The Imaginary m Geometry 21 



in fact, 



x=- — e'*)' (see fig. 21). 



If the first line, with all its elements, be rotated positively 

 through a quadrant, any element (x, y) will be brought into 

 coincidence with the element (y, — x) belonging to the second. 

 We are led then to the following definition. 



Two lines having the same center are perpendicular if by a 

 rotation of one of them through a quadrant it can be brought into 

 coincidence with the other. In particular, an /-line or a /-line is 

 perpendicular to itself. Similarly, two lines are parallel if by a 

 shift without turning one can be brought wholly into coincidence 

 with the other. In particular, two /-lines are parallel as are also 

 two /-lines. 



Two lines not having the same center are perpendicular if one 

 is parallel to or perpendicular to the other. In particular, two 

 /-lines are perpendicular as are also two /-lines. 



In fact, two /-lines make any real angle you please with each 

 other, since an /-line comes into coincidence with itself for any 

 turn whatsoever. The same is true of two /-lines. Analytically, if 



y = ± ix, 



then also 



x sin <f> — y cos </> = ± i(x cos $ — y sin <f>) . 



By no real shift and turn can an /-line be brought into coinci- 

 dence with a /-line. No more can a line of slope e** be so 

 brought into coincidence with one of slope e~ i4> when both<£ and <$> x 

 lie between o and it. If the red vectors corresponding to ele- 

 ments of the one so lie that to one looking along them the center 

 of the line is on the left, then for the vectors belonging to the 

 other the center is on the right. It is, indeed, an exception when 

 one imaginary line can by a real turn be brought into coincidence 



y = mx and y = nx, 



the tangent of the angle through which the left one must be turned 



21 



