Bonaventura Cavalieri

Bonaventura Francesco Cavalieri (1598–1647) was an Italian mathematician, astronomer and priest, born in Milan. He studied theology at the monastery of San Gerolamo and geometry at the University of Pisa, where he became acquainted with Galileo, whose belief in heliocentrism he shared, then considered heretical by the Church. He maintained an extensive correspondence with him, comprising at least 112 letters. Cavalieri is best known for two fundamental contributions that foreshadow integral calculus. The first is the method of indivisibles, set out in Geometria indivisibilium continuorum nova quadam ratione promota in 1635: this method regards lines as composed of infinitely many points, surfaces as formed of infinitely many lines, and solids as formed of infinitely many surfaces, making it possible to calculate areas and volumes without explicitly resorting to limiting processes. The second is Cavalieri's principle, already stated by Liu Hui in 3rd-century China: two solids have equal volumes if every pair of parallel cross-sections at equal distances from a reference plane has equal areas. Cavalieri was aware of the paradoxes to which his method could lead; Torricelli would propose an improved version. His method was a decisive step towards differential geometry and the integral calculus of Newton and Leibniz. He also contributed to the study of conic sections (1632), plane and spherical trigonometry for astronomical use (1635), and provided rigorous proofs of the theorems attributed to Guldin.