Mr. J. Cockle on a new Imaginary in Algebra. 41 



Is J then identical with either of the ordinary square roots 

 of unity? No. In what respect does it differ from those 

 roots ? This I proceed to show, and as follows. Unity may 

 be regarded as the square, either of positive, or of negative 

 unity : and, as we regard it from one or the other point of 

 view, we must write its square roots thus: — 



+ ^(+T?, or - V{^^, 

 both of which are included in the expression 



Let us now reverse the signs under the radical ; then the last 

 expression becomes 



This last is a contradictory and, consequently, impossible ex- 

 pression, which takes the following two impossible forms, 



+ v/(_i)2 and - \/(TT? 

 which I shall represent by/ and/' respectively. Now, if we 

 square + V'(+l)2, it becomes (+1)S the contradiction being 

 eliminated by involution. Hence we infer that 



and it is not difficult to see that^',/ and/' are values of im- 

 possible square roots of unity*. To indicate, however, that 

 the discrepancy between the signs without and within the 

 radical cannot be eliminated by merely changing the sign pre- 

 fixed to the radical, I shall use additional brackets, and sup- 

 pose that the following equations hold; viz. 



±y- ± [+ ^^r^, and +/'= ± [- -/(+!?]. 



Seeing thus the contradictory nature of the symbols//,/', 

 we must not be surprised at finding ourselves, very early in 

 our inquiries on the subject, face to face with such a result as 

 the equation (3.) given above. We see, however, that con- 

 tradictions may vanish and available results follow. 



S.Ofa certain Anomalous Result. 

 In the case of the equation (3.), we have seen that there is an 

 anomaly, inherent in the very supposition of impossible quan- 

 tity, which does not occur in treating of real or unreal quan- 



* I think that the following relations hold, viz, — 



I shall not here attempt to discuss the relation of/ and/' to j. The 

 former quantities are only introduced here to illustrate what is meant by 

 an impossible square-root of unity. 



