THE AIM AND ACHIEVEMENTS OF SCIENTIFIC METHOD. 103 



none of which are present in the " resultant." The apparent 

 further difficulty that it would be, in practice, impossible to 

 effect the vector summation of the " notice " taken by a particle 

 of all the other particles in the universe, disappears when we 

 adopt a formula, like Newton's Law of Gravitation, which 

 makes the rate of change of momentum exhibited by two 

 particles at a given moment a function of the distance between 

 them of such a character that for great distances it becomes 

 negligible. 



45. 



It is of more importance to note with Mr. Russell,* that in 

 the present state of mathematical / theory, it is no longer 

 possible to think of a particle as possessing a " velocity " 

 or an "acceleration," or (consequently) a rate of change 

 of momentum. Beturning to our single pair of particles^- 

 A and B, moving towards one another along a straight line, the 

 Objective facts are that at the moments ti, t 2 , and t s , A 

 occupies the points PI, P 2 , and P 3 ; while B occupies the points 

 pi, p2, and p s . Now suppose that? r seconds after A leaves the 

 point PI, it occupies a point between PI and P 2 distant B from PI. 

 Then the fraction S/r will have a definite numerical value 

 which it is customary to call the average velocity of A 

 during the time r. It is clear that if we make this time 

 shorter repeatedly we shall obtain a compact series of these 

 fractions. Under these circumstances it will be possible, as a 

 rule, to specify a number, Vi, which is the limit of this series of 

 fractions ; that is such a number that it is impossible that any 

 number should lie between it and the compact series of fractions. 

 This number is the velocity of the mass particle at the point 

 PI. Thus defined, it is obvious that the velocity is nothing 

 that can be thought of as a state of A. If in a similar way the 

 velocity of the particle B at the point p\ be supposed to be deter- 

 mined, and to have the value v\ t then the doctrine of mass which 



* Op. cit., p. 473. 



