MATHEMATICAL CONCEPTS OF THE MATERIAL WOULD. 479 



IXHpR is the statement that there exists a point and a plane, such that the 

 plane does not possess the point. The definition in symbols is 



IXHpR. = :( a p, d) . pep\e R . dift(;;i)-p Df 



X H^> R is the statement that there exist four points a, b, c, d, such that the three- 

 dimensional space U R (abcd) contains the whole class of points. The definition in 

 symbols is, 



XHpR. =.(aa,6, c, d). ~R-(;;;) cU H (abcd) Df 



XI Hp R is some statement which secures the continuity (in CANTOR'S sense) of the 

 points on a line. The axiom need not be given here, since there will be no reasoning 

 in this memoir connected with it. 



XII Up R is the statement that, if a be any plane and a a line contained in it, then 

 there exists a point c in a, such that there is not more than one line, possessing c and 

 contained in the plane a and not intersecting a. The definition in symbols is 



XII Hp R . =.'.. ple H . a e lin K n cls'a . D O a : (g;c) : c e a : 



I, I' e lin K n cls'a . c e I n V . I n a = A . V n = A . =>,, v . I = I' Df 



Of these axioms, IX Hp R secures that space is of three dimensions at least, and 

 X Hp R secures that it is of three dimensions at most, and XII Hp R is the 

 "Euclidean" axiom. From these twelve axioms the whole of geometry* can be 

 deduced. The well-known parabolic (i.e., Euclidean) definition of distance (not given 

 here) assumes an important significance, and all the usual metrical properties follow. 



The Extraneous Relations. Nothing could be more beautiful than the above issue 

 of the classical concept, if only we limit ourselves to the consideration of an 

 unchanging world of space. Unfortunately, it is a changing world to which the 

 complete concept must apply, and the intrusion at this stage into the classical concept 

 of the necessity of providing for change can only spoil a harmonious and complete 

 whole. Owing to the fact that the instants of time are not members of the field of 

 the essential relation, the time relation and the essential relation have (so to speak) 

 no point of contact. To remedy this, another subdivision of the class of objective 

 reals is conceived, namely, the class of particles (where the particles are the ultimate 

 entities composing the fundamental " stuff" which moves in space). These particles 

 must form part of the fields of a class of extraneous relations. Each such extraneous 

 relation is conceived as a triadic relation, which in any particular instance holds 

 between a particle, a point of space, and an instant of time. Also the field of each 

 such extraneous relation only possesses one particle, and no particle belongs to the 

 field of two such relations. Thus each extraneous relation is peculiar to one particle. 

 Also, as has been pointed out by RUSSELL, t to whom the above analysis of these 

 extraneous relations of the classical concept is in substance due, the impenetrability 



* Of. VEBLEN, loc. at. 



t Of. 'Principles of Mathematics,' vol. I., 440. 



