MATHEMATICAL CONCEPTS OF THE MATERIAL WORLD. 489 



Then the definitions of Concept I. will be assumed to apply to K<. For example, 

 the punctual line joining the points A and B is the class of points which is the logical 

 sum of R t ; (;AB) and R, ; (A;B) and R t ; (AB;) together with A and B themselves. Its 

 symbol is R ( AB. The definition in symbols is 



KvAB = By ( ; AB) u BY ( A ; B) u BY ( AB ; ) u i' A u I'B Df 



It will follow (cf. *1'23'31) from the axioms that a punctual line is the class of those 

 points with some member of as sole common member. The other definitions can be 

 managed in like manner, only in the symbolism a suffix to a suffix will be avoided by 

 writing A R( ; (ABC), and so on, instead of A R ; (ABC), and so on. 



The Axioms. The earlier axioms have to be modified from those of Concept I., but 

 the later axioms are simply those of Concept I. with the R of that concept replaced 

 by the R ( of Concept IV. 



IHpE. = :* e T.D t .OcR ; (;;;;0 Df 



IIHpE. = .a!R : ( [Y.e., 3 !T] Df 



IIIHpR. = .aHpR Df (cf. *1-51) 



IVHpR. = .HpR Df (c/*l-52) 



VHpR. = . y HpR Df (cf. *l-53) 



VIHpR.=.SHpR Df (cf. *1'54) 



VII Hp R . = . intpnt Hp R Df (cf. *1'41) 



VIII Hp E. = .-. (A, B, C).-. A, B, CeR,-(;;;). A^B. A^C.B^C. 



3 ! ( A n B n C) . D : R ( ; ( ABC) . V . R< : (BCA) . V . IV (CAB) Df 

 IXHpR.= :(A, B): A, BeE,'(;;;). A ^ B . D . gr !R ; (AB;) Df 



XHpR . = : (A, B, C, D, E) : A, ; (ABC) . R e ; (BCD) . R ( ; (CEA) . => . 



a !{R ( 'DEnR e ; (A;B)} Df 



XIHpR . = : JtT .=H . ta D) .^eple R( . DeE,'(;;;) -p Df 



XII Hp R . = . (3 A, B, C, D) . R t ; ( ; ; ; ) c U ut (ABCD) Df 



XIII Hp R . = . the axiom of continuity, cf. XII Hp R of Concept I. Df 



XIV Hp R . =.'. ple Ri . a e lin Ki n cls'a . D a> . : (gC) : C e a : 



I, V e lin R( n cls'a . C e I n V . I n a = A . I' n a = A . ot.v I = V 



Note that only I Hp R and XI Hp R require the hypothesis t e T ; in all the other 

 axioms there is a hypothesis which can only be true when t e T. For the purpose of 

 comparison with the axioms of Concept I., the following propositions are required : 



VOL. ccv. A. 3 R 



