394? ALGORITHM OF IMAGINARY QUANTITIES. 



when we are required to extract the square root of o*, 

 it is either + a, or — a; but if it be required to extract 

 the square root of — a*, we must necesearily be stop- 

 ' ped in our progress, for there being no previous conven- 



tion, that — o* shall represent the square of any quan- 

 tity ; it follows, that, when such a case arises, we are 

 not able to assign to it any particular root, and we are thus 

 presented with the first and the mo^t sinople form of ima- 

 ginary quantiiies. These expressions, however, though 

 they have no definite value, yet, considered as inaihema- 

 tical symbols, ought to be subject to certain rules, as well 

 as other symbols, which are the representatives of real 

 quantities; the rules, however, which are laid d«wn for 

 operating in the latter case, frequently require certain md- 

 Difficulties <3>fications before they can be applied to the former, and 

 from applying most of the dliBculties, which have occurred with regard to 

 wa"Tntended ^ *^^ algorithm of imaginaries, have arisen in making rules 



for particular general, which were first intended to answer only in parti- 

 cases. , "^ ^ 

 cular cases. 



Theambigu- We have seen above, that -/<»* rr t a^ that is to say, it 

 ity does not has an ambiguous root, which may be taken either at + a, 

 or — o ; but this ambiguity has not place, if we know how 

 the quantity a* was generated, and have occasion afterwards 

 to retrace the steps of our operation : we cannot, for in- 

 stance, say, that V _« X — a ^ ^c* = i a ; or that 



exist m cer- 

 tain cases 



* -h a X 4- a =: '^ «* = ih a for the square root of a \ti 

 both these cases is determined ; that is, when considered 

 ^.vith regard to its generation, it has but one root; whereas 

 its origin not been known, we must have prefixed the 

 ambiguous sign to the root a, and for this obvious reason, 

 that, we know riot, when a* is unconditionally assumed, whe- 

 ther it be the representative of [-\- c)*, or of { — a)*; these 

 being both expressed by the same symbol c*. In fact, 

 there is no ambiguity in the extraction of the roots of 



Caused by ex- quantities, except in those cases in which we are unac- 



pressuig dif- -^ i -ii .1 - • , . , 



ferent quanti- qu^unted with their generation ; an<d in these it must ne- 



ties by one cessarily arise, because we have agreed to represent the 



powers of different quantities by the same symbol, 



Thisillus There are, for instance, three different quantities, which, 



being 



