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{{redirect|Cavalerius|the lunar crater|Cavalerius (crater)}}
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[[File:Bonaventura Cavalieri.jpeg|thumb|Bonaventura Francesco Cavalieri]]
[[File:IMG 4064 - Milano, Palazzo di Brera - Cavalieri, Bonaventura - Foto Giovanni Dall'Orto 19-jan 2007.jpg|thumb|Monument to Cavalieri in [[Milan]].]]
'''Bonaventura Francesco Cavalieri''' (in [[Latin]], '''Cavalerius''') (1598 &ndash; November 30, 1647) was an [[Italy|Italian]] [[mathematics|mathematician]].  He is known for his work on the problems of [[optics]] and [[motion (physics)|motion]], work on the precursors of [[infinitesimal calculus]], and the introduction of [[logarithms]] to Italy.  [[Cavalieri's principle]] in [[geometry]] partially anticipated [[integral calculus]].


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Born in [[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.<ref>{{cite journal|last=Eves|first=Howard|date=1981, edition consulted 1998|title=Slicing it Thin|work=Mathematical Recreations: A Collection in Honour of Martin Gardner|pages=100|publisher=Dover}}</ref> He published eleven books, his first being published in 1632. He worked on the problems of [[optics]] and [[motion (physics)|motion]]. His [[astronomy|astronomical]] and [[astrology|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]].<ref>[http://galileo.rice.edu/Catalog/NewFiles/cavaleri.html Cavalieri, Bonaventura], at The Galileo Project</ref>
 
Cavalieri's first book was ''Lo Specchio Ustorio, overo, Trattato delle settioni coniche'', or ''The Burning Mirror, or a Treatise on Conic Sections''.<ref>[http://books.google.com/books?hl=en&lr=&id=ZX64AAAAIAAJ&oi=fnd&pg=PA1&dq=%22lo+specchio+ustorio%22&ots=CzX83kjdXO&sig=bZmh3XzXN2mWZdIa32OGlyypwEk Lo Specchio Ustorio, overo, Trattato delle settioni coniche]</ref>  In this book he developed the theory of mirrors shaped into [[parabola]]s, [[hyperbola]]s, and [[ellipse]]s, 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.<ref>Stargazer, the Life and Times of the Telescope, by Fred Watson, [http://books.google.com/books?id=2LZZginzib4C&pg=PA134&dq=gregory+cassegrain+mersenne p. 135]</ref>
 
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 <math>y=x^n</math> – an early [[integral]] – which is known as [[Cavalieri's quadrature formula]].
 
 
Cavalieri is known  for [[Cavalieri's principle]], which states that the [[volume]]s 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&ndash;525) of [[China]].<ref name="needham volume 3 143">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'(《[[:zh:缀术|缀术]]》). This principle was also worked out by [[Shen Kuo]] in the 11th century.</ref>)  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]].<ref>"[http://eom.springer.de/I/i050950.htm Infinitesimal calculus]", Encyclopaedia of Mathematics</ref>
 
Cavalieri also constructed a [[hydraulics|hydraulic pump]] for his monastery and published tables of [[logarithm]]s, emphasizing their practical use in the fields of astronomy and [[geography]].  He died at [[Bologna]].
 
Cavalieri takes place alongside [[Isaac Newton|Newton]], [[Gottfried Leibniz|Leibniz]], [[Blaise Pascal|Pascal]], [[John Wallis|Wallis]] and [[Colin Maclaurin|MacLaurin]] as one of those who in the  17th and 18th centuries "redefine[d] the mathematical object".<ref>{{fr icon}} [[Gilles-Gaston Granger]], ''Formes, opérations, objets'', Vrin, 1994, p. 365 [http://books.google.ca/books?id=EuIjwu5avPcC&pg=PA365&lpg=PA365#v=onepage&f=false Online quotation]</ref>
 
The [[lunar crater]] [[Cavalerius (crater)|Cavalerius]] is named for Cavalieri.
 
== Bibliography ==
* [http://books.google.ca/books?id=zRwAAAAAQAAJ&printsec=frontcover&hl=fr#v=onepage&q&f=false Elogj di Galileo Galilei e di Bonaventura Cavalieri] by Giuseppe Galeazzi, Milan, 1778
* [http://books.google.ca/books?id=I-eaNAAACAAJ&dq=%22Bonaventura+Cavalieri%22&hl=fr&sa=X&ei=eJMzT-XkHcT50gHK9P3WAg&redir_esc=y Bonaventura Cavalieri] by Antonio Favaro, vol. 31 of ''Amici e corrispondenti di Galileo Galilei'', C. Ferrari, 1915, 767 p.
 
==See also==
*[[List of Roman Catholic scientist-clerics]]
 
== Notes ==
<references/>
 
== References ==
*[http://galileo.rice.edu/Catalog/NewFiles/cavaleri.html The Galileo Project: Cavalieri]
 
==External links==
* Online texts by Cavalieri:
** {{la icon}} ''[http://echo.mpiwg-berlin.mpg.de/ECHOdocuView/ECHOzogiLib?url=/mpiwg/online/permanent/library/05TCTFNR/pageimg&start=1&pn=1&mode=imagepath Geometria Indivisibilibus]'' (Online 1653 edition, 543 p.)
** {{it icon}} '' [http://books.google.ca/books?id=ZX64AAAAIAAJ&ots=CzX83kjdXO&dq=%22lo%20specchio%20ustorio%22&lr&pg=PP5#v=onepage&q&f=false Lo specchio ustorio: overo, Trattato delle settioni coniche...]''
** {{la icon}} ''[http://books.google.ca/books?id=_742AAAAMAAJ&printsec=frontcover&sa=X&ei=vIwzT9O6Lau10AGFlNG7Ag&redir_esc=y#v=onepage&q&f=false Directorium generale uranometricum]''
** {{it icon}} ''[http://books.google.ca/books?id=zpIoAAAAcAAJ&hl=fr&pg=PR3#v=onepage&q&f=false Sfera astronomica]''
** {{it icon}} See also [[Wikisource]]
* Biographies:
** {{MacTutor Biography|id=Cavalieri}}
** [http://www.bookrags.com/research/bonaventura-cavalieri-scit-03123/ Short biography on bookrags.com]
** {{la icon}} {{cite journal
| last1 = Fabroni
| first1 = Angelo
| authorlink1 = Angelo Fabroni| editor-first = | editor-link = | date =
| year = 1778
| title = Bonaventura Cavalerius
| journal = Vitae Italorum doctrina excellentium qui saeculis XVII. et XVIII. floruerunt | volume = I
| pages = 262–301
| location = Pisa | publisher =
| language = | format = | id =
| oclc = | pmid = | pmc = | bibcode = | doi =
| accessdate = 2012-02-08
| url = http://books.google.ca/books?id=wVkwAAAAYAAJ&pg=PA262#v=onepage&q&f=false
}}
* Modern mathematical or historical research:
** [http://eom.springer.de/I/i050950.htm Infinitesimal Calculus] On its historical development, in [[Encyclopaedia of Mathematics]], [[Michiel Hazewinkel]] ed.
** {{de icon}} [http://www.dividano.de/prinzip-von-cavalieri.html More information about the method of Cavalieri]
** [http://researchspace.csir.co.za/dspace/bitstream/10204/5267/1/Grobler5_2011.pdf Cavalieri Integration]
 
{{Authority control|VIAF=44324467}}
 
{{wikisourcelang|it|Autore:Bonaventura_Cavalieri|Bonaventura Cavalieri}}
 
{{Persondata <!-- Metadata: see [[Wikipedia:Persondata]]. -->
| NAME              = Cavalieri, Bonaventura
| ALTERNATIVE NAMES =
| SHORT DESCRIPTION = Italian mathematician and astronomer
| DATE OF BIRTH    = 1598
| PLACE OF BIRTH    =
| DATE OF DEATH    = November 30, 1647
| PLACE OF DEATH    =
}}
{{DEFAULTSORT:Cavalieri, Bonaventura}}
[[Category:1598 births]]
[[Category:1647 deaths]]
[[Category:16th-century Italian mathematicians]]
[[Category:17th-century Italian mathematicians]]
[[Category:Italian astronomers]]
[[Category:University of Pisa alumni]]
[[Category:Roman Catholic cleric–scientists]]

Latest revision as of 06:42, 9 December 2014

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