The Imaginary in Geometry 



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the bearer of P , and whose arms constitute the involution of lines 

 through O. At the same time J" 01" becomes a right angle 

 whose revolution gives the circular involution or ray of center 

 and cuts out the involution or element J'"XI" Y on z. 



Just as a red vector is the symbol of an involution on a line, so 

 the red vector of 6 is the symbol of an involution about the vertex 

 of 6 for center, the involution, viz., whose rays make the angles 

 — 0" , o, 6", tt/2 with the terminal line of 9'. In rectangular 

 coordinates these would be the two arms of an angle and the two 

 bisectors of the angle. A glance at fig. 31 shows that this in- 

 volution cuts out on the perpendicular to the terminal line of 6' 



Fig. 34. 



through the tip P" of 0" the involution — P" , o, P" L 00 or — sh 

 6", o, sh#", 00 whose symbol is P or sin 0". Sin 6 is the pro- 

 jection on the y-axis and cos S the projection on the A--axis of that 



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