Bonaventura Francesco Cavalieri (; 1598 – 30 November 1647) was an Italian mathematician and a Jesuate. 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 in Milan, Cavalieri joined the Jesuates order (not to be confused with the Jesuits) 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 geometry at the University of Pisa. There he came under the tutelage of Benedetto Castelli, who probably introduced him to Galileo Galilei. In 1617, he briefly joined the Medici court in 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 but was turned down. 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.

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. The aim of Lo Specchio Ustorio was to address the question of how Archimedes 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 parabolas, hyperbolas, and ellipses, 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.

thumb|Geometrical figures from Lo Speccio Ustorio, used in proofs of properties of parabolic reflecting surfaces.|alt=Two illustrations from Lo Speccio Ustorio, demonstrating two principles of reflection of light on the surface of a parabola.

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. 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.

An immediate application of the method of indivisibles is 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 used by Zu Gengzhi (480–525) of China, in the specific case of calculating the volume of the sphere.)

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 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.

The lunar crater Cavalerius is named after Cavalieri.

See also

  • Evangelista Torricelli
  • Stefano degli Angeli
  • Cavalieri's quadrature formula

References

Sources

  • The Galileo Project: Cavalieri

Further reading

  • 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.
  • 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