Manchester Memoirs, Vol. lix. (191 5), No. 13. 29 



Approximate Methods. 



Many constructions, involving the use of a pair of 

 compasses and a ruler only, have been employed for the 

 purpose of approximately trisecting an angle or a circular 

 arc. Reference will now be made to some of these 

 methods. The following table shows the results of calcu- 

 lations based on the various constructions. The angle to 

 be trisected is /3 radians, and the theoretical result of the 

 construction for /3/$ is a radians. The error a — /3/3 is given 

 in radians, neglecting powers of /3/3 which are at least two 

 higher than the power stated. The actual values in 

 degrees, minutes and seconds corresponding to a are given 

 in the cases where /3 corresponds to 90° and 45° respec- 

 tively. It is to be noticed that if an angle be bisected 

 and then the approximate trisection construction be ap- 

 plied to one of the halves, one of the trisecting lines will 

 also be a trisecting line of the whole angle. Thus any 

 angle up to 90° may be trisected with an actual error not 



Method. 



„ 3 



_Lrror = a - 



Approximation to 



1/3 °f 90° ( = 30°)- 



Approximation to 

 i/3 of 45° ( = 15°). 



A 



i@' 



3 6 ° 5 2 ' 



t if 



15 43 20 



B 



-;(!)' 



29° 40' 30" 



J 4° 59' 3o" 



C 



*0 



3° u 55' 2 °" 



° f ft 



15 9 12" 



D 



-m 



T //3\ 7 



26 34 



14 38' 20" 



E 



— [ ( ) 



2 9 U 54' 44""6 



14 59' 59" '88 



F 



648X3/ 



30° 0' o"'93 



1500 OO I S 



