciences as far as those results exist in common, and without which the latter science would degenerate into a science expressing the arbitrary combinations of symbols only, whose results would, in the first instance at least, be altogether separated from arithmetic, and therefore from arithmetical computation. Upon this view of the principles of symbolical algebra, it will follow that its operations are determined by the definitions of arithmetical algebra, as far as they proceed in common, and by the "principle of the permanence of equivalent forms" in all other cases, which those definitions cannot comprehend: it will follow therefore that in all such cases, the meaning of the operation or of the result obtained, whenever such a meaning can be assigned, must be determined in conformity with the conditions which it must satisfy, and consequently must vary with every variation of those conditions: upon this principle we shall be enabled to give a consistent interpretation to symbolical expressions or results such as +a anda, considered with reference to each other, where a denotes a line, a force, a period of time, and various other concrete magnitudes, as well as to the operations called ddition and subtraction, multiplication and division, denoted by ub the usual signs, when applied to such quantities: but in fou for innumerable other instances, it will be found that th results obtained will admit of no interpretation whatever or have hitherto failed to receive it. The results therefore of symbolical algebra, which ar not common to arithmetical algebra, are generalizations o form, and not necessary consequences of the definitions which are totally inapplicable in such cases. It is quit true indeed that writers on algebra have not hithert remarked the character of the transition from one class of results to the other, and have treated them both a equally consequences of the fundamental definitions of arith metic or arithmetical algebra: and we are consequently presented with forms of demonstration, which though really applicable to specific values of the symbols only, are tacitly extended to all values whatsoever: such are the usua proofs which are given of the indifference of the order o succession of the factors of a product', of the rule fo the multiplication and division of fractions, of the rule for finding the greatest common measure of two quantities, whether numbers or algebraical expressions3, of the form of the product of powers of the same symbol', as well as others which might be mentioned: whilst the rules for performing the fundamental operations, for the incorporation of signs, and the meaning of fractional powers of a symbol, are little more than assumptions, of which not even the form of a demonstration is attempted to be given. In all such works, some few of very recent date excepted, it has been presumed that these conclusions could be deduced or these generalizations made in perfect consistency with the arithmetical definitions of the operations employed, or, in other words, without any change in the meaning of such operations, however different may be the quantities which are subjected to them: the consequence of this fundamental error has been, that no sufficient demonstration or adequate exposition has been given of the principles of 1 Wood's Algebra, Art. 75. 3 Ib. Art. 90, 91, 92, and Example. Ib. Art. 73, 74, &c. 7 lb. Art. 57. 2 lb. Art. 105 and 108. the epres esult umb Li or, i that science which is of all others the most general in its applications. A student who is not only familiar with the results of arithmetical algebra, but likewise with the limitations which it imposes, will be in a condition to comprehend and appreciate the whole extent of the legitimate conclusions which it furnishes: and though he may find himself perpetually brought in contact as it were, by the generality of the form of the symbols which he employs and of the rules of the operations to which they are submitted, with results, which the definitions of arithmetical algebra will hat not authorize, yet he will thus acquire a habit of observing lumb there not merely what is within, but what is without the just and proper boundaries of the science: and what is still more and important, as a preparation for the study of symbolical letter algebra, he will be thus enabled to appreciate at once the origin and the full extent of the "principle of the perma2nence of equivalent forms," which assumes a knowledge of thee the rules of arithmetical as the basis of those of symbolical ables algebra, without which the identity of the results of the order two sciences, as far as they proceed in common, could not For, be secured. I have considered the above mentioned reasons from sufficient to justify me in separating the exposition and orde study of the two sciences from each other: and I feel subt satisfied that the adoption of such a course will be found form materially to contribute to the most secure, if not to the most expeditious acquisition of distinct and general views indof the principles of this most important science. as Fo The present volume, though strictly elementary as far as the theory and operations of arithmetical algebra are concerned, will be found to comprehend many other subjects, which cannot be considered as equally necessary as introductory to the study of symbolical algebra: such are some parts of the discussion and exposition of the theory of arithmetical operations, of the theory of continued fractions, of the extraction of simple and compound four roots, of the theory of ratios and proportions and various subt questions of commercial arithmetic connected with them, of ce the solution of indeterminate problems of the first degree, forn b and of several important propositions in the theory numbers. Such an arrangement of subjects is perfect consistent with their philosophical order of dependence upo each other, though it may not be altogether consister with their order of difficulty: but I have endeavoured the end of the Table of Contents, to point out thos articles which a student, at his first reading of this volum may pass over, without any material sacrifice of propos tions, the knowledge of which may be necessary to th complete understanding and demonstration of those whic follow. |