510 DR. A. N. WHITEHEAD ON 



B |in ( ABC) . = . A, B, C e pnt K( - OO R( . (ga, x, u, v, w) . a e A n B n C . x e cogrd R( ' a . 



u, v, t(;eR : (a;???) . ~Ri n -(ui>wi) . [ ABC] persp R( [> R( t> R( i{> m ] Df 



Si nce ( c f. *27'43) x in the above definition is distinct from a, three collinear 

 (i.e., on a) points, A, B, C, cannot directly take their point-order from three inter- 

 points which they themselves may severally contain (cf. however *2T5l). The 

 point-order of A, B, and C must arise from the order communicated (in a sense) to a 

 copunctual pencil of three punctual lines by three interpoints contained respectively 

 in points in these lines, and all three interpoints possessing an objective real (x) in 

 common. The punctual lines of this pencil must possess A, B, and C respectively. 

 This intervention of a pencil for the communication of point-order is necessary for 

 the comparison of the orders of different ranges. If the apparently simpler plan is 

 adopted, inextricable difficulties seem to arise. Also it will be remembered that not 

 every point will necessarily contain an interpoint. 



*20'61. Definition. A Punctual Plane is a figure which is either the punctual 

 associate of some plane, or is the class OO R( . The class of punctual planes is denoted 

 by pple,. The definition in symbols is 



pple m = ass u ,"ple E( u i' oo Rt Df 



Note. This definition is only convenient for Euclidean geometry. 



*2072. Definition. A figure is called Punctually Coplanar if there is a punctual 

 plane containing it. The symbol copple u( !w will denote that u is a punctually coplanar 

 figure at the instant t. In symbols, 



copple K( !w . = . (3^) . p e pple R( . u e cls'p Df 



Note. This definition should be compared with that of cople R( !?< in *3'43. 



*21. General Deductions. 



*21'01. Proposition. All the general deductions in the theory of dimensions, 

 namely, *4 to *8, hold. 



The following propositions, dependent on the special definition of homaloty, also 

 hold :- 



*2ril. Proposition. O R < is the class R ; (;;;;t). In symbols, 



Proof. Cf. *20'12. 



Note. If t is not an instant of time, the classes O Ri and R ; (;;;;) are both the null 

 class, and are thus identical. Accordingly the hypothesis, <eT, is not required in 

 this proposition. A similar explanation of the absence of the hypothesis, t e T, holds 

 for many other propositions. 



