The Imaginary in Geometry 15 



tor, as of an ordinary point, is merely to move it so that it always 

 remains upon the line determined by itself and the origin. 



It is interesting to observe how a line of imaginary slope passes 

 over into one of real slope as the center moves out to infinity. 

 For example, consider 



y= J (- r + ^ 

 approaching 



y=i, 



as k becomes infinite. 



For all values of k the vector (o, i) belongs to the line. This 

 vector is tangent to the ellipse 



and has its blue point on 



+ £) 2 , 2 



Moreover, all vectors drawn from tangency to the first ellipse 

 and terminated by the second belong to the line or its conjugate. 

 Indeed, all vectors belonging to the line are drawn from tan- 

 gencv to some ellipse 



k- ' J \ ' k 

 with axes k -\-l and 1 -\- k/l, and terminated by 



T 2 +r= 2 



with axes (k + I) V 2 and ( 1 -f -j j V 2 - 



Among these is the vector /, /( 1 -f t ) • Thus for all values 



of k and of / the vector I /, /( 1 + -r ) belongs to the line. As k 



-••K) 



becomes large this becomes more and more nearly (/, /) which be- 

 longs to y = i. 



15 



