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{{Other uses|Square the Circle (disambiguation){{!}}Square the Circle}}
This is a preview for the new '''MathML rendering mode''' (with SVG fallback), which is availble in production for registered users.
{{multiple image
  | width    = 250
  | image1    = Squaring the circle.svg
  | caption1  = Squaring the circle: the areas of this square and this circle are equal. In 1882, it was proven that this figure cannot be constructed in a finite number of steps with an idealized [[compass and straightedge]].
  | image2    = Hipocrat arcs.svg
  | caption2  = Some apparent partial solutions gave false hope for a long time. In this figure, the shaded figure is the [[Lune of Hippocrates]]. Its area is equal to the area of the triangle {{math|ABC}} (found by [[Hippocrates of Chios]]).
  }}
{{Pi box}}
'''Squaring the circle''' is a problem proposed by [[classical antiquity|ancient]] [[geometers]]. It is the challenge of constructing a [[square (geometry)|square]] with the same area as a given [[circle]] by using only a finite number of steps with [[compass and straightedge]]. More abstractly and more precisely, it may be taken to ask whether specified [[axiom]]s of [[Euclidean geometry]] concerning the existence of lines and circles entail the existence of such a square.


In 1882, the task was proven to be impossible, as a consequence of the [[Lindemann–Weierstrass theorem]] which proves that [[pi]] ({{pi}}) is a [[Transcendental number|transcendental]], rather than an algebraic irrational number; that is, it is not the [[root of a function|root]] of any [[polynomial]] with rational coefficients. It had been known for some decades before then that the construction would be impossible if pi were transcendental, but pi was not proven transcendental until 1882. Approximate squaring to any given non-perfect accuracy, in contrast, is possible in a finite number of steps, since there are rational numbers arbitrarily close to {{pi}}.
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The expression "squaring the circle" is sometimes used as a metaphor for trying to do the impossible.<ref>{{cite web|last=Ammer|first=Christine|title=Square the Circle. Dictionary.com. The American Heritage® Dictionary of Idioms|url=http://dictionary.reference.com/browse/square%20the%20circle|publisher=Houghton Mifflin Company|accessdate=16 April 2012}}</ref>
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The term ''[[numerical integration|quadrature]] of the circle'' is sometimes used synonymously or may refer to approximate or numerical methods for finding the area of a circle.
'''MathML'''
:<math forcemathmode="mathml">E=mc^2</math>


==History==
<!--'''PNG''' (currently default in production)
Methods to approximate the area of a given circle with a square were known already to [[Babylonian mathematics|Babylonian mathematicians]]. The Egyptian [[Rhind papyrus]] of 1800BC gives the area of a circle as (64/81)&nbsp;{{math|''d''}}<sup>&nbsp;2</sup>, where {{math|''d''}} is the diameter of the circle, and pi approximated to 256/81, a number that appears in the older [[Moscow Mathematical Papyrus]] and used for volume approximations (i.e. [[hekat (volume unit)|hekat]]). [[Indian mathematics|Indian mathematicians]] also found an approximate method, though less accurate, documented in the ''[[Sulba Sutras]]''.<ref>{{Cite web|author=O'Connor, John J. and Robertson, Edmund F. |year=2000|url=http://www-groups.dcs.st-and.ac.uk/~history/HistTopics/Indian_sulbasutras.html|title=The Indian Sulbasutras|work=MacTutor History of Mathematics archive|publisher=St Andrews University}}</ref> [[Archimedes]] showed that the value of pi lay between 3&nbsp;+&nbsp;1/7 (approximately 3.1429) and 3&nbsp;+&nbsp;10/71 (approximately 3.1408). See [[Numerical approximations of π]] for more on the history.
:<math forcemathmode="png">E=mc^2</math>


The first known Greek to be associated with the problem was [[Anaxagoras]], who worked on it while in prison. [[Hippocrates of Chios]] squared certain [[Lune (mathematics)|lunes]], in the hope that it would lead to a solution — see [[Lune of Hippocrates]]. [[Antiphon the Sophist]] believed that inscribing regular polygons within a circle and doubling the number of sides will eventually fill up the area of the circle, and since a polygon can be squared, it means the circle can be squared. Even then there were skeptics—[[Eudemus]] argued that magnitudes cannot be divided up without limit, so the area of the circle will never be used up.<ref>{{Cite book| last = Heath | first = Thomas | year = 1981 | title = History of Greek Mathematics | publisher = Courier Dover Publications| isbn = 0-486-24074-6}}</ref> The problem was even mentioned in [[Aristophanes]]'s play ''[[The Birds (play)|The Birds]]''.
'''source'''
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It is believed that [[Oenopides]] was the first Greek who required a plane solution (that is, using only a compass and straightedge). [[James Gregory (astronomer and mathematician)|James Gregory]] attempted a proof of its impossibility in ''Vera Circuli et Hyperbolae Quadratura'' (The True Squaring of the Circle and of the Hyperbola) in 1667. Although his proof was incorrect, it was the first paper to attempt to solve the problem using algebraic properties of pi. It was not until 1882 that [[Ferdinand von Lindemann]] rigorously proved its impossibility.
<span style="color: red">Follow this [https://en.wikipedia.org/wiki/Special:Preferences#mw-prefsection-rendering link] to change your Math rendering settings.</span> You can also add a [https://en.wikipedia.org/wiki/Special:Preferences#mw-prefsection-rendering-skin Custom CSS] to force the MathML/SVG rendering or select different font families. See [https://www.mediawiki.org/wiki/Extension:Math#CSS_for_the_MathML_with_SVG_fallback_mode these examples].


[[Image:Quadrature of Circle Cajori 1919.png|right|thumb|A partial history by [[Florian Cajori]] of attempts at the problem.<ref>{{cite book|author=Florian Cajori|title=A History of Mathematics|edition=2nd|page= 143|location= New York|publisher= The Macmillan Company|year=1919}}</ref>]] The famous Victorian-age mathematician, logician and author, [[Lewis Carroll|Charles Lutwidge Dodgson]] (better known under the pseudonym "Lewis Carroll") also expressed interest in debunking illogical circle-squaring theories. In one of his diary entries for 1855, Dodgson listed books he hoped to write including one called "Plain Facts for Circle-Squarers". In the introduction to "A New Theory of Parallels", Dodgson recounted an attempt to demonstrate logical errors to a couple of circle-squarers, stating:<ref>{{Cite book
==Demos==
|author=Martin Gardner
|title=The Universe in a Handkerchief
|isbn=0-387-94673-X
|publisher=Springer
|year=1996
}}</ref>


<blockquote>The first of these two misguided visionaries filled me with a great ambition to do a feat I have never heard of as accomplished by man, namely to convince a circle squarer of his error! The value my friend selected for Pi was 3.2: the enormous error tempted me with the idea that it could be easily demonstrated to BE an error. More than a score of letters were interchanged before I became sadly convinced that I had no chance.
Here are some [https://commons.wikimedia.org/w/index.php?title=Special:ListFiles/Frederic.wang demos]:
</blockquote>


Perhaps the most famous and effective ridiculing of circle squaring appears in [[Augustus de Morgan]]'s [[A Budget of Paradoxes]] published posthumously by his widow in 1872. Originally published as a series of articles in the ''Athenæum'', he was revising them for publication at the time of his death. Circle squaring was very popular in the nineteenth century, but hardly anyone indulges in it today and it is believed that de Morgan's work helped bring this about.<ref>{{citation|first=Underwood|last=Dudley|title=A Budget of Trisections|publisher=Springer-Verlag|year=1987|isbn=0-387-96568-8|pages=xi-xii}} Reprinted as ''The Trisectors''.</ref>


==Impossibility==
* accessibility:
The solution of the problem of squaring the circle by compass and straightedge demands construction of the number <math>\scriptstyle \sqrt{\pi}</math>, and the impossibility of this undertaking follows from the fact that pi is a [[transcendental number|transcendental]]
** Safari + VoiceOver: [https://commons.wikimedia.org/wiki/File:VoiceOver-Mac-Safari.ogv video only], [[File:Voiceover-mathml-example-1.wav|thumb|Voiceover-mathml-example-1]], [[File:Voiceover-mathml-example-2.wav|thumb|Voiceover-mathml-example-2]], [[File:Voiceover-mathml-example-3.wav|thumb|Voiceover-mathml-example-3]], [[File:Voiceover-mathml-example-4.wav|thumb|Voiceover-mathml-example-4]], [[File:Voiceover-mathml-example-5.wav|thumb|Voiceover-mathml-example-5]], [[File:Voiceover-mathml-example-6.wav|thumb|Voiceover-mathml-example-6]], [[File:Voiceover-mathml-example-7.wav|thumb|Voiceover-mathml-example-7]]
([[algebraic number|non-algebraic]] and therefore [[constructible number|non-constructible]]) number. If the problem of the quadrature of the circle is solved using only compass and straightedge, then an algebraic value of pi would be found, which is impossible. [[Johann Heinrich Lambert]] conjectured that pi was transcendental in 1768 in the same paper in which he proved its irrationality, even before the existence of transcendental numbers was proven. It was not until 1882 that [[Ferdinand von Lindemann]] proved its transcendence.
** [https://commons.wikimedia.org/wiki/File:MathPlayer-Audio-Windows7-InternetExplorer.ogg Internet Explorer + MathPlayer (audio)]
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** From our testing, ChromeVox and JAWS are not able to read the formulas generated by the MathML mode.


The transcendence of pi implies the impossibility of exactly "circling" the square, as well as of squaring the circle.
==Test pages ==


It is possible to construct a square with an area arbitrarily close to that of a given circle. If a rational number is used as an approximation of pi, then squaring the circle becomes possible, depending on the values chosen. However, this is only an approximation and does not meet the constraints of the ancient rules for solving the problem. Several mathematicians have demonstrated workable procedures based on a variety of approximations.
To test the '''MathML''', '''PNG''', and '''source''' rendering modes, please go to one of the following test pages:
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Bending the rules by allowing an infinite number of compass-and-straightedge operations or by performing the operations on certain [[non-Euclidean space]]s also makes squaring the circle possible. For example, although the circle cannot be squared in [[Euclidean space]], it can be in [[Gauss–Bolyai–Lobachevsky space]]. Indeed, even the preceding phrase is  overoptimistic.<ref>{{Cite journal| doi = 10.1007/BF03024895  | last = Jagy  | first = William C. | title = Squaring circles in the hyperbolic plane | journal = Mathematical Intelligencer  | volume = 17  | issue = 2  | pages = 31–36  | year = 1995  | url = http://zakuski.math.utsa.edu/~jagy/papers/Intelligencer_1995.pdf  | format = [[Portable Document Format|PDF]]  | postscript = <!--None--> }}</ref><ref>{{Cite book
*[[Inputtypes|Inputtypes (private Wikis only)]]
  | last = Greenberg
*[[Url2Image|Url2Image (private Wikis only)]]
  | first = Marvin Jay
==Bug reporting==
  | title =  Euclidean and Non-Euclidean Geometries
If you find any bugs, please report them at [https://bugzilla.wikimedia.org/enter_bug.cgi?product=MediaWiki%20extensions&component=Math&version=master&short_desc=Math-preview%20rendering%20problem Bugzilla], or write an email to math_bugs (at) ckurs (dot) de .
  | publisher = W H Freeman
  | year = 2008
  | pages = 520–528
  | edition = Fourth
  | isbn =  0-7167-9948-0
  | postscript = <!--None--> }}</ref>  There are no squares as such in the hyperbolic plane, although there are regular quadrilaterals, meaning quadrilaterals with all sides congruent and all angles congruent (but these angles are strictly smaller than right angles).
There exist, in the hyperbolic plane, (countably) infinitely many pairs of constructible circles and constructible regular quadrilaterals of equal area.
However, there is no method for starting with a regular quadrilateral and constructing the circle of equal area, and there is no method for starting with a circle and constructing a regular quadrilateral of equal area (even when the circle has small enough radius such that a regular quadrilateral of equal area exists).
 
==Modern approximative constructions==
Though squaring the circle is an impossible problem using only compass and straightedge, approximations to squaring the circle can be given by constructing lengths close to pi.
It takes only minimal knowledge of elementary geometry to convert any given rational approximation of pi into a corresponding compass-and-straightedge construction, but constructions made in this way tend to be very long-winded in comparison to the accuracy they achieve. After the exact problem was proven unsolvable, some mathematicians applied their ingenuity to finding elegant approximations to squaring the circle, defined roughly and informally as constructions that are particularly simple among other imaginable constructions that give similar precision.
 
Among the modern approximate constructions was one by [[E. W. Hobson]] in 1913.<ref>{{Cite book|last=Hobson|first= Ernest William |year=1913|title=Squaring the Circle: A History of the Problem|publisher= Cambridge University Press}} Reprinted by Merchant Books in 2007.</ref> This was a fairly accurate construction which was based on constructing the approximate value of 3.14164079..., which is accurate to 4 decimals (i.e. it differs from pi by about {{Val|4.8|e=-5}}).
 
Indian mathematician [[Srinivasa Ramanujan]] in 1913, [[C. D. Olds]] in 1963, [[Martin Gardner]] in 1966, and [[Benjamin Bold]] in 1982 all gave geometric constructions for
 
:<math>\tfrac{355}{113} = 3.1415929203539823008\dots</math>
 
which is accurate to six decimal places of pi.
 
[[Image:Kochanski-1.svg|right|thumb|[[Adam Adamandy Kochański|Kochański's]] approximate construction]]
Srinivasa Ramanujan in 1914 gave a ruler-and-compass construction which was equivalent to taking the approximate value for pi to be
 
:<math>\left(9^2 + \frac{19^2}{22}\right)^{1/4} = \sqrt[4]{\frac{2143}{22}} = 3.1415926525826461252\dots</math>
 
giving a remarkable eight decimal places of pi.
 
In 1991, [[Robert Dixon (mathematician)|Robert Dixon]] gave constructions for
 
:<math>\frac{6}{5} (1 + \varphi)\text{ and }\sqrt{{40 \over 3} - 2 \sqrt{3}\  }</math>
 
([[Kochański's approximation]]), though these were only accurate to four decimal places of pi.
 
==Squaring or quadrature as integration==
The problem of finding the area under a curve, known as [[Integral|integration]] in [[calculus]], or [[numerical quadrature|quadrature]] in [[numerical analysis]], was known as ''squaring'' before the invention of calculus.  Since the techniques of calculus were unknown, it was generally presumed that a squaring should be done via geometric constructions, that is, by compass and straightedge.  For example [[Isaac Newton|Newton]] wrote to [[Henry Oldenberg|Oldenberg]] in 1676 "I believe M. Leibnitz will not dislike the Theorem towards the beginning of my letter pag. 4 for '''squaring Curve lines''' Geometrically" (emphasis added).<ref>{{cite book|url=http://books.google.com/?id=OVPJ6c9_kKgC&pg=PA259&vq=squaring&dq=newton+squaring-curves+date:0-1923|title=Correspondence of Sir Isaac Newton and Professor Cotes: Including letters of other eminent men|author1=Cotes|first1=Roger|year=1850}}</ref>  After Newton and [[Gottfried Leibniz|Leibniz]] invented calculus, they still referred to this integration problem as squaring a curve.
 
==Claims of circle squaring==
===Connection with the longitude problem===
The mathematical proof that the [[numerical integration|quadrature]] of the circle is impossible using only compass and straightedge has not proved to be a hindrance to the many people who have invested years in this problem anyway.  Having squared the circle is a famous [[crank (person)|crank]] assertion. (''See also'' [[pseudomathematics]].) In his old age, the English philosopher [[Thomas Hobbes]] convinced himself that he had succeeded in squaring the circle.
 
During the 18th and 19th century, the notion that the problem of squaring the circle was somehow related to the [[Longitude prize|longitude problem]] seems to have become prevalent among would-be circle squarers. Using "cyclometer" for circle-squarer,  [[Augustus de Morgan]] wrote in 1872:
 
<blockquote>
[[Jean Etienne Montucla|Montucla]] says, speaking of France, that he finds three notions prevalent among cyclometers: 1. That there is a large reward offered for success; 2. That the longitude problem depends on that success; 3. That the solution is the great end and object of geometry. The same three notions are equally prevalent among the same class in England. No reward has ever been offered by the government of either country.<ref>{{cite book|author=[[Augustus de Morgan]] |year=1872|title=[[A Budget of Paradoxes]]|page= 96}}</ref></blockquote>
 
Although from 1714 to 1828 the British government did indeed sponsor a £20,000 prize for finding a solution to the longitude problem, exactly why the connection was made to squaring the circle is not clear; especially since two non-geometric methods (the astronomical [[Lunar distance (navigation)|method of lunar distances]] and the mechanical [[marine chronometer|chronometer]]) had been found by the late 1760s. De Morgan goes on to say that "[t]he longitude problem in no way depends upon perfect solution; existing approximations are sufficient to a point of accuracy far beyond what can be wanted." In his book, de Morgan also mentions receiving many threatening letters from would-be circle squarers, accusing him of trying to "cheat them out of their prize".
 
===Other modern claims===
Even after it had been proved impossible, in 1894, amateur mathematician Edwin J. Goodwin claimed that he had developed a method to square the circle. The technique he developed did not accurately square the circle, and provided an incorrect area of the circle which essentially redefined pi as equal to 3.2. Goodwin then proposed the [[Indiana Pi Bill]] in the Indiana state legislature allowing the state to use his method in education without paying royalties to him. The bill passed with no objections in the state house, but the bill was tabled and never voted on in the Senate, amid increasing ridicule from the press.
 
==In literature==
The problem of squaring the circle has been mentioned by poets such as [[Dante]] and [[Alexander Pope]], with varied [[metaphor]]ical meanings.
 
The character [[Meton of Athens]] in the play [[The Birds (play)|The Birds]] by [[Aristophanes]] (first performed in 414 BC) mentions squaring the circle.<ref>{{citation
| last = Amati | first = Matthew
| issue = 3
| journal = [[The Classical Journal]]
| jstor = 10.5184/classicalj.105.3.213
| pages = 213–222
| title = Meton's star-city: Geometry and utopia in Aristophanes' ''Birds''
| volume = 105
| year = 2010}}.</ref>
 
Dante's ''Paradise'' canto XXXIII lines 133–135 contain the verses:
<poem style="margin-left: 2em">
As the geometer his mind applies
To square the circle, nor for all his wit
Finds the right formula, howe'er he tries</poem>
For Dante, squaring the circle represents a task beyond human comprehension, which he compares to his own inability to comprehend Paradise.<ref>{{citation
| last1 = Herzman | first1 = Ronald B.
| last2 = Towsley | first2 = Gary B.
| journal = Traditio
| jstor = 27831895
| pages = 95–125
| title = Squaring the circle: ''Paradiso'' 33 and the poetics of geometry
| volume = 49
| year = 1994}}.</ref>
 
By 1742, when [[Alexander Pope]] published the fourth book of his [[Dunciad]], attempts at circle-squaring had come to be seen as "wild and fruitless":<ref>{{citation
| last = Schepler | first = Herman C.
| journal = [[Mathematics Magazine]]
| jstor = 3029832
| mr = 0037596
| pages = 165–170, 216–228, 279–283
| title = The chronology of pi
| volume = 23
| year = 1950}}.</ref>
<poem style="margin-left: 2em">Mad Mathesis alone was unconfined,
Too mad for mere material chains to bind,
Now to pure space lifts her ecstatic stare,
Now, running round the circle, finds it square.</poem>
 
Similarly, the [[Gilbert and Sullivan]] comic opera ''[[Princess Ida]]'' features a song which satirically lists the impossible goals of the women's university run by the title character, such as finding [[perpetual motion]].  One of these goals is "And the circle – they will square it/Some fine day."<ref>{{citation
| last = Dolid | first = William A.
| issue = 2
| journal = The Shaw Review
| jstor = 40682600
| pages = 52–56
| title = Vivie Warren and the Tripos
| volume = 23
| year = 1980}}. Dolid contrasts Vivie Warren, a fictional female mathematics student in ''[[Mrs. Warren's Profession]]'' by [[George Bernard Shaw]], with the satire of college women presented by Gilbert and Sullivan. He writes that "Vivie naturally knew better than to try to square circles."</ref>
 
The [[sestina]], a poetic form first used in the 12th century by [[Arnaut Daniel]], has been said to square the circle in its use of a square number of lines (six stanzas of six lines each) with a circular scheme of six repeated words. {{harvtxt|Spanos|1978}} writes that this form invokes a symbolic meaning in which the circle stands for heaven and the square stands for the earth.<ref>{{citation|title=The Sestina: An Exploration of the Dynamics of Poetic Structure|first=Margaret|last=Spanos|journal=Speculum|volume=53|issue=3|year=1978|pages=545–557|jstor=2855144}}.</ref>
A similar metaphor was used in "Squaring The Circle", a 1908 short story by [[O. Henry]], about a long-running family feud. In the title of this story, the circle represents the natural world, while the square represents the city, the world of man.<ref>{{citation|title=Twentieth-century American literature|first=Harold|last=Bloom|authorlink=Harold Bloom|publisher=Chelsea House Publishers|year=1987|isbn=9780877548034|page=1848|quotation=Similarly, the story "Squaring the Circle" is permeated with the integrating image: nature is a circle, the city a square.}}</ref>
 
In [[James Joyce|James Joyce's]] novel ''[[Ulysses_(novel)|Ulysses]]'', Leopold Bloom dreams of becoming wealthy by squaring the circle, unaware that the quadrature of the circle had been proved impossible 22 years earlier and that the British government had never offered a reward for its solution.<ref>{{citation
| last = Pendrick | first = Gerard
| issue = 1
| journal = [[James Joyce Quarterly]]
| jstor = 25473619
| pages = 105–107
| title = Two notes on "Ulysses"
| volume = 32
| year = 1994}}.</ref>
 
==See also==
* The two other classical problems of antiquity were [[doubling the cube]] and [[trisecting the angle]], described in the [[compass and straightedge]] article. Unlike squaring the circle, these two problems can be solved by the slightly more powerful construction method of [[origami]], as described at [[mathematics of paper folding]].
* For a more modern related problem, see [[Tarski's circle-squaring problem]].
* The [[Indiana Pi Bill]], an 1897 attempt in the [[Indiana state legislature]] to dictate a solution to the problem by legislative fiat.
* [[Squircle]], a mathematical shape with properties between those of a square and those of a circle.
* [[Squared circle]], a [[professional wrestling]] ring.
 
==References==
{{Reflist|2}}
 
==External links==
{{Wikisource}}
* ''[http://www-groups.dcs.st-and.ac.uk/~history/HistTopics/Squaring_the_circle.html Squaring the circle]'' at the [[MacTutor History of Mathematics archive]]
* ''[http://www.cut-the-knot.org/impossible/sq_circle.shtml Squaring the Circle]'' at [[cut-the-knot]]
* ''[http://mathworld.wolfram.com/CircleSquaring.html Circle Squaring]'' at [[MathWorld]], includes information on procedures based on various approximations of pi
* "[http://mathdl.maa.org/convergence/1/?pa=content&sa=viewDocument&nodeId=1207&bodyId=1357 Squaring the Circle]" at "[http://mathdl.maa.org/convergence/1/ Convergence]"
* [http://mathdl.maa.org/convergence/1/?pa=content&sa=viewDocument&nodeId=1203&bodyId=1593 The Quadrature of the Circle and Hippocrates' Lunes] at [http://mathdl.maa.org/convergence/1/ Convergence]
* [http://www.song-of-songs.net/Squaring_the_Circle.html ''How to Unroll a Circle''] Pi accurate to eight decimal places, using straightedge and compass.
* [http://www.gresham.ac.uk/event.asp?PageId=45&EventId=624 ''Squaring the Circle and Other Impossibilities''], lecture by [[Robin Wilson (mathematician)|Robin Wilson]], at [[Gresham College]], 16 January 2008 (available for download as text, audio or video file).
* {{cite web|last=Grime|first=James|title=Squaring the Circle|url=http://www.numberphile.com/videos/squaring_circle.html|work=Numberphile|publisher=[[Brady Haran]]}}
 
{{Greek mathematics}}
{{Use dmy dates|date=September 2010}}
 
{{DEFAULTSORT:Squaring The Circle}}
[[Category:Pi]]
[[Category:Euclidean plane geometry]]
[[Category:Mathematical problems]]
[[Category:History of geometry]]
[[Category:Compass and straightedge constructions]]

Latest revision as of 23:52, 15 September 2019

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