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Title: Action
Original Title: Action
Volume and Page: Vol. 1 (1751), pp. 119–120
Author: Jean Le Rond D'Alembert
Translator: John S.D. Glaus [The Euler Society, restinn@roadrunner.com]
Original Version (ARTFL): Link
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URL: http://hdl.handle.net/2027/spo.did2222.0000.685
Citation (MLA): D'Alembert, Jean Le Rond. "Action." The Encyclopedia of Diderot & d'Alembert Collaborative Translation Project. Translated by John S.D. Glaus. Ann Arbor: Michigan Publishing, University of Michigan Library, 2006. Web. [fill in today's date in the form 18 Apr. 2009 and remove square brackets]. <http://hdl.handle.net/2027/spo.did2222.0000.685>. Trans. of "Action," Encyclopédie ou Dictionnaire raisonné des sciences, des arts et des métiers, vol. 1. Paris, 1751.
Citation (Chicago): D'Alembert, Jean Le Rond. "Action." The Encyclopedia of Diderot & d'Alembert Collaborative Translation Project. Translated by John S.D. Glaus. Ann Arbor: Michigan Publishing, University of Michigan Library, 2006. http://hdl.handle.net/2027/spo.did2222.0000.685 (accessed [fill in today's date in the form April 18, 2009 and remove square brackets]). Originally published as "Action," Encyclopédie ou Dictionnaire raisonné des sciences, des arts et des métiers, 1:119–120 (Paris, 1751).

Action is a term which is used sometimes in Mechanics to designate the effort which is made by a body or the power exercised against another power even if at times it is the result of that effort.

It is in order to conform to the common language of Mechanics and Physics that we provide this dual definition. Since if we are asked what must be understood by action , only by providing a clear definition, will we be able to reply that it is the motion that a body really produces or which is produced by another which is what would be produced if nothing impeded it. See Motion.

In effect all power is nothing else than a body which is actually in motion or which tends to be in motion. That is to say that it would move if nothing impeded it. See Power. However in a body where there is actual motion or which tends to move, we are not able to see clearly that the motion which it possesses or which it would have if there were nothing to prevent its movement; thus the action of a body does not occur unless there is motion. So we should not attach to the word action any other idea than that of actual motion or a simple tendency to motion since it gives rise to the idea that there is some imagined metaphysical nonsense which resides in the body and then no one will have a clear understanding. It is due to this misunderstanding that we attribute the famous question of live forces which, according to appearances should never had been the object of a dispute if we had been willing to observe that the only precise and distinct notion that can be given to the word force is reduced to its effect, that is to say the motion that it produces or tends to produce. See Force [Force (Grammar. Literature), Force (Iconology), Conservation of Kinetic Energy, Force of Inertia, Kinetic Energy, Force, Resulting force, Force of motion, Moving force, Force of Water (Hydraulics)] .

Quantity of action is the term given by Mr. de Maupertuis in the Mémoires of the Academy of Sciences of Paris in 1744 and those of the Academy of Berlin in 1746. It was determined as the product of mass times the speed over the distance which it travels. Mr. de Maupertuis was only able to discover this general law through the changes that occur in the state of a body as the quantity of action necessary to produce this change is the least possible. He was happily able to apply this principle in his research of the laws of refraction, laws of impact, laws of equilibrium, etc., and he was even able bring this reasoning to the most sublime ends concerning the existence of a first being.. The two works that we have mentioned of Mr. de Maupertuis should focus the attention of all philosophers and we urge them to read these works and they will note that the author manages to ally the metaphysics of final causes ( See Final causes) with the fundamental truths of mechanics, as well as the dependency of the laws of impact of elastic and inelastic bodies which until now had been separate laws; and to reduce to a same principle the laws of motion and those of equilibrium.

The first Mémoire in which Mr. de Maupertuis provides an idea of his principle is on 15 April 1744 and at the end of the same year Professor Euler [page 120 ill.] published his excellent book: Methodus inveniendi lineas curvas maximi vel minimi proprietate gaudentes . (A method for finding curved lines enjoying properties of maximum or minimum) In the supplement which had been added, this famous Geometer proved that the curves of the trajectories that bodies describe due to central forces, the speed multiplied by the variable of the curve is always a minimum. This theory is an elegant application of the principle of Mr. de Maupertuis' planetary motion.

In the same Mémoire of 15 April 1744 that we have just quoted and one can see how Mr. de Maupertuis' thoughts on the laws of refraction led him to the theory in question. We are aware of the principle that Mr. de Fermat and after him Mr. Leibniz used to explain the laws of refraction. These great Geometers had assumed that a ray of light which travels from one point to another while traveling through two separate medias does so at different speeds and must travel in the shortest time which is possible. And as a rust of this principle that proved geometrically that this ray would not travel from one point to another in a straight line, but having arrived onto the surface which separates the two mediums, it must change directions in such a way that the sine of its incidence is the sine of its refraction as the speed through its first medium is the speed through the second from which the deduced the well-known law of reciprocal sines. See Sines, Refraction, etc.

This explanation although very ingenious is subject to great difficulty, and that is when the ray of light arrives at a perpendicular in the medias where its speed is less and by consequence is more resistant which appears to be contrary to all the mechanical explanations that have been given until now regarding the refraction of bodies and in particular those of the refraction of light.

The explanation amongst others that were thought of by Mr. Newton and the most satisfactory that has been given until now coincides perfectly with the relationship between sines by attributing the rays refraction to the attractive force of the medium from which it stands that the denser medias at which the attraction is the strongest must necessarily attract them at the perpendicular which is what has been confirmed by experiments. However the attraction within the media would be unable to approach the ray at the perpendicular without increasing its speed as can be easily proved. Therefore following Mr. Newton refraction must take place when approaching the perpendicular when the speed increases, which is contrary to the laws of Mssrs Fermat and Leibniz.

Mr. de Maupertuis attempted to reconcile Newton's explanation with metaphysical principles, Instead of assuming like Mssrs. Fermat and Leibniz that a ray of light travels from one point to another in the least time possible, he assumed that a ray of light travelled from one point to another in such a way that the least amount of action was used. He wrote that this quantity of action is an expense that nature dispenses. Through this philosophical principle he found that not only are the sines in ratio to one another, but that the are also in a ratio to the inverse proportion of their velocities, (which agrees with Mr. Newton) and not directly as assumed Mssrs. Fermat and Leibniz.

It is particularly interesting that so many philosophers who have written on refraction had not been able to imagine such a simple manner of reconciling metaphysics with mechanics which requires nothing more than a slight adjustment to the mathematics established by Mr. Fermat. In effect, following time as a variable, that is to say space divided by speed must be a minimum: in such a way that one assigns E as the space travelled in the first medium with speed V , and e the space travelled in the second medium speed v , we will obtain E/V + e/v = a minimum, that is to say de/v + dE/V + de/v = 0 . It is now easy to see that the sines of incidence and of refraction are as though dE to -de , from where is arrives that these sines are in direct proportion to the speeds V, v which is assumed by Mr. Fermat. However for these sines to be in direct inverse proportion of their speeds, it would only be necessary to assume that V dE + v de = 0 , which provides ExV + exv = to a minimum and that is Mr. de Maupertuis' principle. See Minimum.

One can find in the Mémoires of the Berlin Academy that we have already quoted, all the other applications that he has done with this principle, which one will come to regard as on of the most general in all Mechanics.

Irrespective of the position that he has taken concerning Metaphysics which has served as his base, Mr. de Maupertuis has given the quantity of action , it would be no less true that the product of space by speed is a minimum in most general laws of nature. This geometric truth due to Mr. de Maupertuis will survive forever and one will be able, if one wishes, to take the word quantity of action to express only an abbreviated form to express the product of space multiplied by speed.