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Euler et D’Alembert. (French) Zbl 0536.01010

Zum Werk Leonhard Eulers, Vortr. Euler-Kolloq., Berlin 1983, 95-117 (1984).
[For the entire collection see Zbl 0527.00001.]
In 1980 there appeared a volume of Euler’s Opera omnia (Zbl 0462.01022) containing the correspondence of Euler with Clairaut, D’Alembert and Lagrange. The author, who edited this volume in collaboration with A. P. Jushkevich, has taken advantage of the new documents and information provided in this volume to offer a new and accurate account of the evolution of the scientific relations between Euler and D’Alembert, which were established in 1746 and then, following various episodes, were progressively weakened and finally ceased to exist after 1766. The paper was first presented as a lecture delivered at an Euler- Colloquium in Berlin in 1983. Following the award of the prize of the Berlin Academy of Sciences to D’Alembert in 1746 for an essay on the winds, Euler entered into correspondence with the young mathematician and over the next five years exchanged views with him on most of the current problems of mathematics. In his first surviving letter to D’Alembert, Euler praised his young correspondent for the parts of his memoir, ”Recherches sur le calcul intégral”, which concern the integration of rational fractions (including the first demonstration of the fundamental theorem of algebra) and certain categories of elliptic integrals but expressed his disagreement with the last part, where D’Alembert claims to demonstrate the real character of the logarithms of negative numbers. All the paradoxes, Euler believed, were resolved by his formula \(\log(\cos \phi +i\sin \phi)=i(\phi \pm 2n\pi).\) By 1752 Euler was no longer interested in discussing D’Alembert’s views on logarithms.
In the same year, D’Alembert’s ”Recherches”, parts of which Euler had earlier praised, gave rise to two more polemics concerning priority. The first related to the reduction of all imaginary quantities to the form \(a+b\sqrt{-1}\) and the second to the possibility for algebraic curves of possessing cusps of second order. An attentive study of the documents leads the author to conclude that, if D’Alembert was the first to publish the precise results on these points, Euler arrived at them simultaneously and independently, although (at least in appearance) he could have profited by the reading of D’Alembert’s memoir two years before its publication. The choice by the Paris Academy of Sciences of a particular case (relating to Jupiter and Saturn) for the prize in 1748 had a considerable influence on the development of celestial mechanics in the eighteenth century. While Euler contributed the prize essay, D’Alembert and Clairaut, who, as members were ineligible for the prize, investigated the analogous problems in the lunar theory. Euler suggested this subject for the prize of the St. Petersburg Academy in 1751. This prize was awarded to Clairaut, who had decided after all that the anomalies could be explained without any modification of the Newtonian law of attraction. D’Alembert withdrew at the last moment but sent this work to the Paris Academy. This was published only in 1754, one year after Euler had published an essay on the same subject. Rivalry between the two mathematicians working on the same problems in the context of prizes for which they were either competitors or judges led to a rapid deterioration of their relations, especially after D’Alembert blamed Euler for the failure of his ”Essai d’une nouvelle théorie des fluides” to win a prize of the Berlin Academy.
This essay has been described as a turning point in mathematical physics, containing for the first time a theory in terms of a field satisfying partial differential equations and among other things, the first statement of the so-called Cauchy-Riemann equations. Following a visit by D’Alembert to the Berlin Academy, where he met Euler in person, relations again became cordial, but after Euler returned to St. Petersburg in 1766, the correspondence came to an end. The author concludes that, although several details in the correspondence confirm D’Alembert’s assertive and impulsive character, Euler was also partly to blame for the deterioration of relations on account of his too rapid exploitation of some of D’Alembert’s discoveries, where he wished perhaps to claim priority for himself, and certain questionable judgements that he made on the work of D’Alembert.
Reviewer: E. J. Aiton

MSC:

01A50 History of mathematics in the 18th century
70-03 History of mechanics of particles and systems

Biographic References:

Euler, L.; D’Alembert, J.