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  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  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 . 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 .
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 , 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.