1815.] On Imaginary Ode Roots. 4^ 



effects, as well as several anomalies, seem to point at something of 

 this nature ; and as opinions formed agreeably to this view of tlie 

 subject will account for most of the galvanic phenomena in a 

 simple and plausible manner, without the aid of mysterious prin- 

 ciples, the subject assumes an highly interesting character, by the 

 increasing probability, that the phenomena of galvanism are most 

 intimately connected with many other important branches of 

 natural philosophy. 



Article VIII. 



Defence of the Opinion that all Numlers have Four Imaginary 

 Cube Roots. By James Lockhart, Esq. 



(To Dr. Thomson.) 

 SIR, 



I AM mucK obliged to Dr. Tiarks, and to your Correspondent 

 N. R. D., for their attention to my late communication. The 

 disagreement of these Gentlemen in respect of the value of the 

 imaginary quantity gives me encouragement to hope that some 

 doubt of the error which they suppose I have made will be excited. 

 Dr. Tiarks affirms that the quantity is nothing but a different form 

 of a well-known root of 64 ; whereas N. R. D. insists that it is a 

 cube root of 8, and not of 64 ; and thus it would appear that the 

 quantity is the square, and the square root of itself also. If im- 

 possible expressions, only a little complicated, universally lead to 

 such difference of sentiment, it will be wise to abandon them alto- 

 gether. Nevertheless, It now becomes me to endeavour to show 

 that I have not made a hasty assertion, and that I was duly ac- 

 quainted with the nature and construction of the quantity in ques- 

 tion ; and for this purpose I resort to the following remarks and 

 demonstration. 



In the general equation x* — I x = c, there are three roots, 

 X the greater, — t the middlemost, and — v the least. The rule 

 promulgated by Cardan gives all the three values, which, however, 

 is denied by some eminent algebraists of the present day. 1 shall 

 place the cube roots in their order under Cardan's binomials ; and I 

 believe that it is the first time of their being so exhibited. 



\/^ + \/-r - -Sr V "^ " \/t- - if 





