2. _The Use of General Symbols._--The employment of algebraical symbols, of which we have just spoken, has been another of the main instruments to which the successes of modern mathematics are owing. And here again the processes by which we obtain our {154} results depend for their evidence upon a fundamental conception,--the conception of _arbitrary symbols_ as the _Signs_ of quantity and its relations; and upon a corresponding axiom, that 'The interpretation of such symbols must be perfectly general.' In this case, as in the last, it was only by degrees that mathematicians were led to a just apprehension of the grounds of their reasoning. For symbols were at first used only to represent numbers considered with regard to their numerical properties; and thus the science of Algebra was formed. But it was found, even in cases belonging to common algebra, that the symbols often admitted of an interpretation which went beyond the limits of the problem, and which yet was not unmeaning, since it pointed out a question closely analogous to the question proposed. This was the case, for example, when the answer was a _negative quantity_; for when Descartes had introduced the mode of representing curves by means of algebraical relations among the symbols of the _co-ordinates_, or distances of each of their points from fixed lines, it was found that negative quantities must be dealt with as not less truly significant than positive ones. And as the researches of mathematicians proceeded, other cases also were found, in which the symbols, although destitute of meaning according to the original conventions of their institution, still pointed out truths which could be verified in other ways; as in the cases in which what are called _impossible quantities_ occur. Such processes may usually be confirmed upon other principles, and the truth in question may be established by means of a demonstration in which no such seeming fallacies defeat the reasoning. But it has also been shown in many such cases, that the process in which some of the steps appear to be without real meaning, does in fact involve a valid proof of the proposition. And what we have here to remark is, that this is not true accidentally or partially only, but that the results of systematic symbolical reasoning must _always_ express general truths, by their nature; and do not, for their justification, require each of the steps of the process to represent {155} some definite operation upon quantity. _The absolute universality of the interpretation of symbols_ is the fundamental principle of their use. This has been shown very ably by Dr. Peacock in his _Algebra_. He has there illustrated, in a variety of ways, this principle: that 'If general symbols express an identity when they are supposed to be of any special nature, they must also express an identity when they are general in their nature.' And thus, this universality of symbols is a principle in addition to those we have already noticed; and