766 PROCEEDINGS OF THE AMERICAN ACADEMY. 



B lies on a line b. The same figure gives 



ab = const. 



where b is a fixed line and a is a line making a constant angle with b 

 (Figure 18). 



Figure 18. 



This amounts to saying that if two sides of a triangle are equal the 

 opposite angles are equal and have values independent of the third side 

 or angle. Thus with every distance AB is associated an angle ab. With 

 any equal distance is associated the same or equal angle, since as pre- 

 viously mentioned the construction for equal distances at the same time 

 makes the angles equal. 



We shall determine the relation between these corresponding dis- 

 tances and angles. For this purpose take two distances AB and AC 

 along the same line and with them the corresponding angles ab and ac 

 (Figure 19). Let FA and AB cut f in P and Q. Draw FQ cutting 

 PB and PC in D and E. 



In the isosceles triangles ABD and ACE 



Z ABD 8 = Z BAD, 



^ The angle ABB is the angle whose first side is AB and second side BD 

 and which does not contain F. Thus in the figure the angle ABD is the 

 angle ABP. The triangle ABD is isosceles in the sense that 



PA = DB. 



We shall later call triangles isosceles or equilateral when the sides described 

 in a definite order around the triangle are equal, i. e. when 



BD = DA. 



