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 ma.s.s, 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 mult.i.tude of circ.u.mstances 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_.

We are therefore obliged in the definition of the equality of the two forces to bring in the principle of the equality of action and reaction; _on this account, this principle must no longer be regarded as an experimental law, but as a definition_.

For recognizing the equality of two forces here, we are then in possession of two rules: equality of two forces which balance; equality of action and reaction. But, as we have seen above, these two rules are insufficient; we are obliged to have recourse to a third rule and to a.s.sume that certain forces, as, for instance, the weight of a body, are constant in magnitude and direction. But this third rule, as I have said, is an experimental law; it is only approximately true; _it is a bad definition_.

We are therefore reduced to Kirchhoff's definition; _force is equal to the ma.s.s multiplied by the acceleration_. This 'law of Newton' in its turn ceases to be regarded as an experimental law, it is now only a definition. But this definition is still insufficient, for we do not know what ma.s.s 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 ma.s.s of _A_ is equal to the action of _B_ upon _A_; in the same way, the product of the acceleration of _B_ by its ma.s.s is equal to the reaction of _A_ upon _B_. As, by definition, action is equal to reaction, the ma.s.ses of _A_ and _B_ are in the inverse ratio of their accelerations. Here we have the ratio of these two ma.s.ses 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 mult.i.tude 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 a.s.sume_ 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 a.s.sume, 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 a.s.sume _the hypothesis of central forces_.

You know that to determine the ma.s.ses 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 ma.s.ses; if _r_ is their distance apart, _m_ and _m'_ their ma.s.ses, _k_ a constant, their attraction will be _kmm'_/_r_^{2}.

What we are measuring then is not ma.s.s, the ratio of force to acceleration, but the attracting ma.s.s; 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 ma.s.ses, that it was equal to _f_/_r_^{2}, 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 ma.s.ses 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 ma.s.ses 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 to the various bodies of a system isolated from all external action will be null. Or, in other words, _the motion of the center of gravity of this system will be rectilinear and uniform_.

There it seems we have a means of defining ma.s.s; the position of the center of gravity evidently depends on the values attributed to the ma.s.ses; it will be necessary to dispose of these values in such a way that the motion of the center of gravity may be rectilinear and uniform; this will always be possible if Newton's third law is true, and possible in general only in a single way.

But there exists no system isolated from all external action; all the parts of the universe are subject more or less to the action of all the other parts. _The law of the motion of the center of gravity is rigorously true only if applied to the entire universe._

But then, to get from it the values of the ma.s.ses, it would be necessary to observe the motion of the center of gravity of the universe. The absurdity of this consequence is manifest; we know only relative motions; the motion of the center of gravity of the universe will remain for us eternally unknown.

Therefore nothing remains and our efforts have been fruitless; we are driven to the following definition, which is only an avowal of powerlessness: _ma.s.ses are coefficients it is convenient to introduce into calculations_.

We could reconstruct all mechanics by attributing different values to all the ma.s.ses. This new mechanics would not be in contradiction either with experience or with the general principles of dynamics (principle of inertia, proportionality of forces to ma.s.ses and to accelerations, equality of action and reaction, rectilinear and uniform motion of the center of gravity, principle of areas).

Only the equations of this new mechanics would be _less simple_. Let us understand clearly: it would only be the first terms which would be less simple, that is those experience has already made us acquainted with; perhaps one could alter the ma.s.ses by small quant.i.ties without the _complete_ equations gaining or losing in simplicity.

Hertz has raised the question whether the principles of mechanics are rigorously true. "In the opinion of many physicists," he says, "it is inconceivable that the remotest experience should ever change anything in the immovable principles of mechanics; and yet, what comes from experience may always be rectified by experience." After what we have just said, these fears will appear groundless.

The principles of dynamics at first appeared to us as experimental truths; but we have been obliged to use them as definitions. It is _by definition_ that force is equal to the product of ma.s.s by acceleration; here, then, is a principle which is henceforth beyond the reach of any further experiment. It is in the same way by definition that action is equal to reaction.

But then, it will be said, these unverifiable principles are absolutely devoid of any significance; experiment can not contradict them; but they can teach us nothing useful; then what is the use of studying dynamics?

This over-hasty condemnation would be unjust. There is not in nature any system _perfectly_ isolated, perfectly removed from all external action; but there are systems _almost_ isolated.

If such a system be observed, one may study not only the relative motion of its various parts one in reference to another, but also the motion of its center of gravity in reference to the other parts of the universe. We ascertain then that the motion of this center of gravity is _almost_ rectilinear and uniform, in conformity with Newton's third law.

That is an experimental truth, but it can not be invalidated by experience; in fact, what would a more precise experiment teach us? It would teach us that the law was only almost true; but that we knew already.

_We can now understand how experience has been able to serve as basis for the principles of mechanics and yet will never be able to contradict them._

ANTHROPOMORPHIC MECHANICS.--"Kirchhoff," it will be said, "has only acted in obedience to the general tendency of mathematicians toward nominalism; from this his ability as a physicist has not saved him. He wanted a definition of force, and he took for it the first proposition that presented itself; but we need no definition of force: the idea of force is primitive, irreducible, indefinable; we all know what it is, we have a direct intuition of it. This direct intuition comes from the notion of effort, which is familiar to us from infancy."

But first, even though this direct intuition made known to us the real nature of force in itself, it would be insufficient as a foundation for mechanics; it would besides be wholly useless. What is of importance is not to know what force is, but to know how to measure it.

Whatever does not teach us to measure it is as useless to mechanics as is, for instance, the subjective notion of warmth and cold to the physicist who is studying heat. This subjective notion can not be translated into numbers, therefore it is of no use; a scientist whose skin was an absolutely bad conductor of heat and who, consequently, would never have felt either sensations of cold or sensations of warmth, could read a thermometer just as well as any one else, and that would suffice him for constructing the whole theory of heat.

Now this immediate notion of effort is of no use to us for measuring force; it is clear, for instance, that I should feel more fatigue in lifting a weight of fifty kilos than a man accustomed to carry burdens.

But more than that: this notion of effort does not teach us the real nature of force; it reduces itself finally to a remembrance of muscular sensations, and it will hardly be maintained that the sun feels a muscular sensation when it draws the earth.

All that can there be sought is a symbol, less precise and less convenient than the arrows the geometers use, but just as remote from the reality.

Anthropomorphism has played a considerable historic role in the genesis of mechanics; perhaps it will still at times furnish a symbol which will appear convenient to some minds; but it can not serve as foundation for anything of a truly scientific or philosophic character.

'THE SCHOOL OF THE THREAD.'--M. Andrade, in his _Lecons de mecanique physique_, has rejuvenated anthropomorphic mechanics. To the school of mechanics to which Kirchhoff belongs, he opposes that which he bizarrely calls the school of the thread.

This school tries to reduce everything to "the consideration of certain material systems of negligible ma.s.s, envisaged in the state of tension and capable of transmitting considerable efforts to distant bodies, systems of which the ideal type is the _thread_."

A thread which transmits any force is slightly elongated under the action of this force; the direction of the thread tells us the direction of the force, whose magnitude is measured by the elongation of the thread.

One may then conceive an experiment such as this. A body _A_ is attached to a thread; at the other extremity of the thread any force acts which varies until the thread takes an elongation [alpha]; the acceleration of the body _A_ is noted; _A_ is detached and the body _B_ attached to the same thread; the same force or another force acts anew, and is made to vary until the thread takes again the elongation [alpha]; the acceleration of the body _B_ is noted. The experiment is then renewed with both _A_ and _B_, but so that the thread takes the elongation [beta]. The four observed accelerations should be proportional. We have thus an experimental verification of the law of acceleration above enunciated.

Or still better, a body is submitted to the simultaneous action of several identical threads in equal tension, and by experiment it is sought what must be the orientations of all these threads that the body may remain in equilibrium. We have then an experimental verification of the law of the composition of forces.

But, after all, what have we done? We have defined the force to which the thread is subjected by the deformation undergone by this thread, which is reasonable enough; we have further a.s.sumed that if a body is attached to this thread, the effort transmitted to it by the thread is equal to the action this body exercises on this thread; after all, we have therefore used the principle of the equality of action and reaction, in considering it, not as an experimental truth, but as the very definition of force.

This definition is just as conventional as Kirchhoff's, but far less general.

All forces are not transmitted by threads (besides, to be able to compare them, they would all have to be transmitted by identical threads). Even if it should be conceded that the earth is attached to the sun by some invisible thread, at least it would be admitted that we have no means of measuring its elongation.

Nine times out of ten, consequently, our definition would be at fault; no sort of sense could be attributed to it, and it would be necessary to fall back on Kirchhoff's.

Why then take this detour? You admit a certain definition of force which has a meaning only in certain particular cases. In these cases you verify by experiment that it leads to the law of acceleration. On the strength of this experiment, you then take the law of acceleration as a definition of force in all the other cases.