MATHEMATICAL CONCEPTS OF THE MATERIAL WORLD. 507 



sufficient condition of a straight line x being a member of it, that x should meet two 

 members of the class in points which are not their point of meeting (if they have a 

 point of meeting). Owing to the fact that " intersection " (as used here) is wider in 

 intension and narrower in extension than the idea of the " meeting " of two punctual 

 lines, two punctual lines may " meet " without the corresponding objective reals 

 " intersecting." The result is that homaloty and flatness have some different 

 properties, for example, cf. *21'21. 



*20'21. Definition. The punctual associate of a class u is the class of those points 

 which have a member in common with u. The punctual associate of u is denoted by 

 ass K j'tt. The definition in symbols is 



ass ltj < u = P {P e pnt B( . 3 ! (P n u)} Df 



Note. The punctual associate of the class i'a, ivhere a is an objective real, will be 

 called the punctual associate of a. Its symbol is ass R( ' t' a. 



*20'22. Definition. A punctual line is a class of points such that there exist two 

 planes, p and q, which are distinct and are such that the class of points is the common 

 subclass of the punctual associates of p and q. The class of punctual lines at any 

 instant t is denoted by lin ju . The symbolic definition is 



lin K( = m{(Kp, q) . p, qeple Rt . p ^ q . m = &ss !U 'p n ass,,/?} Df 



Note. Those punctual lines which are not " lines at infinity " (to be explained 

 later) will be proved as the result of the axioms to be the punctual associates of the 

 various objective reals. 



*20'23. Dejinition. The point, if there is one and one only, which contains a class 

 u is called the dominant point of u. The dominant point of u is denoted by K( . The 



symbolic definition is 



u Kt = (tp) {p e pnt H , . u e cls'p} Df 



Note. The idea of a dominant point obtains its importance from the fact that, 

 according to the axioms given below, each interpoint is contained in one and only one 

 point. 



*20'231. Dejinition. The nonsecant part of u is that subclass of u of which no 

 member is a member of any interpoint which is a subclass of u. The nonsecant part 

 of u is denoted by nsc K( ' u. The symbolic definition is 



nsc K( ' u = x {x e u . -*- (3^) v e intpnt K( n els' u . xev} Df 



Note. This definition takes its importance from the fact that (assuming the 

 subsequent axioms) a point in general consists of a nonsecant part and of a part 

 made up of interpoints contained in it. Either the interpoints or the nonsecant 

 part may be wholly absent. 



*20 < 232. Definition. A class of points is called a Figure. 



3 T 2 



