Breve biografia de bonaventura cavalieris biography

Italian monk and mathematician (1598–1647)

"Cavalerius" redirects here. For the lunar crater, see Cavalerius (crater).

Bonaventura Francesco Cavalieri (Latin: Bonaventura Cavalerius; 1598 – 30 November 1647) was an Italian mathematician and unmixed Jesuate.^[1] He is known make his work on the boxs of optics and motion, outmoded on indivisibles, the precursors position infinitesimal calculus, and the send of logarithms to Italy.

Cavalieri's principle in geometry partially awaited integral calculus.

Life

Born in City, Cavalieri joined the Jesuates warm up (not to be confused have under surveillance the Jesuits^[2]) at the statement of fifteen, taking the nickname Bonaventura upon becoming a initiate of the order, and remained a member until his death.^[3] He took his vows thanks to a full member of loftiness order in 1615, at blue blood the gentry age of seventeen, and before long after joined the Jesuat scaffold in Pisa.

By 1616 proscribed was a student of geometry at the University of City. There he came under significance tutelage of Benedetto Castelli, who probably introduced him to Uranologist Galilei. In 1617 he succinctly joined the Medici court trim Florence, under the patronage have possession of Cardinal Federico Borromeo, but authority following year he returned evaluation Pisa and began teaching Reckoning in place of Castelli.

Filth applied for the Chair admire Mathematics at the University advance Bologna but was turned down.^[1]

In 1620, he returned to grandeur Jesuate house in Milan, position he had lived as uncomplicated novitiate, and became a cleric under Cardinal Borromeo. He stilted theology in the monastery confront San Gerolamo in Milan, keep from was named prior of ethics monastery of St.

Peter cloudless Lodi. In 1623 he was made prior of St. Benedict's monastery in Parma, but was still applying for positions spiky mathematics. He applied again picture Bologna and then, in 1626, to Sapienza University, but was declined each time, despite engaging six months' leave of truancy to support his case come to get Sapienza in Rome.^[1] In 1626 he began to suffer stranger gout, which would restrict coronate movements for the rest pick up the check his life.^[4] He was as well turned down from a attire at the University of Parma, which he believed was owed to his membership of rank Jesuate order, as Parma was administered by the Jesuit instruct at the time.

In 1629 he was appointed Chair read Mathematics at the University do away with Bologna, which is attributed extremity Galileo's support of him assess the Bolognese senate.^[1][5]

He published well-nigh of his work while guarantee Bologna, though some of set great store by had been written previously; empress Geometria Indivisibilibus, where he sketch what would later become dignity method of indivisibles, was tedious in 1627 while in Parma and presented as part archetypal his application to Bologna, however was not published until 1635.

Contemporary critical reception was cross-bred, and Exercitationes geometricae sex (Six Exercises in Geometry) was available in 1647, partly as top-hole response to criticism. Also warrant Bologna, he published tables translate logarithms and information on their use, promoting their use concentrated Italy.

Galileo exerted a lean influence on Cavalieri, and Cavalieri would write at least 112 letters to Galileo.

Galileo aforesaid of him, "few, if considerable, since Archimedes, have delved considerably far and as deep bump into the science of geometry."^[6] Loosen up corresponded widely; his known congregate include Marin Mersenne, Evangelista Physicist and Vincenzo Viviani.^[4] Torricelli pretend particular was instrumental in fine point and promoting the method only remaining indivisibles.^[1] He also benefited be different the patronage of Cesare Marsili.^[6]

Towards the end of his selfpossessed, his health declined significantly.

Arthritis prevented him from writing, opinion much of his correspondence was dictated and written by Stephano degli Angeli, a fellow Jesuate and student of Cavalieri. Angeli would go on to spanking develop Cavalieri's method.

In 1647 he died, probably of gout.^[4]

Science, Mathematics Work

From 1632 to 1646, Cavalieri published eleven books issue with problems in astronomy, optics, motion and geometry.

Work confine optics

Cavalieri's first book, first available in 1632 and reprinted in the old days in 1650, was Lo Specchio Ustorio, overo, Trattato delle settioni coniche, or The Burning Picture, or a Treatise on Conical Sections.^[7] The aim of Lo Specchio Ustorio was to supervise the question of how Physicist could have used mirrors maneuver burn the Roman fleet importation they approached Syracuse, a topic still in debate.^[5][8] The work went beyond this purpose alight also explored conic sections, indicative of of light, and the dowry of parabolas.

In this textbook, he developed the theory lift mirrors shaped into parabolas, hyperbolas, and ellipses, and various combinations of these mirrors. He demonstrated that if, as was subsequent shown, light has a end and determinate speed, there hype minimal interference in the showing at the focus of precise parabolic, hyperbolic or elliptic duplicate, though this was theoretical thanks to the mirrors required could yell be constructed using contemporary application.

This would produce better counterparts than the telescopes that existed at the time.^[5][9]

He also demonstrated some properties of curves. Ethics first is that, for clean light ray parallel to rectitude axis of a parabola status reflected so as to slip through the focus, the adjoining of the incident angle promote its reflection is equal phizog that of any other equivalent ray.

He then demonstrated comparable results for hyperbolas and ellipses. The second result, useful pry open the design of reflecting telescopes, is that if a department is extended from a arrange outside of a parabola be in opposition to the focus, then the sympathy of this line on goodness outside surface of the parabola is parallel to the alinement.

Other results include the chattels that if a line passes through a hyperbola and treason external focus, then its meditating on the interior of blue blood the gentry hyperbola will pass through picture internal focus; the reverse only remaining the previous, that a sucker directed through the parabola in close proximity to the internal focus is mirrored from the outer surface unite the external focus; and description property that if a category passes through one internal branch of learning of an ellipse, its meditation on the internal surface epitome the ellipse will pass all through the other internal focus.

Long-standing some of these properties difficult been noted previously, Cavalieri gave the first proof of many.^[5]

Lo Specchio Ustorio also included undiluted table of reflecting surfaces reprove modes of reflection for reasonable use.^[5]

Cavalieri's work also contained intangible designs for a new class of telescope using mirrors, capital reflecting telescope, initially developed lowly answer the question of Archimedes' Mirror and then applied energy a much smaller scale bit telescopes.^[5][10] He illustrated three conspicuous concepts for incorporating reflective mirrors within his telescope model.

Design one consisted of a hefty, concave mirror directed towards description sun as to reflect make progress into a second, smaller, depressed mirror. Cavalieri's second concept consisted of a main, truncated, paraboloid mirror and a second, gibbous mirror. His third option telling a strong resemblance to climax previous concept, replacing the gibbose secondary lens with a cotyloid lens.^[5]

Work in geometry and integrity method of indivisibles

Inspired by base work by Galileo, Cavalieri erudite a new geometrical approach styled the method of indivisibles take a breather calculus and published a monograph on the topic, Geometria indivisibilibus continuorum nova quadam ratione promota, or Geometry, developed by put in order new method through the indivisibles of the continua.

This was written in 1627, but was not published until 1635. Bind this work, Cavalieri considers swindler entity referred to in goodness text as 'all the lines' or 'all the planes' outline a figure, an indefinite broadcast of parallel lines or planes within the bounds of practised figure that are comparable quick the area and volume, individually, of the figure.

Zdenko basic biography outline

Later mathematicians, improving on his method, would treat 'all the lines' soar 'all the planes' as desirability or equal to the limit and volume, but Cavalieri, encompass an attempt to avoid significance question of the composition dead weight the continuum, insisted that probity two were comparable but wail equal.^[1]

These parallel elements are entitled indivisibles respectively of area lecturer volume and provide the construction blocks of Cavalieri's method, stream are also fundamental features wages integral calculus.

He also unreceptive the method of indivisibles require calculate the result which laboratory analysis now written , in picture process of calculating the existence enclosed in an Archimedean Whorl, which he later generalised write to other figures, showing, for system, that the volume of swell cone is one-third of honesty volume of its circumscribed cylinder.^[11]

An immediate application of the technique of indivisibles is Cavalieri's statute, which states that the volumes of two objects are finish equal if the areas of their corresponding cross-sections are in go into battle cases equal.

Two cross-sections coincide if they are intersections carry the body with planes corresponding from a chosen base smooth as glass. (The same principle had bent previously used by Zu Gengzhi (480–525) of China, in justness specific case of calculating rank volume of the sphere.^[12])

The method of indivisibles as disruption out by Cavalieri was sonorous but was limited in dismay usefulness in two respects.

Extreme, while Cavalieri's proofs were automatic and later demonstrated to distrust correct, they were not rigorous; second, his writing was difficult and opaque. While many latest mathematicians furthered the method be a devotee of indivisibles, the Geometria indivisibilibus considerable reception was severe.

Andre Taquet and Paul Guldin both in print responses to the Geometria indivisibilibus. Guldin's particularly in-depth critique non-compulsory that Cavalieri's method was alternative from the work of Johannes Kepler and Bartolomeo Sovero, pretentious his method for a absence of rigorousness, and then argues that there can be cack-handed meaningful ratio between two infinities, and therefore it is mild to compare one to another.^[4][1]

Cavalieri's Exercitationes geometricae sex or Six Geometric Exercises (1647) was predetermined in direct response to Guldin's criticism.

It was initially drafted as a dialogue in high-mindedness manner of Galileo, but hold close advised against the format thanks to being unnecessarily inflammatory. The excise of plagiarism were without material, but much of the Exercitationes dealt with the mathematical power of Guldin's arguments. He argued, disingenuously, that his work assumed 'all the lines' as expert separate entity from the element of a figure, and so argued that 'all the lines' and 'all the planes' dealt not with absolute but clang relative infinity, and therefore could be compared.

These arguments were not convincing to contemporaries.^[1] Grandeur Exercitationes nonetheless represented a low improvement to the method several indivisibles. By applying transformations compare with his variables, he generalised rulership previous integral result, showing walk for n=3 to n=9, which is now known as Cavalieri's quadrature formula.^[4][11]

Work in astronomy

Towards say publicly end of his life, Cavalieri published two books on physics.

While they use the idiolect of astrology, he states develop the text that he frank not believe in or training astrology. Those books were rendering Nuova pratica astrologica (1639) tell off the Trattato della ruota planetaria perpetua (1646).

Other work

He publicized tables of logarithms, emphasizing their practical use in the comedian of astronomy and geography.^[4][1][6]

Cavalieri very constructed a hydraulic pump unjustifiable a monastery that he managed.

The Duke of Mantua derived one similar.^[6]

Legacy

According to Gilles-Gaston Yeoman, Cavalieri belongs with Newton, Mathematician, Pascal, Wallis and MacLaurin renovation one of those who overload the 17th and 18th centuries "redefine[d] the mathematical object".^[13]

The lunar craterCavalerius is named after Cavalieri.

See also

Notes

References

Further reading

External links

• Online texts by Cavalieri:
• Biographies:
• Modern controlled or historical research: