Obviously the smaller the interval δs, the closer will be this approximation. Suppose, then, that we diminish δs till it “becomes zero.” It might appear now that when δs 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 therefore 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.
We therefore think of the stone as moving in the immediate vicinity of the point in the sense already discussed. We say that the “immediate vicinity” is an interval such that any point in it, p1, approximates to the arbitrary point p which we are considering within any standard of approximation: that is, no point in the interval is further away from p than a certain number expressing the standard of approximation, and this can be any number, however small. We say the same thing about the interval of time. That is to say, we make the intervals as small as we like: they can be smaller than any interval which will cause an error in our deduced velocity, no matter how small this error may be.
The limit of the velocity of a stone falling past a point in its path is, therefore, that velocity towards which the mean velocities approximate within any standard of approximation, when we regard the interval of space as being the immediate vicinity of the point, and the interval of time as being the time in the immediate vicinity of the moment when the stone passes the point. The limit of the velocity is not δs/δt but ds/dt, dt and ds being, not finite intervals of time and space, but “differentials.” We determine this limit by the methods of the differential calculus.
FREQUENCY DISTRIBUTIONS AND PROBABILITY
Let the reader keep a note of the number of trumps held by himself and partner in a large number of games of whist (the cards being cut for trump). In 200 hands he may get such results as the following:
No. of trumps in his own and partner’s hands—0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13.
No. of times this hand was held—0, 0, 0, 1, 9, 29, 53, 52, 35, 14, 6, 1, 0, 0.
He should note also the number of times that trumps were spades, clubs, diamonds, and hearts: he will get some such results as the following: spades, 46; clubs, 53; diamonds, 51; hearts, 50.