Translation:Investigation of the letter of Leibniz

by Leonhard Euler

Mr. de Maupertuis, President of the Royal Academy, has shown (by several very convincing arguments) that the action is always minimized, not only at equilibrium but also in the motions of bodies under external forces; this remarkable principle of least action expresses the most general law of Nature. Professor König, however, has tried in several ways to destroy this great discovery. First, he does not believe that it pertains to equilibrium problems, which he suggests corresponds not to the minimum of the action, but rather to where the action is zero; to support his case, he gives several examples in which minimization of the action results in zero action. However, this criticism is irrelevant, since it is recognized in the calculus of minima and maxima that a minimized quantity disappears entirely. Although König's "zero action principle" may apply in certain cases, it cannot be extended to all cases; for there are an infinite number of cases in which the action is minimized, but not zero. Hence, beyond any doubt, Nature's goal is to minimize the action, rather than set it to zero. If we consider the example of the "chain curve" (the catenary), where the total action is defined as the distance of the center of gravity of the chain to the center of the Earth, clearly this distance cannot be zero; rather, and most effectively, it should adopt the smallest possible value. It is indeed true that the force of gravity, given no resistance, would pull the chain to the center of the Earth, and that the only equilibrium position of the chain in that case would be at the center of the earth. However, since the chain is impeded at its points of suspension, the effect of gravity is limited to making the distance between the centers of gravity of the chain and the Earth as small as possible. Of course we agree with Mr. König that the action is indeed zero in those cases where it is true; however, this is not always possible, as we may see from the brachistochrone problem; rather, the action is to be minimized. It is as though Nature wishes to achieve some effect, and approaches it as closely as possible. Not only is the truth of this principle evidence, but also the reasoning upon which it is based, and to destroy entirely the objections of Mr. König who, far from striking a blow against this principle, has wonderfully helped to confirm it. It is a specious criticism of the principle of least action to note that there are certain cases in which the action disappears entirely, since the action can assuredly not become smaller than zero. Nevertheless, this objection might have some merit, if the action were reduced to zero in every case of equilibrium, as Mr. König seems to insinuate; but that cannot be proven, since there is an infinite number of cases in which the action is clearly not zero, but rather adopts the smallest possible value. In addition to the catenary problem, Mr. D. Bernoulli and Mr. Euler have shown that the elastic curves of every type and other figures adopted by flexible bodies (when the forces acting on them are in equilibrium) may be found by a method of maxima and minima applied to a formula that always contains the action, which is to be minimized and not set to zero.

Mr. König also criticized Maupertuis' excellent principle of least action for motion and, although he did not attack its validity, his remarks require serious attention. Being unable to shake the foundations of this principle, König now seeks to rob Mr. de Maupertuis of the glory of its invention and give it to Mr. Leibniz. For this purpose, he quotes from a letter allegedly written by Mr. Leibniz to Mr. Hermann, as follows

The action is not at all what you imagine, since time is involved. The action is rather the product of the mass and the time, or the time multiplied by the kinetic energy. I have noticed that, in changes of motion, the action is either minimized or maximimized. One can derive from that several important results, such as the paths followed by bodies attracted to one or several centers of force. I would like to address these topics (among others) in the second part of my Dyanmics, which I have suppressed, due to the biased reception of the first part.

It follows from this passage that Mr. Leibniz not only knew the sublime principal of least action, but also was able to derive the paths of bodies attracted by one or several centers of force. This raised the first suspicions about this passage, all the more since the letter itself (from which this quote was taken) has never been published. Therefore, it became necessary to determine whether this letter was indeed written by Mr. Leibniz, by identifying the place where the original existed; otherwise, such evidence is worthless, especially in a case like this one, which concerns a discovery of such importance. For these reasns, it became utterly necessary to prove the authenticity of the quotation. First, the published letters of Mr. Leibniz were carefully reviewed, letters in which his sublime thoughts on every type of science can be appreciated and admired; however, Mr. Leibniz did not reveal this admirable principle of least action to any other friend. In particular, Leibniz had a long and intimate friendship with the renowned Mr. Johann Bernoulli, with whom he often conversed about dynamics; although their long correspondence contains several discussions of kinetic energy and the correct way of calculating the action, it does not contain any indication that Mr. Leibniz ever thought of the principle of least action. Considering that Mr. Leibniz hid nothing from Mr. Bernoulli anything that could support his theory of kinetic energy or extend its application, one cannot imagine any reason to keep this least action principle secret.

Moreover, concerning the determination of paths followed by bodies attracted to one or several centers of force, the method of maxima and minima necessary to determine these paths had not been sufficiently developed to carry out that calculation, even if it had been known that the action ought to be minimized. And although the glory of Mr. Leibniz rests on several discoveries of the first rank, the principle of least action would be equal to them all; it cannot be believed that he would impart this discovery to no one except Mr. Hermann.

Since these considerations had already weakened the credibility of this quotation, President de Maupertuis (who proposed the principle of least action as his own invention) felt it necessary to search for proof of his priority, to protect himself from every suspicion of plagiarism. Although there is no trace of the least action principle in the published writings of Mr. Leibniz, one must hinder the potential accusation that he had taken his principle from the same letter of Mr. Leibniz, which perchance might have come into his hands. Since no one is more concerned with the credibility of the letter than Mr. de Maupertuis, he began to write to Mr. König himself, asking him (in a friendly letter dated 28 May 1751) to show the original of the letter and establish its authenticity. Mr. König did not reply until June 26th. He claimed that the letter had been communicated to him by the famous Henzi, who was decapitated in Berne three years ago for inciting sedition against the State, and that Henzi, being much attached to the study, had collected letters attributed to Leibiz and would have published them, had he not been prevented by fate. At the same time, M. König sent to Mr. de Maupertuis a copy of the entire letter that he had quoted, which was dated 16 October 1707. The quotation was found near the end, but with a slight change of wording; instead of the original wording

The action is rather the product of the mass and the time, or the time multiplied by the kinetic energy.

which contains a clear contradiction, the letter itself contained the correct wording

The action is rather the product of the mass, distance and speed, or the time multiplied by the kinetic energy.

This difference in wording, which could not have been a simple printer's error, increased considerably the suspicions surrounding this fragment. Although the entire letter could not be rejected, it seemed plausible that certain phrases and even entire sentences had been introduced, and that one did not make them match sufficiently with the remaining text. On top of this suspicion, it seemed regrettable that the credibility of this fragment depended entirely on the testimony of a man who had lost his head; and that such testimoney was insufficient to establish its authenticity. Nevertheless, Mr. de Maupertuis felt that he should not confine himself to Mr. König's response; and, given that no one's papers are guarded more carefully than those of an enemy of the State, he asked the Marquis de Paulmy, Ambassador of SMTC in Switzerland, to use his influence to have the papers left behind by Henzi in Berne investigated. A careful investigation showed not only that Mr. Henzi had no letters of Leibniz, but not even the smallest sign that he had ever had even one such letter in his possession.

On October 7th of last year (1751), President de Maupertuis reported to the Royal Academy everything that had occurred up to that point, with the intention that the affair be handled according to the proper forms, and that it be ended with suitable measures. The Academy judged immediately that, in a question that concerns such an important discovery, one ought to conduct a careful examination of what pertains to the great Leibniz and to what its members have a right. The Academy charged Mr. Formey, in his post as Secretary, to write to Mr. König a letter (which was written the following day) to request again in a friendly way that he confirm the authenticity of the fragment he had alleged, and to produce the original letter of Leibniz within a month. However, Mr. König had used delaying tactics in this affair, and to prevent him from claiming that the Academy's letter had been lost, the Academy also sent a copy of its letter to Mr. de Hellen, the chargé des affaires of his Majesty for the General States, who testified that the letter was delivered on the 19th of October. On the 11th of December, at which point over a month had passed without any response, the Academy decided to reiterate its request. One is certain that Mr. König received that second letter, along with a letter written by the President, who delivered it himself on the 6th of January of this year (1752).

During this time, the King himself, the Protector of the Academy, sent a letter to the Magistrate in Berne, requesting him to search for the letter (reputedly) of Leibniz among all possible papers of Henzi. This search was carried out by assistants to the Magistrate himself, with the result that absolutely nothing was found.

Before the second letter of the Academy to Mr. König had arrived, Mr. de Maupertuis himself received a letter dated the 10th of December, which he read to the Academy on the 23rd of the same month. Although the ;etter was full of politeness, Mr. König seemed far off from producing the original letter of Mr. Leibniz, or even to state the place where it existed. Instead he tried to divert the attention of the Academy from this question, and to implicate other controversies foreign to the question at hand. He complained a lot about the injustice to which he was being subjected, expecting the freedom that scholars enjoy by custom, as though it were a injust to demand of a scholar that he provide proof of a letter that he made public.

The same day that Mr. de Maupertuis read his own response to this letter, in which he defended the right of the Academy to establish the authenticity of writings that helped to determine who deserved credit for certain discoveries, and that there was nothing wrong in demanding of Mr. König that he produce the original letter of Mr. Leibniz; no other controversy was involved and nothing else was reuiqred of him except that he establish the authenticity of the letter he had cited, which should not be refused by anyone, particularly in their own defence. In the same letter, Mr. de Maupertuis reported to Mr. König the two searches that had been carried out in vain in Berne, as ordered by the King and by the Ambassador of France. This reply was endorsed unanimously by the Academy was delivered on the 6th of January of this year (1752), along with the second letter of the Academy to Mr. König.

The reply to these letters was dated the 15th of February. In this letter, as in its predecessor, Mr. König complained bitterly about what he regarded as a violation of his rights as a scholar of letters with the involvement of the Academy's authority in a controversy of literature. He continued to dissimulate concerning the state of the question, and did not respond at all to the one request made of him, that he confirm the authenticity of the fragment he had quoted by producing the original letter of Mr. Leibniz. He looked everywhere for an escape. Sometimes he would say that the words of Leibniz did not remove anything from the discovery of the principle of least action, that Mr. de Maupertuis preserved the right to attribute it to himself, since the letter was completely unknown up to the present and that Mr. Leibniz had not revealed or extended his ideas on the subject. Sometimes he would claim that Mr. Leibniz was even opposed to this principle, having perhaps had a different principle in mind, where Mr. König tried to cast doubt on the principle itself, rather than dealing with the question at hand, the authenticity of the letter. Yet in the end, he seemed to be moved to action, saying that he had written to a friend to ask him to look for the original, and that he was still expecting his reply.

It should be noted that, in the same letter, Mr. König appealed to his first publication where, seeking to show that Mr. Leibniz had known of this discovery for a long time, he said he had suppressed this discovery. (unclear translation) Although this assertion would have to be similarly proven, and would require a certain authority to give it credibility in the province of letters, it seems implausible that Mr. Leibniz, tired of the criticism of evil judges, would not want to publish the second part of his Dynamics, to the irreparable harm of science, so that the ignorant would not regard his principles as monstrous chimeras. However, one sees well enough that this suppression has little to do with the present affair, which is restricted to demanding of Mr. König that he justify his quotation by producing the original letter.

At the same time, additional searches were carried out in Basel, where Mr. Hermann died, to dig up the letters which he had received from Mr. Leibniz. The results of these searches showed clearly that such letters had long been in the hands of Mr. König, and that it would be useless work to look for them in other places. No others have been found, and there is not yet any reason to suspect that some have been lost.

The Magistrate in Basel was required by the King to examine in detail all the papers left behind by Mr. Hermann and preserved by his heirs. Mr. Johann Bernoulli, Professor of Mathematics, who was charged with this commission, found only three letters of Mr. Leibniz, which contained no mention of the principle of least action.

In a letter dated the 12th of March, Mr. König said that he had searched in vain, with the help of a friend, for this letter of Leibniz among the papers left behind by Mr. Hermann and he sent at the same time the reply he had received on the subject. He added that the attention he was giving to this reply was the reason causing him to treat the main question only lightly, but that he believed that his demonstrated diligence in searching for the original letter ought to satisfy the Academy.

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.

Approved by the assembled Academy on Thursday, 13 April 1752

Present:

Mr. de Keith, Curator
Mr. de Redern, Curator

Mr. de Marschall, Honorary
Mr. de Cagnony, Honorary

Mr. Eller, Director
Mr. Heinius, Director
Mr. Euler, Director

Mr. Formey, Secret. perpet.

Ordinaries

Mr. Pelloutier
Mr. Pott
Mr. Spraegel
Mr. Küstler
Mr. M. M. Ludolff
Mr. Becmann
Mr. Gleditsch
Mr. C. L. Ludolff
Mr. de Beausobre
Mr. Kies
Mr. Sulzer
Mr. la Lande, Assoc. ext.

Mr. Hesse, stranger
Mr. Hirzel, stranger

Curator de Keith remitted to the Academy the following letter from President de Maupertuis, which the Secretary read.

Sir,

Since the Academy has heard today the summary of Mr. Euler on the subject of the letter cited by Mr. König and has pronounced its judgment on the affair, I have the honor of reporting that I was in no way implicated. Since I do not wish any reparations from Mr. König, I ask the Academy only to continue to ask for verification, i.e., evidence for the authenticity of the letter of Mr. Leibniz that Mr. König cited. I have the honor of being, etc.

Maupertuis

The Secretary assembled for the Bureau all the documents pertaining to the affair between Mr. König and the Academy, gave a summary and read in their entirety certain documents that had never been communicated to the Academy (such as the letter from Mr. König to Mr. Formey of 15 February 1752, a letter of Mr. de Maupertuis to Mr. König of 9 February 1752 and the reply of Mr. König to that letter of 12 March 1752), to which he attached the latest evidence concerning the letters of Mr. Leibniz, sent to SM by MM the Magistrates of Basel.

Director Euler then read a report in Latin, where he alleged that all the evidence (both concrete and that arrived at by reasoning) showed that the passage alleged by Mr. König to have been quoted from a letter of Mr. Leibniz was suspicious on its own and clearly false given the circumstances.

Upon this point, Curator de Keith polled all the present members of the Academy, with the following unanimous result:

that the passage published by Mr. König in the Acta Eruditorum of Leipzig as a quotation from a letter of Mr. Leibniz written in French to Mr. Hermann, appears to be a forgery and, consequently, does not carry even a shadow of authority concerning the legitimate rights of the Academy members involved in this affair, to take credit for the principles developed from their meditations and research (and for everything that may be derived from them) in reports adopted by the Academy and in its other publications; and that the conclusions drawn by Mr. Euler in his report are just and worthy, with all the force of the terms in which they were expressed. After considering the letter of Mr. de Maupertuis read at the beginning of the meeting, the Academy does not wish to press the matter further and concludes its deliberations concerning Mr. König, as it is authorized to do for such proceedings.

Keith.
Redern.
Eller.
Heinius.
Euler.
Formey Secr. perp.