Suppose that we wish to find the rate of variation of volume for a pressure change in the immediate vicinity of the value b1, that is, the rate of variation as the pressure changes from a little less than b1 to a little more than b1. If we find the point b on the curve corresponding to b1, and if we then draw a line ff1, touching the curve at the point b, we shall obtain the angle off1. It might appear now that the tangent of this angle, that is, the ratio of1/of, would give us a measure of the rate of variation of volume.
But the reasoning would be faulty. The line ff1 only touches the curve, it does not coincide with an element of the curve. Also at the point b1 the pressure has a certain definite value, and there is no change. At the corresponding point b11 the volume also has a certain definite value, and there is no change. There can therefore be no rate of variation. The value of the tangent does not give us a measure of the rate of variation: it gives us the limit to the rate of variation, when the pressure is changing in the immediate vicinity of b1.
We must stick to the notion of a pressure change in the immediate vicinity of b1. What do we mean by “immediate vicinity”? We mean that we are thinking of a range of pressure-values in which the particular pressure-value b1 is contained, but not as an end-point. We mean also that we choose a definite standard of approximation to the value b1, so that any pressure-value within our interval differs from b1 by less than this standard of approximation. It means further that, no matter how small is the number representing this standard of approximation, any pressure-value within the interval will differ from b1 by less than this number. This is what we really mean when we say that the interval we are thinking about is an “infinitely small one.”
Now corresponding to this interval of pressure-values in the immediate vicinity of b1, there will be an interval of volume-values in the immediate vicinity of b11, and, as before, any one of these volume-values will differ from b11 by less than any number representing a standard of approximation to b11. We then find the point on the curve corresponding to both b1 and b11, that is b, and we draw the line ff1, and find the tangent of the angle which this line makes with op. The value of this tangent is the limit of the rate of variation of the volume of the gas when the pressure undergoes a change in the immediate vicinity of b1.
“Rate of variation” is a function of the argument “pressure.” This function has the limit l for a value of its argument b1, when, as the argument varies in the immediate vicinity of b1, the value of the function approximates to l within any standard whatever of approximation.[36]
We should not, of course, find the rate of variation of volume of the gas by this means. We should calculate the value of the differential co-efficient dv/dp from the equation pv=k(1 + at): this would be −k 1 + at/p2. But the reasoning involved in the methods of the calculus are those which we have attempted to outline. We try to avoid the terms “infinitely small,” “infinitely near,” “infinitely small quantities,” and so on, by the device of standards of approximation. It may appear to the non-mathematical reader that all this is rather to be regarded as “quibbling,” but the success of the methods of mathematical physics should convince him that such is not the case. He should also reflect that clear and definite ideas on the fundamental concepts of the science are just as necessary in speculative biology as they are in mathematics.
(Another example.)
Let us consider the case of a stone failing from a state of rest. Observations will show that when the stone has fallen for one second it has traversed a space of 16 feet; at the end of two seconds it has fallen through 64 feet; and at the end of three seconds the space traversed is 144 feet. From these and similar data we can deduce the velocity of motion of the stone as it passes any point in its path.
The velocity is the space traversed in a certain time s/t. If we take any easily observable space (say five feet) on either side of the point chosen, and then determine the times when the stone was at the extremities of this interval, and divide the interval of space by the interval of time, we shall obtain the average velocity of motion of the stone over this fraction of the whole path chosen. But the velocity did not vary in a constant manner during this interval (as we see by considering the spaces traversed during the first three seconds of the fall). Therefore our average velocity does not accurately represent the velocity of the stone as it passes the point at the middle of the path chosen.
We therefore reduce the length of the path more and more so as to make the average velocity approximate closer and closer to the velocity near the middle portion of the path. In this way we find the ratio δs/δt, where δs is a very small interval of path containing the point chosen, but not as an end-point, and δt 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 δs is still a finite interval, and δt 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 approximates to the velocity, as the arbitrary point was passed, within a certain standard of approximation.