Algèbre des fonctions elliptiques et géométrie des ovales cartésiennes. (Algebra of elliptic functions and geometry of Cartesian ovals).

*(French)*Zbl 1030.01022Summary: Research on Cartesian ovals over the course of the 19th century attest to the revival of geometrical methods and illustrate a competition between these methods and analytic calculations. In particular, they played a part in the relations between the algebra of elliptic functions and the geometry of curves, which mathematicians saw in terms of application or of interpretation of one field in terms of another. In 1850, W. Roberts and A. Genocchi obtained rectifications of ovals with arcs of ellipses through formal computations; some ten years later, Mannheim and Darboux proved them again using geometrical reasoning. Deep relations between elliptic functions and Cartesian ovals were also established in 1867, with the geometrical proofs of the addition theorem of elliptic functions given by Darboux and Laguerre. When Darboux proved the orthogonality of systems of homofocal ovals, he also showed that ovals provide a geometrical interpretation of the addition theorem, and that they constitute the algebraic form of the integral solution. Laguerre, on the other hand, proved the addition theorem with the help of anallagmatic curves using Poncelet’s theorem on inscribed and circumscribed polygons in two conics. Works on the representation of elliptic functions provide yet another point of view. In the 1880s, A. G. Greenhill proved that the elliptic functions of Jacobi and Weierstrass could be represented by bicircular quartics, a special case of which are ovals (JFM 18.0400.01). In particular, he used the elliptic formula to prove the orthogonality of systems of homofocal ovals. In her paper of 1913 (JFM 44.0527.02), Clara L. Bacon both established geometrical properties of ovals from Weierstrass’s function and interpreted geometrically the algebra of elliptic functions with the help of ovals.