320 INTRODUCTION TO PHILOSOPHY OF SCIENCE 



latter of which is essentially involved in the notion of 

 "time." Hence, motion may be defined as the correlation of 

 two continuous linear series, as these notions were defined in 

 Chapter XIII, one of which is space and the other of which 

 is time. But not every correlation of space and time is 

 motion. By virtue of the notion of particle, by which the 

 correlation is achieved, certain types are excluded. Ab- 

 stractly, four general types would be possible: taking the 

 two series in the order space-time, the correlation of the 

 points of the former with the instants of the latter may be 

 many-one, many-many, one-one, or one-many. But by 

 virtue of the definition of "particle" the first two are impossi- 

 ble; a particle cannot be at two different points at the same 

 instant. A many-one correlation of space and time defines 

 instantaneous extension, and a many-many correlation de- 

 fines enduring extension. The one-one and the one-many 

 correlations then define, respectively, motion and rest. A 

 particle is said to be in motion at an instant if it is possible 

 to find an interval of time including that instant such that 

 at every instant within the interval the particle is at a dif- 

 ferent point in space. A particle is said to be at rest at an 

 instant if it is possible to find an interval of time including 

 that instant such that at every instant within the interval 

 the particle is at the same point in space. 1 



Two features of this definition may be called to attention. 

 The first is the fact that a correlation of this kind gives the 

 desired continuity to motion. "In a continuous motion . . . 

 at any given instant the moving body occupies a certain 

 position, and at other instants it occupies other positions; 

 the interval between any two instants and between any two 

 positions is always finite, but the continuity of the motion is 

 shown in the fact that, however near together we take the 

 two positions and the two instants, there are an infinite 

 number of positions still nearer together, which are occupied 



1 Compare Russell's definitions of these notions in his Principles of Mathematics 

 (pp. 472-473). Difficulties with regard to end-points, i.e., points of change from 

 motion to rest, or from rest to motion, can be handled by introducing the proper 

 technicalities into the definition. 



