Autograph letter signed, with important scientific content concerning the ‘vis viva’ controversy, from Basel, dated 27 July 1728, to Gabriel Cramer, ‘Professor of Mathematics,’ presently in London.

An important autograph letter from Johann Bernoulli (1667-1748), then one of the elder scientific statesmen of Europe, and still one of its greatest mathematicians, to his gifted student Gabriel Cramer (1704-52), who despite his youth had been appointed to the chair of mathematics at Geneva the previous year, but was now traveling through Europe and England making the acquaintance of the leading mathematicians of the day. The letter concerns the problem of vis viva (‘forces vives’, ‘living force’), one of the most controversial topics of the day. This was the question of whether it is (to use modern terminology) momentum (‘quantity of motion’, mass x velocity) or kinetic energy (‘living force’, mass x velocity2) which is the true measure of the ‘force’ between colliding bodies in motion. As with so many other issues, this controversy pitted the supporters of Leibniz against those of Newton. Bernoulli had recently published a major contribution to the dispute, Discours sur les Loix de la Communication du Mouvement (1727), supporting the Leibnizian position, in which he presented an analysis of vis viva in terms of balls moved by releasing compressed springs. This was attacked by the young English Newtonian Benjamin Robins (1707-51) in May 1728 in an article in The Present State of the Republick of Letters, in which he gave a detailed discussion of the impact of elastic bodies. This article won Robins many admirers in England. In the present letter, Bernoulli refutes Robins’ article, and writes that an experiment proposed by another English Newtonian, James Jurin (1684-1750), and carried out by the London instrument maker George Graham (1673-1751), involving dropping a lead weight onto an elastic plate, ‘prouve rien contre la théorie des forces vives’. Bernoulli also responds to a misunderstanding by Cramer of a point in his Discours which Cramer had raised in an earlier letter. In a postscript, Bernoulli conveys the compliments of his nephew, the mathematician Nicolas Bernoulli (1687-1759). Letters by Bernoulli are rare on the market, particularly those with significant scientific content.


Monsieur Robert Caille

Marchand Banquier, pour faire tenir à Monsieur Cramer, Professeur en mathematique presant à Londres


Ce mot de lettre n’est que pour vous don[n]er avis que je vous ecrivis jeudi passé une reponse à la votre du 22. Juin, que j’ai adressée à Mr. de Mairan à Paris. Elle contient quelques reflexions generales sur la piece de Mr. Robins, et une reponse à l’objection tirée de l’experience avec la plaque de cuivre par laquelle étant en oscillation on laisse tomber un poids de plomb. J’ai fait voir que cette experience ne prouve rien contre la theorie des forces vives, et qu’elle est semblable à celle qu’on feroit avec deux corps sans ressort dont l’un en mouvement choqueroit directement l’autre en repos, auquel si le choquant étoit egal, ne lui com[m]uniqueroit que la moitié de la vitesse et iroit avec lui après le choc de compagnie, en sorte que la moitié de la force vive paroitra étre perdue. Je vous avois aussi ecrit une tres grande lettre datée du 23. Mai en reponse à vos deux precedentes du 10. Mars et 15 Avril, mais de laquelle vous ne faites pas mention dans votre derniere du 22 Juin, auquel temps vous prairés [pourriez] déjà avoir reçû la mienne; ce silence me mettant en peine, je vous prie de m’en tirer au plutot pour savoir si en fin elle vous a été rendue : vous y aurés trouvé bien des choses pour la confirmation de la theorie des forces vives et une ample solution à votre difficulté, qui consistoit à me demander, d’où vient que c’est l’increment de la vitesse, et non pas celui de la force vive, qui dans un temps infiniment petit est proportionel à ce temps et à la pression: c’est-à-dire, pourquoi il faut faire du = pdx/u, et non pas df = pdx/u ? 

Je finis en vous temoignant que je suis toujours avec la plus parfait consideration Monsieur votre tres humble et tres obeissant serviteur

J Bernoulli

Bale, ce 27. Juillet 1728

  1. S. Mon neveu vous fait ses compliments ; il y a quelques semaines qu’il vous a ecrit une lettre sous l’adresse de Mr. Caille : dont je me sers aussi toujours en vous ecrivant.


Mr Robert Caille

Merchant Banker, for the attention of Mr. Cramer, Professor in mathematics present in London


This brief letter is only meant to give notice that I wrote you last Thursday a reply to your letter of 22 June which I addressed to Monsieur de Mairan in Paris. It contains some general reflections on Mr. Robins' essay, and a response to the objection derived from the experiment with an oscillating copper plate on which a lead weight is dropped. I have shown that this experiment proves nothing against the theory of the live force, and that it is similar to that which would be made with two inelastic bodies, one of which in motion would directly collide with the other at rest. If the colliding body was equal to the one at rest, it would convey to it only half the speed and go with it after the collision, so that half the force will appear to be lost. I also wrote you a very long letter dated 23 May in reply to your two previous ones of 10 March and 15 April, which you do not mention in your last letter of 22 June, though at that time you may have already received mine; this silence is distressing and I beg you to put an end to it and let me know if eventually it was delivered to you: you will have found [in it] many things confirming the theory of live forces and an ample solution to your difficulty, which consisted in asking me, how come that it is the increment of speed, and not that of the live force, which in an infinitely small time interval is proportional to this interval and to pressure; that is to say, why should we do du = pdx/u, not df = pdx/u?

I am ending this letter with the renewed assurance that I remain, with the most perfect consideration

Your very humble and very obedient servant

J Bernoulli

Basel, 27 July 1728

P.S. My nephew sends you his compliments; a few weeks ago, he wrote you a letter, care of Mr. Caille, whose address I also always use when writing to you.

“The vis viva problem, first formulated by Leibniz in the 1680s, centred on how to define quantity of motion and, to some extent, how to define force. Descartes had argued that the quantity of motion (or force) in the universe must remain constant, and that its measurement was the product of the quantity of matter and velocity (mv). Leibniz disagreed with Descartes’ assumption and argued that mv2 (what he called vis viva) defined the quantity of motion. (For clarity today, we would characterize the controversy as a confusion between conservation of momentum and conservation of kinetic energy, but in the mid-eighteenth century there was no concept of energy as we now understand it, nor was there agreement about what constituted force).

“In 1717, the correspondence between Leibniz and Samuel Clarke was published, which revealed that the Newtonians were the allies of the Cartesians on this particular point … The importance of the vis viva controversy was further revealed by the fact that in 1724 the Paris Académie des Sciences set the communication of motion as the subject of their prize competition. Although he did not win, Jean Bernoulli contributed a widely acclaimed essay supporting Leibniz [Discours sur les Loix de la Communication du Mouvement]” (Correspondence of James Jurin 1684-1750, p. 40).

“In his prize essay, Bernoulli started by denying the very existence of hard bodies ‘in the vulgar sense’, and went on to model the force of collision by analogy with the compression and release of springs. He predicated his analysis on the fundamental elasticity of matter, following Leibniz. Hardness, or ‘rigidity’, was equivalent to perfect elasticity, said Bernoulli. By looking at the motion imparted by springs to rigid bodies, he was able to show that the force of the spring was proportional to the square of the velocity it gave to the body. He presented his ideal springs as a thought experiment, to give a measure of concreteness to the abstract consideration of force and motion. Bernoulli framed his systematic and comprehensive analysis of collisions as a demonstration of the true measure of living force, or the force of motion – something that he claimed Leibniz had only proved indirectly. His demonstration used the integral calculus to obtain the equation ½v2 = pdx (for unit mass) [where p is the pressure of the spring]. From what he called ‘the familiar law of acceleration’, according to which pressure equals mdv/dt, he deduced that the vis viva produced or destroyed by the action of a spring (or by the elastic surface of a body) is proportional to the distance through which the spring extends, or is compressed” (Terrall, pp. 192-3).

In May 1798 the English mathematician and military engineer Benjamin Robins published a point-by-point refutation of Bernoulli’s prize essay (The Present State of the Republick of Letters, Vol. 1, Article XXIII, pp. 357-372), to which Bernoulli refers in the present letter. Robins concludes (p. 372): “I think, I have proved, that nothing Mr. Bernoulli has urged in defence of Mr. Leibniz’s opinion, is in any way conclusive; that many parts of his discourse are contradictory; and that all his determinations of the laws of motion are wrong, since they are by him applied to bodies, which only perfectly restore themselves; whereas they are true in none but such as restore themselves in the same time they were compressed.”

“The Swiss mathematician Gabriel Cramer wrote Jurin an extensive letter in January 1729, once again arguing the case for vis viva. Cramer, who resided in London in the late 1720s and became acquainted with Jurin at that time, had visited s’Gravesande in Leyden on his return to Geneva and had corresponded with Bernoulli. Cramer became convinced of the correctness of vis viva” (Correspondence of James Jurin 1684-1750, pp. 40-41).

In his January 1729 letter Cramer gives further detail on the topics touched upon in the offered letter, in particular the experiment in which a heavy weight is dropped on to an elastic plate (though the plate is made of leather rather than copper). He writes: “I received two or three Letters from Mr Bernoulli about the remarks of Mr Robins upon his Discourse, and about the Experiment made under your direction by Mr Graham against his theory. About this, he says it appears plainly that Mr Robins has mistaken his sense in many places, whether on purpose or no, he does not care to determine; that he lends him ridiculous opinions which he never had; and refutes absurdities which Mr Bernoulli never wrote … About the Experiment he writes me so: ‘In the experiment made with a sheet of leather, on which a smooth lead weight is gently dropped when it is at the base of its oscillation: I will tell you, Sir, that earlier I had a similar experiment in mind; Mr Euler, presently at Petersburg, could testify to this, but before I could carry out the experiment, I soon recognized that it was not at all suitable to decide the controversy; thus I neglected it in the hope that it was bound to show what you say it has. I am surprised that this affair has troubled you so much that you have not found a suitable response in this area. The objection is similar to that made for two bodies without elasticity. It is clear that in these cases half of the vis viva is consumed during the compression without being restored, because the vis viva is communicated either to the surrounding matter, or to the small internal particles of the two bodies, in order to make them vibrate. When the lead weight suddenly joins the sheet of leather and the two bodies are no longer separated, they are forced to move together; thus they present a case similar to the two non-elastic bodies A and B, combined by impact into one body, which has no other difference than this: that the body A united to the body B pushes it before itself, here the sheet of leather having received the weight carries it with itself. But perhaps you will tell me that between the sheet & the lead there is no conceivable compression. But I respond that the fibers of the 2 surfaces interweave, that the fibers of leather in between the two can contract and expand a little so that there is no space created by the compression of the bodies A & B; from which it follows that half of the vis viva is used in expanding these fibers & thus causing a vibration in the small particles of leather & lead, & the other half is conserved in the shared movement of the total mass’ (ibid., pp. 376-7).

In modern terms, we would say that when a body of mass m travelling with velocity v impacts inelastically with a second body of the same mass initially at rest, the total momentum mv is conserved; since the total mass of the two bodies after the collision is 2m, the velocity of the combined body must be v/2. The total kinetic energy before the collision is ½mv2; after the collision it is ½(2m)(v/2)2 = ¼mv2. The energy lost, we would say, is transformed into sound or heat in the collision, essentially in agreement with Bernoulli who says that it resides in the vibrations of the small particles of the bodies and the surrounding matter.

Gabriel Cramer was appointed to the chair of mathematics at the Académie de Calvin in Geneva, jointly with Giovanni Ludovico Calandrini, in 1727. This appointment provided that the men share both the position’s duties and its salary. It was also provided that they might take turns traveling for two or three years ‘to perfect their knowledge,’ provided the one who remained in Geneva performed all the duties and received all the pay” (DSB). “In May 1727 Cramer was given his first opportunity to travel, visiting Basel, Leiden, London and Paris. He lost no time in establishing his first professional scientific connection with a group of Leibnizian mathematicians in Basel, known as the Bernoulli circle. Cramer spent six months there studying principally with Johann (I) Bernoulli. This was a singular honor since the elder Bernoulli accepted only the most promising students. In Basel Cramer met other members of the Bernoulli circle: Leonhard Euler, and Daniel and Nicholas Bernoulli, establishing lifelong friendships with each. Johann Bernoulli and his brother Jacob (I) (1654-1705) can be regarded as the major avenue by which Leibnizian physics and mathematics were introduced into Switzerland. Evidence of the high esteem in which Cramer was held by Johann Bernoulli was his selection as editor of the two elder Bernoullis’ works. According to Jacob Vernet, the Genevan theologian who wrote Cramer’s ‘Eloge historique’ in the Nouvelle Bibliotheque Germanique, … Bernoulli had urged Cramer to write a work defending vis viva, but Cramer had refused saying that he did not want to begin his career with a polemical work. Vernet reported that when Cramer visited London he did, however, propose several experiments designed to throw light on the famous question” (Dawson, pp. 65-66).

Dawson, Nature’s Enigma, 1987. Terrall, ‘Vis viva revisited,’ History of Science 42 (2004), pp. 189-209.

Two pages on a single sheet (230 x 170 mm), folded for posting, corner torn from wax seal, small tears in the vertical fold repaired with japanese paper.

Item #4709

Price: $18,500.00