MATHEMATICS 33 



impossible quantities, yet the results of the facts and 

 laws thereby discovered (like those of arithmetic) corre- 

 spond with the facts and laws of real or possible 

 existences. They express abstract truths which have 

 real applications or would have them could the im- 

 possible conditions sometimes supposed really exist. Thus 

 even the absurd and impossible quantity expressed by the 

 symbol J x has its relation with reality. It is really 

 impossible in itself, since there is no quantity which, 

 being multiplied by itself, gives a negative product. Yet 

 it has its relation with reality, inasmuch as it can be used 

 as if it were a real quantity, and all the laws and relations 

 relating to real quantities can be applied to it. Thus : 



Thus we may investigate laws concerning space, 

 motion, pressure, &c., apart from certain conditions 

 which in fact always exist, but which may be tem- 

 porarily disregarded. 



The results so arrived at will be absolutely true, 

 though of course they will not correspond with the 

 phenomena of the world about us, till we take into 

 consideration the conditions which before had been 

 purposely left out of the calculation. These being 

 correctly restored and added, the results will correspond 

 with the realities of experience. 



The truths and processes of algebra may be tested by 

 selecting any numbers as representatives of the alge- 

 braic symbols (which latter are valid for all numbers) 

 and treating them similarly. This translates the results 

 into arithmetic, and arithmetical results may then be 

 tested by experiments with corresponding numbers of 

 material bodies. 



Thus as an example of the correspondence of alge- 



C 



