
	Where results make sense

Topic: Method of exhaustion

	Method of exhaustion - Wikipedia, the free encyclopedia
		The method of exhaustion 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.
		The method of exhaustion is seen as a precursor to the methods of calculus.
		Archimedes used the method of exhaustion as a way to calculate π by filling the circle with a polygon of a greater and greater number of sides.
	en.wikipedia.org /wiki/Method_of_exhaustion (409 words)

	Proof by exhaustion - Wikipedia, the free encyclopedia
		Proof by exhaustion, also known as the brute force method, is a method of mathematical proof in which the statement to be proved is split into a finite number of cases, and each case is proved separately.
		In contrast, the method of exhaustion of Eudoxus of Cnidus was a geometrical and essentially rigorous way of calculating mathematical limits.
		The first proof of the four colour theorem was a proof by exhaustion with 1,936 cases.
	en.wikipedia.org /wiki/Proof_by_exhaustion (462 words)

	Calculus history
		Archimedes used the method of exhaustion to find an approximation to the area of a circle.
		His method consisted of thinking of areas as sums of lines, another crude form of integration, but Kepler had little time for Greek rigour and was rather lucky to obtain the correct answer after making two cancelling errors in this work.
		Barrow gave a method of tangents to a curve where the tangent is given as the limit of a chord as the points approach each other known as Barrow's differential triangle.
	www-gap.dcs.st-and.ac.uk /~history/HistTopics/The_rise_of_calculus.html (1696 words)

	Ohio Department of Taxation (Site not responding. Last check: 2007-10-09)
		In summary the method proposed provides that all fully depreciated write-offs be added back at cost by year of acquisition for all years including, a ten year period preceding the year in which the twenty per cent (20%) utility value is reached on the regular 302 method.
		This revised method for determining the true value of items of furniture, fixtures and equipment, which have been written off the book records, but which are still on hand and in use or held for use, does not, under any circumstances, penalize a person recording disposals of assets.
		The exhaustion concept assumes that items of property which have aged to the "floor" of the 10% annual allowance are disposed of, thereafter, at a uniform rate of 10% per year.
	tax.ohio.gov /divisions/personal_property/county_auditor_bulletins/bulletin_No_124.stm (585 words)

	Method of exhaustion - (Site not responding. Last check: 2007-10-09)
		The method of exhaustion is a way of finding the area or volume of a shape that is not easily defined in terms of traditional shapes.
		The process, the creation of which is credited to Eudoxus, is performed by estimating the area or volume using a number of known shapes which, when put together, provide a reasonable estimation of the volume or area.
		The method of exhaustion is often associated with Archimedes.
	psychcentral.com /psypsych/Method_of_exhaustion (217 words)

	PlanetMath: method of exhaustion
		The method of exhaustion is calculating an area by approximating it by the areas of a sequence of polygons.
		derivation of a definite integral formula using the method of exhaustion
		This is version 2 of method of exhaustion, born on 2003-08-29, modified 2005-03-04.
	planetmath.org /encyclopedia/MethodOfExhaustion.html (80 words)

	History of Calculus (Site not responding. Last check: 2007-10-09)
		Of greatest importance is Eudoxus' Method of Exhaustion (4th century BC) which was applied with great ingenuity by Archimedes (3rd century BC) to find the area of a parabolic segment and the surface area and volume of a sphere.
		Also important is Archimedes' "Method", discovered only recently (1906), in which he used the concept of elements of a figure (lines made up of points, areas made up of lines, solids made up of planes) to discover important results.
		Fermat's method of finding maximum and minimum values of a polynomial function is the equivalent, and very nearly identical to, our modern method of finding the derivative of a function and setting it equal to zero.
	www.southwestern.edu /~sawyerc/cal1/history_of_calculus.htm (1767 words)

	Sphæra issue no. 6: article 12
		Proposition 18 of Book XII of the Elements states that 'spheres are to one another in the triplicate ratio of their respective diameters' and is one of several applications of the so-called 'method of exhaustion' found there.
		Largely on evidence from Archimedes, the method of exhaustion is attributed to Eudoxus of Cnidus, who was a member of Plato's Academy from around 370 B.C. until his death some twenty or so years later.
		The method is a reductio ad absurdum and was put to use by the Greeks as the method of proof of a wide range of different results.
	www.mhs.ox.ac.uk /sphaera/issue6/articl12.htm (1030 words)

	[No title]
		This method of exhaustion has many applications, including proving the formula for the area of a circle, and that the volume of a pyramid is 1/3 that of a prism, and calculating the area of a segment of a parabola.
		His method utilized the idea that, if 2 three dimensional figures are situated between two planes, and the areas of the cross sections of the two figures are equal on every plane that lies between and parallel to the two planes, then the volumes of the two solids are equal.
		Fermat’s method of beginning with an equation and working towards finding the points that describe it allowed for the discovery of a number of new curves. Fermat also developed a way of calculating a relative maximum or minimum of a function, which is analogous to differentiation.
	www.duke.edu /~ltr/Calculus_Paper.doc (4431 words)

	Free Essay The Development Of Calculus
		The method of exhaustion is named so because one must think of the areas measured as if they are expanding so that they account for more and more of the required area.
		Using these methods, he showed that the integral of xn from 0 to a was an+1/(n + 1) by showing the result for a number of values of n and inferring the general result.
		Barrow gave a method of calulating tangents to a curve where the tangent is given as the limit of a chord as the points approach each other.
	www.echeat.com /essay.php?t=27890 (1916 words)

	HM/MA 004, Assignment 10, due Friday, 15 November 2002
		The method of exhaustion was that which the ancients used in their difficult researches, and particularly in the theory of curved lines and surfaces, and in the evaluation of areas and volumes which they enclosed.
		The method of first and last ratios, or limits, also takes its origin from the method of exhaustion; and strictly speaking it is only a development and simplification of that.
		It is always the method of exhaustion of the ancients, more or less simplified, more or less conveniently adapted to the needs of calculation, and finally reduced to a regular algorithm.
	www.brown.edu /Departments/History_Mathematics/HM0004/assignments/assignment_11-15.html (1069 words)

	Euclidís Elements, by far his most famous and important work, is a comprehensive collection of the mathematical ...
		The method of exhaustion is a modern term that came into use during the seventeen century and refers to the approximation of a figure using a sequence of inscribed figures within it.
		This method provided the ability to determine areas and volumes bounded by curves without the use of limits and is considered to be the predecessor of integral calculus (Aulie 1).
		Eudoxus, through his work in extending the theory of proportion and inventing the method of exhaustion, played a significant role in Euclid’s development of the Elements, considered to be one of the greatest works in history.
	jwilson.coe.uga.edu /EMT668/EMAT6680.F99/Wise/essay7/essay7.htm (1902 words)

	mielec2.html
		By 1629, Cavalieri developed a method of indivisibles which became a factor in the development of the integral calculus.
		Cavalieri developed his theory of indivisibles from Archimedes' method of exhaustion and incorporated Kepler's theory of infinitesionally small geometric quantities.
		Cavalieri's Principle is a method for computing the volume of solid regions, much like the present-day integration (except Cavalieri's Principle is not as systematic as integration).
	www.ms.uky.edu /~carl/ma330/project2/mielec21.html (1022 words)

	Barnaba Bienkowski History of Computing Project
		Despite this Eudoxus was able to (in the 5'th century B.C.) compute by indirect reasoning, and prove by the method of exhaustion, the area of a circle to be
		The view at the time was that using methods not simplistic enough was a sort of ignorance that constituted a mathematical error.
		Furthemore he practically invented differentiation by noting that outcomes of an equation based on the inputs of an equation are a almost an equation themselves the closer the difference between the two is equal to zero.
	www.cs.rit.edu /~bjb5557/Leibniz.html (1959 words)

	Hip Mathematicians (Site not responding. Last check: 2007-10-09)
		First to prove method of exhaustion: shows area of circle proportional to square of radius, volume of pyramid proportional to area of base and height.
		Archimedes' area under parabolic segment (uses an approximation by triangles, which in this problem is more elegant but less universal than approximation by rectangles) used method of exhaustion extensively, as did his work on the volume and surface area of the sphere.
		Archimedes' area under parabolic segment (approx by triangles--more elegant but less universal than approx by rectangles) used method of exhaustion extensively, as did his volume of sphere/surface area of sphere.
	www.stanford.edu /~nfiori/mathematicians (749 words)

	Archimedes's method
		It came to light about one hundred years ago, and was brought to the attention of J.L. Heiberg, perhaps the greatest student of the primary sources for classical Greek geometry, who studied it in Istanbul.
		It provides a glimpse into the thinking which led Archimedes to many of his famous results, including the determination of the area of a parabola, the area and volume of a sphere, and the volume of an ellipsoid.
		From The Method of Archimedes (the translation is that of T.L. Heath):
	www.cut-the-knot.org /pythagoras/Archimedes.shtml (747 words)

	[No title] (Site not responding. Last check: 2007-10-09)
		First by the method of dissection, later by exhaustion.
		(Method of discovering area and volume formulas from slicing figures into "lamina" and comparing their areas (lengths) and how they balance according to a.
		We have studied three examples of this Method, two for computing the volume of spheres and one for the quadrature of the parabola.
	www.math.uiuc.edu /~gfrancis/math306/studyguide (436 words)

	Renaissance and 17th Century Mathematics
		To accomplish this, it was necessary to replace mechanical means of computation (abacus) with algorithms for carrying out the calculations, to systematize a place value method for writing numbers with an appropriate notation, and to develop an effective notation for expressing equations.
		All these tasks were accomplished between 1450 and 1700, allowing for the introduction of negative and imaginary numbers, for the development of calculus, and the first use of functions.
		Nicolas of Cusa (1401-1464) extends his ideas of continuum by stating that a circle is the limit of a polygon having an infinite number of sides.
	www-personal.umich.edu /~pberman/renmath.html (971 words)

	Eudoxus of Cnidus
		Although ideas similar to the Method of Exhaustion was known before him, he is the one responsible for formulating this early method of Integration.
		The founder of celestial mechanics, Eudoxus invented a method to calculate the distance of the sun and the moon.
		Eudoxus and an Introduction to the method of exhaustion
	www.mlahanas.de /Greeks/Eudoxus.htm (1495 words)

	Mathematical Masterpieces: Teaching with Original Sources
		An additional feature of the method is that suddenly value judgments need to be made: there is good and bad mathematics, there are elegant proofs and clumsy ones, and of course plenty of mistakes and unsubstantiated assertions which need to be examined critically.
		The Greek method of exhaustion for computing areas and volumes, pioneered by Eudoxus, reached its pinnacle in the work of Archimedes during the third century BC.
		By the early seventeenth century the Greek method of exhaustion was being transformed into Cavalieri's method of indivisibles, the precursor of Leibniz's infinitesimals and of Newton's fluxions.
	math.nmsu.edu /~history/masterpieces/masterpieces.html (2084 words)

	Eudoxus (Site not responding. Last check: 2007-10-09)
		Another remarkable contribution to mathematics made by Eudoxus was his early work on integration using his method of exhaustion.
		Eudoxus was able to make Antiphon's theory into a rigorous one, applying his methods to give rigorous proofs of many theorems, including the volumes of cones and pyramids.
		Archimedes went on to use Eudoxus's method of exhaustion to prove a remarkable collection of theorems.
	www.stetson.edu /~efriedma/periodictable/html/Xe.html (525 words)

	Calculation of Pi
		All of the early attempts to find the value of pi were made through completely practical methods, such as comparing area of circles with rectangles, and as such all the early attempts found only very approximate values of pi.
		He came up with the first theoretical method of approximating pi, all based around regular polygons and circles.
		The premise is simple; draw a circle, then draw a regular polygon (for example an octagon) inside the circle, so that the vertices of the octagon touch the circle.
	www.bath.ac.uk /~ma3slt/Calculation_of_Pi.html (573 words)

	Eudoxus of Cnidus
		The Method of Exhaustion unquestionably helped resolve number of loose ends then extant.
		The volume of every cone is one third of the cylinder on the same base and with the same height.
		Curiously, the proof is by the method of slabs, familiar to all freshmen.
	www.math.tamu.edu /~don.allen/history/eudoxus/eudoxus.html (759 words)

	Intro to Calculus (Site not responding. Last check: 2007-10-09)
		One of the most famous Greek mathematicians, Archimedes, used the "method of exhaustion" to approximate the area of a circle.
		This work to find unknown areas is one of the earliest forms of Integration and forms the basis for one of the branches of Calculus known as Integral Calculus.
		These methods can be used to solve many problems beyond geometric area and volume questions.
	www.coolschool.bc.ca /lor/CALC12/unit1/U01L01.htm (409 words)

	Gregory Saint Vincent, S.J.
		From this word arose the name of "method of exhaustion," as applied to the method of Euclid and Archimedes.
		In his Opus geometricum (1649) he proposed a new ingenious method of approaching the problem of infinitesimals and he gave his propositions a direct rigorous demonstration instead of the reductio ad absurdum argument used previously.
		It is practically the same fundamental principle as today's present method of finding a volume of a solid of integration.
	www.faculty.fairfield.edu /jmac/sj/scientists/vincent.htm (1296 words)

	Archimedes and Method of Exhaustion information (Site not responding. Last check: 2007-10-09)
		Author: matthew r johnson I am interested in finding out about the method of exhaustion used by Archimedes to find the area of a sphere.
		We talked a little about this in my calculus class, but did not have the time to here about this early version of calculus before it became "discovered." I would like to know about the notation used by Archimedes, what he discovered, and where to find more about this interesting early math history.
		This is reasonable since we are trained to read modern form, much of which did not exist in the time of Archimedes.
	www.newton.dep.anl.gov /newton/askasci/1995/math/MATH015.HTM (192 words)

Try your search on: Qwika (all wikis)