reasoning immediately connected with them, or in their remoter consequences, which did not admit of a simple and uniform explanation, we should very properly hesitate before we acceded to any innovations in those principles or in their exposition: for under such circumstances, the perfect union and attachment of the parts of the fabric would furnish the best evidence of the sufficiency of the foundations: but it is the admitted existence of difficulties in the consequences of the principles of Algebra, as they are commonly stated, both immediate and remote, which naturally, and indeed necessarily, induces us to suspect the existence likewise of imperfections or inaccuracies in the principles themselves: a suspicion which becomes confirmed when it appears, after the most careful examination of them, that the difficulties in question are not referable to their imperfect developement. Algebra has always been considered as merely such a modification of Arithmetic as arose from the use of symbolical language, and the operations of one science have been transferred to the other without any statement of an extension of their meaning and application: thus symbols are assumed to be the general and unlimited representatives of every species of quantity: the operations of Addition and Subtraction in their simple arithmetical sense, are assumed to be denoted by the signs + and -, and to be used in connecting such symbols with each other: Multiplication and Division, two inverse operations in Arithmetic, are supposed to be equally applicable to all quantities which symbols may denote, without any necessary modification of their meaning: but at the same time that the primitive assumption of such signs and operations is thus carefully limited in the extent of their signification, there is no such limitation imposed upon the extent of their application: thus it is not considered necessary that the operations of Addition and Subtraction should be confined to quantities of the same kind, or that the quantities subtracted should be less than the quantities from which they are subtracted: and when the violation of this restriction, which would appear to be rendered necessary by the primitive meaning of those operations, has led to the independent existence of the signs and, as an assumption which is also necessary in order to preserve the assumed universality of the values of the symbols and of the possibility of the operations which they designate, it is not considered that by this additional usage of them, we have altogether abandoned the definitions of those operations in practice, though we have retained them in name: for the consequences of those operations, and of the assumptions connected with them, must be determined by the fundamental rules for performing them, which are independent of each other, or whose necessary connection is dependent upon their assumed universality only: and the imposition of the names of Addition and Subtraction upon such operations, and even their immediate derivation from a science in which their meaning and applications are perfectly understood and strictly limited, can exercise no influence upon the results of a science, which regards the combinations of signs and symbols only, according to determinate laws, which are altogether independent of the specific values of the symbols themselves. It is this immediate derivation of Algebra from Arithmetic, and the close connection which it has been attempted to preserve between those sciences, which has led to the formation of the opinion, that one is really founded upon the other: there is one sense, which we shall afterwards examine, in which this opinion is true: but in the strict and proper sense in which we speak of the principles of a demonstrative science, which constitute the foundation of its propositions, it would appear from what we have already stated, that such an opinion would cease to be maintainable: in order however to establish this conclusion more comp.ely, it may be proper to exhibit at some length the successive transitions which are made from the principles and operations of Arithmetic to those of Algebra, in order to shew that their connection is not necessary but conventional, and that Arithmetic can only be considered as a Science of Suggestion, to which the principles and operations of Algebra are adapted, but by which they are neither limited nor determined. In our first transition from Arithmetic to Algebra, we consider symbols as the general representatives of numbers, and the signs of operation and other modes of combining them as designating operations with arithmetical names and arithmetical meanings: but in the very first applications of such operations, the mere use of general symbols renders the proper limitation of their values, which is necessary in order to prevent the exhibition or performance of impossible operations or of such as have no prototypes in Arithmetic, extremely difficult and embarrassing, inasmuch as such limitations can very rarely be conveyed to the eye or to the mind by the symbols themselves: thus a-(a+b) would obviously express an impossible operation in such a system of Algebra; but if a + b was replaced by a single symbol c, the expression a-c, though equally impossible with a-(a+b), would cease to express it. The assumption however of the independent existence of the signs + and - removes this limitation, and renders the performance of the operation denoted by equally possible in all cases and it is this assumption which effects the separation of arithmetical and symbolical Algebra, and which renders it necessary to establish the principles of this science upon a basis of their own: for the assumption in question can result from no process of reasoning from the principles or operations of Arithmetic, and if considered as a generalization of them, it is not the last result of a series of propositions connected with them: it must be considered therefore as an independent principle, which is suggested as a means of evading a difficulty which results from the application of arithmetical operations to general symbols. It is the admission of this principle, in whatever manner we are led to it, which makes it necessary to consider symbols not merely as the general representatives of numbers, but of every species of quantity, and likewise to give a form to the definitions of the operations of Algebra, which must render them independent of any subordinate science: for in the first place the symbols, whatever they denote, must be unlimited in value, and it is only by their ceasing to be abstract numbers that we shall be enabled to interpret the affections which the signs + or - (or any other signs) essentially attached to them may be supposed to express : and in the second place, in framing the definitions of algebraical operations, to which symbols thus affected are subjected, we must necessarily omit every condition which is in any way connected with their specific value or representation: in other words, the definitions of those operations must regard the laws of their combination only: thus the operations denoted by + and - must regard the affectation of symbols (with their proper signs + and b whether accompanied or not by any other signs of affection which they are capable of receiving) by them, according to an assumed law for the concurrence of those signs: and the operations denoted by x and ÷, or equivalent modes of denoting them, must regard in the first place the result of the combinations of the symbols, and in the second place the result of the combination of the peculiar signs which belong to them. Again, in order that such operations may possess an invariable meaning and character, when the symbols with their proper signs, which are submitted to them, are the same, we shall suppose them independent of any mere accident of position, or order of succession; or, in other words, when any number of such operations are to be performed and of symbols to be combined by means of them, we shall suppose the results to be the same, in whatever order those operations succeed each other. If we should rest satisfied with such assumed rules for the combinations of symbols and of signs by such operations, which are perfectly independent of any interpretation of their meaning, or of their relation to each other, we should retain in the results obtained all the symbols which were incorporated, without possessing the power of any further simplification it is as a first step to effect such further reduction of the results, and in order to define the symbolical relation of pairs of those operations to each other, that we assume the operation denoted by + to be the inverse of that which is denoted by and conversely; and the operation denoted by x to be the inverse of that which is denoted by, or conversely: or, in other words, we consider a + b − b and a-b + b, a × b÷b or a÷b × bor ab b - , to be identical in signification with the simple symbol a. |