294 . G. H. KNIBBS. 



sentation — consequently the lines, if really bent, — as indicated 

 by arrows — are not bent in the third dimension of space, but if at 

 all in some other dimensions. 1 These others, being conceptually 

 perpendicular to each of the other three, have, it might be sup- 

 posed, the effect of altering the plane angle from what it would 

 have been, had the lines been uniquely straight. If such a view 

 be correct, uniquely straight lines must be straight with respect 

 to all possible dimensions, because the effect, if any, will be the 

 same if the curvature be in any one of the supposititious n 

 dimensions including the first three. 



Curved space, to affect the angles of a triangle must, however, 

 obviously be curved in all dimensions, since its radius of curvature 

 will be / N i ,~a\ 



and if any quantity is infinite, p is infinite, unless some other 

 quantity is zero, a case the consideration of which can be set aside 

 as unnecessary. 2 Consequently even though ri-dimensional space is 

 only the boundary of a space of (n+\) dimensions, if any dimen- 

 sion is without curvature, angles between geodesies in that space 

 are not affected and will total 7r (n ± 2). Hence the tests 

 proposed to be applied by Lobatchewsky, Helmholtz, and others, 

 would fail, if space is not curved in each dimension of space. 



The invalid nature of the supposition that "actual" space may 

 possibly be curved, may be further realised as follows: — With O 

 as origin, let the values of the coordinates of the terminal P of a 

 line OSP, Fig. 23, be x, 0, ; S being the middle point. If the 

 line be curved in any direction, in the plane xy let us suppose, 

 that is if it be bent in the direction + y only, it will on rotating 

 be bent successively in the directions +z, —y, - z, that is to say 

 the existence of the curvature will be revealed in other directions 



1 That is to say, Fig. 22 cannot be a picture of the lines, since the 

 difference between the curves and the lines AB, BC etc., are in the 

 supposititious dimensions. 



2 In the cylinder and cone the surface is straight in one direction 

 curved at right angles thereto. Hence p = ^{p^ ) = oo : consequently 

 the angles are not affected. 



