[3] He took his vows as a full member of the order in 1615, at the age of seventeen, and shortly after joined the Jesuat house in Pisa.

In 1617 he briefly joined the Medici court in Florence, under the patronage of Cardinal Federico Borromeo, but the following year he returned to Pisa and began teaching Mathematics in place of Castelli.

[1] In 1620, he returned to the Jesuate house in Milan, where he had lived as a novitiate, and became a deacon under Cardinal Borromeo.

In 1629 he was appointed Chair of Mathematics at the University of Bologna, which is attributed to Galileo's support of him to the Bolognese senate.

Arthritis prevented him from writing, and much of his correspondence was dictated and written by Stephano degli Angeli, a fellow Jesuate and student of Cavalieri.

[4] From 1632 to 1646, Cavalieri published eleven books dealing with problems in astronomy, optics, motion and geometry.

Cavalieri's first book, first published in 1632 and reprinted once in 1650, was Lo Specchio Ustorio, overo, Trattato delle settioni coniche, or The Burning Mirror, or a Treatise on Conic Sections.

[5][8] The book went beyond this purpose and also explored conic sections, reflections of light, and the properties of parabolas.

[5] Inspired by earlier work by Galileo, Cavalieri developed a new geometrical approach called the method of indivisibles to calculus and published a treatise on the topic, Geometria indivisibilibus continuorum nova quadam ratione promota, or Geometry, developed by a new method through the indivisibles of the continua.

[1] These parallel elements are called indivisibles respectively of area and volume and provide the building blocks of Cavalieri's method, and are also fundamental features of integral calculus.

(The same principle had been previously used by Zu Gengzhi (480–525) of China, in the specific case of calculating the volume of the sphere.

First, while Cavalieri's proofs were intuitive and later demonstrated to be correct, they were not rigorous; second, his writing was dense and opaque.

While many contemporary mathematicians furthered the method of indivisibles, the Geometria indivisibilibus critical reception was severe.

Guldin's particularly in-depth critique suggested that Cavalieri's method was derived from the work of Johannes Kepler and Bartolomeo Sovero, attacked his method for a lack of rigorousness, and then argues that there can be no meaningful ratio between two infinities, and therefore it is meaningless to compare one to another.

[4][1] Cavalieri's Exercitationes geometricae sex or Six Geometric Exercises (1647) was written in direct response to Guldin's criticism.

It was initially drafted as a dialogue in the manner of Galileo, but correspondents advised against the format as being unnecessarily inflammatory.

Those books were the Nuova pratica astrologica (1639) and the Trattato della ruota planetaria perpetua (1646).

He published tables of logarithms, emphasizing their practical use in the fields of astronomy and geography.

[6] According to Gilles-Gaston Granger, Cavalieri belongs with Newton, Leibniz, Pascal, Wallis and MacLaurin as one of those who in the 17th and 18th centuries "redefine[d] the mathematical object".