**Bonaventure Francesco Cavalieri ( Italian: Bonaventura Francesco Cavalieri , lat. Cavalerius** ), ( 1598 - November 30, 1647 ) - Italian mathematician , the forerunner of mathematical analysis , the most vivid and influential representative of the " geometry of the indivisible ." The principles and methods put forward by him made it possible even before the discovery of mathematical analysis to successfully solve many problems of an analytical nature.

Bonaventure Cavalieri | |
---|---|

Date of Birth | 1598 |

Place of Birth | Milan |

Date of death | November 30, 1647 |

Place of death | |

A country | Italy |

Scientific field | Maths |

Place of work | University of Bologna |

Alma mater | University of Pisa |

supervisor | Benedetto Castelli |

Famous students | and |

Known as | The author of the indivisible method |

## Content

## Biography

Cavalieri was born in Milan , at an early age was tonsured a monk and belonged to the Jesuit Order of Blessed Jerome . He studied mathematics in Pisa under the guidance of a follower and friend of Galileo Benedetto Castelli . Through Castelli, Cavalieri met Galileo , who then lived in nearby Florence.

At the end of 1621, Cavalieri had already made significant progress in developing the method of indivisibles , and in correspondence with Galileo he discussed the question of the permissibility of decomposing figures into infinitesimal elements.

When the Department of Mathematics in Bologna was vacated in 1629 , Cavalieri presented a manuscript of finished work on the geometry of indivisibles. His candidacy was warmly supported by Galileo, who characterized the young scientist as a "rival of Archimedes ."

Cavalieri worked as a professor at the University of Bologna for the rest of his life. Pope Urban VIII, who favored him , appointed him rector of the monastery.

The last years of Cavalieri were overshadowed by the severe form of gout , from which he died prematurely at the age of 49 years.

In 1632, Cavalieri introduced the designation “log.” For the logarithm . Before him, Kepler used the notation “Log.” ^{[2]} .

Cavalieri owns several works on trigonometry , logarithms , geometric optics , etc., but the main work of his life was the treatise " *Geometry, developed in a new way using indivisible continuous* " ( 1635 ) and serving as its continuation of " *Six geometric studies* " ( 1647 ) .

In honor of Cavalieri named a crater on the moon .

## The indivisible method

Comparison of the areas of planar figures of Cavalieri reduces to comparing “all lines”, which can be imagined as sections of figures that are straight, which move, but remain all the time parallel to some guide - the *regulation* . Similarly, to compare the volumes of bodies, flat sections taken in their entirety are introduced.

The technique of applying the method in planimetry was usually as follows: a figure of a known area was selected, the cross-sections of which can be compared with the cross-sections of the investigated. If the lengths of the sectional sections from each pair were in a ratio of, say, 1: 2, it was concluded that the same ratio was true for the squares of the figures, which immediately yielded the result. Similarly, in the case of three-dimensional bodies.

The main pillar of the new geometry, Cavalieri considered the theorem:

The figures relate to each other, like all their lines, taken according to any regulation, and the bodies - like all their planes, taken according to any regulation.

It follows that to find the relationship between two flat or solid figures, it is enough to find the relationship between all the indivisibles of both figures by any rule.

Note that sometimes Cavalieri and his followers used curved sections in the decomposition.

Cavalieri offered numerous examples of the successful application of the indivisible method, both for known bodies and new ones (for example, a hyperboloid of revolution ). He gave an example of a paradox that can lead to incorrect conclusions due to the unsuccessful choice of indivisible sections. But he did not give a clear rule to avoid mistakes.

The power and relative simplicity of the new method made an extremely strong impression on contemporary mathematicians. Entire generations of eminent mathematicians have studied with Cavalieri.

## Proceedings in the Russian translation

*Bonaventure Cavalieri.*Geometry set forth in a new way using indivisible continuous. Translation with an introductory article and notes by S. Ya. Lurie. M. - L .: Publishing. technical and theoretical literature, 1940, 414 p.

## Notes

- ↑ Cavalieri Bonaventure // Great Soviet Encyclopedia : [in 30 vol.] / Ed. A. M. Prokhorov - 3rd ed. - M .: Soviet Encyclopedia , 1969.
- ↑
*Cajori F.*A History of Mathematical Notations. Vol. 2 (1929 reprint) . - NY: Cosimo, Inc., 2007 .-- P. ss. - xii + 392 p. - ISBN 978-1-60206-713-4 .

## Literature

- History of Mathematics / Edited by A.P. Yushkevich , in three volumes. - M .: Science, 1970. - T. II (Mathematics of the XVII century).
- Shawl . A historical review of the origin and development of geometric methods . Ch. 2, § 5. M., 1883.

## Links

*John J. O'Connor*and*Edmund F. Robertson*. Cavalieri, Bonaventure (English) - biography in the MacTutor archive. (eng.)- More informations about the method of Cavalieri
- Infinitesimal Calculus , an article on its historical development, in Encyclopaedia of Mathematics, Michiel Hazewinkel ed. (eng.)