266 G. H. KNIBBS. 



off by boundaries, does not necessarily imply discontinuity. 1 The 

 point relations corresponding to (4) become simply 

 Magnitudes of Generated Quanta. 



Summa- 

 tion. 



(1) 



(2) 



(3) 



(4) 



etc. 



1 



O 1 



O 2 



3 



4 



etc- 



00 



l 1 



O 1 



2 



3 



etc. 



oo 2 



oo 1 



l 2 



O 1 



O' 2 



etc. 



oo 3 



oo 2 



oo 1 



l 3 



O 1 



etc. 



oo 4 



oo 3 



oo 2 



oo ] 



l 4 



etc. 



etc. 



etc. 



etc. 



etc. 



etc. 







.(5) 



which completely defines the purely numerical relationships of 

 different dimensions, their infinities and infinitesimals. That mere 

 numerical analyses of spatial properties are liable to lead to 

 erroneous conclusions, is obvious on comparing (5) with (4). Thus 

 if with Helmholtz, Lie, Poincare, etc., we define the matter of 

 ordinary geometry a 'numerical multiplicity' or 'manifoldness' 

 ( Zahlenmannigfaltigkeil) 2 of three dimensions, it ought to be 

 understood that this applies only to its formal or purely numerical 

 representation, and not to its spatial properties which are to be 

 interpreted therefrom. 



In order to guard against erroneous deductions, this distinction 

 between spatial quantities and mere numbers, i.e. between vectors, 

 rotors, etc., and scalars and abstract numbers generally, should be 

 carefully preserved. 



Remembering that the geometrical figure l 4 yzw may be com- 

 pletely written l 4 whenever the particular axes do not need 

 specification, the quanta X, Y, Z, W,...N, of regular figures, 

 parallelograms, parallelepipeds, etc., whose sides parallel to the 

 several axes may be expressed by the numbers x, y, z, w...etc. t 



1 Biemann distinguishes between continuous or discrete manifoldness, 

 calling the spatial specialisations of the former points, and of the latter 

 elements; Op. cit. 1, § 1. In English is not the reverse use preferable? 

 A discrete manifold is strictly not space, but a distribution in space. 



2 In general, number imperfectly represents quantity. The science of 

 number, says Clifford, is founded on the hypothesis of the distinctness of 

 things ; but the science ot quantity on the totally different hypothesis 

 of continuity. Lectures 1, 337. 



