PRINCIPLE OF CONTINUITY IN THE THEORY OF SPACE. 259 



And instead of we may substitute 1 throughout for units of 

 successive dimensions, and go for infinities of the successive 

 dimensions. 



Similarly if we start with the line px, move it qy, then move 

 the generated surface rz, etc., we obtain 



{L(^x > $v)— > r z ] > s w j >etc. =pqrs...\l Tiyf ^..(\a) 



in which pqrs... is merely scalar, 1 and n, the number of factors is 

 equal to the number of suffixes; that is to say, n is the number 

 of dimensions. 



5. General laws of flaxional generation. — From what has pre- 

 ceded it is evident that an ^-dimensional geometrical figure can 

 be generated, by a single motion into the nth dimension, of an 

 (n — l)-dimensional figure, by two successive motions of an in - 2) 

 dimensional figure, one being in the direction of the (n - 1) the 

 axis, and the other in that of the n-th axis of the higher spaces; 

 and similarly by a greater number of specifically-different successive 

 motions, when the generatrix is of lower dimension. It is further 

 evident that, at least in homaloidal space, the relative order of 

 the generative motions is immaterial. 



The results may be presented in the following form : — 

 A geometrical figure of n dimensions can be generated by k 

 successive 2 motions or displacements of a generatrix of (n-k) 

 dimensions, parallel to k independent axes, not included in the 

 space-axes of the generatrix. 



as already shewn, not an absolute zero. If such a zero had no duration, 

 an infinity of such could not be conceived as implying duration at all. 

 More crudely, infinitesimal motion may be apprehended as the incipience 

 of motion, the state of the generatrix at the instant of commencing to 

 move, or rather during the infinitesimally small time when it commences 

 to move. 



1 A scalar, or vector "divided by" a parallel vector is a purely abstract 

 number, a mere ratio, px is to be understood as p x l x , where l x is the 

 vector unit, and p is an abstract number or a scalar. As scalars have no 

 spatial properties, pqr etc. or xyz etc. have no other essential meaning 

 than that they are the product of abstract numbers. 



2 It is essential that the motions be successive, and not simultaneous : 

 a point for example moving in the direct x and y generates a line instead 

 of the surface xy : it is the path x, moving the distance y, or the path y, 

 moving the distance x, that generates the surface. 



