Editing Pi (section)

=== Adoption of the symbol {{pi}} ===
{{Multiple image
| image1            = William Jones, the Mathematician.jpg
| caption1          = The earliest known use of the Greek letter {{pi}} to represent the ratio of a circle's circumference to its diameter was by Welsh mathematician [[William Jones (mathematician)|William Jones]] in 1706
| caption2          = [[Leonhard Euler]] popularized the use of the Greek letter {{pi}} in works he published in 1736 and 1748.
| total_width       = 300
| image2            = Leonhard Euler.jpg
| align             = right
}}

The first recorded use of the symbol {{pi}} in circle geometry is in [[William Oughtred|Oughtred's]] ''Clavis Mathematicae'' (1648),<ref>{{cite book |url=https://archive.org/details/bub_gb_ddMxgr27tNkC |title=Clavis Mathematicæ |last=Oughtred |author-link=William Oughtred |first=William |date=1648 |publisher=Thomas Harper |location=London |page=[https://archive.org/details/bub_gb_ddMxgr27tNkC/page/n220 69] |language=la |trans-title=The key to mathematics}} (English translation: {{Cite book |url=https://books.google.com/books?id=S50yAQAAMAAJ&pg=PA99 |title=Key of the Mathematics |last=Oughtred |first=William |author-link=William Oughtred |date=1694 |publisher=J. Salusbury |language=en}})</ref>{{sfn|Arndt|Haenel|2006|p=166}}
where the [[Greek letters]] {{pi}} and ''δ'' were combined into the fraction {{tmath|\tfrac \pi \delta}} for denoting the ratios [[semiperimeter]] to [[semidiameter]] and perimeter to diameter, that is, what is  presently denoted as {{pi}}.<ref name=firstPi>{{Cite book |url=https://books.google.com/books?id=KTgPAAAAQAAJ&pg=PP3 |title=Theorematum in libris Archimedis de sphaera et cylindro declarario |last=Oughtred |first=William |author-link=William Oughtred |date=1652 |publisher=Excudebat L. Lichfield, Veneunt apud T. Robinson |language=la |quote={{math|''δ''.''π''}} :: semidiameter. semiperipheria}}</ref><ref name="Cajori-2007">{{Cite book |url=https://books.google.com/books?id=bT5suOONXlgC&pg=PA9 |title=A History of Mathematical Notations: Vol. II |last=Cajori |first=Florian |author-link=Florian Cajori |date=2007 |publisher=Cosimo, Inc. |isbn=978-1-60206-714-1 |pages=8–13 |language=en |quote=the ratio of the length of a circle to its diameter was represented in the fractional form by the use of two letters ... J.A. Segner ... in 1767, he represented {{math|3.14159...}} by {{math|''δ'':''π''}}, as did Oughtred more than a century earlier}}</ref><ref name="Smith-1958">{{Cite book |url=https://books.google.com/books?id=uTytJGnTf1kC&pg=PA312 |title=History of Mathematics |last=Smith |first=David E. |author-link=David Eugene Smith |date=1958 |publisher=Courier Corporation |isbn=978-0-486-20430-7 |page=312 |language=en}}</ref><ref>{{Cite journal |last=Archibald |first=R. C. |author-link=Raymond C. Archibald |date=1921 |title=Historical Notes on the Relation {{math |1=''e''<sup>−(''π''/2)</sup> = ''i''<sup>''i''</sup>}} |jstor=2972388 |journal=The American Mathematical Monthly |volume=28 |issue=3 |pages=116–121 |doi=10.2307/2972388 |quote=It is noticeable that these letters are ''never'' used separately, that is, {{pi}} is ''not'' used for 'Semiperipheria'}}</ref> (Before then, mathematicians sometimes used letters such as ''c'' or ''p'' instead.{{sfn|Arndt|Haenel|2006|p=166}}) [[Isaac Barrow|Barrow]] likewise used the same notation,<ref>{{Cite book |chapter-url=https://archive.org/stream/mathematicalwor00whewgoog#page/n405/mode/1up |title=The mathematical works of Isaac Barrow |last=Barrow |first=Isaac |author-link=Isaac Barrow |date=1860 |publisher=Cambridge University press |others=Harvard University |editor-last=Whewell |editor-first=William |page=381 |language=la |chapter=Lecture XXIV}}</ref> while [[David Gregory (mathematician)|Gregory]] instead used <math display=inline>\frac \pi \rho</math> to represent {{math|6.28...&thinsp;}}.<ref>{{Cite journal |last=Gregorius |first=David |date=1695 |title=Ad Reverendum Virum D. Henricum Aldrich S.T.T. Decanum Aedis Christi Oxoniae |jstor=102382 |journal=Philosophical Transactions |language=la |volume=19 |issue=231 |pages=637–652 |doi=10.1098/rstl.1695.0114 |bibcode=1695RSPT...19..637G |doi-access=free |url=https://archive.org/download/crossref-pre-1909-scholarly-works/10.1098%252Frstl.1684.0084.zip/10.1098%252Frstl.1695.0114.pdf}}</ref>{{r|Smith-1958}}

The earliest known use of the Greek letter {{pi}} alone to represent the ratio of a circle's circumference to its diameter was by Welsh mathematician [[William Jones (mathematician)|William Jones]] in his 1706 work ''{{lang|la|Synopsis Palmariorum Matheseos|italic=unset}}; or, a New Introduction to the Mathematics''.{{r|jones}}{{sfn|Arndt|Haenel|2006|p=165|ps=: A facsimile of Jones' text is in {{harvnb|Berggren|Borwein|Borwein|1997|pp=108–109}}.}} The Greek letter appears on p.&nbsp;243 in the phrase "<math display=inline>\tfrac12</math> Periphery ({{pi}})", calculated for a circle with radius one. However, Jones writes that his equations for {{pi}} are from the "ready pen of the truly ingenious Mr. [[John Machin]]", leading to speculation that Machin may have employed the Greek letter before Jones.{{sfn|Arndt|Haenel|2006|p=166}} Jones' notation was not immediately adopted by other mathematicians, with the fraction notation still being used as late as 1767.{{r|Cajori-2007}}<ref>{{Cite book |url=https://books.google.com/books?id=NmYVAAAAQAAJ&pg=PA282 |title=Cursus Mathematicus |last=Segner |first=Joannes Andreas |date=1756 |publisher=Halae Magdeburgicae |page=282 |language=la |access-date=15 October 2017 |archive-url=https://web.archive.org/web/20171015150340/https://books.google.co.uk/books?id=NmYVAAAAQAAJ&pg=PA282 |archive-date=15 October 2017 |url-status=live}}</ref>

[[Euler]] started using the single-letter form beginning with his 1727 ''Essay Explaining the Properties of Air'', though he used {{math|1=''π'' = 6.28...}}, the ratio of periphery to radius, in this and some later writing.<ref>{{Cite journal |last=Euler |first=Leonhard |author-link=Leonhard Euler |date=1727 |title=Tentamen explicationis phaenomenorum aeris |url=http://eulerarchive.maa.org/docs/originals/E007.pdf#page=5 |journal=Commentarii Academiae Scientiarum Imperialis Petropolitana |language=la |volume=2 |page=351 |id=[http://eulerarchive.maa.org/pages/E007.html E007] |quote=Sumatur pro ratione radii ad peripheriem, {{math|I : π}} |access-date=15 October 2017 |archive-url=https://web.archive.org/web/20160401072718/http://eulerarchive.maa.org/docs/originals/E007.pdf#page=5 |archive-date=1 April 2016 |url-status=live}} [http://www.17centurymaths.com/contents/euler/e007tr.pdf#page=3 English translation by Ian Bruce] {{Webarchive|url=https://web.archive.org/web/20160610172054/http://www.17centurymaths.com/contents/euler/e007tr.pdf#page=3 |date=10 June 2016 }}: "{{mvar|π}} is taken for the ratio of the radius to the periphery [note that in this work, Euler's {{pi}} is double our {{pi}}.]" {{pb}} {{Cite book |url=https://books.google.com/books?id=3C1iHFBXVEcC&pg=PA139 |title=Lettres inédites d'Euler à d'Alembert |last=Euler |first=Leonhard |author-link=Leonhard Euler |series=Bullettino di Bibliografia e di Storia delle Scienze Matematiche e Fisiche |year=1747 |editor-last=Henry |editor-first=Charles |volume=19 |publication-date=1886 |page=139 |language=fr |id=[http://eulerarchive.maa.org/pages/E858.html E858] |quote=Car, soit π la circonference d'un cercle, dout le rayon est {{math|{{=}} 1}}}} English translation in {{Cite journal |last=Cajori |first=Florian |author-link=Florian Cajori |date=1913 |title=History of the Exponential and Logarithmic Concepts |jstor=2973441 |journal=The American Mathematical Monthly |volume=20 |issue=3 |pages=75–84 |doi=10.2307/2973441 |quote=Letting {{pi}} be the circumference (!) of a circle of unit radius}}</ref>   Euler first used {{nowrap|1={{pi}} = 3.14...}} in his 1736 work ''[[Mechanica]]'',<ref>{{Cite book |last=Euler |first=Leonhard |author-link=Leonhard Euler |title=Mechanica sive motus scientia analytice exposita. (cum tabulis) |date=1736 |publisher=Academiae scientiarum Petropoli |volume=1 |page=113 |language=la |chapter=Ch. 3 Prop. 34 Cor. 1 |id=[http://eulerarchive.maa.org/pages/E015.html E015] |quote=Denotet {{math|1 : ''π''}} rationem diametri ad peripheriam |chapter-url=https://books.google.com/books?id=jgdTAAAAcAAJ&pg=PA113}} [http://www.17centurymaths.com/contents/euler/mechvol1/ch3a.pdf#page=26 English translation by Ian Bruce] {{Webarchive|url=https://web.archive.org/web/20160610183753/http://www.17centurymaths.com/contents/euler/mechvol1/ch3a.pdf#page=26|date=10 June 2016}} : "Let {{math|1 : ''π''}} denote the ratio of the diameter to the circumference"</ref> and continued in his widely read 1748 work {{lang|la|[[Introductio in analysin infinitorum]]|italic=yes}} (he wrote: "for the sake of brevity we will write this number as {{pi}}; thus {{pi}} is equal to half the circumference of a circle of radius {{math|1}}").<ref>{{Cite book |url=http://gallica.bnf.fr/ark:/12148/bpt6k69587/f155 |title=Leonhardi Euleri opera omnia. 1, Opera mathematica. Volumen VIII, Leonhardi Euleri introductio in analysin infinitorum. Tomus primus / ediderunt Adolf Krazer et Ferdinand Rudio |last=Euler |first=Leonhard |date=1922 |author-link=Leonhard Euler |publisher=B. G. Teubneri |location=Lipsae |pages=133–134 |language=la |id=[http://eulerarchive.maa.org/pages/E101.html E101] |access-date=15 October 2017 |archive-url=https://web.archive.org/web/20171016022758/http://gallica.bnf.fr/ark:/12148/bpt6k69587/f155 |archive-date=16 October 2017 |url-status=live}}</ref> Because Euler corresponded heavily with other mathematicians in Europe, the use of the Greek letter spread rapidly, and the practice was universally adopted thereafter in the [[Western world]],{{sfn|Arndt|Haenel|2006|p=166}} though the definition still varied between {{math|3.14...}} and {{math|6.28...}} as late as 1761.<ref>{{Cite book |url=https://books.google.com/books?id=P-hEAAAAcAAJ&pg=PA374 |title=Cursus Mathematicus: Elementorum Analyseos Infinitorum Elementorum Analyseos Infinitorvm |last=Segner |first=Johann Andreas von |date=1761 |publisher=Renger |page=374 |language=la |quote=Si autem {{pi}} notet peripheriam circuli, cuius diameter eſt {{math|2}}}}</ref>