It thus appears, that if we consider four of the five symbols in the equation as arbitrary and independent, the fifth is necessarily dependent upon them and its value determined. ization of first dis 131. Our great object in the very lengthened dis- Generalcussion which we have just concluded, has been to point equivalent out the distinction between the science of Algebra when forms when the symbols considered with reference to its own principles, and when in which considered with reference to its applications, and to shew they were in what manner and to what extent the assumptions which covered regulate the combinations of general and arbitrary symbols were not general. in Algebra were suggested, and their interpretation limited by other and subordinate sciences: the principles which determine the connection between these sciences being once established, we shall be fully prepared to consider to what extent we can consider equivalent forms suggested or investigated upon the principles of a subordinate science, as equally true when expressed in general symbols. Thus the principles of Arithmetical Algebra lead to the equation a" × am=a"+m, when n and m were whole numbers: it was the conversion of this conclusion in one science into an assumption in the other, which lead to the same equation, a" × a"=a"+m, when n and m were general symbols. If, however, we had commenced with the assumption that there existed some equivalent form for a" x a", when n and m were general symbols; and if we had discovered and proved that this form in Arithmetical Algebra was a", where n and m were such quantities as Arithmetical Algebra recognizes, then we might infer that such likewise must be the equivalent form in Symbolical Algebra: for this form can undergo no change, according to the assumptions which we have made, from any change in the nature of Law of the permanence its symbols, and must therefore continue the same when the symbols are numbers: if, therefore, we discover this form in any one case, we discover it for all others. 132. Let us again recur to this principle or law of of equiva- the permanence of equivalent forms, and consider it when lent forms stated in the form of a direct and converse proportion. stated and proved. "Whatever form is Algebraically equivalent to another, when expressed in general symbols, must be true, whatever those symbols denote." Conversely, if we discover an equivalent form in Arithmetical Algebra or any other subordinate science, when the symbols are general in form though specific in their nature, the same must be an equivalent form, when the symbols are general in their nature as well as in their form. The direct proposition must be true, since the laws of the combinations of symbols by which such equivalent forms are deduced, have been assumed without any reference to their specific nature, and the forms themselves, therefore, are equally independent. The converse proposition must likewise be true for the following reasons: If there be an equivalent form when the symbols are general in form and in their nature also, it must coincide with the form discovered and proved in the subordinate science, where the symbols are general in form but specific in their nature: for in passing from the first to the second, no change in its form can take place by the first proposition. Secondly, we may assume such an equivalent form in general symbols, since the laws of the combinations of symbols are assumed in such a manner as to coincide strictly with the corresponding laws in subordinate sciences, such as Arithmetical Algebra: the conclusions, therefore, so far as their form is concerned, are necessarily the same in both; and the Algebraical equivalence which exists in one case must exist likewise in the other. importance. 133. The principle which is expressed in these pro- Its great positions, which we have named the law of the permanence of algebraical forms, is one of the greatest importance, and merits the most profound and careful consideration: it points out the proper object of those demonstrations in Algebra, which have reference to the research of equivalent forms, and shews why they may be safely generalized, even though they may be obtained by the aid of specific values of the symbols: if a general equivalent form be assumed to exist, it is clearly sufficient if we can discover it in any one of its states of existence, corresponding to the different specific values of its general symbols: and if we commence by detecting its existence for specific values of the symbols, we may generalize the symbols, since the same form continues to be equivalent for all algebraical operations. 134. Thus, we may assume the existence of an equivalent form for (1+x)”, when n is a general symbol, and we may discover it when n is a whole number: or we may commence by the discovery of the equivalent form of (1+x)", when n is a whole number, and subsequently assume the existence of it, when n is a general symbol: in the first case, assuming its existence, it is necessarily the form discovered: in the second case, the law we have mentioned and the reasoning by which it is established, would shew that the form discovered is algebraically equivalent, when n is a general symbol. An example. superseded 135. It is no objection to the use of this principle, Not to be that in consequence of its extreme abstraction and gener- by other ality, it requires a great and painful effort of the to apprehend thoroughly the evidence of its truth, mind methods of investiga unless tion. Principle of symmetri cal combi nations. it can be shewn that there are other and more easily intelligible, though less general, methods of arriving at similar conclusions; but a very little consideration will shew, that the province of demonstration is extremely limited when general symbols are employed, since we must reject altogether the aid derived from the specific values of the symbols: we are thus confined to the assumed laws of their combination and incorporation with each other, and consequently demonstration can only extend to cases which those laws comprehend: it is for this reason that there can be no demonstration, independently of additional assumptions, of the existence of an equivalent form for (1 + x)", when n is a general symbol. 136. Amongst the most important aids for the discovery and determination of equivalent forms, whether their general existence can be assumed or proved, may be mentioned the principle of symmetrical combinations, of which use has been made in many examples in the last chapter: it may be considered as such a modification of the principle of sufficient reason as is applicable to the science of Algebra, and in its most general form may be stated as follows: "When several events are equally likely to happen, if one can be shewn to happen, the others must: and if one of them can be shewn not to happen, neither can the others happen." When we speak of events as equally likely to happen, we mean that they are similarly circumstanced; and when we speak of one event as necessarily determining the others, we merely say that the same premises must necessarily lead to the same conclusion. Example. 137. Thus, in our determination of the product or equivalent form of (x+a) (x+b) (x+c), we observed that the general symbols a, b, c were equally involved in each of the factors a+a, x + b and x+c: and we concluded from the general principle above-mentioned, that if a was a term of the product, ba2 and ca2 must be equally so: and that if aba was a term of the product, aca and bea must be equally so: for the same reasons which determined the existence of ax and abæ, must determine that of b and c in one case, and that of aer and be in the other: if we deny these consequences, there is an end of all certainty in our reasoning from the same causes to the same effects. species of 138. Amongst other processes of reasoning which are Different employed in the discovery of equivalent forms, we may Induction. mention Induction, of which there are several species, presenting very different degrees of evidence, from what is merely probable to what may be considered as demon strative. Induction from 139. The first and most humble of these species of Induction, is that by which we draw a general conclusion insulated from the observation of particular facts. In physical science, such inductions are useful as a means of classifying facts, and as furnishing indications of general truths, which other methods may establish. In Algebra, or rather in Arithmetical Algebra, it may suggest the existence of an equivalent form, but no dependence can be placed upon the generalization of conclusions thus deduced, unless other methods can be invented for their demonstration.* As an example of the distrust with which such generalizations should be viewed, may be mentioned the formula x2+ x +41 discovered by Euler, which represents prime numbers for all values of x, from 0 to 39: it would be a false induction, however, to infer from so many concurrent facts, that such would generally be the case, for if x=40 or 41, the number expressed by the formula is clearly divisible by 41. Another and a more remarkable example occurs in the formula 2"+1, which Fermat asserted was always a prime number, but which Euler shewed not to be so, when icas 32. In the theory of numbers, we meet with numerous examples of a similar nature. facts. |