In the case of scales, the standard “weights” are altered just as much as the body to be weighed, so that the result is the same everywhere; but the result is the “mass,” not the “weight.” A standard “weight” has the same mass everywhere, but not the same “weight”; it is in fact a unit of mass, not of weight. For theoretical purposes, mass, which is almost invariable for a given body, is much more important than weight, which varies according to circumstances. Mass may be regarded, to begin with, as “quantity of matter”; we shall see that this view is not strictly correct, but it will serve as a starting point for subsequent refinements. For theoretical purposes, a mass is defined as being determined by the amount of force required to produce a given acceleration: The more massive a body is, the greater will be the force required to alter its velocity by a given amount in a given time. It takes a more powerful engine to make a long train attain a speed of ten miles an hour at the end of the first half-minute, than it does to make a short train do so. Or we may have circumstances where the force is the same for a number of different bodies; in that case, if we can measure the accelerations produced in them, we can tell the ratios of their masses: the greater the mass, the smaller the acceleration. We may take, in illustration of this method, an example which is important in connection with relativity. Radio-active bodies emit beta-particles (electrons) with enormous velocities. We can observe their path by making them travel through water vapor and form a cloud as they go. We can at the same time subject them to known electric and magnetic forces, and observe how much they are bent out of a straight line by these forces. This makes it possible to compare their masses. It is found that the faster they travel, the greater is their mass, as measured by the stationary observer; the increase is greatest as applied to their mass as measured by the effect of a force in the line of motion. In regard to forces at right angles to the line of motion, there is a change of mass with velocity in the same proportion as the changes of length and time. It is known otherwise that, apart from the effect of motion, all electrons have the same mass. All this was known before the theory of relativity was invented, but it showed that the traditional conception of mass had not quite the definiteness that had been ascribed to it. Mass used to be regarded as “quantity of matter,” and supposed to be quite invariable. Now mass was found to be relative to the observer, like length and time, and to be altered by motion in exactly the same proportion. However, this could be remedied. We could take the “proper mass,” the mass as measured by an observer who shares the motion of the body. This was easily inferred from the measured mass, by taking the same proportion as in the case of lengths and times.