Preface to Treatise on Algebra.
The work which I have now the honour of presenting to the public, was written with a view of conferring upon Algebra the character of a demonstrative science, by making its first principles co-extensive with the conclusions which were founded upon them: and it was in consequence of the very particular examination of those principles to which I was led in the course of the enquiry, that I have felt myself compelled to depart so very widely from the form under which they are commonly exhibited. The object which I propose to effect is undoubtedly one of great importance, and of no small difficulty, inasmuch as it brought me into immediate contact with the discussion of many subjects of dispute and controversy, which have not hitherto been settled upon satisfactory grounds: and though I am very sensible of the great responsibility which I incur by an attempt of this nature, accompanied as it is by the proposal of so many innovations, yet I shall be perfectly satisfied if I may be considered as having succeeded in removing any difficulties or imperfections from the elements of this beautiful and most comprehensive science.
If the first principles of Algebra had been consistent with themselves, or had led to no difficulties either in the reasoning immediately connected with them, or in their remote 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 development.
Algebra has always been considered as merely such a modification of Arithmetic as arose from the use of symbolic 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 operation, 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 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 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 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 completely, 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 show 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 with 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 generalisation of them, it is not the last result in a series of propositions connected with them: it must be considered therefore as an independent principle, which is suggested as a means of ending a difficulty which results from the application of arithmetical operations to general symbols.