264 G. H. KNIBBS. 



(g) a surface of infinite length the other dimension being unity; 

 (h) a surface infinite in both dimensions. Volume elements, (i) a 

 linear volume-zero, i e. l x y z ; (j) a surface volume-zero, i.e. l* y 8 ; 

 (k) a parallelepipedic unit volume 1 ; (I) a volume of infinite length 

 but of parallelogrammic unit dimensions. Jfih di?nensional 

 elements, (m) a linear 4th dimensional zero, (u) a surface 4th 

 dimensional-zero, i.e. lx y O zw ; (o) a volume 4th dimensional-zero, 

 i.e. l| yz w ; (p) an orthogonal or oblique 4th dimensional quantum, 

 according as the axes are orthogonal or oblique. Throughout the 

 series (4) the division has been made in such a manner as to shew 

 that the equivalence (horizontally) is dependent upon the extension 

 of the several quantities infinitesimally into the higher dimensions, 

 if the generated forms are to be continuous. 



9. Zeros and infinities of successive orders. — The scheme of 

 summational generation outlined, indicates that in rigorous 

 mathematical thought, the conception of different orders of 

 infinitesimals and of infinities 2 is essential, not merely as a purely 

 formal numerical artifice, but as really representing the develop- 

 mental or generative processes, without which conceptual continuity 

 of geometrical figures would not be a possibility, Hence linear, 

 surface, and volume infinities, etc., are properly distinguished as to 

 their dimensions and also as to their powers. To develope the 

 unit of any n-dimensional quantity 3 from its proper zero, i.e. from 

 the zero which is (a first order) infinitesimal in every dimension* 

 it is necessary to multiply not merely by infinity but by the nth 

 order of infinity, that is to say, as in (3a) 



OS x oo" = is 



1 If axes are orthogonal, and units are also equal, substitute " cubic " 

 for "parallelepipedic." 



2 According to Riemann, (op. cit., Cap. 3, § 3, see Nature vin., p. 36) 

 questions about the infinitely great are useless in respect of the interpre- 

 tation of Nature, but not questions about the infinitely small. Once it is 

 admitted, however, that we are compelled to deal with a greater range 

 than O-l-oo viz. U - l n - cx n such a statement is logically defenceless, 

 because magnitude is essentially relative. The interpretation of the 

 infinitely great may sometimes be difficult, but that does not justify the 

 dictum. 



3 Or n-ply extended magnitude. See Riernann's treatise, op. cit. Cap.- 

 1, § 1. 



