In 1738 she published Propositiones Philosophicae a series of essays on philosophy and natural science: 191 philosophical theses which Agnesi defended with specially invited audiences at her home.
Agnesi concentrated her efforts on studying religious books and learning mathematics. She wrote a commentary on de L'Hôpital's Traite analytique des section coniques, never been published. A monk, Ramiro Rampinelli, a mathematician who had been a professor at both Rome and Bologna, became a frequent visitor to the Agnesi house. With Rampinelli's help Agnesi studied Reyneau's calculus text Analyse démontrée (1708). Rampinelli encouraged Agnesi to write a book on differential calculus, in Italian as a teaching text, privately printed. Rampinelli's teacher, Riccati, read the final draft of Agnesi's book and made suggestions. On feb. 1, 1747, Riccati offered Agnesi his some of his earlier work on integration for inclusion in her book. Agnesi included the work with proper acknowledgement to Riccati. A report on it made by a committee of the Académie des Sciences in Paris states:-
It took much skill and sagacity to reduce, as the author has done, to almost uniform methods these discoveries scattered among the works of modern mathematicians and often presented by methods very different from each other. Order, clarity and precision reign in all parts of this work. ... We regard it as the most complete and best made treatise.Pope Benedict XIV wrote to Agnesi saying that he had studied mathematics when he was young and could see that her work would bring credit to Italy and to the Academy of Bologna. Soon after this he appointed Agnesi to the position of honorary reader at the University of Bologna. Then Agnesi was approached by the president of the Academy of Bologna and three other professors of the Academy and invited to accept the chair of mathematics at the University of Bologna. In October [Agnesi] received a papal rescript confirming her appointment. She had already devoted herself to a holy, retired life; while her name remained on the rolls of the university for forty-five years, she never went to Bologna. She chose to inhabit rooms of her home away from where the rest of the family lived, helping old women who were ill. Agnesi spent all her money on this charitable work and she died in total poverty in the poorhouse of which she had been the director.
The treatise Instituzioni analitiche ad uso della gioventù italiana contains no original mathematics by Agnesi. The book includes a discussion of the cubic curve now know as the 'witch of Agnesi'. There has been much argument over the reason why the curve is called a 'witch'. The curve was discussed by the great (amateur) mathematician, Pierre Fermat and, in 1703, a construction for the curve was given by Grandi.