FIRST ORDER DIFFERENTIAL EQUATIONS: EARLY HISTORY FOLLOWING JACOPO RICCATI AND GABRIELE MANFREDI

n 1722 the Italian mathematician Jacopo Ric- cati (1676–1754) held a course of lectures on differential equations, which remained unpublished until 1761. They finally emerged, under the title Della separazione delle in- determinate nelle equazioni differenziali del primo grado e della riduzione delle equazioni differenziali del secondo e di altri gradi ulteriori, in the first of four tomes of his Opera Omnia. This treatise, following the Leibniz (1646–1716) tradition of seeking solutions finitely expressed in terms of known functions, focuses on the separation of variables, to integrate a broad class of ordinary differential equations, of first or higher order. Its first part, entitled Dei metodi inventati da varj celebri Autori per separare le indeterminate nelle equazioni differenziali del primo grado, makes a survey of all relevant integrating methods, known at that time. These are thanks to Gabriele Manfredi (1681–1761), Jakob Bernoulli (1654–1705), his younger brother Johann (1667– 1748), Vincenzo Riccati (1707–1775), son of Jacopo, and Jacopo himself: our paper considers those of them, which in our opinion are the smartest and most ingenious.