506 DR. A. N. WHITEHEAD ON 



to *3'42) that the geometry cannot be of less than three dimensions. Hence in this 

 concept geometry of three dimensions occupies a position of unique simplicity. 



The points at infinity, here called cogredie.nl points, are points in exactly the same 

 sense as the other points. They are defined by a property not hitherto taken as 

 fundamental. The properties of cogredient points play an essential part in the 

 construction of a relation which assigns an order to the points on any straight line. 



*20. Definitions. 



*20'11. Definition. An objective real p is doubly secant with a class u at an 

 instant t if there exist two objective reals, members of u (x and y, say), which are 

 both intersected by p at the instant t, and are such that there exists no interpoint on 

 p of which x and y are both members. The symbol (uu) M \p will denote that p is 

 doubly secant with u at the instant t. The symbolic definition is 



(uu) w \p . = : fax, y) . x ^ y . x, yen n K ; (p;;;<) . - far) . v eR ; (p???) . x, yev Df 



*20'12. Definition. A class u is homalous at an instant t, either when a necessary 

 and sufficient condition, that x should be a member of u, is that x should be doubly 

 secant with u, or when u is a unit class contained in R ; ( ;;;;). The symbol /AR ( !M 

 will denote that u lias the property of homaloty at the instant t. The symbolic 

 definition is 



p. ]u lu . = .'. xeu . = x . (uu) Ht \x : V : ue 1 n cls'B/( ;;;;*) Df 



This property (p. Rt ) of homaloty will now be taken as the special value of <f>, to 

 which the theory of dimensions will be applied. The common /i Rr subregion for u is 

 denoted, according to the definition of *3'12, by cm M '. But a suifix to a suffix will 

 be avoided by using the simpler symbol cm lu 'u, and similarly for the other entities 

 defined in *3. Thus the following symbols are also defined, namely, 



O Ht , equiv H( X prm R( , dim lt( , ax, u , mx R( , u Rt 'v, conc H( , ple H ,, 

 pnt u( , cople H( h, copnt H ,!w. 



With regard to the nomenclature, the term " ^-equivalence" should be 

 particularized into " homaloty-equivalence," and "<-prime" into " homaloty-prime," 

 and so on. But, except where confusion is likely to occur, the term " homaloty " will 

 be dropped ; and the terms " equivalence," " prime," " dimensions," " axial," 

 " maximal," " concurrence of u with v," " self-concurrence," " plane," " point," 

 " coplanar," " copunctual" will be used in the senses defined in *3, with < particu- 

 larized into homality. 



Elucidatory Note. This definition of homaloty should be compared with the 

 definition of the flatness of a class of punctual lines which has been used in the 

 elucidatory notes of *3. Thus a class of punctual lines is flat, either when it is a 

 unit class whose single member is a straight line, or when it is a necessary and 



