When Giovanni Borelli met Galileo Galilei in Florence in 1640, he may also have crossed path with the young Vincenzio Viviani (1622–1703), last disciple of the great astronomer. He could not know, though, that he would later have a rather turbulent relationship with he who claimed to be the true heir of the Galileian tradition. From the priority dispute on the reconstruction and translation of Apollonius’ Conics, to their collaboration in matters of experimental physics at the short-lived Accademia del Cimento, to the eighteenth-century retrospective construction of a rivalry, let us briefly retrace the lineaments of an early modern collegial relationship.