ALGORITHM OP llffAGINAET QUANTITIES. Ip^ 



VI. 



On the Algorithm of Imaginary Quantities. In a Letter from, 

 a Corresvondeiit. 



A. 



a Correspondent. 

 To W. NICHOLSON, Esq. 

 SIR, 



S neither of the papers which you did me the honour Defective algo. 

 to insert in your Journal, "On the Defective Algorithm "'^"^°^^'".^*" 

 of Imaginary Quantities*," has been noticed by any of your ties, 

 mathematical correspondents, 1 now propose to fulfil my 

 promise, viz. " That if no answer appeared within three 

 months, I would then, through the medium of your Jour- 

 nalj give my own explanation of the difficulty." I cannot, 

 however, but regret, that, amongst the many able advo- 

 cates for the introduction of these quantities into ma- 

 thematical investigations, no one of them should have 

 stepped forward to explain an anomaly attending such in- 

 troduction, which seems to militate so much against the 

 boasted certainty of results, deduced from mathematical 

 investigations. 



The defect pointed out in my former letters is, I believe, Rjssults from 



a new case, which has not been noticed by any of the ""*S'"^'^y 



* . . quantities un* 



writers on the subject ; and, unless it can be satisfactorily certain. 



answered, it is impossible to deny, that uncertainty and 

 doubt must necessarily attend all results obtained through 

 the medium of imaginary quantities. I am sorry, through 

 the backwardness of your other correspondents, it has fallen 

 to me to enter upon this subject; I will, however, do my 

 best to explain the anomaly in question; at the same 

 time I cannot but again repeat my regret, that it has not 

 been taken up by some one more competent to the task. 



First then, let us consider the origin of these imaginary Origin of ima= 

 expressions, which may be traced to the very first princi- gina^y expres- 

 pies of algebra. We are taught in multiplication, that 

 ( + c 1^5 aud { — fl)% are both to be represented by «* ; and 

 -Consequentlyi when the converse of this operation arises, of 



* Vol. XXIX, p. 254, and XXX, p. 209. 



Vol. XXXI.— March, 1812. O wh«» 



