Editing Pi (section)

=== Polygon approximation era ===
[[File:Domenico-Fetti Archimedes 1620.jpg|alt=A painting of a man studying|thumb|[[Archimedes]] developed the polygonal approach to approximating {{pi}}.]]
[[File:Archimedes pi.svg|thumb|alt=diagram of a hexagon and pentagon circumscribed outside a circle|{{pi}} can be estimated by computing the perimeters of circumscribed and inscribed polygons.]]

The first recorded algorithm for rigorously calculating the value of {{pi}} was a geometrical approach using polygons, devised around 250&nbsp;BC by the Greek mathematician [[Archimedes]], implementing the [[method of exhaustion]].{{sfn|Arndt|Haenel|2006|p=170}} This polygonal algorithm dominated for over 1,000 years, and as a result {{pi}} is sometimes referred to as Archimedes's constant.{{sfn|Arndt|Haenel|2006|pp=175, 205}} Archimedes computed upper and lower bounds of {{pi}} by drawing a regular hexagon inside and outside a circle, and successively doubling the number of sides until he reached a 96-sided regular polygon. By calculating the perimeters of these polygons, he proved that {{math|{{sfrac|223|71}} < {{pi}} < {{sfrac|22|7}}}} (that is, {{math|3.1408 < {{pi}} < 3.1429}}.<ref name=life-of-pi>{{cite book |last=Borwein |first=Jonathan M. |author-link=Jonathan Borwein |editor1-last=Sidoli |editor1-first=Nathan |editor2-last=Van Brummelen |editor2-first=Glen |contribution=The life of {{pi}}: from Archimedes to ENIAC and beyond |doi=10.1007/978-3-642-36736-6_24 |location=Heidelberg |mr=3203895 |pages=531–561 |publisher=Springer |title=From Alexandria, through Baghdad: Surveys and studies in the ancient Greek and medieval Islamic mathematical sciences in honor of J. L. Berggren |year=2014 |isbn=978-3-642-36735-9}}</ref> Archimedes' upper bound of {{math|{{sfrac|22|7}}}} may have led to a widespread popular belief that {{pi}} is equal to {{math|{{sfrac|22|7}}}}.{{sfn|Arndt|Haenel|2006|p=171}} Around 150&nbsp;AD, Greco-Roman scientist [[Ptolemy]], in his ''[[Almagest]]'', gave a value for {{pi}} of 3.1416, which he may have obtained from Archimedes or from [[Apollonius of Perga]].{{sfn|Arndt|Haenel|2006|p=176}}{{sfn|Boyer|Merzbach|1991|p=168}}<!--may be suspect--> Mathematicians using polygonal algorithms reached 39 digits of {{pi}} in 1630, a record only broken in 1699 when infinite series were used to reach 71 digits.<ref name="ArPI">{{harvnb|Arndt|Haenel|2006|pp=15–16, 175, 184–186, 205}}. Grienberger achieved 39 digits in 1630; Sharp 71 digits in 1699.</ref>

In [[ancient China]], values for {{pi}} included 3.1547 (around 1&nbsp;AD), <math>\sqrt{10}</math> (100&nbsp;AD, approximately 3.1623), and {{math|{{sfrac|142|45}}}} (3rd century, approximately 3.1556).{{sfn|Arndt|Haenel|2006|pp=176–177}} Around 265&nbsp;AD, the [[Cao Wei]] mathematician [[Liu Hui]] created a [[Liu Hui's π algorithm|polygon-based iterative algorithm]], with which he constructed a 3,072-sided polygon to approximate {{pi}} as&nbsp;3.1416.{{sfn|Boyer|Merzbach|1991|p=202}}{{sfn|Arndt|Haenel|2006|p=177}} Liu later invented a faster method of calculating {{pi}} and obtained a value of 3.14 with a 96-sided polygon, by taking advantage of the fact that the differences in area of successive polygons form a geometric series with a factor of&nbsp;4.{{sfn|Boyer|Merzbach|1991|p=202}} Around 480&nbsp;AD, [[Zu Chongzhi]] calculated that <math>3.1415926 < \pi < 3.1415927</math> and suggested the approximations <math display="inline">\pi \approx \frac{355}{113} = 3.14159292035...</math> and <math display="inline">\pi \approx \frac{22}{7} = 3.142857142857...</math>, which he termed the ''[[milü]]'' ('close ratio') and ''yuelü'' ('approximate ratio') respectively, iterating with Liu Hui's algorithm up to a 12,288-sided polygon. With a correct value for its seven first decimal digits, Zu's result remained the most accurate approximation of {{pi}} for the next 800 years.{{sfn|Arndt|Haenel|2006|p=178}}

The Indian astronomer [[Aryabhata]] used a value of 3.1416 in his ''[[Āryabhaṭīya]]'' (499&nbsp;AD).{{sfn|Arndt|Haenel|2006|p=179}} Around 1220, [[Fibonacci]] computed 3.1418 using a polygonal method devised independently of Archimedes.{{sfn|Arndt|Haenel|2006|p=180}} Italian author [[Dante]] apparently employed the value <math display="inline">3+\frac{\sqrt{2}}{10} \approx 3.14142</math>.{{sfn|Arndt|Haenel|2006|p=180}}

The Persian astronomer [[Jamshīd al-Kāshī]] produced nine [[sexagesimal]] digits, roughly the equivalent of 16 decimal digits, in 1424, using a polygon with <math display="inline">3\times 2^{28}</math> sides,<ref>{{cite journal |first1=Mohammad K. |last1=Azarian |title=al-Risāla al-muhītīyya: A Summary |journal=Missouri Journal of Mathematical Sciences |volume=22 |issue=2 |year=2010 |pages=64–85 |doi=10.35834/mjms/1312233136 |doi-access=free}} {{pb}} {{cite web |last1=O'Connor |first1=John J. |last2=Robertson |first2=Edmund F. |year=1999 |title=Ghiyath al-Din Jamshid Mas'ud al-Kashi |work=[[MacTutor History of Mathematics archive]] |url=http://www-history.mcs.st-and.ac.uk/history/Biographies/Al-Kashi.html |access-date=11 August 2012 |url-status=live |archive-url=https://web.archive.org/web/20110412192025/http://www-history.mcs.st-and.ac.uk/history/Biographies/Al-Kashi.html |archive-date=12 April 2011}}</ref> which stood as the world record for about 180 years.{{sfn|Arndt|Haenel|2006|p=182}} French mathematician [[François Viète]] in 1579 achieved nine digits with a polygon of <math display="inline">3\times 2^{17}</math> sides.{{sfn|Arndt|Haenel|2006|p=182}} Flemish mathematician [[Adriaan van Roomen]] arrived at 15 decimal places in 1593.{{sfn|Arndt|Haenel|2006|p=182}} In 1596, Dutch mathematician [[Ludolph van Ceulen]] reached 20 digits, a record he later increased to 35 digits (as a result, {{pi}} was called the "Ludolphian number" in Germany until the early 20th century).{{sfn|Arndt|Haenel|2006|pp=182–183}} Dutch scientist [[Willebrord Snellius]] reached 34 digits in 1621,{{sfn|Arndt|Haenel|2006|p=183}} and Austrian astronomer [[Christoph Grienberger]] arrived at 38 digits in 1630 using 10<sup>40</sup> sides.<ref>{{cite book |first=Christophorus |last=Grienbergerus |author-link=Christoph Grienberger |language=la |year=1630 |title=Elementa Trigonometrica |url=http://librarsi.comune.palermo.it/gesuiti2/06.04.01.pdf |archive-url=https://web.archive.org/web/20140201234124/http://librarsi.comune.palermo.it/gesuiti2/06.04.01.pdf |archive-date=1 February 2014}} His evaluation was 3.14159 26535 89793 23846 26433 83279 50288 4196 < {{pi}} < 3.14159 26535 89793 23846 26433 83279 50288 4199.</ref> [[Christiaan Huygens]] was able to arrive at 10 decimal places in 1654 using a slightly different method equivalent to [[Richardson extrapolation]].<ref>{{cite book |last=Brezinski |first=C. |contribution=Some pioneers of extrapolation methods |date=2009 |url=https://www.worldscientific.com/doi/10.1142/9789812836267_0001 |title=The Birth of Numerical Analysis |pages=1–22 |publisher=World Scientific |doi=10.1142/9789812836267_0001 |isbn=978-981-283-625-0 |editor1-first=Adhemar |editor1-last=Bultheel |editor1-link=Adhemar Bultheel |editor2-first=Ronald |editor2-last=Cools}} {{pb}} {{Cite journal |last=Yoder |first=Joella G. |author-link=Joella Yoder |date=1996 |title=Following in the footsteps of geometry: The mathematical world of Christiaan Huygens |journal=De Zeventiende Eeuw |volume=12 |pages=83–93 |url=https://www.dbnl.org/tekst/_zev001199601_01/_zev001199601_01_0009.php |via=[[Digital Library for Dutch Literature]]}}</ref>