Pierre Louis Maupertuis
Pierre-Louis Moreau de Maupertuis (July 17, 1698 – July 27, 1759) was a French mathematician, philosopher and man of letters. He became the Director of the Académie des Sciences, and the first President of the Berlin Academy of Science, at the invitation of Frederick the Great. Maupertuis made an expedition to Lapland to determine the shape of the earth. He is often credited with having invented the principle of least action.
- I must now explain what I mean by the quantity of action. A certain action is necessary for the carrying of a body from one point to another: this action depends on the velocity which the body has and the space which it describes; but it is neither the velocity nor the space taken separately. The quantity of action varies directly as the velocity and the length of path described; it is proportional to the sum of the spaces, each being multiplied by the velocity with which the body describes it. It is this quantity of action which is here the true expense (dépense) of nature, and which she economizes as much as possible in the motion of light.
- Nature always uses the simplest means to accomplish its effects.
- Formulation of the principle of least action, as stated in Mémoires de l'académie royale des sciences (Accord between different laws of Nature that seemed incompatible), 1748, 417-426 (15 April 1744).
- We cannot doubt that all things are regulated by a supreme Being, who, while he has imprinted on matter forces which show his power, has destined it to execute effects which mark his wisdom... Let us calculate the motion of bodies, but let us also consult the designs of the Intelligence which makes them move.
- Les Oeuvres De Mr. De Maupertuis (1752) vol. iv p. 22; as quoted by Philip Edward Bertrand Jourdain, The Principle of Least Action (1913) p. 6.
Accord de différentes loix de la nature qui avoient jusqu’ici paru incompatibles (1744)
- The most beautiful discoveries since the Renaissance, indeed since the beginnings of all science, are the laws governing light, whether moving through a uniform medium, or being reflected from an opaque surface, or changing direction upon entering another transparent medium.
- The ancient Greeks knew the laws that govern the propagation of light in a uniform medium and upon its reflection. However, the law governing the refraction of light as it passes from one transparent medium to another was unknown until the last century. Snell discovered it, Descartes tried to explain it and Fermat criticized his explanation. Since then, many great geometers have researched the problem, although no one has yet found a way of harmonizing the law of refraction with more fundamental laws that Nature must obey.
- The first law is the same for both light and material bodies; they both move in a straight line, as long as they are not deflected by an outside force.
The second law is also the same as that governing the reflection of an elastic ball from an impenetrable surface. Mechanics shows that such a ball is reflected from such a surface so that its angle of reflection equals its angle of incidence, as observed for light.
But the third law still requires a plausible explanation. The passage of light from one medium to another exhibits behavior that is totally different from a ball moving through different media.
- After meditating deeply on this topic, it occurred to me that light, upon passing from one medium to another, has to make a choice, whether to follow the path of shortest distance (the straight line) or the path of least time. But why should it prefer time over space? Light cannot travel both paths at once, yet how does it decide to take one path over another? Rather than taking either of these paths per se, light takes the path that offers a real advantage: light takes the path that minimizes its action.
- Now I have to define what I mean by "action". When a material body is transported from one point to another, it involves an action that depends on the speed of the body and on the distance it travels. However, the action is neither the speed nor the distance taken separately; rather, it is proportional to the sum of the distances travelled multiplied each by the speed at which they were travelled.
- The refraction of light agrees with the grand principle that Nature always uses the simplest means to accomplish its effects. From this principle, can be derived whenever light passes from one medium to another, the ratio of the sine of the angle of refraction to the sine of the angle of refraction equals the inverse ratio of the speeds at which light moves in each medium.
But this "budget", this expense of action that Nature minimizes in the refraction of light, is it also minimized in the direct propagation and reflection of light? Yes, it always has the smallest possible value.
- Having discovered the true principle, I then derived all the laws that govern the motion of light, those concerning its direct propagation, its reflection and its refraction. I reserve for particular members of our Assembly the geometrical demonstration of my theory.
I know the distaste that many mathematicians have for final causes applied to physics, a distaste that I share up to some point. I admit, it is risky to introduce such elements; their use is dangerous, as shown by the errors made by Fermat and Leibniz in following them. Nevertheless, it is perhaps not the principle that is dangerous, but rather the hastiness in taking as a basic principle that which is merely a consequence of a basic principle.
- Let us calculate the motion of bodies, but also consult the plans of the Intelligence that makes them move.
It seems that the ancient philosophers made the first attempts at this sort of science, in looking for metaphysical relationships between numbers and material bodies. When they said that God occupies himself with geometry, they surely meant that He unites in that science the works of His power with the perspectives of His wisdom.
From the all too few ancient geometers who undertook such studies, we have little that is intelligible or well-founded. The perfection which geometry has acquired since their time puts us in a better position to succeed, and may more than compensate for the advantages that those great minds had over us.
Les Loix du Mouvement et du Repos, déduites d'un Principe Métaphysique (1746)
- On the 15th of April 1744, I described the principle upon which the following work is based, in the public assembly of the Royal Academy of Sciences of Paris, as reported in the Acts of that academy.
At the end of the same year, Professor Euler published his excellent book Methodus inveniendi lineas curvas maximi minimive proprietate gaudentes. In a supplement to his book, this illustrious geometer showed that, in the trajectory of a particle acted on by a central force, the velocity multiplied by the line element of the trajectory is minimized.
This observation gave me great pleasure, as a beautiful application of my principle to the motion of the planets, which is determined by this principle.
From the same principle, I will now try to derive higher and more important truths.
- It is interesting to note that Newton was not impressed by Descartes' great argument for God's existence derived from the idea of a perfect Being, nor by other metaphysical arguments that we have mentioned; yet Newton's own arguments for God's existence from the uniformity and suitability of different parts of the universe would not have seemed like proofs to Descartes.
- Despite the disorder observed in Nature, one finds enough traces of the wisdom and power of its Author that one cannot fail to recognize Him.
- One should not be deceived by philosophical works that pretend to be mathematical, but are merely dubious and murky metaphysics. Just because a philosopher can recite the words lemma, theorem and corollary doesn't mean that his work has the certainty of mathematics. That certainty does not derive from big words, or even from the method used by geometers, but rather from the utter simplicity of the objects considered by mathematics.
- The supreme Being is everywhere; but He is not equally visible everywhere. Let us seek Him in the simplest things, in the most fundamental laws of Nature, in the universal rules by which movement is conserved, distributed or destroyed; and let us not seek Him in phenomena that are merely complex consequences of these laws.
- Everything is so arranged that the blind logic of mathematics executes the will of the most enlightened and free Mind.
- It is only mental habit that prevents us from realizing how miraculous it is that motion can be passed from one body to another. Once our eyes have opened, nothing is so striking. For those who have never thought about it, it doesn't seem mysterious; by contrast, those who have meditated on it may despair of ever understanding it.
- Research into motion was not to the liking (or perhaps not within the scope) of the ancients, so that we may consider it as a completely new science. How could the ancients have discovered the laws of moiton, given that some philosophers reduced all their speculations about motion to sophistic disputes, whereas others denied that motion existed at all?
- A true philosopher does not engage in vain disputes about the nature of motion; rather, he wishes to know the laws by which it is distributed, conserved or destroyed, knowing that such laws is the basis for all natural philosophy.
- The elements that make up all other bodies, these must be bodies that are perfectly inelastic, undeformable and unchangeable.
- After so many great men have worked on this subject, I almost do not dare to say that I have discovered the universal principle upon which all these laws are based, a principle that covers both elastic and inelastic collisions and describes the motion and equilibrium of all material bodies.
This is the principle of least action, a principle so wise and so worthy of the supreme Being, and intrinsic to all natural phenomena; one observes it at work not only in every change, but also in every constancy that Nature exhibits. In the collision of bodies, motion is distributed such that the quantity of action is as small as possible, given that the collision occurs. At equilibrium, the bodies are arranged such that, if they were to undergo a small movement, the quantity of action would be smallest.
The laws of motion and equilibrium derived from this principle are exactly those observed in Nature. We may admire the applications of this principle in all phenomena: the movement of animals, the growth of plants, the revolutions of the planets, all are consequences of this principle. The spectacle of the universe seems all the more grand and beautiful and worthy of its Author, when one considers that it is all derived from a small number of laws laid down most wisely. Only thus can we gain a fitting idea of the power and wisdom of the supreme Being, not from some small part of creation for which we know neither the construction, usage, nor its relationship to other parts. What satisfaction for the human spirit in contemplating these laws of motion and equilibrium for all bodies in the universe, and in finding within them proof of the existence of Him who governs the universe!
- When a change occurs in Nature, the quantity of action necessary for that change is as small as possible.
- The quantity of action is the product of the mass of the bodies times their speed and the distance they travel. When a body is transported from one place to another, the action is proportional to the mass of the body, to its speed and to the distance over which it is transported.
Quotes about Maupertuis
- According to Du Bois Reymond, Maupertuis's teleological tendencies showed themselves early in his career in speculations as to what grounds the Creator could have had for preferring the law of the inverse square to all other possible laws of attraction. ...
Maupertuis read to the Paris Academy on the 20th of February, 1740, a memoir entitled: "Loi du Repos des Corps." He began by remarking that demonstrations a priori of such principles as that of the conservation of vis viva "cannot apparently be given by physics; they seem to belong to some higher science." ...
Maupertuis's first enunciation of the law of the least quantity of action was in a memoir read to the French Academy on April 15th, 1744, entitled "Accord de différentes Loix de la Nature qui avoient jusqu'ici paru incompatibles." The laws in question appear to be those of the reflection and of the refraction of light. When a ray of light in a uniform medium travels from one point to another, either without meeting an obstacle or with meeting a reflecting surface, nature leads it by the shortest path and in the shortest time. But when a ray is refracted by passing from a uniform medium to one of different density, the ray neither describes the shortest space nor does it take the shortest time about it. As Fermat showed, the time would be the shortest if light moved more quickly in rarer media, but Newton proved that, as Descartes had believed, light moves more quickly in denser media. Maupertuis's discovery was that light neither takes always the shortest path nor always that path which it describes in the shortest time, but "that for which the quantity of action is the least."
- After having worked in the theory of light and gravitation, he announced, in 1744, a new minimum principle, the Principle of Least Action, from which he claimed he could deduce the behavior of light and masses in motion. The principle asserts that nature always behaves so as to minimize an integral known technically as action, and amounting to the integral of the product of mass, velocity, and distance traversed by a moving object. From this principle he deduced the Newtonian laws of motion. With sometimes suitable and sometimes questionable interpretation of the quantities involved, Maupertuis managed to show that optical phenomena, too, could be deduced from this principle. Hence, to an extent at least, he succeeded in uniting the optics of the eighteenth century and mechanical phenomena. ...
Maupertuis advocated his principle for theological reasons. ...He ...proclaimed his principle to be not only a universal law of nature but also the first scientific proof of the existence of God, for it was "so wise a principle as to be worthy only of a Supreme Being.
- Morris Kline, Mathematics and the Physical World (1959) Ch. 25: From Calculus to Cosmic Planning, p. 438.
- Maupertuis really had no principle, properly speaking, but only a vague formula, which was forced to do duty as the expression of different familiar phenomena not really brought under one conception. ...Maupertuis' performance, though it had been unfavorably criticized by all mathematicians, is, nevertheless, sort of invested with a sort of historical halo. It would seem almost as if something of the pious faith of the church had crept into mechanics. However, the mere endeavor to gain a more extensive view... was not altogether without results. Euler, at least, if not also Gauss, was stimulated by the attempt of Maupertuis.