PRINCIPLE OF CONTINUITY IN THE THEORY OF SPACE. 253 



subject of generation. Consequently, the dimensions of geometrical 

 figures are limited to the conceivable dimensions of the space in 

 which they may be supposed to exist, but can never transcend it. 1 



The rigorous conception of the pure or dimensionless point is 

 that it can exist in space, can define position, but is itself absolutely 

 without spatial magnitude. 2 



This is brought into conceptual relief by considering that the 

 interpolation of any number of points, represented by the symbol 

 oo , between two terminals, separated, let us suppose, by a unit 

 distance in linear or 1-dimensional space, leaves oo + 1 linear 

 elements or parts of the unit, each susceptible, conceptually, of 

 similar subdivision, no matter how great the number may be. If 

 we suppose oo to denote an infinite number 3 the concept remains 

 unchanged, for there is nothing whatever to suggest that as the 

 number increases there is any tendency to generic change. Thus 

 we may say there are oo + 1 infinitesimal linear-elements divided 

 by oo dimensionless points. It is not the summation of the 

 infinite series of points 4 which constitutes the unit-line, but the 

 summation of the infinite number of infinitesimal lines; 5 which 

 those points conceptually separate; the essential difference between 

 these two ideas being that in the one case there is a discrete series 

 or assemblage of dimensionless points, infinitesimally approximating 

 to continuity, i.e. conceptually discontinuous, in the other there 

 is a series of infinitesimal but not dimensionless lines, schematically 



1 See Hermann Schubert — The fourth dimension. Monist, Vol. in., 

 pp. 402-449, April 1893. Also, On the nature of mathematical know- 

 ledge. Monist, Vol. vi., pp. 294 - 305, 1895-6. 



3 Point-contact, is therefore coincidence, if the points be dimensionless. 

 Kant, 'Eaum und Zeit sind quanta continua. . . . Der Eaum besteht also 

 nur aus Eaumen. . . . Punkte sind . . . nur Grenzen, d. i. blosse stellen 

 ihrer Einschraukung ; stellen aber setzen jederzeit jene Anschauungen, 

 die sich beschranken oder bestimmen sollen, voraus, und aus blossen 

 stellen, als aus Bestandteilen, die noch vor dem Eaume oder der Zeit 

 gegeben werden konnten, kann weder Eaum noch Zeit zusammengesetzt 

 werden/ Kritik d. rein. Vernunft. Elementarlehre, Buch n., 2, 3, § 2. 



s Greater than any assignable or finite number. 



1 See Kant, preceding footnote. 



5 Cavalieri's indivisibles, must be conceived as schematically indivisible, 

 rather than as conceptually indivisible. 



