To sum up, this law, verified experimentally in some particular cases, may unhesitatingly be extended to the most general cases, since we know that in these general cases experiment no longer is able either to confirm or to contradict it.
The Law of Acceleration.—The acceleration of a body is equal to the force acting on it divided by its mass. Can this law be verified by experiment? For that it would be necessary to measure the three magnitudes which figure in the enunciation: acceleration, force and mass.
I assume that acceleration can be measured, for I pass over the difficulty arising from the measurement of time. But how measure force, or mass? We do not even know what they are.
What is mass? According to Newton, it is the product of the volume by the density. According to Thomson and Tait, it would be better to say that density is the quotient of the mass by the volume. What is force? It is, replies Lagrange, that which moves or tends to move a body. It is, Kirchhoff will say, the product of the mass by the acceleration. But then, why not say the mass is the quotient of the force by the acceleration?
These difficulties are inextricable.
When we say force is the cause of motion, we talk metaphysics, and this definition, if one were content with it, would be absolutely sterile. For a definition to be of any use, it must teach us to measure force; moreover that suffices; it is not at all necessary that it teach us what force is in itself, nor whether it is the cause or the effect of motion.
We must therefore first define the equality of two forces. When shall we say two forces are equal? It is, we are told, when, applied to the same mass, they impress upon it the same acceleration, or when, opposed directly one to the other, they produce equilibrium. This definition is only a sham. A force applied to a body can not be uncoupled to hook it up to another body, as one uncouples a locomotive to attach it to another train. It is therefore impossible to know what acceleration such a force, applied to such a body, would impress upon such another body, if it were applied to it. It is impossible to know how two forces which are not directly opposed would act, if they were directly opposed.
It is this definition we try to materialize, so to speak, when we measure a force with a dynamometer, or in balancing it with a weight. Two forces F and F´, which for simplicity I will suppose vertical and directed upward, are applied respectively to two bodies C and C´; I suspend the same heavy body P first to the body C, then to the body C´; if equilibrium is produced in both cases, I shall conclude that the two forces F and F´ are equal to one another, since they are each equal to the weight of the body P.
But am I sure the body P has retained the same weight when I have transported it from the first body to the second? Far from it; I am sure of the contrary; I know the intensity of gravity varies from one point to another, and that it is stronger, for instance, at the pole than at the equator. No doubt the difference is very slight and, in practise, I shall take no account of it; but a properly constructed definition should have mathematical rigor; this rigor is lacking. What I say of weight would evidently apply to the force of the resiliency of a dynamometer, which the temperature and a multitude of circumstances may cause to vary.
This is not all; we can not say the weight of the body P may be applied to the body C and directly balance the force F. What is applied to the body C is the action A of the body P on the body C; the body P is submitted on its part, on the one hand, to its weight; on the other hand, to the reaction R of the body C on P. Finally, the force F is equal to the force A, since it balances it; the force A is equal to R, in virtue of the principle of the equality of action and reaction; lastly, the force R is equal to the weight of P, since it balances it. It is from these three equalities we deduce as consequence the equality of F and the weight of P.