REPORT ON CERTAIN BRANCHES OF ANALYSIS. 197 



In the concurrence of the signs + and — , in whatever man- 

 ner used, if two Hke signs come together, whether + and -f , or 



— and — , they are replaced by the single sign + ; and when 

 two unlike signs come together, whether + and — , or — and + , 

 they are replaced by the single sign — . 



When diiFerent operations are performed or indicated, it is 

 indifferent in what order they succeed each other. 



The operations called multiplication and division are de- 

 noted by the signs x and -r-, or more frequently by a conven- 

 tional position of the quantities or symbols with respect to 

 each other : thus, the product of a and b is denoted by a x 6, 

 a . b, or a b; the quotient of a divided by b is denoted by 



a -T- b, or by -- . 



•' b 



The operations of multiplication and division are the inverse 

 of each other. 



In the concurrence of the signs + and — in multiplication or 

 division, if two like signs come together, whether + and -|- , or 



— and — , they are replaced by the single sign + ; and if two un- 

 like signs come together, whether + and — , or — and +, they 

 are replaced by the single sign — . 



When different operations succeed each other, it is not indif- 

 rent in what order they are taken. 



We arrive at all these rules, when the operations are defined 

 and when the symbols are numbers, by deductions, not from 

 each other, but from the definitions themselves : in other words, 

 these conclusions are not dependent upon each other, but upon 

 the definitions only. In the absence, therefore, of such defini- 

 tions of the meaning of the operations which these signs or 

 forms of notation indicate, they become assumptions, which are 

 independent of each other, and which serve to define, or rather 

 to interpret* the operations, when the specific nature of the 

 symbols is known ; and which also identify/ the results of those 

 operations tvith the corresponding results in arithmetical alge- 

 bra, when the symbols are numbers and when the operations are 

 arithmetical operations. 



The rules of symbolical combination which are thus assumed 



* To define, is to assign beforehand the meaning or conditions of a term or 

 operation ; to interpret, is to determine the meaning of a term or operation 

 conformably to definitions or to conditions previously given or assigned. It is 

 for this reason, that we define operations in arithmetic and arithmetical alge- 

 bra conformably to their popular meaning, and we interpret them in symboli- 

 cal algebra conformably to the symbolical conditions to which they are sub- 

 ject. 



