206 THIRD REPORT — 1833. 



was not equally permanent likewise. In assuming, therefore, the 

 existence of such a permanent series, our symbolical conclu- 

 sions are necessarily consistent with each other, and it is the 

 interpretation of the operations which produce them, which 

 must be made in conformity with them. It is true that we can 

 extract the square or the cube root of 1 + x, and we can also 

 determine the corresponding series by the processes of arith- 

 metical algebra ; and we likewise interpret (1 -f- xy and (1-1- xy 

 to mean the square and the cube root of 1 -)- x, in conformity 

 with the general principle of indices. The coincidence of the 



series for (1 -i- *) and (1-1- xY, whether produced by the 

 processes of arithmetical algebra, or deduced by the principle 

 of the permanence of equivalent forms from the series for 

 (1 -f- x)", would be a proof of the correctness of our interpreta- 

 tion, not a condition of the truth of the general principle itself. 



In order to distinguish more accurately the precise limits of 

 hypothesis and of proof in the establishment of the fundamental 

 propositions of symbolical algebra, it may be expedient to re- 

 state, at this point in the progress of our inquiry, the order in 

 which the hypotheses and the demonstrations succeed each 

 other. 



We are supposed to be in possession of a science of arith- 

 metical algebra whose symbols denote numbers or arithmetical 

 quantities only, and whose laws of combination are capable of 

 strict demonstration, without the aid of any principle which is 

 not furnished by our knowledge of common arithmetic. 



The symbols in arithmetical algebra, though general in form, 

 are not general in value, being subject to limitations, which are 

 necessary in many cases, in order to secure the practicability 

 or possibility of the operations to be performed. In order to 

 effect the transition from arithmetical to symbolical algebra, we 

 now make the following hypotheses : 



(1.) The symbols are unlimited, both in value and in repre- 

 sentation. 



(2.) The operations upon them, whatever they may be, are 

 possible in all cases. 



(3.) The laws of combination of the symbols are of such a 

 kind as to coincide vniversally with those in arithmetical algebra 

 when the symbols are arithmetical quantities, and when the 

 operations to which they are subject are called by the same 

 names as in arithmetical algebra. 



The most general expression of this last condition, and of its 

 connexion with the first hypothesis, is the law of the perma- 



