THE THEORY OF CONTINUITY 135 



The difficulty to imagination lies chiefly, I think, in 

 keeping out the suggestion of infinitesimal distances and 

 times. Suppose we halve a given distance, and then 

 halve the half, and so on, we can continue the process as 

 long as we please, and the longer we continue it, the 

 smaller the resulting distance becomes. This infinite 

 divisibility seems, at first sight, to imply that there are 

 infinitesimal distances, i.e. distances so small that any 

 finite fraction of an inch would be greater. This, how- 

 ever, is an error. The continued bisection of our distance, 

 though it gives us continually smaller distances, gives us 

 always finite distances. If our original distance was an 

 inch, we reach successively half an inch, a quarter of an 

 inch, an eighth, a sixteenth, and so on ; but every one 

 of this infinite series of diminishing distances is finite. 

 " But," it may be said, " in the end the distance will grow 

 infinitesimal." No, because there is no end. The 

 process of bisection is one which can, theoretically, be 

 carried on for ever, without any last term being attained. 

 Thus infinite divisibility of distances, which must be 

 admitted, does not imply that there are distances so small 

 that any finite distance would be larger. 



It is easy, in this kind of question, to fall into an 

 elementary logical blunder. Given any finite distance, 

 we can find a smaller distance ; this may be expressed in 

 the ambiguous form " there is a distance smaller than any 

 finite distance." But if this is then interpreted as mean- 

 ing " there is a distance such that, whatever finite distance 

 may be chosen, the distance in question is smaller," then 

 the statement is false. Common language is ill adapted 

 to expressing matters of this kind, and philosophers who 

 have been dependent on it have frequently been misled 

 by it. 



In a continuous motion, then, we shall say that at any 



