262 G. H. KNIBBS. 



the difficulty even in linear quantities of conceiving an infinity of 

 infinities is transcended, there can be no further conceptual 

 difficulty in admitting infinities of any orders. A similar remark 

 applies to zeros of any order. 



8. Summational generation by means of the n-dimensional point. 

 — The essential difference between mere multiplication, of a 

 geometrical infinitesimal of any dimension, by an abstract number, 

 and the apparent numerical identity of essentially different oper- 

 ations upon the dimensionless point, may be elucidated by regard- 

 ing inductively, the developments conceivable in successive 

 dimensions of space. Assuming a series of axes x, y, z, w, etc., 

 not necessarily orthogonal, let first an infinite number of linear 

 zeros be taken having dimension in the direction x only. These 

 may be summed in two ways, (a) by addition along the axis x } 

 (b) by addition in the direction y. In the former case we get as 

 result l x , in the latter X , since they are dimensionless in the 

 direction y. If further oo 2 of these zeros be taken, we get oo x by 

 the former addition, and still only X by the latter. Suppose, 

 however, we start with an infinite number of 2-dimensional zeros, 

 xy . Addition along the axis x gives l x , along y, gives l y , the 

 resultant lines not being purely linear, but fully represented by 

 the surface symbols l x y = xy ; l y x = 0J^ = xy ; that is we 

 obtain a line l x of infinitesimal breadth y , or a line l y of breadth 

 X . It is also a surface zero, but one that is infinitely greater 

 than x y , that is to say, oo xy = xy . With such infinitesimals 

 as these, any surface can be generated summationally, but only 

 by operating in the xy plane. Thus an infinite number of l x y 

 units by addition along the x axis give oo x y an infinite line of 

 infinitesimal breadth, by addition along the y axis l x l y i.e. l xy or 

 a unit of surface, and vice versa for l y x . Addition in the direc- 

 tion of the z axis produces no result whatever, because these 

 elements are dimensionless in that direction. Thus we may state 

 with generality that : — 



All summational operations with infinitesimals are significant 

 only for axial directions corresponding with their dimensions, but 

 not otherwise. 



