definition is still insufficient, for we do not know what mass is. It enables us doubtless to calculate the relation of two forces applied to the same body at different instants; it teaches us nothing about the relation of two forces applied to two different bodies.
To complete it, it is necessary to go back anew to Newton's third law (equality of action and reaction), regarded again, not as an experimental law, but as a definition. Two bodies A and B act one upon the other; the acceleration of A multiplied by the mass of A is equal to the action of B upon A; in the same way, the product of the acceleration of B by its mass is equal to the reaction of A upon B. As, by definition, action is equal to reaction, the masses of A and B are in the inverse ratio of their accelerations. Here we have the ratio of these two masses defined, and it is for experiment to verify that this ratio is constant.
That would be all very well if the two bodies A and B alone were present and removed from the action of the rest of the world. This is not at all the case; the acceleration of A is not due merely to the action of B, but to that of a multitude of other bodies C, D,... To apply the preceding rule, it is therefore necessary to separate the acceleration of A into many components, and discern which of these components is due to the action of B.
This separation would still be possible, if we should assume that the action of C upon A is simply adjoined to that of B upon A, without the presence of the body C modifying the action of B upon A; or the presence of B modifying the action of C upon A; if we should assume, consequently, that any two bodies attract each other, that their mutual action is along their join and depends only upon their distance apart; if, in a word, we assume the hypothesis of central forces.
You know that to determine the masses of the celestial bodies we use a wholly different principle. The law of gravitation teaches us that the attraction of two bodies is proportional to their masses; if r is their distance apart, m and m´ their masses, k a constant, their attraction will be kmm´/r2.
What we are measuring then is not mass, the ratio of force to acceleration, but the attracting mass; it is not the inertia of the body, but its attracting force.
This is an indirect procedure, whose employment is not theoretically indispensable. It might very well have been that attraction was inversely proportional to the square of the distance without being proportional to the product of the masses, that it was equal to f/r2, but without our having f = kmm´.
If it were so, we could nevertheless, by observation of the relative motions of the heavenly bodies, measure the masses of these bodies.
But have we the right to admit the hypothesis of central forces? Is this hypothesis rigorously exact? Is it certain it will never be contradicted by experiment? Who would dare affirm that? And if we must abandon this hypothesis, the whole edifice so laboriously erected will crumble.
We have no longer the right to speak of the component of the acceleration of A due to the action of B. We have no means of distinguishing it from that due to the action of C or of another body. The rule for the measurement of masses becomes inapplicable.
What remains then of the principle of the equality of action and reaction? If the hypothesis of central forces is rejected, this principle should evidently be enunciated thus: the geometric resultant of all the forces applied