THE GEOMETRY OF MOTION 237 



planetary theory is a striking example of the value of 

 such a synthesis. 



In this chapter we have seen how the relative position, 

 velocity, and acceleration of points may be defined, de- 

 scribed, and measured. We have been gleaning wholly 

 in the conceptual field of geometrical ideals. We have 

 next to ask how these conceptions may be applied to 

 describe our perceptual experience of change in the world 

 of phenomena. How are these three factors, position, 

 velocity, and acceleration, related to each other in that 

 ideal dance of corpuscles to which we reduce the physical 

 universe, in that atomic waltz by aid of which we describe 

 and assume our sense - impressions ? How do we con- 

 ceive the relative position of these corpuscles to change ? 

 How are their speeds and directions of motion varying ? 

 Does experience show us that relative position produces 

 a definite speed, or a definite spurt and shunt ? The 

 answer to these questions lies in the so-called properties 

 of matter and in the laws of motion which will be the 

 topics of our two following chapters. 



SUMMARY 



1. All the notions by aid of which we describe and measure change are 

 geometrical, and thus are not real perceptual limits. They are forms dis- 

 tinguishing and classifying the contents of our perceptual experience under 

 the mixed mode of motion. The principal of these forms are point-motion, 

 spin of a rigid body and strain. Motion is found to be relative, never 

 absolute ; for example, it is meaningless to speak of the motion of a point 

 without reference to what system the motion of the point is considered with 

 regard to. 



2. An analysis of point-motion leads us to the conceptions of velocity and 

 acceleration, the first as a proper measure of the manner in which position is 

 instantaneous changing, the second as a proper measure of how velocity itself 

 is changing. It is found that a motion is fully determined, or theoretically 

 a complete description of the path and position at each instant of time may 

 be deduced, when the velocity in any one position and the acceleration for 

 all positions are given. 



3. The parallelogram law as the general rule for combining motions is 

 the foundation of the synthesis by which complex motions are constructed out 

 of simple motions. 



