The method of exhaustion (; ) is a method of finding the

area Area is the quantity that expresses the extent of a Region (mathematics), region on the plane (geometry), plane or on a curved surface (mathematics), surface. The area of a plane region or ''plane area'' refers to the area of a shape or planar ...

of a

shape A shape or figure is a graphical representation of an object or its external boundary, outline, or external surface, as opposed to other properties such as color, texture, or material type. A plane shape or plane figure is constrained to lie ...

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 call ...

polygon In geometry, a polygon () is a plane figure that is described by a finite number of straight line segments connected to form a closed '' polygonal chain'' (or ''polygonal circuit''). The bounded plane region, the bounding circuit, or the two t ...

s whose

s converge to the area of the containing

. 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 and mathematics, 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. Proof by contradiction is also known ...

, known as '' reductio ad absurdum''. 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 An antiphon (Greek ἀντίφωνον, ἀντί "opposite" and φωνή "voice") is a short chant in Christian ritual, sung as a refrain. The texts of antiphons are the Psalms. Their form was favored by St Ambrose and they feature prominently ...

, 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 (; grc, Εὔδοξος ὁ Κνίδιος, ''Eúdoxos ho Knídios''; ) was an ancient Greek astronomer, mathematician, scholar, and student of Archytas and Plato. All of his original works are lost, though some fragments a ...

, 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. It is the world's most populous country, with a population exceeding 1.4 billion, slightly ahead of India. China spans the equivalent of five time zones an ...

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, originally called infinitesimal calculus or "the calculus of infinitesimals", is the mathematical study of continuous change, in the same way that geometry is the study of shape, and algebra is the study of generalizations of arit ...

. The development of analytical geometry and rigorous

integral calculus In mathematics, an integral assigns numbers to functions in a way that describes displacement, area, volume, and other concepts that arise by combining infinitesimal data. The process of finding integrals is called integration. Along with di ...

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 In geometry, Cavalieri's principle, a modern implementation of 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 pl ...

, also termed the ''

method of indivisibles In geometry, Cavalieri's principle, a modern implementation of 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 p ...

'' which eventually evolved into the

infinitesimal In mathematics, an infinitesimal number is a quantity that is closer to zero than any standard real number, but that is not zero. The word ''infinitesimal'' comes from a 17th-century Modern Latin coinage ''infinitesimus'', which originally refer ...

calculus of Roberval, Torricelli, Wallis,

Leibniz Gottfried Wilhelm (von) Leibniz . ( – 14 November 1716) was a German polymath active as a mathematician, philosopher, scientist and diplomat. He is one of the most prominent figures in both the history of philosophy and the history of mathem ...

, and others.

Euclid

Euclid Euclid (; grc-gre, Εὐκλείδης; 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 ...

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 a Greek mathematician, physicist, engineer, astronomer, and inventor from the ancient city of Syracuse in Sicily. Although few details of his life are known, he is regarded as one of the leading scientists i ...

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 points in a plane that are at a given distance from a given point, the centre. Equivalently, it is the curve traced out by a point that moves in a plane so that its distance from a given point is con ...

with a

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 (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 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'' ( el, Τετραγωνισμὸς παραβολῆς) is a treatise on geometry, written by Archimedes in the 3rd century BC and addressed to his Alexandrian acquaintance Dositheus. It contains 24 propositions rega ...

); * 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 In mathematics, a spiral is a curve which emanates from a point, moving farther away as it revolves around the point. Helices Two major definitions of "spiral" in the American Heritage Dictionary are:infinite geometric series (for the first time);

References

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

History

Euclid

Archimedes

See also

References