Colorado College Studies. 



geometric foriiis, upou wliicli is reared the edifice of ana- 

 lytical geometry. In the customary treatment of the latter, 

 while imaginary values of the variables appear frequently 

 in the algebraic work, and are recognized as of high im- 

 portance to the theory of curves (witness the "circular 

 points at infinity"), they are excluded from geometric in- 

 terpretation. This is not necessary. Imaginaries may be 

 admitted, with "real" quantities, into the constructions as 

 well as the arguments of analytical geometry, if we are 

 content to lay aside that habit of associating each axis 

 with a single variable, which has resulted in the names 

 "axis of X" and "axis of Y" respectively. Let us regard 

 these lines rather as an axis of reals and an axis of imagi- 

 naries, and then express the usual Cartesian method of 

 locating a point by its coordinates, by saying that the point 

 (x, y) means the point at the extremity of the vector x + n'. 

 When X and y are real, this will locate the point just as 

 Descartes does, but if either coordinate be imaginary, the 

 point has still a definite position in the plane. Thus if 

 x=5— 2iandY=4+3iwefindx + /Y=5— 2t+4/-3=2+22, 

 and the point is situated 2 units to the right of one axis, 

 and 2 above the other, just as if its coordinates were 2, 2, 



Thus, when imaginaries are admitted, any point of a 

 plane serves for the geometric representation of an infinite 

 number of sets of coordinates, all distinct from one an- 

 other, and we can no longer infer from the position alone 

 of a point what are its coordinates. If we are informed 

 by some other means what is the imaginary part of each 

 coordinate, then this knowledge, combined with that of the 

 position of the point, suffices to determine the real part, 

 and hence the coordinates in full. Thus in the instance 

 given, if we know that x + n'=2 + 2/, and that the imagi- 

 nary part of x is —2/, we find by subtraction that the im- 

 aginary part of /y is 4/, hence that the real part of y is 4. 

 In like manner may be found the real part of X. 



An equation between the variables, such as x^+Y^=r\ 

 serves to restrict the number of sets of coordinates belong- 

 ing to any one point; never depriving any point, however. 



