APPENDIX 34& 



closer and closer to the velocity near the middle portion 

 of the path. In this way we find the^ ratio ^, where 



8s is a very small interval of path containing the point 

 chosen, but not as an end-point, and St is a very small 

 interval of time. Perhaps this average velocity may 

 be near enough for our purposes, but perhaps it may 

 not. The interval of path Ss is still a finite interval, 

 and St is still a finite time, and so long as these values 

 are finite ones the velocity deduced from them remains 

 a mean one. All that we can say is that it approxi- 

 mates to the velocity, as the arbitrary point was passed, 

 within a certain standard of approximation. 



Obviously the smaller the interval Ss, the closer will 

 be this approximation . Suppose , then , that we diminish 

 8s till it " becomes zero." It might appear now that 

 when Ss coincides with the point chosen we shall 

 obtain the velocity of the stone at this point. But if 

 there is no interval of path, and no interval of time, 

 there can be no velocity, which is an interval of path 

 divided by an interval of time ; and if the stone is " at 

 the point," it does not move at all. We must stick to 

 the idea of intervals of space and time, and yet we must 

 think of these intervals as being so small that no error 

 whatever is involved in regarding the mean velocity 

 deduced from them as the " true velocity." We there- 

 fore think of the point as being placed in an interval 

 of path, but not at an end-point of this interval. We 

 think of the velocity as a mean one, but we must have 

 a standard of approximation, so that we may be able 

 to say that the mean velocity approximates to the 

 " actual " or limiting velocity of the stone as it passes 

 the point, within this standard of approximation. 

 The smaller we make the interval, the closer will the 

 mean velocity approximate to the limiting velocity. 



