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Howard Eves received the George Polya Award for the article Two Surprising Theorems on Cavalieri Congruence.

In this article he wrote a brief biography of Cavalieri (1598-1647): "Cavalieri was one of the most influential mathematicians of his time, and the author of a number of works on trigonometry, geometry, optics, astronomy and astrology. He was among the first to recognize the great value of logarithms and was largely responsible for their introduction into Italy. But his greatest contribution to mathematics was a treatise, Geometria indivisibilibus, published in its first form in 1635, devoted to the precalculus method of indivisibles."

"Cavalieri's treatise on the method of indivisibles is voluable and not clearly written, and it is not easy to learn from it precisely what Cavalieri meant by an 'indivisible'. It seems that an indivisible of a given planar piece is a chord of the piece, and a planar piece can be considered as made up of an infinite parallel set of such indivisibles. Similarly, it seems that an indivisible of a given solid is a planar section of that solid, and a solid can be considered as made up of an infinite parallel set of this kind of indivisibles. Now, Cavalieri argued, if we slide each member of a parallel set of indivisibles of some planar piece along its own axis, so that the endpoints of the indivisibles still trace a continuous boundary, then the area of the new planar piece so formed is the same as that of the original planar piece, inasmuch as the two pieces are made up of the same indivisibles. A similar sliding of the members of a parallel set of indivisibles of a given solid will yield another solid having the same volume as the original one." (H. Eves)

"Accepting Cavalieri's principles as intuitively evident, one can solve many problems in mensuration that normally require the more advanced techniques of the calculus. Let us define two planar pieces that can be placed so that they cut off equal segments on each member of a family of parallel lines, or two solids that can be placed so that they intersect equiareal sections on each members of a family of parallel planes, to be Cavalieri congruent. To find the unknown area of some planar piece, or the unknown volume of some solid, one tries to find a comparaison figure of easily found area or volumen that is Cavalieri congruent to the given figure." (H. Eves)

Cavalieri: The volume of a sphere
Using Cavalieri's Principle we can calculate the volume of a sphere.

One of the purposes of Eves' article is to prove something that is difficult to believe, that there exists a polyhedron to which a given sphere is Cavalieri congruent. "A sphere is so round, and a polyhedron is so angular and made up of nothing but planes! Nevertheless, we will prove that there exists a polyhedron (actually a tetrahedron) to which a given sphere is Cavalieri congruent." (H. Eves)

We have seen that the areas of the sections in one sphere are:

Sections in the sphere
We want to calculate the surface area of sections of a sphere using the Pythagorean Theorem. We also study the relation with the Geometric Mean and the Right Triangle Altitude Theorem.

Sections in a sphere and Geometric mean: calculation of the surface area of a section| matematicasVisuales

The corresponding areas of the sections in the Howard Eves's tetrahedron are equal

Sections in Howard Eves's tetrahedron
In his article 'Two Surprising Theorems on Cavalieri Congruence' Howard Eves describes an interesting tetrahedron. In this page we calculate its cross-section areas and its volume.
Howard Eves: Surprising Cavalieri congruence between a tetrahedron and a sphere: formula | matematicasvisuales
Howard Eves: Surprising Cavalieri congruence between a tetrahedron and a sphere: | matematicasVisuales

The volume of Howard Eves' tetrahedron is:

Sections in Howard's Eves tetrahedron: area of the section in the sphere and area of the section of the tetrahedron | matematicasVisuales

Therefore, Howard Eves claims that

"Theorem 2. There exist a tetrahedron to witch a give sphere is Cavalieri congruent."

Then, the volume of the sphere is the same as the volume of the tetrahedron:

REFERENCES

Howard Eves, mathematician and historian of Mathematics, received the George Polya Award for the article Two Surprising Theorems on Cavalieri Congruence.

MORE LINKS

Pythagoras Theorem: Euclid's demonstration
Demonstration of Pythagoras Theorem inspired in Euclid.
The volume of the tetrahedron
The volume of a tetrahedron is one third of the prism that contains it.
Sections on a tetrahedron
Special sections of a tetrahedron are rectangles (and even squares). We can calculate the area of these cross-sections.
Sections in the sphere
We want to calculate the surface area of sections of a sphere using the Pythagorean Theorem. We also study the relation with the Geometric Mean and the Right Triangle Altitude Theorem.
Leonardo da Vinci:Drawing of a SEPTUAGINTA made to Luca Pacioli's De divina proportione.
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.
The World
Basic world map in a sphere. Latitude and longitude
Kepler: The Area of a Circle
Kepler used an intuitive infinitesimal approach to calculate the area of a circle.
Plane developments of geometric bodies: Tetrahedron
The first drawing of a plane net of a regular tetrahedron was published by Dürer in his book 'Underweysung der Messung' ('Four Books of Measurement'), published in 1525 .
Archimedes and the area of an ellipse: an intuitive approach
In his book 'On Conoids and Spheroids', Archimedes calculated the area of an ellipse. We can see an intuitive approach to Archimedes' ideas.
Archimedes and the area of an ellipse: Demonstration
In his book 'On Conoids and Spheroids', Archimedes calculated the area of an ellipse. It si a good example of a rigorous proof using a double reductio ad absurdum.
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.
Plane developments of geometric bodies (1): Nets of prisms
We study different prisms and we can see how they develop into a plane net. Then we explain how to calculate the lateral surface area.
Plane developments of geometric bodies (3): Cylinders
We study different cylinders and we can see how they develop into a plane. Then we explain how to calculate the lateral surface area.
Plane developments of geometric bodies (5): Pyramid and pyramidal frustrum
Plane net of pyramids and pyramidal frustrum. How to calculate the lateral surface area.
Plane developments of geometric bodies (7): Cone and conical frustrum
Plane developments of cones and conical frustum. How to calculate the lateral surface area.
Plane developments of geometric bodies: Dodecahedron
The first drawing of a plane net of a regular dodecahedron was published by Dürer in his book 'Underweysung der Messung' ('Four Books of Measurement'), published in 1525 .
Leonardo da Vinci:Drawing of a SEPTUAGINTA made to Luca Pacioli's De divina proportione.
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.