770 PROCEEDINGS OF THE AMERICAN ACADEMY. 



It is to be noted in particular that the three angles of a triangle are 

 not functionally related. The reason that the angles of a triangle in 

 Euclidean geometry are so related is because angle is there of zero di- 

 mension in distance. The three angles of a triangle being homogeneous 

 functions of zero dimensions in the sides are functions of two ratios 



-, -, and hence must be functionally related. 

 c c 



From (11) by division we get 



Hence — = — = — (17) 



abc ^ ^ 



a set of relations similar to the sine proportions in trigonometry. In 

 this system angle often replaces sine of the angle in ordinary trig- 

 onometry 



31. Area. To determine the sides and angles of a triangle a direction 

 around the triangle must be given. In speaking of the triangle ABC 



Figure 21. 



we shall assume this direction to be A, B, C. The lengths of the sides 

 are there AB, BC, CA and the angles ab, be, ca where a is opposite A, 

 b opposite B, etc. Quantities such as area connected with a triangle 

 may depend on this direction of description. 



We define the line area of a triangle as a scalar quantity, determined 

 by three points taken in a definite order, such that triangles having a 

 vertex in common and their bases on the same line have areas propor- 

 tional to the lengths of their bases. 



Take two triangles ABC and AB'C having the same vertex A and 

 bases BC, B'C on the same line (Figure 21). 



