Bonaventura Cavalieri (1598–1647) was a prominent mathematician and a pivotal figure in the Galilei-Castelli mathematical school. Originally named Francesco Cavalieri in Milan, he adopted the name Bonaventura upon joining the Jesuati order at the age of fifteen. The Jesuati, distinct from the Jesuits, was an order established in 1360 and disbanded in 1668, recognized for its charitable efforts during the Black Death and its members’ unique practice of vociferously invoking the name of Jesus in prayers.
Cavalieri’s foray into mathematics commenced in Pisa, where he became acquainted with Benedetto Castelli, a Benedictine monk and a mathematical protégé of Galileo. Castelli familiarized Cavalieri with geometry, and his talent swiftly became evident; he even took over for Castelli in university lectures. Galileo, taken by Cavalieri’s understanding of geometry, adopted him as a mentee. A significant correspondence developed, but only a handful of Galileo’s communications to Cavalieri have survived.
Cavalieri’s professional journey was riddled with obstacles. Despite various rejections for academic positions in Bologna, Parma, and Rome, he persisted and ultimately secured a professorship in mathematics at Bologna in 1629, a role he held until his passing. His appointment was aided by Galileo’s influence over significant individuals.
Cavalieri’s main contributions were in the realms of geometry and mathematical optics. His inaugural book, “Lo Specchio Ustorio,” explored the optical characteristics of conic sections and visionary designs for reflecting telescopes. His research preceded Newton’s reflecting telescope but faced skepticism akin to Galileo’s opposition to Keplerian telescopes.
In 1635, Cavalieri unveiled his method of indivisibles, an early iteration of integral calculus, in “Geometria indivisibilibus continuorum nova quadam ratione promota.” This technique, analogous to contemporary infinitesimals, was critical in the advancement of calculus ideas. Cavalieri’s Principle expressed the concept of volume comparison through the equality of sectional areas.
Cavalieri’s indivisibles drew criticism from Jesuit mathematicians André Tacquet and Paul Guldin, who challenged the employment of actual infinity. Guldin accused him of copying ideas, hinting at influences from Bartolomeo Sovero and Johannes Kepler. Amidst tensions, Cavalieri defended his theories in “Exercitationes geometricae sex,” honing his explanation of indivisibles.
Cavalieri also promoted John Napier’s logarithms in Italy, authoring works that combined logarithms with trigonometry and their uses in astronomy. Although the impact of his optical theories was limited during his lifetime, Cavalieri’s method of indivisibles played a crucial role in the development of calculus, highlighting his lasting influence in mathematics.