Cavalieri: Volume of a sphere
Bonaventura Cavalieri(15981647) 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. LiuZu 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 ZuGeng's axiom vs Cavalieri's theory by JiHuan 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(because we know how to calculate the volume of a cylinder minus the volume of a cone), then we get the volume of the hemisphere.
We need to probe that the area of the disc is equal to the area of the annulus (for each section): 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): Interesting application of Cavalieri's Principle: in the video Without integration, why is the volume of a paraboloid half of its inscribing cylinder? Mathemaniac developed an idea from Yehuda Simcha Waldman to calculate the volume of a paraboloid. Yehuda Simcha made an interactive application showing his idea.
REFERENCES Carl B. Boyer  The History of the Calculus and its Conceptual Development  Dover Publications
C.H. Edwards  The Historical Development of the Calculus (p. 102) SpringerVerlag
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We study a kind of polyhedra inscribed in a sphere, in particular the Campanus' sphere that was very popular during the Renaissance.
Leonardo da Vinci made several drawings of polyhedra for Luca Pacioli's book 'De divina proportione'. Here we can see an adaptation of the Campanus' sphere.
In his article 'Two Surprising Theorems on Cavalieri Congruence' Howard Eves describes an interesting tetrahedron. In this page we calculate its crosssection areas and its volume.
Howard Eves's tetrahedron is Cavalieri congruent with a given sphere. You can see that corresponding sections have the same area. Then the volumen of the sphere is the same as the volume of the tetrahedron. And we know how to calculate this volumen.
Special sections of a tetrahedron are rectangles (and even squares). We can calculate the area of these crosssections.
Archimedes show us in 'The Method' how to use the lever law to discover the area of a parabolic segment.
In his book 'On Conoids and Spheroids', Archimedes calculated the area of an ellipse. We can see an intuitive approach to Archimedes' ideas.
