THE GEOMETRY OF MOTION 233 



" mechanical determinism " of nature ; wrongly interpreted, 

 it is the foundation of that crude materialism which 

 pictures the universe as an aggregate of objective material 

 bodies, enforcing for all eternity certain motions on each 

 other, and a perception of those motions upon us. What 

 the proposition exactly tells us is this : that a motion is 

 fully determined, that is, can be conceptually described, 

 either by giving the path and the time to each position of 

 the path, or by giving the velocity in any one position 

 and the acceleration in all positions. We are really 

 dealing with two different modes of describi7ig motion, 

 either of which can be deduced from the other, but neither 

 of which explains why the motion takes place, or can be 

 said to " determine " it in the sense of the materialists. 



S 17, — The Relativity of Motion. Its Synthesis from 

 Simple Co mponen ts 



There still remains a matter to which it is needful to 

 draw the reader's attention. The whole motion of our 

 point P (Fig. 9, p. 210) has been considered relative to a 

 point O and a particular frame. We started with a position 

 relative to O, and it follows that the velocity and acceler- 

 ation we have been discussing describe changes of motion 

 relative to O and its frame also. Thus absolute velocity 

 and absolute acceleration are seen to be as meaningless as 

 absolute position. If the points O and P were botJi to 

 have their motions accelerated in the same manner the 

 relative path would not be changed — any more than the 

 map (Fig. 9) is changed by our moving about, in any 

 manner we please, the page on which it is printed. But 

 the fact that all motion is relative leads us at once to the 

 very natural question : How are we to pass from the 

 motion of a point relative to one pole O to motion 

 relative to a second pole O', the bearing being measured 

 with regard to the same frame. We must look at this 

 point somewhat closely, for it involves some important 

 consequences. 



Let us suppose the motion of P relative to O known, 



