# Bonaventura Cavalieri

# Bonaventura Cavalieri

**1598-1647**

**Italian Mathematician**

Praised by thinkers ranging from his friend Galileo (1564-1642) to twentieth-century writer Isaac Asimov, Bonaventura Cavalieri is best known for his work on the concept of indivisibles. This laid the foundation for the development of the infinitesimal, and with it calculus as conceived by Sir Isaac Newton (1642-1727).

Cavalieri's true first name is not known: Bonaventura was a religious name adopted when he joined a monastery at age 17. As for the circumstances of his birth, it is known only that he was born in Milan in 1598. In his mid-teens, Cavalieri joined the Jesuatis, an order under the Augustinian rule not to be confused with the Jesuits, and in 1615 took minor orders at a Milan monastery.

The following year found him at another monastery in Pisa, where he came under the influence of Benedetto Castelli, a former student of Galileo. Castelli inspired in his young friend an interest in geometry, and eventually introduced him to Galileo himself. Cavalieri became a devoted protegee of the latter, and sent him more than a hundred letters over the course of the years that followed.

Cavalieri was still only 23 years old when he received ordination as a deacon under Cardinal Federigo Borromeo (1564-1631). Again, he had the good fortune of positive association: Borromeo encouraged Cavalieri's scholarship, and helped him obtain a position as teacher of theology at the monastery of San Girolamo in Milan. Cavalieri first began work on his method of indivisibles at San Girolamo, and continued this after receiving an appointment as prior of St. Peter's at Lodi.

Despite his youth, Cavalieri was soon struck down with an attack of gout, which occurred while on a visit back to Milan. As a result, he was confined to his bed for several months, and during this time wrote much of the book that would appear in 1635 as *Geometrica.*

Meanwhile, in 1628, Cavalieri took an interest in a teaching position at the University of Bologna—Europe's first university, established 470 years before. He sought and received a glowing letter of recommendation from Galileo, who informed the patron of the university that "few, if any, since Archimedes have delved as far and as deep into the science of geometry" as Cavalieri. Not only did Cavalieri receive the appointment, which he held until his death, but his order appointed him prior of Bologna's Church of Santa Maria della Mascarella. During the nearly two decades that remained for him, he published 11 books.

Drawing on ideas first explored by Archimedes (c. 287-212 b.c.), but barely considered in the intervening centuries, Cavalieri proposed that indivisibles could be used for the determination of area, volume, and center of gravity. Later, Evangelista Torricelli (1608-1647), who improved on the idea, wrote that "the geometry of the invisible was, indeed, the mathematical briar brush, the so-called royal road, and one that Cavalieri first opened and laid out for the public as a device of marvelous invention." It would later prove, as Asimov observed more than four centuries later, "a stepping-stone toward . . . the development of the calculus by Newton, which is the dividing line between classical and modern mathematics."

Cavalieri also put forth what came to be known as Cavalieri's Theorem. The latter states that for two solids of equal altitude, if sections made by planes parallel to and at equal distance from the bases always have a given ratio, then the volumes of the two solids will have the same ratio. Furthermore, Cavalieri developed a general proof of Guldin's theorem, which concerns the area of a surface and the volume of rotating solids. He died in Bologna on November 30, 1547.

**JUDSON KNIGHT**

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**Bonaventura Cavalieri**