Method of exhaustion - AbsoluteAstronomy.com

The method of exhaustion is a method of finding the area

Area

Area is a quantity that expresses the extent of a two-dimensional surface or shape in the plane. Area can be understood as the amount of material with a given thickness that would be necessary to fashion a model of the shape, or the amount of paint necessary to cover the surface with a single coat...

of a shape

Shape

The shape of an object located in some space is a geometrical description of the part of that space occupied by the object, as determined by its external boundary – abstracting from location and orientation in space, size, and other properties such as colour, content, and material...

by inscribing inside it a sequence of polygon

Polygon

In geometry a polygon is a flat shape consisting of straight lines that are joined to form a closed chain orcircuit.A polygon is traditionally a plane figure that is bounded by a closed path, composed of a finite sequence of straight line segments...

s whose area

Area

s converge to the area of the containing shape

Shape

. If the sequence is correctly constructed, the difference in area between the n-th 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 , 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 Greek 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 in 1647 by Grégoire de Saint-Vincent

Grégoire de Saint-Vincent

Grégoire de Saint-Vincent , a Jesuit, was a mathematician who discovered that the area under a rectangular hyperbola is the same over [a,b] as over [c,d] when a/b = c/d...

in Opus geometricum quadraturae circuli et sectionum .

The method of exhaustion typically required a form of proof by contradiction, known as reductio ad absurdum
Reductio ad absurdum
In logic, proof by contradiction is a form of proof that establishes the truth or validity of a proposition by showing that the proposition's being false would imply a contradiction...

. 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 focused on limits, functions, derivatives, integrals, and infinite series. This subject constitutes a major part of modern mathematics education. It has two major branches, differential calculus and integral calculus, which are related by the fundamental theorem...

. The development of analytical geometry and rigorous integral calculus in the 17th-19th centuries (in particular a rigorous definition of limit

Limit (mathematics)

In mathematics, the concept of a "limit" is used to describe the value that a function or sequence "approaches" as the input or index approaches some value. The concept of limit allows mathematicians to define a new point from a Cauchy sequence of previously defined points within a complete metric...

) subsumed the method of exhaustion so that it is no longer explicitly used to solve problems. An important early intermediate step was Cavalieri's principle

Cavalieri's principle

In geometry, Cavalieri's principle, sometimes called the method of indivisibles, named after Bonaventura Cavalieri, is as follows:* 2-dimensional case: Suppose two regions in a plane are included between two parallel lines in that plane...

, also termed the "method of indivisibles", which was a bridge between the method of exhaustion and full-fledged integral calculus.

Use by Euclid

Euclid

Euclid , fl. 300 BC, also known as Euclid of Alexandria, was a Greek mathematician, often referred to as the "Father of Geometry". He was active in Alexandria during the reign of Ptolemy I...

used the method of exhaustion to prove the following six propositions in the book 12 of Elements

Euclid's Elements

Euclid's Elements is a mathematical and geometric treatise consisting of 13 books written by the Greek mathematician Euclid in Alexandria c. 300 BC. It is a collection of definitions, postulates , propositions , and mathematical proofs of the propositions...

.
Proposition 2

The area of a circle is proportional to the square of its diameter.

Proposition 5

The volume of a tetrahedron of the same height is proportional to the triangular area of the base each other.

Proposition 10

The volume of a cone is a third of the volume of the corresponding cylinder which has the same base and height.

Proposition 11

The volume of a cone (or cylinder) of the same height is proportional to the area of the base.

Proposition 12

The volume of a cone (or cylinder) that is the similar to another is proportional to the cube of the ratio of the diameters of the bases.

Proposition 18

The volume of a sphere is proportional to the cube of its diameter.

Use by Archimedes

Archimedes

Archimedes of Syracuse was a Greek mathematician, 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. Among his advances in physics are the foundations of hydrostatics, statics and an...

used the method of exhaustion as a way to compute the area inside a circle by filling the circle

Circle

A circle is a simple shape of Euclidean geometry consisting of those points in a plane that are a given distance from a given point, the centre. The distance between any of the points and the centre is called the radius....

with a polygon

Polygon

of a greater area 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 π as the number of polygon sides becomes large, proving that the area inside the circle of radius r is πr², π being defined as the ratio of the circumference to the diameter.
Incidentally, he also provided the celebrated bounds 3+10/71 < π < 3 + 1/7 comparing the perimeters of the circle with the perimeters of the inscribed and circumscribed 96-sided regular polygons.

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).

Use by Euclid

Use by Archimedes

See also