The method of exhaustion (Latin: methodus exhaustionibus; French: méthode des anciens) is a method of finding the area of a shape by inscribing inside it a sequence of polygons whose areas converge 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 method of exhaustion typically required a form of proof by contradiction, 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, and then proving that assertion false, and then assuming that it is less than the second area, and proving that assertion false, too.

History

Gregory of Saint Vincent

The idea originated in the late 5th century BC with Antiphon, although it is not entirely clear how well he understood it.[1] The theory was made rigorous a few decades later by Eudoxus of Cnidus, who used it to calculate areas and volumes. It was later reinvented in China by Liu Hui in the 3rd century AD in order to find the area of a circle.[2] 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. 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 calculus of Roberval, Torricelli, Wallis, Leibniz, and others.

Euclid

Euclid 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.[3]

Proposition 5: The volumes of two tetrahedra of the same height are proportional to the areas of their triangular bases.[4]

Proposition 10: The volume of a cone is a third of the volume of the corresponding cylinder which has the same base and height.[5]

Proposition 11: The volume of a cone (or cylinder) of the same height is proportional to the area of the base.[6]

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.[7]

Proposition 18: The volume of a sphere is proportional to the cube of its diameter.[8]

Archimedes

Main article: Pi

Archimedes used the method of exhaustion to compute the area inside a circle

Archimedes used the method of exhaustion as a way to compute the area inside a circle by filling the circle with a 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 (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[9]

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 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 Quadrature of the Parabola

Trapezoidal rule

Pythagorean Theorem

References

"Antiphon (480 BC-411 BC)". www-history.mcs.st-andrews.ac.uk.

Dun, Liu. 1966. "A comparison of Archimedes' and Liu Hui's studies of circles." Pp. 279–87 in Chinese Studies in the History and Philosophy of Science and Technology 179, edited by D. Fan, and R. S. Cohen. Kluwer Academic Publishers. ISBN 0-7923-3463-9. p. 279.

"Euclid's Elements, Book XII, Proposition 2". aleph0.clarku.edu.

"Euclid's Elements, Book XII, Proposition 5". aleph0.clarku.edu.

"Euclid's Elements, Book XII, Proposition 10". aleph0.clarku.edu.

"Euclid's Elements, Book XII, Proposition 11". aleph0.clarku.edu.

"Euclid's Elements, Book XII, Proposition 12". aleph0.clarku.edu.

"Euclid's Elements, Book XII, Proposition 18". aleph0.clarku.edu.

Smith, David E (1958). History of Mathematics. New York: Dover Publications. ISBN 0-486-20430-8.

vte

Ancient Greek and Hellenistic mathematics (Euclidean geometry)

Mathematicians

(timeline)

Anaxagoras Anthemius Archytas Aristaeus the Elder Aristarchus Apollonius Archimedes Autolycus Bion Bryson Callippus Carpus Chrysippus Cleomedes Conon Ctesibius Democritus Dicaearchus Diocles Diophantus Dinostratus Dionysodorus Domninus Eratosthenes Eudemus Euclid Eudoxus Eutocius Geminus Heliodorus Heron Hipparchus Hippasus Hippias Hippocrates Hypatia Hypsicles Isidore of Miletus Leon Marinus Menaechmus Menelaus Metrodorus Nicomachus Nicomedes Nicoteles Oenopides Pappus Perseus Philolaus Philon Philonides Porphyry Posidonius Proclus Ptolemy Pythagoras Serenus Simplicius Sosigenes Sporus Thales Theaetetus Theano Theodorus Theodosius Theon of Alexandria Theon of Smyrna Thymaridas Xenocrates Zeno of Elea Zeno of Sidon Zenodorus

Treatises

Almagest Archimedes Palimpsest Arithmetica Conics (Apollonius) Catoptrics Data (Euclid) Elements (Euclid) Measurement of a Circle On Conoids and Spheroids On the Sizes and Distances (Aristarchus) On Sizes and Distances (Hipparchus) On the Moving Sphere (Autolycus) Euclid's Optics On Spirals On the Sphere and Cylinder Ostomachion Planisphaerium Sphaerics The Quadrature of the Parabola The Sand Reckoner

Problems

Angle trisection Doubling the cube Squaring the circle Problem of Apollonius

Concepts/Definitions

Circles of Apollonius

Apollonian circles Apollonian gasket Circumscribed circle Commensurability Diophantine equation Doctrine of proportionality Golden ratio Greek numerals Incircle and excircles of a triangle Method of exhaustion Parallel postulate Platonic solid Lune of Hippocrates Quadratrix of Hippias Regular polygon Straightedge and compass construction Triangle center

Results

In Elements

Angle bisector theorem Exterior angle theorem Euclidean algorithm Euclid's theorem Geometric mean theorem Greek geometric algebra Hinge theorem Inscribed angle theorem Intercept theorem Pons asinorum Pythagorean theorem Thales's theorem Theorem of the gnomon

Apollonius

Apollonius's theorem

Other

Aristarchus's inequality Crossbar theorem Heron's formula Irrational numbers Menelaus's theorem Pappus's area theorem Ptolemy's inequality Ptolemy's table of chords Ptolemy's theorem Spiral of Theodorus

Centers

Cyrene Library of Alexandria Platonic Academy

Other

Ancient Greek astronomy Greek numerals Latin translations of the 12th century Neusis construction

Undergraduate Texts in Mathematics

Graduate Studies in Mathematics

Hellenica World - Scientific Library

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