Bonaventura Cavalieri was born in Milan around 1598. He joined the Jesuit Order in 1615 and the following year moved to Pisa where he became a pupil of Benedetto Castelli (1577/8-1643), who introduced him to Galileo. Cavalieri, a mathematician, established himself in Rome from 1626 on, where he substituted Castelli at the University during the latter's frequent absences. Young Cavalieri aspired to a lecturer's post but in Pisa was passed over in favor of Niccolò Aggiunti (1600-1635). In 1628, when a position in Bologna opened up at the death of Giovanni Antonio Magini (1555-1617), Cavalieri requested Galileo's backing, which succeeded in landing him the job. He dedicated his logarithmic tables, printed under the title Directorium generale uranometricum (Bologna, 1632), to the Bolognese Senate, which embraced it enthusiastically, and he followed up in the same year with Lo specchio ustorio, overo trattato delle settioni coniche (Bologna, 1632).

In his work Geometria indivisibilibus continuorum nova quadam rati one promota (Bologna, 1635), dedicated to Giovanni Ciampoli (1589-1643), he anticipated Leibnitz's infinitesimal calculus and by a stroke of genius applied indivisibles to Archimedes' spiral. Given its extremely technical nature, the book, though much appreciated in Galileo's circle and considered a masterpiece by later generations, was not understood completely by his contemporaries. Galileo himself, crushed in those years by his condemnation, was unable to devote due attention to Cavalieri's work.

Cavalieri entered into a controversy with the Jesuit Paul Guldin (1577-1643), who among other things accused him of having plagiarized Kepler and of having contradicted Galileo in certain particulars without sufficient warrant, to which Cavalieri replied with his Trigonometr ia plana e sphaerica linearis et logaritmica (Bologna, 1643) and with the third of his Exercitationes geometricae sex (Bologna, 1647), succeeding also in proving with his indivisibles a theorem which Guldin himself had conceived but, being deceased by then, never fully demonstrated.