200 THE GRAMMAR OF SCIENCE 



Thus rigidity is very much like the spherical surfaces of 

 geometry. The latter correspond accurately to nothing 

 whatever in our perceptual experience, and we cannot even 

 conceive a continuous surface as a limit to be reached in 

 perception. Both, however, are alike valuable bases of 

 classification. By replacing real bodies by ideal rigid 

 bodies we are able, although neglecting their changes of 

 form, to • classify and describe a wide range of our per- 

 ceptions of motion. To classify other perceptions, how- 

 ever, we conceive the same bodies not to be rigid, but 

 to be varying in form ; we actually measure the very 

 changes in shape, which we purposely neglected in another 

 branch of our survey of the physical universe. 



^ 4. — On Change of Aspect or Rotation 



Even when we have transferred our moving body from 



the perceptual to the conceptual sphere by postulating its 



rigidity, we shall still find the notions of aspect and spin 



involve further geometrical conceptions. Let us consider 



our rigid body capable of turning about a point, the 



question then arises, How can we distinguish one aspect 



from a second ? Clearly, the notion of direction involves 



that of a line, but the change in direction in one \me will 



not be sufficient to describe change of aspect. For if C 



(Fig. 4) represent the fixed point about which the body 



rotates, and A be another definite point of the body, the 



line CA may take up a new position CA' ; but the change 



in position of CA to CA' does not fully determine the 



aspect of the body, for there is nothing to fix how much 



the body may have been turned about the line CA while 



it was moving into the position CA'. We are compelled, 



therefore, to take a second point B, and a second direction 



CB ; then if we state the new position CB' taken by CB 



as well as the new position CA' of CA, we shall have 



absolutely determined the change of aspect of the body. 



The reader will very easily convince himself that in giving 



the new positions of two definite points A and B of the 



rigid body we have absolutely fixed its position. It is 



