en>Yobot: Tagging using AWB (10703)

2015-01-07T10:20:20Z

Tagging using AWB (10703)

en>Molly-in-md: Removed the "orphan" tag from June 2008, because at least one article points to this one.

2012-10-08T23:46:29Z

Removed the "orphan" tag from June 2008, because at least one article points to this one.

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			{{italic title}}
			'''''Measurement of a Circle''''' ([[Greek language\|Greek]]: {{lang\|grc\|Κύκλου μέτρησις}}, ''Kuklou metrēsis'') is a [[treatise]] that consists of three propositions by [[Archimedes]]. The treatise is only a fraction of what was a longer work.<ref name=History>{{Citation\| last = Heath\| first = Thomas Little\| year = 1921\| title = A History of Greek Mathematics\| isbn = 0-543-96877-4\| url = http://books.google.com/?id=zGIYbEtzD-QC&printsec=frontcover\| accessdate = 2008-06-30\| publisher = Adamant Media Corporation\| location = Boston}}</ref><ref name=Brit>{{cite web\| title=Archimedes\| publisher=Encyclopædia Britannica\| year= 2008\| accessdate=2008-06-30\| url=http://www.britannica.com/EBchecked/topic/32808/Archimedes}}</ref>

			==Propositions==
			===Proposition one===
			[[File:Area of circle and triangle.svg\|thumb\|The circle and the triangle are equal in area.]]
			Proposition one states:
			<blockquote>The area of any circle is equal to a right-angled triangle in which one of the sides about the right angle is equal to the radius, and the other to the circumference, of the circle.</blockquote>
			Any [[circle]] with a [[circumference]] ''c'' and a [[radius]] ''r'' is equal in [[area]] with a [[right triangle]] with the two [[Cathetus\|legs]] being ''c'' and ''r''. This proposition is proved by the [[method of exhaustion]].<ref name=Works>{{Citation\| last = Heath\| first = Thomas Little\| year = 1897\| title = The Works of Archimedes\| publication-place = Cambridge University\| pages = ''lxxvii ''; 50\| url = http://www.archive.org/details/worksofarchimede029517mbp \| accessdate = 2008-06-30}}</ref>

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			Proposition two states:
			<blockquote>The area of a circle is to the square on its diameter as 11 to 14.</blockquote>
			This proposition could not have been placed by Archimedes, for it relies on the outcome of the third proposition.<ref name=Works/>

			===Proposition three===
			[[File:Archimedes pi.svg\|thumb\|Examples of how Archimedes calculated pi. Archimedes used a 96-sided polygon to find his estimate.]]
			Proposition three states:
			<blockquote>The ratio of the circumference of any circle to its diameter is greater than <math>3\tfrac{10}{71}</math> but less than <math>3\tfrac{1}{7}</math>.</blockquote>
			This approximates what we now call the [[mathematical constant]] [[π]]. He found these bounds on the value of π by [[inscribe\|inscribing]] and [[circumscribe\|circumscribing]] a circle with two [[Similarity (geometry)\|similar]] 96-sided [[regular polygon]]s.<ref name=manual>{{Citation\| last=Heath\| first=Thomas Little\| year=1931\| title=A Manual of Greek Mathematics\| page=146\| isbn=0-486-43231-9\| url=http://books.google.com/?id=_HZNr_mGFzQC&printsec=frontcover\| publisher=Dover Publications\| location=Mineola, N.Y.}}</ref>

			====Approximation to square roots====
			This proposition also contains accurate approximations to the [[square root of 3]] (one larger and one smaller) and other larger non-perfect [[square root]]s; however, Archimedes gives no explanation as to how he found these numbers.<ref name=Brit/>
			He gives the upper and lower bounds to √3 as&emsp;
			<math>\tfrac{1351}{780} > \sqrt{3} > \tfrac{265}{153}\,.</math><ref name=Works/> However, these bounds are familiar from the
			study of [[Pell's equation]] and the convergents of an associated [[continued fraction]], leading to much speculation as to
			how much of this number theory might have been accessible to Archimedes. Discussion of this approach goes back at least to
			[[Thomas Fantet de Lagny]], FRS (compare [[Chronology of computation of π]]) in 1723, but was treated more explicitly by
			[[Hieronymus Georg Zeuthen]]. In the early 1880s, [[Friedrich Otto Hultsch]] (1833–1906) and [[Karl Heinrich Hunrath]]
			(b. 1847) noted how the bounds could be found quickly by means of simple binomial bounds on square roots close to a perfect
			square modelled on Elements II.4, 7; this method is favoured by [[Thomas Little Heath]]. Although only one route to the bounds is mentioned,
			in fact there are two others, making the bounds almost inescapable however the method is worked. But the bounds can also be produced by an iterative geometrical construction suggested by Archimedes' [[Stomachion]] in the setting of the regular
			dodecagon. In this case, the task is to give rational approximations to the tangent of π/12.

			==References==
			{{reflist}}

			[[Category:Ancient Greek mathematical works]]
			[[Category:Works by Archimedes]]
			[[Category:Euclidean geometry]]

en>Avicennasis: Fixing deprecated parameters

2011-10-07T21:50:55Z

Fixing deprecated parameters

New page

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Generalized quaternion interpolation - Revision history

en>Yobot: Tagging using AWB (10703)

en>Molly-in-md: Removed the "orphan" tag from June 2008, because at least one article points to this one.

en>Avicennasis: Fixing deprecated parameters