CAXDV 



TRISKCTION OF AN ANGI.K. 



45 



Since angles remain unclianged in tlie process of inversion, these 

 results are already proved by what has preceded. 



A similar process of inversion cannot be applied to the fourth 

 method, because in that case the solution is performed by taking a 

 fixed distance on a line which does not pass through the center of 

 inversion. 



If it were desirable to have a solution adapted to practical purposes, 

 the first and second of the preceding series coidd be used with great 

 facilitv, since the curve by means .of which the results are there 

 obtained can be readily constructed. The accompanying diagram 

 (fig XI) represents a simple instrument which the writer has invented 

 for this purpose and used in his classes for the last four years. It is 

 adapted only to black-board work, but an instrument suitable for 

 drafting purposes could be made on the same princi])le. 



AC is a movable crank, attached to 

 the bars CD and PE. 



AC=OC=AP. 



As AC revolves about C the bar PE, 

 by means of a slot, moves along and 

 turns around a fixed pin at O. 



Thus as A describes the directric ^ 

 circle, P describes the outer branch 

 of the limacon of which CD is the 

 axis. 



This particular case of the limacon 

 possesses many other interesting prop- 

 erties which are in no way connected 

 with the solution of the problem dis- 

 cussed in this paper. 



Fiy. XL 



