side of the purchase; the money one gains the other loses; the goods the seller loses the buyer gains. The force with which a horse tows a boat is at each pull exactly equal to the force with which the boat drags the horse. The boat moves forward, and the horse has hist motion forward neutralized to an equivalent amount. If we do not see the horse actually dragged backwards, as we see the boat dragged forwards, this is because the horse has a surplus of force over and above that expended on the pull. If I give a beggar who has nothing threepence out of my last sixpence, I have beggared myself to the same amount that I have enriched him; he has reacted on my fortunes in the exact ratio to my action on his; but if I give him threepence out of my pocketful of money, although there has been the same ratio between the action and reaction, its effect on my fortunes is trifling. The horse is rich in force, and partly expends it in pulls at the boat; each pull diminishes the store; and after a while the horse will be unable to drag the boat forward, because the boat will drag him backward by an equal pull.

Suppose a body, A, moving with a velocity, 12, overtakes another body, B, moving with a velocity, 6; the result of their meeting will be a combination of these velocities in respect to any third body on which they may impinge, but a redistribution of their velocities in respect to themselves. They will both, if of equal mass, have a velocity, 9.. A has lost 3, which B has gained; the sum remains as before, 18: between A and B there has been equality of action and reaction.

27. Since Force is realized in Motion, the Laws of Force are the Laws of Motion; and it is on this ground that modern science hopes to reduce the whole universe to Molecular Dynamics. The idea is resisted by many, because they persist in imagining something else

in Force, something which is the Cause of Motion, and that Cause inscrutable, or at any rate different from the effect.*

* I will merely touch here on the separation of Velocity from Motion, and the disputes respecting the reality or the artificiality of the proportionality of Force and Velocity. D'Alembert thinks the law ought to be banished altogether, for he says "it is grounded only on the vague and obscure axiom that the effect must be proportional to its cause [which we shall presently see to be an identical proposition], and whether true or doubtful, clear or obscure, it is useless to Mechanics, and therefore should be banished." (D'ALEMBERT, Traité de Dynamique, Paris, 1796; Discours Prelim., p. xi.) Daniel Bernouilli regarded the principle as contingent, because we, being in ignorance of the nature of cause and its manner of acting, cannot say whether the effect is really proportional, or whether it is not some function of that cause. Poisson at first regarded it as an hypothesis: "Car de ce que nous entendons par le rapport numérique des forces nous ne pouvons rien conclure relativement aux vitesses qu'elles produisent. Nous disons par exemple qu'une force est double d'une autre quand la première est formée par la réunion de deux forces égales à la seconde agissant simultanément et dans le même sens sur un point matériel; or il ne s'ensuit pas nécessairement que cette force double doive communiquer au mobile une vitesse précisément double de celle que la force simple lui communiquerait dans le même temps. La vitesse communiquée à une mobile par une force qui agit sur lui pendant un temps déterminé est une fonction du nombre qui représente l'intensité de cette force; le peu de données que nous avons sur la nature des forces [always this recurring fallacy!] ne nous permet pas de déterminer à priori la forme de cette fonction; nous sommes done obligés pour résoudre les problèmes de la dynamique de partir d'une supposition; et nous choisissons la plus simple en regardant la vitesse comme proportionelle à la force." This passage This passage I extract from the first edition (p. 278) of the Mécanique; but if we turn to the third edition, issued twenty years later, we see that he no longer follows Bernouilli in declaring it to be an hypothesis, but regards it "comme une conséquence nécessaire de ce que les vitesses imprimées par des forces quelconques dans des intervalles de temps infiniment petits sont toujours infiniment petites, et de ce qu'en même temps les déplacements des mobiles sont aussi infiniment petits" (I. 215).

Laplace regards the proportionality as an induction from experience ; Force, he says, being only known by the space which it causes a body to describe in a given time, we naturally take this space as its measure. We cannot know à priori that force is proportional to velocity, because we are ignorant of the nature of force; and there would be no contradic

ence.

THE MANIFESTATIONS OF FORCE.

we

or

28. We have now reached a point of view from which a distinct conception of Force may be had. Setting aside the metempirical conception of Force as something apart from and independent of its manifestations, a Noumenon of which the observed actions are Phenomena, see that, both in the ordinary and the scientific acceptations of the term, it means simply the activity of ExistMatter and Force are two abstract expressions for Agent and Activity. When we speak of physical forces, vital forces, mental forces, social forces, we mean ought to mean-the actions of molecular agents, organic agents, social agents. When we speak of the force (energy) of a projectile, we mean the mass, and its velocity as related to the work which will be effected on some other body when the projectile strikes it. A prospective glance at the effect sees the projectile energy as a cause; this cause is measured by its effect on the object struck, the co-operation of which is left out of account. And as we thus abstract one term of the relation, we are also led to abstract the relation itself from both the related terms, and view this cause apart from its conditions, this action apart from the agents acting, and then declare it to be the manifestation of some unknown or even unknowable Cause. A serious error. There is nothing in the action but the action itself; what preceded the action, what lies outside it, is not this force, but another. A thing is what it does. What it is and does may be determined by sometion in supposing the force equal to the square of the velocity. He then shows how the principle of proportionality follows as a consequence from the principle of the independence of motions (the second law of Motion discovered by Galileo). (LAPLACE, Exposition du Système du Monde, 6th ed., 1836, I. 278.) Duhamel also rejects the idea of its being an hypothesis, and says it is the foundation of dynamics verified by experience. (Cours de Mécanique de l'Ecole Polytechnique, 1845, I. 239.)

thing which acts on it, since all things are related; but the particular group of relations specified is that group, and not another.

29. Consider a single example: Among the observed actions of liquids and gases there is one named Diffusion, and the cause is said to be a "diffusive force," - obviously a mere translation of the observed fact. This cause is the effect, and the maximum of effect (the maximum of diffusion) is at the moment of contact and at the surface of contact of the two liquids or gases; that is to say, where the rate of change per unit of distance is greatest. The law or fact being that difference of rate determines the diffusion, we see at once that the rapidity of the diffusion will gradually diminish with the diminishing rate of change, the exchange is a function of the rate. When, therefore, we are told that the same force varies in intensity according to the differences of rate, we are simply restating the fact that the cause, being identical with the effect, varies with the effect: the force which is the so-called manifestation of this cause varies with the varying conditions of its existence; but to call this varying force the same force is to say that changing relations are the same relations. It is the same force in the sense of being an action of the same kind.

30. It is obvious that if Force is Activity, the forces are infinite, since the combinations of agents are infinite. Nevertheless, the purpose of science being classification and reduction of multiplicity to unity, all our efforts are directed towards a systematic selection and classification of actions under well-marked groups. Out of the infinite variety of incessant changes we select those that have constant characters, and these are our finite forces; numerous as they are, they admit of being grouped under a few heads. They may be compared to the tones which we distinguish from noises. Noises are the irregular

mingling of vibrations; tones are the regular recurrence of vibrations: both are ultimately reducible by analysis to simple pendulum movements of the air, but the definite and constant character of the periodically recurrent vibrations detaches them from the indefinite and variable noises, so that out of them Music is constructed, as out of the forces of Science. The tones and the forces are measurable, because their relations are constant. We recognize two tones as in unison, however various their clang, when their periodic vibrations are numerically identical; and we say two forces are identical, however various their accompanying phenomena, when their mechanical relations are expressible by the same quantities.

Whether all forces are manifestations of one Force, is a metaphysical question. Its answer will depend on, 1o, whether we choose to disregard all Diversity, as if it were not equally with Identity a fact of Feeling; or, 2o, whether we are considering the universe as it is reflected in Thought, or as it is reflected in Feeling.

Into the many deeply interesting details which Science has collected respecting the actions of material objects I cannot enter here, but in the Appendix will be found some remarks on the metaphysical question of "Action at a distance." (Appendix C.)