Pierre Louis Maupertuis and the Principle of Least Action

Two years ago, I published my second post, entitled The Best of All Possible Worlds. Therein, I quoted from Gottfried Wilhelm Leibniz’s (1646-1716) correspondence with Samuel Clarke (1675-1729), in which Leibniz was critiquing Isaac Newton’s (1643-1727) Philosophiæ Naturalis Principia Mathematica.

Sir Isaac Newton and his followers also have a very odd opinion concerning the work of God. According to them, God Almighty needs to wind up his watch from time to time, otherwise it would cease to move. He had not, it seems, sufficient foresight to make it a perpetual motion. No, the machine of God’s making is so imperfect, according to these gentlemen, that he is obliged to clean it now and then by an extraordinary concourse, and even to mend it, as a clockmaker mends his work, who must consequently be so much the more unskillful a workman as he is more often obliged to mend his work and to set it right. According to the opinion, the same force and vigor remains always in the world and only passes from one part of matter to another agreeably to the laws of nature and the beautiful pre-established order. I hold that when God works miracles, he does not do it in order to supply the wants of nature, but those of grace. Whoever thinks otherwise must needs have a very mean notion of the wisdom and power of God. [pp.320-321]¹

I continued on as follows:

All of a sudden, Leibniz’s “fundamental assumption that God has chosen the best of all possible worlds” [Leibniz, Theodicy, p.168] takes on a clear meaning, beyond the simple moral arguments that Voltaire parodied in his Candide. The Universe is a perpetual motion “machine”, and since it has already been proven that it is impossible to construct a perpetual motion mechanical device, the logical conclusion is that the Universe cannot simply be understood through the principles of mechanics, nor will there ever be a Heat Death, as predicted by theoreticians working with the Second Law of Thermodynamics.

And I concluded with «À suivre» (To follow).

Little did I know that it would take me two years to continue on with my examination of Leibniz’s ideas in science, as I was focussing on 17th-century developments. With respect to mathematics, I did write about Leibniz and the infinitesimal calculus. See My Writings on the Infinitesimal Calculus.

I did mention Leibniz in one of my more recent posts, Pierre de Fermat's Principle of Least Time:

Fermat’s principle of least time was picked up by Gottfried Wilhelm Leibniz (1646-1716), who wrote in his New Essays on Human Understanding, which was finished in 1704, but published posthumously, in 1765:

…the maxim that nature acts by the shortest way, or at least by the most determinate way, is sufficient by itself to explain almost the whole of optics, including the optics of reflection and refraction, i.e. the whole of what goes on ‘outside us’ in the actions of light. [Leibniz², p.423]

At the time of my writing, I was under the assumption that the Principle of Least Time was a special case of the Principle of Least Action, which played a key rôle in the development of 18th century physics and mathematics, and that Leibniz was responsible for the latter principle. As an example of the influence of that principle, consider the beginning of a summary biography of Joseph-Louis Lagrange (1736-1813):

Lagrange was born in Turin in 1736. He rapidly became attracted to algebraic aspects of mathematics, and adopted positions involving the algebrisation of branches of the subject: the calculus founded on the Taylor-series expansion of a function, and also enriched by the calculus of variations on his own highly algebraic form with the δ operator; dynamics founded as far as possible on the principle of least action, and then reduced to statics via d’Alembert’s principle and the principle of virtual velocities.³ [my emphasis]

Pierre Louis Maupertuis

But when I started to look into this in more detail, I was surprised to find out that Pierre Louis Maupertuis (1698-1759), president of the Berlin Academy of Science, is considered today to have first stated the Principle of Least Action in 1744, whose key paragraphs are below:

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. Hence, the action increases linearly with the speed of the body and with the distance travelled.

This action is the true expense of Nature, which she manages to make as small as possible in the motion of light.⁴

I note that Maupertuis (incorrectly) takes sides with Descartes (1596-1650) and Newton, against Fermat (1607-1665) and Leibniz, assuming that light travels faster through denser media.⁵

Johann Samuel König

In my search, I came across the Wikipedia entry for “Maupertuis’s principle”⁶. Therein I read that a Swiss mathematician, Johann Samuel König (1712-1757), published a note in 1751 that he found a 1707 letter written by Leibniz to Jacob Hermann (1678-1733), clearly demonstrating that Leibniz had primacy over Maupertuis. König was challenged to present the original letter, which he could not. He had copied the original, which was in the hands of Samuel Henzi (1701-1749), who had been condemned to death for participating in a anti-monarchical coup attempt in Bern, and it is quite possible that the original letter was burnt along with parts of Henzi’s estate.

Leonhard Euler

So then the Berlin Academy of Science took a virulent turn against König, who at the time was working as librarian in Orange, the Netherlands, and Leonhard Euler (1707-1783) penned an astonishingly vicious letter against König, whose concluding paragraph reads as follows:

These matters are such that they may be reported. The quotation is intrinsicially suspicious; and Mr. König, after learning that the original letter of Mr. Leibniz could not be found in the papers of Henzi (to whom he had referred), has not produced the original, nor has he been able to identify the place where it is preserved. Hence, it is assuredly obvious that his cause is bad, and that this quotation is a forgery, either to put Mr. de Maupertuis in the wrong, or to exaggerate (as if by pious fraud) the praises of the great Leibniz, who without question does not need such help. Having weighed these considerations appropriately, the Academy does not hestitate to declare this quotation fraudulent and to strip it of any credibility it might have ever possessed.⁷ [my emphasis]

Voltaire

Voltaire was in Frederick the Great’s (1712-1786) Sanssouci court [Sanssouci is a suburb of Berlin] at the same time. In fact, it was Voltaire who had recommended Maupertuis to Frederick for the position of President of the Berlin Academy of Science. Voltaire also knew König very well, as the latter had taught Émilie du Chatelet (1706-1749), Voltaire’s long-time lover, about the calculus and the works of Leibniz. Du Chatelet’s translation of Newton’s Principia into French was considered to be the reference until the 21st century.

Voltaire was scandalized about the treatment of König, a known Leibniz scholar who had spent much of his life sifting through the Leibniz archives in Hannover, and published in 1752-1753 a scathing satirical series of tracts, as only Voltaire could write, mocking Maupertuis and Frederick, under the title Histoire du Docteur Akakia et du Natif de St Malo⁸ [History of Doctor Akakia and the native of St-Malo], Maupertuis being a native of St-Malo. Voltaire’s tracts were burnt in public, he was held for three weeks, and then expelled from Prussia.

So who was right? If we read the “Maupertuis’s principle” Wikipedia entry, everything is simple:

In 1751, Maupertuis's priority for the principle of least action was challenged in print (Nova Acta Eruditorum of Leipzig) by an old acquaintance, Johann Samuel Koenig, who quoted a 1707 letter purportedly from Gottfried Wilhelm Leibniz to Jakob Hermann that described results similar to those derived by Leonhard Euler in 1744.

Maupertuis and others demanded that Koenig produce the original of the letter to authenticate its having been written by Leibniz. Leibniz died in 1716 and Hermann in 1733, so neither could vouch for Koenig. Koenig claimed to have the letter copied from the original owned by Samuel Henzi, and no clue as to the whereabouts of the original, as Henzi had been executed in 1749 for organizing the Henzi conspiracy for overthrowing the aristocratic government of Bern. Subsequently, the Berlin Academy under Euler's direction declared the letter to be a forgery and that Maupertuis, could continue to claim priority for having invented the principle. Curiously Voltaire got involved in the quarrel by composing Diatribe du docteur Akakia (“Diatribe of Doctor Akakia”) to satirize Maupertuis’s scientific theories (not limited to the principle of least action). While this work damaged Maupertuis’s reputation, his claim to priority for least action remains secure. [my emphasis]

Attached to the last sentence is a reference to a 1942 article by Jerome Fee⁹, which I had a look at. Therein, I found this remarkable paragraph:

There are two cardinal points which must be understood in connection with this famous episode. The first is, that Maupertuis’s title to fame for the discovery of least action was not even challenged. Had Koenig succeeded in proving the authenticity of his copy, it would not have detracted from the brilliance of Maupertuis’s achievement. As he was the first to publish, his place in history is secure. Newton and Liebnitz, for example, both discovered the integral calculus; the two men were, therefore, of equal genius in this respect. Newton, however, was first to publish, and consequently deserves to be mentioned first in connection with this branch of mathematics. [my emphasis]

Of course, this is complete nonsense. Leibniz published his Nova Methodus pro Maximis et Minimis [New Method for maxima and minima] in 1684, while Newton’s De Methodis Serierum et Fluxionum [Method of Fluxions] was only published posthumously in 1736. If Fee could not even get these well-known facts correct, why should I consider what he wrote about the Maupertuis affair to be of value?

Willy Kabitz

A copy of Leibniz’s letter to Hermann was found by Willy Kabitz (1876-1942) in 1913. So König should have been exonerated, but in fact the debate rages on, even today. I came across a 2016 article published in Annalen der Physik by Harmut Hecht, whose title says everything: “Gottfried Wilhelm Leibniz and the origin of the principle of least action—a never ending story.”¹⁰

This debate about who first proposed the Principle of Least Action is fascinating, and does raise some questions. For me, most important is why Euler would have gone out of his way to defend Maupertuis and accuse König of forgery. And, of course, reading Voltaire’s scathing words is always fun.

But, in my opinion, this argument over primacy hides the essential truth: that the Principle of Least Action follows naturally from Leibniz’s teleology. This will be the subject of my next post, in which I will look at a small text published in 1928 by Adolf Kneser (1862-1930), Das Prinzip der Kleinsten Wirkung von Leibniz bis zur Gegenwart¹¹ [The Principle of Least Action from Leibniz to the Present].

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1. Roger Ariew and Daniel Garber, eds. G.W. Leibniz: Philosophical Essays, Hacket Publishing Company, 1989.

2. G.W. Leibniz. New Essays on Human Understanding. Translated and edited by Peter Remnant and Jonathan Bennett. Cambridge University Press, 1996.

3. Themistocles M. RASSIAS. Eigenvalues of the Laplacian. p.325. In Mauro Francaviglia, editor. Mechanics, Analysis and Geometry: 200 years after Lagrange. North-Holland, 1991.

4. Pierre Louis Moreau de Maupertuis. Accord between different laws of Nature that seemed incompatible (1744). Translated from French by Wikisource. https://en.wikisource.org/wiki/Translation:Accord_between_different_laws_of_Nature_that_seemed_incompatible

5. P. Radelet-de Grave. La diatribe du Docteur Akakia, Médecin du pape. Rapport UCL-IPT-97-10, Université Catholique de Louvain.

6. Maupertuis’s principle. https://en.wikipedia.org/wiki/Maupertuis%27s_principle

7. Leonhard Euler. Investigation of the letter, allegedly written by Leibniz (1752). Translated from French by Wikisource. https://en.wikisource.org/wiki/Translation:Investigation_of_the_letter_of_Leibniz

8. Voltaire. Histoire du Docteur Akakia et du Natif de St Malo. https://www.monsieurdevoltaire.com/article-facetie-histoire-du-docteur-akakia-partie-1-111976122.html

9. Jerome Fee. Maupertuis and the principle of least action. American Scientist 30(2):149-158, 1942.

10. Hartmut Hecht. Gottfried Wilhelm Leibniz and the origin of the principle of least action—a never ending story. Annalen der Physik 528 (9-10):641-646, 2016.

11. Adolf Kneser. Das Prinzip der Kleinsten Wirkung von Leibniz bis zur Gegenwart. Springer Fachmedien Wiesbaden, 1928.