Difference between revisions of "Method of exhaustion"
Karl Jones (Talk | contribs) (Created page with "The '''method of exhaustion''' (''methodus exhaustionibus'', or ''méthode des anciens'') is a method of finding the area of a shape by inscribing inside it a sequence...") |
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| − | The '''method of exhaustion''' (''methodus exhaustionibus'', or ''méthode des anciens'') is a method of finding the [[area]] of a [[shape]] by inscribing inside it a sequence of [[Polygon|polygons] whose areas converge to the area of the containing shape. | + | The '''method of exhaustion''' (''methodus exhaustionibus'', or ''méthode des anciens'') is a method of finding the [[area]] of a [[shape]] by inscribing inside it a sequence of [[Polygon|polygons]] whose areas converge to the area of the containing shape. |
== Description == | == Description == | ||
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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. | 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. | + | 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). | 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). | ||
Latest revision as of 08:53, 28 April 2016
The method of exhaustion (methodus exhaustionibus, or 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.
Description
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, 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.
Proof by exhaustion
This article is about the method of finding the area of a shape using limits.
For the method of proof, see Proof by exhaustion.
