136 SCIENTIFIC METHOD IN PHILOSOPHY 



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 at instants that are also still nearer together. 

 The moving body never jumps from one position to 

 another, but always passes by a gradual transition through 

 an infinite number of intermediaries. At a given instant, 

 it is where it is, like Zeno's arrow ; l but we cannot say 

 that it is at rest at the instant, since the instant does not 

 last for a finite time, and there is not a beginning and 

 end of the instant with an interval between them. Rest 

 consists in being in the same position at all the instants 

 throughout a certain finite period, however short ; it does 

 not consist simply in a body's being where it is at a given 

 instant. This whole theory, as is obvious, depends upon 

 the nature of compact series, and demands, for its full 

 comprehension, that compact series should have become 

 familiar and easy to the imagination as well as to deliberate 

 thought. 



What is required may be expressed in mathematical 

 language by saying that the position of a moving body 

 must be a continuous function of the time. To define 

 accurately what this means, we proceed as follows. 

 Consider a particle which, at the moment /, is at the 



point P. Choose now any 

 P i P ^ 2 Q small portion P x P 2 of the 



path of the particle, this 

 portion being one which contains P. We say then 

 that, if the motion of the particle is continuous at the 



1 See next lecture. 



