The method of exhaustionCreation The method of exhaustion was a techniqueused from ancient Greek mathematicians to solve results that are now solvedwith the use of limits. It amounts to an early form of integral calculus and itwas created with the purpose of finding the area of a shape by engraving insideit a series of polygons whose areas unite to the area of the containing shape.The difference in area between the nth etched polygon and that of the coveringshape will become smaller as n becomes large, if we execute the sequence withthe correct way. While the space between the incised polygon and the involvedshape becomes extremely small, then the possible values for the area of the coveringshape are methodically “exhausted” by the lower bound polygonal areasconsecutively recognized by the sequence members. The idea of the method ofexhaustion was firstly originated with Antiphon of Athens in the 5thcentury BC. The method of exhaustion is considered as a forerunner to themethods of modern calculus. Between the period of 17th and 19th century, thedevelopment of analytical geometry and rigorous integral calculus (morespecifically in the sector of the limit definition) classified the method ofexhaustion so that it is no longer used today in order to solve geometricalproblems.
Bryson Anancient Greek mathematician and sophist named Bryson of Heraclea that was bornaround 450 BCE, was the first to engrave a polygon inside a circle, discoverthe polygon’s area, twofold the number of sides of the polygon, and repeat theprocess, leading to a lower bound approximation for the area of circle. Lateron, Bryson used the same process in pursuance of polygons circumscribing acircle, resulting in a higher certain approximation for the area of a circle.Bryson after all these calculations was able to almost accurate ? and furtherplace lower and upper bounds on ?’s real value.
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Unfortunately, due to the difficultyof the method, Bryson was only able to compute ? to a few digits. We willprobably never know faithfully who was first to find out that the ratio betweenthe area of a circle and the area of a square having side length equal to thatof the circle’s radius.Archimedes Archimedes,one of the greatest mathematicians of all time was always working to produceformulas because we wanted to calculate the areas of regular shapes. Forinstance, he wanted to estimate the area of a circle.
To achieve that, he designeda polygon outside the circle and a smaller one inside it. Each time he enclosedI bigger polygon in the circle from both side approximating the area of thecircle more closely. This is the method of exhaustion and Archimedes was one ofits first exponents of this method. With this method he managed to discover thearea of a parabolic segment, the volume of a paraboloid, the tangent to aspiral and also a proof that the volume of a sphere is 2/3 the volume of acircumscribing cylinder. As for the area of a circle, the way Archimedes statedhis proposition was the area equals to the area of a triangle whose height andbase equals to its radius and to its circumference respectively: (1/2)(r*2?r)=2?r^2. But there is something delicate here. We have never seen a referencesimilar with this before in Greek mathematics talking about the length of acurve opposed to the length of a polygon.
In the present, the length of a curveis defined to be a limit. In fact there are curves with infinite length butArchimedes is restricting them to a countable value. This was a wise decisiondone by him because limits were discovered many years later in about 1820.Until then, Archimedes method seemed to be the best choice for those years.Convexity Archimedesdoesn’t need to know much information regarding the length of the curves, sincea circle is a relatively a simple one. His axiom is concerned only with arestricted class of curved paths which are called convex paths. These kinds ofpaths can be described by examining whether something is convex or not. Here’sone way to distinguish the convex paths from the others.
Convex paths bulge outwhile the others have dimples.