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



pretation. Supposing, then, / and its interpretation to be ad- 

 mitted into geometrical inquiries, the question comes, how can 

 j enter into such inquiries when it can never enter into the 

 rational equations in which such inquiries usually result? 

 The answer is, that geometrical conditions are not necessarily 

 reducible to the form of rational equations. Consider, by way 

 of example, the equation (5.). This equation, after reduction 

 to the usual form, may be resolved into congeneric surd fac- 

 tors ; and, the geometrical meanings of a and x being lines, 

 we may express the evanescence of those factors in geometrical 

 language. Such evanescence may be possible or impossible; 

 if the latter, then^* forces itself into our investigations, and the 

 next point is, how to interpret its occurrence. I think that the 

 following are the considerations which ought to guide us in 

 this interpretation. Imagine three points A, B, and C. If 

 these points are in a straight line, their relations may be re- 

 presented by real quantities, and the only determined space 

 before us is a line. But suppose that these points are required 

 to fulfill conditions inconsistent with the hypothesis of their 

 being in the same straight line; then (as will be clearly seen 

 on referring to the above problem) a. plane is determined, and 

 unreal quantities introduced ; and, supposing that a problem 

 respecting three points admits of solution, the most general 

 geometrical entity that can be determined by it is a plane. If 

 then we arrive at an impossible quantity, as the result of our 

 geometrical inquiries respecting the possibility of a supposed 

 relation between three points, we may be sure that the relation 

 cannot exist. Were the relation possible, it would be possible 

 in a plane ; and i is quite adequate to express (in combination 

 with real quantities) any possible relative position of three 

 points. Let us take a step further, and suppose that we are 

 discussing the position of four points A, B, C, D. Take AB 

 as the primitive axis, and let AB = 2a; then the expression 

 p + ig may be made to represent either C or D (or both, pro- 

 vided that all the four points are in the same plane). Thus 

 we shall have A represented by zero, B by 2a, C by c + z>, 

 and D by d + if. But, suppose that D is out of the plane of 

 A, B and C, then, from what has been before observed, d+if 

 will cease to represent D, for i means perpendicularity to AB 

 in the plane of A, B and C. Hence, in any inquiry respecting 

 four points, the occurrence of j would not be conclusive as to 

 the impossibility of the problem, but only as to the fact that 

 the points cannot be in the same plane. (I assume, of course, 

 that, in forming the condition or conditions of the problem, all 

 the points are represented by different values of the expression 

 p-\-iq.) In such a case, then, j would (provided the problem 



