Demo One Bonaventura Cavalieri
(1598-1647)
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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. He
published eleven books, his first being published in 1632. He
worked on the problems of optics and motion. His astronomical
and 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."
Bonaventura Cavalieri
(1598-1647)
He developed a method of indivisibles which became a
factor in the development of the integral calculus.
Cavalieri's first book was Lo Specchio Ustorio, overo, Trattato delle settioni coniche,
or The Burning Mirror, or a Treatise on Conic Sections.[1] In this book he developed
he theory of mirrors shaped into parabolas, hyperbolas, and ellipses, 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.

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.

Cavalieri is known for 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 discovered by Zu Gengzhi (480–525) of China.[3]) 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 ([3]).

Cavalieri also constructed a hydraulic pump for his monastery and published tables
of logarithms, emphasizing their practical use in the fields of astronomy and
geography. He died at Bologna.

The lunar crater Cavalerius is named for the Latin name of Bonaventura Cavalieri.
Coins illustrating Cavalieri's
principle
Cavalieri’s principle says that if two
three-dimensional figures have the
same height and have the same cross-
sectional area at every level, they
have the same volume.
Volume = Bh, where B is the area of a
cross-section and h is the height of the
solid.
If two three-dimensional figures have the same height and
cross-sectional area at every level, they have the same
volume.
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