MATHEMATICAL CONCEPTS OF THE MATERIAL WORLD. 509 



perspective with the trio of dominant points u' Rt , t/ H< , u/ Kt , then the interpoint relation 

 (R in ), if it arranges either trio of interpoints in an interpoint order, arranges both 

 trios of interpoiuts in the same interpoint order (i.e., either uvw and u'v'w', or vwu 

 and t/u/u', or so on), and (2) there exist three interpoints u, v, w on a in the 

 interpoint order uvw, and three interpoints u', v', w' on c such that R( , v ut , w m are in 

 perspective with ' 1/ R( , w' R( . The symbol cogrd R( 'a denotes the class of objective 

 reals cogredient with a. The symbolic definition is 



cogrd R< 'a = x {(u, v, w, u', v', iv') : u, v, w c R : (a ? ? ? t) . u', v', w' e R ; ( ? ? ? t) . 



persp R( OV R X R J . => . R hl : 



persp R( [u'^v'^w'^} Df 



. The class cogrd R( 'a does not include a itself (of. *27'43). It will be noticed 

 that universal preservation of order by ranges in perspective on a pair of lines is a 

 characteristic of a pair of parallel lines in Euclidean space, and of nonsecant lines in 

 hyperbolic space. The choice of this property for the definition of parallelism (or 

 nonsecancy) arises from the facts that (l) any two coplanar objective reals are 

 copunctual (according to the subsequent axioms), so that the property of nonsecancy 

 (in its ordinary acceptation) is not available, (2) we do not wish to make "cogredience" 

 synonymous with " nonintersection " (using " intersection " in the special sense here 

 defined), as this would impose an unnecessary limitation on the concept. The idea of 

 cogredience is an essential element in the definition of a relation which, with the aid 

 of axioms, distributes the points in any punctual line into an order. 



*20'42. Definition. A Cogredient Point is the class of objective reals cogredient 

 with some objective real a, together with a itself. The symbol QO IM denotes the class 

 of cogredient points at the instant t. The definition in symbols is 



OO K( = it {(3) . a e O R4 . u = I'a u cogrd K( 'ct} Df 



Note. In the case of Euclidean geometry, which is the only case considered here, 

 each cogredient point is a point according to the definition of *3'42. The present 

 definition would be very inconvenient, unless this were the case. The symbol o K( is 

 reminiscent of the fact that the cogredient points are the points at infinity. 



*20;51. Definition. The Point-Ordering Relation is a tetradic relation holding 

 between three points and an instant of time. Its symbol is R p ,,, and R |m : (ABC) is 

 defined to mean that, at the instant t, (1) A, B, C are non-cogredient points upon the 

 same objective real, a say, and (2) there exist an objective real x and three 

 interpoints u, v, w on x such that (i) x is cogredient with a, and (ii) u, v, w are in the 

 interpoint order uvw, and (iii) A, B, C are in perspective with the dominant points 

 u Rt > v xt> w m- The definition in symbols is 



