498 DE. A. N. WHITEHEAD ON 



*6'25. Proposition. If v is a (-prime and has a ^-dimension number, then the 

 cardinal number of v is less than, or equal to, dim+'v. In symbols, 



h : (ExXdim/w) . v e prm^, . D . Nc'v = dim/ v 



Proof. Of. *3'21. 



*6'26. Proposition. If v is <-axial and is ^-equivalent to u, then the cardinal 

 number of v is equal to dim,,,' u. In symbols, 



i" : v ax^, n equiv/ u . D . Nc' v = dim/ u 

 Proof. Cf. *3'21'22. 



*8. On (^-Concurrences. 



*8'21. Proposition. If u is contained in w, then the ^-concurrence of u with v is 

 contained in the ^-concurrence of w with v. In symbols, 



i~ : u c.w . D . ftj,' v c. Wj v 

 Proof. Cf. *3'31. 



*8'22. Proposition. If t' is contained in w, then the ^-concurrence of u with w is 

 contained in the (^-concurrence of u with v. In symbols, 



h : v c w . D . ?/ w c ?7/ v 



Proof Cf. *3'31. 



*10. Geometrical Properties. A property < is called geometrical if it satisfies the 

 five axioms (X, yu, i>, TT, />) H^; $ stated below. The axiom v Up <$> takes the special 

 form for three dimensions. It is to be noticed that three dimensions is the lowest 

 number for which a <-point (cf. *3'42) can be defined. The reasoning can be applied 

 to higher dimensions, only more elaborate inductions and an extra axiom are required. 

 Other axioms and definitions are wanted to enable all the propositions of projective 

 geometry to be proved. These will not be considered here as such an investigation 

 would involve some repetition when we come to Concept V. The class O^, is the class 

 of straight lines of the geometry, conceived as simple unities. The class put^, is the 

 class of points, each point being a class of lines. The class ple^ is the class of planes, 

 each plane being a class of lines. 



*10'1. XH/ ( is the statement that 0^, has the properly (f>. In symbols, 



XHp^. = .^!O + Df 



*10'2. p. Hp (f> is the statement that, ifx is any member of Of, the unit clans L'X has 

 the property <$>. In symbols, 



p.Kp<l>. = :ajeO t .3 s .^!i c oj Df 



*10'3. vHp<f> is the statement that the ^-dimension number of 0^ is three. In 

 symbols, 



O, = 3 Df 



