MATHEMATICAL CONCEPTS OF THE MATERIAL WORLD. 511 



*21'21. Proposition. An objective real, which is doubly secant with the common 

 subregion for u, is a member of the common subregion for u. In symbols, 



I" : w = cm R( 'w . (ww) Rt \x . z> . xecm Rt 'u 



Proof. Cf. *3'12 and *20'11. 



Note. The converse is not in general true, namely, that, if u is a class of objective 

 reals, and x is a member of cm Bt 'u, then a; is doubly secant with cm R( 'w. Nor does 

 this converse follow from subsequent axioms. In the absence of this converse 

 proposition the properties of homaloty differ from those of " flatness" for classes of 

 punctual lines. For if u is a class of punctual lines, and (f> stands for the property of 

 flatness, then cm^'u is flat. 



*21'31. Proposition. The proposition p. Hp< is true when p Rt is substituted for <f>. 

 In symbols, 



Proof. Cf. *10-2 and *20'12. 



*21'41. Proposition. If a is an objective real cogredient with c, then c is 

 cogredient with a. In symbols, 



I" : a e cogrd R( ' a . = . c e cogrd m ' a 

 Proof. Cf. *20'41. 



*21'51. Proposition. If u, v, w be three interpoints, possessing the same objective 

 real, and with dominant points u m , v Rt , w Rt , then R pn ; (u Rt v Rt iv m t) implies ^-^(uviut). 



In symbols, 



|- .'. u, v, w e K ; (a ???). D : R,, n ; (u Rt v Rt w Rt t) . D . ~R- m ; (uv-wt) 



Proof. -By definition (cf. *20'5l) ^ v ^(u Kt v m w Rt t) implies (l) the existence of an 

 objective real x, cogredient with a, and also of three interpoints, u', v', w', all 

 possessing x, and (2) that ~R, in -(u'v'iv't}, and (3) that u', v', w' are contained in 

 dominant points u' Rt , t/ Rt , i' Rt in perspective with u Rt , v Ri , iv m . Hence by the 

 definition of cogredience (cf. *20'41) also R in ; (uvwt) holds. 



*22. The Axioms. Just as in Concept III., the axiom of persistence (cf. *22'l) 

 does not enter into the geometrical reasoning, but it is essential to the physical side 

 of the concept. 



*22'1. I Hp II is the statement that, if t be an instant of time, O is contained in 

 O R( . In symbols, 



= :eT.D ( .OcO R< Df 



The next four axioms, viz. (II.-V.)HpTl, are the axioms of order. They have 

 already been explained in *l'5r52'53'54. 



*22'21. IIHpR=aHpIl Df 



*22-22. IIIHpR = H2>K Df 



*22'23. IVHpR = yHpR Df 



*22-24. VHpR=8HpR Df 



