THE CONCEPTUAL WORLD 25 



which was compelled to live in a tube, the sides of 

 which it fitted closely, so that it could move only in one 

 direction up and down. A parasite, living attached 

 to some fixed object, and the movements of which were 

 represented only by the growth of its tissues, could 

 not form any idea of space ; and the " higher " forms 

 of geometry, that is, space of four or more dimensions, 

 present no clear notion to our minds, even although 

 we regard the operations included in mathematics 

 of this kind as pure symbolism, because we cannot 

 relate this imaginary space to any form of bodily 

 exertion. Geometry, then, represents the manner in 

 which our bodily exertion cuts up the homogeneous 

 medium in which \ve live. 



Motion, whether it be that of our own body in 

 controlled muscular activity, or that imaginary motion 

 of the environment which we call giddiness, or a sensibly 

 perceived motion of some part of the environment, 

 that is, a motion which we can compensate by some 

 actual or imaginary change in the position of our own 

 body produced by our own exertion, is an intuitively 

 felt change, and is incapable of intellectual representa- 

 tion. It is not clearly conceived either in ancient or 

 in modern geometry. Euclidean geometry is, as we 

 have seen, based directly on our intuition of bodily 

 exertion, but it is essentially static in treatment. 

 Let it be admitted that we can draw a straight line 

 of any length and in any direction, and so on ; then 

 we regard these straight lines, etc., as motionless, 

 abstract things, and we proceed to discuss their relation- 

 ships. Cartesian geometry, and the methods of the 

 infinitesimal calculus, do not treat of real motion, and 

 the concept, if it is introduced at all, is introduced 

 illegitimately and surreptitiously. Consider what we 

 do when we " plot a. curve." Let the latter be a 



