method of exhaustion
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The method of exhaustion () is a method of finding the
area Area is the measure of a region's size on a surface. The area of a plane region or ''plane area'' refers to the area of a shape or planar lamina, while '' surface area'' refers to the area of an open surface or the boundary of a three-di ...
of a
shape A shape is a graphics, graphical representation of an object's form or its external boundary, outline, or external Surface (mathematics), surface. It is distinct from other object properties, such as color, Surface texture, texture, or material ...
by inscribing inside it a
sequence In mathematics, a sequence is an enumerated collection of objects in which repetitions are allowed and order matters. Like a set, it contains members (also called ''elements'', or ''terms''). The number of elements (possibly infinite) is cal ...
of
polygon In geometry, a polygon () is a plane figure made up of line segments connected to form a closed polygonal chain. The segments of a closed polygonal chain are called its '' edges'' or ''sides''. The points where two edges meet are the polygon ...
s (one at a time) whose
area Area is the measure of a region's size on a surface. The area of a plane region or ''plane area'' refers to the area of a shape or planar lamina, while '' surface area'' refers to the area of an open surface or the boundary of a three-di ...
s converge to the area of the containing
shape A shape is a graphics, graphical representation of an object's form or its external boundary, outline, or external Surface (mathematics), surface. It is distinct from other object properties, such as color, Surface texture, texture, or material ...
. 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 method of exhaustion typically required a form of
proof by contradiction In logic, proof by contradiction is a form of proof that establishes the truth or the validity of a proposition by showing that assuming the proposition to be false leads to a contradiction. Although it is quite freely used in mathematical pr ...
, known as ''
reductio ad absurdum In logic, (Latin for "reduction to absurdity"), also known as (Latin for "argument to absurdity") or ''apagogical argument'', is the form of argument that attempts to establish a claim by showing that the opposite scenario would lead to absur ...
''. 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, proving that assertion false, assuming it is less than the second area, then proving that assertion false, too.


History

The idea originated in the late 5th century BC with Antiphon, although it is not entirely clear how well he understood it. The theory was made rigorous a few decades later by
Eudoxus of Cnidus Eudoxus of Cnidus (; , ''Eúdoxos ho Knídios''; ) was an Ancient Greece, ancient Greek Ancient Greek astronomy, astronomer, Greek mathematics, mathematician, doctor, and lawmaker. He was a student of Archytas and Plato. All of his original work ...
, who used it to calculate areas and volumes. It was later reinvented in
China China, officially the People's Republic of China (PRC), is a country in East Asia. With population of China, a population exceeding 1.4 billion, it is the list of countries by population (United Nations), second-most populous country after ...
by
Liu Hui Liu Hui () was a Chinese mathematician who published a commentary in 263 CE on ''Jiu Zhang Suan Shu ( The Nine Chapters on the Mathematical Art).'' He was a descendant of the Marquis of Zixiang of the Eastern Han dynasty and lived in the state ...
in the 3rd century AD in order to find the area of a circle. The first use of the term was in 1647 by Gregory of Saint Vincent in ''Opus geometricum quadraturae circuli et sectionum''. The method of exhaustion is seen as a precursor to the methods 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 development of analytical geometry and rigorous integral calculus in the 17th-19th centuries subsumed the method of exhaustion so that it is no longer explicitly used to solve problems. An important alternative approach was Cavalieri's principle, also termed the '' method of indivisibles'' which eventually evolved into the
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 " ...
calculus of Roberval, Torricelli, Wallis, Leibniz, and others.


Euclid

Euclid Euclid (; ; BC) was an ancient Greek mathematician active as a geometer and logician. Considered the "father of geometry", he is chiefly known for the '' Elements'' treatise, which established the foundations of geometry that largely domina ...
used the method of exhaustion to prove the following six propositions in the 12th book of his '' Elements''. Proposition 2: The area of circles is proportional to the square of their diameters. Proposition 5: The volumes of two tetrahedra of the same height are proportional to the areas of their triangular bases. 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 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.


Archimedes

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 ...
used the method of exhaustion as a way to compute the area inside a circle by filling the
circle A circle is a shape consisting of all point (geometry), points in a plane (mathematics), plane that are at a given distance from a given point, the Centre (geometry), centre. The distance between any point of the circle and the centre is cal ...
with a sequence of
polygon In geometry, a polygon () is a plane figure made up of line segments connected to form a closed polygonal chain. The segments of a closed polygonal chain are called its '' edges'' or ''sides''. The points where two edges meet are the polygon ...
s with an increasing number of sides and a corresponding increase in area. The quotients formed by the area of these polygons 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 (C/d). He also provided the bounds 3 + 10''/''71 < ''π'' < 3 + 10''/''70, (giving a range of 1''/''497) by comparing the perimeters of the circle with the perimeters of the
inscribed An inscribed triangle of a circle In geometry, an inscribed planar shape or solid is one that is enclosed by and "fits snugly" inside another geometric shape or solid. To say that "figure F is inscribed in figure G" means precisely the same th ...
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
quadrature of the parabola ''Quadrature of the Parabola'' () is a treatise on geometry, written by Archimedes in the 3rd century BC and addressed to his Alexandrian acquaintance Dositheus. It contains 24 propositions regarding parabolas, culminating in two proofs showing t ...
); * 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 of the same radius and height equal to this 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 an infinite geometric series (for the first time);


See also

* ''
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, ...
'' * '' The Quadrature of the Parabola'' * Trapezoidal rule *
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 ...


References

{{Ancient Greek mathematics Volume Euclidean geometry Integral calculus History of mathematics 5th century BC in Greece Greek mathematics