The Imaginary in Geometry 9 



them P 1 and P 2 do not belong to a line of real slope. Let the 

 equation of the line be 



ax -\- by = 1 . 



In this equation, since the line does not have a real slope, we 



must have 



a', a" I 



r+o. 



b', b" \ 



We say that a and b must not be similar complex quantities. 



Since the given vectors satisfy the equation so also does P z 

 given by 



( P*i + 1*2, Py\ + ?y 2 ) f where p + q = 1. 



If /> and g are real the blue point P 3 " is on the line joining 

 the blue points of P Y " and P 2 " , while the black point P 3 ' is on the 



iB 



Fig. 9. 



line joining the black points F/ and P/. In fact, as p varies 

 from 1 to o to 00 to — co to 1 again, the black point describes 

 a black line and the blue point a corresponding blue line, so as 

 always to keep the segment of these lines included between the 

 vectors P x and P. 2 divided in the ratio 1 q : p. We can say that 

 the double segment of the line P 1 P. 2 is divided in the ratio q : p. 

 Suppose, however, that the ratio in which we divide the line 

 is complex q : p. We can so take p and q that p -{- q = I and 

 therefore p'-\-q'=i and />"-}- q" = 0. The division will then 

 be found connected with that in the ratios of q' : p' and q' -\- q" '• 



1 The row of blue points is projective to the row of black ones, but in 

 addition the infinite points of the rows are corresponding. Thus the 

 rows are similar. 



