HERMANN VON HELMHOLTZ 



conceivable circumstances. The Euclidean method 

 of proof is to establish the congruence of lines, 

 angles, plane figures, solids, etc., and this is done 

 by applying the one figure to the other, without 

 changing their form or dimensions. But the method 

 of establishing congruence implies mechanical move- 

 ments, and by mechanical movements we acquire 

 experience. If so, every proof of congruence rests 

 upon experience. 



To illustrate this point of view, Helmholtz imagines 

 beings with perceptions like our own living in worlds 

 differing from our own. The mind can readily con- 

 ceive beings of two dimensions living on a plane 

 surface, and so confined to it, that they had no 

 power of perceiving anything outside this sur- 

 face. The geometry of such beings would show 

 that the movement of a point described a line ; and 

 that of a line described a surface j but they could 

 not even imagine the form produced by the surface 

 moving out of itself, so as to describe say a 

 sphere or a cone. Living on an infinite plane, 

 their geometry would be like planimetry. Again, 

 we can conceive of intelligent beings living not on 

 a plane but on the surface of a sphere. Their 

 shortest line would then be an arc of the great 

 circle passing through its ends. On a plane there 

 could be only one shortest line between any two 

 points, and in general this is also true of two points 

 on a spherical surface, except when the two points 

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