PKINCIPLE OF CONTINUITY IN THE THEORY OF SPACE. 265 



Inasmuch as an n-dimensional unit can be divided by an infinity 

 of the nth order, and as, moreover, scale-differences do not con- 

 ceptually limit our thought, there is no conceptual limitation to a 

 similar division of any infinitesimal or zero quantity. 1 Hence 

 numerically, the quantities of ordinary mathematical conceptions 

 range at least between 



'0\ i 1 , oo 1 and 0°°, 1", oo 00 , 

 or, for ordinary tridimensional space, the higher limit must be at 

 least 3 , l 3 , oo 3 , and it is now easy to see this last is only an 

 artificial restriction. 



1 0. Spatial continuity and its numerical expression. — Although, 

 as has been shewn, it is logically essential to distinguish between 

 a dimensionless point and an infinitesimal of any dimension, it is 

 possible to employ the former in a perfectly definite and rigorous 

 manner to define the development of geometrical figures; that is 

 to say the dimensionless point may be taken as a sign or mark of 

 the infinitesimal, its locus in space being to the first order of 

 infinitesimal identical therewith, but not exhaustively so. 2 



Space being essentially a continuum, and a plenum, the substitu- 

 tion of dimensionless and therefore discrete points, in a mere 

 numerical scheme for determining its quanta, i.e. portions marked 



1 Or we may realise the inherent simplicity of the conception in this 

 way: — We have no difficulty in conceiving that a finite line is a singly- 

 infinite continuous series or group of infinitesimal lines; a finite plane area, 

 is a doubly infinite continuous group of infinitesimal areas ; a finite volume 

 is a triply infinite continuous group of infinitesimal volumes; and generally 

 a finite quantity of the nth dimension is an n-ply infinite continuous group 

 of an infinitesimal of n-dimensions. Since these facts are numerically 

 representable, there is no conceptual difficulty in extending our conception 

 of infinity and infinitesimals to the higher orders of such quantities, 

 hence n , oo n represent not merely numerical abstractions, but real 

 ideas. 



2 This may be pictured as the point occupying the centre of the infini- 

 tesimal, which latter, in relation to it, can be regarded, loosely of course, 

 as absolutely infinite. In Cayley's expositions of higher geometry, (e.g. 

 the " Sixth Memoir on Quantics/' Phil. Trans. Vol. 149, 1859, pp. 61 - 90) 

 and in Henrici's, the geometry is essentially discrete point geometry, and 

 a row, range, or assemblage of points is distinguishable from the base, or 

 line, surface etc., in which they lie, only when account is taken of higher 

 orders of infinitesimals. Thus the equation, (* & a?, y) m = may be 

 variously interpreted, as soon as account is taken of the nature of the zero. 



