PRINCIPLE OF CONTINUITY IN THE THEORY OF SPACE. 261 



A linear unit divided into m parts may be reconstituted by m 

 elements of length l/m; a surface unit, each unit side of which is 

 divided into m parts, by ra 2 elements of surface-area each l 2 /ra 2 ; 

 and similarly a unit quantum of n dimensions, may be made up of 

 m n elements, each of l n /ra n quantum. 1 If now zero be defined as 

 unity divided by infinity, that is if zeros and infinities are pure 

 reciprocals, viz. 



n =^—\ 1* see n oo n : oo u ^— : etc.. ..(3), 



where n may have any value whatever. Making m infinite the 

 index and dimension are throughout identical, that is to say, we 

 must admit the relations : — 



°» = ^; in = }i; ^-^ etc.. ..(3a) 



The purely numerical relationships of n-dimensional zeros, of 

 zero-value in each dimension, absolutely involves the admission 

 of the idea of different orders of numerical zero, infinity, etc., as 

 conceptually necessary; in fact no less necessary than that of a 

 first order infinity. Apart however from the mere numerical 

 development implied in spatial relationships, the ordinary linear 

 conceptions involve at least two orders of infinity, for a finite line 

 is conceptually admitted to be capable of subdivision into infinitely 

 small parts, and on the other hand to increase to infinity. Con- 

 sequently x = dx, = 1, = oo imply a range of oo 2 at least. 2 Once 



the doctrine of limits has been developed, in which the object of study 

 has been the relation of quantities at the point of their evanescence, or on 

 the other hand when they become infinitely great. There is no escape 

 from the conception of the differential, and the doctrine of limits does not 

 materially avoid this particular difficulty. Certain writers, e.g. Todhunter, 

 see his Differential Calculus, Art. 26, treat the differential fraction ex- 

 pressing the ratio of quantities at the point of evanishing, as a "whole," 

 instead of as the real ratio of actual infinitesimals, or * indivisibles.' That 

 is to say dy\dx is not to be regarded as a real relation between the zero 

 quantity dy and. the zero dx, but as an undecomposable ratio representing 

 the limit of their real ratios at the point of vanishing of the quantities 

 themselves. This introduces logical difficulty, when afterwards we find 

 dx and dy on opposite sides of an equation, or when we are to integrate 

 say/ (x) dx. 



1 The use of an index with unity defines its dimension. 



* In fact dxlx = xjcc x, that is to say where the infinitesimal and infinite 

 are admitted as concepts at all, the commonest conception involves at 

 least a second order of linear infinity. 



