View source for Measurement of a Circle

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'''''Measurement of a Circle''''' or '''''Dimension of the Circle''''' ([[Greek language|Greek]]: {{lang|grc|Κύκλου μέτρησις}}, ''Kuklou metrēsis'')<ref>{{Cite journal|last=Knorr|first=Wilbur R.|date=1986-12-01|title=Archimedes' dimension of the circle: A view of the genesis of the extant text|journal=Archive for History of Exact Sciences|language=en|volume=35|issue=4|pages=281–324|doi=10.1007/BF00357303|issn=0003-9519}}</ref> is a [[treatise]] that consists of three propositions by [[Archimedes]], ca. 250 BCE.<ref>{{Cite journal|last=Lit|first=L.W.C. (Eric)  van|date=|title=Naṣīr al-Dīn al-Ṭūsī's Version of The Measurement of the Circle of Archimedes from his Revision of the Middle Books|url=https://www.academia.edu/1348933|journal=Tarikh-e Elm|language=en|volume=|pages=|quote=The ''measurement of the circle'' was written by Archimedes (ca. 250 B.C.E.)|via=}}</ref><ref>{{Cite book|url=https://books.google.com/books?id=_poUuMFvA3oC&lpg=PA153&pg=PA153|title=The Ancient Tradition of Geometric Problems|last=Knorr|first=Wilbur Richard|date=1986|publisher=[[Courier Corporation]]|isbn=9780486675329|location=|pages=|language=en|quote=Most accounts of Archimedes' works assign this writing to a time relatively late in his career.  But this view is the consequence of a plain misunderstanding.}}</ref> 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 = 978-0-543-96877-7| url = https://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:
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.
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>{{Citation| last = Heath| first = Thomas Little| year = 1897| title = The Works of Archimedes| location = Cambridge University| pages = ''lxxvii ''; 50| url = https://archive.org/details/worksofarchimede029517mbp | accessdate = 2008-06-30| publisher = Cambridge University Press.}}</ref>

===Proposition two===
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>{{Citation| last = Heath| first = Thomas Little| year = 1897| title = The Works of Archimedes| location = Cambridge University| pages = ''lxxvii ''; 50| url = https://archive.org/details/worksofarchimede029517mbp | accessdate = 2008-06-30| publisher = Cambridge University Press.}}</ref>

===Proposition three===

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]] [[&pi;]]. 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=978-0-486-43231-1| url=https://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 {{radic|3}} as {{sfrac|1351|780}} > {{radic|3}} > {{sfrac|265|153}}.<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 &pi;]]) 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==
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{{Archimedes}}
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[[Category:Ancient Greek mathematical works]]
[[Category:Works by Archimedes]]
[[Category:Euclidean geometry]]