PRINCIPLE OF CONTINUITY IN THE THEORY OF SPACE. 275 



since the range from the first order infinitesimal to the first order 

 infinity necessarily transcends practical requirements. In testing 

 the conceptual validity of spatial theories however, these things 

 are no longer unimportant. That the infinitesimal calculus affords 

 a clear illustration of the essential difference between infinitesimal 

 approximation and absolute identity, may be readily illustrated by 

 considering the difference between a curve that is the path of a 

 moving point, and one which is merely a range of discrete points. 

 Let for example £ and 77 denote the coordinates, with the same 

 axial directions, of any point in the curve 



y = a + bx + car 2 + dx 3 + etc (15) 



from some point x, y as origin, in the curve itself: then we shall 

 have 



|=5+_C£ + 2)£ 2 rt-ect (16) 



B being the differential coefficient of (15), C half the differential 

 coefficient of B, etc. 1 If £ = 1 , the term CO 1 is an infinitesimal of 

 the first order compared with B, and therefore has a value of 

 which no numerical account can be taken, so long as B is finitely 

 expressed, but has a finite value if the right hand member of (16) 

 be multiplied by infinity, shewing that the tangent to the curve 

 at the point £ = 1 is not absolutely the true tangent at the origin, 

 but really differs therefrom infinitesimally . The tangent at the 

 point £ = 2 evidently more closely approximates 2 to the tangent 

 at the origin, since the C term is infinitely reduced. The difference 

 is of course conceptual only, since it can only be expressed sym- 

 bolically, and not numerically. Nevertheless once it is recognised 

 that there is no escape from admitting different orders of infinity 

 and therefore of infinitesimals, as conceptual entities, the distinction 

 is not unimportant in discriminating what are really geometrical 

 forms in space, and the space itself in which these forms are con- 

 ceived to exist, and of which they constitute quanta. 



The admission of higher orders of infinitesimals renders intel- 

 ligible the conception, that although quantities may be reduced 



1 B=dyjdx ; C = $, d 2 yjdx 2 ; D = hd 3 yjdx 3 ; etc. 2 Is infinitely closer. 



