PRINCIPLE OF CONTINUITY IN THE THEORY OF SPACE. 295 



in space, than that in which it originally existed. Similarly if 

 the coordinates of P be x, 0, 0, and of S, \x, y = 0, z = 0, w = h, 

 w being at right angles to each of the other dimensions, rotation 

 will cause the curvature to appear in the xy and xz planes, (See 

 § 11), so that h can be measured, although (supposititiously) it was 

 unrevealed while in the fourth dimension. 



If it be objected that the curvature appears because the plane, 

 vertical to the line x, is the yz plane, it may be remarked that w 

 is in every sense as much at right angles to the xy plane, as to the xz, 

 and if, on rotating, the position of the line does not vary, whatever 

 curvature in the tv axis may mean, it is something which cannot, 

 as evidenced by that test, affect the angles of inclination in 

 tridimensional space: which is equivalent to affirming that the 

 supposititious curvature has no existence. 



It is now obvious that tridimensional elliptic and hyperbolic 

 space cannot "actually" exist, if a line joining any two points can 

 be rotated in a plane perpendicular thereto without varying its 

 position in a surface perpendicular to the line. This is essentially 

 identical with the following proposition: — A line is uniquely or 

 absolutely straight? if, when orthogonally projected in tridimen- 

 sional space on any two planes perpendicular to one another, its 

 projection in both is a straight line. 



Mathematicians who allege that our space may be "curved 

 space," do not imply that it is "visibly curved" in any plane, but 

 that the curvature would appear in the "excess" or "defect" of 

 the three angles of a triangle, with respect to the homaloidal value 

 7r, a fact inexplicable in homaloidal space, and demanding for its 

 interpretation the assumption of a curvature in what to us is an 

 imaginary dimension of space. If visibly curved, the line is 

 neither a straight line, nor the shortest distance between the 

 points. 2 We may conclude therefore that geometrically, elliptic 



1 Straight in every spatial dimension. 



2 It is perhaps necessary to say that it is not pretended that lines are 

 merely refracted, we refer to this later. 



