

Neio Protractor for Trisecting Angles 



245 



the centre at A, be drawn a semi-circle (which in the figure co- 

 incides with the outer circumference of the protractor) whose 

 diameter passes through A and C and is thus coincident with one 

 leg of the given angle ; and the points A and D be joined by a 

 straight line : then whenever of a radius prolonged from C the 

 distance A'C intercepted between the straight line AD and the 

 circumference of the semi-circle is equal to the original radius 

 AC, the arc AE (cut off by a straight line joining A / and C) is 

 one-third of the original arc AD ; or what is the same, the angle 

 A'CA is one-third of the given angle ACD. 



A New Protractor for Trisecting Angles. 



These relations are very manifest ; for, assuming the construc- 

 tion as given and supposing the triangles to be completed by join- 

 ing the points AA' and CC (which are in the figure left to be 

 imagined), we have AA / C / and ACE, two equal and similar isos- 

 celes triangles ; and because isosceles, with the side A'C equal to 



AA 



or AC, that is, to the radius in question. Also ACE being an 



isosceles triangle, C'AE is likewise isosceles and similar, for both 

 have A EC a common angle at their base. The angle EAC 

 therefore is equal to the angle ECA. But the angle EAC 7 (or 

 iiAD) at the circumference is subtended by the same arc ED 

 which subtends the angle ECD at the centre ; and is therefore 

 one half of this last. The angle ACE (the equivalent of L'AD) 

 J s then one half of ECD ; and so is one-third of the given angle 

 ACD, which is the sum of its double and itself, q. e. d. 



In the device of the protractor, I have done no more than ren- 

 der permanent the essential parts of the construction. In point 

 °f fact, it is quite easy, with a little tact in manipulation, to dis- 

 pense with apparatus at all more than a graduated scale and a 

 P&ir of compasses, against whose point held at C, the scale re- 

 volves until the number of its divisions intercepted between the 

 chord AD and the semi-circle previously described, equals the 



number assumed originally as radius. But as the management 



i 



