PRINCIPLE OF CONTINUITY IN THE THEORY OF SPACE. 257 



only of two points, being fixed, in such a way that there will be 

 no spatial displacement of any part of it, the line "lies evenly 

 between its extreme points," i.e. it is uniquely straight. The 

 complete test of straightness requires that the line be free to 

 rotate about the line joining its terminals, the last being fixed : 

 then whatever its shape the mean position is a uniquely straight 

 line. 1 



The normal geometry of any number of dimensions presupposes 

 straight axes in the euclidean sense, i.e. that the space is homa- 

 loidal in every direction. When other suppositions are made the 

 geometry is merely specialised. 



4. Fluxional generation by means of diinensionless point. — The 

 generation of an ^-dimensional geometrical figure, in conceptual 

 space of the same number of dimensions, by continuous motion 2 or 

 displacement therein, of an (n - 1) dimensional generatrix, presents 

 no conceptual difficulty. 3 We may say that, in general, motion of 



1 In a two-dimensional space the line cannot remain thereon if it be 

 not plane : and if the line be not straight. Consider the rotation of a 

 segment of a small circle on a spherical surface ; rotation about its 

 terminals would give two positions on the surface the mean of which 

 would be a great circle thereon. The lengths of several segments, of 

 different radii, the radii of the circles of which they formed part, and the 

 distances between the opposite positions, would enable Helmholtz's 

 intelligent sphere-dwellers to ascertain the dimensions and form of their 

 world, and they could arrive at the theory of the unique straight line, of 

 a homaloid of two dimensions, and would know the phsenomena involved 

 the supposition of a curvature, the radius of which was (to them) 

 "imaginary." 



* According to Henrici it appears impossible to avoid the introduction 

 of the idea of motion in geometry. Would it not be more rigorous to 

 qualify this by adding "if it is to explain the development of geometrical 

 figures, consistently with the principle of continuity, with the diinen- 

 sionless point as generatrix." See Encyc. Brit. 9° Edit, x., 376. 



3 It may, however, be thought by some open to doubt whether psycho- 

 logically we are able to distinguish between successive apparitions at 

 dimensionless points infinitesimally separated, and continuous motion 

 along the range or path in which the points lie. It might be urged that 

 in representing the path of a continuously moving generatrix, a point 

 say, we leave as it were in imagination a trail of points in order to fix our 

 ideas, and that this exhausts the motion of a path, provided the points 

 are conceived to be separated by infinitesimal distances ; for — it may be 

 alleged — the infinitesimal marks the limit beyond which mental repre- 

 sentation fails, or we may conceive of continuity as apparent, as in the 

 kinetoscope. The fact that we can conceptually distinguish between, as 

 well as symbolically represent and operate upon, different classes of zeros,. 



0— Dec. 4, 1901. 



