34 Gee AND ADAMSON, Trisecting an Angle. 

 Therefore 



Also 



hence 



But 



3 = -, -), neglecting (-"i 



2V3/ * * v 3 / 



sin - = sin 







neglecting / - j 



7 = ■ 



3 3^3^ \3 



X -r /S\3 



a=0+d-y= ' + d -_ + -(_) 



3 3 3W 



- i7 .13 



= - + -(-) neglecting (- 



= - + (-) neglecting (-) 



3 648 W s ^3' 



In addition to the preceding methods an angle, say 

 A OB, may be trisected approximately by repeated bisec- 

 tions, thus : — 



Bisect A OB by OP lt 



„ AOP x by OB,, 

 „ ^(9^ 2 by OP„ 

 ,', AOP 3 by (9/^, and so on. 



The bisecting lines ultimately become the trisecting 

 lines of angle A OB. 



Proof. — Let angle AOB=fi, then 

 P i OB = AOP x = $l2 



AOP, = fi/4 

 BOP 3 = ±{BOPJ = i{BOA) - i(AOP. 2 ) = BOP 1 -[3lS 



AOP,= l(AOP 3 ) = i(AOP l ) + i(P 1 OP.,) = AOP + i3li6 



:. p 2 op 4 = pi 16. 



Similarly : — 

 i?aP g = \{BOP,) = J(i?<2^ 2 ) - i(^ 2 <9^ 4 ) = BOP 3 - /3/ 3 2 



AOP, = i(^(9^ 5 ) = i(^OP 3 ) + i(^ 3 ^ 5 ) = ^(9^ 4 + /3/6 4 



:.P 4 OP, = (3l6 4 



