Conservation of kinetic energy. The conservation of live forces or kinetic energy is a principle in Mechanics that Mr. Huyghens appears to have been the first to notice and one of which Mr. Bernoulli and other geometers have allowed for due to the pervasiveness of its usage in the solution to problems in Dynamics. Here is this principle which is contained in the two following laws:

1. If bodies act one against the other, either by pulling on threads or inelastic rods, by pushing or by impact, as long as in this last case it has perfect elasticity, the sum of the product of the masses multiplied by the square of the speeds will always be a constant quantity.

2. If the bodies should be animated by whatever forces, the sum of the product of the masses by the squares of the speeds at every moment is equal to the sum of the product of the masses multiplied by the square of the initial speeds. Plus the squares of the speeds that the bodies would have acquired if they had been animated by the same forces and were able to move freely on the line that it has inscribed.

We have said either by being pushed or through impact and we distinguish between impulse and impact because the conservation of kinetic energy takes place in the movement of bodies which are being pushed, insofar as these movements change only by minute degrees or infinitely small amounts rather than if it occurs in elastic bodies which impact each other, even in the case where a spring would act in a indivisible time and would pass them and would allow them to continue without any gradual motion from one to the other.

Mr. Huyghens appears to have been the first to notice that this law of the conservation of kinetic energy was contained in the impact of elastic bodies. It also appears that he was familiar with the law of the conservation of kinetic energy during the motion of bodies which are animated by forces. Since the principle which he uses to solve the problems within the centers of oscillation, is nothing other than the second law expressed differently. In his discussion regarding the communication of motion of which we have spoken, Mr. Johann Bernoulli has developed and extended this discovery of Mr. Huyghens and he did not forget to use this to prove his opinion concerning the measurement of force . He believes this principle to be very useful, since within the mutual action of the two bodies, it is almost never the sum of the product of the masses multiplied by the speed which would make a constant, but that the sum of the product of the mass times the square of the speed. Descartes believed that the same quantity of force would always be in existence in the universe and by consequence, he falsely assumed that motion could not be lost since he believed that force was proportional to the quantity of motion. This philosopher’s ideas might not have been so far from admitting that the measurement of kinetic energy by the square of its speed, if this idea had come to mind. However if we are mindful as to what we have said regarding the meaning that we attach to the word force , it appears that this new proof is in favor of kinetic e nergy , from which nothing new is presented, or from which it is a fact and a truth known to everyone.

In my treatise of Dynamics published in 1743, I proved the principle of kinetic energy under all possible circumstances and I also showed that it is dependent

on another principle that when forces come into equilibrium, the virtual speeds at the points at which they are applied are estimated following the direction of these forces and are in inverse proportion to these same forces. This last principle has been recognized for a long time by mathematicians as the principle foundation of equilibrium or at least as a necessary consequence of equilibrium.

In his excellent work entitled Hydrodynamica , Daniel Bernoulli was the first to apply the principle of kinetic energy to fluid motion, however without providing a proof. In 1744, I published in Paris, a treatise of equilibrium and fluid motion in which I believe to have been the first to prove the conservation of kinetic energy. It is up to the scientists to determine whether I succeeded. I also believe to have proved that M. Daniel Bernoulli made use of the principle of the conservation of kinetic energy in certain cases when he should not have. They are those moments when fluid speed or a part of the fluid changes abruptly and without a gradual decrease. That is to say without decreasing in infinitely small amounts. Since the principle of the conservation of kinetic energy never occurs when bodies act on each other move suddenly from one movement to another, without passing through the intermarry motion by degrees, unless the bodies are perfectly elastic. Still, under these circumstances the change does not occur except by infinitely small degrees, which allows it to fall under the general rule. See Hydrodynamics and Fluid.

In the Mémoires of the Academy of Sciences of 1742, Mr. Clairaut had also provided a proof of a particular type concerning the principle of kinetic energy and I must admit in reference to this topic that even though Mr. Clairaut’s mémoire was printed in the 1742 volume and that my treatise on Dynamics only appeared in 1743, this mémoire and this treatise were both presented on the same day at the Academy.

One can see by the different mémoires spread throughout the volumes of the Scientific Academies of Paris, Berlin and Petersburg, how the principle of kinetic energy facilitates the solution of a great number of problems in Dynamics. We believe that there was a time when it would have been very difficult to resolve a number of these problems without using this principle and if I appear biased towards my own work it should be said that I was the first to provide for a general and direct method in my treatise on Dynamics to solve all the questions imaginable of this type, without using the principle of the conservation of kinetic energy or any other indirect or secondary principle. This in no way should prevent me from suggesting the utility of these last principles to facilitate or better yet abridge in certain cases the solutions, especially when there might be reason to demonstrate these same principles.

In what concerns the relation of kinetic energy to action, We have noted with the word Cosmology, that those modern partisans of kinetic energy had imagined action as the product of the mass by the square of speed and distance, or that which comes to the same thing which is the product of the mass by the square of the speed and time since in uniform movement as it is supposed here, distance is the product of speed by time. See Speed.

We have also mentioned with the words Action [Action, Action (Ethics)] and Cosmology that the definition taken by action of itself is absolutely arbitrary; however we fear that the modern partisans of kinetic energy pretended nothing more than to attach to this definition some reality which they call action. Since according to them the instantaneous force

of a body in motion is the product of the mass times the square of the speed and they appear to have interpreted action as equal to product of kinetic energy by time. One can read a quite poorly written mémoire on the subject by the late Professor Wolf and inserted in Volume I of the Acta of Petersburg, one may come away being convinced that the professor believed that he had determined the true nature of action; however it is easy to see that when we are willing to look closer at this notion as a definition of a word and that it is nothing more than a chimera by itself and within the partisan principles of kinetic energy.

1. In itself because within the uniformity of movement of a body there is no resistance to overcome, nor consequentially of action properly defined.

2. Within the principles of kinetic energy , since according to them, kinetic energy is the one which consumes itself, or is supposed to be able to consume itself in its exercise. There is therefore no true action except when this force is truly consumed when directed against obstacles, furthermore in that case, according to the defenders of kinetic energy that the time is irrelevant since it is in the matter of things that when a force is greater for it to be longer to diminish. Why do they wish to time as a consideration of action? Action should not be part of their principles except as kinetic energy when it acts against obstacles. The way in which it is considered should have nothing to do with its measurement, since according to them this force is only seen as proportional to the square of the speed; insofar as it is supposed that this force will diminish incrementally by the obstacles against which they act.

Let us recognize that this definition of action, provided for by the partisans of kinetic energy, is purely arbitrary and conforms even less to their own principles. In regards to those like M. de Maupertuis, who did not take part in the dispute concerning live forces, one cannot object to their definition of action, especially since they appear to have given it as the definition as a noun. M. de Maupertuis has mentioned himself on page 26 of the first volume of his newly completed works published in Lyon; What I have defined as Action, might have been better called force, but having found that this word established by Wolf and Leibniz as an expression of the same idea and finding that it applied well to the term, I did not wish to change it. These words appear to indicate that Mr. de Maupertuis, even though he believed that action could be represented by the product of the square of the speed and time, believes that another meaning could be attached to this notion at the same time, to which we will add something relative to the articles of Action [1] [2] and Cosmology, since when he views the suggested action under this point of view as the expenditure of nature. This word expenditure must not be interpreted in a metaphysical and rigorous sense but rather in the purely mathematical sense, that is to say for a mathematical quantity which in many cases is equal to a minimum.

For the same reasons, I believe that one can equally well adopt all other definitions of action, for example the one provided by M. d’Arcy in the Mémoires of the Academy of Sciences in 1747 and 1752, insofar (this does not contradict Mr. d’Arcy’s principles in any way) as we regard this definition as a simple noun definition. We can agree with M. d’Arcy that the action of one system with two equal bodies which moves in opposite directions at equal speeds is nil, since the action which would provide for the equilibrium to the sum of these actions; however we are able to look as well at this system of action in another way as thought the sum were of independent actions and consequentially as real. Thus we are able to see as truly real the action of two cannon balls being shot in opposite directions. With the rest M. d’Arcy has remarked with reason that the conservation of action, interpreted in the way in which he provides it, takes place generally in the movement of bodies which act one on the other and he had advantageously used this principle to ease the solution of some problems in Dynamics. [*]

As an idea that we normally attach to the word action supposes some resistance to overcome, and that we are only able to have the idea of action through its effects, I felt that I was able to define action in the Encyclopedie as a motion which a body tends to produce or which tends to be produced within another body. An author unknown to me insinuates in the 1753 Mémoires of the Berlin Academy that the following definition is vague. I am not aware if he intended to criticize me on this matter, in any way, I invited him to provide us with a mathematical definition of action, that is a more accurate and precise not the metaphysical idea of the word action, which is a chimera, but the popularized version of this idea.

Everything that we have just said concerning action has a necessary relation to the word force and can be seen as a supplement to the words Action [1] [2] and Cosmology to which we refer..

Some thoughts on the nature of dead forces and their different kinds: While adopting as a simple noun definition the idea that the defenders of live forces provide us for dead forces (potential energy); the ones which cease to exist as soon as their effect has stopped, which is what occurs when two rigid bodies collide directly from opposite directions with equal speeds. The second type of dead force contains those which dissipate and renew themselves in each instant, in such a way that if one was to eliminate the obstacle, they would act in their full and actual effect; as is the effect of two bound springs with one acting upon the other and as well the effect of gravity. See the end of article Equilibrium, (Mechanics) from which we have noted that the word equilibrium does not provide for a correct understanding of the mutual action concerning this last type of dead force (potential energy).

This distinction between the dead forces has provided us with a opportunity to make another; and that is where the dead force is such that it would produce a finite speed, if there were no obstacle to prevent it; or in which the obstacle is removed and the result would be in incrementally small speeds or to say it more exactly that the body would begin its movement from a dead stop and would increase to this speed by degrees. The first case would be one with two equal bodies which collide or which push each other or which pull themselves in opposite directions by equal and finite speeds; the second happens to be that of a weighted body applied to a horizontal plane. If this plane is removed the body will drop as it begins its descent to zero speed, and the action of gravity will increase the speed at every moment, at least this is what is presumed to happen. See Acceleration and Descent.

From these points those Physicists had concluded that the force of percussion was infinitely greater that the force of gravity, since the first is to the second as a speed which has ended is to one which is infinitely small or more properly at zero; and by this they have explained why an enormous weight which has been dropped onto a nail partially inserted into a table does not drive the nail forward whereas sometimes a small striking force will cause the same effect. See the article Percussion.

^* I believe that I had explained with great accuracy the question of least action in the article on Cosmology. The intended criticism that appears to have been leveled against me in the 1752 Mémoires of the Academy would disappear entirely if one would carefully read this article and under the word Final Causes. For example, I mentioned the action of the lever in the article Cosmology, and I expressed myself in such a way that the application and use of the principle does not contain a more general inclusiveness; and to the word Final Causes, I have noticed that the path for reflection is often (and not always) a maximum in concave mirrors. [An additional note from the translator: This note (*) is worthy of a lengthy paper explaining the elusive and ethereal connections between metaphysics and physics, between science coming of age and the quest for the philosopher’s stone, the statute of empirical experimentation over old wives tales. The years 1749-1753 were particularly brutal years for some people: Euler, König, de Maupertuis, Frederick II and Voltaire in particular. D’Alembert enjoyed the safety (somewhat) of Paris and remained aloof from the adventures of the principle of least action which was eventually to turn into a circus of the actions of the least principled. This episode of Cosmology and Final Causes gives rise to an attempt at the tenuous connection between the new Physics and the modernization of Metaphysics. The last reference concerning the reflection of concave mirrors is one of the principle studies in Optics that mechanically reveals the principle of least action.]