34 



THE SEVEN FOLLIES OF SCIENCE 



describe a short arc crossing the arc ADEB in D, and join 

 CD. The angle DCB will be 6o,'and as the angle ACB 

 is 90, the angle ACD must be 30, or one-third part of 

 the whole. In the same way lay off the angle ACE of 

 60, and ECB must be 30, and the remainder DCE must 

 also be 30. The angle ACB is therefore easily divided 



A 



C 



B 



Fig. 4. 



into three equal parts, or in other words, it is trisected. 

 And with a slight modification of the method, the same 

 may be done with an angle of 45, and with some others. 

 These however are only special cases, and the very essence 

 of a geometrical solution of any problem is that it shall be 

 applicable to all cases so that we require a method by 

 which any angle may be divided into three equal parts by 

 a pure Euclidian construction. The ablest mathematicians 

 declare that the problem cannot be solved by such means, 

 and De Morgan gives the following reasons for this conclu- 

 sion : " The trisector of an angle, if he demand attention 

 from any mathematician, is bound to produce from his con- 

 struction, an expression for the sine or cosine of the third 

 part of any angle, in terms of the sine or cosine of the 

 angle itself, obtained by the help of no higher than the 



