Let us turn now to a different matter. The new law of composition of velocities and the resistance of a velocity-limit equal to that of light may be expressed in a different language from that we have hitherto used. Up to this we have spoken only of velocities and movements. Let us see how these things look when we at the same time examine the particular qualities of the moving objects, of bodies, of matter.
Everybody knows that the characteristic feature of matter is what we call inertia. If matter is at rest, a force is needed to set it in motion. If it is in motion, it needs a force to stop it. It needs one to accelerate the movement and one to alter the direction. This resistance which matter offers to the forces which tend to modify its condition of rest or movement is what we call inertia. But different bodies may offer a different degree of resistance to these forces. If a force is applied to an object, it will give it a certain acceleration. But the same force applied to another object will, as a rule, give it a different acceleration. A race-horse making a supreme effort will get along much more quickly under a small jockey than under a man of fifteen stone. A draught-horse will run more quickly if the cart it draws is empty than if it is full of goods. You can start a perambulator with a push that would be useless in the case of a heavy truck.
When a locomotive with a few coaches suddenly starts, the velocity imparted to the train during the first second is what we call its acceleration. If the same locomotive starts, in the same conditions, with a much longer train, we see that the acceleration is less. Hence the idea, introduced into science by Newton, of the mass of bodies, which is the measure of their inertia.
If in our example the locomotive produces in the second case an acceleration only half as great, we express this by saying that the mass of the second train is double that of the first. If we find that the acceleration produced by the locomotive is the same for three trucks loaded with wheat as for a single truck loaded with metal, we see that the two trains are equal in mass.
In a word, the masses of bodies are conventional data defined by the fact that they are proportional to the accelerations caused by one and the same force. To put it differently, the mass of a body is the quotient of the force which acts upon it by the acceleration given to it. Poincaré used to say picturesquely: “Masses are coefficients which it is convenient to use in calculations.”
If there is one property of bodies which comes within the range of our senses, a property of which every man has some sort of instinct or intuition, it is mass. Yet careful analysis shows us that we are unable to define it otherwise than by disguised conventions. Poincaré’s definition seems paradoxical in its admission of powerlessness. But it is correct. Mass is only a “coefficient,” a conventional outcome of our weakness!
Nevertheless, something remained upon which we thought we could base, if not our craving for certainty—genuine men of science gave up the idea of certainty long ago—at least our desire for accuracy of deduction in our classification of phenomena. We believed in the constancy of mass, of this convenient and so clearly defined coefficient.
Here again, unfortunately, we have to recant—or, perhaps, we should say fortunately, as there is no pleasure like that of novelty.
The older mechanics taught us that mass is constant in one and the same body, and is therefore independent of the velocity which the body acquires. From which it followed, as we have already explained, that, if a force continues to act, the velocity acquired at the end of a second will be doubled at the end of two seconds, tripled at the end of three seconds, and so on indefinitely.
But we have just seen that the velocity increases less during the second second than during the first, and so on, continuously diminishing until, when the velocity of light is attained, that of the moving body can increase no further, whatever force may act upon it.