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Method of exhaustion
 
 
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The method of exhaustion is a method of finding the area
Area

Area is a quantity expressing the two-dimensional size of a defined part of a surface, typically a region bounded by a closed curve. The term surface area refers to the total area of the exposed surface of a 3-dimensional solid, such as the sum of the areas of the exposed sides of a polyhedron....
 of a shape
Shape

The shape of an object located in some space is the part of that space occupied by the object, as determined by its external boundary ? abstracting from other properties such as colour, content, and material composition, as well as from the object's other spatial properties ....
 by inscribing inside it a sequence of polygon
Polygon

In geometry a polygon is traditionally a plane Shape that is bounded by a closed curve path or circuit, composed of a finite sequence of straight line segments ....
s whose areas converge
Convergence

In the absence of a more specific context, convergence denotes the approach toward a definite value, as time goes on; or to a definite point, a common view or opinion, or toward a fixed or equilibrium point state....
 to the area of the containing shape. If the sequence is correctly constructed, the difference in area between the nth polygon and the containing shape will become arbitrarily small as n becomes large. As this difference becomes arbitrarily small, the possible values for the area of the shape are systematically "exhausted" by the lower bound areas successively established by the sequence members.






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The method of exhaustion is a method of finding the area
Area

Area is a quantity expressing the two-dimensional size of a defined part of a surface, typically a region bounded by a closed curve. The term surface area refers to the total area of the exposed surface of a 3-dimensional solid, such as the sum of the areas of the exposed sides of a polyhedron....
 of a shape
Shape

The shape of an object located in some space is the part of that space occupied by the object, as determined by its external boundary ? abstracting from other properties such as colour, content, and material composition, as well as from the object's other spatial properties ....
 by inscribing inside it a sequence of polygon
Polygon

In geometry a polygon is traditionally a plane Shape that is bounded by a closed curve path or circuit, composed of a finite sequence of straight line segments ....
s whose areas converge
Convergence

In the absence of a more specific context, convergence denotes the approach toward a definite value, as time goes on; or to a definite point, a common view or opinion, or toward a fixed or equilibrium point state....
 to the area of the containing shape. If the sequence is correctly constructed, the difference in area between the nth polygon and the containing shape will become arbitrarily small as n becomes large. As this difference becomes arbitrarily small, the possible values for the area of the shape are systematically "exhausted" by the lower bound areas successively established by the sequence members. The idea originated with Antiphon
Antiphon (person)

Antiphon the Sophist lived in Athens probably in the last two decades of the 5th century BC. There is an ongoing controversy over whether he is one and the same with Antiphon of the Athenian deme Rhamnus in Attica, Greece , the earliest of the ten Attic orators....
, although it is not entirely clear how well he understood it. The theory was made rigorous by Eudoxus
Eudoxus of Cnidus

Eudoxus of Cnidus was a Ancient Greece astronomer, mathematician, scholar and student of Plato. Since all his own works are lost, our knowledge of him is obtained from secondary sources, such as Aratus's poem on astronomy....
. The first use of the term was by Gregorie de Saint-Vincent in Opus geometricum guadraturae circuli et sectionum coni in 1647.

The method of exhaustion typically required a form of proof by contradiction, known as reductio ad absurdum
Reductio ad absurdum

Reductio ad absurdum , also known as an apagogical argument, reductio ad impossibile, or proof by contradiction, is a type of logical argument where one assumes a claim for the sake of argument and derives an absurd or ridiculous outcome, and then concludes that the original claim must have been wrong as it led to an abs...
. This amounts to finding an area of a region by first comparing it to the area of a second region (which can be “exhausted” so that its area becomes arbitrarily close to the true area). The proof involves assuming that the true area is greater than the second area, and then proving that assertion false, and then assuming that it is less than the second area, and proving that assertion false, too.

The method of exhaustion is seen as a precursor to the methods of calculus
Calculus

Calculus is a branch of mathematics that includes the study of limit , derivatives, integrals, and infinite series, and constitutes a major part of modern university education....
. The development of analytical geometry and rigorous integral calculus
Calculus

Calculus is a branch of mathematics that includes the study of limit , derivatives, integrals, and infinite series, and constitutes a major part of modern university education....
 in the 17th-19th centuries (in particular a rigorous definition of limit
Limit

A limit can be:* Limit , including:** Limit of a function** Limit of a sequence** One-sided limit** Limit superior and limit inferior** Net ...
) subsumed the method of exhaustion so that it is no longer explicitly used to solve problems.

Archimedes
Archimedes

Archimedes of Syracuse was a Greek mathematics, physicist, engineer, inventor, and astronomer. Although few details of his life are known, he is regarded as one of the leading scientists in classical antiquity....
 used the method of exhaustion as a way to calculate p
P

P is the sixteenth letter of the modern Latin alphabet. Its name in English language is pronounced pee ....
 by filling the circle with a polygon of a greater and greater number of sides. The quotient formed by the area of this polygon divided by the square of the circle radius can be made arbitrarily close to the actual value of p as the number of polygon sides becomes large.

Other results he obtained with the method of exhaustion included

  • The area bounded by the intersection of a line and a parabola is 4/3 that of the triangle having the same base and height;
  • The area of an ellipse is proportional to a rectangle having sides equal to its major and minor axes;
  • The volume of a sphere is 4 times that of a cone having a base and height of the same radius;
  • The volume of a cylinder having a height equal to its diameter is 3/2 that of a sphere having the same diameter;
  • The area bounded by one spiral rotation and a line is 1/3 that of the circle having a radius equal to the line segment length;
  • Use of the method of exhaustion also led to the successful evaluation of a geometric series (for the first time).


See also

  • The Quadrature of the Parabola
    The Quadrature of the Parabola

    The Quadrature of the Parabola is a treatise on geometry, written by Archimedes in the 3rd century B.C. Written as a letter to his friend Dositheus, the work presents 24 propositions regarding parabolas, culminating in a proof that the area of a parabolic segment is 4/3 that of a certain inscribe triangle....