200 THIRD REPORT 1833. 



formed, in which the symbols and the conventional operations 

 to which they were required to be subjected would be perfectly 

 general both in value and application. If, however, in the con- 

 sti-uction of such a system, we looked to the assumption of such 

 rules of operation or of combination only, as would be sufficient, 

 and not more than sufficient, for deducing equivalent forms, 

 without any reference to any subordinate science, we shovdd be 

 altogether without any means of interpreting either our opera- 

 tions or their results, and the science thus formed would be 

 one of symbols only, admitting of no applications whatever. It 

 is for this reason that we adopt a subordinate science as a sci- 

 ence of suggestion, and we frame our assumptions so that our 

 results shall be the same as those of that science, when the 

 symbols and the operations upon them become identical like- 

 wise : and in as much as arithmetic is the science of calculation, 

 comprehending all sciences which are reducible to measure and 

 to number ; and in as much as arithmetical algebra is the imme- 

 diate form which arithmetic takes when its digits are replaced 

 by symbols and when the fundamental operations of arithmetic 

 are applied to them, those symbols being general in form, 

 though specific in value, it is most convenient to assume it as 

 the subordinate science, which our system of symbolical algebra 

 must be required to comprehend in all its parts. The principle 

 of the permanence of equivalent forms is the most general ex- 

 pression of this law, in as much as its truth is absolutely neces- 

 sary to the identity of the results of the two sciences, when the 

 symbols in both denote the same things and are subject to the 

 same conditions. It was with reference to this principle that 

 the fundamental assumptions respecting the operations of ad- 

 dition, subtraction, multiplication and division were said to be 

 suggested by the ascertained rules of the operations bearing 

 the same names in arithmetical algebra. The independent use 

 of the signs -}- and — , and of other signs of affection, was an as- 

 sumption requisite to satisfy the still more general principle of 

 symbolical algebra, that its symbols should be unlimited in value 

 and representation, and the operations to which they are sub- 

 ject unlimited in their application. 



In arithmetical algebra, the definitions of the operations de- 

 termine the rules ; in symbolical algebra^ the rules determine 

 the meaning of the operations, or more properly speaking, they 

 furnish the means of interpreting them : but the rules of the 

 former science are invariably the same as those of the latter, 

 in as much as the rules of the latter are assumed with this view, 

 and merely differ from the former in the universality of their 

 applications : and in order to secure this universality of their 



