Bonaventura Francesco Cavalieri (; 1598 – 30 November 1647) was an Italian

mathematician A mathematician is someone who uses an extensive knowledge of mathematics in their work, typically to solve mathematical problems. Mathematicians are concerned with numbers, data, quantity, mathematical structure, structure, space, Mathematica ...

and a Jesuate. He is known for his work on the problems of

optics Optics is the branch of physics that studies the behaviour and properties of light, including its interactions with matter and the construction of optical instruments, instruments that use or Photodetector, detect it. Optics usually describes t ...

and

motion In physics, motion is when an object changes its position with respect to a reference point in a given time. Motion is mathematically described in terms of displacement, distance, velocity, acceleration, speed, and frame of reference to an o ...

, work on indivisibles, the precursors of

infinitesimal calculus Calculus is the mathematical study of continuous change, in the same way that geometry is the study of shape, and algebra is the study of generalizations of arithmetic operations. Originally called infinitesimal calculus or "the calculus of ...

, and the introduction of

logarithms In mathematics, the logarithm of a number is the exponent by which another fixed value, the base, must be raised to produce that number. For example, the logarithm of to base is , because is to the rd power: . More generally, if , the ...

to Italy.

Cavalieri's principle In geometry, Cavalieri's principle, a modern implementation of the method of indivisibles, named after Bonaventura Cavalieri, is as follows: * 2-dimensional case: Suppose two regions in a plane are included between two parallel lines in that pl ...

geometry Geometry (; ) is a branch of mathematics concerned with properties of space such as the distance, shape, size, and relative position of figures. Geometry is, along with arithmetic, one of the oldest branches of mathematics. A mathematician w ...

partially anticipated

integral calculus In mathematics, an integral is the continuous analog of a sum, which is used to calculate areas, volumes, and their generalizations. Integration, the process of computing an integral, is one of the two fundamental operations of calculus,Int ...

Life

Born in

Milan Milan ( , , ; ) is a city in northern Italy, regional capital of Lombardy, the largest city in Italy by urban area and the List of cities in Italy, second-most-populous city proper in Italy after Rome. The city proper has a population of nea ...

, Cavalieri joined the Jesuates order (not to be confused with the

Jesuits The Society of Jesus (; abbreviation: S.J. or SJ), also known as the Jesuit Order or the Jesuits ( ; ), is a religious order (Catholic), religious order of clerics regular of pontifical right for men in the Catholic Church headquartered in Rom ...

) at the age of fifteen, taking the name Bonaventura upon becoming a novice of the order, and remained a member until his death. He took his vows as a full member of the order in 1615, at the age of seventeen, and shortly after joined the Jesuat house in Pisa. By 1616 he was a student of

at the University of Pisa. There he came under the tutelage of

Benedetto Castelli Benedetto Castelli (1578 – 9 April 1643), born Antonio Castelli, was an Italians, Italian mathematician. Benedetto was his name in religion on entering the Benedictine Order in 1595. Life Born in Brescia, Castelli studied at the University of ...

, who probably introduced him to

Galileo Galilei Galileo di Vincenzo Bonaiuti de' Galilei (15 February 1564 – 8 January 1642), commonly referred to as Galileo Galilei ( , , ) or mononymously as Galileo, was an Italian astronomer, physicist and engineer, sometimes described as a poly ...

. In 1617 he briefly joined the

Medici The House of Medici ( , ; ) was an Italian banking family and political dynasty that first consolidated power in the Republic of Florence under Cosimo de' Medici and his grandson Lorenzo "the Magnificent" during the first half of the 15th ...

court in

Florence Florence ( ; ) is the capital city of the Italy, Italian region of Tuscany. It is also the most populated city in Tuscany, with 362,353 inhabitants, and 989,460 in Metropolitan City of Florence, its metropolitan province as of 2025. Florence ...

, under the patronage of Cardinal Federico Borromeo, but the following year he returned to Pisa and began teaching Mathematics in place of Castelli. He applied for the Chair of Mathematics at the

University of Bologna The University of Bologna (, abbreviated Unibo) is a Public university, public research university in Bologna, Italy. Teaching began around 1088, with the university becoming organised as guilds of students () by the late 12th century. It is the ...

but was turned down. In 1620, he returned to the Jesuate house in Milan, where he had lived as a novitiate, and became a deacon under Cardinal Borromeo. He studied

theology Theology is the study of religious belief from a Religion, religious perspective, with a focus on the nature of divinity. It is taught as an Discipline (academia), academic discipline, typically in universities and seminaries. It occupies itse ...

in the

monastery A monastery is a building or complex of buildings comprising the domestic quarters and workplaces of Monasticism, monastics, monks or nuns, whether living in Cenobitic monasticism, communities or alone (hermits). A monastery generally includes a ...

of San Gerolamo in Milan, and was named prior of the monastery of St. Peter in Lodi. In 1623 he was made prior of St. Benedict's monastery in Parma, but was still applying for positions in mathematics. He applied again to Bologna and then, in 1626, to Sapienza University, but was declined each time, despite taking six months' leave of absence to support his case to Sapienza in Rome. In 1626 he began to suffer from gout, which would restrict his movements for the rest of his life.J J O'Connor and E F Robertson, Bonaventura Francesco Cavalieri, ''MacTutor History of Mathematics'', (University of St Andrews, Scotland, July 2014) He was also turned down from a position at the University of Parma, which he believed was due to his membership of the Jesuate order, as Parma was administered by the Jesuit order at the time. In 1629 he was appointed Chair of Mathematics at the University of Bologna, which is attributed to Galileo's support of him to the Bolognese senate. He published most of his work while at Bologna, though some of it had been written previously; his ''Geometria Indivisibilibus'', where he outlined what would later become the method of indivisibles, was written in 1627 while in Parma and presented as part of his application to Bologna, but was not published until 1635. Contemporary critical reception was mixed, and ''Exercitationes geometricae sex'' (Six Exercises in Geometry) was published in 1647, partly as a response to criticism. Also at Bologna, he published tables of logarithms and information on their use, promoting their use in Italy. Galileo exerted a strong influence on Cavalieri, and Cavalieri would write at least 112 letters to Galileo. Galileo said of him, "few, if any, since

Archimedes Archimedes of Syracuse ( ; ) was an Ancient Greece, Ancient Greek Greek mathematics, mathematician, physicist, engineer, astronomer, and Invention, inventor from the ancient city of Syracuse, Sicily, Syracuse in History of Greek and Hellenis ...

, have delved as far and as deep into the science of geometry." He corresponded widely; his known correspondents include

Marin Mersenne Marin Mersenne, OM (also known as Marinus Mersennus or ''le Père'' Mersenne; ; 8 September 1588 – 1 September 1648) was a French polymath whose works touched a wide variety of fields. He is perhaps best known today among mathematicians for ...

Evangelista Torricelli Evangelista Torricelli ( ; ; 15 October 160825 October 1647) was an Italian people, Italian physicist and mathematician, and a student of Benedetto Castelli. He is best known for his invention of the barometer, but is also known for his advances i ...

and

Vincenzo Viviani Vincenzo Viviani (April 5, 1622 – September 22, 1703) was an Italian mathematician and scientist. He was a pupil of Torricelli and Galileo. Torricelli in particular was instrumental in refining and promoting the method of indivisibles. He also benefited from the patronage of Cesare Marsili.Cavalieri, Bonaventura
at The Galileo Project Towards the end of his life, his health declined significantly. Arthritis prevented him from writing, and much of his correspondence was dictated and written by Stephano degli Angeli, a fellow Jesuate and student of Cavalieri. Angeli would go on to further develop Cavalieri's method. In 1647 he died, probably of gout.

Science, Mathematics Work

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

Work in optics

Cavalieri's first book, first published in 1632 and reprinted once in 1650, was , or ''The Burning Mirror, or a Treatise on Conic Sections''. The aim of ''Lo Specchio Ustorio'' was to address the question of how

could have used mirrors to burn the Roman fleet as they approached Syracuse, a question still in debate. The book went beyond this purpose and also explored conic sections, reflections of light, and the properties of parabolas. In this book, he developed the theory of mirrors shaped into

parabola In mathematics, a parabola is a plane curve which is Reflection symmetry, mirror-symmetrical and is approximately U-shaped. It fits several superficially different Mathematics, mathematical descriptions, which can all be proved to define exactl ...

hyperbola In mathematics, a hyperbola is a type of smooth function, smooth plane curve, curve lying in a plane, defined by its geometric properties or by equations for which it is the solution set. A hyperbola has two pieces, called connected component ( ...

s, and

ellipse In mathematics, an ellipse is a plane curve surrounding two focus (geometry), focal points, such that for all points on the curve, the sum of the two distances to the focal points is a constant. It generalizes a circle, which is the special ty ...

s, and various combinations of these mirrors. He demonstrated that if, as was later shown, light has a finite and determinate speed, there is minimal interference in the image at the focus of a parabolic, hyperbolic or elliptic mirror, though this was theoretical since the mirrors required could not be constructed using contemporary technology. This would produce better images than the telescopes that existed at the time. He also demonstrated some properties of curves. The first is that, for a light ray parallel to the axis of a parabola and reflected so as to pass through the focus, the sum of the incident angle and its reflection is equal to that of any other similar ray. He then demonstrated similar results for hyperbolas and ellipses. The second result, useful in the design of reflecting telescopes, is that if a line is extended from a point outside of a parabola to the focus, then the reflection of this line on the outside surface of the parabola is parallel to the axis. Other results include the property that if a line passes through a hyperbola and its external focus, then its reflection on the interior of the hyperbola will pass through the internal focus; the reverse of the previous, that a ray directed through the parabola to the internal focus is reflected from the outer surface to the external focus; and the property that if a line passes through one internal focus of an ellipse, its reflection on the internal surface of the ellipse will pass through the other internal focus. While some of these properties had been noted previously, Cavalieri gave the first proof of many. ''Lo Specchio Ustorio'' also included a table of reflecting surfaces and modes of reflection for practical use. Cavalieri's work also contained theoretical designs for a new type of telescope using mirrors, a

reflecting telescope A reflecting telescope (also called a reflector) is a telescope that uses a single or a combination of curved mirrors that reflect light and form an image. The reflecting telescope was invented in the 17th century by Isaac Newton as an alternati ...

, initially developed to answer the question of Archimedes' Mirror and then applied on a much smaller scale as telescopes. 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.

Work in geometry and the method of indivisibles

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, , or ''Geometry, developed by a new method through the indivisibles of the continua''. This was written in 1627, but was not published until 1635. In this work, Cavalieri considers an entity referred to in the text as 'all the lines' or 'all the planes' of a figure, an indefinite number of parallel lines or planes within the bounds of a figure that are comparable to the area and volume, respectively, of the figure. Later mathematicians, improving on his method, would treat 'all the lines' and 'all the planes' as equivalent or equal to the area and volume, but Cavalieri, in an attempt to avoid the question of the composition of the continuum, insisted that the two were comparable but not equal. These parallel elements are called indivisibles respectively of area and volume and provide the building blocks of Cavalieri's method, and are also fundamental features of

. He also used the method of indivisibles to calculate the result which is now written

\int_^ x^2dx = 1/3

, in the process of calculating the area enclosed in an

Archimedean Spiral The Archimedean spiral (also known as Archimedes' spiral, the arithmetic spiral) is a spiral named after the 3rd-century BC Ancient Greece, Greek mathematician Archimedes. The term ''Archimedean spiral'' is sometimes used to refer to the more gene ...

, which he later generalised to other figures, showing, for instance, that the volume of a cone is one-third of the volume of its circumscribed cylinder. An immediate application of the method of indivisibles is

, which states that the

volume Volume is a measure of regions in three-dimensional space. It is often quantified numerically using SI derived units (such as the cubic metre and litre) or by various imperial or US customary units (such as the gallon, quart, cubic inch) ...

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 used by Zu Gengzhi (480–525) of

China China, officially the People's Republic of China (PRC), is a country in East Asia. With population of China, a population exceeding 1.4 billion, it is the list of countries by population (United Nations), second-most populous country after ...

, in the specific case of calculating the volume of the sphere.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 Shen Kuo (; 1031–1095) or Shen Gua, courtesy name Cunzhong (存中) and Art name#China, pseudonym Mengqi (now usually given as Mengxi) Weng (夢溪翁),Yao (2003), 544. was a Chinese polymath, scientist, and statesman of the Song dynasty (960� ...

in the 11th century.) The method of indivisibles as set out by Cavalieri was powerful but was limited in its usefulness in two respects. First, while Cavalieri's proofs were intuitive and later demonstrated to be correct, they were not rigorous; second, his writing was dense and opaque. While many contemporary mathematicians furthered the method of indivisibles, the critical reception was severe. Andre Taquet and Paul Guldin both published responses to the Guldin's particularly in-depth critique suggested that Cavalieri's method was derived from the work of

Johannes Kepler Johannes Kepler (27 December 1571 – 15 November 1630) was a German astronomer, mathematician, astrologer, Natural philosophy, natural philosopher and writer on music. He is a key figure in the 17th-century Scientific Revolution, best know ...

and Bartolomeo Sovero, attacked his method for a lack of rigorousness, and then argues that there can be no meaningful ratio between two infinities, and therefore it is meaningless to compare one to another. Cavalieri's ''Exercitationes geometricae sex'' or ''Six Geometric Exercises'' (1647) was written in direct response to Guldin's criticism. It was initially drafted as a

dialogue Dialogue (sometimes spelled dialog in American and British English spelling differences, American English) is a written or spoken conversational exchange between two or more people, and a literature, literary and theatrical form that depicts suc ...

in the manner of Galileo, but correspondents advised against the format as being unnecessarily inflammatory. The charges of plagiarism were without substance, but much of the ''Exercitationes'' dealt with the mathematical substance of Guldin's arguments. He argued, disingenuously, that his work regarded 'all the lines' as a separate entity from the area of a figure, and then argued that 'all the lines' and 'all the planes' dealt not with absolute but with relative infinity, and therefore could be compared. These arguments were not convincing to contemporaries. The ''Exercitationes'' nonetheless represented a significant improvement to the method of indivisibles. By applying transformations to his variables, he generalised his previous integral result, showing that

\int_^ x^n dx = 1/(n+1)

for n=3 to n=9, which is now known as Cavalieri's quadrature formula.

Work in astronomy

Towards the end of his life, Cavalieri published two books on

astronomy Astronomy is a natural science that studies celestial objects and the phenomena that occur in the cosmos. It uses mathematics, physics, and chemistry in order to explain their origin and their overall evolution. Objects of interest includ ...

. While they use the language of

astrology Astrology is a range of Divination, divinatory practices, recognized as pseudoscientific since the 18th century, that propose that information about human affairs and terrestrial events may be discerned by studying the apparent positions ...

, he states in the text that he did not believe in or practice

. Those books were the (1639) and the (1646).

Other work

He published tables of

logarithm In mathematics, the logarithm of a number is the exponent by which another fixed value, the base, must be raised to produce that number. For example, the logarithm of to base is , because is to the rd power: . More generally, if , the ...

s, emphasizing their practical use in the fields of astronomy and

geography Geography (from Ancient Greek ; combining 'Earth' and 'write', literally 'Earth writing') is the study of the lands, features, inhabitants, and phenomena of Earth. Geography is an all-encompassing discipline that seeks an understanding o ...

. Cavalieri also constructed a hydraulic pump for a monastery that he managed. The Duke of Mantua obtained one similar.

Legacy

According to Gilles-Gaston Granger, Cavalieri belongs with Newton,

Leibniz Gottfried Wilhelm Leibniz (or Leibnitz; – 14 November 1716) was a German polymath active as a mathematician, philosopher, scientist and diplomat who is credited, alongside Sir Isaac Newton, with the creation of calculus in addition to many ...

, Pascal, Wallis and MacLaurin as one of those who in the 17th and 18th centuries "redefine the mathematical object". Gilles-Gaston Granger, ''Formes, opérations, objets'', Vrin, 1994, p. 36
Online quotation
/ref> The

lunar crater Lunar craters are impact craters on Earth's Moon. The Moon's surface has many craters, all of which were formed by impacts. The International Astronomical Union currently recognizes 9,137 craters, of which 1,675 have been dated. History The wo ...

Cavalerius is named after Cavalieri.

Notes

References

The Galileo Project: Cavalieri

External links

* * Online texts by Cavalieri: ** '
Lo specchio ustorio: overo, Trattato delle settioni coniche...
' (1632) **
Directorium generale uranometricum
' (1632) **
Geometria indivisibilibus
' (1653) **
Sfera astronomica
' (1690) * Biographies: ** *
Short biography on bookrags.com
** * Modern mathematical or historical research: *
Infinitesimal Calculus
On its historical development, in Encyclopaedia of Mathematics, Michiel Hazewinkel ed. **
More information about the method of Cavalieri
*
Cavalieri Integration
{{DEFAULTSORT:Cavalieri, Bonaventura 1598 births 1647 deaths 17th-century Italian mathematicians 17th-century Italian astronomers University of Pisa alumni Catholic clergy scientists Members of the Lincean Academy