Bonaventura Cavalieri Biography

Bonaventura Cavalieri refined early Greek work on the concept of indivisibles. His work served as a stepping stone to the concept of infinitesimals and was the foundation of Isaac Newton's development of the calculus. Cavalieri was born in Milan, Italy, in 1598. His actual birth date is uncertain; even his first name is unknown, because he entered a monastic order at an early age and adopted the first name Bonaventura as his religious name.

Cavalieri entered the Jesuatis (a Roman Catholic order founded on the rule of St. Augustine and not to be confused with the Jesuits or Society of Jesus) and took minor orders in 1615 at a monastery in Milan. In 1616, he transferred to a monastery in Pisa, where he met Benedetto Castelli, a former pupil of Galileo. Castelli introduced Cavalieri to the study of geometry and later introduced him to Galileo himself. From that point on, Cavalieri considered Galileo his mentor and teacher. During their long association, Cavalieri wrote more than a hundred letters to Galileo, which have survived in the national edition of the Le opere di Galileo Galilei.

Cavalieri's entry into religious life was typical of the route many aspiring scholars took in that era. Monasteries were centers of learning and operated the most comprehensive libraries; the Roman Catholic Church wielded tremendous power over education and the dissemination of information. (For instance, Galileo was forced by the Inquisition to repudiate his 1632 treatise that clarified the Copernican theory on the movement of the Earth around the sun. The Church did not officially lift the ban on treatises that treated as fact such celestial movement until 1828.) As a cleric, Cavalieri had access to the classic texts of Euclid, Archimedes, and Apollonius, and his status as a monk served as an introduction to the finest minds of his time.

One Theory, Indivisible

In 1621, Cavalieri was ordained as a deacon to Cardinal Federigo Borromeo, whose regard for scholarship and appreciation of mathematics encouraged Cavalieri. The cardinal's esteem for Cavalieri's abilities made it possible for him to teach theology at the monastery of San Girolamo in Milan while still in his early twenties. It was during this time (1620 to 1623), that Cavalieri first began his work on the method of indivisibles. From 1623 to 1629 he served as prior of St. Peter's at Lodi and at the Jesuati monastery in Parma. On a trip to Milan, he fell ill with an attack of gout and was confined to Milan for several months. Perhaps the forced relaxation from his duties as prior gave him the opportunity to concentrate on his works in mathematics, because he told Galileo and Cardinal Borromeo that he had completed his Geometriain December of 1627.

In 1628, Cavalieri sought Galileo's help in securing a teaching post at the University of Bologna. Galileo wrote to a patron of the institution that "few, if any, since Archimedes, have delved as far and as deep into the science of geometry." Cavalieri won the academic appointment and was named the first chair in mathematics in 1629, a post he held until his death.

At the same time, his order appointed him prior of the Church of Santa Maria della Mascarella in the city, combining his responsibilities to the Church with convenient access to the university. Cavalieri was able to pursue his academic and theological ambitions, and this period was a fruitful one. While in Bologna, he published 11 books, including Geometria in 1635.

Cavalieri's inspiration was Archimedes, who first proposed the notion--unexamined for centuries--that indivisibles could be used to determine areas, volumes, and centers of gravity. Cavalieri developed a rational system that employed indivisibles to determine area and volume, a method that made calculations easier and quicker than the ancient Greek method of exhaustion. Cavalieri's Theorem states that if two solids have equal altitudes, and if sections made by planes parallel to the bases and at equal distances from the bases always has a given ratio, then volumes of the solids have the same ratio to each other. Cavalieri also refined a general proof of Guldin's theoremrelated to the area of a surface and the volume of rotating solids.

Cavalieri's work had tremendous impact on the works of his contemporaries. Torricelli, who expanded on Cavalieri's concepts, wrote "the geometry of indivisible was, indeed, the mathematical briar bush, the so-called royal road, and one that Cavalieri first opened and laid out for the public as a device of marvelous invention." Pierre de Fermat, Blaise Pascal, Issac Barrow and Newton were influenced by Cavalieri's work, which, according to Isaac Asimov, was "a stepping-stone toward . . . the development of the calculus by Newton, which is the dividing line between classical and modern mathematics."