Bonaventura Cavalieri(1598-1647) was an italian mathematician. He was a precursor of infinitesimal calculus. Cavalieri, Kepler and other mathematicians, who lived during the century preceding Newton and Leibniz, invented and used intuitive infinitesimal methods to solve area and volume problems.

Twenty years after the publication of Kepler's Stereometria Doliorum, Cavalieri wrote a very popular book: Geometria indivisibilibus (1635).

In this book, the Italian mathematician used what is now known as Cavalieri's Principle: If two solids have equal altitudes, and if sections made by planes parallel to the bases and at equal distances from them are always in a given ratio, then the volumes of the solids are also in this ratio.

Cavalieri's Principle is also call the method of indivisibles. "Cavalieri made the notion of the indivisible the basis of a geometrical method of demonstration. He didn't explained precisely what he understood by the word indivisible, which he employed to characterize the infinitesimal elements used in his method. Cavalieri conceived of a surface as made up an indefinite number of equidistant parallel lines and of a solid as composed of parallel equidistant planes, these elements being designated the indivisibles of the surface and of the volume respectively." (C.H. Edwards)

Zu Geng, born about 450, was a chinese mathematician who used what is now know as the Principle of Liu Hui and Zu Geng to calculate the volume of a sphere. Liu-Zu theory is equivalent to Cavalieri's Principle. Then, chinese mathematicians had used this principle for more than one millennium before Cavalieri. You can read a biography of Zu Geng in MacTutor and the article Zu-Geng's axiom vs Cavalieri's theory by Ji-Huan He.

A well known application of Cavalieri's Principle is used to calculate the volume of a sphere. We can compare the area of a section of an hemisphere and the area of a section of a body that is a cylinder minus a cone. This two areas are equal. Then the two bodies have the same volume. It is very easy to calculate the volume of the second body, then we get the volume of the hemisphere.

For each altitude h, the area of the disc is:

and the area of the annulus is:

Then, for each altitude the two sections have equal area. Using Cavalieri's Principle we can deduce:

Using the formulas for the volume of a cylinder and of a cone we can write the volume of an hemisphere:

Then, the volume of a sphere of radius R is (as Archimedes already knew, 1800 years before):

LINKS

	Kepler: Surface and volume of a sphere Kepler studied the volume and surface of the sphere. He thought the volume of the sphere as made up of small cones, then he sum all of these cones and get a relation between the surface of a sphere en its volume.
	Kepler: The Area of a Circle Kepler used an intuitive infinitesimal approach to calculate the area of a circle.
	Kepler: The volume of a wine barrel Kepler was one mathematician who contributed to the origin of integral calculus. He used infinitesimal techniques for calculating areas and volumes.
	Kepler: The best proportions for a wine barrel Studying the volume of a barrel, Kepler solved a problem about maxima in 1615.
	Surprising Cavalieri congruence between a sphere and a tetrahedronn We show a sphere and the Howard Eves's tetrahedron with congruent sections.
	Archimedes' Method to calculate the area of a parabolic segment Archimedes show us in 'The Method' how to use the lever law to discover the area of a parabolic segment.