The Nodding Sphere and the Bird's Beak: D'Alembert's Dispute with Euler

Author(s): 
Robert E. Bradley

1749 was a typically productive year in the middle period of Leonhard Euler’s career.  Among the many articles he prepared for publication that year, five were destined for the journal of the Berlin Academy, where he was the director of the mathematics department.  However, four of these provoked controversy with Jean le Rond d’Alembert, who believed that one of the articles was mistaken and the other three contained results that he himself had first discovered and for which Euler had not given him due credit.

The purpose of this article is to explain this controversy between Euler and d’Alembert and to provide links to English translations of all four of the offending articles, as well as of the brief notice that Euler inserted in the 1750 volume of the Academy’s journal, where he acknowledged d’Alembert’s priority for two of the papers in question.

Leonhard Euler

 

Jean le Rond d'Alembert

 

If you want to go directly to the translations, here are the links:

"On the Controversy between Messrs. Leibniz and Bernoulli concerning Logarithms of Negative and Imaginary Numbers"

"Research on the Precession of the Equinoxes and on the Nutation of the Earth's Axis"

"On the Cuspidal Point of the second kind of Monsieur le Marquis de l'Hospital"

"Investigations on the Imaginary Roots of Equations"

"Notice on the Subject of Recherches sur la précession des équinoxes"

The Nodding Sphere and the Bird's Beak: D'Alembert's Dispute with Euler - Introduction

Author(s): 
Robert E. Bradley

1749 was a typically productive year in the middle period of Leonhard Euler’s career.  Among the many articles he prepared for publication that year, five were destined for the journal of the Berlin Academy, where he was the director of the mathematics department.  However, four of these provoked controversy with Jean le Rond d’Alembert, who believed that one of the articles was mistaken and the other three contained results that he himself had first discovered and for which Euler had not given him due credit.

The purpose of this article is to explain this controversy between Euler and d’Alembert and to provide links to English translations of all four of the offending articles, as well as of the brief notice that Euler inserted in the 1750 volume of the Academy’s journal, where he acknowledged d’Alembert’s priority for two of the papers in question.

Leonhard Euler

 

Jean le Rond d'Alembert

 

If you want to go directly to the translations, here are the links:

"On the Controversy between Messrs. Leibniz and Bernoulli concerning Logarithms of Negative and Imaginary Numbers"

"Research on the Precession of the Equinoxes and on the Nutation of the Earth's Axis"

"On the Cuspidal Point of the second kind of Monsieur le Marquis de l'Hospital"

"Investigations on the Imaginary Roots of Equations"

"Notice on the Subject of Recherches sur la précession des équinoxes"

The Nodding Sphere and the Bird's Beak: D'Alembert's Dispute with Euler - Euler and D'Alembert

Author(s): 
Robert E. Bradley

Mathematicians are generally familiar with Leonhard Euler (1707-1783); see the Dictionary of Scientific Biography [1, p. IV.467-484] for information on his life and achievements, or shorter biographies by Dunham [3, p. xix-xxviii] and Calinger [2, p. 486-489]. Euler was perhaps the most productive mathematician of all time. He was certainly the most productive and influential mathematician of his century, and his role in the development of modern physics is almost as significant.

The name of Jean le Rond d'Alembert (1717-1783) is less well known to mathematicians, although he is certainly better known than Euler to those in the humanities. He was an editor of the first encyclopedia, and a member of a loosely-defined group of French philosophers and men of letters known as les philosophes, who were very influential in the European Enlightenment. However, d'Alembert was a successful mathematician and physicist in his own right, especially during his younger days. Perhaps his best-known achievement was the solution of the wave equation. Once again, the Dictionary of Scientific Biography [1, p. I.110-117] is a good source on his life. The book by Hankins [4] is more substantial and gives a sympathetic account of d'Alembert, who had a tendency to be quarrelsome and vain.

D'Alembert wrote his first letter to Euler in the summer of 1746, shortly after he won the Berlin Academy's prize for that year. His essay, On the General Cause of Winds [5] had been chosen for the prize by a jury chaired by Euler, who was the director of the mathematics section at that academy. For a period of almost five years, the two men carried on a lively and amicable dialogue through their correspondence, most of which has been collected in Euler's Opera Omnia [6]. However, d'Alembert suffered a great disappointment in the Berlin Prize competition for 1750. The jury, once again chaired by Euler, determined that none of the entries submitted for that year's prize competition were worthy, and so the prize was remanded to 1752. D'Alembert declined to enter the 1752 competition and instead published his 1750 essay himself. It appeared as Essay on a New Theory of Fluid Resistance in Paris early in 1752 [7].

In December 1750, the young astronomer Augustin Nathanael Grischow (1726-1760) was dismissed from the Berlin Academy. Grischow, not to be confused with his father Augustin (1683-1749), also an astronomer with the Berlin Academy, had been one of the three judges of the 1750 competition. He was also an acquaintance of d'Alembert. No doubt humiliated by the Academy's actions, he made trouble for his former colleagues by revealing to d'Alembert and others in Parisian society his version of the events that had led to the rejection of all the entries in that competition. Whatever may have actually happened behind closed doors, d'Alembert came away with the belief that Euler had recognized his entry and convinced Grischow and the other judge that the paper, which they considered to be the front-runner, had not sufficiently answered the question set for the competition.

The Berlin competition, like other prize competitions of this time, involved anonymous entries, identitfied only by a motto or dévise. It would not have been difficult for Euler to identify d'Alembert's distinctive mathematical style, so the story has at least some credibility. In any case, d'Alembert believed that he had been treated unfairly, and broke off his correspondence with Euler in an angry letter of September 10, 1751. Those who read French may wish to consult [6, p. 27-28, 311-314] for more on this controversy.

This might have marked the end of d'Alembert's relationship with Euler. However, later in the same year the Berlin Academy's journal for the year 1749 appeared and caused d'Alembert even more grief. Hankins describes it as follows [4, p. 50]:

 [The volume] appeared with a series of articles by Euler including one on the precession of the equinoxes and three others on subjects that d'Alembert had enthusiatically described to Euler in their correspondence. D'Alembert quickly took alarm. All of his work was being stolen! Even his most important contribution, his book on the equinoxes, had not received a single mention from Euler.

D'Alembert was now thoroughly annoyed. Just at the time when his work with the Encyclopédie was achieving so much success, Euler apparently was not only obstructing his efforts, but also borrowing his ideas and claiming them as his own.

Actually, Euler published five memoirs in this volume of the Histoires of the Berlin Academy. This was typical of his productivity at that stage in his career. Only one of these articles, a piece on the parallax of the moon [8], was unproblematic as far as d'Alembert was concerned. As noted by Hankins, Euler and d'Alembert had discussed matters relevant to the contents of the other four papers in their correspondence between 1746 and 1750. Although D'Alembert felt that he had not been duly cited in three of these papers, his concerns with the fourth one were of a very different nature.

The Nodding Sphere and the Bird's Beak: D'Alembert's Dispute with Euler - Logarithms of Negative Numbers

Author(s): 
Robert E. Bradley

Euler's paper "On the controversy between Messrs. Leibniz and Bernoulli concerning logarithms of negative and imaginary numbers" [9] is the first publication in which the riddle of logarithms of negative numbers is solved. An English translation of this article by Stacy Langton is available by clicking here. Although d'Alembert, like Johann (I) Bernoulli before him, believed that \(\ln{\left(-x\right)} = \ln{x}\) for positive real numbers \(x,\) Euler showed that \(\ln{\left(-x\right)} = \ln{x}+i(2n+1)\pi\) for any integer \(n.\) Euler and d'Alembert had debated this at great length in their early correspondence, and although Euler was eventually able to persuade his correspondent that the logarithm is a complex-valued multi-function, d'Alembert never abandoned his assertion that \(\ln{x}\) was a value of \(\ln{\left(-x\right)}.\)

In response to Euler's article, d'Alembert addressed a memoir to the Berlin Academy in June of 1752, in which he attacked Euler's position and attempted to prove that ln(-x)=lnx, or at least persuade the reader that it was possible. The Academy never published the paper, presumably on account of both its mathematical content and polemical tone. Instead, d'Alembert published the piece in the first volume of his Opuscules [10], his self-published collection of mathematical papers, which appeared in 8 volumes between 1761 and 1780.

Since d'Alembert disagreed with Euler's conclusions about logarithms in [9], it's not quite fair for Hankins to characterize d'Alembert's concerns about this paper as involving a matter of his priority in a discovery. On the other hand, priority was precisely d'Alembert's concern with regards to Euler's three other pieces in the 1749 volume. To that end, d'Alembert addressed a second essay to the Berlin Academy in June of 1752. Called "Observations on several memoirs printed in the Academy's volume for 1749," [6, p. 337-346] this essay made the case that various results in Euler's papers [11, 12, 13]  were first proved by d'Alembert, and revealed to Euler through letters and pieces already in print.

The Nodding Sphere and the Bird's Beak: D'Alembert's Dispute with Euler - Precession of the Equinoxes

Author(s): 
Robert E. Bradley

D'Alembert's strongest case was the one against Euler's paper on the precession of the equinoxes and the nutation of the earth's axis [11]. An English translation of this article by Steven Jones is available by clicking here for the html version and here for the pdf version. The precession of the equinoxes is a phenomenon that has been known since classical times. The earth's axis is not in fact stationary and instead traces out a large circle with respect to the fixed stars, rather like a top spinning on an oblique axis. The period of the precession is about 26,000 years and it will significantly alter the location of the north celestial pole in the millennia to come. In 1748, the British Astronomer Royal James Bradley announced his discovery of another disturbance in the earth's axis of rotation, a nodding motion or "nutation" with an 18 year cycle. D'Alembert had set himself the task of explaining both phenomena in strictly mechanical terms, as a consequence of Newton's inverse-square law of gravitation. He eventually cracked the problem, and published his book-length solution [14] in the middle of 1749.

Euler had also been working on the problem of precession and nutation, but had not been able to solve it. He received a copy of d'Alembert's book late in the summer of 1749. In his "Observations" essay, sent to the Academy in June 1752, d'Alembert reported receiving a letter from Euler, dated January 3, 1750, in which Euler acknowledged receiving the book [6, p. 338]. Euler also said that he had not really been able to follow d'Alembert's argument, but that after he had read it, he saw the big picture and was able to give his own solution to the problem. It was this solution that Euler published in the Berlin Academy's 1749 volume. Euler's solution is certainly shorter (36 pages as opposed to d'Alembert's 184) and more comprehensible. Indeed, d'Alembert's mathematical writings were notorious for poor organization and impenetrability. More importantly, Euler's solution was far more general, and led him to an important paper the following year on general principles governing the motion of rigid bodies [15], a problem he had been working on since 1734 at least. So although Euler owed much to d'Alembert in his solution of the precession and equinox problem, there is also much in his paper [11] that is novel; all of this is explained in careful detail in a recent paper by Curtis Wilson [16].

Nevertheless, Euler ought to have acknowledged at the outset of the paper that he was only presenting an alternate solution to a problem that had already been solved by d'Alembert. It was a serious lapse of academic etiquette to have neglected this. In addition, the records show that Euler didn't actually present his results on the problem to the Berlin Academy until March 5, 1750, so it was ethically questionable for him to have inserted the article in the Academy's volume for 1749. In any case, Euler recognized the validity of D'Alembert's priority claim and inserted a brief notice [17] to this effect in the next volume of Academy's journal, published in translation here. In this notice, Euler acknowledges that he had written his paper only after he had read d'Alembert's book, and that he "makes no pretense to the glory that is due to he that first resolved this important question."

The majority of D'Alembert's "Observations" essay is occupied with his priority claim on the problem of precession and nutation. However, he also demanded that his priority be recognized for two other papers. Euler capitulated in one case but not in the other.

The Nodding Sphere and the Bird's Beak: D'Alembert's Dispute with Euler - The Bird's Beak

Author(s): 
Robert E. Bradley

Euler's paper "On the cuspidal point of the second kind of Mr. the Marquis of l'Hôpital" [12] gives a resolution of a curious question in the theory of equations. My translation of this article is available here. In his 1696 calculus text [18] L'Hospital gave the construction of a curve that doubled back on itself and so has a cusp in which the two branches are both concave in the same direction; he called this a cuspidal point of the second kind and it later came to be called the bird's beak (le bec d'oiseau). This is in contrast with the more familiar cusp of the first kind, such as the one at the origin in the graph of y2 = x3 or y = ± x3/2, where the two branches are concave in opposite directions: one upwards and one downwards. L'Hopital gave only a geometric construction: he did not provide the equation of a curve with a cusp of the second kind. Then in 1740, the French mathematician Jean Paul de Gua de Malves published a flawed proof that no algebraic curve could have such a cusp [19].

When Euler's textbook Introductio in analysin infinitorum [20] appeared in 1748, it contained a counterexample to Gua de Malves' claim, that is an algebraic equation whose graph is endowed with a cusp of the second kind. Euler's example is

y4 - x3 - 4x2y - 2xy2 + x2 = 0,

which may also be written as y = x ± x3/4. In the same year, an article by d'Alembert appeared in the Berlin Academy's journal for the year 1746. In this piece [21], d'Alembert also gives an example of a curve with a cusp of the second kind: y = x2± x5/2. Since d'Alembert had submitted his article to the Academy in December 1746, he believed that he had been the first to discover such a curve. Therefore he was upset not to have received mention of priority in Euler's article [12].

What d'Alembert did not know was that Euler had finished writing his book in 1744 or possibly even 1743 and it had languished at the printer's in Lausanne for many years. Furthermore, surviving letters that Euler had exchanged with Gabriel Cramer in 1744 clearly demonstrate to us that he had discovered his example long before d'Alembert. However, Cramer had died in January 1752, so it would not have been easy for Euler to establish his priority. Instead, he quietly acquiesced to d'Alembert's demand for credit and inserted a brief mention of the bird's beak at the end of his Notice concerning the precession of the equinoxes [17]. Euler chose his words carefully: he didn't acknowledge that d'Alembert had been the first to make the discovery, only that he "was the first to give an account of the nature of those curves that have a cuspidal point of the second kind."

The Nodding Sphere and the Bird's Beak: D'Alembert's Dispute with Euler - The Fundamental Theorem of Algebra

Author(s): 
Robert E. Bradley

D'Alembert received no satisfaction on his third and final priority claim, and for a good reason. This dispute centered on Euler's paper "Research on imaginary roots of equations" [13]. In this paper, Euler attempted to prove the fundamental theorem of algebra (FTA): that every real polynomial may be decomposed into real linear and irreducible quadratic factors. A translation of this article by Todd Doucet is available here. This paper is also discussed in some detail by Dunham [3, p. 111-119]. D'Alembert had attempted to prove the FTA in the same article in which he mentioned the cusp of the second kind [21]. By modern standards, neither proof of the FTA is adequate: the first rigorous proof of the theorem is due to Gauss; for more on this, see Dunham [3, p. 119-124].

Although Euler ceded priority to d'Alembert in [17] for both of the previous issues, he made no admission with respect to the FTA. At least part of the reason for this is clear: Euler had already credited d'Alembert in the original article! In §64 of the paper [13, p. 257-258] he explicitly cites d'Alembert's results in [21] and says that he is simply trying to give an alternate proof without recourse to infinitesimals. In his "Observations" d'Alembert seems to be insulted at Euler's suggestion that an alternate proof might be called for. So although d'Alembert wrote this essay as a plea for the recognition of his priority, his actual arguments in the final portion amount largely to amplifying and defending his mathematical arguments in [21]. Therefore, this portion of the essay has more in common with his essay on logarithms than with the rest of the essay.

The Berlin Academy never published d'Alembert's "Observations" essay. About two thirds of the essay consisted of a thorough documentation of the similarities between D'Alembert's book on precession and nutation and Euler's paper. Since Euler was prepared to admit d'Alembert's priority, it's not clear what purpose this would have served to the readers of the Berlin journal. Euler presumably felt that by ceding priority to D'Alembert for both this discovery and that of the cusp of the second kind, he was giving d'Alembert the satisfaction he craved, while simultaneously sparing the Academy from having controversy aired in the pages of its journal. We will probably never understand why Euler failed to credit d'Alembert with the first solution of the problem of precession and nutation in [11]. It may have been a careless oversight or there may have been a darker motive, but it seems unlikely that Euler was trying to take credit for the discovery, since almost anyone at that time who was interested in astronomy would have been aware of d'Alembert's triumph. Whatever the reason for Euler's oversight, there was a cost to be paid for it: not only did he have to make a public apology, but for the sake of a speedy resolution he also had to cede priority for the bird's beak, a result that really did belong to him.

The Nodding Sphere and the Bird's Beak: D'Alembert's Dispute with Euler - References

Author(s): 
Robert E. Bradley
  1. Dictionary of Scientific Biography, Gillispie, C. G., ed., New York: Scribner, 1972.
  2. Calinger, R., Classics of Mathematics, Englewood Cliffs, NJ: Prentice Hall, 1995.
  3. Dunham, W., Euler: The Master of Us All, Washington, DC: Math. Assoc. America, 1999.
  4. Hankins, T. L., Jean d'Alembert: Science and the Enlightenment, Oxford: Clarendon, 1970.
  5. D'Alembert, J.  Réflexions sur la cause générale des vents, Berlin and Paris, 1747.
  6. Euler, L., Opera Omnia, Series IVA, vol. 5 (correspondence with Clairaut, d'Alembert and Lagrange), eds.  A. P. Juskevic, R. Taton, Basel: Birkhäuser, 1980.
  7. D'Alembert, J., Essai d'une nouvelle théorie de la résistance des fluides, Paris, 1752.
  8. Euler, L., "De la parallaxe de la lune ... dans l'hypothese de la terre spheroidique," Mem. acad. sci. Berlin 5 (1749), 1751, p. 326-338. Reprinted in Euler, L., Opera Omnia, Series II, vol. 30, p. 140-150.
  9. Euler, L., "De la controverse entre Mrs. Leibniz et Bernoulli sur les logarithmes des nombres negatifs et imaginaires," Mem. acad. sci. Berlin 5 (1749), 1751, p. 139-179. Reprinted in Euler, L., Opera Omnia, Series I, vol. 17, p. 195-232.
  10. D'Alembert, J., Opuscules mathématiques, vol. I, Paris: David, 1761.
  11. Euler, L., "Recherches sur la precession des equinoxes et sur la nutation de l'axe de la terre," Mem. acad. sci. Berlin 5 (1749), 1751, p. 289-325. Reprinted in Euler, L., Opera Omnia, Series II, vol. 29, p. 92-123.
  12. Euler, L., "Sur le point de rebroussement de la seconde espece de M. le Marquis de l'Hopital," Mem.  acad. sci. Berlin 5 (1749), 1751, p. 203-221. Reprinted in Euler, L. Opera Omnia, Series I, vol. 27, p. 236-252.
  13. Euler, L., "Recherches sur les racines imaginaires des equations," Mem. acad. sci. Berlin 5 (1749), 1751, p. 222-288. Reprinted in Euler, L.  Opera Omnia, Series I, vol. 6, p. 78-150.
  14. D'Alembert, J., Recherches sur la précession des équinoxes et sur la nutation de l'axe de la terre dans le système newtonien, Paris: David, 1749.
  15. Euler, L., "Découverte d'un nouveau principe de Mécanique," Mem. acad. sci. Berlin 6 (1750), 1752, p. 185-217. Reprinted in Euler, L., Opera Omnia, Series II, vol. 5, p. 81 - 108.
  16. Wilson, C. "D'Alembert versus Euler on the precession of the equinoxes and the mechanics of rigid bodies," Arch. Hist. Exact Sci. 37, 1987, p. 233-273.
  17. Euler, L., "Avertissement au sujet des recherches sur la precession des equinoxes," Mem.  acad. sci. Berlin 6 (1750), 1752, p. 412. Reprinted in Euler, L., Opera Omnia, Series II, vol. 29, p. 124.
  18. L'Hospital, G. de, Analyse des infiniment petits, pour l'intelligence des lignes courbes, Paris: Imprimerie royale, 1696.
  19. Gua de Malves, J. P. de, Usages de l'analyse de Descartes pour découvrir, sans le secours du Calcul Differentiel, les Propriétés, ou affectations principales des lignes géometriques de tous les ordres,, Paris: Briassaon, 1740.
  20. Euler, L., Introductio in analysin infinitorum, 2 vols., Lausanne: Bousquet, 1748. Reprinted in Euler, L., Opera Omnia, ser. I, vols. 8-9.
  21. D'Alembert, J., "Recherches sur le calcul intégral," Mem. acad. sci. Berlin 2 (1746), 1748, p. 182-224.