Permit me to employ mathematical language a moment to express the same thought under another form. Suppose we observe _n_ molecules and ascertain that their 3_n_ coordinates satisfy a system of 3_n_ differential equations of the fourth order (and not of the second order as the law of inertia would require). We know that by introducing 3_n_ auxiliary variables, a system of 3_n_ equations of the fourth order can be reduced to a system of 6_n_ equations of the second order. If then we suppose these 3_n_ auxiliary variables represent the coordinates of _n_ invisible molecules, the result is again in conformity with the law of inertia. 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.