752 PROCEEDINGS OF THE AMERICAN ACADEMY. 



for collinear points 



AB + BC = AC. (7) 



16. Distance on a line. The definition of distance can be satisfied 

 for points on a given line in two ways. 



(1) There may exist two points C, D on the line such that CD is 

 finite and not zero. For every number X and point A of CD there 

 exists an unique point B on CD such that 



A - B = A (C - D), 



Hence there exists an unique point on the line at any given distance 

 from the point A. 



(2) There may exist two distinct points C, D on the line such that 

 CD equals zero. In this case the above equation shows that the 

 distance between any two points on the line CD (but not on f) is zero. 

 For the point P in which CD cuts f, X is infinite. Hence 



AP= 00 • 



is indeterminate. We assume that this distance may have any value 

 whatever. Here again we find an unique point at a distance not zero 

 from A. 



Along most lines our distance will be of the first type. Along 

 certain lines, however (analogous to minimal lines in metric geometry), 

 distance will be of the second type. 



17. Distance in the plane. Along each line through a point A is 

 a single point at a given distance from A. There is a certain locus 

 of these points. We assume that this locus is an analytic curve. 

 Since it cuts each line through A in a single point it is then a straight 

 line. Thus the locus of points at a given distance from a given point 

 is a straight line. 



We assume that there exist distances AB not zero. Let 



where ^ is a constant not zero and A, B points not on f. If A is held 

 fixed B describes a line b cutting f in P. If B is held fixed A describes 

 a line a cutting f in a point Q. Distance along any line through A 

 except AP is of type (1). Along AP it is of type (2). Similarly 

 distance along every line through B except BQ is of type (1). Let 

 the intersection of AP and BQ be F. Through any point Ai except F 

 there passes a line AiA or AiB along which distance is of type (1). 

 Therefore, for any such point Ai there exists a line of points b, such 



