Bonaventura Cavalieri

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Bonaventura Cavalieri
Bonaventura Cavalieri.jpeg
Cavalieri led the way to integral calculus
Bonaventura Francesco Cavalieri

Milan, Italy
DiedNovember 30, 1647(1647-11-30) (aged 48–49)
Bologna, Italy
Alma materUniversity of Pisa
Known forCavalieri's principle
Cavalieri's quadrature formula

Bonaventura Francesco Cavalieri (Latin: Cavalerius; 1598 – 30 November 1647) was an Italian mathematician and a Jesuate.[1] 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.


Born in Milan, Cavalieri joined the Jesuates order (not to be confused with the Jesuits) at the age of fifteen and remained a member until his death.[2] He studied theology in the monastery of San Gerolamo in Milan, and geometry at the University of Pisa.

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.[3]

Cavalieri's first book was Lo Specchio Ustorio, overo, Trattato delle settioni coniche, or The Burning Mirror, or a Treatise on Conic Sections.[4] 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.[5]

Cavalieri's work also contained theoretical designs for a new type of telescope using mirrors, a reflecting telescope.[6][7] He illustrated three different concepts for incorporating reflective mirrors within his telescope model. Plan one consisted of a large, concave mirror directed towards the sun as to reflect light into a second, smaller, convex mirror. Cavalieri's second concept consisted of a main, truncated, paraboloid mirror and a second, convex mirror. His third option illustrated a strong resemblance to his previous concept, replacing the convex secondary lens with a concave lens.[6]

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 (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.[8]) 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 in Bologna.


Monument to Cavalieri by Giovanni Antonio Labus, Palazzo di Brera, Milan, 1844

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".[9]

The lunar crater Cavalerius is named for Cavalieri.

See also[edit]


  1. ^ Amir Alexander (2014). Infinitesimal: How a Dangerous Mathematical Theory Shaped the Modern World. Scientific American / Farrar, Straus and Giroux. ISBN 978-0374176815.
  2. ^ Eves, Howard (1998). David A. Klarner (ed.). "Slicing it Thin". Mathematical Recreations: A Collection in Honour of Martin Gardner. Dover: 100. ISBN 0-486-40089-1.
  3. ^ Cavalieri, Bonaventura, at The Galileo Project
  4. ^ Lo Specchio Ustorio, overo, Trattato delle settioni coniche
  5. ^ Stargazer, the Life and Times of the Telescope, by Fred Watson, p. 135
  6. ^ a b Ariotti, Piero E. (September 1975). "Bonaventura Cavalieri, Marin Mersenne, and the Reflecting Telescope". Isis. 66 (3): 303–321. doi:10.1086/351471. ISSN 0021-1753.
  7. ^ Eves, Howard (March 1991). "Two Surprising Theorems on Cavalieri Congruence". The College Mathematics Journal. 22 (2): 118–124. doi:10.2307/2686447. ISSN 0746-8342. JSTOR 2686447.
  8. ^ 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.
  9. ^ (in French) Gilles-Gaston Granger, Formes, opérations, objets, Vrin, 1994, p. 365 Online quotation


Further reading[edit]

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