because this is the quantity for which the law analogous to conservation holds. And when the velocity is very great, it gives a better measure of energy than the traditional formula. The traditional formula must therefore be regarded as an approximation, of which the new formula gives the exact version. In this way, energy and measured mass become identified. I come now to the notion of “action,” which is less familiar to the general public than energy, but has become more important in relativity physics, as well as in the theory of quanta.[10] (The quantum is a small amount of action.) The word “action” is used to denote energy multiplied by time. That is to say, if there is one unit of energy in a system, it will exert one unit of action in a second, 100 units of action in 100 seconds, and so on; a system which has 100 units of energy will exert 100 units of action in a second, and 10,000 in 100 seconds, and so on. “Action” is thus, in a loose sense, a measure of how much has been accomplished: it is increased both by displaying more energy and by working for a longer time. Since energy is the same thing as measured mass, we may also take action to be measured mass multiplied by time. In classical mechanics, the “density” of matter in any region is the mass divided by the volume; that is to say, if you know the density in a small region, you discover the total amount of matter by multiplying the density by the volume of the small region. In relativity mechanics, we always want to substitute space-time for space; therefore a “region” must no longer be taken to be merely a volume, but a volume lasting for a time; a small region will be a small volume lasting for a small time. It follows that, given the density, a small region in the new sense contains, not a small mass merely, but a small mass multiplied by a small time, that is to say, a small amount of “action.” This explains why it is to be expected that “action” will prove of fundamental importance in relativity mechanics. And so in fact it is. [10] On this subject, see the present author’s _A.B.C. of Atoms_, chaps. VI and XIII. All the laws of dynamics have been put together into one principle, called “The Principle of Least Action.” This states that, in passing from one state to another, a body chooses a route involving less action than any slightly different route—again a law of cosmic laziness. The principle is subject to certain limitations, which have been pointed out by Eddington,[11] but it remains one of the most comprehensive ways of stating the purely formal part of mechanics. The fact that the quantum is a unit of action seems to show that action is also fundamental in the empirical structure of the world. But at present there is no bridge connecting the quantum with the theory of relativity. [11] _Op. cit._ § 60. CHAPTER XI: IS THE UNIVERSE FINITE?