# Bonaventura Cavalieri

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**Bonaventura Francesco Cavalieri** (in Latin, **Cavalerius**) (1598 – 30 November 1647) was an Italian mathematician. He is known for his work on the problems of optics and motion, work on indivisibles, the precursors of infinitesimal calculus, and the introduction of logarithms to Italy. Cavalieri's principle in geometry partially anticipated integral calculus.

## Life

Born at Milan, Cavalieri studied theology in the monastery of San Gerolamo in Milan and geometry at the University of Pisa as a member of the Jesuates order.^{[1]} He published eleven books, his first being published in 1632. He worked on the problems of optics and motion. His astronomical and astrological work remained marginal to these main interests, though his last book, *Trattato della ruota planetaria perpetua* (1646), was dedicated to the former. He was introduced to Galileo Galilei through academic and ecclesiastical contacts. Galileo exerted a strong influence on Cavalieri encouraging him to work on his new method and suggesting fruitful ideas, and Cavalieri would write at least 112 letters to Galileo. Galileo said of Cavalieri, "few, if any, since Archimedes, have delved as far and as deep into the science of geometry." He also benefited from the patronage of Cesare Marsili.^{[2]}

Cavalieri's first book was *Lo Specchio Ustorio, overo, Trattato delle settioni coniche*, or *The Burning Mirror, or a Treatise on Conic Sections*.^{[3]} In this book he developed the theory of mirrors shaped into parabolas, hyperbolas, and ellipses, and various combinations of these mirrors. The work was purely theoretical since the needed mirrors could not be constructed with the technologies of the time, a limitation well understood by Cavalieri.^{[4]}

Building on the classic method of exhaustion, Cavalieri developed a geometrical approach to calculus and published a treatise on the topic, *Geometria indivisibilibus continuorum nova quadam ratione promota* (*Geometry, developed by a new method through the indivisibles of the continua,* 1635). In this work, an area is considered as constituted by an indefinite number of parallel segments and a volume as constituted by an indefinite number of parallel planar areas. Such elements are called indivisibles respectively of area and volume and provide the building blocks of Cavalieri's method. As an application, he computed the areas under the curves – an early integral – which is known as Cavalieri's quadrature formula.

Cavalieri is known for Cavalieri's principle, which states that the volumes of two objects are equal if the areas of their corresponding cross-sections are in all cases equal. Two cross-sections correspond if they are intersections of the body with planes equidistant from a chosen base plane. (The same principle had been previously discovered by Zu Gengzhi (480–525) of China.^{[5]}) Cavalieri developed a "method of the indivisibles," which he used to determine areas and volumes. It was a significant step on the way to modern infinitesimal calculus.^{[6]}

Cavalieri also constructed a hydraulic pump for his monastery and published tables of logarithms, emphasizing their practical use in the fields of astronomy and geography. He died at Bologna.

## Legacy

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".^{[7]}

The lunar crater Cavalerius is named for Cavalieri.

## Bibliography

- Elogj di Galileo Galilei e di Bonaventura Cavalieri by Giuseppe Galeazzi, Milan, 1778
- Bonaventura Cavalieri by Antonio Favaro, vol. 31 of
*Amici e corrispondenti di Galileo Galilei*, C. Ferrari, 1915, 767 p.

## See also

## Notes

- ↑ {{#invoke:Citation/CS1|citation |CitationClass=journal }}
- ↑ Cavalieri, Bonaventura, at The Galileo Project
- ↑ Lo Specchio Ustorio, overo, Trattato delle settioni coniche
- ↑ Stargazer, the Life and Times of the Telescope, by Fred Watson, p. 135
- ↑ Needham, Joseph (1986).
*Science and Civilization in China: Volume 3; Mathematics and the Sciences of the Heavens and the Earth*. Taipei: Caves Books, Ltd. Page 143.) and was first documented in his book 'Zhui Su'(《缀术》). This principle was also worked out by Shen Kuo in the 11th century. - ↑ "Infinitesimal calculus", Encyclopaedia of Mathematics
- ↑
- REDIRECT Template:Link language Gilles-Gaston Granger,
*Formes, opérations, objets*, Vrin, 1994, p. 365 Online quotation

- REDIRECT Template:Link language Gilles-Gaston Granger,

## References

## External links

- Online texts by Cavalieri:
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*Geometria Indivisibilibus*(Online 1653 edition, 543 p.)

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*Sfera astronomica*

- REDIRECT Template:Link language See also Wikisource

- Biographies:
- Template:MacTutor Biography
- Short biography on bookrags.com
- {{#invoke:Citation/CS1|citation

|CitationClass=journal }}

- Modern mathematical or historical research:
- Infinitesimal Calculus On its historical development, in Encyclopaedia of Mathematics, Michiel Hazewinkel ed.
- Template:De icon More information about the method of Cavalieri
- Cavalieri Integration

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