Manfredi was one of a group of young men at the University who became interested in the techniques of
Cartesian geometry and
differential calculus, and who engaged in experiments and astronomical observation. Others were his brother Eustachio,
Vittorio Francesco Stancari and
Giuseppe Verzaglia. Of these, Gabriele Manfredi developed the most advanced understanding of mathematics. Eustachio Manfredi became more interested in astronomy, but Gabriele persisted with mathematics, studying the works of
Leibniz and of
Johann and
Jacob Bernoulli on
infinitesimal calculus. After graduating, Gabriele went to Rome at the end of 1702, where he became librarian to Cardinal
Pietro Ottoboni, a historian, antiquarian and astronomer. He helped Ottoboni build a sundial at
Santa Maria degli Angeli e dei Martiri and helped in the work of reforming the
Gregorian calendar. He continued to study mathematics, including differential and integral calculus and logarithmic curves. In 1707 he returned to Bologna where he published his best known work on first-order differential equations. This was the first European work on differential equations. The book, praised by
Leibniz and
Johann Bernoulli and favourably reviewed by the
Acta Eruditorum, quickly gained a great reputation throughout Europe. It was regarded as the equivalent in the subject of the integral calculus to
Guillaume de l'Hôpital’s ''
Analyse des infiniment petits pour l'intelligence des lignes courbes'' (1696) in the subject of the differential calculus. Despite this, Manfredi was not given a senior position in the university. He made further contributions to the theory of calculus, although his main contribution after 1715 was as a teacher. ==Later career==