SPACE AND TIME 171 



fully cut and polished, and its faces appear at first as true 

 planes. But we find that a small body placed upon one 

 of its faces does not slide off when the cube is slightly 

 tilted. The face of the cube must, after all, be rough, 

 there are hollows and projections in it which catch those 

 of the superposed body ; our plane again appears delusive. 

 Or we may take one of Whitworth's wonderful metal 

 planes obtained by rubbing the faces of three pieces of 

 metal upon each other. Here again a powerful micro- 

 scope reveals to us that we are still dealing with a surface 

 having ridges and hollows. 



The fact remains, that however great the care we take 

 in the preparation of a plane surface, either a microscope 

 or other means can be found of sufficient power to show 

 that it is not a plane surface. It is precisely the same 

 with a straight line ; however accurate it appears at first 

 to be, exact methods of investigation invariably show it 

 to be widely removed from the conceptual straight line of 

 geometry. It is a race between our power of representing 

 a straight line or plane and our power of creating instru- 

 ments which demonstrate that the sameness and continuity 

 of the geometrical conceptions are wanting. Absolutely 

 perfect instruments could probably only be constructed if 

 we were already in possession of a true geometrical line 

 or plane, but the instruments we can make appear invari- 

 ably to win the race. Our experience gives us no reason 

 to suppose that luitJi any amount of care we could obtain a 

 perceptual straight line or plane, the elements of ^vJiich luould 

 on indefinite magnification satisfy the condition of ultimate 

 sameness involved in the geometrical definitiojis. We are 

 thus forced to conclude that the geometrical definitions 

 are the results of processes which may be started, but the 

 limits of which can never be reached in perception ; they 

 are pure conceptions having no correspondence with any 

 possible perceptual experience. What we have said of 

 straight lines and planes holds equally of all geometrically 

 defined curves and surfaces. The fundamental conceptions 

 of geometry are only ideal symbols which enable us to 

 form an approximate, but in no sense absolute, analysis 



