768 PROCEEDINGS OF THE AMERICAN ACADEMY. 



Angle is therefore of dimension minus one in distance. In fact, since 

 angle and distance are dual if there is to be a dimensional relation be- 

 tween them the first should be of the same dimensions in the second 

 that the second is in the first. The dimensions could then be only 

 1 or —1 in distance. 



Now ZBAD= ^ 



AB 

 and BD = Ali = AB. 



Hence ZBAK = =. 



AB^ 



We may consider the line BD as a circle of radius AB and center A. 

 (The same line may be considered as a circle in an infinite number of 

 ways but to each center corresponds a unique length of radius. Any 

 point on AF may be taken as center of the same circle BD.) Then 

 the angle at the center is given by 



30. Triangle relations. Analogy with Euclidean geometry leads 

 us to expect relations between the sides and angles of a triangle. We 

 shall now determine these relations. 



Let ABO be the given triangle. Draw the lines as indicated in fig- 

 ure 20. Denote the length of the sides BC, CA, AB by a, h, c re- 

 spectively and the angles CAB, ABC, BCA by A, B, C respectively. 



Then a 4- i + c = BC -f CA -t- AB = BE -I- DA + AB == DE. 

 Also = = (EA I DB) = (QS [ PR) = (BH | AB) = = 



Therefore A=^ = .M=, = ^^'. dD 



Similarly £ = „" ^ (12) 



a6 • <^^> 



