[2] Around 1614, Arrighetti assisted Galileo in replying to an attack by Giorgio Coresio on his views of the behaviour of bodies in water.
[3] Arrighetti maintained a close interest in Galileo’s experiments in physics and his mathematical work (notably on the flow of water in both straight and twisting channels).
[4][3] Arrighetti was a member of the main Florentine academies and delivered some notable orations at the Accademia della Crusca in praise of Filippo Salviati and of Cosimo II, Grand Duke of Tuscany.
[5] It was probably as a result of a petition from Arrighetti that Prince Leopoldo de' Medici and Grand Duke Ferdinando II decided to refound that Platonic Academy which had been the pride of fifteenth-century Florence.
He also had wide literary interests and wrote an unpublished work, Oeconomicus in which he emphaised the importance of teaching being conducted in a secluded or composed environment as well as comic monologues ('cicalate') 'on the cucumber' and 'on the cake.'