The idea connoted by the term weight is the pull which the earth exerts on the mass of a body; thus, when we say that an iron ball weighs six pounds, we mean that the earth pulls it downwards with a force equal to the weight of six pounds. That the weight of a given lump of matter is not a constant but a dependent quantity may be seen from a number of considerations. Its weight in vacuo, for instance, is different from its weight in air, and this latter differs considerably from its weight in water or in oil. Again, if we take our experimental ball down the shaft of a mine, the spring-balance used to measure the pull of the earth on it will not record six pounds but something less; and the further we descend, the less will the spring-balance be found to register. At a depth of two thousand miles below the surface, the ball would be found to have lost half its weight; and at a depth of four thousand, all its weight. At the earth's center a "box of weights" would still be called a "box of weights," though neither the box itself nor its enclosed standards singly or collectively would have any weight whatever. It has been shown experimentally that two masses weigh slightly less when placed one above the other than when placed side by side, because in the latter case their common mass-center is measurably nearer to the center of the earth. Every mother knows that when a boy is sent to buy a pound of candy, it is the mass of the sweet stuff that makes him happy, and not its weight, for this acts more like an incumbrance while he is bringing it home. Of course, weight is every day used, and correctly, as a measure of mass, for every student of mechanics writes without the least hesitation,
W=Mg.
by which he simply means that the weight of a body is directly proportional to its mass (M), which is constant wherever the body may be taken, and to the intensity of gravity (g), which varies slightly with geographical position. As both scale-pans of an ordinary balance are equally affected by the local value of g, it follows that equilibrium is established only when the two masses—that of the body and that of the standards—are themselves equal: hence weighing is in reality only a process of comparing masses, i.e., a process of "massing."
If we bring our experimental ball to the top of a hill or to the summit of a mountain or aloft in a balloon, we find the pull on the registering spring growing less and less as we go higher and higher, from which we naturally conclude that if we could go far enough out into circumterrestrial space, say, towards the moon, the ball would lose its weight entirely; it would cease to stretch the spring of the measuring balance, its weight vanishing at a definite, calculable distance from the earth's center. If carried beyond that point the ball would come under the moon's preponderating attraction and would begin to depress anew the index of the balance until at the surface of our satellite it would be found to weigh exactly one pound. If transferred to the planet Mars the ball would weigh two pounds, and if to the surface of the giant planet Jupiter, sixteen pounds. But while its weight thus changes continually, its mass or quantity of matter, the stuff of which it is made, remains constant all the while, being equally unaffected by such variables as motion, position or even temperature.
Returning from celestial space to our more congenial terrestrial surroundings, we find a similar inconstancy in the weight of the ball as we travel from the equator toward either pole, the weight being least at the equator and slightly greater at either end of our axis of rotation. This change is fully accounted for by the spheroidal figure of the earth and its motion of rotation, in virtue of which, while going from the equator toward the pole, our distance from the center of attraction undergoes a slight diminution, as does also the component of the local centrifugal force, which is in opposition to gravity.
From all this, it will be seen that the weight of a body is more of the nature of an accidental rather than an essential property of matter, whereas its mass is a necessary and unvarying property. Hence we speak with propriety of the conservation of mass just as we speak with equal propriety of the conservation of energy; but we may never speak or write of the conservation of weight. The mass of our iron ball is precisely the same away from the surface of the earth as it is anywhere on the surface, whether a thousand miles below the surface or a thousand miles above it; and the same it would be found in any part of the solar system or of the starry universe to which it might be taken.
Since weight is nothing else than the pull which the earth exerts on a body, it follows that, big and massive as our planet is, it must, nevertheless, be weightless; for it cannot with any degree of propriety be said to pull itself. It is incapable of producing even an infinitesimal change in the position of its mass-center, or center of gravity, as this centroid is sometimes called. The earth attracts itself with no force whatever; but is attracted and governed in its annual movement by the sun, the central controlling body of our system, while the moon and planets play only the part of petty disturbers.
It would, however, be right to speak of the weight of the earth relatively to the sun; for the sun attracts the mass of our planet with a certain definite force, readily calculable from the familiar formula for central force, viz., mv2/r., in which m is the mass of the earth, v its orbital velocity and r its distance from the sun. Supplying the numbers, the weight of the earth relatively to the sun, comes out to be
3,000000,000000,000000 or 3×10^{18} tons weight,
or, in words, three million million million tons weight.