THE CONCEPTUAL WORLD 19 



and right and left, occurs; and we have to recognise 

 that in it there is something fundamental, as funda- 

 mental as the intuitive knowledge that we possess of 

 the direction of right and left. It is because we can 

 move in such a way that any of our motions, no 

 matter how complex, can be resolved into the com- 

 ponents of backward and forward, right and left, 

 and up and down, these directions all being at right 

 angles to each other, that we speak of our movements 

 as three-dimensional ones. Our geometry is founded, 

 therefore, on concepts derived from our modes of 

 activity ; and there is nothing in the universe, apart 

 from our own activity, that makes this the only 

 geometry possible to us. Euclidean geometry does 

 not depend on the constitution of the external universe, 

 but on the nature of the organism itself. 



There is a little Infusorian which lives, in its adult 

 phase, on the surface of the spherical ova of fishes. 

 These ova float freely in sea water, and the Infusorian 

 crawls on their surfaces, moving about by means of 

 ciliary appendages. It does not swim about in the 

 water, but adheres closely to the surface of the ovum 

 on which it lives. Let us suppose that it is an in- 

 telligent animal and that it is able to construct a 

 geometry of its own ; if so, this geometry would be 

 very different from our own. 



It would be a two-dimensional geometry, for the 

 animal can move backward and forward, and right 

 and left, but not up and down ; it is a stereotropic 

 organism, as Jacques Loeb would say, that is, it is 

 compelled by its organisation to apply its body closely 

 to the surface on which it lives. But its two-dimen- 

 sional geometry would, on this account, be different 

 from ours. Our straight lines are really the directions 

 in which we move from one point to another point in 



