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vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Probability"> <div class="vector-toc-text"> <span class="vector-toc-numb">2</span> <span>Probability</span> </div> </a> <button aria-controls="toc-Probability-sublist" class="cdx-button cdx-button--weight-quiet cdx-button--icon-only vector-toc-toggle"> <span class="vector-icon mw-ui-icon-wikimedia-expand"></span> <span>Toggle Probability subsection</span> </button> <ul id="toc-Probability-sublist" class="vector-toc-list"> <li id="toc-Priority_regarding_the_Poisson_distribution" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Priority_regarding_the_Poisson_distribution"> <div class="vector-toc-text"> <span class="vector-toc-numb">2.1</span> <span>Priority regarding the Poisson distribution</span> </div> </a> <ul id="toc-Priority_regarding_the_Poisson_distribution-sublist" class="vector-toc-list"> </ul> </li> </ul> </li> <li id="toc-De_Moivre's_formula" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#De_Moivre's_formula"> <div class="vector-toc-text"> <span class="vector-toc-numb">3</span> <span>De Moivre's formula</span> </div> </a> <ul id="toc-De_Moivre's_formula-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Stirling's_approximation" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Stirling's_approximation"> <div class="vector-toc-text"> <span class="vector-toc-numb">4</span> <span>Stirling's approximation</span> </div> </a> <ul id="toc-Stirling's_approximation-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-See_also" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#See_also"> <div class="vector-toc-text"> <span class="vector-toc-numb">5</span> <span>See also</span> </div> </a> <ul id="toc-See_also-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Notes" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Notes"> <div class="vector-toc-text"> <span class="vector-toc-numb">6</span> <span>Notes</span> </div> </a> <ul id="toc-Notes-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-References" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#References"> <div class="vector-toc-text"> <span class="vector-toc-numb">7</span> <span>References</span> </div> </a> <ul id="toc-References-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Further_reading" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Further_reading"> <div class="vector-toc-text"> <span class="vector-toc-numb">8</span> <span>Further reading</span> </div> </a> <ul id="toc-Further_reading-sublist" class="vector-toc-list"> </ul> </li> </ul> </div> </div> </nav> </div> </div> <div class="mw-content-container"> <main id="content" class="mw-body"> <header class="mw-body-header vector-page-titlebar"> <nav aria-label="Contents" class="vector-toc-landmark"> <div id="vector-page-titlebar-toc" class="vector-dropdown vector-page-titlebar-toc vector-button-flush-left" > <input type="checkbox" id="vector-page-titlebar-toc-checkbox" role="button" aria-haspopup="true" data-event-name="ui.dropdown-vector-page-titlebar-toc" class="vector-dropdown-checkbox " aria-label="Toggle the table of contents" > <label id="vector-page-titlebar-toc-label" for="vector-page-titlebar-toc-checkbox" class="vector-dropdown-label cdx-button cdx-button--fake-button cdx-button--fake-button--enabled cdx-button--weight-quiet cdx-button--icon-only " aria-hidden="true" ><span class="vector-icon mw-ui-icon-listBullet mw-ui-icon-wikimedia-listBullet"></span> <span class="vector-dropdown-label-text">Toggle the table of contents</span> </label> <div class="vector-dropdown-content"> <div id="vector-page-titlebar-toc-unpinned-container" class="vector-unpinned-container"> </div> </div> </div> </nav> <h1 id="firstHeading" class="firstHeading mw-first-heading"><span class="mw-page-title-main">Abraham de Moivre</span></h1> <div id="p-lang-btn" class="vector-dropdown mw-portlet mw-portlet-lang" > <input type="checkbox" id="p-lang-btn-checkbox" role="button" aria-haspopup="true" data-event-name="ui.dropdown-p-lang-btn" class="vector-dropdown-checkbox mw-interlanguage-selector" aria-label="Go to an article in another language. Available in 53 languages" > <label id="p-lang-btn-label" for="p-lang-btn-checkbox" class="vector-dropdown-label cdx-button cdx-button--fake-button cdx-button--fake-button--enabled cdx-button--weight-quiet cdx-button--action-progressive mw-portlet-lang-heading-53" aria-hidden="true" ><span class="vector-icon mw-ui-icon-language-progressive mw-ui-icon-wikimedia-language-progressive"></span> <span class="vector-dropdown-label-text">53 languages</span> </label> <div class="vector-dropdown-content"> <div class="vector-menu-content"> <ul class="vector-menu-content-list"> <li class="interlanguage-link interwiki-ar mw-list-item"><a href="https://ar.wikipedia.org/wiki/%D8%A3%D8%A8%D8%B1%D8%A7%D9%87%D8%A7%D9%85_%D8%AF%D9%8A_%D9%85%D9%88%D8%A7%D9%81%D8%B1" title="أبراهام دي موافر – Arabic" lang="ar" hreflang="ar" data-title="أبراهام دي موافر" data-language-autonym="العربية" data-language-local-name="Arabic" class="interlanguage-link-target"><span>العربية</span></a></li><li class="interlanguage-link interwiki-az mw-list-item"><a href="https://az.wikipedia.org/wiki/Abraham_de_Muavr" title="Abraham de Muavr – Azerbaijani" lang="az" hreflang="az" data-title="Abraham de Muavr" data-language-autonym="Azərbaycanca" data-language-local-name="Azerbaijani" class="interlanguage-link-target"><span>Azərbaycanca</span></a></li><li class="interlanguage-link interwiki-azb mw-list-item"><a href="https://azb.wikipedia.org/wiki/%D8%A7%D8%A8%D8%B1%D8%A7%D9%87%D8%A7%D9%85_%D8%AF%D9%88_%D9%85%D9%88%D8%A7%D9%88%D8%B1" title="ابراهام دو مواور – South Azerbaijani" lang="azb" hreflang="azb" data-title="ابراهام دو مواور" data-language-autonym="تۆرکجه" data-language-local-name="South Azerbaijani" class="interlanguage-link-target"><span>تۆرکجه</span></a></li><li class="interlanguage-link interwiki-bn mw-list-item"><a href="https://bn.wikipedia.org/wiki/%E0%A6%86%E0%A6%AC%E0%A7%8D%E0%A6%B0%E0%A6%BE%E0%A6%86%E0%A6%AE_%E0%A6%A6%E0%A7%8D%E0%A6%AF_%E0%A6%AE%E0%A7%8B%E0%A6%AF%E0%A6%BC%E0%A6%BE%E0%A6%AD%E0%A7%8D%E0%A6%B0%E0%A7%8D%E2%80%8C" title="আব্রাআম দ্য মোয়াভ্র্‌ – Bangla" lang="bn" hreflang="bn" data-title="আব্রাআম দ্য মোয়াভ্র্‌" data-language-autonym="বাংলা" data-language-local-name="Bangla" class="interlanguage-link-target"><span>বাংলা</span></a></li><li class="interlanguage-link interwiki-be mw-list-item"><a href="https://be.wikipedia.org/wiki/%D0%90%D0%B1%D1%80%D0%B0%D1%85%D0%B0%D0%BC_%D0%B4%D1%8D_%D0%9C%D1%83%D0%B0%D1%9E%D1%80" title="Абрахам дэ Муаўр – Belarusian" lang="be" hreflang="be" data-title="Абрахам дэ Муаўр" data-language-autonym="Беларуская" data-language-local-name="Belarusian" class="interlanguage-link-target"><span>Беларуская</span></a></li><li class="interlanguage-link interwiki-bg mw-list-item"><a href="https://bg.wikipedia.org/wiki/%D0%90%D0%B1%D1%80%D0%B0%D1%85%D0%B0%D0%BC_%D0%B4%D1%8C%D0%BE_%D0%9C%D0%BE%D0%B0%D0%B2%D1%8A%D1%80" title="Абрахам дьо Моавър – Bulgarian" lang="bg" hreflang="bg" data-title="Абрахам дьо Моавър" data-language-autonym="Български" data-language-local-name="Bulgarian" class="interlanguage-link-target"><span>Български</span></a></li><li class="interlanguage-link interwiki-ca mw-list-item"><a href="https://ca.wikipedia.org/wiki/Abraham_de_Moivre" title="Abraham de Moivre – Catalan" lang="ca" hreflang="ca" data-title="Abraham de Moivre" data-language-autonym="Català" data-language-local-name="Catalan" class="interlanguage-link-target"><span>Català</span></a></li><li class="interlanguage-link interwiki-cs mw-list-item"><a href="https://cs.wikipedia.org/wiki/Abraham_de_Moivre" title="Abraham de Moivre – Czech" lang="cs" hreflang="cs" data-title="Abraham de Moivre" data-language-autonym="Čeština" data-language-local-name="Czech" class="interlanguage-link-target"><span>Čeština</span></a></li><li class="interlanguage-link interwiki-da mw-list-item"><a href="https://da.wikipedia.org/wiki/Abraham_de_Moivre" title="Abraham de Moivre – Danish" lang="da" hreflang="da" data-title="Abraham de Moivre" data-language-autonym="Dansk" data-language-local-name="Danish" class="interlanguage-link-target"><span>Dansk</span></a></li><li class="interlanguage-link interwiki-de mw-list-item"><a href="https://de.wikipedia.org/wiki/Abraham_de_Moivre" title="Abraham de Moivre – German" lang="de" hreflang="de" data-title="Abraham de Moivre" data-language-autonym="Deutsch" data-language-local-name="German" class="interlanguage-link-target"><span>Deutsch</span></a></li><li class="interlanguage-link interwiki-el mw-list-item"><a href="https://el.wikipedia.org/wiki/%CE%91%CE%B2%CF%81%CE%B1%CE%AC%CE%BC_%CE%BD%CF%84%CE%B5_%CE%9C%CE%BF%CF%85%CE%AC%CE%B2%CF%81" title="Αβραάμ ντε Μουάβρ – Greek" lang="el" hreflang="el" data-title="Αβραάμ ντε Μουάβρ" data-language-autonym="Ελληνικά" data-language-local-name="Greek" class="interlanguage-link-target"><span>Ελληνικά</span></a></li><li class="interlanguage-link interwiki-es mw-list-item"><a href="https://es.wikipedia.org/wiki/Abraham_de_Moivre" title="Abraham de Moivre – Spanish" lang="es" hreflang="es" data-title="Abraham de Moivre" data-language-autonym="Español" data-language-local-name="Spanish" class="interlanguage-link-target"><span>Español</span></a></li><li class="interlanguage-link interwiki-eo mw-list-item"><a href="https://eo.wikipedia.org/wiki/Abraham_de_Moivre" title="Abraham de Moivre – Esperanto" lang="eo" hreflang="eo" data-title="Abraham de Moivre" data-language-autonym="Esperanto" data-language-local-name="Esperanto" class="interlanguage-link-target"><span>Esperanto</span></a></li><li class="interlanguage-link interwiki-eu mw-list-item"><a href="https://eu.wikipedia.org/wiki/Abraham_de_Moivre" title="Abraham de Moivre – Basque" lang="eu" hreflang="eu" data-title="Abraham de Moivre" data-language-autonym="Euskara" data-language-local-name="Basque" class="interlanguage-link-target"><span>Euskara</span></a></li><li class="interlanguage-link interwiki-fa mw-list-item"><a href="https://fa.wikipedia.org/wiki/%D8%A7%D8%A8%D8%B1%D8%A7%D9%87%D8%A7%D9%85_%D8%AF%D9%88_%D9%85%D9%88%D8%A7%D9%88%D8%B1" title="ابراهام دو مواور – Persian" lang="fa" hreflang="fa" data-title="ابراهام دو مواور" data-language-autonym="فارسی" data-language-local-name="Persian" class="interlanguage-link-target"><span>فارسی</span></a></li><li class="interlanguage-link interwiki-fr mw-list-item"><a href="https://fr.wikipedia.org/wiki/Abraham_de_Moivre" title="Abraham de Moivre – French" lang="fr" hreflang="fr" data-title="Abraham de Moivre" data-language-autonym="Français" data-language-local-name="French" class="interlanguage-link-target"><span>Français</span></a></li><li class="interlanguage-link interwiki-gl mw-list-item"><a href="https://gl.wikipedia.org/wiki/Abraham_de_Moivre" title="Abraham de Moivre – Galician" lang="gl" hreflang="gl" data-title="Abraham de Moivre" data-language-autonym="Galego" data-language-local-name="Galician" class="interlanguage-link-target"><span>Galego</span></a></li><li class="interlanguage-link interwiki-ko mw-list-item"><a href="https://ko.wikipedia.org/wiki/%EC%95%84%EB%B8%8C%EB%9D%BC%EC%95%94_%EB%93%9C%EB%AC%B4%EC%95%84%EB%B8%8C%EB%A5%B4" title="아브라암 드무아브르 – Korean" lang="ko" hreflang="ko" data-title="아브라암 드무아브르" data-language-autonym="한국어" data-language-local-name="Korean" class="interlanguage-link-target"><span>한국어</span></a></li><li class="interlanguage-link interwiki-hy mw-list-item"><a href="https://hy.wikipedia.org/wiki/%D4%B1%D5%A2%D6%80%D5%A1%D5%B0%D5%A1%D5%B4_%D5%A4%D5%A8_%D5%84%D5%B8%D6%82%D5%A1%D5%BE%D6%80" title="Աբրահամ դը Մուավր – Armenian" lang="hy" hreflang="hy" data-title="Աբրահամ դը Մուավր" data-language-autonym="Հայերեն" data-language-local-name="Armenian" class="interlanguage-link-target"><span>Հայերեն</span></a></li><li class="interlanguage-link interwiki-hr mw-list-item"><a href="https://hr.wikipedia.org/wiki/Abraham_de_Moivre" title="Abraham de Moivre – Croatian" lang="hr" hreflang="hr" data-title="Abraham de Moivre" data-language-autonym="Hrvatski" data-language-local-name="Croatian" class="interlanguage-link-target"><span>Hrvatski</span></a></li><li class="interlanguage-link interwiki-it mw-list-item"><a href="https://it.wikipedia.org/wiki/Abraham_de_Moivre" title="Abraham de Moivre – Italian" lang="it" hreflang="it" data-title="Abraham de Moivre" data-language-autonym="Italiano" data-language-local-name="Italian" class="interlanguage-link-target"><span>Italiano</span></a></li><li class="interlanguage-link interwiki-he mw-list-item"><a href="https://he.wikipedia.org/wiki/%D7%90%D7%91%D7%A8%D7%94%D7%9D_%D7%93%D7%94_%D7%9E%D7%95%D7%90%D7%91%D7%A8" title="אברהם דה מואבר – Hebrew" lang="he" hreflang="he" data-title="אברהם דה מואבר" data-language-autonym="עברית" data-language-local-name="Hebrew" class="interlanguage-link-target"><span>עברית</span></a></li><li class="interlanguage-link interwiki-kn mw-list-item"><a href="https://kn.wikipedia.org/wiki/%E0%B2%85%E0%B2%AC%E0%B3%8D%E0%B2%B0%E0%B2%B9%E0%B2%BE%E0%B2%82_%E0%B2%A1%E0%B2%BF_%E0%B2%AE%E0%B3%8A%E0%B2%AF%E0%B3%8D%E0%B2%B5%E0%B3%8D%E2%80%8C%E0%B2%B0%E0%B3%8D" title="ಅಬ್ರಹಾಂ ಡಿ ಮೊಯ್ವ್‌ರ್ – Kannada" lang="kn" hreflang="kn" data-title="ಅಬ್ರಹಾಂ ಡಿ ಮೊಯ್ವ್‌ರ್" data-language-autonym="ಕನ್ನಡ" data-language-local-name="Kannada" class="interlanguage-link-target"><span>ಕನ್ನಡ</span></a></li><li class="interlanguage-link interwiki-ka mw-list-item"><a href="https://ka.wikipedia.org/wiki/%E1%83%90%E1%83%91%E1%83%A0%E1%83%90%E1%83%90%E1%83%9B_%E1%83%93%E1%83%94_%E1%83%9B%E1%83%A3%E1%83%90%E1%83%95%E1%83%A0%E1%83%98" title="აბრაამ დე მუავრი – Georgian" lang="ka" hreflang="ka" data-title="აბრაამ დე მუავრი" data-language-autonym="ქართული" data-language-local-name="Georgian" class="interlanguage-link-target"><span>ქართული</span></a></li><li class="interlanguage-link interwiki-ht mw-list-item"><a href="https://ht.wikipedia.org/wiki/Abraham_de_Moivre" title="Abraham de Moivre – Haitian Creole" lang="ht" hreflang="ht" data-title="Abraham de Moivre" data-language-autonym="Kreyòl ayisyen" data-language-local-name="Haitian Creole" class="interlanguage-link-target"><span>Kreyòl ayisyen</span></a></li><li class="interlanguage-link interwiki-hu mw-list-item"><a href="https://hu.wikipedia.org/wiki/Abraham_de_Moivre" title="Abraham de Moivre – Hungarian" lang="hu" hreflang="hu" data-title="Abraham de Moivre" data-language-autonym="Magyar" data-language-local-name="Hungarian" class="interlanguage-link-target"><span>Magyar</span></a></li><li class="interlanguage-link interwiki-mk mw-list-item"><a href="https://mk.wikipedia.org/wiki/%D0%90%D0%B1%D1%80%D0%B0%D1%85%D0%B0%D0%BC_%D0%B4%D0%B5_%D0%9C%D0%BE%D0%B0%D0%B2%D1%80" title="Абрахам де Моавр – Macedonian" lang="mk" hreflang="mk" data-title="Абрахам де Моавр" data-language-autonym="Македонски" data-language-local-name="Macedonian" class="interlanguage-link-target"><span>Македонски</span></a></li><li class="interlanguage-link interwiki-mr mw-list-item"><a href="https://mr.wikipedia.org/wiki/%E0%A4%86%E0%A4%AC%E0%A5%8D%E0%A4%B0%E0%A4%BE%E0%A4%AE_%E0%A4%A6_%E0%A4%AE%E0%A5%8D%E0%A4%B5%E0%A4%BE%E0%A4%B5%E0%A5%8D%E0%A4%B0" title="आब्राम द म्वाव्र – Marathi" lang="mr" hreflang="mr" data-title="आब्राम द म्वाव्र" data-language-autonym="मराठी" data-language-local-name="Marathi" class="interlanguage-link-target"><span>मराठी</span></a></li><li class="interlanguage-link interwiki-arz mw-list-item"><a href="https://arz.wikipedia.org/wiki/%D8%A7%D8%A8%D8%B1%D8%A7%D9%87%D8%A7%D9%85_%D8%AF%D9%89_%D9%85%D9%88%D8%A7%D9%81%D8%B1" title="ابراهام دى موافر – Egyptian Arabic" lang="arz" hreflang="arz" data-title="ابراهام دى موافر" data-language-autonym="مصرى" data-language-local-name="Egyptian Arabic" class="interlanguage-link-target"><span>مصرى</span></a></li><li class="interlanguage-link interwiki-ms mw-list-item"><a href="https://ms.wikipedia.org/wiki/Abraham_de_Moivre" title="Abraham de Moivre – Malay" lang="ms" hreflang="ms" data-title="Abraham de Moivre" data-language-autonym="Bahasa Melayu" data-language-local-name="Malay" class="interlanguage-link-target"><span>Bahasa Melayu</span></a></li><li class="interlanguage-link interwiki-mn mw-list-item"><a href="https://mn.wikipedia.org/wiki/%D0%90%D0%B1%D1%80%D0%B0%D1%85%D0%B0%D0%BC_%D0%B4%D0%B5_%D0%9C%D1%83%D0%B0%D0%B2%D1%80" title="Абрахам де Муавр – Mongolian" lang="mn" hreflang="mn" data-title="Абрахам де Муавр" data-language-autonym="Монгол" data-language-local-name="Mongolian" class="interlanguage-link-target"><span>Монгол</span></a></li><li class="interlanguage-link interwiki-nl mw-list-item"><a href="https://nl.wikipedia.org/wiki/Abraham_de_Moivre" title="Abraham de Moivre – Dutch" lang="nl" hreflang="nl" data-title="Abraham de Moivre" data-language-autonym="Nederlands" data-language-local-name="Dutch" class="interlanguage-link-target"><span>Nederlands</span></a></li><li class="interlanguage-link interwiki-ja mw-list-item"><a href="https://ja.wikipedia.org/wiki/%E3%82%A2%E3%83%96%E3%83%A9%E3%83%BC%E3%83%A0%E3%83%BB%E3%83%89%E3%83%BB%E3%83%A2%E3%82%A2%E3%83%96%E3%83%AB" title="アブラーム・ド・モアブル – Japanese" lang="ja" hreflang="ja" data-title="アブラーム・ド・モアブル" data-language-autonym="日本語" data-language-local-name="Japanese" class="interlanguage-link-target"><span>日本語</span></a></li><li class="interlanguage-link interwiki-no mw-list-item"><a href="https://no.wikipedia.org/wiki/Abraham_de_Moivre" title="Abraham de Moivre – Norwegian Bokmål" lang="nb" hreflang="nb" data-title="Abraham de Moivre" data-language-autonym="Norsk bokmål" data-language-local-name="Norwegian Bokmål" class="interlanguage-link-target"><span>Norsk bokmål</span></a></li><li class="interlanguage-link interwiki-nn mw-list-item"><a href="https://nn.wikipedia.org/wiki/Abraham_de_Moivre" title="Abraham de Moivre – Norwegian Nynorsk" lang="nn" hreflang="nn" data-title="Abraham de Moivre" data-language-autonym="Norsk nynorsk" data-language-local-name="Norwegian Nynorsk" class="interlanguage-link-target"><span>Norsk nynorsk</span></a></li><li class="interlanguage-link interwiki-pms mw-list-item"><a href="https://pms.wikipedia.org/wiki/Abraham_De_Moivre" title="Abraham De Moivre – Piedmontese" lang="pms" hreflang="pms" data-title="Abraham De Moivre" data-language-autonym="Piemontèis" data-language-local-name="Piedmontese" class="interlanguage-link-target"><span>Piemontèis</span></a></li><li class="interlanguage-link interwiki-pl mw-list-item"><a href="https://pl.wikipedia.org/wiki/Abraham_de_Moivre" title="Abraham de Moivre – Polish" lang="pl" hreflang="pl" data-title="Abraham de Moivre" data-language-autonym="Polski" data-language-local-name="Polish" class="interlanguage-link-target"><span>Polski</span></a></li><li class="interlanguage-link interwiki-pt mw-list-item"><a href="https://pt.wikipedia.org/wiki/Abraham_de_Moivre" title="Abraham de Moivre – Portuguese" lang="pt" hreflang="pt" data-title="Abraham de Moivre" data-language-autonym="Português" data-language-local-name="Portuguese" class="interlanguage-link-target"><span>Português</span></a></li><li class="interlanguage-link interwiki-ro mw-list-item"><a href="https://ro.wikipedia.org/wiki/Abraham_de_Moivre" title="Abraham de Moivre – Romanian" lang="ro" hreflang="ro" data-title="Abraham de Moivre" data-language-autonym="Română" data-language-local-name="Romanian" class="interlanguage-link-target"><span>Română</span></a></li><li class="interlanguage-link interwiki-ru mw-list-item"><a href="https://ru.wikipedia.org/wiki/%D0%9C%D1%83%D0%B0%D0%B2%D1%80,_%D0%90%D0%B1%D1%80%D0%B0%D1%85%D0%B0%D0%BC_%D0%B4%D0%B5" title="Муавр, Абрахам де – Russian" lang="ru" hreflang="ru" data-title="Муавр, Абрахам де" data-language-autonym="Русский" data-language-local-name="Russian" class="interlanguage-link-target"><span>Русский</span></a></li><li class="interlanguage-link interwiki-skr mw-list-item"><a href="https://skr.wikipedia.org/wiki/%D8%A7%D8%A8%D8%B1%D8%A7%DB%81%D8%A7%D9%85_%DA%88%DB%8C_%D9%85%D9%88%D8%A7%D9%81%D8%B1" title="ابراہام ڈی موافر – Saraiki" lang="skr" hreflang="skr" data-title="ابراہام ڈی موافر" data-language-autonym="سرائیکی" data-language-local-name="Saraiki" class="interlanguage-link-target"><span>سرائیکی</span></a></li><li class="interlanguage-link interwiki-simple mw-list-item"><a href="https://simple.wikipedia.org/wiki/Abraham_de_Moivre" title="Abraham de Moivre – Simple English" lang="en-simple" hreflang="en-simple" data-title="Abraham de Moivre" data-language-autonym="Simple English" data-language-local-name="Simple English" class="interlanguage-link-target"><span>Simple English</span></a></li><li class="interlanguage-link interwiki-sk mw-list-item"><a href="https://sk.wikipedia.org/wiki/Abraham_de_Moivre" title="Abraham de Moivre – Slovak" lang="sk" hreflang="sk" data-title="Abraham de Moivre" data-language-autonym="Slovenčina" data-language-local-name="Slovak" class="interlanguage-link-target"><span>Slovenčina</span></a></li><li class="interlanguage-link interwiki-sl mw-list-item"><a href="https://sl.wikipedia.org/wiki/Abraham_de_Moivre" title="Abraham de Moivre – Slovenian" lang="sl" hreflang="sl" data-title="Abraham de Moivre" data-language-autonym="Slovenščina" data-language-local-name="Slovenian" class="interlanguage-link-target"><span>Slovenščina</span></a></li><li class="interlanguage-link interwiki-sr mw-list-item"><a href="https://sr.wikipedia.org/wiki/%D0%90%D0%B1%D1%80%D0%B0%D0%BC_%D0%B4%D0%B5_%D0%9C%D0%BE%D0%B0%D0%B2%D1%80" title="Абрам де Моавр – Serbian" lang="sr" hreflang="sr" data-title="Абрам де Моавр" data-language-autonym="Српски / srpski" data-language-local-name="Serbian" class="interlanguage-link-target"><span>Српски / srpski</span></a></li><li class="interlanguage-link interwiki-fi mw-list-item"><a href="https://fi.wikipedia.org/wiki/Abraham_de_Moivre" title="Abraham de Moivre – Finnish" lang="fi" hreflang="fi" data-title="Abraham de Moivre" data-language-autonym="Suomi" data-language-local-name="Finnish" class="interlanguage-link-target"><span>Suomi</span></a></li><li class="interlanguage-link interwiki-sv mw-list-item"><a href="https://sv.wikipedia.org/wiki/Abraham_de_Moivre" title="Abraham de Moivre – Swedish" lang="sv" hreflang="sv" data-title="Abraham de Moivre" data-language-autonym="Svenska" data-language-local-name="Swedish" class="interlanguage-link-target"><span>Svenska</span></a></li><li class="interlanguage-link interwiki-tr mw-list-item"><a href="https://tr.wikipedia.org/wiki/Abraham_de_Moivre" title="Abraham de Moivre – Turkish" lang="tr" hreflang="tr" data-title="Abraham de Moivre" data-language-autonym="Türkçe" data-language-local-name="Turkish" class="interlanguage-link-target"><span>Türkçe</span></a></li><li class="interlanguage-link interwiki-uk mw-list-item"><a href="https://uk.wikipedia.org/wiki/%D0%90%D0%B1%D1%80%D0%B0%D1%85%D0%B0%D0%BC_%D0%B4%D0%B5_%D0%9C%D1%83%D0%B0%D0%B2%D1%80" title="Абрахам де Муавр – Ukrainian" lang="uk" hreflang="uk" data-title="Абрахам де Муавр" data-language-autonym="Українська" data-language-local-name="Ukrainian" class="interlanguage-link-target"><span>Українська</span></a></li><li class="interlanguage-link interwiki-ur mw-list-item"><a href="https://ur.wikipedia.org/wiki/%D8%A7%D8%A8%D8%B1%D8%A7%DB%81%D9%85_%DA%88%DB%8C_%D9%85%D9%88%D8%A7%D9%81%D8%B1" title="ابراہم ڈی موافر – Urdu" lang="ur" hreflang="ur" data-title="ابراہم ڈی موافر" data-language-autonym="اردو" data-language-local-name="Urdu" class="interlanguage-link-target"><span>اردو</span></a></li><li class="interlanguage-link interwiki-vi mw-list-item"><a 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</div> </div> <div id="bodyContent" class="vector-body" aria-labelledby="firstHeading" data-mw-ve-target-container> <div class="vector-body-before-content"> <div class="mw-indicators"> </div> <div id="siteSub" class="noprint">From Wikipedia, the free encyclopedia</div> </div> <div id="contentSub"><div id="mw-content-subtitle"></div></div> <div id="mw-content-text" class="mw-body-content"><div class="mw-content-ltr mw-parser-output" lang="en" dir="ltr"><div class="shortdescription nomobile noexcerpt noprint searchaux" style="display:none">French mathematician (1667–1754)</div> <p class="mw-empty-elt"> </p> <style data-mw-deduplicate="TemplateStyles:r1257001546">.mw-parser-output .infobox-subbox{padding:0;border:none;margin:-3px;width:auto;min-width:100%;font-size:100%;clear:none;float:none;background-color:transparent}.mw-parser-output .infobox-3cols-child{margin:auto}.mw-parser-output .infobox .navbar{font-size:100%}@media screen{html.skin-theme-clientpref-night .mw-parser-output .infobox-full-data:not(.notheme)>div:not(.notheme)[style]{background:#1f1f23!important;color:#f8f9fa}}@media screen and (prefers-color-scheme:dark){html.skin-theme-clientpref-os .mw-parser-output .infobox-full-data:not(.notheme) div:not(.notheme){background:#1f1f23!important;color:#f8f9fa}}@media(min-width:640px){body.skin--responsive .mw-parser-output .infobox-table{display:table!important}body.skin--responsive .mw-parser-output .infobox-table>caption{display:table-caption!important}body.skin--responsive .mw-parser-output .infobox-table>tbody{display:table-row-group}body.skin--responsive .mw-parser-output .infobox-table tr{display:table-row!important}body.skin--responsive .mw-parser-output .infobox-table th,body.skin--responsive .mw-parser-output .infobox-table td{padding-left:inherit;padding-right:inherit}}</style><table class="infobox biography vcard"><tbody><tr><th colspan="2" class="infobox-above" style="font-size:125%;"><div class="fn">Abraham de Moivre</div><div class="honorific-suffix" style="font-size: 77%; font-weight: normal;"><span class="nobold noexcerpt nowraplinks" style="font-size:;"><span style="font-size: 100%;"><a href="/wiki/Fellow_of_the_Royal_Society" title="Fellow of the Royal Society">FRS</a></span></span></div></th></tr><tr><td colspan="2" class="infobox-image"><span class="mw-default-size" typeof="mw:File/Frameless"><a href="/wiki/File:Abraham_de_moivre.jpg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/1/1b/Abraham_de_moivre.jpg/220px-Abraham_de_moivre.jpg" decoding="async" width="220" height="280" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/1/1b/Abraham_de_moivre.jpg/330px-Abraham_de_moivre.jpg 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/1/1b/Abraham_de_moivre.jpg/440px-Abraham_de_moivre.jpg 2x" data-file-width="523" data-file-height="666" /></a></span></td></tr><tr><th scope="row" class="infobox-label">Born</th><td class="infobox-data"><span style="display:none">(<span class="bday">1667-05-26</span>)</span>26 May 1667<br /><div style="display:inline" class="birthplace"><a href="/wiki/Vitry-le-Fran%C3%A7ois" title="Vitry-le-François">Vitry-le-François</a>, <a href="/wiki/Kingdom_of_France" title="Kingdom of France">Kingdom of France</a></div></td></tr><tr><th scope="row" class="infobox-label">Died</th><td class="infobox-data">27 November 1754<span style="display:none">(1754-11-27)</span> (aged 87)<br /><div style="display:inline" class="deathplace"><a href="/wiki/London" title="London">London</a>, England</div></td></tr><tr><th scope="row" class="infobox-label">Alma mater</th><td class="infobox-data"><a href="/wiki/Academy_of_Saumur" title="Academy of Saumur">Academy of Saumur</a><br /><a href="/wiki/Coll%C3%A8ge_d%27Harcourt" class="mw-redirect" title="Collège d'Harcourt">Collège d'Harcourt</a><span class="noprint" style="font-size:85%; font-style: normal;"> [<a href="https://fr.wikipedia.org/wiki/Coll%C3%A8ge_d%27Harcourt" class="extiw" title="fr:Collège d'Harcourt">fr</a>]</span></td></tr><tr><th scope="row" class="infobox-label">Known for</th><td class="infobox-data"><a href="/wiki/De_Moivre%27s_formula" title="De Moivre's formula">De Moivre's formula</a><br /><a href="/wiki/De_Moivre%27s_law" title="De Moivre's law">De Moivre's law</a><br /><a href="/wiki/Martingale_(probability_theory)#Examples_of_submartingales_and_supermartingales" title="Martingale (probability theory)">De Moivre's martingale</a><br /><a href="/wiki/De_Moivre%E2%80%93Laplace_theorem" title="De Moivre–Laplace theorem">De Moivre–Laplace theorem</a><br /><a href="/wiki/Inclusion%E2%80%93exclusion_principle" title="Inclusion–exclusion principle">Inclusion–exclusion principle</a><br /><a href="/wiki/Generating_function" title="Generating function">Generating function</a></td></tr><tr><td colspan="2" class="infobox-full-data"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1257001546"><b>Scientific career</b></td></tr><tr><th scope="row" class="infobox-label">Fields</th><td class="infobox-data category"><a href="/wiki/Mathematics" title="Mathematics">Mathematics</a></td></tr><tr style="display:none"><td colspan="2"> </td></tr></tbody></table> <p><b>Abraham de Moivre</b> <a href="/wiki/Fellow_of_the_Royal_Society" title="Fellow of the Royal Society">FRS</a> (<style data-mw-deduplicate="TemplateStyles:r1177148991">.mw-parser-output .IPA-label-small{font-size:85%}.mw-parser-output .references .IPA-label-small,.mw-parser-output .infobox .IPA-label-small,.mw-parser-output .navbox .IPA-label-small{font-size:100%}</style><span class="IPA-label IPA-label-small">French pronunciation:</span> <span class="IPA nowrap" lang="fr-Latn-fonipa"><a href="/wiki/Help:IPA/French" title="Help:IPA/French">[abʁaam<span class="wrap"> </span>də<span class="wrap"> </span>mwavʁ]</a></span>; 26 May 1667 – 27 November 1754) was a French mathematician known for <a href="/wiki/De_Moivre%27s_formula" title="De Moivre's formula">de Moivre's formula</a>, a formula that links <a href="/wiki/Complex_number" title="Complex number">complex numbers</a> and <a href="/wiki/Trigonometry" title="Trigonometry">trigonometry</a>, and for his work on the <a href="/wiki/Normal_distribution" title="Normal distribution">normal distribution</a> and <a href="/wiki/Probability_theory" title="Probability theory">probability theory</a>. </p><p>He moved to <a href="/wiki/England" title="England">England</a> at a young age due to the religious persecution of <a href="/wiki/Huguenot" class="mw-redirect" title="Huguenot">Huguenots</a> in France which reached a climax in 1685 with the <a href="/wiki/Edict_of_Fontainebleau" title="Edict of Fontainebleau">Edict of Fontainebleau</a>.<sup id="cite_ref-mactutor_1-0" class="reference"><a href="#cite_note-mactutor-1"><span class="cite-bracket">[</span>1<span class="cite-bracket">]</span></a></sup> He was a friend of <a href="/wiki/Isaac_Newton" title="Isaac Newton">Isaac Newton</a>, <a href="/wiki/Edmond_Halley" title="Edmond Halley">Edmond Halley</a>, and <a href="/wiki/James_Stirling_(mathematician)" title="James Stirling (mathematician)">James Stirling</a>. Among his fellow Huguenot exiles in England, he was a colleague of the editor and translator <a href="/wiki/Pierre_des_Maizeaux" title="Pierre des Maizeaux">Pierre des Maizeaux</a>. </p><p>De Moivre wrote a book on <a href="/wiki/Probability_theory" title="Probability theory">probability theory</a>, <i><a href="/wiki/The_Doctrine_of_Chances" title="The Doctrine of Chances">The Doctrine of Chances</a></i>, said to have been prized by gamblers. De Moivre first discovered <a href="/wiki/Binet%27s_formula" class="mw-redirect" title="Binet's formula">Binet's formula</a>, the <a href="/wiki/Closed-form_expression" title="Closed-form expression">closed-form expression</a> for <a href="/wiki/Fibonacci_numbers" class="mw-redirect" title="Fibonacci numbers">Fibonacci numbers</a> linking the <i>n</i>th power of the <a href="/wiki/Golden_ratio" title="Golden ratio">golden ratio</a> <i>φ</i> to the <i>n</i>th Fibonacci number. He also was the first to postulate the <a href="/wiki/Central_limit_theorem" title="Central limit theorem">central limit theorem</a>, a cornerstone of probability theory. </p> <meta property="mw:PageProp/toc" /> <div class="mw-heading mw-heading2"><h2 id="Life">Life</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Abraham_de_Moivre&action=edit&section=1" title="Edit section: Life"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <figure class="mw-default-size" typeof="mw:File/Thumb"><a href="/wiki/File:Moivre_-_Doctrine_of_chances,_1761_-_722666.tif" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/0/0a/Moivre_-_Doctrine_of_chances%2C_1761_-_722666.tif/lossy-page1-220px-Moivre_-_Doctrine_of_chances%2C_1761_-_722666.tif.jpg" decoding="async" width="220" height="279" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/0/0a/Moivre_-_Doctrine_of_chances%2C_1761_-_722666.tif/lossy-page1-330px-Moivre_-_Doctrine_of_chances%2C_1761_-_722666.tif.jpg 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/0/0a/Moivre_-_Doctrine_of_chances%2C_1761_-_722666.tif/lossy-page1-440px-Moivre_-_Doctrine_of_chances%2C_1761_-_722666.tif.jpg 2x" data-file-width="3668" data-file-height="4660" /></a><figcaption><i>Doctrine of chances</i>, 1756</figcaption></figure> <div class="mw-heading mw-heading3"><h3 id="Early_years">Early years</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Abraham_de_Moivre&action=edit&section=2" title="Edit section: Early years"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>Abraham de Moivre was born in <a href="/wiki/Vitry-le-Fran%C3%A7ois" title="Vitry-le-François">Vitry-le-François</a> in <a href="/wiki/Champagne_(province)" title="Champagne (province)">Champagne</a> on 26 May 1667. His father, Daniel de Moivre, was a surgeon who believed in the value of education. Though Abraham de Moivre's parents were Protestant, he first attended Christian Brothers' Catholic school in Vitry, which was unusually tolerant given religious tensions in France at the time. When he was eleven, his parents sent him to the Protestant Academy at <a href="/wiki/Sedan,_Ardennes" title="Sedan, Ardennes">Sedan</a>, where he spent four years studying <a href="/wiki/Greek_language" title="Greek language">Greek</a> under Jacques du Rondel. The Protestant <a href="/wiki/Academy_of_Sedan" title="Academy of Sedan">Academy of Sedan</a> had been founded in 1579 at the initiative of Françoise de Bourbon, the widow of Henri-Robert de la Marck. </p><p>In 1682 the Protestant Academy at <a href="/wiki/Sedan,_Ardennes" title="Sedan, Ardennes">Sedan</a> was suppressed, and de Moivre enrolled to study logic at <a href="/wiki/Saumur" title="Saumur">Saumur</a> for two years. Although mathematics was not part of his course work, de Moivre read several works on mathematics on his own, including Éléments des mathématiques by the French Oratorian priest and mathematician <a href="/wiki/Jean_Prestet" title="Jean Prestet">Jean Prestet</a> and a short treatise on games of chance, <i>De Ratiociniis in Ludo Aleae</i>, by <a href="/wiki/Christiaan_Huygens" title="Christiaan Huygens">Christiaan Huygens</a> the Dutch physicist, mathematician, astronomer and inventor. In 1684, de Moivre moved to Paris to study physics, and for the first time had formal mathematics training with private lessons from <a href="/wiki/Jacques_Ozanam" title="Jacques Ozanam">Jacques Ozanam</a>. </p><p>Religious persecution in France became severe when <a href="/wiki/King_Louis_XIV" class="mw-redirect" title="King Louis XIV">King Louis XIV</a> issued the <a href="/wiki/Edict_of_Fontainebleau" title="Edict of Fontainebleau">Edict of Fontainebleau</a> in 1685, which revoked the <a href="/wiki/Edict_of_Nantes" title="Edict of Nantes">Edict of Nantes</a>, that had given substantial rights to French Protestants. It forbade Protestant worship and required that all children be baptised by Catholic priests. De Moivre was sent to Prieuré Saint-Martin-des-Champs, a school that the authorities sent Protestant children to for indoctrination into Catholicism. </p><p>It is unclear when de Moivre left the Prieure de Saint-Martin and moved to England, since the records of the Prieure de Saint-Martin indicate that he left the school in 1688, but de Moivre and his brother presented themselves as Huguenots admitted to the Savoy Church in London on 28 August 1687. </p> <div class="mw-heading mw-heading3"><h3 id="Middle_years">Middle years</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Abraham_de_Moivre&action=edit&section=3" title="Edit section: Middle years"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>By the time he arrived in London, de Moivre was a competent mathematician with a good knowledge of many of the standard texts.<sup id="cite_ref-mactutor_1-1" class="reference"><a href="#cite_note-mactutor-1"><span class="cite-bracket">[</span>1<span class="cite-bracket">]</span></a></sup> To make a living, de Moivre became a private tutor of <a href="/wiki/Mathematics" title="Mathematics">mathematics</a>, visiting his pupils or teaching in the coffee houses of London. De Moivre continued his studies of mathematics after visiting the <a href="/wiki/Earl_of_Devonshire" title="Earl of Devonshire">Earl of Devonshire</a> and seeing Newton's recent book, <a href="/wiki/Philosophi%C3%A6_Naturalis_Principia_Mathematica" title="Philosophiæ Naturalis Principia Mathematica"><i>Principia Mathematica</i></a>. Looking through the book, he realised that it was far deeper than the books that he had studied previously, and he became determined to read and understand it. However, as he was required to take extended walks around London to travel between his students, de Moivre had little time for study, so he tore pages from the book and carried them around in his pocket to read between lessons. </p><p>According to a possibly apocryphal story, Newton, in the later years of his life, used to refer people posing mathematical questions to him to de Moivre, saying, "He knows all these things better than I do."<sup id="cite_ref-2" class="reference"><a href="#cite_note-2"><span class="cite-bracket">[</span>2<span class="cite-bracket">]</span></a></sup> </p><p>By 1692, de Moivre became friends with <a href="/wiki/Edmond_Halley" title="Edmond Halley">Edmond Halley</a> and soon after with <a href="/wiki/Isaac_Newton" title="Isaac Newton">Isaac Newton</a> himself. In 1695, Halley communicated de Moivre's first mathematics paper, which arose from his study of <a href="/wiki/Newton%27s_notation" class="mw-redirect" title="Newton's notation">fluxions</a> in the <i>Principia Mathematica</i>, to the <a href="/wiki/Royal_Society" title="Royal Society">Royal Society</a>. This paper was published in the <i>Philosophical Transactions</i> that same year. Shortly after publishing this paper, de Moivre also generalised Newton's noteworthy <a href="/wiki/Binomial_theorem" title="Binomial theorem">binomial theorem</a> into the <a href="/wiki/Multinomial_theorem" title="Multinomial theorem">multinomial theorem</a>. The <a href="/wiki/Royal_Society" title="Royal Society">Royal Society</a> became apprised of this method in 1697, and it elected de Moivre a Fellow on 30 November 1697. </p><p>After de Moivre had been accepted, Halley encouraged him to turn his attention to astronomy. In 1705, de Moivre discovered, intuitively, that "the centripetal force of any planet is directly related to its distance from the centre of the forces and reciprocally related to the product of the diameter of the evolute and the cube of the perpendicular on the tangent." In other words, if a planet, M, follows an elliptical orbit around a focus F and has a point P where PM is tangent to the curve and FPM is a right angle so that FP is the perpendicular to the tangent, then the centripetal force at point P is proportional to FM/(R*(FP)<sup>3</sup>) where R is the radius of the curvature at M. The mathematician <a href="/wiki/Johann_Bernoulli" title="Johann Bernoulli">Johann Bernoulli</a> proved this formula in 1710. </p><p>Despite these successes, de Moivre was unable to obtain an appointment to a chair of mathematics at any university, which would have released him from his dependence on time-consuming tutoring that burdened him more than it did most other mathematicians of the time. At least a part of the reason was a bias against his French origins.<sup id="cite_ref-3" class="reference"><a href="#cite_note-3"><span class="cite-bracket">[</span>3<span class="cite-bracket">]</span></a></sup><sup id="cite_ref-4" class="reference"><a href="#cite_note-4"><span class="cite-bracket">[</span>4<span class="cite-bracket">]</span></a></sup><sup id="cite_ref-5" class="reference"><a href="#cite_note-5"><span class="cite-bracket">[</span>5<span class="cite-bracket">]</span></a></sup> </p><p>In November 1697 he was elected a <a href="/wiki/Fellow_of_the_Royal_Society" title="Fellow of the Royal Society">Fellow of the Royal Society</a><sup id="cite_ref-mactutor_1-2" class="reference"><a href="#cite_note-mactutor-1"><span class="cite-bracket">[</span>1<span class="cite-bracket">]</span></a></sup> and in 1712 was appointed to a commission set up by the society, alongside MM. Arbuthnot, Hill, Halley, Jones, Machin, Burnet, Robarts, Bonet, Aston, and Taylor to review the claims of Newton and Leibniz as to who discovered calculus. The full details of the controversy can be found in the <a href="/wiki/Leibniz_and_Newton_calculus_controversy" class="mw-redirect" title="Leibniz and Newton calculus controversy">Leibniz and Newton calculus controversy</a> article. </p><p>Throughout his life de Moivre remained poor. It is reported that he was a regular customer of <a href="/wiki/Old_Slaughter%27s_Coffee_House" title="Old Slaughter's Coffee House">old Slaughter's Coffee House</a>, St. Martin's Lane at Cranbourn Street, where he earned a little money from playing chess. </p> <div class="mw-heading mw-heading3"><h3 id="Later_years">Later years</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Abraham_de_Moivre&action=edit&section=4" title="Edit section: Later years"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>De Moivre continued studying the fields of probability and mathematics until his death in 1754 and several additional papers were published after his death. As he grew older, he became increasingly <a href="/wiki/Lethargic" class="mw-redirect" title="Lethargic">lethargic</a> and needed longer sleeping hours. It is a common claim that De Moivre noted he was sleeping an extra 15 minutes each night and correctly calculated the date of his death as the day when the sleep time reached 24 hours, 27 November 1754.<sup id="cite_ref-Cajori-History_6-0" class="reference"><a href="#cite_note-Cajori-History-6"><span class="cite-bracket">[</span>6<span class="cite-bracket">]</span></a></sup> On that day he did in fact die, in London and his body was buried at <a href="/wiki/St_Martin-in-the-Fields" title="St Martin-in-the-Fields">St Martin-in-the-Fields</a>, although his body was later moved. The claim of him predicting his own death, however, has been disputed as not having been documented anywhere at the time of its occurrence.<sup id="cite_ref-hsmstex_7-0" class="reference"><a href="#cite_note-hsmstex-7"><span class="cite-bracket">[</span>7<span class="cite-bracket">]</span></a></sup> </p> <div class="mw-heading mw-heading2"><h2 id="Probability">Probability</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Abraham_de_Moivre&action=edit&section=5" title="Edit section: Probability"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <style data-mw-deduplicate="TemplateStyles:r1236090951">.mw-parser-output .hatnote{font-style:italic}.mw-parser-output div.hatnote{padding-left:1.6em;margin-bottom:0.5em}.mw-parser-output .hatnote i{font-style:normal}.mw-parser-output .hatnote+link+.hatnote{margin-top:-0.5em}@media print{body.ns-0 .mw-parser-output .hatnote{display:none!important}}</style><div role="note" class="hatnote navigation-not-searchable">See also: <a href="/wiki/De_Moivre%E2%80%93Laplace_theorem" title="De Moivre–Laplace theorem">de Moivre–Laplace theorem</a></div> <p>De Moivre pioneered the development of analytic geometry and the theory of probability by expanding upon the work of his predecessors, particularly Christiaan Huygens and several members of the Bernoulli family. He also produced the second textbook on probability theory, <a href="/wiki/The_Doctrine_of_Chances" title="The Doctrine of Chances"><i>The Doctrine of Chances: a method of calculating the probabilities of events in play</i></a>. (The first book about games of chance, <i>Liber de ludo aleae</i> (<i>On Casting the Die</i>), was written by <a href="/wiki/Girolamo_Cardano" class="mw-redirect" title="Girolamo Cardano">Girolamo Cardano</a> in the 1560s, but it was not published until 1663.) This book came out in four editions, 1711 in Latin, and in English in 1718, 1738, and 1756. In the later editions of his book, de Moivre included his unpublished result of 1733, which is the first statement of an approximation to the binomial distribution in terms of what we now call the normal or <a href="/wiki/Gaussian_function" title="Gaussian function">Gaussian function</a>.<sup id="cite_ref-8" class="reference"><a href="#cite_note-8"><span class="cite-bracket">[</span>8<span class="cite-bracket">]</span></a></sup> This was the first method of finding the probability of the occurrence of an error of a given size when that error is expressed in terms of the variability of the distribution as a unit, and the first identification of the calculation of <a href="/wiki/Probable_error" title="Probable error">probable error</a>. In addition, he applied these theories to gambling problems and <a href="/wiki/Actuarial_table" class="mw-redirect" title="Actuarial table">actuarial tables</a>. </p><p>An expression commonly found in probability is <i>n</i>! but before the days of calculators calculating <i>n</i>! for a large <i>n</i> was time-consuming. In 1733 de Moivre proposed the formula for estimating a factorial as <i>n</i>! = <i>cn</i><sup>(<i>n</i>+1/2)</sup><i>e</i><sup>−<i>n</i></sup>. He obtained an approximate expression for the constant <i>c</i> but it was <a href="/wiki/James_Stirling_(mathematician)" title="James Stirling (mathematician)">James Stirling</a> who found that c was <span class="nowrap">√<span style="border-top:1px solid; padding:0 0.1em;">2<span class="texhtml mvar" style="font-style:italic;">π</span></span></span>.<sup id="cite_ref-9" class="reference"><a href="#cite_note-9"><span class="cite-bracket">[</span>9<span class="cite-bracket">]</span></a></sup> </p><p>De Moivre also published an article called "Annuities upon Lives" in which he revealed the normal distribution of the mortality rate over a person's age. From this he produced a simple formula for approximating the revenue produced by annual payments based on a person's age. This is similar to the types of formulas used by insurance companies today. </p> <div class="mw-heading mw-heading3"><h3 id="Priority_regarding_the_Poisson_distribution">Priority regarding the Poisson distribution</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Abraham_de_Moivre&action=edit&section=6" title="Edit section: Priority regarding the Poisson distribution"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>Some results on the <a href="/wiki/Poisson_distribution" title="Poisson distribution">Poisson distribution</a> were first introduced by de Moivre in <i>De Mensura Sortis seu; de Probabilitate Eventuum in Ludis a Casu Fortuito Pendentibus</i> in Philosophical Transactions of the Royal Society, p. 219.<sup id="cite_ref-JKK157_10-0" class="reference"><a href="#cite_note-JKK157-10"><span class="cite-bracket">[</span>10<span class="cite-bracket">]</span></a></sup> As a result, some authors have argued that the Poisson distribution should bear the name of de Moivre.<sup id="cite_ref-11" class="reference"><a href="#cite_note-11"><span class="cite-bracket">[</span>11<span class="cite-bracket">]</span></a></sup><sup id="cite_ref-12" class="reference"><a href="#cite_note-12"><span class="cite-bracket">[</span>12<span class="cite-bracket">]</span></a></sup> </p> <div class="mw-heading mw-heading2"><h2 id="De_Moivre's_formula"><span id="De_Moivre.27s_formula"></span>De Moivre's formula</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Abraham_de_Moivre&action=edit&section=7" title="Edit section: De Moivre's formula"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>In 1707, de Moivre derived an equation from which one can deduce: </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \cos x={\tfrac {1}{2}}(\cos(nx)+i\sin(nx))^{1/n}+{\tfrac {1}{2}}(\cos(nx)-i\sin(nx))^{1/n}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>cos</mi> <mo>⁡</mo> <mi>x</mi> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mfrac> <mn>1</mn> <mn>2</mn> </mfrac> </mstyle> </mrow> <mo stretchy="false">(</mo> <mi>cos</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mi>n</mi> <mi>x</mi> <mo stretchy="false">)</mo> <mo>+</mo> <mi>i</mi> <mi>sin</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mi>n</mi> <mi>x</mi> <mo stretchy="false">)</mo> <msup> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <mi>n</mi> </mrow> </msup> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mfrac> <mn>1</mn> <mn>2</mn> </mfrac> </mstyle> </mrow> <mo stretchy="false">(</mo> <mi>cos</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mi>n</mi> <mi>x</mi> <mo stretchy="false">)</mo> <mo>−</mo> <mi>i</mi> <mi>sin</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mi>n</mi> <mi>x</mi> <mo stretchy="false">)</mo> <msup> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <mi>n</mi> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \cos x={\tfrac {1}{2}}(\cos(nx)+i\sin(nx))^{1/n}+{\tfrac {1}{2}}(\cos(nx)-i\sin(nx))^{1/n}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ef48a808b32996fb0a68f4e303008e8d9fa4159f" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.171ex; width:61.555ex; height:3.676ex;" alt="{\displaystyle \cos x={\tfrac {1}{2}}(\cos(nx)+i\sin(nx))^{1/n}+{\tfrac {1}{2}}(\cos(nx)-i\sin(nx))^{1/n}}"></span></dd></dl> <p>which he was able to prove for all positive <a href="/wiki/Integer" title="Integer">integers</a> <i>n</i>.<sup id="cite_ref-13" class="reference"><a href="#cite_note-13"><span class="cite-bracket">[</span>13<span class="cite-bracket">]</span></a></sup><sup id="cite_ref-14" class="reference"><a href="#cite_note-14"><span class="cite-bracket">[</span>14<span class="cite-bracket">]</span></a></sup> In 1722, he presented equations from which one can deduce the better known form of <a href="/wiki/De_Moivre%27s_Formula" class="mw-redirect" title="De Moivre's Formula">de Moivre's Formula</a>: </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle (\cos x+i\sin x)^{n}=\cos(nx)+i\sin(nx).\,}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">(</mo> <mi>cos</mi> <mo>⁡</mo> <mi>x</mi> <mo>+</mo> <mi>i</mi> <mi>sin</mi> <mo>⁡</mo> <mi>x</mi> <msup> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msup> <mo>=</mo> <mi>cos</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mi>n</mi> <mi>x</mi> <mo stretchy="false">)</mo> <mo>+</mo> <mi>i</mi> <mi>sin</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mi>n</mi> <mi>x</mi> <mo stretchy="false">)</mo> <mo>.</mo> <mspace width="thinmathspace" /> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle (\cos x+i\sin x)^{n}=\cos(nx)+i\sin(nx).\,}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ab4857121e23068ff095a8a6abcb189b8b1ee4c2" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:39.654ex; height:2.843ex;" alt="{\displaystyle (\cos x+i\sin x)^{n}=\cos(nx)+i\sin(nx).\,}"></span><sup id="cite_ref-15" class="reference"><a href="#cite_note-15"><span class="cite-bracket">[</span>15<span class="cite-bracket">]</span></a></sup><sup id="cite_ref-16" class="reference"><a href="#cite_note-16"><span class="cite-bracket">[</span>16<span class="cite-bracket">]</span></a></sup></dd></dl> <p>In 1749 Euler proved this formula for any real n using <a href="/wiki/Euler%27s_formula" title="Euler's formula">Euler's formula</a>, which makes the proof quite straightforward.<sup id="cite_ref-17" class="reference"><a href="#cite_note-17"><span class="cite-bracket">[</span>17<span class="cite-bracket">]</span></a></sup> This formula is important because it relates <a href="/wiki/Complex_numbers" class="mw-redirect" title="Complex numbers">complex numbers</a> and <a href="/wiki/Trigonometry" title="Trigonometry">trigonometry</a>. Additionally, this formula allows the derivation of useful expressions for cos(<i>nx</i>) and sin(<i>nx</i>) in terms of cos(<i>x</i>) and sin(<i>x</i>). </p> <div class="mw-heading mw-heading2"><h2 id="Stirling's_approximation"><span id="Stirling.27s_approximation"></span>Stirling's approximation</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Abraham_de_Moivre&action=edit&section=8" title="Edit section: Stirling's approximation"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>De Moivre had been studying probability, and his investigations required him to calculate binomial coefficients, which in turn required him to calculate factorials.<sup id="cite_ref-18" class="reference"><a href="#cite_note-18"><span class="cite-bracket">[</span>18<span class="cite-bracket">]</span></a></sup><sup id="cite_ref-19" class="reference"><a href="#cite_note-19"><span class="cite-bracket">[</span>19<span class="cite-bracket">]</span></a></sup> In 1730 de Moivre published his book <i>Miscellanea Analytica de Seriebus et Quadraturis</i> [Analytic Miscellany of Series and Integrals], which included tables of log (<i>n</i>!).<sup id="cite_ref-20" class="reference"><a href="#cite_note-20"><span class="cite-bracket">[</span>20<span class="cite-bracket">]</span></a></sup> For large values of <i>n</i>, de Moivre approximated the coefficients of the terms in a binomial expansion. Specifically, given a positive integer <i>n</i>, where <i>n</i> is even and large, then the coefficient of the middle term of (1 + 1)<sup><i>n</i></sup> is approximated by the equation:<sup id="cite_ref-21" class="reference"><a href="#cite_note-21"><span class="cite-bracket">[</span>21<span class="cite-bracket">]</span></a></sup><sup id="cite_ref-22" class="reference"><a href="#cite_note-22"><span class="cite-bracket">[</span>22<span class="cite-bracket">]</span></a></sup> </p> <dl><dd><dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {n \choose n/2}={\frac {n!}{(({\frac {n}{2}})!)^{2}}}\approx 2^{n}{\frac {2{\frac {21}{125}}{(n-1)}^{n-{\frac {1}{2}}}}{n^{n}}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow> <mrow class="MJX-TeXAtom-OPEN"> <mo maxsize="2.047em" minsize="2.047em">(</mo> </mrow> <mfrac linethickness="0"> <mi>n</mi> <mrow> <mi>n</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <mn>2</mn> </mrow> </mfrac> <mrow class="MJX-TeXAtom-CLOSE"> <mo maxsize="2.047em" minsize="2.047em">)</mo> </mrow> </mrow> </mrow> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi>n</mi> <mo>!</mo> </mrow> <mrow> <mo stretchy="false">(</mo> <mo stretchy="false">(</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>n</mi> <mn>2</mn> </mfrac> </mrow> <mo stretchy="false">)</mo> <mo>!</mo> <msup> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mrow> </mfrac> </mrow> <mo>≈</mo> <msup> <mn>2</mn> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msup> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mn>2</mn> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>21</mn> <mn>125</mn> </mfrac> </mrow> <msup> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">(</mo> <mi>n</mi> <mo>−</mo> <mn>1</mn> <mo stretchy="false">)</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mn>2</mn> </mfrac> </mrow> </mrow> </msup> </mrow> <msup> <mi>n</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msup> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {n \choose n/2}={\frac {n!}{(({\frac {n}{2}})!)^{2}}}\approx 2^{n}{\frac {2{\frac {21}{125}}{(n-1)}^{n-{\frac {1}{2}}}}{n^{n}}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/998880ef9369b42757a2b3b863b64f2f1abf6ea3" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.005ex; width:40.205ex; height:8.343ex;" alt="{\displaystyle {n \choose n/2}={\frac {n!}{(({\frac {n}{2}})!)^{2}}}\approx 2^{n}{\frac {2{\frac {21}{125}}{(n-1)}^{n-{\frac {1}{2}}}}{n^{n}}}}"></span></dd></dl></dd></dl> <p>On June 19, 1729, <a href="/wiki/James_Stirling_(mathematician)" title="James Stirling (mathematician)">James Stirling</a> sent to de Moivre a letter, which illustrated how he calculated the coefficient of the middle term of a binomial expansion (<i>a</i> + <i>b</i>)<sup><i>n</i></sup> for large values of <i>n</i>.<sup id="cite_ref-23" class="reference"><a href="#cite_note-23"><span class="cite-bracket">[</span>23<span class="cite-bracket">]</span></a></sup><sup id="cite_ref-24" class="reference"><a href="#cite_note-24"><span class="cite-bracket">[</span>24<span class="cite-bracket">]</span></a></sup> In 1730, Stirling published his book <i>Methodus Differentialis</i> [The Differential Method], in which he included his series for log(<i>n</i>!):<sup id="cite_ref-25" class="reference"><a href="#cite_note-25"><span class="cite-bracket">[</span>25<span class="cite-bracket">]</span></a></sup> </p> <dl><dd><dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \log _{10}(n+{\frac {1}{2}})!\approx \log _{10}{\sqrt {2\pi }}+n\log _{10}n-{\frac {n}{\ln 10}},}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>log</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>10</mn> </mrow> </msub> <mo>⁡</mo> <mo stretchy="false">(</mo> <mi>n</mi> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mn>2</mn> </mfrac> </mrow> <mo stretchy="false">)</mo> <mo>!</mo> <mo>≈</mo> <msub> <mi>log</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>10</mn> </mrow> </msub> <mo>⁡</mo> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mn>2</mn> <mi>π</mi> </msqrt> </mrow> <mo>+</mo> <mi>n</mi> <msub> <mi>log</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>10</mn> </mrow> </msub> <mo>⁡</mo> <mi>n</mi> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>n</mi> <mrow> <mi>ln</mi> <mo>⁡</mo> <mn>10</mn> </mrow> </mfrac> </mrow> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \log _{10}(n+{\frac {1}{2}})!\approx \log _{10}{\sqrt {2\pi }}+n\log _{10}n-{\frac {n}{\ln 10}},}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7cc1be6f1a451b3e2a9db91848e2bf5f60daf1e4" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.005ex; width:46.528ex; height:5.343ex;" alt="{\displaystyle \log _{10}(n+{\frac {1}{2}})!\approx \log _{10}{\sqrt {2\pi }}+n\log _{10}n-{\frac {n}{\ln 10}},}"></span></dd></dl></dd></dl> <p>so that for large <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle n}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>n</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle n}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a601995d55609f2d9f5e233e36fbe9ea26011b3b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.395ex; height:1.676ex;" alt="{\displaystyle n}"></span>, <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle n!\approx {\sqrt {2\pi }}\left({\frac {n}{e}}\right)^{n}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>n</mi> <mo>!</mo> <mo>≈</mo> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mn>2</mn> <mi>π</mi> </msqrt> </mrow> <msup> <mrow> <mo>(</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>n</mi> <mi>e</mi> </mfrac> </mrow> <mo>)</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle n!\approx {\sqrt {2\pi }}\left({\frac {n}{e}}\right)^{n}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e825a2d6be07a8cab114dab66e8a4668255bdae2" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:15.795ex; height:4.843ex;" alt="{\displaystyle n!\approx {\sqrt {2\pi }}\left({\frac {n}{e}}\right)^{n}}"></span>. </p><p>On November 12, 1733, de Moivre privately published and distributed a pamphlet – <i>Approximatio ad Summam Terminorum Binomii</i> (<i>a</i> + <i>b</i>)<sup><i>n</i></sup> <i>in Seriem expansi</i> [Approximation of the Sum of the Terms of the Binomial (<i>a</i> + <i>b</i>)<sup><i>n</i></sup> expanded into a Series] – in which he acknowledged Stirling's letter and proposed an alternative expression for the central term of a binomial expansion.<sup id="cite_ref-26" class="reference"><a href="#cite_note-26"><span class="cite-bracket">[</span>26<span class="cite-bracket">]</span></a></sup> </p> <div class="mw-heading mw-heading2"><h2 id="See_also">See also</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Abraham_de_Moivre&action=edit&section=9" title="Edit section: See also"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <ul><li><a href="/wiki/Root_of_unity" title="Root of unity">De Moivre number</a></li> <li><a href="/wiki/Quintic_function#Roots_of_a_solvable_quintic" title="Quintic function">De Moivre quintic</a></li> <li><a href="/wiki/Economic_model" title="Economic model">Economic model</a></li> <li><a href="/wiki/Gaussian_integral" title="Gaussian integral">Gaussian integral</a></li> <li><a href="/wiki/Poisson_distribution" title="Poisson distribution">Poisson distribution</a></li></ul> <div class="mw-heading mw-heading2"><h2 id="Notes">Notes</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Abraham_de_Moivre&action=edit&section=10" title="Edit section: Notes"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <style data-mw-deduplicate="TemplateStyles:r1239543626">.mw-parser-output .reflist{margin-bottom:0.5em;list-style-type:decimal}@media screen{.mw-parser-output .reflist{font-size:90%}}.mw-parser-output .reflist .references{font-size:100%;margin-bottom:0;list-style-type:inherit}.mw-parser-output .reflist-columns-2{column-width:30em}.mw-parser-output .reflist-columns-3{column-width:25em}.mw-parser-output .reflist-columns{margin-top:0.3em}.mw-parser-output .reflist-columns ol{margin-top:0}.mw-parser-output .reflist-columns li{page-break-inside:avoid;break-inside:avoid-column}.mw-parser-output .reflist-upper-alpha{list-style-type:upper-alpha}.mw-parser-output .reflist-upper-roman{list-style-type:upper-roman}.mw-parser-output .reflist-lower-alpha{list-style-type:lower-alpha}.mw-parser-output .reflist-lower-greek{list-style-type:lower-greek}.mw-parser-output .reflist-lower-roman{list-style-type:lower-roman}</style><div class="reflist"> <div class="mw-references-wrap mw-references-columns"><ol class="references"> <li id="cite_note-mactutor-1"><span class="mw-cite-backlink">^ <a href="#cite_ref-mactutor_1-0"><sup><i><b>a</b></i></sup></a> <a href="#cite_ref-mactutor_1-1"><sup><i><b>b</b></i></sup></a> <a href="#cite_ref-mactutor_1-2"><sup><i><b>c</b></i></sup></a></span> <span class="reference-text"><style data-mw-deduplicate="TemplateStyles:r1238218222">.mw-parser-output cite.citation{font-style:inherit;word-wrap:break-word}.mw-parser-output .citation q{quotes:"\"""\"""'""'"}.mw-parser-output .citation:target{background-color:rgba(0,127,255,0.133)}.mw-parser-output .id-lock-free.id-lock-free a{background:url("//upload.wikimedia.org/wikipedia/commons/6/65/Lock-green.svg")right 0.1em center/9px no-repeat}.mw-parser-output .id-lock-limited.id-lock-limited a,.mw-parser-output .id-lock-registration.id-lock-registration a{background:url("//upload.wikimedia.org/wikipedia/commons/d/d6/Lock-gray-alt-2.svg")right 0.1em center/9px no-repeat}.mw-parser-output .id-lock-subscription.id-lock-subscription a{background:url("//upload.wikimedia.org/wikipedia/commons/a/aa/Lock-red-alt-2.svg")right 0.1em center/9px no-repeat}.mw-parser-output .cs1-ws-icon a{background:url("//upload.wikimedia.org/wikipedia/commons/4/4c/Wikisource-logo.svg")right 0.1em center/12px no-repeat}body:not(.skin-timeless):not(.skin-minerva) .mw-parser-output .id-lock-free a,body:not(.skin-timeless):not(.skin-minerva) .mw-parser-output .id-lock-limited a,body:not(.skin-timeless):not(.skin-minerva) .mw-parser-output .id-lock-registration a,body:not(.skin-timeless):not(.skin-minerva) .mw-parser-output .id-lock-subscription a,body:not(.skin-timeless):not(.skin-minerva) .mw-parser-output .cs1-ws-icon a{background-size:contain;padding:0 1em 0 0}.mw-parser-output .cs1-code{color:inherit;background:inherit;border:none;padding:inherit}.mw-parser-output .cs1-hidden-error{display:none;color:var(--color-error,#d33)}.mw-parser-output .cs1-visible-error{color:var(--color-error,#d33)}.mw-parser-output .cs1-maint{display:none;color:#085;margin-left:0.3em}.mw-parser-output .cs1-kern-left{padding-left:0.2em}.mw-parser-output .cs1-kern-right{padding-right:0.2em}.mw-parser-output .citation .mw-selflink{font-weight:inherit}@media screen{.mw-parser-output .cs1-format{font-size:95%}html.skin-theme-clientpref-night .mw-parser-output .cs1-maint{color:#18911f}}@media screen and (prefers-color-scheme:dark){html.skin-theme-clientpref-os .mw-parser-output .cs1-maint{color:#18911f}}</style><cite id="CITEREFO'ConnorRobertson" class="citation cs2">O'Connor, John J.; <a href="/wiki/Edmund_F._Robertson" title="Edmund F. Robertson">Robertson, Edmund F.</a>, <a rel="nofollow" class="external text" href="https://mathshistory.st-andrews.ac.uk/Biographies/De_Moivre.html">"Abraham de Moivre"</a>, <i><a href="/wiki/MacTutor_History_of_Mathematics_Archive" title="MacTutor History of Mathematics Archive">MacTutor History of Mathematics Archive</a></i>, <a href="/wiki/University_of_St_Andrews" title="University of St Andrews">University of St Andrews</a></cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=bookitem&rft.atitle=Abraham+de+Moivre&rft.btitle=MacTutor+History+of+Mathematics+Archive&rft.pub=University+of+St+Andrews&rft.aulast=O%27Connor&rft.aufirst=John+J.&rft.au=Robertson%2C+Edmund+F.&rft_id=https%3A%2F%2Fmathshistory.st-andrews.ac.uk%2FBiographies%2FDe_Moivre.html&rfr_id=info%3Asid%2Fen.wikipedia.org%3AAbraham+de+Moivre" class="Z3988"></span></span> </li> <li id="cite_note-2"><span class="mw-cite-backlink"><b><a href="#cite_ref-2">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFBellhouse2011" class="citation book cs1">Bellhouse, David R. (2011). <i>Abraham De Moivre: Setting the Stage for Classical Probability and Its Applications</i>. London: Taylor & Francis. p. 99. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-1-56881-349-3" title="Special:BookSources/978-1-56881-349-3"><bdi>978-1-56881-349-3</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Abraham+De+Moivre%3A+Setting+the+Stage+for+Classical+Probability+and+Its+Applications&rft.place=London&rft.pages=99&rft.pub=Taylor+%26+Francis&rft.date=2011&rft.isbn=978-1-56881-349-3&rft.aulast=Bellhouse&rft.aufirst=David+R.&rfr_id=info%3Asid%2Fen.wikipedia.org%3AAbraham+de+Moivre" class="Z3988"></span></span> </li> <li id="cite_note-3"><span class="mw-cite-backlink"><b><a href="#cite_ref-3">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFCoughlinZitarelli1984" class="citation book cs1">Coughlin, Raymond F.; <a href="/wiki/David_E._Zitarelli" title="David E. Zitarelli">Zitarelli, David E.</a> (1984). <i>The ascent of mathematics</i>. McGraw-Hill. p. 437. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/0-07-013215-1" title="Special:BookSources/0-07-013215-1"><bdi>0-07-013215-1</bdi></a>. <q>Unfortunately, because he was not British, De Moivre was never able to obtain a university teaching position</q></cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=The+ascent+of+mathematics&rft.pages=437&rft.pub=McGraw-Hill&rft.date=1984&rft.isbn=0-07-013215-1&rft.aulast=Coughlin&rft.aufirst=Raymond+F.&rft.au=Zitarelli%2C+David+E.&rfr_id=info%3Asid%2Fen.wikipedia.org%3AAbraham+de+Moivre" class="Z3988"></span></span> </li> <li id="cite_note-4"><span class="mw-cite-backlink"><b><a href="#cite_ref-4">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFJungnickelMcCormmach1996" class="citation book cs1"><a href="/wiki/Christa_Jungnickel" title="Christa Jungnickel">Jungnickel, Christa</a>; <a href="/wiki/Russell_McCormmach" title="Russell McCormmach">McCormmach, Russell</a> (1996). <a rel="nofollow" class="external text" href="https://books.google.com/books?id=eiDoN-rg8I8C&pg=PA52"><i>Cavendish</i></a>. Memoirs of the American Philosophical Society. Vol. 220. American Philosophical Society. p. 52. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/9780871692207" title="Special:BookSources/9780871692207"><bdi>9780871692207</bdi></a>. <q>Well connected in mathematical circles and highly regarded for his work, he still could not get a good job. Even his conversion to the Church of England in 1705 could not alter the fact that he was an alien.</q></cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Cavendish&rft.series=Memoirs+of+the+American+Philosophical+Society&rft.pages=52&rft.pub=American+Philosophical+Society&rft.date=1996&rft.isbn=9780871692207&rft.aulast=Jungnickel&rft.aufirst=Christa&rft.au=McCormmach%2C+Russell&rft_id=https%3A%2F%2Fbooks.google.com%2Fbooks%3Fid%3DeiDoN-rg8I8C%26pg%3DPA52&rfr_id=info%3Asid%2Fen.wikipedia.org%3AAbraham+de+Moivre" class="Z3988"></span></span> </li> <li id="cite_note-5"><span class="mw-cite-backlink"><b><a href="#cite_ref-5">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFTanton2005" class="citation book cs1">Tanton, James Stuart (2005). <a rel="nofollow" class="external text" href="https://books.google.com/books?id=MfKKMSuthacC&pg=PA122"><i>Encyclopedia of Mathematics</i></a>. Infobase Publishing. p. 122. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/9780816051243" title="Special:BookSources/9780816051243"><bdi>9780816051243</bdi></a>. <q>He had hoped to receive a faculty position in mathematics but, as a foreigner, was never offered such an appointment.</q></cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Encyclopedia+of+Mathematics&rft.pages=122&rft.pub=Infobase+Publishing&rft.date=2005&rft.isbn=9780816051243&rft.aulast=Tanton&rft.aufirst=James+Stuart&rft_id=https%3A%2F%2Fbooks.google.com%2Fbooks%3Fid%3DMfKKMSuthacC%26pg%3DPA122&rfr_id=info%3Asid%2Fen.wikipedia.org%3AAbraham+de+Moivre" class="Z3988"></span></span> </li> <li id="cite_note-Cajori-History-6"><span class="mw-cite-backlink"><b><a href="#cite_ref-Cajori-History_6-0">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFCajori1991" class="citation book cs1"><a href="/wiki/Florian_Cajori" title="Florian Cajori">Cajori, Florian</a> (1991). <i>History of Mathematics</i> (5 ed.). <a href="/wiki/American_Mathematical_Society" title="American Mathematical Society">American Mathematical Society</a>. p. 229. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/9780821821022" title="Special:BookSources/9780821821022"><bdi>9780821821022</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=History+of+Mathematics&rft.pages=229&rft.edition=5&rft.pub=American+Mathematical+Society&rft.date=1991&rft.isbn=9780821821022&rft.aulast=Cajori&rft.aufirst=Florian&rfr_id=info%3Asid%2Fen.wikipedia.org%3AAbraham+de+Moivre" class="Z3988"></span></span> </li> <li id="cite_note-hsmstex-7"><span class="mw-cite-backlink"><b><a href="#cite_ref-hsmstex_7-0">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite class="citation web cs1"><a rel="nofollow" class="external text" href="http://hsm.stackexchange.com/questions/333/did-abraham-de-moivre-really-predict-his-own-death">"Biographical details - Did Abraham de Moivre really predict his own death?"</a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=unknown&rft.btitle=Biographical+details+-+Did+Abraham+de+Moivre+really+predict+his+own+death%3F&rft_id=http%3A%2F%2Fhsm.stackexchange.com%2Fquestions%2F333%2Fdid-abraham-de-moivre-really-predict-his-own-death&rfr_id=info%3Asid%2Fen.wikipedia.org%3AAbraham+de+Moivre" class="Z3988"></span></span> </li> <li id="cite_note-8"><span class="mw-cite-backlink"><b><a href="#cite_ref-8">^</a></b></span> <span class="reference-text">See: <ul><li>Abraham De Moivre (12 November 1733) "Approximatio ad summam terminorum binomii (a+b)<sup>n</sup> in seriem expansi" (self-published pamphlet), 7 pages.</li> <li>English translation: A. De Moivre, <i>The Doctrine of Chances</i> … , 2nd ed. (London, England: H. Woodfall, 1738), <a rel="nofollow" class="external text" href="https://books.google.com/books?id=PII_AAAAcAAJ&pg=PA235">pp. 235–243</a>.</li></ul> </span></li> <li id="cite_note-9"><span class="mw-cite-backlink"><b><a href="#cite_ref-9">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFPearson1924" class="citation journal cs1">Pearson, Karl (1924). "Historical note on the origin of the normal curve of errors". <i>Biometrika</i>. <b>16</b> (3–4): 402–404. <a href="/wiki/Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1093%2Fbiomet%2F16.3-4.402">10.1093/biomet/16.3-4.402</a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=Biometrika&rft.atitle=Historical+note+on+the+origin+of+the+normal+curve+of+errors&rft.volume=16&rft.issue=3%E2%80%934&rft.pages=402-404&rft.date=1924&rft_id=info%3Adoi%2F10.1093%2Fbiomet%2F16.3-4.402&rft.aulast=Pearson&rft.aufirst=Karl&rfr_id=info%3Asid%2Fen.wikipedia.org%3AAbraham+de+Moivre" class="Z3988"></span></span> </li> <li id="cite_note-JKK157-10"><span class="mw-cite-backlink"><b><a href="#cite_ref-JKK157_10-0">^</a></b></span> <span class="reference-text">Johnson, N.L., Kotz, S., Kemp, A.W. (1993) <i>Univariate Discrete distributions</i> (2nd edition). Wiley. <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/0-471-54897-9" title="Special:BookSources/0-471-54897-9">0-471-54897-9</a>, p157</span> </li> <li id="cite_note-11"><span class="mw-cite-backlink"><b><a href="#cite_ref-11">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFStigler1982" class="citation journal cs1">Stigler, Stephen M. (1982). "Poisson on the poisson distribution". <i>Statistics & Probability Letters</i>. <b>1</b>: 33–35. <a href="/wiki/Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1016%2F0167-7152%2882%2990010-4">10.1016/0167-7152(82)90010-4</a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=Statistics+%26+Probability+Letters&rft.atitle=Poisson+on+the+poisson+distribution&rft.volume=1&rft.pages=33-35&rft.date=1982&rft_id=info%3Adoi%2F10.1016%2F0167-7152%2882%2990010-4&rft.aulast=Stigler&rft.aufirst=Stephen+M.&rfr_id=info%3Asid%2Fen.wikipedia.org%3AAbraham+de+Moivre" class="Z3988"></span></span> </li> <li id="cite_note-12"><span class="mw-cite-backlink"><b><a href="#cite_ref-12">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFHaldde_MoivreMcClintock1984" class="citation journal cs1">Hald, Anders; de Moivre, Abraham; McClintock, Bruce (1984). "A. de Moivre:'De Mensura Sortis' or'On the Measurement of Chance'<span class="cs1-kern-right"></span>". <i>International Statistical Review/Revue Internationale de Statistique</i>. <b>1984</b> (3): 229–262. <a href="/wiki/JSTOR_(identifier)" class="mw-redirect" title="JSTOR (identifier)">JSTOR</a> <a rel="nofollow" class="external text" href="https://www.jstor.org/stable/1403045">1403045</a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=International+Statistical+Review%2FRevue+Internationale+de+Statistique&rft.atitle=A.+de+Moivre%3A%27De+Mensura+Sortis%27+or%27On+the+Measurement+of+Chance%27&rft.volume=1984&rft.issue=3&rft.pages=229-262&rft.date=1984&rft_id=https%3A%2F%2Fwww.jstor.org%2Fstable%2F1403045%23id-name%3DJSTOR&rft.aulast=Hald&rft.aufirst=Anders&rft.au=de+Moivre%2C+Abraham&rft.au=McClintock%2C+Bruce&rfr_id=info%3Asid%2Fen.wikipedia.org%3AAbraham+de+Moivre" class="Z3988"></span></span> </li> <li id="cite_note-13"><span class="mw-cite-backlink"><b><a href="#cite_ref-13">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFMoivre1707" class="citation journal cs1 cs1-prop-foreign-lang-source">Moivre, Ab. de (1707). "Aequationum quarundam potestatis tertiae, quintae, septimae, nonae, & superiorum, ad infinitum usque pergendo, in termimis finitis, ad instar regularum pro cubicis quae vocantur Cardani, resolutio analytica" [Of certain equations of the third, fifth, seventh, ninth, & higher power, all the way to infinity, by proceeding, in finite terms, in the form of rules for cubics which are called by Cardano, resolution by analysis.]. <i>Philosophical Transactions of the Royal Society of London</i> (in Latin). <b>25</b> (309): 2368–2371. <a href="/wiki/Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1098%2Frstl.1706.0037">10.1098/rstl.1706.0037</a>. <a href="/wiki/S2CID_(identifier)" class="mw-redirect" title="S2CID (identifier)">S2CID</a> <a rel="nofollow" class="external text" href="https://api.semanticscholar.org/CorpusID:186209627">186209627</a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=Philosophical+Transactions+of+the+Royal+Society+of+London&rft.atitle=Aequationum+quarundam+potestatis+tertiae%2C+quintae%2C+septimae%2C+nonae%2C+%26+superiorum%2C+ad+infinitum+usque+pergendo%2C+in+termimis+finitis%2C+ad+instar+regularum+pro+cubicis+quae+vocantur+Cardani%2C+resolutio+analytica&rft.volume=25&rft.issue=309&rft.pages=2368-2371&rft.date=1707&rft_id=info%3Adoi%2F10.1098%2Frstl.1706.0037&rft_id=https%3A%2F%2Fapi.semanticscholar.org%2FCorpusID%3A186209627%23id-name%3DS2CID&rft.aulast=Moivre&rft.aufirst=Ab.+de&rfr_id=info%3Asid%2Fen.wikipedia.org%3AAbraham+de+Moivre" class="Z3988"></span> <ul><li>English translation by Richard J. Pulskamp (2009)</li></ul> On p. 2370 de Moivre stated that if a series has the form <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle ny+{\tfrac {1-nn}{2\times 3}}ny^{3}+{\tfrac {1-nn}{2\times 3}}{\tfrac {9-nn}{4\times 5}}ny^{5}+{\tfrac {1-nn}{2\times 3}}{\tfrac {9-nn}{4\times 5}}{\tfrac {25-nn}{6\times 7}}ny^{7}+\cdots =a}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>n</mi> <mi>y</mi> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mfrac> <mrow> <mn>1</mn> <mo>−</mo> <mi>n</mi> <mi>n</mi> </mrow> <mrow> <mn>2</mn> <mo>×</mo> <mn>3</mn> </mrow> </mfrac> </mstyle> </mrow> <mi>n</mi> <msup> <mi>y</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msup> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mfrac> <mrow> <mn>1</mn> <mo>−</mo> <mi>n</mi> <mi>n</mi> </mrow> <mrow> <mn>2</mn> <mo>×</mo> <mn>3</mn> </mrow> </mfrac> </mstyle> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mfrac> <mrow> <mn>9</mn> <mo>−</mo> <mi>n</mi> <mi>n</mi> </mrow> <mrow> <mn>4</mn> <mo>×</mo> <mn>5</mn> </mrow> </mfrac> </mstyle> </mrow> <mi>n</mi> <msup> <mi>y</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>5</mn> </mrow> </msup> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mfrac> <mrow> <mn>1</mn> <mo>−</mo> <mi>n</mi> <mi>n</mi> </mrow> <mrow> <mn>2</mn> <mo>×</mo> <mn>3</mn> </mrow> </mfrac> </mstyle> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mfrac> <mrow> <mn>9</mn> <mo>−</mo> <mi>n</mi> <mi>n</mi> </mrow> <mrow> <mn>4</mn> <mo>×</mo> <mn>5</mn> </mrow> </mfrac> </mstyle> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mfrac> <mrow> <mn>25</mn> <mo>−</mo> <mi>n</mi> <mi>n</mi> </mrow> <mrow> <mn>6</mn> <mo>×</mo> <mn>7</mn> </mrow> </mfrac> </mstyle> </mrow> <mi>n</mi> <msup> <mi>y</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>7</mn> </mrow> </msup> <mo>+</mo> <mo>⋯</mo> <mo>=</mo> <mi>a</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle ny+{\tfrac {1-nn}{2\times 3}}ny^{3}+{\tfrac {1-nn}{2\times 3}}{\tfrac {9-nn}{4\times 5}}ny^{5}+{\tfrac {1-nn}{2\times 3}}{\tfrac {9-nn}{4\times 5}}{\tfrac {25-nn}{6\times 7}}ny^{7}+\cdots =a}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ec3a9c46eee39b68bced090959c48fbe48a0a640" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.338ex; width:62.068ex; height:3.843ex;" alt="{\displaystyle ny+{\tfrac {1-nn}{2\times 3}}ny^{3}+{\tfrac {1-nn}{2\times 3}}{\tfrac {9-nn}{4\times 5}}ny^{5}+{\tfrac {1-nn}{2\times 3}}{\tfrac {9-nn}{4\times 5}}{\tfrac {25-nn}{6\times 7}}ny^{7}+\cdots =a}"></span> , where <i>n</i> is any given odd integer (positive or negative) and where <i>y</i> and <i>a</i> can be functions, then upon solving for <i>y</i>, the result is equation (2) on the same page: <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle y={\tfrac {1}{2}}{\sqrt[{n}]{a+{\sqrt {aa-1}}}}+{\tfrac {1}{2}}{\sqrt[{n}]{a-{\sqrt {aa-1}}}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>y</mi> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mfrac> <mn>1</mn> <mn>2</mn> </mfrac> </mstyle> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mroot> <mrow> <mi>a</mi> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mi>a</mi> <mi>a</mi> <mo>−</mo> <mn>1</mn> </msqrt> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </mroot> </mrow> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mfrac> <mn>1</mn> <mn>2</mn> </mfrac> </mstyle> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mroot> <mrow> <mi>a</mi> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mi>a</mi> <mi>a</mi> <mo>−</mo> <mn>1</mn> </msqrt> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </mroot> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle y={\tfrac {1}{2}}{\sqrt[{n}]{a+{\sqrt {aa-1}}}}+{\tfrac {1}{2}}{\sqrt[{n}]{a-{\sqrt {aa-1}}}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/5e00e33900370cd1603acc07dd2af9559c28250c" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.671ex; width:39.995ex; height:4.843ex;" alt="{\displaystyle y={\tfrac {1}{2}}{\sqrt[{n}]{a+{\sqrt {aa-1}}}}+{\tfrac {1}{2}}{\sqrt[{n}]{a-{\sqrt {aa-1}}}}}"></span>. If <i>y</i> = cos x and <i>a</i> = cos nx , then the result is <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \cos x={\tfrac {1}{2}}(\cos(nx)+i\sin(nx))^{1/n}+{\tfrac {1}{2}}(\cos(nx)-i\sin(nx))^{1/n}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>cos</mi> <mo>⁡</mo> <mi>x</mi> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mfrac> <mn>1</mn> <mn>2</mn> </mfrac> </mstyle> </mrow> <mo stretchy="false">(</mo> <mi>cos</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mi>n</mi> <mi>x</mi> <mo stretchy="false">)</mo> <mo>+</mo> <mi>i</mi> <mi>sin</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mi>n</mi> <mi>x</mi> <mo stretchy="false">)</mo> <msup> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <mi>n</mi> </mrow> </msup> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mfrac> <mn>1</mn> <mn>2</mn> </mfrac> </mstyle> </mrow> <mo stretchy="false">(</mo> <mi>cos</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mi>n</mi> <mi>x</mi> <mo stretchy="false">)</mo> <mo>−</mo> <mi>i</mi> <mi>sin</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mi>n</mi> <mi>x</mi> <mo stretchy="false">)</mo> <msup> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <mi>n</mi> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \cos x={\tfrac {1}{2}}(\cos(nx)+i\sin(nx))^{1/n}+{\tfrac {1}{2}}(\cos(nx)-i\sin(nx))^{1/n}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ef48a808b32996fb0a68f4e303008e8d9fa4159f" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.171ex; width:61.555ex; height:3.676ex;" alt="{\displaystyle \cos x={\tfrac {1}{2}}(\cos(nx)+i\sin(nx))^{1/n}+{\tfrac {1}{2}}(\cos(nx)-i\sin(nx))^{1/n}}"></span><br /> <ul><li>In 1676, <a href="/wiki/Isaac_Newton" title="Isaac Newton">Isaac Newton</a> found the relation between two chords that were in the ratio of n to 1; the relation was expressed by the series above. The series appears in a letter — <i>Epistola prior D. Issaci Newton, Mathescos Professoris in Celeberrima Academia Cantabrigiensi; …</i> — of 13 June 1676 from Isaac Newton to Henry Oldenburg, secretary of the Royal Society; a copy of the letter was sent to <a href="/wiki/Gottfried_Wilhelm_Leibniz" title="Gottfried Wilhelm Leibniz">Gottfried Wilhelm Leibniz</a>. See p. 106 of: <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFBiotLefort1856" class="citation book cs1 cs1-prop-foreign-lang-source">Biot, J.-B.; Lefort, F., eds. (1856). <a rel="nofollow" class="external text" href="https://books.google.com/books?id=BHBtAAAAMAAJ&pg=PA106"><i>Commercium epistolicum J. Collins et aliorum de analysi promota, etc: ou …</i></a> (in Latin). Paris, France: Mallet-Bachelier. pp. 102–112.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Commercium+epistolicum+J.+Collins+et+aliorum+de+analysi+promota%2C+etc%3A+ou+%E2%80%A6&rft.place=Paris%2C+France&rft.pages=102-112&rft.pub=Mallet-Bachelier&rft.date=1856&rft_id=https%3A%2F%2Fbooks.google.com%2Fbooks%3Fid%3DBHBtAAAAMAAJ%26pg%3DPA106&rfr_id=info%3Asid%2Fen.wikipedia.org%3AAbraham+de+Moivre" class="Z3988"></span></li> <li>In 1698, de Moivre derived the same series. See: <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFde_Moivre1698" class="citation journal cs1">de Moivre, A. (1698). <a rel="nofollow" class="external text" href="https://doi.org/10.1098%2Frstl.1698.0034">"A method of extracting roots of an infinite equation"</a>. <i>Philosophical Transactions of the Royal Society of London</i>. <b>20</b> (240): 190–193. <a href="/wiki/Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<span class="id-lock-free" title="Freely accessible"><a rel="nofollow" class="external text" href="https://doi.org/10.1098%2Frstl.1698.0034">10.1098/rstl.1698.0034</a></span>. <a href="/wiki/S2CID_(identifier)" class="mw-redirect" title="S2CID (identifier)">S2CID</a> <a rel="nofollow" class="external text" href="https://api.semanticscholar.org/CorpusID:186214144">186214144</a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=Philosophical+Transactions+of+the+Royal+Society+of+London&rft.atitle=A+method+of+extracting+roots+of+an+infinite+equation&rft.volume=20&rft.issue=240&rft.pages=190-193&rft.date=1698&rft_id=info%3Adoi%2F10.1098%2Frstl.1698.0034&rft_id=https%3A%2F%2Fapi.semanticscholar.org%2FCorpusID%3A186214144%23id-name%3DS2CID&rft.aulast=de+Moivre&rft.aufirst=A.&rft_id=https%3A%2F%2Fdoi.org%2F10.1098%252Frstl.1698.0034&rfr_id=info%3Asid%2Fen.wikipedia.org%3AAbraham+de+Moivre" class="Z3988"></span>; see p 192.</li> <li>In 1730, de Moivre explicitly considered the case where the functions are cos θ and cos nθ. See: <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFMoivre1730" class="citation book cs1 cs1-prop-foreign-lang-source">Moivre, A. de (1730). <a rel="nofollow" class="external text" href="https://books.google.com/books?id=bMl5NAAACAAJ&pg=PP5"><i>Miscellanea Analytica de Seriebus et Quadraturis</i></a> (in Latin). London, England: J. Tonson & J. Watts. p. 1.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Miscellanea+Analytica+de+Seriebus+et+Quadraturis&rft.place=London%2C+England&rft.pages=1&rft.pub=J.+Tonson+%26+J.+Watts&rft.date=1730&rft.aulast=Moivre&rft.aufirst=A.+de&rft_id=https%3A%2F%2Fbooks.google.com%2Fbooks%3Fid%3DbMl5NAAACAAJ%26pg%3DPP5&rfr_id=info%3Asid%2Fen.wikipedia.org%3AAbraham+de+Moivre" class="Z3988"></span> From p. 1: <i>"Lemma 1. Si sint l & x cosinus arcuum duorum A & B, quorum uterque eodem radio 1 describatur, quorumque prior sit posterioris multiplex in ea ratione quam habet numerus n ad unitatem, tunc erit <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x={\tfrac {1}{2}}{\sqrt[{n}]{l+{\sqrt {ll-1}}}}+{\tfrac {1}{2}}{\tfrac {1}{\sqrt[{n}]{l+{\sqrt {ll-1}}}}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>x</mi> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mfrac> <mn>1</mn> <mn>2</mn> </mfrac> </mstyle> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mroot> <mrow> <mi>l</mi> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mi>l</mi> <mi>l</mi> <mo>−</mo> <mn>1</mn> </msqrt> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </mroot> </mrow> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mfrac> <mn>1</mn> <mn>2</mn> </mfrac> </mstyle> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mfrac> <mn>1</mn> <mroot> <mrow> <mi>l</mi> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mi>l</mi> <mi>l</mi> <mo>−</mo> <mn>1</mn> </msqrt> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </mroot> </mfrac> </mstyle> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle x={\tfrac {1}{2}}{\sqrt[{n}]{l+{\sqrt {ll-1}}}}+{\tfrac {1}{2}}{\tfrac {1}{\sqrt[{n}]{l+{\sqrt {ll-1}}}}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/3059585fa99ce0feebf306b7ec4da490e1e21a0f" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.171ex; width:32.465ex; height:6.676ex;" alt="{\displaystyle x={\tfrac {1}{2}}{\sqrt[{n}]{l+{\sqrt {ll-1}}}}+{\tfrac {1}{2}}{\tfrac {1}{\sqrt[{n}]{l+{\sqrt {ll-1}}}}}}"></span>."</i> (If l and x are cosines of two arcs A and B both of which are described by the same radius 1 and of which the former is a multiple of the latter in that ratio as the number n has to 1, then it will be [true that] <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x={\tfrac {1}{2}}{\sqrt[{n}]{l+{\sqrt {ll-1}}}}+{\tfrac {1}{2}}{\tfrac {1}{\sqrt[{n}]{l+{\sqrt {ll-1}}}}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>x</mi> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mfrac> <mn>1</mn> <mn>2</mn> </mfrac> </mstyle> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mroot> <mrow> <mi>l</mi> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mi>l</mi> <mi>l</mi> <mo>−</mo> <mn>1</mn> </msqrt> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </mroot> </mrow> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mfrac> <mn>1</mn> <mn>2</mn> </mfrac> </mstyle> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mfrac> <mn>1</mn> <mroot> <mrow> <mi>l</mi> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mi>l</mi> <mi>l</mi> <mo>−</mo> <mn>1</mn> </msqrt> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </mroot> </mfrac> </mstyle> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle x={\tfrac {1}{2}}{\sqrt[{n}]{l+{\sqrt {ll-1}}}}+{\tfrac {1}{2}}{\tfrac {1}{\sqrt[{n}]{l+{\sqrt {ll-1}}}}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/3059585fa99ce0feebf306b7ec4da490e1e21a0f" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.171ex; width:32.465ex; height:6.676ex;" alt="{\displaystyle x={\tfrac {1}{2}}{\sqrt[{n}]{l+{\sqrt {ll-1}}}}+{\tfrac {1}{2}}{\tfrac {1}{\sqrt[{n}]{l+{\sqrt {ll-1}}}}}}"></span>.) So if arc A = n × arc B, then l = cos A = cos nB and x = cos B. Hence <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \cos B={\tfrac {1}{2}}(\cos(nB)+{\sqrt {-1}}\sin(nB))^{1/n}+{\tfrac {1}{2}}(\cos(nB)+{\sqrt {-1}}\sin(nB))^{-1/n}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>cos</mi> <mo>⁡</mo> <mi>B</mi> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mfrac> <mn>1</mn> <mn>2</mn> </mfrac> </mstyle> </mrow> <mo stretchy="false">(</mo> <mi>cos</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mi>n</mi> <mi>B</mi> <mo stretchy="false">)</mo> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mo>−</mo> <mn>1</mn> </msqrt> </mrow> <mi>sin</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mi>n</mi> <mi>B</mi> <mo stretchy="false">)</mo> <msup> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <mi>n</mi> </mrow> </msup> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mfrac> <mn>1</mn> <mn>2</mn> </mfrac> </mstyle> </mrow> <mo stretchy="false">(</mo> <mi>cos</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mi>n</mi> <mi>B</mi> <mo stretchy="false">)</mo> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mo>−</mo> <mn>1</mn> </msqrt> </mrow> <mi>sin</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mi>n</mi> <mi>B</mi> <mo stretchy="false">)</mo> <msup> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <mn>1</mn> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <mi>n</mi> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \cos B={\tfrac {1}{2}}(\cos(nB)+{\sqrt {-1}}\sin(nB))^{1/n}+{\tfrac {1}{2}}(\cos(nB)+{\sqrt {-1}}\sin(nB))^{-1/n}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/cbd1bc7a32608f1dd4c92e401e04280ba6862857" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.171ex; width:73.213ex; height:3.676ex;" alt="{\displaystyle \cos B={\tfrac {1}{2}}(\cos(nB)+{\sqrt {-1}}\sin(nB))^{1/n}+{\tfrac {1}{2}}(\cos(nB)+{\sqrt {-1}}\sin(nB))^{-1/n}}"></span></li></ul> See also: <ul><li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFCantor1898" class="citation book cs1 cs1-prop-foreign-lang-source">Cantor, Moritz (1898). <a rel="nofollow" class="external text" href="https://babel.hathitrust.org/cgi/pt?id=chi.11625415&view=1up&seq=644"><i>Vorlesungen über Geschichte der Mathematik</i></a> [<i>Lectures on the History of Mathematics</i>]. Bibliotheca mathematica Teuberiana, Bd. 8-9 (in German). Vol. 3. Leipzig, Germany: B.G. Teubner. p. 624.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Vorlesungen+%C3%BCber+Geschichte+der+Mathematik&rft.place=Leipzig%2C+Germany&rft.series=Bibliotheca+mathematica+Teuberiana%2C+Bd.+8-9&rft.pages=624&rft.pub=B.G.+Teubner&rft.date=1898&rft.aulast=Cantor&rft.aufirst=Moritz&rft_id=https%3A%2F%2Fbabel.hathitrust.org%2Fcgi%2Fpt%3Fid%3Dchi.11625415%26view%3D1up%26seq%3D644&rfr_id=info%3Asid%2Fen.wikipedia.org%3AAbraham+de+Moivre" class="Z3988"></span></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFBraunmühl1901" class="citation journal cs1 cs1-prop-foreign-lang-source">Braunmühl, A. von (1901). <a rel="nofollow" class="external text" href="https://babel.hathitrust.org/cgi/pt?id=hvd.32044102937091&view=1up&seq=112">"Zur Geschichte der Entstehung des sogenannten Moivreschen Satzes"</a> [On the history of the origin of the so-called Moivre theorem]. <i>Bibliotheca Mathematica</i>. 3rd series (in German). <b>2</b>: 97–102.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=Bibliotheca+Mathematica&rft.atitle=Zur+Geschichte+der+Entstehung+des+sogenannten+Moivreschen+Satzes&rft.volume=2&rft.pages=97-102&rft.date=1901&rft.aulast=Braunm%C3%BChl&rft.aufirst=A.+von&rft_id=https%3A%2F%2Fbabel.hathitrust.org%2Fcgi%2Fpt%3Fid%3Dhvd.32044102937091%26view%3D1up%26seq%3D112&rfr_id=info%3Asid%2Fen.wikipedia.org%3AAbraham+de+Moivre" class="Z3988"></span>; see p. 98.</li></ul> </span></li> <li id="cite_note-14"><span class="mw-cite-backlink"><b><a href="#cite_ref-14">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFSmith1959" class="citation cs2">Smith, David Eugene (1959), <a rel="nofollow" class="external text" href="https://books.google.com/books?id=3TSKAAAAQBAJ&pg=PA444"><i>A Source Book in Mathematics, Volume 3</i></a>, Courier Dover Publications, p. 444, <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/9780486646909" title="Special:BookSources/9780486646909"><bdi>9780486646909</bdi></a></cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=A+Source+Book+in+Mathematics%2C+Volume+3&rft.pages=444&rft.pub=Courier+Dover+Publications&rft.date=1959&rft.isbn=9780486646909&rft.aulast=Smith&rft.aufirst=David+Eugene&rft_id=https%3A%2F%2Fbooks.google.com%2Fbooks%3Fid%3D3TSKAAAAQBAJ%26pg%3DPA444&rfr_id=info%3Asid%2Fen.wikipedia.org%3AAbraham+de+Moivre" class="Z3988"></span></span> </li> <li id="cite_note-15"><span class="mw-cite-backlink"><b><a href="#cite_ref-15">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFMoivre1722" class="citation journal cs1 cs1-prop-foreign-lang-source">Moivre, A. de (1722). <a rel="nofollow" class="external text" href="https://zenodo.org/record/1432196/files/article.pdf">"De sectione anguli"</a> [Concerning the section of an angle] <span class="cs1-format">(PDF)</span>. <i>Philosophical Transactions of the Royal Society of London</i> (in Latin). <b>32</b> (374): 228–230. <a href="/wiki/Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1098%2Frstl.1722.0039">10.1098/rstl.1722.0039</a>. <a href="/wiki/S2CID_(identifier)" class="mw-redirect" title="S2CID (identifier)">S2CID</a> <a rel="nofollow" class="external text" href="https://api.semanticscholar.org/CorpusID:186210081">186210081</a><span class="reference-accessdate">. Retrieved <span class="nowrap">6 June</span> 2020</span>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=Philosophical+Transactions+of+the+Royal+Society+of+London&rft.atitle=De+sectione+anguli&rft.volume=32&rft.issue=374&rft.pages=228-230&rft.date=1722&rft_id=info%3Adoi%2F10.1098%2Frstl.1722.0039&rft_id=https%3A%2F%2Fapi.semanticscholar.org%2FCorpusID%3A186210081%23id-name%3DS2CID&rft.aulast=Moivre&rft.aufirst=A.+de&rft_id=https%3A%2F%2Fzenodo.org%2Frecord%2F1432196%2Ffiles%2Farticle.pdf&rfr_id=info%3Asid%2Fen.wikipedia.org%3AAbraham+de+Moivre" class="Z3988"></span> <ul><li>English translation by <a rel="nofollow" class="external text" href="http://cerebro.xu.edu/math/Sources/Moivre/de_%20sectione_anguli.pdf">Richard J. Pulskamp (2009)</a> <a rel="nofollow" class="external text" href="https://web.archive.org/web/20201128125816/http://cerebro.xu.edu/math/Sources/Moivre/de_%20sectione_anguli.pdf">Archived</a> 28 November 2020 at the <a href="/wiki/Wayback_Machine" title="Wayback Machine">Wayback Machine</a></li></ul> From p. 229:<br /> <i>"Sit </i>x<i> sinus versus arcus cujuslibert.</i><br /> <i>[Sit] </i>t<i> sinus versus arcus alterius.</i><br /> <i>[Sit] 1 radius circuli.</i><br /> <i>Sitque arcus prior ad posteriorum ut 1 ad </i>n<i>, tunc, assumptis binis aequationibus quas cognatas appelare licet,</i> <i>1 – 2</i>z<sup><i>n</i></sup> + <i>z</i><sup>2<i>n</i></sup> = – 2<i>z</i><sup><i>n</i></sup><i>t</i> <i>1 – 2</i>z<i> + </i>zz<i> = – 2</i>zx<i>.</i> <i>Expunctoque </i>z<i> orietur aequatio qua relatio inter </i>x<i> & </i>t<i> determinatur."</i><br /> (Let <i>x</i> be the versine of any arc [i.e., <i>x</i> = 1 – cos θ ].<br /> [Let] <i>t</i> be the versine of another arc.<br /> [Let] 1 be the radius of the circle.<br /> And let the first arc to the latter [i.e., "another arc"] be as 1 to <i>n</i> [so that <i>t</i> = 1 – cos <i>nθ</i>], then, with the two equations assumed which may be called related, 1 – 2<i>z</i><sup><i>n</i></sup> + <i>z</i><sup>2<i>n</i></sup> = –2<i>z</i><sup><i>n</i></sup><i>t</i> 1 – 2<i>z</i> + <i>zz</i> = – 2<i>zx</i>. And by eliminating <i>z</i>, the equation will arise by which the relation between <i>x</i> and <i>t</i> is determined.)<br /> That is, given the equations 1 – 2<i>z</i><sup><i>n</i></sup> + <i>z</i><sup>2<i>n</i></sup> = – 2<i>z</i><sup><i>n</i></sup> (1 – cos <i>n</i>θ) 1 – 2<i>z</i> + <i>zz</i> = – 2<i>z</i> (1 – cos θ),<br /> use the <a href="/wiki/Quadratic_formula" title="Quadratic formula">quadratic formula</a> to solve for <i>z</i><sup><i>n</i></sup> in the first equation and for <i>z</i> in the second equation. The result will be: <i>z</i><sup><i>n</i></sup> = cos <i>n</i>θ ± <i>i</i> sin <i>n</i>θ and <i>z</i> = cos θ ± <i>i</i> sin θ , whence it immediately follows that (cos θ ± <i>i</i> sin θ)<sup><i>n</i></sup> = cos <i>n</i>θ ± <i>i</i> sin <i>n</i>θ.<br /> See also: <ul><li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFSmith1959" class="citation book cs1">Smith, David Eugen (1959). <a rel="nofollow" class="external text" href="https://archive.org/details/in.ernet.dli.2015.165707/page/n151/mode/2up/"><i>A Source Book in Mathematics</i></a>. Vol. 2. New York City, New York, USA: Dover Publications Inc. pp. 444–446.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=A+Source+Book+in+Mathematics&rft.place=New+York+City%2C+New+York%2C+USA&rft.pages=444-446&rft.pub=Dover+Publications+Inc.&rft.date=1959&rft.aulast=Smith&rft.aufirst=David+Eugen&rft_id=https%3A%2F%2Farchive.org%2Fdetails%2Fin.ernet.dli.2015.165707%2Fpage%2Fn151%2Fmode%2F2up%2F&rfr_id=info%3Asid%2Fen.wikipedia.org%3AAbraham+de+Moivre" class="Z3988"></span> see p. 445, footnote 1.</li></ul> </span></li> <li id="cite_note-16"><span class="mw-cite-backlink"><b><a href="#cite_ref-16">^</a></b></span> <span class="reference-text">In 1738, de Moivre used trigonometry to determine the nth roots of a real or complex number. See: <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFMoivre1738" class="citation journal cs1 cs1-prop-foreign-lang-source">Moivre, A. de (1738). "De reductione radicalium ad simpliciores terminos, seu de extrahenda radice quacunque data ex binomio <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle a+{\sqrt {+b}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>a</mi> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mo>+</mo> <mi>b</mi> </msqrt> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle a+{\sqrt {+b}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/cb99a4409bcf8b57dad77ee619cf26ca39c25b21" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:8.812ex; height:3.009ex;" alt="{\displaystyle a+{\sqrt {+b}}}"></span>, vel <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle a+{\sqrt {-b}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>a</mi> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mo>−</mo> <mi>b</mi> </msqrt> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle a+{\sqrt {-b}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7477a1a9626d9b0205a2f36eed0212f9f25e0ccc" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:8.812ex; height:3.009ex;" alt="{\displaystyle a+{\sqrt {-b}}}"></span>. Epistola" [On the reduction of radicals to simpler terms, or on extracting any given root from a binomial, <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle a+{\sqrt {+b}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>a</mi> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mo>+</mo> <mi>b</mi> </msqrt> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle a+{\sqrt {+b}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/cb99a4409bcf8b57dad77ee619cf26ca39c25b21" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:8.812ex; height:3.009ex;" alt="{\displaystyle a+{\sqrt {+b}}}"></span> or <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle a+{\sqrt {-b}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>a</mi> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mo>−</mo> <mi>b</mi> </msqrt> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle a+{\sqrt {-b}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7477a1a9626d9b0205a2f36eed0212f9f25e0ccc" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:8.812ex; height:3.009ex;" alt="{\displaystyle a+{\sqrt {-b}}}"></span>. A letter.]. <i>Philosophical Transactions of the Royal Society of London</i> (in Latin). <b>40</b> (451): 463–478. <a href="/wiki/Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1098%2Frstl.1737.0081">10.1098/rstl.1737.0081</a>. <a href="/wiki/S2CID_(identifier)" class="mw-redirect" title="S2CID (identifier)">S2CID</a> <a rel="nofollow" class="external text" href="https://api.semanticscholar.org/CorpusID:186210174">186210174</a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=Philosophical+Transactions+of+the+Royal+Society+of+London&rft.atitle=De+reductione+radicalium+ad+simpliciores+terminos%2C+seu+de+extrahenda+radice+quacunque+data+ex+binomio+MATH+RENDER+ERROR%2C+vel+MATH+RENDER+ERROR.+Epistola.&rft.volume=40&rft.issue=451&rft.pages=463-478&rft.date=1738&rft_id=info%3Adoi%2F10.1098%2Frstl.1737.0081&rft_id=https%3A%2F%2Fapi.semanticscholar.org%2FCorpusID%3A186210174%23id-name%3DS2CID&rft.aulast=Moivre&rft.aufirst=A.+de&rfr_id=info%3Asid%2Fen.wikipedia.org%3AAbraham+de+Moivre" class="Z3988"></span> From p. 475: <i>"Problema III. Sit extrahenda radix, cujus index est n, ex binomio impossibli <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle a+{\sqrt {-b}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>a</mi> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mo>−</mo> <mi>b</mi> </msqrt> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle a+{\sqrt {-b}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7477a1a9626d9b0205a2f36eed0212f9f25e0ccc" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:8.812ex; height:3.009ex;" alt="{\displaystyle a+{\sqrt {-b}}}"></span>. … illos autem negativos quorum arcus sunt quadrante majores."</i> (Problem III. Let a root whose index [i.e., degree] is n be extracted from the complex binomial <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle a+{\sqrt {-b}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>a</mi> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mo>−</mo> <mi>b</mi> </msqrt> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle a+{\sqrt {-b}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7477a1a9626d9b0205a2f36eed0212f9f25e0ccc" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:8.812ex; height:3.009ex;" alt="{\displaystyle a+{\sqrt {-b}}}"></span>. Solution. Let its root be <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x+{\sqrt {-y}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>x</mi> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mo>−</mo> <mi>y</mi> </msqrt> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle x+{\sqrt {-y}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f433fe7a2ca320e7039f8b08281b4b342d8eea99" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.338ex; width:9.457ex; height:3.509ex;" alt="{\displaystyle x+{\sqrt {-y}}}"></span>, then I define <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\sqrt[{n}]{aa+b}}=m}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mroot> <mrow> <mi>a</mi> <mi>a</mi> <mo>+</mo> <mi>b</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </mroot> </mrow> <mo>=</mo> <mi>m</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\sqrt[{n}]{aa+b}}=m}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/0315afa9bba9d895ca5fed35bc4cb255cf7e3283" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:13.372ex; height:3.009ex;" alt="{\displaystyle {\sqrt[{n}]{aa+b}}=m}"></span>; I also define <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\tfrac {n+1}{n}}=p}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mfrac> <mrow> <mi>n</mi> <mo>+</mo> <mn>1</mn> </mrow> <mi>n</mi> </mfrac> </mstyle> </mrow> <mo>=</mo> <mi>p</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\tfrac {n+1}{n}}=p}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/1c5e5ad6c5ade15ce66df6a1a5855c5a1b0a5c88" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:8.191ex; height:3.509ex;" alt="{\displaystyle {\tfrac {n+1}{n}}=p}"></span> [Note: should read: <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\tfrac {n+1}{2}}=p}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mfrac> <mrow> <mi>n</mi> <mo>+</mo> <mn>1</mn> </mrow> <mn>2</mn> </mfrac> </mstyle> </mrow> <mo>=</mo> <mi>p</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\tfrac {n+1}{2}}=p}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b806da21ff7037d3a9a539e45da2d952ca8e8370" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.171ex; width:8.191ex; height:3.676ex;" alt="{\displaystyle {\tfrac {n+1}{2}}=p}"></span> ], draw or imagine a circle, whose radius is <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\sqrt {m}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mi>m</mi> </msqrt> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\sqrt {m}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/1a9b7591c91d2c7318b06857157fb8a1401447f8" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:3.976ex; height:3.009ex;" alt="{\displaystyle {\sqrt {m}}}"></span>, and assume in this [circle] some arc A whose cosine is <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\tfrac {a}{{m}^{p}}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mfrac> <mi>a</mi> <msup> <mrow class="MJX-TeXAtom-ORD"> <mi>m</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>p</mi> </mrow> </msup> </mfrac> </mstyle> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\tfrac {a}{{m}^{p}}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/bf75d34a139fff300597ce4bb47d27affda5edce" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.171ex; width:3.114ex; height:3.176ex;" alt="{\displaystyle {\tfrac {a}{{m}^{p}}}}"></span> ; let C be the entire circumference. Assume, [measured] at the same radius, the cosines of the arcs <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\tfrac {A}{n}},{\tfrac {C-A}{n}},{\tfrac {C+A}{n}},{\tfrac {2C-A}{n}},{\tfrac {2C+A}{n}},{\tfrac {3C-A}{n}},{\tfrac {3C+A}{n}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mfrac> <mi>A</mi> <mi>n</mi> </mfrac> </mstyle> </mrow> <mo>,</mo> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mfrac> <mrow> <mi>C</mi> <mo>−</mo> <mi>A</mi> </mrow> <mi>n</mi> </mfrac> </mstyle> </mrow> <mo>,</mo> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mfrac> <mrow> <mi>C</mi> <mo>+</mo> <mi>A</mi> </mrow> <mi>n</mi> </mfrac> </mstyle> </mrow> <mo>,</mo> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mfrac> <mrow> <mn>2</mn> <mi>C</mi> <mo>−</mo> <mi>A</mi> </mrow> <mi>n</mi> </mfrac> </mstyle> </mrow> <mo>,</mo> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mfrac> <mrow> <mn>2</mn> <mi>C</mi> <mo>+</mo> <mi>A</mi> </mrow> <mi>n</mi> </mfrac> </mstyle> </mrow> <mo>,</mo> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mfrac> <mrow> <mn>3</mn> <mi>C</mi> <mo>−</mo> <mi>A</mi> </mrow> <mi>n</mi> </mfrac> </mstyle> </mrow> <mo>,</mo> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mfrac> <mrow> <mn>3</mn> <mi>C</mi> <mo>+</mo> <mi>A</mi> </mrow> <mi>n</mi> </mfrac> </mstyle> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\tfrac {A}{n}},{\tfrac {C-A}{n}},{\tfrac {C+A}{n}},{\tfrac {2C-A}{n}},{\tfrac {2C+A}{n}},{\tfrac {3C-A}{n}},{\tfrac {3C+A}{n}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/386e65c8977c12b93f41a04eb93313c488f3b4fa" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:39.138ex; height:3.676ex;" alt="{\displaystyle {\tfrac {A}{n}},{\tfrac {C-A}{n}},{\tfrac {C+A}{n}},{\tfrac {2C-A}{n}},{\tfrac {2C+A}{n}},{\tfrac {3C-A}{n}},{\tfrac {3C+A}{n}}}"></span>, etc.<br /> until the multitude [i.e., number] of them [i.e., the arcs] equals the number n; when this is done, stop there; then there will be as many cosines as values of the quantity <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>x</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle x}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/87f9e315fd7e2ba406057a97300593c4802b53e4" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.33ex; height:1.676ex;" alt="{\displaystyle x}"></span>, which is related to the quantity <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle y}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>y</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle y}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b8a6208ec717213d4317e666f1ae872e00620a0d" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.155ex; height:2.009ex;" alt="{\displaystyle y}"></span>; this [i.e., <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle y}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>y</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle y}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b8a6208ec717213d4317e666f1ae872e00620a0d" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.155ex; height:2.009ex;" alt="{\displaystyle y}"></span>] will always be <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle m-xx}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>m</mi> <mo>−</mo> <mi>x</mi> <mi>x</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle m-xx}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e7da9d27d3dd38d390e5eb345a92aa05258555ce" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:7.54ex; height:2.176ex;" alt="{\displaystyle m-xx}"></span>.<br /> It is not to be neglected, although it was mentioned previously, [that] those cosines whose arcs are less than a right angle must be regarded as positive but those whose arcs are greater than a right angle [must be regarded as] negative.)<br /> See also: <ul><li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFBraunmühl1903" class="citation book cs1 cs1-prop-foreign-lang-source">Braunmühl, A. von (1903). <a rel="nofollow" class="external text" href="https://books.google.com/books?id=uB0PAAAAIAAJ&pg=PA76"><i>Vorlesungen über Geschichte der Trigonometrie</i></a> [<i>Lectures on the history of trigonometry</i>] (in German). Vol. 2. Leipzig, Germany: B.G. Teubner. pp. 76–77.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Vorlesungen+%C3%BCber+Geschichte+der+Trigonometrie&rft.place=Leipzig%2C+Germany&rft.pages=76-77&rft.pub=B.G.+Teubner&rft.date=1903&rft.aulast=Braunm%C3%BChl&rft.aufirst=A.+von&rft_id=https%3A%2F%2Fbooks.google.com%2Fbooks%3Fid%3DuB0PAAAAIAAJ%26pg%3DPA76&rfr_id=info%3Asid%2Fen.wikipedia.org%3AAbraham+de+Moivre" class="Z3988"></span></li></ul> </span></li> <li id="cite_note-17"><span class="mw-cite-backlink"><b><a href="#cite_ref-17">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFEuler1749" class="citation journal cs1 cs1-prop-foreign-lang-source">Euler (1749). <a rel="nofollow" class="external text" href="https://archive.org/details/euler-e170/page/n37/mode/2up">"Recherches sur les racines imaginaires des equations"</a> [Investigations into the complex roots of equations]. <i>Mémoires de l'académie des sciences de Berlin</i> (in French). <b>5</b>: 222–288.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=M%C3%A9moires+de+l%27acad%C3%A9mie+des+sciences+de+Berlin&rft.atitle=Recherches+sur+les+racines+imaginaires+des+equations&rft.volume=5&rft.pages=222-288&rft.date=1749&rft.au=Euler&rft_id=https%3A%2F%2Farchive.org%2Fdetails%2Feuler-e170%2Fpage%2Fn37%2Fmode%2F2up&rfr_id=info%3Asid%2Fen.wikipedia.org%3AAbraham+de+Moivre" class="Z3988"></span> See pp. 260–261: "<i>Theorem XIII. §. 70. De quelque puissance qu'on extraye la racine, ou d'une quantité réelle, ou d'une imaginaire de la forme M + N √-1, les racines seront toujours, ou réelles, ou imaginaires de la même forme M + N √-1.</i>" (Theorem XIII. §. 70. For any power, either a real quantity or a complex [one] of the form <i>M</i> + <i>N</i> √−1, from which one extracts the root, the roots will always be either real or complex of the same form <i>M</i> + <i>N</i>√−1.)</span> </li> <li id="cite_note-18"><span class="mw-cite-backlink"><b><a href="#cite_ref-18">^</a></b></span> <span class="reference-text">De Moivre had been trying to determine the coefficient of the middle term of (1 + 1)<sup><i>n</i></sup> for large <i>n</i> since 1721 or earlier. In his pamphlet of November 12, 1733 – "Approximatio ad Summam Terminorum Binomii (<i>a</i> + <i>b</i>)<sup><i>n</i></sup> in Seriem expansi" [Approximation of the Sum of the Terms of the Binomial (<i>a</i> + <i>b</i>)<sup><i>n</i></sup> expanded into a Series] – de Moivre said that he had started working on the problem 12 years or more ago: <i>"Duodecim jam sunt anni & amplius cum illud inveneram; … "</i> (It is now a dozen years or more since I found this [i.e., what follows]; … ). <ul><li>(Archibald, 1926), p. 677.</li> <li>(de Moivre, 1738), p. 235.</li></ul> De Moivre credited Alexander Cuming (ca. 1690 – 1775), a Scottish aristocrat and member of the Royal Society of London, with motivating, in 1721, his search to find an approximation for the central term of a binomial expansion. (de Moivre, 1730), p. 99.</span> </li> <li id="cite_note-19"><span class="mw-cite-backlink"><b><a href="#cite_ref-19">^</a></b></span> <span class="reference-text">The roles of de Moivre and Stirling in finding Stirling's approximation are presented in: <ul><li>Gélinas, Jacques (24 January 2017) <a rel="nofollow" class="external text" href="https://arxiv.org/pdf/1701.06689.pdf">"Original proofs of Stirling's series for log (<i>N</i>!)"</a> arxiv.org</li> <li>Lanier, Denis; Trotoux, Didier (1998). "La formule de Stirling" [Stirling's formula] Commission inter-IREM histoire et épistémologie des mathématiques (ed.). <i>Analyse & démarche analytique : les neveux de Descartes : actes du XIème Colloque inter-IREM d'épistémologie et d'histoire des mathématiques, Reims, 10 et 11 mai 1996</i> [Analysis and analytic reasoning: the "nephews" of Decartes: proceedings of the 11th inter-IREM colloquium on epistemology and the history of mathematics, Reims, 10–11 May 1996] (in French). Reims, France: IREM [Institut de Rercherche sur l'Enseignement des Mathématiques] de Reims. pp. 231–286.</li></ul> </span></li> <li id="cite_note-20"><span class="mw-cite-backlink"><b><a href="#cite_ref-20">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFMoivre1730" class="citation book cs1">Moivre, A. de (1730). <a rel="nofollow" class="external text" href="https://books.google.com/books?id=bMl5NAAACAAJ&pg=PA103"><i>Miscellanea Analytica de Seriebus et Quadraturis</i></a> [<i>Analytical Miscellany of Series and Quadratures [i.e., Integrals]</i>]. London, England: J. Tonson & J. Watts. pp. 103–104.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Miscellanea+Analytica+de+Seriebus+et+Quadraturis&rft.place=London%2C+England&rft.pages=103-104&rft.pub=J.+Tonson+%26+J.+Watts&rft.date=1730&rft.aulast=Moivre&rft.aufirst=A.+de&rft_id=https%3A%2F%2Fbooks.google.com%2Fbooks%3Fid%3DbMl5NAAACAAJ%26pg%3DPA103&rfr_id=info%3Asid%2Fen.wikipedia.org%3AAbraham+de+Moivre" class="Z3988"></span></span> </li> <li id="cite_note-21"><span class="mw-cite-backlink"><b><a href="#cite_ref-21">^</a></b></span> <span class="reference-text"><a rel="nofollow" class="external text" href="https://books.google.com/books?id=bMl5NAAACAAJ&pg=PA102">From p. 102 of (de Moivre, 1730)</a>: <i>"Problema III. Invenire Coefficientem Termini medii potestatis permagnae & paris, seu invenire rationem quam Coefficiens termini medii habeat ad summam omnium Coefficientium. … ad 1 proxime."</i><br /> (Problem 3. Find the coefficient of the middle term [of a binomial expansion] for a very large and even power [<i>n</i>], or find the ratio that the coefficient of the middle term has to the sum of all coefficients.<br /> Solution. Let <i>n</i> be the degree of the power to which the binomial <i>a</i> + <i>b</i> is raised, then, setting [both] <i>a</i> and <i>b</i> = 1, the ratio of the middle term to its power (<i>a</i> + <i>b</i>)<sup><i>n</i></sup> or 2<sup><i>n</i></sup> [Note: the sum of all the coefficients of the binomial expansion of (1 + 1)<sup>n</sup> is 2<sup><i>n</i></sup>.] will be nearly as <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\frac {2(n-1)^{n-{\frac {1}{2}}}}{{n}^{n}}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mn>2</mn> <mo stretchy="false">(</mo> <mi>n</mi> <mo>−</mo> <mn>1</mn> <msup> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mn>2</mn> </mfrac> </mrow> </mrow> </msup> </mrow> <msup> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msup> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\frac {2(n-1)^{n-{\frac {1}{2}}}}{{n}^{n}}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/db73dad6bd5fa33e42b614659612c1309949472c" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:13.206ex; height:6.843ex;" alt="{\displaystyle {\frac {2(n-1)^{n-{\frac {1}{2}}}}{{n}^{n}}}}"></span> to 1.<br /> But when some series for an inquiry could be determined more accurately [but] had been neglected due to lack of time, I then calculate by re-integration [and] I recover for use the particular quantities [that] had previously been neglected; so it happened that I could finally conclude that the ratio [that's] sought is approximately <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\frac {2{\frac {21}{125}}{(n-1)}^{n-{\frac {1}{2}}}}{n^{n}}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mn>2</mn> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>21</mn> <mn>125</mn> </mfrac> </mrow> <msup> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">(</mo> <mi>n</mi> <mo>−</mo> <mn>1</mn> <mo stretchy="false">)</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mn>2</mn> </mfrac> </mrow> </mrow> </msup> </mrow> <msup> <mi>n</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msup> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\frac {2{\frac {21}{125}}{(n-1)}^{n-{\frac {1}{2}}}}{n^{n}}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/39f64bb1a193a153bdb10be159abed9c592a46a9" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:16.508ex; height:7.176ex;" alt="{\displaystyle {\frac {2{\frac {21}{125}}{(n-1)}^{n-{\frac {1}{2}}}}{n^{n}}}}"></span> or <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\frac {2{\frac {21}{125}}{(1-{\frac {1}{n}})}^{n}}{\sqrt {n-1}}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mn>2</mn> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>21</mn> <mn>125</mn> </mfrac> </mrow> <msup> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">(</mo> <mn>1</mn> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mi>n</mi> </mfrac> </mrow> <mo stretchy="false">)</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msup> </mrow> <msqrt> <mi>n</mi> <mo>−</mo> <mn>1</mn> </msqrt> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\frac {2{\frac {21}{125}}{(1-{\frac {1}{n}})}^{n}}{\sqrt {n-1}}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/0c7ae7f40a7023cf0515963bfd2d91c1487d1cc8" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.838ex; width:14.154ex; height:7.509ex;" alt="{\displaystyle {\frac {2{\frac {21}{125}}{(1-{\frac {1}{n}})}^{n}}{\sqrt {n-1}}}}"></span> to 1.)<br /> The approximation <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\frac {{2}{\frac {21}{125}}{(n-1)}^{n-{\frac {1}{2}}}}{{n}^{n}}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>21</mn> <mn>125</mn> </mfrac> </mrow> <msup> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">(</mo> <mi>n</mi> <mo>−</mo> <mn>1</mn> <mo stretchy="false">)</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mn>2</mn> </mfrac> </mrow> </mrow> </msup> </mrow> <msup> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msup> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\frac {{2}{\frac {21}{125}}{(n-1)}^{n-{\frac {1}{2}}}}{{n}^{n}}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/fbbbd289b59717e9efc415a9e7cd687805f96eb1" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:16.508ex; height:7.176ex;" alt="{\displaystyle {\frac {{2}{\frac {21}{125}}{(n-1)}^{n-{\frac {1}{2}}}}{{n}^{n}}}}"></span> is derived on pp. 124-128 of (de Moivre, 1730).</span> </li> <li id="cite_note-22"><span class="mw-cite-backlink"><b><a href="#cite_ref-22">^</a></b></span> <span class="reference-text">De Moivre determined the value of the constant <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \textstyle 2{\frac {21}{125}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mstyle displaystyle="false" scriptlevel="0"> <mn>2</mn> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>21</mn> <mn>125</mn> </mfrac> </mrow> </mstyle> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \textstyle 2{\frac {21}{125}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ab7e56997031bb55b79c5527712fd546c731b623" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.338ex; width:4.465ex; height:3.676ex;" alt="{\displaystyle \textstyle 2{\frac {21}{125}}}"></span> by approximating the value of a series by using only its first four terms. De Moivre thought that the series converged, but the English mathematician <a href="/wiki/Thomas_Bayes" title="Thomas Bayes">Thomas Bayes</a> (ca. 1701–1761) found that the series actually diverged. From pp. 127-128 of (de Moivre, 1730): <i>"Cum vero perciperem has Series valde implicatas evadere, … conclusi factorem 2.168 seu <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\textstyle 2{\frac {21}{125}},\ldots }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mn>2</mn> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>21</mn> <mn>125</mn> </mfrac> </mrow> <mo>,</mo> <mo>…</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\textstyle 2{\frac {21}{125}},\ldots }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ce187ec12a6049f2b2ba9b51ef8289bd00488a9a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.338ex; width:8.222ex; height:3.676ex;" alt="{\textstyle 2{\frac {21}{125}},\ldots }"></span> "</i> (But when I conceived [how] to avoid these very complicated series — although all of them were perfectly summable — I think that [there was] nothing else to be done, than to transform them to the infinite case; thus set m to infinity, then the sum of the first rational series will be reduced to 1/12, the sum of the second [will be reduced] to 1/360; thus it happens that the sums of all the series are achieved. From this one series <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \textstyle {\frac {1}{12}}-{\frac {1}{360}}+{\frac {1}{1260}}-{\frac {1}{1680}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mstyle displaystyle="false" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mn>12</mn> </mfrac> </mrow> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mn>360</mn> </mfrac> </mrow> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mn>1260</mn> </mfrac> </mrow> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mn>1680</mn> </mfrac> </mrow> </mstyle> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \textstyle {\frac {1}{12}}-{\frac {1}{360}}+{\frac {1}{1260}}-{\frac {1}{1680}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/59746808375a963473c32ecb2b9908bebe392056" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.338ex; width:22.551ex; height:3.676ex;" alt="{\displaystyle \textstyle {\frac {1}{12}}-{\frac {1}{360}}+{\frac {1}{1260}}-{\frac {1}{1680}}}"></span>, etc., one will be able to discard as many terms as it will be one's pleasure; but I decided [to retain] four [terms] of this [series], because they sufficed [as] a sufficiently accurate approximation; now when this series be convergent, then its terms decrease with alternating positive and negative signs, [and] one may infer that the first term 1/12 is larger [than] the sum of the series, or the first term is larger [than] the difference that exists between all positive terms and all negative terms; but that term should be regarded as a hyperbolic [i.e., natural] logarithm; further, the number corresponding to this logarithm is nearly 1.0869 [i.e., ln(1.0869) ≈ 1/12], which if multiplied by 2, the product will be 2.1738, and so [in the case of a binomial being raised] to an infinite power, designated by <i>n</i>, the quantity <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \textstyle {\frac {{2.1738}{(n-1)}^{n-{\frac {1}{2}}}}{n^{n}}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mstyle displaystyle="false" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2.1738</mn> </mrow> <msup> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">(</mo> <mi>n</mi> <mo>−</mo> <mn>1</mn> <mo stretchy="false">)</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mn>2</mn> </mfrac> </mrow> </mrow> </msup> </mrow> <msup> <mi>n</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msup> </mfrac> </mrow> </mstyle> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \textstyle {\frac {{2.1738}{(n-1)}^{n-{\frac {1}{2}}}}{n^{n}}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/0e2503d1cea0758bde3906b6ea1754de1697122f" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.171ex; width:13.276ex; height:5.676ex;" alt="{\displaystyle \textstyle {\frac {{2.1738}{(n-1)}^{n-{\frac {1}{2}}}}{n^{n}}}}"></span> will be larger than the ratio that the middle term of the binomial has to the sum of all terms, and proceeding to the remaining terms, it will be discovered that the factor 2.1676 is just smaller [than the ratio of the middle term to the sum of all terms], and similarly that 2.1695 is greater, in turn that 2.1682 sinks a little bit below the true [value of the ratio]; considering which, I concluded that the factor [is] 2.168 or <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\textstyle 2{\frac {21}{125}},\ldots }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mn>2</mn> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>21</mn> <mn>125</mn> </mfrac> </mrow> <mo>,</mo> <mo>…</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\textstyle 2{\frac {21}{125}},\ldots }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ce187ec12a6049f2b2ba9b51ef8289bd00488a9a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.338ex; width:8.222ex; height:3.676ex;" alt="{\textstyle 2{\frac {21}{125}},\ldots }"></span> Note: The factor that de Moivre was seeking, was: <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\frac {2e}{\sqrt {2\pi }}}=2.16887\ldots }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mn>2</mn> <mi>e</mi> </mrow> <msqrt> <mn>2</mn> <mi>π</mi> </msqrt> </mfrac> </mrow> <mo>=</mo> <mn>2.16887</mn> <mo>…</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\frac {2e}{\sqrt {2\pi }}}=2.16887\ldots }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7f52bc0dbec9088b060b13416760e723e2d6cfe4" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.838ex; width:19.097ex; height:6.176ex;" alt="{\displaystyle {\frac {2e}{\sqrt {2\pi }}}=2.16887\ldots }"></span> (Lanier & Trotoux, 1998), p. 237. <ul><li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFBayes1763" class="citation journal cs1">Bayes, Thomas (31 December 1763). "A letter from the late Reverend Mr. Bayes, F.R.S. to John Canton, M.A. and F.R.S.". <i>Philosophical Transactions of the Royal Society of London</i>. <b>53</b>: 269–271. <a href="/wiki/Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1098%2Frstl.1763.0044">10.1098/rstl.1763.0044</a>. <a href="/wiki/S2CID_(identifier)" class="mw-redirect" title="S2CID (identifier)">S2CID</a> <a rel="nofollow" class="external text" href="https://api.semanticscholar.org/CorpusID:186214800">186214800</a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=Philosophical+Transactions+of+the+Royal+Society+of+London&rft.atitle=A+letter+from+the+late+Reverend+Mr.+Bayes%2C+F.R.S.+to+John+Canton%2C+M.A.+and+F.R.S.&rft.volume=53&rft.pages=269-271&rft.date=1763-12-31&rft_id=info%3Adoi%2F10.1098%2Frstl.1763.0044&rft_id=https%3A%2F%2Fapi.semanticscholar.org%2FCorpusID%3A186214800%23id-name%3DS2CID&rft.aulast=Bayes&rft.aufirst=Thomas&rfr_id=info%3Asid%2Fen.wikipedia.org%3AAbraham+de+Moivre" class="Z3988"></span></li></ul> </span></li> <li id="cite_note-23"><span class="mw-cite-backlink"><b><a href="#cite_ref-23">^</a></b></span> <span class="reference-text">(de Moivre, 1730), pp. 170–172.</span> </li> <li id="cite_note-24"><span class="mw-cite-backlink"><b><a href="#cite_ref-24">^</a></b></span> <span class="reference-text">In Stirling's letter of June 19, 1729 to de Moivre, Stirling stated that he had written to Alexander Cuming <i>"quadrienium circiter abhinc"</i> (about four years ago [i.e., 1725]) about (among other things) approximating, by using Isaac Newton's method of differentials, the coefficient of the middle term of a binomial expansion. Stirling acknowledged that de Moivre had solved the problem years earlier: <i>" … ; respondit Illustrissimus vir se dubitare an Problema a Te aliquot ante annos solutum de invenienda Uncia media in quavis dignitate Binonii solvi posset per Differentias."</i> ( ... ; this most illustrious man [Alexander Cuming] responded that he doubted whether the problem solved by you several years earlier, concerning the behavior of the middle term of any power of the binomial, could be solved by differentials.) Stirling wrote that he had then commenced to investigate the problem, but that initially his progress was slow. <ul><li>(de Moivre, 1730), p. 170.</li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFZabell2005" class="citation book cs1">Zabell, S.L. (2005). <a rel="nofollow" class="external text" href="https://books.google.com/books?id=6fd1IeWpdiYC&pg=PA113"><i>Symmetry and Its Discontents: Essays on the History of Inductive Probability</i></a>. New York City, New York, USA: Cambridge University Press. p. 113. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/9780521444705" title="Special:BookSources/9780521444705"><bdi>9780521444705</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Symmetry+and+Its+Discontents%3A+Essays+on+the+History+of+Inductive+Probability&rft.place=New+York+City%2C+New+York%2C+USA&rft.pages=113&rft.pub=Cambridge+University+Press&rft.date=2005&rft.isbn=9780521444705&rft.aulast=Zabell&rft.aufirst=S.L.&rft_id=https%3A%2F%2Fbooks.google.com%2Fbooks%3Fid%3D6fd1IeWpdiYC%26pg%3DPA113&rfr_id=info%3Asid%2Fen.wikipedia.org%3AAbraham+de+Moivre" class="Z3988"></span></li></ul> </span></li> <li id="cite_note-25"><span class="mw-cite-backlink"><b><a href="#cite_ref-25">^</a></b></span> <span class="reference-text">See: <ul><li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFStirling1730" class="citation book cs1 cs1-prop-foreign-lang-source">Stirling, James (1730). <a rel="nofollow" class="external text" href="https://archive.org/details/bub_gb_71ZHAAAAYAAJ/page/n145/mode/2up"><i>Methodus Differentialis …</i></a> (in Latin). London: G. Strahan. p. 137.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Methodus+Differentialis+%E2%80%A6&rft.place=London&rft.pages=137&rft.pub=G.+Strahan&rft.date=1730&rft.aulast=Stirling&rft.aufirst=James&rft_id=https%3A%2F%2Farchive.org%2Fdetails%2Fbub_gb_71ZHAAAAYAAJ%2Fpage%2Fn145%2Fmode%2F2up&rfr_id=info%3Asid%2Fen.wikipedia.org%3AAbraham+de+Moivre" class="Z3988"></span> From p. 137: <i>"Ceterum si velis summam quotcunque Logarithmorum numerorum naturalam 1, 2, 3, 4, 5, &c. pone z–n esse ultimum numerorum, existente n = ½ ; & tres vel quatuor Termini hujus Seriei <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle zl,z-az-{\frac {a}{24z}}+{\frac {7a}{2880z^{3}}}-{}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>z</mi> <mi>l</mi> <mo>,</mo> <mi>z</mi> <mo>−</mo> <mi>a</mi> <mi>z</mi> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>a</mi> <mrow> <mn>24</mn> <mi>z</mi> </mrow> </mfrac> </mrow> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mn>7</mn> <mi>a</mi> </mrow> <mrow> <mn>2880</mn> <msup> <mi>z</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msup> </mrow> </mfrac> </mrow> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle zl,z-az-{\frac {a}{24z}}+{\frac {7a}{2880z^{3}}}-{}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4ed0939e845aa4444458bd8e617ba11a1dd75258" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.171ex; width:29.463ex; height:5.509ex;" alt="{\displaystyle zl,z-az-{\frac {a}{24z}}+{\frac {7a}{2880z^{3}}}-{}}"></span> </i>[Note: l,z = log(z)]<i> additi Logarithmo circumferentiae Circuli cujus Radius est Unitas, id est, huic 0.39908.99341.79 dabunt summam quaesitam, idque eo minore labore quo plures Logarithmi sunt summandi."</i> (Furthermore, if you want the sum of however many logarithms of the natural numbers 1, 2, 3, 4, 5, etc., set z–n to be the last number, n being ½ ; and three or four terms of this series <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle z\log(z)-az-{\frac {a}{24z}}+{\frac {7a}{2880z^{3}}}-{}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>z</mi> <mi>log</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mi>z</mi> <mo stretchy="false">)</mo> <mo>−</mo> <mi>a</mi> <mi>z</mi> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>a</mi> <mrow> <mn>24</mn> <mi>z</mi> </mrow> </mfrac> </mrow> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mn>7</mn> <mi>a</mi> </mrow> <mrow> <mn>2880</mn> <msup> <mi>z</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msup> </mrow> </mfrac> </mrow> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle z\log(z)-az-{\frac {a}{24z}}+{\frac {7a}{2880z^{3}}}-{}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/bbd03874aa6335b8fa7066bf6af915079245b37d" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.171ex; width:32.904ex; height:5.509ex;" alt="{\displaystyle z\log(z)-az-{\frac {a}{24z}}+{\frac {7a}{2880z^{3}}}-{}}"></span> added to [half of] the logarithm of the circumference of a circle whose radius is unity [i.e., ½ log(2<span class="texhtml mvar" style="font-style:italic;">π</span>)] – that is, [added] to this: 0.39908.99341.79 – will give the sum [that's] sought, and the more logarithms [that] are to be added, the less work it [is].) Note: <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle a=0.434294481903252}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>a</mi> <mo>=</mo> <mn>0.434294481903252</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle a=0.434294481903252}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b5b905872d3ce09631298147b2d43b11a421e7d3" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:23.574ex; height:2.176ex;" alt="{\displaystyle a=0.434294481903252}"></span> (See p. 135.) = 1/ln(10).</li> <li>English translation: <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFStirling1749" class="citation book cs1">Stirling, James (1749). <a rel="nofollow" class="external text" href="https://books.google.com/books?id=j2xbAAAAQAAJ&pg=PA125"><i>The Differential Method</i></a>. Translated by Holliday, Francis. London, England: E. Cave. p. 121.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=The+Differential+Method&rft.place=London%2C+England&rft.pages=121&rft.pub=E.+Cave&rft.date=1749&rft.aulast=Stirling&rft.aufirst=James&rft_id=https%3A%2F%2Fbooks.google.com%2Fbooks%3Fid%3Dj2xbAAAAQAAJ%26pg%3DPA125&rfr_id=info%3Asid%2Fen.wikipedia.org%3AAbraham+de+Moivre" class="Z3988"></span> [Note: The printer incorrectly numbered the pages of this book, so that page 125 is numbered as "121", page 126 as "122", and so forth until p. 129.]</li></ul> </span></li> <li id="cite_note-26"><span class="mw-cite-backlink"><b><a href="#cite_ref-26">^</a></b></span> <span class="reference-text">See: <ul><li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFArchibald1926" class="citation journal cs1 cs1-prop-foreign-lang-source">Archibald, R.C. (October 1926). "A rare pamphlet of Moivre and some of his discoveries". <i>Isis</i> (in English and Latin). <b>8</b> (4): 671–683. <a href="/wiki/Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1086%2F358439">10.1086/358439</a>. <a href="/wiki/S2CID_(identifier)" class="mw-redirect" title="S2CID (identifier)">S2CID</a> <a rel="nofollow" class="external text" href="https://api.semanticscholar.org/CorpusID:143827655">143827655</a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=Isis&rft.atitle=A+rare+pamphlet+of+Moivre+and+some+of+his+discoveries&rft.volume=8&rft.issue=4&rft.pages=671-683&rft.date=1926-10&rft_id=info%3Adoi%2F10.1086%2F358439&rft_id=https%3A%2F%2Fapi.semanticscholar.org%2FCorpusID%3A143827655%23id-name%3DS2CID&rft.aulast=Archibald&rft.aufirst=R.C.&rfr_id=info%3Asid%2Fen.wikipedia.org%3AAbraham+de+Moivre" class="Z3988"></span></li> <li>An English translation of the pamphlet appears in: <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFMoivre1738" class="citation book cs1">Moivre, Abraham de (1738). <a rel="nofollow" class="external text" href="https://books.google.com/books?id=PII_AAAAcAAJ&pg=PA235"><i>The Doctrine of Chances …</i></a> (2nd ed.). London, England: Self-published. pp. 235–243.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=The+Doctrine+of+Chances+%E2%80%A6&rft.place=London%2C+England&rft.pages=235-243&rft.edition=2nd&rft.pub=Self-published&rft.date=1738&rft.aulast=Moivre&rft.aufirst=Abraham+de&rft_id=https%3A%2F%2Fbooks.google.com%2Fbooks%3Fid%3DPII_AAAAcAAJ%26pg%3DPA235&rfr_id=info%3Asid%2Fen.wikipedia.org%3AAbraham+de+Moivre" class="Z3988"></span></li></ul> </span></li> </ol></div></div> <div class="mw-heading mw-heading2"><h2 id="References">References</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Abraham_de_Moivre&action=edit&section=11" title="Edit section: References"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <ul><li>See de Moivre's <i>Miscellanea Analytica</i> (London: 1730) pp 26–42.</li> <li><a href="/wiki/H._J._R._Murray" title="H. J. R. Murray">H. J. R. Murray</a>, 1913. <i>History of Chess</i>. Oxford University Press: p 846.</li> <li>Schneider, I., 2005, "The doctrine of chances" in <a href="/wiki/Ivor_Grattan-Guinness" title="Ivor Grattan-Guinness">Grattan-Guinness, I.</a>, ed., <i>Landmark Writings in Western Mathematics</i>. Elsevier: pp 105–20</li></ul> <div class="mw-heading mw-heading2"><h2 id="Further_reading">Further reading</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Abraham_de_Moivre&action=edit&section=12" title="Edit section: Further reading"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <style data-mw-deduplicate="TemplateStyles:r1235681985">.mw-parser-output .side-box{margin:4px 0;box-sizing:border-box;border:1px solid #aaa;font-size:88%;line-height:1.25em;background-color:var(--background-color-interactive-subtle,#f8f9fa);display:flow-root}.mw-parser-output .side-box-abovebelow,.mw-parser-output .side-box-text{padding:0.25em 0.9em}.mw-parser-output .side-box-image{padding:2px 0 2px 0.9em;text-align:center}.mw-parser-output .side-box-imageright{padding:2px 0.9em 2px 0;text-align:center}@media(min-width:500px){.mw-parser-output .side-box-flex{display:flex;align-items:center}.mw-parser-output .side-box-text{flex:1;min-width:0}}@media(min-width:720px){.mw-parser-output .side-box{width:238px}.mw-parser-output .side-box-right{clear:right;float:right;margin-left:1em}.mw-parser-output .side-box-left{margin-right:1em}}</style><style data-mw-deduplicate="TemplateStyles:r1237033735">@media print{body.ns-0 .mw-parser-output .sistersitebox{display:none!important}}@media screen{html.skin-theme-clientpref-night .mw-parser-output .sistersitebox img[src*="Wiktionary-logo-en-v2.svg"]{background-color:white}}@media screen and (prefers-color-scheme:dark){html.skin-theme-clientpref-os .mw-parser-output .sistersitebox img[src*="Wiktionary-logo-en-v2.svg"]{background-color:white}}</style><div class="side-box side-box-right plainlinks sistersitebox"><style data-mw-deduplicate="TemplateStyles:r1126788409">.mw-parser-output .plainlist ol,.mw-parser-output .plainlist ul{line-height:inherit;list-style:none;margin:0;padding:0}.mw-parser-output .plainlist ol li,.mw-parser-output .plainlist ul li{margin-bottom:0}</style> <div class="side-box-flex"> <div class="side-box-image"><span class="noviewer" typeof="mw:File"><span><img alt="" src="//upload.wikimedia.org/wikipedia/commons/thumb/4/4c/Wikisource-logo.svg/38px-Wikisource-logo.svg.png" decoding="async" width="38" height="40" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/4/4c/Wikisource-logo.svg/57px-Wikisource-logo.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/4/4c/Wikisource-logo.svg/76px-Wikisource-logo.svg.png 2x" data-file-width="410" data-file-height="430" /></span></span></div> <div class="side-box-text plainlist"><a href="/wiki/Wikisource" title="Wikisource">Wikisource</a> has original works by or about:<br /><b style="text-align: center;"><i><a href="https://en.wikisource.org/wiki/en:Author:Abraham_de_Moivre" class="extiw" title="s:en:Author:Abraham de Moivre">Abraham de Moivre</a></i></b></div></div> </div> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1235681985"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1237033735"><div class="side-box side-box-right plainlinks sistersitebox"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1126788409"> <div class="side-box-flex"> <div class="side-box-image"><span class="noviewer" typeof="mw:File"><span><img alt="" src="//upload.wikimedia.org/wikipedia/en/thumb/4/4a/Commons-logo.svg/30px-Commons-logo.svg.png" decoding="async" width="30" height="40" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/en/thumb/4/4a/Commons-logo.svg/45px-Commons-logo.svg.png 1.5x, //upload.wikimedia.org/wikipedia/en/thumb/4/4a/Commons-logo.svg/59px-Commons-logo.svg.png 2x" data-file-width="1024" data-file-height="1376" /></span></span></div> <div class="side-box-text plainlist">Wikimedia Commons has media related to <span style="font-weight: bold; font-style: italic;"><a href="https://commons.wikimedia.org/wiki/Category:Abraham_de_Moivre" class="extiw" title="commons:Category:Abraham de Moivre">Abraham de Moivre</a></span>.</div></div> </div> <ul><li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite class="citation web cs1"><a rel="nofollow" class="external text" href="https://web.archive.org/web/20071219233914/http://euler.ciens.ucv.ve/English/mathematics/demoivre.html">"de Moivre, Abraham"</a>. Archived from <a rel="nofollow" class="external text" href="http://euler.ciens.ucv.ve/English/mathematics/demoivre.html">the original</a> on 19 December 2007<span class="reference-accessdate">. Retrieved <span class="nowrap">15 June</span> 2002</span>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=unknown&rft.btitle=de+Moivre%2C+Abraham&rft_id=http%3A%2F%2Feuler.ciens.ucv.ve%2FEnglish%2Fmathematics%2Fdemoivre.html&rfr_id=info%3Asid%2Fen.wikipedia.org%3AAbraham+de+Moivre" class="Z3988"></span></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite class="citation encyclopaedia cs1"><span class="cs1-ws-icon" title="s:Encyclopædia Britannica, Ninth Edition/Abraham Demoivre"><a class="external text" href="https://en.wikisource.org/wiki/Encyclop%C3%A6dia_Britannica,_Ninth_Edition/Abraham_Demoivre">"Abraham Demoivre" </a></span>. <i><a href="/wiki/Encyclop%C3%A6dia_Britannica" title="Encyclopædia Britannica">Encyclopædia Britannica</a></i>. Vol. VII (9th ed.). 1878. p. 60.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=bookitem&rft.atitle=Abraham+Demoivre&rft.btitle=Encyclop%C3%A6dia+Britannica&rft.pages=60&rft.edition=9th&rft.date=1878&rfr_id=info%3Asid%2Fen.wikipedia.org%3AAbraham+de+Moivre" class="Z3988"></span></li> <li><a rel="nofollow" class="external text" href="http://www.mathpages.com/home/kmath642/kmath642.htm">The Doctrine of Chance</a> at MathPages.</li> <li><a rel="nofollow" class="external text" href="https://wayback.archive-it.org/all/20080221191829/http://archimede.mat.ulaval.ca/pages/genest/publi/StatSci-2007.pdf">Biography (PDF)</a>, <i><a href="/wiki/Matthew_Maty" title="Matthew Maty">Matthew Maty</a>'s Biography of Abraham De Moivre, Translated, Annotated and Augmented</i>.</li> <li><a rel="nofollow" class="external text" href="https://web.archive.org/web/20181108224526/https://www.york.ac.uk/depts/maths/histstat/demoivre.pdf">de Moivre, On the Law of Normal Probability </a></li></ul> <div class="navbox-styles"><style data-mw-deduplicate="TemplateStyles:r1129693374">.mw-parser-output .hlist dl,.mw-parser-output .hlist ol,.mw-parser-output .hlist ul{margin:0;padding:0}.mw-parser-output .hlist dd,.mw-parser-output .hlist dt,.mw-parser-output .hlist li{margin:0;display:inline}.mw-parser-output .hlist.inline,.mw-parser-output .hlist.inline dl,.mw-parser-output .hlist.inline ol,.mw-parser-output .hlist.inline ul,.mw-parser-output .hlist dl dl,.mw-parser-output .hlist dl ol,.mw-parser-output .hlist dl ul,.mw-parser-output .hlist ol dl,.mw-parser-output .hlist ol ol,.mw-parser-output .hlist ol ul,.mw-parser-output .hlist ul dl,.mw-parser-output .hlist ul ol,.mw-parser-output .hlist ul ul{display:inline}.mw-parser-output .hlist .mw-empty-li{display:none}.mw-parser-output .hlist dt::after{content:": "}.mw-parser-output .hlist dd::after,.mw-parser-output .hlist li::after{content:" · ";font-weight:bold}.mw-parser-output .hlist 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ul{margin-top:0}.mw-parser-output .div-col li,.mw-parser-output .div-col dd{page-break-inside:avoid;break-inside:avoid-column}</style></div><div role="navigation" class="navbox" aria-labelledby="Sir_Isaac_Newton" style="padding:3px"><table class="nowraplinks mw-collapsible autocollapse navbox-inner" style="border-spacing:0;background:transparent;color:inherit"><tbody><tr><th scope="col" class="navbox-title" colspan="2"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1129693374"><style data-mw-deduplicate="TemplateStyles:r1239400231">.mw-parser-output .navbar{display:inline;font-size:88%;font-weight:normal}.mw-parser-output .navbar-collapse{float:left;text-align:left}.mw-parser-output .navbar-boxtext{word-spacing:0}.mw-parser-output .navbar ul{display:inline-block;white-space:nowrap;line-height:inherit}.mw-parser-output .navbar-brackets::before{margin-right:-0.125em;content:"[ "}.mw-parser-output .navbar-brackets::after{margin-left:-0.125em;content:" ]"}.mw-parser-output .navbar li{word-spacing:-0.125em}.mw-parser-output .navbar a>span,.mw-parser-output .navbar a>abbr{text-decoration:inherit}.mw-parser-output .navbar-mini abbr{font-variant:small-caps;border-bottom:none;text-decoration:none;cursor:inherit}.mw-parser-output .navbar-ct-full{font-size:114%;margin:0 7em}.mw-parser-output .navbar-ct-mini{font-size:114%;margin:0 4em}html.skin-theme-clientpref-night .mw-parser-output .navbar li a abbr{color:var(--color-base)!important}@media(prefers-color-scheme:dark){html.skin-theme-clientpref-os .mw-parser-output .navbar li a abbr{color:var(--color-base)!important}}@media print{.mw-parser-output .navbar{display:none!important}}</style><div class="navbar plainlinks hlist navbar-mini"><ul><li class="nv-view"><a href="/wiki/Template:Isaac_Newton" title="Template:Isaac Newton"><abbr title="View this template">v</abbr></a></li><li class="nv-talk"><a href="/wiki/Template_talk:Isaac_Newton" title="Template talk:Isaac Newton"><abbr title="Discuss this template">t</abbr></a></li><li class="nv-edit"><a href="/wiki/Special:EditPage/Template:Isaac_Newton" title="Special:EditPage/Template:Isaac Newton"><abbr title="Edit this template">e</abbr></a></li></ul></div><div id="Sir_Isaac_Newton" style="font-size:114%;margin:0 4em"><a href="/wiki/Isaac_Newton" title="Isaac Newton">Sir Isaac Newton</a></div></th></tr><tr><th scope="row" class="navbox-group" style="width:1%;vertical-align:top;">Publications</th><td class="navbox-list-with-group navbox-list navbox-odd hlist" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><i><a href="/wiki/Method_of_Fluxions" title="Method of Fluxions">Fluxions</a></i> (1671)</li> <li><i><a href="/wiki/De_motu_corporum_in_gyrum" title="De motu corporum in gyrum">De Motu</a></i> (1684)</li> <li><i><a href="/wiki/Philosophi%C3%A6_Naturalis_Principia_Mathematica" title="Philosophiæ Naturalis Principia Mathematica">Principia</a></i> (1687)</li> <li><i><a href="/wiki/Opticks" title="Opticks">Opticks</a></i> (1704)</li> <li><i><a href="/wiki/The_Queries" class="mw-redirect" title="The Queries">Queries</a></i> (1704)</li> <li><i><a href="/wiki/Arithmetica_Universalis" title="Arithmetica Universalis">Arithmetica</a></i> (1707)</li> <li><i><a href="/wiki/De_analysi_per_aequationes_numero_terminorum_infinitas" title="De analysi per aequationes numero terminorum infinitas">De Analysi</a></i> (1711)</li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%;vertical-align:top;">Other writings</th><td class="navbox-list-with-group navbox-list navbox-even hlist" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><i><a href="/wiki/Quaestiones_quaedam_philosophicae" title="Quaestiones quaedam philosophicae">Quaestiones</a></i> (1661–1665)</li> <li>"<a href="/wiki/Standing_on_the_shoulders_of_giants" title="Standing on the shoulders of giants">standing on the shoulders of giants</a>" (1675)</li> <li><i><a href="/wiki/Notes_on_the_Jewish_Temple" title="Notes on the Jewish Temple">Notes on the Jewish Temple</a></i> (c. 1680)</li> <li>"<a href="/wiki/General_Scholium" title="General Scholium">General Scholium</a>" (1713; <i>"<a href="/wiki/Hypotheses_non_fingo" title="Hypotheses non fingo">hypotheses non fingo</a>"</i> )</li> <li><i><a href="/wiki/The_Chronology_of_Ancient_Kingdoms_Amended" title="The Chronology of Ancient Kingdoms Amended">Ancient Kingdoms Amended</a></i> (1728)</li> <li><i><a href="/wiki/An_Historical_Account_of_Two_Notable_Corruptions_of_Scripture" title="An Historical Account of Two Notable Corruptions of Scripture">Corruptions of Scripture</a></i> (1754)</li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%;vertical-align:top;">Contributions</th><td class="navbox-list-with-group navbox-list navbox-odd hlist" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Calculus" title="Calculus">Calculus</a> <ul><li><a href="/wiki/Fluxion" title="Fluxion">fluxion</a></li></ul></li> <li><a href="/wiki/Impact_depth" title="Impact depth">Impact depth</a></li> <li><a href="/wiki/Inertia" title="Inertia">Inertia</a></li> <li><a href="/wiki/Newton_disc" title="Newton disc">Newton disc</a></li> <li><a href="/wiki/Newton_polygon" title="Newton polygon">Newton polygon</a> <ul><li><a href="/wiki/Newton%E2%80%93Okounkov_body" title="Newton–Okounkov body">Newton–Okounkov body</a></li></ul></li> <li><a href="/wiki/Newton%27s_reflector" title="Newton's reflector">Newton's reflector</a></li> <li><a href="/wiki/Newtonian_telescope" title="Newtonian telescope">Newtonian telescope</a></li> <li><a href="/wiki/Newton_scale" title="Newton scale">Newton scale</a></li> <li><a href="/wiki/Newton%27s_metal" title="Newton's metal">Newton's metal</a></li> <li><a href="/wiki/Spectrum" title="Spectrum">Spectrum</a></li> <li><a href="/wiki/Structural_coloration" title="Structural coloration">Structural coloration</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%;vertical-align:top;"><a href="/wiki/Newtonianism" title="Newtonianism">Newtonianism</a></th><td class="navbox-list-with-group navbox-list navbox-even hlist" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Bucket_argument" title="Bucket argument">Bucket argument</a></li> <li><a href="/wiki/Newton%27s_inequalities" title="Newton's inequalities">Newton's inequalities</a></li> <li><a href="/wiki/Newton%27s_law_of_cooling" title="Newton's law of cooling">Newton's law of cooling</a></li> <li><a href="/wiki/Newton%27s_law_of_universal_gravitation" title="Newton's law of universal gravitation">Newton's law of universal gravitation</a> <ul><li><a href="/wiki/Post-Newtonian_expansion" title="Post-Newtonian expansion">post-Newtonian expansion</a></li> <li><a href="/wiki/Parameterized_post-Newtonian_formalism" title="Parameterized post-Newtonian formalism">parameterized</a></li> <li><a href="/wiki/Gravitational_constant" title="Gravitational constant">gravitational constant</a></li></ul></li> <li><a href="/wiki/Newton%E2%80%93Cartan_theory" title="Newton–Cartan theory">Newton–Cartan theory</a></li> <li><a href="/wiki/Schr%C3%B6dinger%E2%80%93Newton_equation" title="Schrödinger–Newton equation">Schrödinger–Newton equation</a></li> <li><a href="/wiki/Newton%27s_laws_of_motion" title="Newton's laws of motion">Newton's laws of motion</a> <ul><li><a href="/wiki/Kepler%27s_laws_of_planetary_motion" title="Kepler's laws of planetary motion">Kepler's laws</a></li></ul></li> <li><a href="/wiki/Newtonian_dynamics" title="Newtonian dynamics">Newtonian dynamics</a></li> <li><a href="/wiki/Newton%27s_method_in_optimization" title="Newton's method in optimization">Newton's method in optimization</a> <ul><li><a href="/wiki/Problem_of_Apollonius" title="Problem of Apollonius">Apollonius's problem</a></li> <li><a href="/wiki/Truncated_Newton_method" title="Truncated Newton method">truncated Newton method</a></li></ul></li> <li><a href="/wiki/Gauss%E2%80%93Newton_algorithm" title="Gauss–Newton algorithm">Gauss–Newton algorithm</a></li> <li><a href="/wiki/Newton%27s_rings" title="Newton's rings">Newton's rings</a></li> <li><a href="/wiki/Newton%27s_theorem_about_ovals" title="Newton's theorem about ovals">Newton's theorem about ovals</a></li> <li><a href="/wiki/Newton%E2%80%93Pepys_problem" title="Newton–Pepys problem">Newton–Pepys problem</a></li> <li><a href="/wiki/Newtonian_potential" title="Newtonian potential">Newtonian potential</a></li> <li><a href="/wiki/Newtonian_fluid" title="Newtonian fluid">Newtonian fluid</a></li> <li><a href="/wiki/Classical_mechanics" title="Classical mechanics">Classical mechanics</a></li> <li><a href="/wiki/Corpuscular_theory_of_light" title="Corpuscular theory of light">Corpuscular theory of light</a></li> <li><a href="/wiki/Leibniz%E2%80%93Newton_calculus_controversy" title="Leibniz–Newton calculus controversy">Leibniz–Newton calculus controversy</a></li> <li><a href="/wiki/Newton%27s_notation" class="mw-redirect" title="Newton's notation">Newton's notation</a></li> <li><a href="/wiki/Rotating_spheres" title="Rotating spheres">Rotating spheres</a></li> <li><a href="/wiki/Newton%27s_cannonball" title="Newton's cannonball">Newton's cannonball</a></li> <li><a href="/wiki/Newton%E2%80%93Cotes_formulas" title="Newton–Cotes formulas">Newton–Cotes formulas</a></li> <li><a href="/wiki/Newton%27s_method" title="Newton's method">Newton's method</a> <ul><li><a href="/wiki/Generalized_Gauss%E2%80%93Newton_method" title="Generalized Gauss–Newton method">generalized Gauss–Newton method</a></li></ul></li> <li><a href="/wiki/Newton_fractal" title="Newton fractal">Newton fractal</a></li> <li><a href="/wiki/Newton%27s_identities" title="Newton's identities">Newton's identities</a></li> <li><a href="/wiki/Newton_polynomial" title="Newton polynomial">Newton polynomial</a></li> <li><a href="/wiki/Newton%27s_theorem_of_revolving_orbits" title="Newton's theorem of revolving orbits">Newton's theorem of revolving orbits</a></li> <li><a href="/wiki/Newton%E2%80%93Euler_equations" title="Newton–Euler equations">Newton–Euler equations</a></li> <li><a href="/wiki/Power_number" title="Power number">Newton number</a> <ul><li><a href="/wiki/Kissing_number" title="Kissing number">kissing number problem</a></li></ul></li> <li><a href="/wiki/Difference_quotient" title="Difference quotient">Newton's quotient</a></li> <li><a href="/wiki/Parallelogram_of_force" title="Parallelogram of force">Parallelogram of force</a></li> <li><a href="/wiki/Puiseux_series" title="Puiseux series">Newton–Puiseux theorem</a></li> <li><a href="/wiki/Absolute_space_and_time#Newton" title="Absolute space and time">Absolute space and time</a></li> <li><a href="/wiki/Luminiferous_aether" title="Luminiferous aether">Luminiferous aether</a></li> <li><a href="/wiki/Finite_difference" title="Finite difference">Newtonian series</a> <ul><li><a href="/wiki/Table_of_Newtonian_series" title="Table of Newtonian series">table</a></li></ul></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%;vertical-align:top;">Personal life</th><td class="navbox-list-with-group navbox-list navbox-odd hlist" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Woolsthorpe_Manor" title="Woolsthorpe Manor">Woolsthorpe Manor</a> (birthplace)</li> <li><a href="/wiki/Cranbury_Park" title="Cranbury Park">Cranbury Park</a> (home)</li> <li><a href="/wiki/Early_life_of_Isaac_Newton" title="Early life of Isaac Newton">Early life</a></li> <li><a href="/wiki/Later_life_of_Isaac_Newton" title="Later life of Isaac Newton">Later life</a></li> <li><a href="/wiki/Isaac_Newton%27s_apple_tree" title="Isaac Newton's apple tree">Apple tree</a></li> <li><a href="/wiki/Religious_views_of_Isaac_Newton" title="Religious views of Isaac Newton">Religious views</a></li> <li><a href="/wiki/Isaac_Newton%27s_occult_studies" title="Isaac Newton's occult studies">Occult studies</a></li> <li><a href="/wiki/Scientific_Revolution" title="Scientific Revolution">Scientific Revolution</a></li> <li><a href="/wiki/Copernican_Revolution" title="Copernican Revolution">Copernican Revolution</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%;vertical-align:top;">Relations</th><td class="navbox-list-with-group navbox-list navbox-even hlist" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Catherine_Barton" title="Catherine Barton">Catherine Barton</a> (niece)</li> <li><a href="/wiki/John_Conduitt" title="John Conduitt">John Conduitt</a> (nephew-in-law)</li> <li><a href="/wiki/Isaac_Barrow" title="Isaac Barrow">Isaac Barrow</a> (professor)</li> <li><a href="/wiki/William_Clarke_(apothecary)" title="William Clarke (apothecary)">William Clarke</a> (mentor)</li> <li><a href="/wiki/Benjamin_Pulleyn" title="Benjamin Pulleyn">Benjamin Pulleyn</a> (tutor)</li> <li><a href="/wiki/Roger_Cotes" title="Roger Cotes">Roger Cotes</a> (student)</li> <li><a href="/wiki/William_Whiston" title="William Whiston">William Whiston</a> (student)</li> <li><a href="/wiki/John_Keill" title="John Keill">John Keill</a> (disciple)</li> <li><a href="/wiki/William_Stukeley" title="William Stukeley">William Stukeley</a> (friend)</li> <li><a href="/wiki/William_Jones_(mathematician)" title="William Jones (mathematician)">William Jones</a> (friend)</li> <li><a class="mw-selflink selflink">Abraham de Moivre</a> (friend)</li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%;vertical-align:top;"><a href="/wiki/Isaac_Newton_in_popular_culture" title="Isaac Newton in popular culture">Depictions</a></th><td class="navbox-list-with-group navbox-list navbox-odd hlist" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Newton_(Blake)" title="Newton (Blake)"><i>Newton</i> by Blake</a> (monotype)</li> <li><a href="/wiki/Newton_(Paolozzi)" title="Newton (Paolozzi)"><i>Newton</i> by Paolozzi</a> (sculpture)</li> <li><i><a href="/wiki/Isaac_Newton_Gargoyle" title="Isaac Newton Gargoyle">Isaac Newton Gargoyle</a></i></li> <li><i><a href="/wiki/Astronomers_Monument" title="Astronomers Monument">Astronomers Monument</a></i></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%;vertical-align:top;"><a href="/wiki/List_of_things_named_after_Isaac_Newton" title="List of things named after Isaac Newton">Namesake</a></th><td class="navbox-list-with-group navbox-list navbox-even hlist" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Newton_(unit)" title="Newton (unit)">Newton (unit)</a></li> <li><a href="/wiki/Newton%27s_cradle" title="Newton's cradle">Newton's cradle</a></li> <li><a href="/wiki/Isaac_Newton_Institute" title="Isaac Newton Institute">Isaac Newton Institute</a></li> <li><a href="/wiki/Institute_of_Physics_Isaac_Newton_Medal" class="mw-redirect" title="Institute of Physics Isaac Newton Medal">Isaac Newton Medal</a></li> <li><a href="/wiki/Isaac_Newton_Telescope" title="Isaac Newton Telescope">Isaac Newton Telescope</a></li> <li><a href="/wiki/Isaac_Newton_Group_of_Telescopes" title="Isaac Newton Group of Telescopes">Isaac Newton Group of Telescopes</a></li> <li><a href="/wiki/XMM-Newton" title="XMM-Newton">XMM-Newton</a></li> <li><a href="/wiki/Sir_Isaac_Newton_Sixth_Form" title="Sir Isaac Newton Sixth Form">Sir Isaac Newton Sixth Form</a></li> <li><a href="/wiki/Statal_Institute_of_Higher_Education_Isaac_Newton" title="Statal Institute of Higher Education Isaac Newton">Statal Institute of Higher Education Isaac Newton</a></li> <li><a href="/wiki/Newton_International_Fellowship" title="Newton International Fellowship">Newton International Fellowship</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%;vertical-align:top;">Categories</th><td class="navbox-list-with-group navbox-list navbox-odd hlist" style="width:100%;padding:0"><div style="padding:0 0.25em"><div class="div-col"> <div class="CategoryTreeTag" data-ct-options="{"mode":20,"hideprefix":20,"showcount":false,"namespaces":false,"notranslations":false}"><div class="CategoryTreeSection"><div class="CategoryTreeItem"><span class="CategoryTreeBullet"><a class="CategoryTreeToggle" data-ct-title="Isaac_Newton" aria-expanded="false"></a> </span> <bdi dir="ltr"><a href="/wiki/Category:Isaac_Newton" title="Category:Isaac Newton">Isaac Newton</a></bdi></div><div class="CategoryTreeChildren" style="display:none"></div></div></div> </div></div></td></tr></tbody></table></div> <div class="navbox-styles"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1129693374"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1236075235"><style 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class="mw-valign-text-top noprint" typeof="mw:File/Frameless"><a href="https://www.wikidata.org/wiki/Q200397#identifiers" title="Edit this at Wikidata"><img alt="Edit this at Wikidata" src="//upload.wikimedia.org/wikipedia/en/thumb/8/8a/OOjs_UI_icon_edit-ltr-progressive.svg/10px-OOjs_UI_icon_edit-ltr-progressive.svg.png" decoding="async" width="10" height="10" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/en/thumb/8/8a/OOjs_UI_icon_edit-ltr-progressive.svg/15px-OOjs_UI_icon_edit-ltr-progressive.svg.png 1.5x, //upload.wikimedia.org/wikipedia/en/thumb/8/8a/OOjs_UI_icon_edit-ltr-progressive.svg/20px-OOjs_UI_icon_edit-ltr-progressive.svg.png 2x" data-file-width="20" data-file-height="20" /></a></span></div></th></tr><tr><th scope="row" class="navbox-group" style="width:1%">International</th><td class="navbox-list-with-group navbox-list navbox-odd" style="width:100%;padding:0"><div style="padding:0 0.25em"><ul><li><span class="uid"><a rel="nofollow" class="external text" href="https://isni.org/isni/000000011070064X">ISNI</a></span></li><li><span class="uid"><a rel="nofollow" class="external text" href="https://viaf.org/viaf/71525671">VIAF</a></span></li><li><span class="uid"><a rel="nofollow" class="external text" href="http://id.worldcat.org/fast/179424/">FAST</a></span></li><li><span class="uid"><a rel="nofollow" class="external text" href="https://id.oclc.org/worldcat/entity/E39PBJtGXhkDt3JFydCDRfRYfq">WorldCat</a></span></li></ul></div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%">National</th><td class="navbox-list-with-group navbox-list navbox-even" style="width:100%;padding:0"><div style="padding:0 0.25em"><ul><li><span class="uid"><a rel="nofollow" class="external text" href="https://d-nb.info/gnd/117586765">Germany</a></span></li><li><span class="uid"><a rel="nofollow" class="external text" href="https://id.loc.gov/authorities/n86837658">United States</a></span></li><li><span class="uid"><a rel="nofollow" class="external text" href="https://catalogue.bnf.fr/ark:/12148/cb130110001">France</a></span></li><li><span class="uid"><a rel="nofollow" class="external text" href="https://data.bnf.fr/ark:/12148/cb130110001">BnF data</a></span></li><li><span class="uid"><span class="rt-commentedText tooltip tooltip-dotted" title="Moivre, Abraham : de"><a rel="nofollow" class="external text" href="https://opac.sbn.it/nome/MILV241767">Italy</a></span></span></li><li><span class="uid"><a rel="nofollow" class="external text" href="https://nla.gov.au/anbd.aut-an49860343">Australia</a></span></li><li><span class="uid"><a rel="nofollow" class="external text" href="https://aleph.nkp.cz/F/?func=find-c&local_base=aut&ccl_term=ica=mub2017952222&CON_LNG=ENG">Czech Republic</a></span></li><li><span class="uid"><a rel="nofollow" class="external text" href="http://catalogo.bne.es/uhtbin/authoritybrowse.cgi?action=display&authority_id=XX1632935">Spain</a></span></li><li><span class="uid"><a rel="nofollow" class="external text" href="http://data.bibliotheken.nl/id/thes/p070811873">Netherlands</a></span><ul><li><span class="uid"><a rel="nofollow" class="external text" href="http://data.bibliotheken.nl/id/thes/p069285675">2</a></span></li></ul></li><li><span class="uid"><a rel="nofollow" class="external text" href="https://authority.bibsys.no/authority/rest/authorities/html/8066654">Norway</a></span></li><li><span class="uid"><a rel="nofollow" class="external text" href="http://katalog.nsk.hr/F/?func=direct&doc_number=000530937&local_base=nsk10">Croatia</a></span></li><li><span class="uid"><a rel="nofollow" class="external text" href="https://libris.kb.se/sq47cnwb0jvv8mn">Sweden</a></span></li><li><span class="uid"><a rel="nofollow" class="external text" href="http://olduli.nli.org.il/F/?func=find-b&local_base=NLX10&find_code=UID&request=987007265403505171">Israel</a></span></li><li><span class="uid"><a rel="nofollow" class="external text" href="https://opac.kbr.be/LIBRARY/doc/AUTHORITY/14176368">Belgium</a></span></li></ul></div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%">Academics</th><td class="navbox-list-with-group navbox-list navbox-odd" style="width:100%;padding:0"><div style="padding:0 0.25em"><ul><li><span class="uid"><a rel="nofollow" class="external text" href="https://ci.nii.ac.jp/author/DA02324943?l=en">CiNii</a></span></li><li><span class="uid"><a rel="nofollow" class="external text" href="https://www.mathgenealogy.org/id.php?id=48320">Mathematics Genealogy Project</a></span></li><li><span class="uid"><a rel="nofollow" class="external text" href="https://zbmath.org/authors/?q=ai:de-moivre.abraham">zbMATH</a></span></li><li><span class="uid"><a rel="nofollow" class="external text" href="https://mathscinet.ams.org/mathscinet/MRAuthorID/126125">MathSciNet</a></span></li></ul></div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%">People</th><td class="navbox-list-with-group navbox-list navbox-even" style="width:100%;padding:0"><div style="padding:0 0.25em"><ul><li><span class="uid"><a rel="nofollow" class="external text" href="https://trove.nla.gov.au/people/1497499">Trove</a></span></li><li><span class="uid"><a rel="nofollow" class="external text" href="https://www.deutsche-biographie.de/pnd117586765.html?language=en">Deutsche Biographie</a></span></li><li><span class="uid"><a rel="nofollow" class="external text" href="https://www.deutsche-digitale-bibliothek.de/person/gnd/117586765">DDB</a></span></li></ul></div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%">Other</th><td class="navbox-list-with-group navbox-list navbox-odd" style="width:100%;padding:0"><div style="padding:0 0.25em"><ul><li><span class="uid"><a rel="nofollow" class="external text" href="https://www.idref.fr/050143808">IdRef</a></span></li></ul></div></td></tr></tbody></table></div>    </div><noscript><img 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