PKINCIPLE OF CONTINUITY IN THE THEORY OF SPACE. 251 



be regarded and understood as really a form, or rather a develop- 

 ment, of euclidean and projective geometry. 



As the subject is of interest to thinkers generally, the intro- 

 duction of the more specialised forms of mathematical expression 

 and symbol, have been studiously avoided. For the same reason 

 the conceptions have been set forth with more fulness than would 

 be deemed necessary by the professional mathematician, to whom 

 therefore apology is not required. 



1. The dimensionless point as generatrix. — The ultimate element 

 from which all geometrical figures may be generated is the dimen- 

 sionless point. Since conceptually this is without magnitude 1 and 

 may be said to be sensibly non-existent, its spatial dimension is 

 appropriately represented by that symbol which in number implies 

 negation, i.e. 0. Consequently a point, per se, may be defined as 

 an entity of zero-dimension. 2 Having no spatial dimensions, even 

 as we shall see, of zero-order, it nevertheless may serve to con- 

 ceptually define position. This type of point we may call the pure 

 or dimensionless point, to distinguish it from a zero or infinitesimal 

 quantity, possessing dimensional properties, which also is often 

 loosely called a point; and is really the monad of geometrical 

 figures. The use of abstract numbers, including 0, to define the 

 dimensions of space, can lead to no ambiguity, so long as the 

 sense in which they are employed is not permitted to vary. The 

 parallelism of dimensional significance in different conceptions of 

 space and spatial quantities, is an independent question. Where- 

 ever parallelism really exists, it may, in practical applications, 

 occasionally relieve to some extent the necessity for rigour of 

 thought ; but in theoretical considerations such relief can not bo 

 permitted. 



2. Summational generation impossible with dimensionless point. 

 — In considering the point as the generatrix of all geometrical 



1 Euclid's definition really exhausts the conception. The point has locus, 

 but no magnitude, not infinitesimal magnitude. 



2 An entity of zero-dimension is not to be confounded with a zero 

 quantity generally, nor is it to be assumed, even in abstract numbers, 

 that zeros are necessarily identical. 



