678 SCIENCE AND HYPOTHESIS 



there is equilibrium in both cases I conclude that the two forces F and 

 F' are equal, for they are both equal to the weight of the body P. But 

 am I certain that the body P has kept its weight when I transferred it 

 from the first body to the second? Far from it. I am certain of the 

 contrary. I know that the magnitude of the weight varies from one 

 point to another, and that it is greater, for instance, at the pole than 

 at the equator. No doubt, the difference is very small, and we neglect 

 it in practice; but a definition must have mathematical rigor; this 

 rigor does not exist. What I say of weight would apply equally to 

 the force of the spring of a dynamometer, which would vary accord- 

 ing to temperature and many other circumstances. Nor is this all. 

 We cannot say that the weight of the body P is applied to the body 

 C and keeps in equilibrium the force F. What is applied to the body 

 C is the action of the body P on the body C. On the other hand, the 

 body P is acted on by its weight, and by the reaction E of the body C 

 on P the forces F and A are equal, because they are in equilibrium; 

 the forces A and R are equal by virtue of the principle of action 

 and reaction; and finally, the force R and the weight P are equal be- 

 cause they are in equilibrium. From these three equalities we deduce 

 the equality of the weight P and the force F. 



Thus we are compelled to bring into our definition of the equality 

 of two forces the principle of the equality of action and reaction; 

 hence this principle can no longer "be regarded as an experimental law 

 but only as a definition. 



To recognize the equality of two forces we are then in possession of 

 two rules: the equality of two forces in equilibrium and the equality 

 of action and reaction. But, as we have seen, these are not sufficient, 

 and we are compelled to have recourse to a third rule, and to admit 

 that certain forces the weight of a body, for instance are con- 

 stant in magnitude and direction. But this third rule is an experi- 

 mental law. It is only approximately true : it is a bad definition. We 

 are therefore reduced to Kirchoff's definition: force is the product of 

 the mass and the acceleration. This law of Newton in its turn 

 ceases to be regarded as an experimental law, it is now only a defini- 

 tion. But as a definition it is insufficient, for we do not know what 

 mass is. It enables us, no doubt, to calculate the ratio of two forces 

 applied at different times to the same body, but it tells us nothing 

 about the ratio of two forces applied to two different bodies. To fill 

 up the gap we must have recourse to Newton's third law, the equality 

 of action and reaction, still regarded not as an experimental law but 

 as a definition. Two bodies, A and B, act on each other ; the accelera- 

 tion of A, multiplied by the mass of A, is equal to the action of B 

 on A ; in the same way the acceleration of B, multiplied by the mass 

 of B, is equal to the -reaction of A on B. As, by definition, the action 

 and the reaction are equal, the masses of A and B are respectively in 



