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geometry Geometry (; ) is a branch of mathematics concerned with properties of space such as the distance, shape, size, and relative position of figures. Geometry is, along with arithmetic, one of the oldest branches of mathematics. A mathematician w ...
, Cavalieri's principle, a modern implementation of the method of indivisibles, named after
Bonaventura Cavalieri Bonaventura Francesco Cavalieri (; 1598 – 30 November 1647) was an Italian mathematician and a Jesuati, Jesuate. He is known for his work on the problems of optics and motion (physics), motion, work on indivisibles, the precursors of infin ...
, is as follows: * 2-dimensional case: Suppose two regions in a plane are included between two parallel lines in that plane. If every line parallel to these two lines intersects both regions in line segments of equal length, then the two regions have equal areas. * 3-dimensional case: Suppose two regions in three-space (solids) are included between two parallel planes. If every plane parallel to these two planes intersects both regions in cross-sections of equal area, then the two regions have equal volumes. Today Cavalieri's principle is seen as an early step towards
integral calculus In mathematics, an integral is the continuous analog of a sum, which is used to calculate areas, volumes, and their generalizations. Integration, the process of computing an integral, is one of the two fundamental operations of calculus,Int ...
, and while it is used in some forms, such as its generalization in
Fubini's theorem In mathematical analysis, Fubini's theorem characterizes the conditions under which it is possible to compute a double integral by using an iterated integral. It was introduced by Guido Fubini in 1907. The theorem states that if a function is L ...
and
layer cake representation In mathematics, the layer cake representation of a non-negative number, negative, real number, real-valued measurable function f defined on a measure space (\Omega,\mathcal,\mu) is the formula :f(x) = \int_0^\infty 1_ (x) \, \mathrmt, for all ...
, 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 The method of exhaustion () is a method of finding the area of a shape by inscribing inside it a sequence of polygons (one at a time) whose areas converge to the area of the containing shape. If the sequence is correctly constructed, the differ ...
, which used limits but did not use
infinitesimal In mathematics, an infinitesimal number is a non-zero quantity that is closer to 0 than any non-zero real number is. The word ''infinitesimal'' comes from a 17th-century Modern Latin coinage ''infinitesimus'', which originally referred to the " ...
s.


History

Cavalieri's principle was originally called the method of indivisibles, the name it was known by in
Renaissance Europe The Renaissance ( , ) is a period of history and a European cultural movement covering the 15th and 16th centuries. It marked the transition from the Middle Ages to modernity and was characterized by an effort to revive and surpass the idea ...
. 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). While Cavalieri's work established the principle, in his publications he denied that the continuum was composed of indivisibles in an effort to avoid the associated paradoxes and religious controversies, and he did not use it to find previously unknown results. In the 3rd century BC,
Archimedes Archimedes of Syracuse ( ; ) was an Ancient Greece, Ancient Greek Greek mathematics, mathematician, physicist, engineer, astronomer, and Invention, inventor from the ancient city of Syracuse, Sicily, Syracuse in History of Greek and Hellenis ...
, 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 ''The Method of Mechanical Theorems'' (), also referred to as ''The Method'', is one of the major surviving works of the ancient Greece, ancient Greek polymath Archimedes. ''The Method'' takes the form of a letter from Archimedes to Eratosthenes, ...
''. In the 5th century AD,
Zu Chongzhi Zu Chongzhi (; 429 – 500), courtesy name Wenyuan (), was a Chinese astronomer, inventor, mathematician, politician, and writer during the Liu Song and Southern Qi dynasties. He was most notable for calculating pi as between 3.1415926 and 3.1415 ...
and his son Zu Gengzhi established a similar method to find a sphere's volume. Neither of the approaches, however, were known in early modern Europe. The transition from Cavalieri's indivisibles to
Evangelista Torricelli Evangelista Torricelli ( ; ; 15 October 160825 October 1647) was an Italian people, Italian physicist and mathematician, and a student of Benedetto Castelli. He is best known for his invention of the barometer, but is also known for his advances i ...
's and
John Wallis John Wallis (; ; ) was an English clergyman and mathematician, who is given partial credit for the development of infinitesimal calculus. Between 1643 and 1689 Wallis served as chief cryptographer for Parliament and, later, the royal court. ...
's
infinitesimal In mathematics, an infinitesimal number is a non-zero quantity that is closer to 0 than any non-zero real number is. The word ''infinitesimal'' comes from a 17th-century Modern Latin coinage ''infinitesimus'', which originally referred to the " ...
s was a major advance in the history of
calculus Calculus is the mathematics, mathematical study of continuous change, in the same way that geometry is the study of shape, and algebra is the study of generalizations of arithmetic operations. Originally called infinitesimal calculus or "the ...
. The indivisibles were entities of
codimension In mathematics, codimension is a basic geometric idea that applies to subspaces in vector spaces, to submanifolds in manifolds, and suitable subsets of algebraic varieties. For affine and projective algebraic varieties, the codimension equals ...
1, so that a plane figure was thought as made out of an infinite number 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/∞.


2-dimensional


Cycloids

N. Reed has shown how to find the area bounded by a
cycloid In geometry, a cycloid is the curve traced by a point on a circle as it Rolling, rolls along a Line (geometry), straight line without slipping. A cycloid is a specific form of trochoid and is an example of a roulette (curve), roulette, a curve g ...
by using Cavalieri's principle. A circle of radius ''r'' can roll in a clockwise direction upon a line below it, or in a counterclockwise direction upon a line above it. A point on the circle thereby traces out two cycloids. When the circle has rolled any particular distance, the angle through which it would have turned clockwise and that through which it would have turned counterclockwise are the same. The two points tracing the cycloids are therefore at equal heights. The line through them is therefore horizontal (i.e. parallel to the two lines on which the circle rolls). Consequently each horizontal cross-section of the circle has the same length as the corresponding horizontal cross-section of the region bounded by the two arcs of cycloids. By Cavalieri's principle, the circle therefore has the same area as that region. Consider the rectangle bounding a single cycloid arch. From the definition of a cycloid, it has width and height , so its area is four times the area of the circle. Calculate the area within this rectangle that lies above the cycloid arch by bisecting the rectangle at the midpoint where the arch meets the rectangle, rotate one piece by 180° and overlay the other half of the rectangle with it. The new rectangle, of area twice that of the circle, consists of the "lens" region between two cycloids, whose area was calculated above to be the same as that of the circle, and the two regions that formed the region above the cycloid arch in the original rectangle. Thus, the area bounded by a rectangle above a single complete arch of the cycloid has area equal to the area of the circle, and so, the area bounded by the arch is three times the area of the circle.


3-dimensional


Cones and pyramids

The fact that the volume of any
pyramid A pyramid () is a structure whose visible surfaces are triangular in broad outline and converge toward the top, making the appearance roughly a pyramid in the geometric sense. The base of a pyramid can be of any polygon shape, such as trian ...
, regardless of the shape of the base, including cones (circular base), 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. In fact, Cavalieri's principle or similar infinitesimal argument is ''necessary'' to compute the volume of cones and even pyramids, which is essentially the content of
Hilbert's third problem The third of Hilbert's problems, Hilbert's list of mathematical problems, presented in 1900, was the first to be solved. The problem is related to the following question: given any two polyhedron, polyhedra of equal volume, is it always possible t ...
– polyhedral pyramids and cones cannot be cut and rearranged into a standard shape, and instead must be compared by infinite (infinitesimal) means. The ancient Greeks used various precursor techniques such as Archimedes's mechanical arguments or
method of exhaustion The method of exhaustion () is a method of finding the area of a shape by inscribing inside it a sequence of polygons (one at a time) whose areas converge to the area of the containing shape. If the sequence is correctly constructed, the differ ...
to compute these volumes.


Paraboloids

Consider a cylinder of radius r and height h, circumscribing a
paraboloid In geometry, a paraboloid is a quadric surface that has exactly one axial symmetry, axis of symmetry and no central symmetry, center of symmetry. The term "paraboloid" is derived from parabola, which refers to a conic section that has a similar p ...
y=h\left(\frac\right)^2 whose apex is at the center of the bottom base of the cylinder and whose base is the top base of the cylinder.
Also consider the paraboloid y=h-h\left(\frac\right)^2, with equal dimensions but with its apex and base flipped. For every height 0\le y\le h, the disk-shaped cross-sectional area \pi\left(\sqrt\,r\right)^2 of the flipped paraboloid is equal to the ring-shaped cross-sectional area \pi r^2-\pi\left(\sqrt\,r\right)^2 of the cylinder part ''outside'' the inscribed paraboloid. Therefore, the volume of the flipped paraboloid is equal to the volume of the cylinder part ''outside'' the inscribed paraboloid. In other words, the volume of the paraboloid is \fracr^2h, half the volume of its circumscribing cylinder.


Spheres

If one knows that the volume of a
cone In geometry, a cone is a three-dimensional figure that tapers smoothly from a flat base (typically a circle) to a point not contained in the base, called the '' apex'' or '' vertex''. A cone is formed by a set of line segments, half-lines ...
is \frac\left(\text \times \text\right), then one can use Cavalieri's principle to derive the fact that the volume of a
sphere A sphere (from Ancient Greek, Greek , ) is a surface (mathematics), surface analogous to the circle, a curve. In solid geometry, a sphere is the Locus (mathematics), set of points that are all at the same distance from a given point in three ...
is \frac\pi r^3, where r is the radius. That is done as follows: Consider a sphere of radius r and a cylinder of radius r and height 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 In mathematics, the Pythagorean theorem or Pythagoras' theorem is a fundamental relation in Euclidean geometry between the three sides of a right triangle. It states that the area of the square whose side is the hypotenuse (the side opposite t ...
, the plane located y units above the "equator" intersects the sphere in a circle of radius \sqrt and area \pi\left(r^2 - y^2\right). The area of the plane's intersection with the part of the cylinder that is ''outside'' of the cone is also \pi\left(r^2 - y^2\right). As can be seen, the area of the circle defined by the intersection with the sphere of a horizontal plane located at any height y equals the area of the intersection of that plane with the part of the cylinder that is "outside" of the cone; thus, applying Cavalieri's principle, it could be said 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 \frac of the volume of the cylinder, thus the volume ''outside'' of the cone is \frac the volume of the cylinder. Therefore the volume of the upper half of the sphere is \frac of the volume of the cylinder. The volume of the cylinder is :\text \times \text=\pi r^2 \cdot r = \pi r^3 ("Base" is in units of ''area''; "height" is in units of ''distance''. .) Therefore the volume of the upper half-sphere is \frac\pi r^3 and that of the whole sphere is \frac \pi r^3.


The napkin ring problem

In what is called the
napkin ring problem In geometry, the napkin-ring problem involves finding the volume of a "band" of specified height around a sphere, i.e. the part that remains after a hole in the shape of a circular cylinder is drilled through the center of the sphere. It is a co ...
, one shows by Cavalieri's principle that when a hole is drilled straight through the centre of a sphere where the remaining band has height h, the volume of the remaining material surprisingly does not depend on the size of the sphere. The cross-section of the remaining ring is a plane annulus, whose area is the difference between the areas of two circles. By the Pythagorean theorem, the area of one of the two circles is \pi \times (r^2-y^2), where r is the sphere's radius and y is the distance from the plane of the equator to the cutting plane, and that of the other is \pi \times \left(r^2 - \left(\frac\right)^2\right). When these are subtracted, the r^2 cancels; hence the lack of dependence of the bottom-line answer upon r.


Generalisation to measures

Let \mu be a measure on \Omega \subset \mathbb R^N. Then Cavalieri's principle would be transcribed for f \colon \Omega \to \mathbb R^+ integrable as\int_\Omega f\,\mathrm d\mu = \int_0^\infty \mu\bigl(\\bigr)\,\mathrm dt \;.For a function f on \Omega with values in \mathbb R, know that it can be rewritten as the difference of two positive functions f = f^+ - f^-, where f^+ and f^- denote the positive and negative parts of f respectively.


See also

*
Fubini's theorem In mathematical analysis, Fubini's theorem characterizes the conditions under which it is possible to compute a double integral by using an iterated integral. It was introduced by Guido Fubini in 1907. The theorem states that if a function is L ...
(Cavalieri's principle is a particular case of Fubini's theorem)


References


External links

* *
Prinzip von Cavalieri

Cavalieri Integration
{{Infinitesimals Geometry Mathematical principles History of calculus Area Volume Zu Chongzhi