194 THIRD REPORT — 1833. 



that the operations of addition, subtraction, multipHcation and 

 division are used in one science and in the other in no sense 

 which the mind may not comprehend by a practicable, though 

 it may not be by a very simple, process of generalization ; that 

 we may be enabled by similar means to conceive both the use 

 and the meaning of the signs + and — , when used independ- 

 ently ; and that though we may be startled and somewhat em- 

 barrassed by the occurrence of impossible quantities, yet that 

 investigations in which they present themselves may generally 

 be conducted by other means, and those difficulties may be 

 evaded which it may not be very easy or very prudent to en- 

 counter directly and openly. 



In reply, however, to such opinions, it ought to be remarked 

 that arithmetic and algebra, under no view of their relation to 

 each other, can be considered as one science, whatever may be 

 the nature of their connexion with each other ; that there is 

 nothing in the nature of the symbols of algebra which can es- 

 sentially confine or limit their signification or value ; that it is 

 an abuse of the term generalization* to apply it to designate 

 the process of mind by which we pass from the meaning of a — b, 

 when a is greater than b, to its meaning when a is less than b, 

 or from that of the product a b, when a and b are abstract num- 

 bers, to its meaning when a and b are concrete numbers of the 

 same or of a different kind ; and similarly in every case where 

 a result is either to be obtained or explained, where no pre- 

 vious definition or explanation can be given of the operation 

 upon which it depends : and even if we should grant the legiti- 

 macy of such generalizations, we do necessarily arrive at a new 

 science much more general than arithmetic, whose principles, 

 however derived, may be considered as the immediate, though 

 not the ultimate foundation of that system of combinations of 

 symbols which constitutes the science of algebra. It is more 

 natural and philosophical, therefore, to assume such principles 

 as independent and ultimate, as far as the science itself is con- 

 cerned, in whatever manner they may have been suggested, so 

 that it may thus become essentially a science of sypibols and 

 their combinations, constructed upon its own rules, which may 



* The operations in arithmetical algebra can be previously defined, whilst 

 those in symbolical algebra, though bearing the same name, cannot : their 

 meaning, however, when the nature of the symbols is known, can be generally, 

 but by no means necessarily, interpreted. The process, therefore, by which we 

 pass from one science to the other is not an ascent from particulars to generals, 

 which is properly called generalization, but one which is essentially arbitrary, 

 though restricted with a specific view to its operations and their results admit- 

 ting of such interpretations as may make its applications most generally useful. 



