**Bonaventura Cavalieri** was an Italian mathematician who developed a method of indivisibles which became a factor in the development of the integral calculus.

- Although they had some success in recruiting, and in particular Cavalieri joined, nevertheless the order eventually failed and was dissolved by pope Clement IX in 1668.
- As a Jesuati, Cavalieri would have always worn sandals and flagellated himself daily.
- In Pisa, Cavalieri was taught mathematics by Benedetto Antonio Castelli, a lecturer in mathematics at the University of Pisa.
- He taught Cavalieri geometry and introduced him to the ideas of Galileo.
- Cavalieri's interest in mathematics had been stimulated by Euclid's Elements and, after meeting Galileo, he considered himself a disciple of the astronomer.
- Galileo did not answer all of them, but sent an occasional letter to Cavalieri; of these all but a very few have disappeared.
- Cavalieri showed such promise that he sometimes took over Castelli's lectures at the university.
- Urbano Diviso, Cavalieri's pupil and first biographer writing about 30 years after Cavalieri's death, claimed that Castelli told Cavalieri to study of mathematics since that would cure him of depression.
- However, there is no other evidence for this claim and certainly some checkable claims in Diviso's account of Cavalieri's life are incorrect.
- In 1619 Cavalieri applied for the chair of mathematics in Bologna, which had become vacant following the death of Giovanni Antonio Magini, but was not successful since he was considered too young for a position of this seniority.
- Cavalieri himself blamed the fact that he was in the Jesuati order as the reason for his lack of success in these applications.
- In 1621 Cavalieri became a deacon and assistant to Cardinal Federico Borromeo at the monastery of San Girolamo in Milan.
- In 1629 Cavalieri was appointed to the chair of mathematics at Bologna.
- In his letter, Galileo said of Cavalieri, "few, if any, since Archimedes, have delved as far and as deep into the science of geometry." 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.
- This was an ideal situation for Cavalieri who now had the peace to undertake mathematics research at the Jesuati convent while teaching mathematics at the university where he could have contacts with other mathematicians.
- Cavalieri's appointment to Bologna had, in the first instance, been for a 3-year trial period but, as we explain below, it was extended.
- Cavalieri's geometry manuscript which had been a factor in his appointment to Bologna, although completed in December 1627, was not published until 1635.
- This theory allowed Cavalieri to find simply and rapidly the area and volume of various geometric figures.
- Now, Cavalieri argued, if we slide each member of a parallel set of indivisibles of some planar piece along its own axis, so that the endpoints of the indivisibles still trace a continuous boundary, then the area of the new planar piece so formed is the same as that of the original planar piece, inasmuch as the two pieces are made up of the same indivisibles.
- Although that is probably the main issue between Cavalieri and Guldin, a more careful reading of the debate will allow us to indicate the existence of other interesting issues ...
- His first point, however, was to accuse Cavalieri of plagiarising Kepler's Stereometria Doliorum (1615) and Sover's Curvi ac Recti Proportio (1630).
- However, Cavalieri's indivisibles are different from Kepler's infinitesimals.
- As to the reference to Sover, Cavalieri, in his defence, pointed out that he wrote his book before Sover's book was published.
- Guldin attacked Cavalieri's indivisibles by arguing that when a surface is generated by rotating a line about the axis, the surface is not just a set of lines.
- If one asks whether Guldin or Cavalieri is right, then the answer must be Cavalieri.
- However, a positive side to Guldin's attack was that Cavalieri improved his exposition publishing Exercitationes geometricae sex (1647) which became the main source for 17th Century mathematicians.
- Cavalieri was also largely responsible for introducing logarithms as a computational tool in Italy through his book Directorium Generale Uranometricum.
- This book of logarithms was published by Cavalieri as part of his successful application to have the position extended.
- Galileo praised Cavalieri for his work on logarithms, in particular the book he wrote entitled A hundred varied problems to illustrate the use of logarithms (1639).
- Cavalieri also wrote on conic sections, trigonometry, optics, astronomy, and astrology.
- Cavalieri's work of interest is his 'Specchio ustorio', printed in 1632 and reprinted in 1650.
- In this work Cavalieri concerned himself with reflecting mirrors for the express purpose of resolving the age-long dispute of how Archimedes allegedly burned the Roman fleet that was besieging Syracuse in 212 B.C. The book, however, goes well beyond the stated purpose and systematically treats the properties of conic sections, reflection of light, sound, heat (and cold!), kinematic and dynamic problems, and the idea of the reflecting telescope.
- Cavalieri's claim that one would obtain a telescope by combining concave mirrors with concave lens have led some historians to claim that Cavalieri invented the reflecting telescope before James Gregory or Isaac Newton.
- Cavalieri did not believe that one could predict the future from astrological considerations, and certainly did not practice astrology.
- Cavalieri corresponded with many mathematicians including Galileo, Mersenne, Renieri, Rocca, Torricelli and Viviani.
- The geometry of indivisibles was indeed, 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 marvellous invention.
- In fact, Torricelli continued to develop the ideas that Cavalieri introduced in Arithmetica infinitorum (1655).
- Perhaps Cavalieri's most famous student was Stefano degli Angeli.
- He studied with Cavalieri at Bologna at a time when Cavalieri was quite old and suffering from arthritis.
- Angeli wrote many of the letters which Cavalieri sent to his fellow mathematicians during his time of study.
- We mentioned Cavalieri's problems with his legs that began around 1629 and also his longstanding problems with gout.
- This was at a time when Galileo was living under house arrest in Arcetri, and Cavalieri spent the summer discussing mathematics with him.
- Returning to Bologna, life became increasingly difficult for Cavalieri.
- His health had not improved and he was being pressed by the university authorities to work on astronomy rather than on mathematics, the topic that Cavalieri loved.
- Cardinal Federico Borromeo offered him a position at the Biblioteca Ambrosiana in Milan, but again Cavalieri chose to remain in Bologna.

Born 1598, Milan, Duchy of Milan, Habsburg Empire (now Italy). Died 30 November 1647, Bologna, Papal States (now Italy).

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