21. But, on the other hand, the energy is not simply proportional to the velocity, for, if it were, the energy of the rifle stock and of the rifle ball would be the same, inasmuch as the rifle stock would gain as much by its superior mass as it would lose by its inferior velocity. Therefore, the energy of a moving body increases with the velocity more quickly than a simple proportion, so that if the velocity be doubled, the energy is more than doubled. Now, in what manner does the energy increase with the velocity? That is the question we have now to answer, and, in doing so, we must appeal to the familiar facts of everyday observation and experience. 22. In the first place, it is well known to artillerymen, that if a ball have a double velocity, its penetrating power or energy is increased nearly fourfold, so that it will pierce through four, or nearly four, times as many deal boards as the ball with only a single velocity--in other words, they will tell us in mathematical language, that the energy varies as the square of the velocity. _Definition of Work._ 23. And now, before proceeding further, it will be necessary to tell our readers how to measure work in a strictly scientific manner. We have defined energy to be the power of doing work, and although every one has a general notion of what is meant by work, that notion may not be sufficiently precise for the purpose of this volume. How, then, are we to measure work? Fortunately, we have not far to go for a practical means of doing this. Indeed, there is a force at hand which enables us to accomplish this measurement with the greatest precision, and this force is gravity. Now, the first operation in any kind of numerical estimate is to fix upon our unit or standard. Thus we say a rod is so many inches long, or a road so many miles long. Here an inch and a mile are chosen as our standards. In like manner, we speak of so many seconds, or minutes, or hours, or days, or years, choosing that standard of time or duration which is most convenient for our purpose. So in like manner we must choose our unit of work, but in order to do so we must first of all choose our units of weight and of length, and for these we will take the _kilogramme_ and the _metre_, these being the units of the metrical system. The kilogramme corresponds to about 15,432·35 English grains, being rather more than two pounds avoirdupois, and the metre to about 39·371 English inches. Now, if we raise a kilogramme weight one metre in vertical height, we are conscious of putting forth an effort to do so, and of being resisted in the act by the force of gravity. In other words, we spend energy and do work in the process of raising this weight. Let us agree to consider the energy spent, or the work done, in this operation as one unit of work, and let us call it the _kilogrammetre_.