MATHEMATICAL CONCEPTS OF THE MATERIAL WORLD. 513 



cogredient points, then there exists at least one point C such that A, B, C are in the 

 point-order ABC at the instant considered. In symbols, 



XIV Up R . = : (A, B, ) : A, B e pnt* - <*, . D . g ! R pn <(AB ; t) Df 



*2272. XV Hp R is the statement that, if A, B, C are three distinct non-cogredient 

 points, on the same objective real, then at the instant considered one of the point- 

 orders ABC, or BCA, or CAB holds. In symbols, 



XVH^R . = .-. (A, B, C, t) .: A, B, C epnt K . oo R< . a !(A n B n C) . 



A^B.B^C.C^A.D: R I>U : (ABCO . v. R,, U ; (BCAO . v. R^CAE*) Df 



The next axiom, XVIHpR, is the well-known "transversal" axiom. 



*2273. XVI HpR is the statement that, if at the instant t the points B, C, D are 

 in the point-order BCD, and the points C, E, A are in the point-order CEA, and the 

 points A, B, C are not collinear, and F lies in the punctual associates botJt of A n B 

 and o/D n E, then the points A, F, B are in the point-order AFB. In symbols, 



XVI HpR . = : (A, B, C, D, E, F, t) : R^BCD*) . R 



AnBnC = A.F ass H( '(A n B) n ass R( '(D n E) . D . R im ; (AFB) Df 



*22'74. As XVII Hp R, an axiom of continuity will be wanted. 



Note. The above axioms are all axioms of geometry, in the sense of "geometry" 

 as defined in the sense definition of it given in "Part I. (i.). But geometry in this 

 Concept V. includes more than does geometry in Concept I. For in Concept I. 

 geometry has only to do with points, punctual lines, and punctual planes ; but in 

 Concept V. geometry has, in addition, to consider the relation of the objective reals 

 (which are all "linear") and of interpoints to the above entities. In this respect, 

 geometry in Concept V. merges into physics more than does geometry in Concept I. 

 Thus the excess of the number of axioms in Concept V. over the number in 

 Concept I. arises from the fact that there is a larger field to be covered. Also, 

 I Hp R is not required in the geometrical reasoning. 



*25. Preliminary Propositions. 



*25'11. Proposition. Assuming (II. -VI) Hp R, all the propositions of the theory 

 of interpoints (cf. *l) hold of the interpoints of this Concept. 



*25'12. Proposition. Assuming (II. -VI.) Hp R, if t be an instant of time, then 

 R , possesses at least four members. In symbols, 



tv. IXHpR . D : eT . D . Nc'O K a 4 



Proof. Cf. *1-61717273 and *2M1 and *25'11. 

 VOL, ccv. A. 3 u 



