Brianchon and Poncelet's joint memoir, the nine-point circle, and beyond (Q2144554)

This informative paper provides an account of Brianchon and Poncelet's joint memoir on equilateral hyperbolas subject to four given conditions, published in 1821 and mostly famous for containing a complete proof of the so-called nine-point circle theorem (for the history of this problem in the second half of the 19th century, see: [\textit{M. A. Vaccaro}, Hist. Math. 51, 26--48 (2020; Zbl 1445.01013)]). The author offers a detailed analysis of the mathematical content of the memoir and an interesting reconstruction of its genesis. In fact, apart from its mathematical relevance, Brianchon and Poncelet's paper is an interesting example of a co-authored work at a time where mathematical collaborations were rare. With respect to this issue, the author concludes that the genesis of the memoir is twofold: the first part derives from Brianchon's studies on hyperbolas as expounded in a memoir written in 1817, the second part was due to Poncelet, and it was actually a draft of another future memoir composed by the latter. The nature of the collaboration between Poncelet and Brianchon is also made more precise: Poncelet and Brianchon never met in person, their contacts were limited to letter exchanges, and their ``joint work'' appears to be ``the fruit of particular circumstances rather than of a close collaboration''.