200 PROCEEDINGS OF THE AMERICAN ACADEMY. 



that in the study of such a geometry angle has to be defined anew. 

 This was done by Beck. 1 A geometry analogous to that in the mini- 

 mum plane was discussed by Phillips and Moore 2 and many of the 

 properties of the triangle ecc. derived analytically by Beck were there 

 obtained synthetically. Such a geometry was also mentioned by 

 Wilson and Lewis 3 and the analytic definition of angle given. 



Phillips and Moore also discussed a second kind of geometry in 

 which the element of arc is not an exact differential. In this paper 

 I shall discuss these two geometries analytically and apply the results 

 to the study of the geometry on the minimum plane and the minimum 

 developable. 



In these two geometries, in order to define distance, and angle, a 

 fundamental line/ and a fundamental point F were assumed. In the 

 first kind the line passed through the point and in the second kind it 

 did not. The equality of two distances on the same line was then 

 defined as follows, let A, B, C, D, be four points on a line cutting/ in P, 

 then AB = CD if A, D and B, C have the same harmonic point with 

 respect to P. The same definition for equality holds for two segments 

 on parallel lines (lines intersecting on/). For any four points on a line, 



AB = XCD, 

 where 



X = (AC | DP) - (BC | DP). 



This also gives a way of comparing distances on two parallel lines. 

 From this definition it is easily seen that there is only one point on a 

 line through A which is at a given distance from A. If we then assume 

 that the locus of points equidistant from A is an analytic curve it is 

 easily shown that it is a straight line. Knowing this locus we can now 

 compare distances on any two lines for on any line through A we can 

 mark off a distance AM which has a given ratio to a given distance. 

 As soon then as we have assumed a unit of length we ,can measure 

 all distances. Distance being defined in terms of double ratios the 

 discussion which follows can be considered as a problem in pure pro- 

 jective geometry. It is to be noted that the above does not give any 

 way of comparing distances on lines through F or on /. The same 



1 Zur Geometrie in der Minimalebene. Archiv der Mathematik und 

 Physik, 20, (1913). 



2 An algebra of plane projective geometry. These Proceedings, 47 (1912). 

 In what follows I shall refer to this paper as A. P. G. 



3 The space-time manifold of relativity. The non-euclidean geometry of 

 mechanics and electrodynamics. These Proceedings, 48 (1913). 



