Cavalieri was born in 1598, in Milan, Italy. The name Bonaventura was not his baptismal name but rather that of his father. It is the name the mathematician adopted when, as a boy, he entered the Jesuati religious order, adherents of the rule of St. Augustine.
Cavalieri was received in the minor orders in Milan in 1615 and in 1616 transferred to the Jesuati monastery in Pisa, where he had the good fortune of meeting the Benedictine monk Benedetto Castelli, who had studied with Galileo at Padua and was at the time a lecturer in mathematics at Pisa. Through him Cavalieri was initiated into the study of geometry. He quickly absorbed the classical works of Euclid, Archimedes, Apollonius, and Pappus, demonstrating such exceptional aptitude that he sometimes substituted for his teacher at the University of Pisa.
Cavalieri was introduced by Castelli to Galileo, whose disciple he always considered himself. He wrote Galileo at least 112 letters, which are included in the national edition of the Opere di Galileo; only two of Galileo’s letters to Cavalieri have come down to us, however. In 1620 Cavalieri returned to Rome under orders of his superiors, and in 1621 he was ordained a deacon to Cardinal Federigo Borromeo, who held Fra’ Bonaventura in great esteem and gladly discussed mathematics with him; the cardinal subsequently wrote a letter commending him to Galileo.
Cavalieri was hardly twenty-one when he taught theology at the monastery of San Girolamo in Milan, attracting attention by his profound knowledge of the subject. During his Milan period (1620-1623) Cavalieri developed his first ideas on the method of indivisibles, his major contribution to mathematics. From 1623 to 1626 he was the prior of St. Peter’s at Lodi. Later he was a guest in Rome of Monsignor Ciampoli, to whom he later dedicated his Geometría. From 1626 to 1629 he was the prior of the monastery of the Jesuati in Parma, hoping in vain to be appointed lecturer in mathematics at the university there. In the autumn of 1626, during a trip from Parma to Milan, he fell ill with the gout, from which he had suffered since childhood and which was to plague him to the end of his life. This illness kept him at Milan for a number of months. On 16 December 1627 he announced to Galileo and Cardinal Borromeo that he had completed his Geometria.
In 1628, learning that a post of lecturer at Bologna had become vacant through the death of the astronomer G. A. Magini, he wrote Galileo for assistance in securing the appointment. Galileo, in 1629, wrote to Cesare Marsili, a gentleman of Bologna and member of the Accademia dei Lincei, who had been commissioned to find a new lecturer in mathematics. In support of his application to the Bologna position, Cavalieri sent Marsili his geometry manuscript and a small treatise on conic sections and their applications in optics. Galileo’s testimonial, as Marsili wrote him, induced the “Gentlemen of the Regiment” to entrust the first chair in mathematics to Cavalieri, who held it continuously from 1629 to his death. At the same time he was appointed prior of a convent of his own order in Bologna, specifically, at the Church of Santa Maria della Mascarella, enabling him to pursue without any impediment both his work in mathematics and his university teaching.
During the period in which Cavalieri taught at Bologna, he published eleven books in that city, including the Geometria (1635). Cavalieri’s theory, as developed in this work and in others subsequently published, relates to an inquiry in infinitesimals, stemming from revived interest in Archimedes’ works, which during the Renaissance were translated from Greek into Latin, with commentaries. The geometry of indivisibles was, in 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.
In Italy such valuable auxiliaries to numerical calculation were introduced by Cavalieri, together with noteworthy developments in trigonometry and applications to astronomy. In his publication, the Centuria, Cavalieri dealt with such topics as the general definition of cylindrical and conical surfaces, formulas to determine the volume of a barrel and the capacity of a vault with pointed arches, and the means of obtaining from the logarithms of two numbers the logarithm of the sum or of the difference, a problem that was subsequently taken up by various mathematicians, Gauss among others. His another publication, Lo specchio ustorio (“The Burning Glass”), contains some interesting historical data on the origin of the theory of the conics among the Greeks; according to Cavalieri, the origins are to be found in the gnomonic requirements. In this work, we find a theory of conics with applications to optics and acoustics. Among the former, we note the idea of the reflecting telescope, of which Cavalieri was the first inventor, preceding Gregory and Newton; determination of the focal length of a lens of uneven sphericity; and explications of the burning glass of Archimedes. In the field of acoustics, Cavalieri attempted the archaeological reconstruction of the resonant vases mentioned by Vitruvius and used in theaters for amplifying sound. In this work, various point-wise constructions of conics appear. More interesting still are the constructions given in the Geometría and in the Exercitationes, obtained by means of projective pencils which antedated the work of Steiner. A delicate question relates to the astrological activities that Cavalieri engaged in by virtue of his office, but, as pointed out by D’Aviso, he was opposed to predictions based upon the position of the stars and states so at the end of his Pratica astrológica.
Cavalieri joined the Jesuates order at the age of fifteen and remained a member until his death.
