38 Mr. J. Cockle 07i a new Imaginary in Algebra. 



hesitate in saying, either that impossibility is incapable of being 

 rendered subservient to the purposes of algebra, or even that 

 it is absolutely uninterpretable. A symbol for impossibility 

 is not only desirable, but actually necessary, provided that we 

 wish to classify with accuracy the various subjects of algebraic 

 research, and to distinguish those which are unreal from those 

 which are impossible. We might adopt an arbitrary symbol 

 to denote impossibility; but a deduced symbol is preferable, 

 for it gives to our investigations the character of true develop- 

 ments of ordinary algebra. Such a deduced symbol I have 

 obtained by means of a surd equation *. The next step is, to 

 ascertain the fundamental properties of the new symbol, its 

 origin and nature being duly considered. As we might ex- 

 pect, a priori, anomalous results offer themselves in the course 

 of our progress with these inquiries; such results require at 

 least an attempt at explanation. The geometrical interpreta- 

 tion (or lather capability of interpretation) of the new symbol 

 is another point not unworthy of consideration, and the same 

 may be said of the employment of that symbol in analytical 

 discussions. I purpose, then, in this paper, to treat, — 1, of the 

 Utility of the new Symbol; 2, of the Value of its Square; 

 3, of a certain Anomalous Result ; 4, of the Interpretation of 

 the Symbol in Geometry ; and, lastly, of its proposed Employ- 

 ment in Analysis. 



1. Of the Utility of the new Symbol. 



I shall show that there are i-elations, which cannot be ex- 

 pressed by means of the ordinary algebraic symbols. Hence, 

 if it be of any importance to express such relations, a new 

 notation must be adopted. That it is of importance, those 

 who value logical precision and accurate classification will, if 

 I am not mistaken, be disposed to admit, on such grounds 

 alone, and quite irrespectively of any ulterior applications of 

 which the new symbol may be susceptible. It will be observed, 

 in what follows, that I have in certain cases employed an ac- 

 cented zero (O'). I have done this in order to distinguish 

 what is, in fact, an absolute negation of existence, from the 



* This symbol possesses the character which we might, almost, have an- 

 ticipated for it a priori. In Universal Arithmetic, a negative quantity is 

 impossible (Peacock, Report, p. 189), but its square is possible. So, even 

 when negative quantities are recognized, a negative square is contradictory, 

 and unreal quantities are impossidle, but their squares are possible. In like 

 manner, the square of a Pure Impossible — of a quantity taken as simulta- 

 neously positive and negative — is to be treated as possible. The contra- 

 diction vanishes on squaring, and there is a striking analogy with the otlier 

 cases. 



