What, then, are these limitations? We note in the first place that the laws of thermodynamics apply to bodies of a certain range of size; or at least the possibility of mathematical investigation (on which, of course, all depends) is limited to “differential elements” of mass, energy, and time. We cannot apply mathematical analysis to bodies, or time-intervals of “finite size,” since the methods of the differential and integral calculus would not strictly be applicable. But molecules are so small (1 cubic centimetre of a gas may contain about 5.4 × 10^{19} of them) that even such a minute part of a body, or liquid, or gas as approximates to the infinitesimally small dimensions required by the calculus, contains an enormous number of molecules. Obviously we cannot investigate the individual molecules. Even if experimental methods could be so applied, such concepts as density, pressure, volume, or temperature would have no meaning. Physics, then, is based on collections of molecules, and the properties of a body are not those of a molecule of the same body. Such concepts as temperature and pressure are _statistical_ ones, and are applied to the mean properties of a large number of molecules. [Illustration: FIG. 11.] We can best illustrate this by considering Maxwell’s famous fiction of the “sorting demons.” Let us imagine a mass of gas contained in a vessel the walls of which do not conduct heat. Let there be a partition in this vessel also of non-conducting material, and let there be an aperture in this partition greater in area than a molecule, but smaller than the mean free path of a molecule. Now this mass of gas has a certain temperature which is proportional to the mean velocity of movement of the molecules. The second law says that heat cannot pass from a cold region in a system to a hot region without work being done on the system from outside, nor can an inequality of temperature be produced in a mass of gas or liquid except under a similar condition. But “conceive a being,” says Maxwell, “whose faculties are so sharpened that he can follow every molecule in its course; such a being, whose attributes are still as essentially finite as our own, would be able to do what is at present impossible to us.”[20] For the temperature of the gas depends on the velocities of the molecules, and in any part of the gas these velocities are very different. Suppose that the demon saw a molecule approach which was moving at a much greater velocity than the mean: he would then open the door in the aperture and let it pass through from - to +. On the other hand, should a molecule moving at a velocity much less than the mean approach he would let it pass from + to -. In this way he would sort out molecules of high from those of low velocity. But the collisions between the molecules in either division of the vessel would continually produce diversity of individual velocity, and in this way the difference of