Cavalieri's principle

In geometry, Cavalieri's principle, a modern implementation of the method of indivisibles, named after Bonaventura Cavalieri, is as follows: In geometry, Cavalieri's principle, a modern implementation of the method of indivisibles, named after Bonaventura Cavalieri, is as follows: Today Cavalieri's principle is seen as an early step towards integral calculus, and while it is used in some forms, such as its generalization in Fubini's theorem, results using Cavalieri's principle can often be shown more directly via integration. In the other direction, Cavalieri's principle grew out of the ancient Greek method of exhaustion, which used limits but did not use infinitesimals. Cavalieri's principle was originally called the method of indivisibles, the name it was known by in Renaissance Europe. Cavalieri developed a complete theory of indivisibles, elaborated in his Geometria indivisibilibus continuorum nova quadam ratione promota (Geometry, advanced in a new way by the indivisibles of the continua, 1635) and his Exercitationes geometricae sex (Six geometrical exercises, 1647). In the 3rd century BC, Archimedes, using a method resembling Cavalieri's principle, was able to find the volume of a sphere given the volumes of a cone and cylinder in his work The Method of Mechanical Theorems. In the 5th century AD, Zu Chongzhi and his son Zu Gengzhi established a similar method to find a sphere's volume. The transition from Cavalieri's indivisibles to Evangelista Torricelli's and John Wallis's infinitesimals was a major advance in the history of the calculus. The indivisibles were entities of codimension 1, so that a plane figure was thought as made out of an infinity of 1-dimensional lines. Meanwhile, infinitesimals were entities of the same dimension as the figure they make up; thus, a plane figure would be made out of 'parallelograms' of infinitesimal width. Applying the formula for the sum of an arithmetic progression, Wallis computed the area of a triangle by partitioning it into infinitesimal parallelograms of width 1/∞. If one knows that the volume of a cone is 1 3 ( base × height ) {displaystyle {frac {1}{3}}left({ ext{base}} imes { ext{height}} ight)} , then one can use Cavalieri's principle to derive the fact that the volume of a sphere is 4 3 π r 3 {displaystyle {frac {4}{3}}pi r^{3}} , where r {displaystyle r} is the radius. That is done as follows: Consider a sphere of radius r {displaystyle r} and a cylinder of radius r {displaystyle r} and height r {displaystyle r} . Within the cylinder is the cone whose apex is at the center of one base of the cylinder and whose base is the other base of the cylinder. By the Pythagorean theorem, the plane located y {displaystyle y} units above the 'equator' intersects the sphere in a circle of area π ( r 2 − y 2 ) {displaystyle pi left(r^{2}-y^{2} ight)} . The area of the plane's intersection with the part of the cylinder that is outside of the cone is also π ( r 2 − y 2 ) {displaystyle pi left(r^{2}-y^{2} ight)} . As we can see, the area of every intersection of the circle with the horizontal plane located at any height y {displaystyle y} equals the area of the intersection of the plane with the part of the cylinder that is 'outside' of the cone; thus, applying Cavalieri's principle, we could say that the volume of the half sphere equals the volume of the part of the cylinder that is 'outside' the cone. The aforementioned volume of the cone is 1 3 {displaystyle {frac {1}{3}}} of the volume of the cylinder, thus the volume outside of the cone is 2 3 {displaystyle {frac {2}{3}}} the volume of the cylinder. Therefore the volume of the upper half of the sphere is 2 3 {displaystyle {frac {2}{3}}} of the volume of the cylinder. The volume of the cylinder is ('Base' is in units of area; 'height' is in units of distance. Area × distance = volume.) Therefore the volume of the upper half-sphere is 2 3 π r 3 {displaystyle {frac {2}{3}}pi r^{3}} and that of the whole sphere is 4 3 π r 3 {displaystyle {frac {4}{3}}pi r^{3}} . The fact that the volume of any pyramid, regardless of the shape of the base, whether circular as in the case of a cone, or square as in the case of the Egyptian pyramids, or any other shape, is (1/3) × base × height, can be established by Cavalieri's principle if one knows only that it is true in one case. One may initially establish it in a single case by partitioning the interior of a triangular prism into three pyramidal components of equal volumes. One may show the equality of those three volumes by means of Cavalieri's principle.