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.