Explicit Solution for a Family of Hermann Riccati Differential Equations

Abstract

This paper deals with a parametric family of ordinary differential equations originated by an ancient geometric problem posed in 1719 by the Swiss mathematician Jakob Hermann (1678-1733) and partially solved by Jacopo Vincenzo Riccati (1676-1754). We translate Riccati approach, called by himself dimidiata separazione (splitter, or halved, separation), in modern terms and we give the full solution of the Hermann problem completing the partial solution provided by Riccati.