GEOMETRICAL AXIOMS. 65 



ever, decide by pure geometry, and without mechanical 

 considerations, whether the coinciding bodies may not 

 both have varied in the same sense. 



If it were useful for any purpose, we might with 

 perfect consistency look upon the space in which we 

 live as the apparent space behind a convex mirror with 

 its shortened and contracted background ; or we might 

 consider a bounded sphere of our space, beyond the 

 limits of which we perceive nothing further, as infinite 

 pseudospherical space. Only then we should have to 

 ascribe to the bodies which appear to us to be solid, and 

 to our own body at the same time, corresponding disten- 

 sions and contractions, and we should have to change 

 our system of mechanical principles entirely ; for even 

 the proposition that every point in motion, if acted upon 

 by no force, continues to move with unchanged velo- 

 city in a straight line, is not adapted to the image of 

 the world in the convex-mirror. The path would in- 

 deed be straight, but the velocity would depend upon 

 the place. 



Thus the axioms of geometry are not concerned 

 with space-relations only but also at the same time 

 with the mechanical deportment of solidest bodies in 

 motion. The notion of rigid geometrical figure might 

 indeed be conceived as transcendental in Kant's sense, 

 namely, as formed independently of actual experience, 

 which need not exactly correspond therewith, any more 



