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_)/_p_^2. 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 δ_s_/δ_t_, where δ_s_ is a very small interval of path containing the point chosen, but not as an end-point, and δ_t_ 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 δ_s_ is still a finite interval, and δ_t_ 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.