REPORT ON CERTAIN BRANCHES OF ANALYSIS. 195 



be applied to arithmetic and to all other sciences by intei'preta- 

 tion : by this means, interpretation -w'tM follow, and woiprecede, 

 the operations of algebra and their results ; an order of suc- 

 cession which a very slight examination of their necessary 

 changes of meaning, corresponding to the changes in the spe- 

 cific values and applications of the symbols involved, will very 

 speedily make manifest. 



But though the science of arithmetic, or of arithmetical al- 

 gebra, does not furnish an adequate foundation for the science 

 of symbolical algebra, it necessarily suggests its principles, or 

 rather its laws of combination ; for in as much as symbolical al- 

 gebra, though arbitrary in the authority of its principles, is not 

 arbitrary in their application, being required to include arith- 

 metical algebra as well as other sciences, it is evident that their 

 rules must be identical with each other, as far as those sciences 

 proceed together in common : the real distinction between them 

 will arise from the supj^osition or assumption that the symbols 

 in symbolical algebra are perfectly general and unlimited both 

 in value and rejjresentation, and that the operations to which 

 they are subject are equally general likewise. Let us now 

 consider some of the consequences of such an assumption. 



A system of symbolical algebra will require the assumption 

 of the independent use of the signs + and — . 



For the general rule in arithmetical algebra* informs us, 

 that the result of the subtraction of 6 + c from a is denoted 

 hy a — b — c, or that a — {b + c) = a — b — c, its application 

 being limited by the necessity of supposing that 6 + e is less 

 than a. The general hypothesis made in symbolical algebra, 

 namely, that symbols are unlimited in value, and that operations 

 are equally applicable in all cases, would necessarily lead us 

 to the conclusion that a — {b + c) = a — b — c for all values 

 of the symbols, and therefore, also, when 6 = a, in which case 

 we have 



a — {a + c) = a — a — c=:— c. 



In a similar manner, also, we find 



a — {a — c) = a — a + c=^ -|-e = cf. 

 We are thus necessarily led to the assumption of the exist- 

 ence of such quantities as — c and + c, or of symbols preceded 



• Whatever general symbolical conclusions are true in arithmetical algebra 

 must be true likewise in symbolical algebra, otherwise one science could not 

 include the other. This is a most important principle, and will be the subject 

 of particular notice hereafter. 



t For it appears from arithmetical algebra that a — a^O, and that a — a 



