The Imaginary in Geometry 



5 



is the .r-projection of ( i, i)re i(b When (i, i)re i<t> gets multiplied 

 by cos 6 -\- i sin 6 all parallel projections are so multiplied. Thus 

 (x' -\- ix")c' 9 is the .r-projection of (i, i)re i( -* +0) . If varies 



Fig. 5. 



uniformly the ends of the red vector of x' -\- ix" execute simple 

 harmonic motions of amplitudes 



2\/-i-' 2 4--r" 2 and 2\/2(.r' 2 + .r" 2 ), 



the phases differing by tt/8. When 6 = — </> or w — ^>, the product 

 (x' -\-ix")c ie is wholly real; when <9 = 71-/2 — </j or 371-2 — <f> the 

 product is wholly imaginary. Thus the cases of the multiplicand 

 wholly real or wholly imaginary can be treated as sub-cases under 

 what we are now considering. 



Suppose that the black vector is finite while the red is infinite. 

 Then, when we multiply by e' e , the projection, parallel to the red 



