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If you disregard the very simplest cases, there is in all of mathematics not a single infinite series whose sum has been rigorously determined. In other words, the most important parts of mathematics stand without a foundation.

In G. F. Simmons, Calculus Gems, New York: McGraw Hill, Inc., 1992, p. 188.

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

## The Bird's Beak

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=Öx ±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."