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class="vector-toc-list-item vector-toc-level-1"> <a class="vector-toc-link" href="#Indian"> <div class="vector-toc-text"> 8 Indian </div> </a> <ul id="toc-Indian-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Islamic_empires" class="vector-toc-list-item vector-toc-level-1"> <a class="vector-toc-link" href="#Islamic_empires"> <div class="vector-toc-text"> 9 Islamic empires </div> </a> <ul id="toc-Islamic_empires-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Maya" class="vector-toc-list-item vector-toc-level-1"> <a class="vector-toc-link" href="#Maya"> <div class="vector-toc-text"> 10 Maya </div> </a> <ul id="toc-Maya-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Medieval_European" class="vector-toc-list-item vector-toc-level-1"> <a class="vector-toc-link" href="#Medieval_European"> <div class="vector-toc-text"> 11 Medieval European </div> </a> <ul id="toc-Medieval_European-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Renaissance" class="vector-toc-list-item vector-toc-level-1"> <a class="vector-toc-link" href="#Renaissance"> <div class="vector-toc-text"> 12 Renaissance </div> </a> <ul id="toc-Renaissance-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Mathematics_during_the_Scientific_Revolution" class="vector-toc-list-item vector-toc-level-1"> <a class="vector-toc-link" href="#Mathematics_during_the_Scientific_Revolution"> <div class="vector-toc-text"> 13 Mathematics during the Scientific Revolution </div> </a> <button aria-controls="toc-Mathematics_during_the_Scientific_Revolution-sublist" class="cdx-button cdx-button--weight-quiet cdx-button--icon-only vector-toc-toggle"> Toggle Mathematics during the Scientific Revolution subsection </button> <ul id="toc-Mathematics_during_the_Scientific_Revolution-sublist" class="vector-toc-list"> <li id="toc-17th_century" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#17th_century"> <div class="vector-toc-text"> 13.1 17th century </div> </a> <ul id="toc-17th_century-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-18th_century" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#18th_century"> <div class="vector-toc-text"> 13.2 18th century </div> </a> <ul id="toc-18th_century-sublist" class="vector-toc-list"> </ul> </li> </ul> </li> <li id="toc-Modern" class="vector-toc-list-item vector-toc-level-1"> <a class="vector-toc-link" href="#Modern"> <div class="vector-toc-text"> 14 Modern </div> </a> <button aria-controls="toc-Modern-sublist" class="cdx-button cdx-button--weight-quiet cdx-button--icon-only vector-toc-toggle"> Toggle Modern subsection </button> <ul id="toc-Modern-sublist" class="vector-toc-list"> <li id="toc-19th_century" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#19th_century"> <div class="vector-toc-text"> 14.1 19th century </div> </a> <ul id="toc-19th_century-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-20th_century" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#20th_century"> <div class="vector-toc-text"> 14.2 20th century </div> </a> <ul id="toc-20th_century-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-21st_century" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#21st_century"> <div class="vector-toc-text"> 14.3 21st century </div> </a> <ul id="toc-21st_century-sublist" class="vector-toc-list"> </ul> </li> </ul> </li> <li id="toc-Future" class="vector-toc-list-item vector-toc-level-1"> <a class="vector-toc-link" href="#Future"> <div class="vector-toc-text"> 15 Future </div> </a> <ul id="toc-Future-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-See_also" class="vector-toc-list-item vector-toc-level-1"> <a class="vector-toc-link" href="#See_also"> <div class="vector-toc-text"> 16 See also </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"> <a class="vector-toc-link" href="#Notes"> <div class="vector-toc-text"> 17 Notes </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"> <a class="vector-toc-link" href="#References"> <div class="vector-toc-text"> 18 References </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"> <a class="vector-toc-link" href="#Further_reading"> <div class="vector-toc-text"> 19 Further reading </div> </a> <button aria-controls="toc-Further_reading-sublist" class="cdx-button cdx-button--weight-quiet cdx-button--icon-only vector-toc-toggle"> Toggle Further reading subsection </button> <ul id="toc-Further_reading-sublist" class="vector-toc-list"> <li id="toc-General" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#General"> <div class="vector-toc-text"> 19.1 General </div> </a> <ul id="toc-General-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Books_on_a_specific_period" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Books_on_a_specific_period"> <div class="vector-toc-text"> 19.2 Books on a specific period </div> </a> <ul id="toc-Books_on_a_specific_period-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Books_on_a_specific_topic" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Books_on_a_specific_topic"> <div class="vector-toc-text"> 19.3 Books on a specific topic </div> </a> <ul id="toc-Books_on_a_specific_topic-sublist" class="vector-toc-list"> </ul> </li> </ul> </li> <li id="toc-External_links" class="vector-toc-list-item vector-toc-level-1"> <a class="vector-toc-link" href="#External_links"> <div class="vector-toc-text"> 20 External links </div> </a> <button aria-controls="toc-External_links-sublist" class="cdx-button cdx-button--weight-quiet cdx-button--icon-only vector-toc-toggle"> Toggle External links subsection </button> <ul id="toc-External_links-sublist" class="vector-toc-list"> <li id="toc-Documentaries" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Documentaries"> <div class="vector-toc-text"> 20.1 Documentaries </div> </a> <ul id="toc-Documentaries-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Educational_material" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Educational_material"> <div class="vector-toc-text"> 20.2 Educational material </div> </a> <ul id="toc-Educational_material-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Bibliographies" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Bibliographies"> <div class="vector-toc-text"> 20.3 Bibliographies </div> </a> <ul id="toc-Bibliographies-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Organizations" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Organizations"> <div class="vector-toc-text"> 20.4 Organizations </div> </a> <ul id="toc-Organizations-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Journals" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Journals"> <div class="vector-toc-text"> 20.5 Journals </div> </a> <ul id="toc-Journals-sublist" class="vector-toc-list"> </ul> </li> </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" > Toggle the table of contents </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">History of mathematics</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 65 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-65" aria-hidden="true" > 65 languages </label> <div class="vector-dropdown-content"> <div class="vector-menu-content"> <ul class="vector-menu-content-list"> <li class="interlanguage-link interwiki-am mw-list-item"><a href="https://am.wikipedia.org/wiki/%E1%8B%A8%E1%88%92%E1%88%B3%E1%89%A5_%E1%89%B3%E1%88%AA%E1%8A%AD" title="የሒሳብ ታሪክ – Amharic" lang="am" hreflang="am" data-title="የሒሳብ ታሪክ" data-language-autonym="አማርኛ" data-language-local-name="Amharic" class="interlanguage-link-target">አማርኛ</a></li><li class="interlanguage-link interwiki-ar mw-list-item"><a href="https://ar.wikipedia.org/wiki/%D8%AA%D8%A7%D8%B1%D9%8A%D8%AE_%D8%A7%D9%84%D8%B1%D9%8A%D8%A7%D8%B6%D9%8A%D8%A7%D8%AA" title="تاريخ الرياضيات – Arabic" lang="ar" hreflang="ar" data-title="تاريخ الرياضيات" data-language-autonym="العربية" data-language-local-name="Arabic" class="interlanguage-link-target">العربية</a></li><li class="interlanguage-link interwiki-ast mw-list-item"><a href="https://ast.wikipedia.org/wiki/Historia_de_les_matem%C3%A1tiques" title="Historia de les matemátiques – Asturian" lang="ast" hreflang="ast" data-title="Historia de les matemátiques" data-language-autonym="Asturianu" data-language-local-name="Asturian" class="interlanguage-link-target">Asturianu</a></li><li class="interlanguage-link interwiki-bn mw-list-item"><a href="https://bn.wikipedia.org/wiki/%E0%A6%97%E0%A6%A3%E0%A6%BF%E0%A6%A4%E0%A7%87%E0%A6%B0_%E0%A6%87%E0%A6%A4%E0%A6%BF%E0%A6%B9%E0%A6%BE%E0%A6%B8" title="গণিতের ইতিহাস – Bangla" lang="bn" hreflang="bn" data-title="গণিতের ইতিহাস" data-language-autonym="বাংলা" data-language-local-name="Bangla" class="interlanguage-link-target">বাংলা</a></li><li class="interlanguage-link interwiki-bg mw-list-item"><a href="https://bg.wikipedia.org/wiki/%D0%98%D1%81%D1%82%D0%BE%D1%80%D0%B8%D1%8F_%D0%BD%D0%B0_%D0%BC%D0%B0%D1%82%D0%B5%D0%BC%D0%B0%D1%82%D0%B8%D0%BA%D0%B0%D1%82%D0%B0" title="История на математиката – Bulgarian" lang="bg" hreflang="bg" data-title="История на математиката" data-language-autonym="Български" data-language-local-name="Bulgarian" class="interlanguage-link-target">Български</a></li><li class="interlanguage-link interwiki-ca mw-list-item"><a href="https://ca.wikipedia.org/wiki/Hist%C3%B2ria_de_les_matem%C3%A0tiques" title="Història de les matemàtiques – Catalan" lang="ca" hreflang="ca" data-title="Història de les matemàtiques" data-language-autonym="Català" data-language-local-name="Catalan" class="interlanguage-link-target">Català</a></li><li class="interlanguage-link interwiki-cv mw-list-item"><a href="https://cv.wikipedia.org/wiki/%D0%9C%D0%B0%D1%82%D0%B5%D0%BC%D0%B0%D1%82%D0%B8%D0%BA%D0%B0_%D0%B8%D1%81%D1%82%D0%BE%D1%80%D0%B8%D0%B9%C4%95" title="Математика историйĕ – Chuvash" lang="cv" hreflang="cv" data-title="Математика историйĕ" data-language-autonym="Чӑвашла" data-language-local-name="Chuvash" class="interlanguage-link-target">Чӑвашла</a></li><li class="interlanguage-link interwiki-cs mw-list-item"><a href="https://cs.wikipedia.org/wiki/D%C4%9Bjiny_matematiky" title="Dějiny matematiky – Czech" lang="cs" hreflang="cs" data-title="Dějiny matematiky" data-language-autonym="Čeština" data-language-local-name="Czech" class="interlanguage-link-target">Čeština</a></li><li class="interlanguage-link interwiki-da badge-Q17559452 badge-recommendedarticle mw-list-item" title="recommended article"><a href="https://da.wikipedia.org/wiki/Matematikkens_historie" title="Matematikkens historie – Danish" lang="da" hreflang="da" data-title="Matematikkens historie" data-language-autonym="Dansk" data-language-local-name="Danish" class="interlanguage-link-target">Dansk</a></li><li class="interlanguage-link interwiki-de badge-Q17437798 badge-goodarticle mw-list-item" title="good article badge"><a href="https://de.wikipedia.org/wiki/Geschichte_der_Mathematik" title="Geschichte der Mathematik – German" lang="de" hreflang="de" data-title="Geschichte der Mathematik" data-language-autonym="Deutsch" data-language-local-name="German" class="interlanguage-link-target">Deutsch</a></li><li class="interlanguage-link interwiki-el mw-list-item"><a href="https://el.wikipedia.org/wiki/%CE%99%CF%83%CF%84%CE%BF%CF%81%CE%AF%CE%B1_%CF%84%CF%89%CE%BD_%CE%BC%CE%B1%CE%B8%CE%B7%CE%BC%CE%B1%CF%84%CE%B9%CE%BA%CF%8E%CE%BD" title="Ιστορία των μαθηματικών – Greek" lang="el" hreflang="el" data-title="Ιστορία των μαθηματικών" data-language-autonym="Ελληνικά" data-language-local-name="Greek" class="interlanguage-link-target">Ελληνικά</a></li><li class="interlanguage-link interwiki-es mw-list-item"><a href="https://es.wikipedia.org/wiki/Historia_de_las_matem%C3%A1ticas" title="Historia de las matemáticas – Spanish" lang="es" hreflang="es" data-title="Historia de las matemáticas" data-language-autonym="Español" data-language-local-name="Spanish" class="interlanguage-link-target">Español</a></li><li class="interlanguage-link interwiki-eo mw-list-item"><a href="https://eo.wikipedia.org/wiki/Historio_de_matematiko" title="Historio de matematiko – Esperanto" lang="eo" hreflang="eo" data-title="Historio de matematiko" data-language-autonym="Esperanto" data-language-local-name="Esperanto" class="interlanguage-link-target">Esperanto</a></li><li class="interlanguage-link interwiki-eu mw-list-item"><a href="https://eu.wikipedia.org/wiki/Matematikaren_historia" title="Matematikaren historia – Basque" lang="eu" hreflang="eu" data-title="Matematikaren historia" data-language-autonym="Euskara" data-language-local-name="Basque" class="interlanguage-link-target">Euskara</a></li><li class="interlanguage-link interwiki-fa mw-list-item"><a href="https://fa.wikipedia.org/wiki/%D8%AA%D8%A7%D8%B1%DB%8C%D8%AE_%D8%B1%DB%8C%D8%A7%D8%B6%DB%8C%D8%A7%D8%AA" title="تاریخ ریاضیات – Persian" lang="fa" hreflang="fa" data-title="تاریخ ریاضیات" data-language-autonym="فارسی" data-language-local-name="Persian" class="interlanguage-link-target">فارسی</a></li><li class="interlanguage-link interwiki-fr mw-list-item"><a href="https://fr.wikipedia.org/wiki/Histoire_des_math%C3%A9matiques" title="Histoire des mathématiques – French" lang="fr" hreflang="fr" data-title="Histoire des mathématiques" data-language-autonym="Français" data-language-local-name="French" class="interlanguage-link-target">Français</a></li><li class="interlanguage-link interwiki-gl mw-list-item"><a href="https://gl.wikipedia.org/wiki/Historia_das_matem%C3%A1ticas" title="Historia das matemáticas – Galician" lang="gl" hreflang="gl" data-title="Historia das matemáticas" data-language-autonym="Galego" data-language-local-name="Galician" class="interlanguage-link-target">Galego</a></li><li class="interlanguage-link interwiki-ko mw-list-item"><a href="https://ko.wikipedia.org/wiki/%EC%88%98%ED%95%99%EC%82%AC" title="수학사 – Korean" lang="ko" hreflang="ko" data-title="수학사" data-language-autonym="한국어" data-language-local-name="Korean" class="interlanguage-link-target">한국어</a></li><li class="interlanguage-link interwiki-ha mw-list-item"><a href="https://ha.wikipedia.org/wiki/Tarihin_lissafi" title="Tarihin lissafi – Hausa" lang="ha" hreflang="ha" data-title="Tarihin lissafi" data-language-autonym="Hausa" data-language-local-name="Hausa" class="interlanguage-link-target">Hausa</a></li><li class="interlanguage-link interwiki-hy mw-list-item"><a href="https://hy.wikipedia.org/wiki/%D5%84%D5%A1%D5%A9%D5%A5%D5%B4%D5%A1%D5%BF%D5%AB%D5%AF%D5%A1%D5%B5%D5%AB_%D5%BA%D5%A1%D5%BF%D5%B4%D5%B8%D6%82%D5%A9%D5%B5%D5%B8%D6%82%D5%B6" title="Մաթեմատիկայի պատմություն – Armenian" lang="hy" hreflang="hy" data-title="Մաթեմատիկայի պատմություն" data-language-autonym="Հայերեն" data-language-local-name="Armenian" class="interlanguage-link-target">Հայերեն</a></li><li class="interlanguage-link interwiki-hi mw-list-item"><a href="https://hi.wikipedia.org/wiki/%E0%A4%97%E0%A4%A3%E0%A4%BF%E0%A4%A4_%E0%A4%95%E0%A4%BE_%E0%A4%87%E0%A4%A4%E0%A4%BF%E0%A4%B9%E0%A4%BE%E0%A4%B8" title="गणित का इतिहास – Hindi" lang="hi" hreflang="hi" data-title="गणित का इतिहास" data-language-autonym="हिन्दी" data-language-local-name="Hindi" class="interlanguage-link-target">हिन्दी</a></li><li class="interlanguage-link interwiki-id mw-list-item"><a href="https://id.wikipedia.org/wiki/Sejarah_matematika" title="Sejarah matematika – Indonesian" lang="id" hreflang="id" data-title="Sejarah matematika" data-language-autonym="Bahasa Indonesia" data-language-local-name="Indonesian" class="interlanguage-link-target">Bahasa Indonesia</a></li><li class="interlanguage-link interwiki-ia mw-list-item"><a href="https://ia.wikipedia.org/wiki/Historia_del_mathematica" title="Historia del mathematica – Interlingua" lang="ia" hreflang="ia" data-title="Historia del mathematica" data-language-autonym="Interlingua" data-language-local-name="Interlingua" class="interlanguage-link-target">Interlingua</a></li><li class="interlanguage-link interwiki-it mw-list-item"><a href="https://it.wikipedia.org/wiki/Storia_della_matematica" title="Storia della matematica – Italian" lang="it" hreflang="it" data-title="Storia della matematica" data-language-autonym="Italiano" data-language-local-name="Italian" class="interlanguage-link-target">Italiano</a></li><li class="interlanguage-link interwiki-he mw-list-item"><a href="https://he.wikipedia.org/wiki/%D7%94%D7%99%D7%A1%D7%98%D7%95%D7%A8%D7%99%D7%94_%D7%A9%D7%9C_%D7%94%D7%9E%D7%AA%D7%9E%D7%98%D7%99%D7%A7%D7%94" title="היסטוריה של המתמטיקה – Hebrew" lang="he" hreflang="he" data-title="היסטוריה של המתמטיקה" data-language-autonym="עברית" data-language-local-name="Hebrew" class="interlanguage-link-target">עברית</a></li><li class="interlanguage-link interwiki-kk mw-list-item"><a href="https://kk.wikipedia.org/wiki/%D0%9C%D0%B0%D1%82%D0%B5%D0%BC%D0%B0%D1%82%D0%B8%D0%BA%D0%B0_%D1%82%D0%B0%D1%80%D0%B8%D1%85%D1%8B" title="Математика тарихы – Kazakh" lang="kk" hreflang="kk" data-title="Математика тарихы" data-language-autonym="Қазақша" data-language-local-name="Kazakh" class="interlanguage-link-target">Қазақша</a></li><li class="interlanguage-link interwiki-gcr mw-list-item"><a href="https://gcr.wikipedia.org/wiki/Istw%C3%A8_di_s%C3%A9_Mat%C3%A9matik" title="Istwè di sé Matématik – Guianan Creole" lang="gcr" hreflang="gcr" data-title="Istwè di sé Matématik" data-language-autonym="Kriyòl gwiyannen" data-language-local-name="Guianan Creole" class="interlanguage-link-target">Kriyòl gwiyannen</a></li><li class="interlanguage-link interwiki-lt mw-list-item"><a href="https://lt.wikipedia.org/wiki/Matematikos_istorija" title="Matematikos istorija – Lithuanian" lang="lt" hreflang="lt" data-title="Matematikos istorija" data-language-autonym="Lietuvių" data-language-local-name="Lithuanian" class="interlanguage-link-target">Lietuvių</a></li><li class="interlanguage-link interwiki-nia mw-list-item"><a href="https://nia.wikipedia.org/wiki/Sejarah_matematika" title="Sejarah matematika – Nias" lang="nia" hreflang="nia" data-title="Sejarah matematika" data-language-autonym="Li Niha" data-language-local-name="Nias" class="interlanguage-link-target">Li Niha</a></li><li class="interlanguage-link interwiki-hu mw-list-item"><a href="https://hu.wikipedia.org/wiki/A_matematika_t%C3%B6rt%C3%A9nete" title="A matematika története – Hungarian" lang="hu" hreflang="hu" data-title="A matematika története" data-language-autonym="Magyar" data-language-local-name="Hungarian" class="interlanguage-link-target">Magyar</a></li><li class="interlanguage-link interwiki-ml mw-list-item"><a href="https://ml.wikipedia.org/wiki/%E0%B4%97%E0%B4%A3%E0%B4%BF%E0%B4%A4%E0%B4%A4%E0%B5%8D%E0%B4%A4%E0%B4%BF%E0%B4%A8%E0%B5%8D%E0%B4%B1%E0%B5%86_%E0%B4%89%E0%B4%A4%E0%B5%8D%E0%B4%AD%E0%B4%B5%E0%B4%82" title="ഗണിതത്തിന്റെ ഉത്ഭവം – Malayalam" lang="ml" hreflang="ml" data-title="ഗണിതത്തിന്റെ ഉത്ഭവം" data-language-autonym="മലയാളം" data-language-local-name="Malayalam" class="interlanguage-link-target">മലയാളം</a></li><li class="interlanguage-link interwiki-mr mw-list-item"><a href="https://mr.wikipedia.org/wiki/%E0%A4%97%E0%A4%A3%E0%A4%BF%E0%A4%A4%E0%A4%BE%E0%A4%9A%E0%A4%BE_%E0%A4%87%E0%A4%A4%E0%A4%BF%E0%A4%B9%E0%A4%BE%E0%A4%B8" title="गणिताचा इतिहास – Marathi" lang="mr" hreflang="mr" data-title="गणिताचा इतिहास" data-language-autonym="मराठी" data-language-local-name="Marathi" class="interlanguage-link-target">मराठी</a></li><li class="interlanguage-link interwiki-ms mw-list-item"><a href="https://ms.wikipedia.org/wiki/Sejarah_matematik" title="Sejarah matematik – Malay" lang="ms" hreflang="ms" data-title="Sejarah matematik" data-language-autonym="Bahasa Melayu" data-language-local-name="Malay" class="interlanguage-link-target">Bahasa Melayu</a></li><li class="interlanguage-link interwiki-my mw-list-item"><a href="https://my.wikipedia.org/wiki/%E1%80%9E%E1%80%84%E1%80%BA%E1%80%B9%E1%80%81%E1%80%BB%E1%80%AC%E1%81%8F_%E1%80%9E%E1%80%99%E1%80%AD%E1%80%AF%E1%80%84%E1%80%BA%E1%80%B8%E1%80%80%E1%80%BC%E1%80%B1%E1%80%AC%E1%80%84%E1%80%BA%E1%80%B8" title="သင်္ချာ၏ သမိုင်းကြောင်း – Burmese" lang="my" hreflang="my" data-title="သင်္ချာ၏ သမိုင်းကြောင်း" data-language-autonym="မြန်မာဘာသာ" data-language-local-name="Burmese" class="interlanguage-link-target">မြန်မာဘာသာ</a></li><li class="interlanguage-link interwiki-nl badge-Q17437796 badge-featuredarticle mw-list-item" title="featured article badge"><a href="https://nl.wikipedia.org/wiki/Geschiedenis_van_de_wiskunde" title="Geschiedenis van de wiskunde – Dutch" lang="nl" hreflang="nl" data-title="Geschiedenis van de wiskunde" data-language-autonym="Nederlands" data-language-local-name="Dutch" class="interlanguage-link-target">Nederlands</a></li><li class="interlanguage-link interwiki-ja mw-list-item"><a href="https://ja.wikipedia.org/wiki/%E6%95%B0%E5%AD%A6%E5%8F%B2" title="数学史 – Japanese" lang="ja" hreflang="ja" data-title="数学史" data-language-autonym="日本語" data-language-local-name="Japanese" class="interlanguage-link-target">日本語</a></li><li class="interlanguage-link interwiki-no badge-Q17437796 badge-featuredarticle mw-list-item" title="featured article badge"><a href="https://no.wikipedia.org/wiki/Matematikkens_historie" title="Matematikkens historie – Norwegian Bokmål" lang="nb" hreflang="nb" data-title="Matematikkens historie" data-language-autonym="Norsk bokmål" data-language-local-name="Norwegian Bokmål" class="interlanguage-link-target">Norsk bokmål</a></li><li class="interlanguage-link interwiki-uz mw-list-item"><a href="https://uz.wikipedia.org/wiki/Matematika_tarixi" title="Matematika tarixi – Uzbek" lang="uz" hreflang="uz" data-title="Matematika tarixi" data-language-autonym="Oʻzbekcha / ўзбекча" data-language-local-name="Uzbek" class="interlanguage-link-target">Oʻzbekcha / ўзбекча</a></li><li class="interlanguage-link interwiki-ps mw-list-item"><a href="https://ps.wikipedia.org/wiki/%D8%AF_%D8%B1%DB%8C%D8%A7%D8%B6%DB%8C%D8%A7%D8%AA%D9%88_%D8%AA%D8%A7%D8%B1%DB%8C%D8%AE%DA%86%D9%87" title="د ریاضیاتو تاریخچه – Pashto" lang="ps" hreflang="ps" data-title="د ریاضیاتو تاریخچه" data-language-autonym="پښتو" data-language-local-name="Pashto" class="interlanguage-link-target">پښتو</a></li><li class="interlanguage-link interwiki-km mw-list-item"><a href="https://km.wikipedia.org/wiki/%E1%9E%94%E1%9F%92%E1%9E%9A%E1%9E%9C%E1%9E%8F%E1%9F%92%E1%9E%8F%E1%9E%B7%E1%9E%93%E1%9F%83%E1%9E%82%E1%9E%8E%E1%9E%B7%E1%9E%8F%E1%9E%9C%E1%9E%B7%E1%9E%91%E1%9F%92%E1%9E%99%E1%9E%B6" title="ប្រវត្តិនៃគណិតវិទ្យា – Khmer" lang="km" hreflang="km" data-title="ប្រវត្តិនៃគណិតវិទ្យា" data-language-autonym="ភាសាខ្មែរ" data-language-local-name="Khmer" class="interlanguage-link-target">ភាសាខ្មែរ</a></li><li class="interlanguage-link interwiki-pl mw-list-item"><a href="https://pl.wikipedia.org/wiki/Historia_matematyki" title="Historia matematyki – Polish" lang="pl" hreflang="pl" data-title="Historia matematyki" data-language-autonym="Polski" data-language-local-name="Polish" class="interlanguage-link-target">Polski</a></li><li class="interlanguage-link interwiki-pt mw-list-item"><a href="https://pt.wikipedia.org/wiki/Hist%C3%B3ria_da_matem%C3%A1tica" title="História da matemática – Portuguese" lang="pt" hreflang="pt" data-title="História da matemática" data-language-autonym="Português" data-language-local-name="Portuguese" class="interlanguage-link-target">Português</a></li><li class="interlanguage-link interwiki-kaa mw-list-item"><a href="https://kaa.wikipedia.org/wiki/Matematika_t%C3%A1riyx%C4%B1" title="Matematika táriyxı – Kara-Kalpak" lang="kaa" hreflang="kaa" data-title="Matematika táriyxı" data-language-autonym="Qaraqalpaqsha" data-language-local-name="Kara-Kalpak" class="interlanguage-link-target">Qaraqalpaqsha</a></li><li class="interlanguage-link interwiki-ro mw-list-item"><a href="https://ro.wikipedia.org/wiki/Istoria_matematicii" title="Istoria matematicii – Romanian" lang="ro" hreflang="ro" data-title="Istoria matematicii" data-language-autonym="Română" data-language-local-name="Romanian" class="interlanguage-link-target">Română</a></li><li class="interlanguage-link interwiki-ru badge-Q17437798 badge-goodarticle mw-list-item" title="good article badge"><a href="https://ru.wikipedia.org/wiki/%D0%98%D1%81%D1%82%D0%BE%D1%80%D0%B8%D1%8F_%D0%BC%D0%B0%D1%82%D0%B5%D0%BC%D0%B0%D1%82%D0%B8%D0%BA%D0%B8" title="История математики – Russian" lang="ru" hreflang="ru" data-title="История математики" data-language-autonym="Русский" data-language-local-name="Russian" class="interlanguage-link-target">Русский</a></li><li class="interlanguage-link interwiki-sq mw-list-item"><a href="https://sq.wikipedia.org/wiki/Historia_e_matematik%C3%ABs_shqiptare" title="Historia e matematikës shqiptare – Albanian" lang="sq" hreflang="sq" data-title="Historia e matematikës shqiptare" data-language-autonym="Shqip" data-language-local-name="Albanian" class="interlanguage-link-target">Shqip</a></li><li class="interlanguage-link interwiki-si mw-list-item"><a href="https://si.wikipedia.org/wiki/%E0%B6%9C%E0%B6%AB%E0%B7%92%E0%B6%AD%E0%B6%BA%E0%B7%9A_%E0%B6%89%E0%B6%AD%E0%B7%92%E0%B7%84%E0%B7%8F%E0%B7%83%E0%B6%BA" title="ගණිතයේ ඉතිහාසය – Sinhala" lang="si" hreflang="si" data-title="ගණිතයේ ඉතිහාසය" data-language-autonym="සිංහල" data-language-local-name="Sinhala" class="interlanguage-link-target">සිංහල</a></li><li class="interlanguage-link interwiki-simple mw-list-item"><a href="https://simple.wikipedia.org/wiki/History_of_mathematics" title="History of mathematics – Simple English" lang="en-simple" hreflang="en-simple" data-title="History of mathematics" data-language-autonym="Simple English" data-language-local-name="Simple English" class="interlanguage-link-target">Simple English</a></li><li class="interlanguage-link interwiki-sd mw-list-item"><a href="https://sd.wikipedia.org/wiki/%D8%B1%D9%8A%D8%A7%D8%B6%D9%8A_%D8%AC%D9%8A_%D8%AA%D8%A7%D8%B1%D9%8A%D8%AE" title="رياضي جي تاريخ – Sindhi" lang="sd" hreflang="sd" data-title="رياضي جي تاريخ" data-language-autonym="سنڌي" data-language-local-name="Sindhi" class="interlanguage-link-target">سنڌي</a></li><li class="interlanguage-link interwiki-sl mw-list-item"><a href="https://sl.wikipedia.org/wiki/Zgodovina_matematike" title="Zgodovina matematike – Slovenian" lang="sl" hreflang="sl" data-title="Zgodovina matematike" data-language-autonym="Slovenščina" data-language-local-name="Slovenian" class="interlanguage-link-target">Slovenščina</a></li><li class="interlanguage-link interwiki-ckb mw-list-item"><a href="https://ckb.wikipedia.org/wiki/%D9%85%DB%8E%DA%98%D9%88%D9%88%DB%8C_%D9%85%D8%A7%D8%AA%D9%85%D8%A7%D8%AA%DB%8C%DA%A9" title="مێژووی ماتماتیک – Central Kurdish" lang="ckb" hreflang="ckb" data-title="مێژووی ماتماتیک" data-language-autonym="کوردی" data-language-local-name="Central Kurdish" class="interlanguage-link-target">کوردی</a></li><li class="interlanguage-link interwiki-sr mw-list-item"><a href="https://sr.wikipedia.org/wiki/%D0%98%D1%81%D1%82%D0%BE%D1%80%D0%B8%D1%98%D0%B0_%D0%BC%D0%B0%D1%82%D0%B5%D0%BC%D0%B0%D1%82%D0%B8%D0%BA%D0%B5" title="Историја математике – Serbian" lang="sr" hreflang="sr" data-title="Историја математике" data-language-autonym="Српски / srpski" data-language-local-name="Serbian" class="interlanguage-link-target">Српски / srpski</a></li><li class="interlanguage-link interwiki-su mw-list-item"><a href="https://su.wikipedia.org/wiki/Sajarah_matematik" title="Sajarah matematik – Sundanese" lang="su" hreflang="su" data-title="Sajarah matematik" data-language-autonym="Sunda" data-language-local-name="Sundanese" class="interlanguage-link-target">Sunda</a></li><li class="interlanguage-link interwiki-fi mw-list-item"><a href="https://fi.wikipedia.org/wiki/Matematiikan_historia" title="Matematiikan historia – Finnish" lang="fi" hreflang="fi" data-title="Matematiikan historia" data-language-autonym="Suomi" data-language-local-name="Finnish" class="interlanguage-link-target">Suomi</a></li><li class="interlanguage-link interwiki-sv mw-list-item"><a href="https://sv.wikipedia.org/wiki/Matematikens_historia" title="Matematikens historia – Swedish" lang="sv" hreflang="sv" data-title="Matematikens historia" data-language-autonym="Svenska" data-language-local-name="Swedish" class="interlanguage-link-target">Svenska</a></li><li class="interlanguage-link interwiki-tl mw-list-item"><a href="https://tl.wikipedia.org/wiki/Kasaysayan_ng_matematika" title="Kasaysayan ng matematika – Tagalog" lang="tl" hreflang="tl" data-title="Kasaysayan ng matematika" data-language-autonym="Tagalog" data-language-local-name="Tagalog" class="interlanguage-link-target">Tagalog</a></li><li class="interlanguage-link interwiki-ta mw-list-item"><a href="https://ta.wikipedia.org/wiki/%E0%AE%95%E0%AE%A3%E0%AE%BF%E0%AE%A4%E0%AE%A4%E0%AF%8D%E0%AE%A4%E0%AE%BF%E0%AE%A9%E0%AF%8D_%E0%AE%B5%E0%AE%B0%E0%AE%B2%E0%AE%BE%E0%AE%B1%E0%AF%81" title="கணிதத்தின் வரலாறு – Tamil" lang="ta" hreflang="ta" data-title="கணிதத்தின் வரலாறு" data-language-autonym="தமிழ்" data-language-local-name="Tamil" class="interlanguage-link-target">தமிழ்</a></li><li class="interlanguage-link interwiki-tt mw-list-item"><a href="https://tt.wikipedia.org/wiki/%D0%9C%D0%B0%D1%82%D0%B5%D0%BC%D0%B0%D1%82%D0%B8%D0%BA%D0%B0_%D1%82%D0%B0%D1%80%D0%B8%D1%85%D1%8B" title="Математика тарихы – Tatar" lang="tt" hreflang="tt" data-title="Математика тарихы" data-language-autonym="Татарча / tatarça" data-language-local-name="Tatar" class="interlanguage-link-target">Татарча / tatarça</a></li><li class="interlanguage-link interwiki-tr mw-list-item"><a href="https://tr.wikipedia.org/wiki/Matematik_tarihi" title="Matematik tarihi – Turkish" lang="tr" hreflang="tr" data-title="Matematik tarihi" data-language-autonym="Türkçe" data-language-local-name="Turkish" class="interlanguage-link-target">Türkçe</a></li><li class="interlanguage-link interwiki-uk mw-list-item"><a href="https://uk.wikipedia.org/wiki/%D0%86%D1%81%D1%82%D0%BE%D1%80%D1%96%D1%8F_%D0%BC%D0%B0%D1%82%D0%B5%D0%BC%D0%B0%D1%82%D0%B8%D0%BA%D0%B8" title="Історія математики – Ukrainian" lang="uk" hreflang="uk" data-title="Історія математики" data-language-autonym="Українська" data-language-local-name="Ukrainian" class="interlanguage-link-target">Українська</a></li><li class="interlanguage-link interwiki-ur mw-list-item"><a href="https://ur.wikipedia.org/wiki/%D8%AA%D8%A7%D8%B1%DB%8C%D8%AE_%D8%B1%DB%8C%D8%A7%D8%B6%DB%8C" title="تاریخ ریاضی – Urdu" lang="ur" hreflang="ur" data-title="تاریخ ریاضی" data-language-autonym="اردو" data-language-local-name="Urdu" class="interlanguage-link-target">اردو</a></li><li class="interlanguage-link interwiki-vi mw-list-item"><a href="https://vi.wikipedia.org/wiki/L%E1%BB%8Bch_s%E1%BB%AD_to%C3%A1n_h%E1%BB%8Dc" title="Lịch sử toán học – Vietnamese" lang="vi" hreflang="vi" data-title="Lịch sử toán học" data-language-autonym="Tiếng Việt" data-language-local-name="Vietnamese" class="interlanguage-link-target">Tiếng Việt</a></li><li class="interlanguage-link interwiki-wuu mw-list-item"><a href="https://wuu.wikipedia.org/wiki/%E6%95%B0%E5%AD%A6%E5%8F%B2" title="数学史 – Wu" lang="wuu" hreflang="wuu" data-title="数学史" data-language-autonym="吴语" data-language-local-name="Wu" class="interlanguage-link-target">吴语</a></li><li class="interlanguage-link interwiki-zh-yue mw-list-item"><a href="https://zh-yue.wikipedia.org/wiki/%E6%95%B8%E5%AD%B8%E5%8F%B2" title="數學史 – Cantonese" lang="yue" hreflang="yue" data-title="數學史" data-language-autonym="粵語" data-language-local-name="Cantonese" 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#aaa;background:#ddddff;text-align:center;;color: var(--color-base)"><a href="/wiki/Areas_of_mathematics" class="mw-redirect" title="Areas of mathematics">Areas</a></div><div class="sidebar-list-content mw-collapsible-content hlist"> <ul><li><a href="/wiki/Number_theory" title="Number theory">Number theory</a></li> <li><a href="/wiki/Geometry" title="Geometry">Geometry</a></li> <li><a href="/wiki/Algebra" title="Algebra">Algebra</a></li> <li><a href="/wiki/Calculus" title="Calculus">Calculus</a> and <a href="/wiki/Mathematical_analysis" title="Mathematical analysis">Analysis</a></li> <li><a href="/wiki/Discrete_mathematics" title="Discrete mathematics">Discrete mathematics</a></li> <li><a href="/wiki/Mathematical_logic" title="Mathematical logic">Logic</a> and <a href="/wiki/Set_theory" title="Set theory">Set theory</a></li> <li><a href="/wiki/Probability" title="Probability">Probability</a></li> <li><a href="/wiki/Statistics" title="Statistics">Statistics</a> and <a href="/wiki/Decision_theory" title="Decision theory">Decision theory</a></li></ul></div></div></td> </tr><tr><td class="sidebar-content"> <div class="sidebar-list mw-collapsible"><div class="sidebar-list-title" style="border-top:1px solid #aaa;background:#ddddff;text-align:center;;color: var(--color-base)">Relationship with sciences</div><div class="sidebar-list-content mw-collapsible-content hlist"> <ul><li><a href="/wiki/Mathematical_physics" title="Mathematical physics">Physics</a></li> <li><a href="/wiki/Mathematical_chemistry" title="Mathematical chemistry">Chemistry</a></li> <li><a href="/wiki/Geomathematics" title="Geomathematics">Geosciences</a></li> <li><a href="/wiki/Computational_mathematics" title="Computational mathematics">Computation</a></li> <li><a href="/wiki/Mathematical_and_theoretical_biology" title="Mathematical and theoretical biology">Biology</a></li> <li><a href="/wiki/Computational_linguistics" title="Computational linguistics">Linguistics</a></li> <li><a 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data-file-height="128" /></a> <a href="/wiki/Portal:Mathematics" title="Portal:Mathematics">Mathematics Portal</a></th></tr><tr><td class="sidebar-navbar"><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:Math_topics_sidebar" title="Template:Math topics sidebar"><abbr title="View this template">v</abbr></a></li><li class="nv-talk"><a href="/wiki/Template_talk:Math_topics_sidebar" title="Template talk:Math topics sidebar"><abbr title="Discuss this template">t</abbr></a></li><li class="nv-edit"><a href="/wiki/Special:EditPage/Template:Math_topics_sidebar" title="Special:EditPage/Template:Math topics sidebar"><abbr title="Edit this template">e</abbr></a></li></ul></div></td></tr></tbody></table> The history of mathematics deals with the origin of discoveries in <a href="/wiki/Mathematics" title="Mathematics">mathematics</a> and the <a href="/wiki/History_of_mathematical_notation" title="History of mathematical notation">mathematical methods and notation of the past</a>. Before the <a href="/wiki/Modern_age" class="mw-redirect" title="Modern age">modern age</a> and the worldwide spread of knowledge, written examples of new mathematical developments have come to light only in a few locales. From 3000 BC the <a href="/wiki/Mesopotamian" class="mw-redirect" title="Mesopotamian">Mesopotamian</a> states of <a href="/wiki/Sumer" title="Sumer">Sumer</a>, <a href="/wiki/Akkad_(region)" class="mw-redirect" title="Akkad (region)">Akkad</a> and <a href="/wiki/Assyria" title="Assyria">Assyria</a>, followed closely by <a href="/wiki/Ancient_Egypt" title="Ancient Egypt">Ancient Egypt</a> and the Levantine state of <a href="/wiki/Ebla" title="Ebla">Ebla</a> began using <a href="/wiki/Arithmetic" title="Arithmetic">arithmetic</a>, <a href="/wiki/Algebra" title="Algebra">algebra</a> and <a href="/wiki/Geometry" title="Geometry">geometry</a> for purposes of <a href="/wiki/Taxation" class="mw-redirect" title="Taxation">taxation</a>, <a href="/wiki/Commerce" title="Commerce">commerce</a>, trade and also in the field of <a href="/wiki/Astronomy" title="Astronomy">astronomy</a> to record time and formulate <a href="/wiki/Calendars" class="mw-redirect" title="Calendars">calendars</a>. The earliest mathematical texts available are from <a href="/wiki/Mesopotamia" title="Mesopotamia">Mesopotamia</a> and <a href="/wiki/Ancient_Egypt" title="Ancient Egypt">Egypt</a> – <a href="/wiki/Plimpton_322" title="Plimpton 322">Plimpton 322</a> (<a href="/wiki/Babylonian_mathematics" title="Babylonian mathematics">Babylonian</a> <abbr title="circa">c.</abbr> 2000 – 1900 BC),<a href="#cite_note-:0-2">[2]</a> the <a href="/wiki/Rhind_Mathematical_Papyrus" title="Rhind Mathematical Papyrus">Rhind Mathematical Papyrus</a> (<a href="/wiki/Egyptian_mathematics" class="mw-redirect" title="Egyptian mathematics">Egyptian</a> c. 1800 BC)<a href="#cite_note-:1-3">[3]</a> and the <a href="/wiki/Moscow_Mathematical_Papyrus" title="Moscow Mathematical Papyrus">Moscow Mathematical Papyrus</a> (Egyptian c. 1890 BC). All of these texts mention the so-called <a href="/wiki/Pythagorean_triple" title="Pythagorean triple">Pythagorean triples</a>, so, by inference, the <a href="/wiki/Pythagorean_theorem" title="Pythagorean theorem">Pythagorean theorem</a> seems to be the most ancient and widespread mathematical development after basic arithmetic and geometry. The study of mathematics as a "demonstrative discipline" began in the 6th century BC with the <a href="/wiki/Pythagoreans" class="mw-redirect" title="Pythagoreans">Pythagoreans</a>, who coined the term "mathematics" from the ancient <a href="/wiki/Greek_language" title="Greek language">Greek</a> μάθημα (mathema), meaning "subject of instruction".<a href="#cite_note-4">[4]</a> <a href="/wiki/Greek_mathematics" title="Greek mathematics">Greek mathematics</a> greatly refined the methods (especially through the introduction of deductive reasoning and <a href="/wiki/Mathematical_rigor" class="mw-redirect" title="Mathematical rigor">mathematical rigor</a> in <a href="/wiki/Mathematical_proof" title="Mathematical proof">proofs</a>) and expanded the subject matter of mathematics.<a href="#cite_note-5">[5]</a> The <a href="/wiki/Ancient_Romans" class="mw-redirect" title="Ancient Romans">ancient Romans</a> used <a href="/wiki/Applied_mathematics" title="Applied mathematics">applied mathematics</a> in <a href="/wiki/Surveying" title="Surveying">surveying</a>, <a href="/wiki/Structural_engineering" title="Structural engineering">structural engineering</a>, <a href="/wiki/Mechanical_engineering" title="Mechanical engineering">mechanical engineering</a>, <a href="/wiki/Bookkeeping" title="Bookkeeping">bookkeeping</a>, creation of <a href="/wiki/Lunar_calendar" title="Lunar calendar">lunar</a> and <a href="/wiki/Solar_calendar" title="Solar calendar">solar calendars</a>, and even <a href="/wiki/Roman_art" title="Roman art">arts and crafts</a>. <a href="/wiki/Chinese_mathematics" title="Chinese mathematics">Chinese mathematics</a> made early contributions, including a <a href="/wiki/Place_value_system" class="mw-redirect" title="Place value system">place value system</a> and the first use of <a href="/wiki/Negative_numbers" class="mw-redirect" title="Negative numbers">negative numbers</a>.<a href="#cite_note-:2-6">[6]</a><a href="#cite_note-7">[7]</a> The <a href="/wiki/Hindu%E2%80%93Arabic_numeral_system" title="Hindu–Arabic numeral system">Hindu–Arabic numeral system</a> and the rules for the use of its operations, in use throughout the world today evolved over the course of the first millennium AD in <a href="/wiki/Indian_mathematics" title="Indian mathematics">India</a> and were transmitted to the <a href="/wiki/Western_world" title="Western world">Western world</a> via <a href="/wiki/Islamic_mathematics" class="mw-redirect" title="Islamic mathematics">Islamic mathematics</a> through the work of <a href="/wiki/Mu%E1%B8%A5ammad_ibn_M%C5%ABs%C4%81_al-Khw%C4%81rizm%C4%AB" class="mw-redirect" title="Muḥammad ibn Mūsā al-Khwārizmī">Muḥammad ibn Mūsā al-Khwārizmī</a>.<a href="#cite_note-8">[8]</a><a href="#cite_note-9">[9]</a> Islamic mathematics, in turn, developed and expanded the mathematics known to these civilizations.<a href="#cite_note-10">[10]</a> Contemporaneous with but independent of these traditions were the mathematics developed by the <a href="/wiki/Maya_civilization" title="Maya civilization">Maya civilization</a> of <a href="/wiki/Mexico" title="Mexico">Mexico</a> and <a href="/wiki/Central_America" title="Central America">Central America</a>, where the concept of <a href="/wiki/Zero" class="mw-redirect" title="Zero">zero</a> was given a standard symbol in <a href="/wiki/Maya_numerals" title="Maya numerals">Maya numerals</a>. Many Greek and Arabic texts on mathematics were <a href="/wiki/Latin_translations_of_the_12th_century" title="Latin translations of the 12th century">translated into Latin</a> from the 12th century onward, leading to further development of mathematics in <a href="/wiki/Middle_Ages" title="Middle Ages">Medieval Europe</a>. From ancient times through the <a href="/wiki/Postclassical_age" class="mw-redirect" title="Postclassical age">Middle Ages</a>, periods of mathematical discovery were often followed by centuries of stagnation.<a href="#cite_note-11">[11]</a> Beginning in <a href="/wiki/Renaissance" title="Renaissance">Renaissance</a> <a href="/wiki/Italy" title="Italy">Italy</a> in the 15th century, new mathematical developments, interacting with new scientific discoveries, were made at an <a href="/wiki/Exponential_growth" title="Exponential growth">increasing pace</a> that continues through the present day. This includes the groundbreaking work of both <a href="/wiki/Isaac_Newton" title="Isaac Newton">Isaac Newton</a> and <a href="/wiki/Gottfried_Wilhelm_Leibniz" title="Gottfried Wilhelm Leibniz">Gottfried Wilhelm Leibniz</a> in the development of infinitesimal <a href="/wiki/Calculus" title="Calculus">calculus</a> during the course of the 17th century. <div> <table class="wikitable zebra"> <caption>Table of numerals </caption> <tbody><tr> <td>European (descended from the West Arabic) </td> <td>0</td> <td>1</td> <td>2</td> <td>3</td> <td>4</td> <td>5</td> <td>6</td> <td>7</td> <td>8</td> <td>9 </td></tr> <tr> <td>Arabic-Indic </td> <td>٠</td> <td>١</td> <td>٢</td> <td>٣</td> <td>٤</td> <td>٥</td> <td>٦</td> <td>٧</td> <td>٨</td> <td>٩ </td></tr> <tr> <td>Eastern Arabic-Indic (Persian and Urdu) </td> <td>۰</td> <td>۱</td> <td>۲</td> <td>۳</td> <td>۴</td> <td>۵</td> <td>۶</td> <td>۷</td> <td>۸</td> <td>۹ </td></tr> <tr> <td>Devanagari (Hindi) </td> <td>०</td> <td>१</td> <td>२</td> <td>३</td> <td>४</td> <td>५</td> <td>६</td> <td>७</td> <td>८</td> <td>९ </td></tr> <tr> <td>Bengali </td> <td>০</td> <td>১</td> <td>২</td> <td>৩</td> <td>৪</td> <td>৫</td> <td>৬</td> <td>৭</td> <td>৮</td> <td>৯ </td></tr> <tr> <td>Chinese </td> <td>零</td> <td>一</td> <td>二</td> <td>三</td> <td>四</td> <td>五</td> <td>六</td> <td>七</td> <td>八</td> <td>九 </td></tr> <tr> <td>Tamil </td> <td>௦</td> <td>௧</td> <td>௨</td> <td>௩</td> <td>௪</td> <td>௫</td> <td>௬</td> <td>௭</td> <td>௮</td> <td>௯ </td></tr></tbody></table> </div> <meta property="mw:PageProp/toc" /> <div class="mw-heading mw-heading2"><h2 id="Prehistoric">Prehistoric</h2>[<a href="/w/index.php?title=History_of_mathematics&action=edit&section=1" title="Edit section: Prehistoric">edit</a>]</div> The origins of mathematical thought lie in the concepts of <a href="/wiki/Number" title="Number">number</a>, <a href="/wiki/Patterns_in_nature" title="Patterns in nature">patterns in nature</a>, <a href="/wiki/Magnitude_(mathematics)" title="Magnitude (mathematics)">magnitude</a>, and <a href="/wiki/Configuration_(geometry)" title="Configuration (geometry)">form</a>.<a href="#cite_note-Boyer_1991_loc=Origins_p._3-12">[12]</a> Modern studies of animal cognition have shown that these concepts are not unique to humans. Such concepts would have been part of everyday life in <a href="/wiki/Hunter-gatherer" title="Hunter-gatherer">hunter-gatherer</a> societies. The idea of the "number" concept evolving gradually over time is supported by the existence of languages which preserve the distinction between "one", "two", and "many", but not of numbers larger than two.<a href="#cite_note-Boyer_1991_loc=Origins_p._3-12">[12]</a> The <a href="/wiki/Ishango_bone" title="Ishango bone">Ishango bone</a>, found near the headwaters of the <a href="/wiki/Nile" title="Nile">Nile</a> river (northeastern <a href="/wiki/Democratic_Republic_of_the_Congo" title="Democratic Republic of the Congo">Congo</a>), may be more than <a href="/wiki/Upper_Paleolithic" title="Upper Paleolithic">20,000</a> years old and consists of a series of marks carved in three columns running the length of the bone. Common interpretations are that the Ishango bone shows either a tally of the earliest known demonstration of <a href="/wiki/Sequence" title="Sequence">sequences</a> of <a href="/wiki/Prime_number" title="Prime number">prime numbers</a><a href="#cite_note-Diaspora-13">[13]</a>[<a href="/wiki/Wikipedia:Verifiability" title="Wikipedia:Verifiability">failed verification</a>] or a six-month lunar calendar.<a href="#cite_note-Marshack-14">[14]</a> Peter Rudman argues that the development of the concept of prime numbers could only have come about after the concept of division, which he dates to after 10,000 BC, with prime numbers probably not being understood until about 500 BC. He also writes that "no attempt has been made to explain why a tally of something should exhibit multiples of two, prime numbers between 10 and 20, and some numbers that are almost multiples of 10."<a href="#cite_note-15">[15]</a> The Ishango bone, according to scholar <a href="/wiki/Alexander_Marshack" title="Alexander Marshack">Alexander Marshack</a>, may have influenced the later development of mathematics in Egypt as, like some entries on the Ishango bone, Egyptian arithmetic also made use of multiplication by 2; this however, is disputed.<a href="#cite_note-16">[16]</a> <a href="/wiki/Predynastic_Egypt" class="mw-redirect" title="Predynastic Egypt">Predynastic Egyptians</a> of the 5th millennium BC pictorially represented geometric designs. It has been claimed that <a href="/wiki/Megalith" title="Megalith">megalithic</a> monuments in <a href="/wiki/England" title="England">England</a> and <a href="/wiki/Scotland" title="Scotland">Scotland</a>, dating from the 3rd millennium BC, incorporate geometric ideas such as <a href="/wiki/Circle" title="Circle">circles</a>, <a href="/wiki/Ellipse" title="Ellipse">ellipses</a>, and <a href="/wiki/Pythagorean_triple" title="Pythagorean triple">Pythagorean triples</a> in their design.<a href="#cite_note-17">[17]</a> All of the above are disputed however, and the currently oldest undisputed mathematical documents are from Babylonian and dynastic Egyptian sources.<a href="#cite_note-18">[18]</a> <div class="mw-heading mw-heading2"><h2 id="Babylonian">Babylonian</h2>[<a href="/w/index.php?title=History_of_mathematics&action=edit&section=2" title="Edit section: Babylonian">edit</a>]</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">Main article: <a href="/wiki/Babylonian_mathematics" title="Babylonian mathematics">Babylonian mathematics</a></div> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1236090951"><div role="note" class="hatnote navigation-not-searchable">See also: <a href="/wiki/Plimpton_322" title="Plimpton 322">Plimpton 322</a></div> <a href="/wiki/Babylonia" title="Babylonia">Babylonian</a> mathematics refers to any mathematics of the peoples of <a href="/wiki/Mesopotamia" title="Mesopotamia">Mesopotamia</a> (modern <a href="/wiki/Iraq" title="Iraq">Iraq</a>) from the days of the early <a href="/wiki/Sumer" title="Sumer">Sumerians</a> through the <a href="/wiki/Hellenistic_period" title="Hellenistic period">Hellenistic period</a> almost to the dawn of <a href="/wiki/Christianity" title="Christianity">Christianity</a>.<a href="#cite_note-19">[19]</a> The majority of Babylonian mathematical work comes from two widely separated periods: The first few hundred years of the second millennium BC (Old Babylonian period), and the last few centuries of the first millennium BC (<a href="/wiki/Seleucid" class="mw-redirect" title="Seleucid">Seleucid</a> period).<a href="#cite_note-Boyer_1991_loc=Mesopotamia_p._26-20">[20]</a> It is named Babylonian mathematics due to the central role of <a href="/wiki/Babylon" title="Babylon">Babylon</a> as a place of study. Later under the <a href="/wiki/Caliphate" title="Caliphate">Arab Empire</a>, Mesopotamia, especially <a href="/wiki/Baghdad" title="Baghdad">Baghdad</a>, once again became an important center of study for <a href="/wiki/Islamic_mathematics" class="mw-redirect" title="Islamic mathematics">Islamic mathematics</a>. <figure class="mw-default-size" typeof="mw:File/Thumb"><a href="/wiki/File:Geometry_problem-Sb_13088-IMG_0593-white.jpg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/7/7b/Geometry_problem-Sb_13088-IMG_0593-white.jpg/220px-Geometry_problem-Sb_13088-IMG_0593-white.jpg" decoding="async" width="220" height="225" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/7/7b/Geometry_problem-Sb_13088-IMG_0593-white.jpg/330px-Geometry_problem-Sb_13088-IMG_0593-white.jpg 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/7/7b/Geometry_problem-Sb_13088-IMG_0593-white.jpg/440px-Geometry_problem-Sb_13088-IMG_0593-white.jpg 2x" data-file-width="1089" data-file-height="1115" /></a><figcaption>Geometry problem on a clay tablet belonging to a school for scribes; <a href="/wiki/Susa" title="Susa">Susa</a>, first half of the 2nd millennium BCE</figcaption></figure> In contrast to the sparsity of sources in <a href="/wiki/Egyptian_mathematics" class="mw-redirect" title="Egyptian mathematics">Egyptian mathematics</a>, knowledge of Babylonian mathematics is derived from more than 400 clay tablets unearthed since the 1850s.<a href="#cite_note-Boyer_1991_loc=Mesopotamia_p._25-21">[21]</a> Written in <a href="/wiki/Cuneiform_script" class="mw-redirect" title="Cuneiform script">Cuneiform script</a>, tablets were inscribed whilst the clay was moist, and baked hard in an oven or by the heat of the sun. Some of these appear to be graded homework.<a href="#cite_note-Boyer_1991_loc=Mesopotamia_p._41-22">[22]</a> The earliest evidence of written mathematics dates back to the ancient <a href="/wiki/Sumer" title="Sumer">Sumerians</a>, who built the earliest civilization in Mesopotamia. They developed a complex system of <a href="/wiki/Metrology" title="Metrology">metrology</a> from 3000 BC that was chiefly concerned with administrative/financial counting, such as grain allotments, workers, weights of silver, or even liquids, among other things.<a href="#cite_note-23">[23]</a> From around 2500 BC onward, the Sumerians wrote <a href="/wiki/Multiplication_table" title="Multiplication table">multiplication tables</a> on clay tablets and dealt with geometrical exercises and <a href="/wiki/Division_(mathematics)" title="Division (mathematics)">division</a> problems. The earliest traces of the Babylonian numerals also date back to this period.<a href="#cite_note-24">[24]</a> <figure class="mw-default-size mw-halign-left" typeof="mw:File/Thumb"><a href="/wiki/File:Plimpton_322.jpg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/c/c2/Plimpton_322.jpg/220px-Plimpton_322.jpg" decoding="async" width="220" height="152" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/c/c2/Plimpton_322.jpg/330px-Plimpton_322.jpg 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/c/c2/Plimpton_322.jpg/440px-Plimpton_322.jpg 2x" data-file-width="1246" data-file-height="863" /></a><figcaption>The Babylonian mathematical tablet <a href="/wiki/Plimpton_322" title="Plimpton 322">Plimpton 322</a>, dated to 1800 BC.</figcaption></figure> Babylonian mathematics were written using a <a href="/wiki/Sexagesimal" title="Sexagesimal">sexagesimal</a> (base-60) <a href="/wiki/Numeral_system" title="Numeral system">numeral system</a>.<a href="#cite_note-Boyer_1991_loc=Mesopotamia_p._25-21">[21]</a> From this derives the modern-day usage of 60 seconds in a minute, 60 minutes in an hour, and 360 (60 × 6) degrees in a circle, as well as the use of seconds and minutes of arc to denote fractions of a degree. It is thought the sexagesimal system was initially used by Sumerian scribes because 60 can be evenly divided by 2, 3, 4, 5, 6, 10, 12, 15, 20 and 30,<a href="#cite_note-Boyer_1991_loc=Mesopotamia_p._25-21">[21]</a> and for scribes (doling out the aforementioned grain allotments, recording weights of silver, etc.) being able to easily calculate by hand was essential, and so a sexagesimal system is pragmatically easier to calculate by hand with; however, there is the possibility that using a sexagesimal system was an ethno-linguistic phenomenon (that might not ever be known), and not a mathematical/practical decision.<a href="#cite_note-Powell_1976_p._418-25">[25]</a> Also, unlike the Egyptians, Greeks, and Romans, the Babylonians had a place-value system, where digits written in the left column represented larger values, much as in the <a href="/wiki/Decimal" title="Decimal">decimal</a> system. The power of the Babylonian notational system lay in that it could be used to represent fractions as easily as whole numbers; thus multiplying two numbers that contained fractions was no different from multiplying integers, similar to modern notation. The notational system of the Babylonians was the best of any civilization until the <a href="/wiki/Renaissance" title="Renaissance">Renaissance</a>, and its power allowed it to achieve remarkable computational accuracy; for example, the Babylonian tablet <a href="/wiki/YBC_7289" title="YBC 7289">YBC 7289</a> gives an approximation of √2 accurate to five decimal places.<a href="#cite_note-Boyer_1991_loc=Mesopotamia_p._27-26">[26]</a> The Babylonians lacked, however, an equivalent of the decimal point, and so the place value of a symbol often had to be inferred from the context.<a href="#cite_note-Boyer_1991_loc=Mesopotamia_p._26-20">[20]</a> By the Seleucid period, the Babylonians had developed a zero symbol as a placeholder for empty positions; however it was only used for intermediate positions.<a href="#cite_note-Boyer_1991_loc=Mesopotamia_p._26-20">[20]</a> This zero sign does not appear in terminal positions, thus the Babylonians came close but did not develop a true place value system.<a href="#cite_note-Boyer_1991_loc=Mesopotamia_p._26-20">[20]</a> Other topics covered by Babylonian mathematics include fractions, algebra, quadratic and cubic equations, and the calculation of <a href="/wiki/Regular_number" title="Regular number">regular numbers</a>, and their <a href="/wiki/Multiplicative_inverse" title="Multiplicative inverse">reciprocal</a> <a href="/wiki/Tuple" title="Tuple">pairs</a>.<a href="#cite_note-27">[27]</a> The tablets also include multiplication tables and methods for solving <a href="/wiki/Linear_equation" title="Linear equation">linear</a>, <a href="/wiki/Quadratic_equation" title="Quadratic equation">quadratic equations</a> and <a href="/wiki/Cubic_equation" title="Cubic equation">cubic equations</a>, a remarkable achievement for the time.<a href="#cite_note-28">[28]</a> Tablets from the Old Babylonian period also contain the earliest known statement of the <a href="/wiki/Pythagorean_theorem" title="Pythagorean theorem">Pythagorean theorem</a>.<a href="#cite_note-29">[29]</a> However, as with Egyptian mathematics, Babylonian mathematics shows no awareness of the difference between exact and approximate solutions, or the solvability of a problem, and most importantly, no explicit statement of the need for <a href="/wiki/Mathematical_proof" title="Mathematical proof">proofs</a> or logical principles.<a href="#cite_note-Boyer_1991_loc=Mesopotamia_p._41-22">[22]</a> <div class="mw-heading mw-heading2"><h2 id="Egyptian">Egyptian</h2>[<a href="/w/index.php?title=History_of_mathematics&action=edit&section=3" title="Edit section: Egyptian">edit</a>]</div> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1236090951"><div role="note" class="hatnote navigation-not-searchable">Main article: <a href="/wiki/Egyptian_mathematics" class="mw-redirect" title="Egyptian mathematics">Egyptian mathematics</a></div> <figure class="mw-default-size mw-halign-right" typeof="mw:File/Thumb"><a href="/wiki/File:Moskou-papyrus.jpg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/f/fd/Moskou-papyrus.jpg/330px-Moskou-papyrus.jpg" decoding="async" width="330" height="158" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/f/fd/Moskou-papyrus.jpg 1.5x" data-file-width="375" data-file-height="180" /></a><figcaption>Image of Problem 14 from the <a href="/wiki/Moscow_Mathematical_Papyrus" title="Moscow Mathematical Papyrus">Moscow Mathematical Papyrus</a>. The problem includes a diagram indicating the dimensions of the truncated pyramid.</figcaption></figure> <a href="/wiki/Egypt" title="Egypt">Egyptian</a> mathematics refers to mathematics written in the <a href="/wiki/Egyptian_language" title="Egyptian language">Egyptian language</a>. From the <a href="/wiki/Hellenistic_period" title="Hellenistic period">Hellenistic period</a>, <a href="/wiki/Greek_language" title="Greek language">Greek</a> replaced Egyptian as the written language of <a href="/wiki/Egyptians" title="Egyptians">Egyptian</a> scholars. Mathematical study in <a href="/wiki/Egypt" title="Egypt">Egypt</a> later continued under the <a href="/wiki/Caliphate" title="Caliphate">Arab Empire</a> as part of <a href="/wiki/Islamic_mathematics" class="mw-redirect" title="Islamic mathematics">Islamic mathematics</a>, when <a href="/wiki/Arabic" title="Arabic">Arabic</a> became the written language of Egyptian scholars. Archaeological evidence has suggested that the Ancient Egyptian counting system had origins in Sub-Saharan Africa.<a href="#cite_note-30">[30]</a> Also, fractal geometry designs which are widespread among Sub-Saharan African cultures are also found in Egyptian architecture and cosmological signs.<a href="#cite_note-31">[31]</a> The most extensive Egyptian mathematical text is the <a href="/wiki/Rhind_papyrus" class="mw-redirect" title="Rhind papyrus">Rhind papyrus</a> (sometimes also called the Ahmes Papyrus after its author), dated to c. 1650 BC but likely a copy of an older document from the <a href="/wiki/Middle_Kingdom_of_Egypt" title="Middle Kingdom of Egypt">Middle Kingdom</a> of about 2000–1800 BC.<a href="#cite_note-Boyer_1991_loc=Egypt_p._11-32">[32]</a> It is an instruction manual for students in arithmetic and geometry. In addition to giving area formulas and methods for multiplication, division and working with unit fractions, it also contains evidence of other mathematical knowledge,<a href="#cite_note-33">[33]</a> including <a href="/wiki/Composite_number" title="Composite number">composite</a> and <a href="/wiki/Prime_number" title="Prime number">prime numbers</a>; <a href="/wiki/Arithmetic_mean" title="Arithmetic mean">arithmetic</a>, <a href="/wiki/Geometric_mean" title="Geometric mean">geometric</a> and <a href="/wiki/Harmonic_mean" title="Harmonic mean">harmonic means</a>; and simplistic understandings of both the <a href="/wiki/Sieve_of_Eratosthenes" title="Sieve of Eratosthenes">Sieve of Eratosthenes</a> and <a href="/wiki/Perfect_number" title="Perfect number">perfect number theory</a> (namely, that of the number 6).<a href="#cite_note-34">[34]</a> It also shows how to solve first order <a href="/wiki/Linear_equation" title="Linear equation">linear equations</a><a href="#cite_note-35">[35]</a> as well as <a href="/wiki/Arithmetic_series" class="mw-redirect" title="Arithmetic series">arithmetic</a> and <a href="/wiki/Geometric_series" title="Geometric series">geometric series</a>.<a href="#cite_note-36">[36]</a> Another significant Egyptian mathematical text is the <a href="/wiki/Moscow_papyrus" class="mw-redirect" title="Moscow papyrus">Moscow papyrus</a>, also from the <a href="/wiki/Middle_Kingdom_of_Egypt" title="Middle Kingdom of Egypt">Middle Kingdom</a> period, dated to c. 1890 BC.<a href="#cite_note-Boyer_1991_loc=Egypt_p._19-37">[37]</a> It consists of what are today called word problems or story problems, which were apparently intended as entertainment. One problem is considered to be of particular importance because it gives a method for finding the volume of a <a href="/wiki/Frustum" title="Frustum">frustum</a> (truncated pyramid). Finally, the <a href="/wiki/Berlin_Papyrus_6619" title="Berlin Papyrus 6619">Berlin Papyrus 6619</a> (c. 1800 BC) shows that ancient Egyptians could solve a second-order <a href="/wiki/Algebraic_equation" title="Algebraic equation">algebraic equation</a>.<a href="#cite_note-38">[38]</a> <div class="mw-heading mw-heading2"><h2 id="Greek">Greek</h2>[<a href="/w/index.php?title=History_of_mathematics&action=edit&section=4" title="Edit section: Greek">edit</a>]</div> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1236090951"><div role="note" class="hatnote navigation-not-searchable">Main article: <a href="/wiki/Greek_mathematics" title="Greek mathematics">Greek mathematics</a></div> <figure class="mw-default-size mw-halign-left" typeof="mw:File/Thumb"><a href="/wiki/File:Pythagorean.svg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/d/d2/Pythagorean.svg/220px-Pythagorean.svg.png" decoding="async" width="220" height="200" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/d/d2/Pythagorean.svg/330px-Pythagorean.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/d/d2/Pythagorean.svg/440px-Pythagorean.svg.png 2x" data-file-width="512" data-file-height="466" /></a><figcaption>The <a href="/wiki/Pythagorean_theorem" title="Pythagorean theorem">Pythagorean theorem</a>. The <a href="/wiki/Pythagoreans" class="mw-redirect" title="Pythagoreans">Pythagoreans</a> are generally credited with the first proof of the theorem.</figcaption></figure> Greek mathematics refers to the mathematics written in the <a href="/wiki/Greek_language" title="Greek language">Greek language</a> from the time of <a href="/wiki/Thales_of_Miletus" title="Thales of Miletus">Thales of Miletus</a> (~600 BC) to the closure of the <a href="/wiki/Platonic_Academy" title="Platonic Academy">Academy of Athens</a> in 529 AD.<a href="#cite_note-39">[39]</a> Greek mathematicians lived in cities spread over the entire Eastern Mediterranean, from Italy to North Africa, but were united by culture and language. Greek mathematics of the period following <a href="/wiki/Alexander_the_Great" title="Alexander the Great">Alexander the Great</a> is sometimes called <a href="/wiki/Hellenistic_period" title="Hellenistic period">Hellenistic</a> mathematics.<a href="#cite_note-40">[40]</a> Greek mathematics was much more sophisticated than the mathematics that had been developed by earlier cultures. All surviving records of pre-Greek mathematics show the use of <a href="/wiki/Inductive_reasoning" title="Inductive reasoning">inductive reasoning</a>, that is, repeated observations used to establish rules of thumb. Greek mathematicians, by contrast, used <a href="/wiki/Deductive_reasoning" title="Deductive reasoning">deductive reasoning</a>. The Greeks used logic to derive conclusions from definitions and axioms, and used <a href="/wiki/Mathematical_rigor" class="mw-redirect" title="Mathematical rigor">mathematical rigor</a> to prove them.<a href="#cite_note-41">[41]</a> Greek mathematics is thought to have begun with <a href="/wiki/Thales_of_Miletus" title="Thales of Miletus">Thales of Miletus</a> (c. 624–c.546 BC) and <a href="/wiki/Pythagoras_of_Samos" class="mw-redirect" title="Pythagoras of Samos">Pythagoras of Samos</a> (c. 582–c. 507 BC). Although the extent of the influence is disputed, they were probably inspired by <a href="/wiki/Egyptian_mathematics" class="mw-redirect" title="Egyptian mathematics">Egyptian</a> and <a href="/wiki/Babylonian_mathematics" title="Babylonian mathematics">Babylonian mathematics</a>. According to legend, Pythagoras traveled to Egypt to learn mathematics, geometry, and astronomy from Egyptian priests. Thales used <a href="/wiki/Geometry" title="Geometry">geometry</a> to solve problems such as calculating the height of <a href="/wiki/Pyramids" class="mw-redirect" title="Pyramids">pyramids</a> and the distance of ships from the shore. He is credited with the first use of deductive reasoning applied to geometry, by deriving four corollaries to <a href="/wiki/Thales%27_Theorem" class="mw-redirect" title="Thales' Theorem">Thales' Theorem</a>. As a result, he has been hailed as the first true mathematician and the first known individual to whom a mathematical discovery has been attributed.<a href="#cite_note-42">[42]</a> Pythagoras established the <a href="/wiki/Pythagoreans" class="mw-redirect" title="Pythagoreans">Pythagorean School</a>, whose doctrine it was that mathematics ruled the universe and whose motto was "All is number".<a href="#cite_note-43">[43]</a> It was the Pythagoreans who coined the term "mathematics", and with whom the study of mathematics for its own sake begins. The Pythagoreans are credited with the first proof of the <a href="/wiki/Pythagorean_theorem" title="Pythagorean theorem">Pythagorean theorem</a>,<a href="#cite_note-44">[44]</a> though the statement of the theorem has a long history, and with the proof of the existence of <a href="/wiki/Irrational_numbers" class="mw-redirect" title="Irrational numbers">irrational numbers</a>.<a href="#cite_note-45">[45]</a><a href="#cite_note-46">[46]</a> Although he was preceded by the <a href="/wiki/Babylonian_mathematics" title="Babylonian mathematics">Babylonians</a>, <a href="/wiki/Indian_mathematics" title="Indian mathematics">Indians</a> and the <a href="/wiki/Chinese_mathematics" title="Chinese mathematics">Chinese</a>,<a href="#cite_note-Nature-47">[47]</a> the <a href="/wiki/Neopythagorean" class="mw-redirect" title="Neopythagorean">Neopythagorean</a> mathematician <a href="/wiki/Nicomachus" title="Nicomachus">Nicomachus</a> (60–120 AD) provided one of the earliest <a href="/wiki/Greco-Roman" class="mw-redirect" title="Greco-Roman">Greco-Roman</a> <a href="/wiki/Multiplication_table" title="Multiplication table">multiplication tables</a>, whereas the oldest extant Greek multiplication table is found on a wax tablet dated to the 1st century AD (now found in the <a href="/wiki/British_Museum" title="British Museum">British Museum</a>).<a href="#cite_note-48">[48]</a> The association of the Neopythagoreans with the Western invention of the multiplication table is evident in its later <a href="/wiki/Middle_Ages" title="Middle Ages">Medieval</a> name: the mensa Pythagorica.<a href="#cite_note-49">[49]</a> <a href="/wiki/Plato" title="Plato">Plato</a> (428/427 BC – 348/347 BC) is important in the history of mathematics for inspiring and guiding others.<a href="#cite_note-50">[50]</a> His <a href="/wiki/Platonic_Academy" title="Platonic Academy">Platonic Academy</a>, in <a href="/wiki/Athens" title="Athens">Athens</a>, became the mathematical center of the world in the 4th century BC, and it was from this school that the leading mathematicians of the day, such as <a href="/wiki/Eudoxus_of_Cnidus" title="Eudoxus of Cnidus">Eudoxus of Cnidus</a> (c. 390 - c. 340 BC), came.<a href="#cite_note-Boyer_1991_loc=The_Age_of_Plato_and_Aristotle_p._88-51">[51]</a> Plato also discussed the foundations of mathematics,<a href="#cite_note-52">[52]</a> clarified some of the definitions (e.g. that of a line as "breadthless length"), and reorganized the assumptions.<a href="#cite_note-53">[53]</a> The <a href="/wiki/Mathematical_analysis" title="Mathematical analysis">analytic method</a> is ascribed to Plato, while a formula for obtaining Pythagorean triples bears his name.<a href="#cite_note-Boyer_1991_loc=The_Age_of_Plato_and_Aristotle_p._88-51">[51]</a> Eudoxus developed the <a href="/wiki/Method_of_exhaustion" title="Method of exhaustion">method of exhaustion</a>, a precursor of modern <a href="/wiki/Integral" title="Integral">integration</a><a href="#cite_note-54">[54]</a> and a theory of ratios that avoided the problem of <a href="/wiki/Incommensurable_magnitudes" class="mw-redirect" title="Incommensurable magnitudes">incommensurable magnitudes</a>.<a href="#cite_note-55">[55]</a> The former allowed the calculations of areas and volumes of curvilinear figures,<a href="#cite_note-56">[56]</a> while the latter enabled subsequent geometers to make significant advances in geometry. Though he made no specific technical mathematical discoveries, <a href="/wiki/Aristotle" title="Aristotle">Aristotle</a> (384–<abbr title="circa">c.</abbr> 322 BC) contributed significantly to the development of mathematics by laying the foundations of <a href="/wiki/Logic" title="Logic">logic</a>.<a href="#cite_note-57">[57]</a> <figure class="mw-default-size mw-halign-right" typeof="mw:File/Thumb"><a href="/wiki/File:P._Oxy._I_29.jpg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/8/8d/P._Oxy._I_29.jpg/220px-P._Oxy._I_29.jpg" decoding="async" width="220" height="134" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/8/8d/P._Oxy._I_29.jpg/330px-P._Oxy._I_29.jpg 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/8/8d/P._Oxy._I_29.jpg/440px-P._Oxy._I_29.jpg 2x" data-file-width="1694" data-file-height="1032" /></a><figcaption>One of the oldest surviving fragments of Euclid's Elements, found at <a href="/wiki/Oxyrhynchus" title="Oxyrhynchus">Oxyrhynchus</a> and dated to circa AD 100. The diagram accompanies Book II, Proposition 5.<a href="#cite_note-58">[58]</a></figcaption></figure> In the 3rd century BC, the premier center of mathematical education and research was the <a href="/wiki/Musaeum" class="mw-redirect" title="Musaeum">Musaeum</a> of <a href="/wiki/Alexandria" title="Alexandria">Alexandria</a>.<a href="#cite_note-59">[59]</a> It was there that <a href="/wiki/Euclid" title="Euclid">Euclid</a> (<abbr title="circa">c.</abbr> 300 BC) taught, and wrote the <a href="/wiki/Euclid%27s_Elements" title="Euclid's Elements">Elements</a>, widely considered the most successful and influential textbook of all time.<a href="#cite_note-Boyer_1991_loc=Euclid_of_Alexandria_p._119-1">[1]</a> The Elements introduced <a href="/wiki/Mathematical_rigor" class="mw-redirect" title="Mathematical rigor">mathematical rigor</a> through the <a href="/wiki/Axiomatic_method" class="mw-redirect" title="Axiomatic method">axiomatic method</a> and is the earliest example of the format still used in mathematics today, that of definition, axiom, theorem, and proof. Although most of the contents of the Elements were already known, Euclid arranged them into a single, coherent logical framework.<a href="#cite_note-Boyer_1991_loc=Euclid_of_Alexandria_p._104-60">[60]</a> The Elements was known to all educated people in the West up through the middle of the 20th century and its contents are still taught in geometry classes today.<a href="#cite_note-61">[61]</a> In addition to the familiar theorems of <a href="/wiki/Euclidean_geometry" title="Euclidean geometry">Euclidean geometry</a>, the Elements was meant as an introductory textbook to all mathematical subjects of the time, such as <a href="/wiki/Number_theory" title="Number theory">number theory</a>, <a href="/wiki/Algebra" title="Algebra">algebra</a> and <a href="/wiki/Solid_geometry" title="Solid geometry">solid geometry</a>,<a href="#cite_note-Boyer_1991_loc=Euclid_of_Alexandria_p._104-60">[60]</a> including proofs that the square root of two is irrational and that there are infinitely many prime numbers. Euclid also <a href="/wiki/Euclid#Other_works" title="Euclid">wrote extensively</a> on other subjects, such as <a href="/wiki/Conic_sections" class="mw-redirect" title="Conic sections">conic sections</a>, <a href="/wiki/Optics" title="Optics">optics</a>, <a href="/wiki/Spherical_geometry" title="Spherical geometry">spherical geometry</a>, and mechanics, but only half of his writings survive.<a href="#cite_note-62">[62]</a> <figure class="mw-default-size mw-halign-left" typeof="mw:File/Thumb"><a href="/wiki/File:Archimedes_pi.svg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/c/c9/Archimedes_pi.svg/260px-Archimedes_pi.svg.png" decoding="async" width="260" height="87" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/c/c9/Archimedes_pi.svg/390px-Archimedes_pi.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/c/c9/Archimedes_pi.svg/520px-Archimedes_pi.svg.png 2x" data-file-width="750" data-file-height="250" /></a><figcaption>Archimedes used the <a href="/wiki/Method_of_exhaustion" title="Method of exhaustion">method of exhaustion</a> to approximate the value of <a href="/wiki/Pi" title="Pi">pi</a>.</figcaption></figure> <a href="/wiki/Archimedes" title="Archimedes">Archimedes</a> (<abbr title="circa">c.</abbr> 287–212 BC) of <a href="/wiki/Syracuse,_Italy" class="mw-redirect" title="Syracuse, Italy">Syracuse</a>, widely considered the greatest mathematician of antiquity,<a href="#cite_note-63">[63]</a> used the <a href="/wiki/Method_of_exhaustion" title="Method of exhaustion">method of exhaustion</a> to calculate the <a href="/wiki/Area" title="Area">area</a> under the arc of a <a href="/wiki/Parabola" title="Parabola">parabola</a> with the <a href="/wiki/Series_(mathematics)" title="Series (mathematics)">summation of an infinite series</a>, in a manner not too dissimilar from modern calculus.<a href="#cite_note-Boyer1991-64">[64]</a> He also showed one could use the method of exhaustion to calculate the value of π with as much precision as desired, and obtained the most accurate value of π then known, 3+<style data-mw-deduplicate="TemplateStyles:r1214402035">.mw-parser-output .sfrac{white-space:nowrap}.mw-parser-output .sfrac.tion,.mw-parser-output .sfrac .tion{display:inline-block;vertical-align:-0.5em;font-size:85%;text-align:center}.mw-parser-output .sfrac .num{display:block;line-height:1em;margin:0.0em 0.1em;border-bottom:1px solid}.mw-parser-output .sfrac .den{display:block;line-height:1em;margin:0.1em 0.1em}.mw-parser-output .sr-only{border:0;clip:rect(0,0,0,0);clip-path:polygon(0px 0px,0px 0px,0px 0px);height:1px;margin:-1px;overflow:hidden;padding:0;position:absolute;width:1px}</style>⁠10/71⁠ < π < 3+<link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1214402035">⁠10/70⁠.<a href="#cite_note-65">[65]</a> He also studied the <a href="/wiki/Archimedes_spiral" class="mw-redirect" title="Archimedes spiral">spiral</a> bearing his name, obtained formulas for the <a href="/wiki/Volume" title="Volume">volumes</a> of <a href="/wiki/Surface_of_revolution" title="Surface of revolution">surfaces of revolution</a> (paraboloid, ellipsoid, hyperboloid),<a href="#cite_note-Boyer1991-64">[64]</a> and an ingenious method of <a href="/wiki/Exponentiation" title="Exponentiation">exponentiation</a> for expressing very large numbers.<a href="#cite_note-66">[66]</a> While he is also known for his contributions to physics and several advanced mechanical devices, Archimedes himself placed far greater value on the products of his thought and general mathematical principles.<a href="#cite_note-67">[67]</a> He regarded as his greatest achievement his finding of the surface area and volume of a sphere, which he obtained by proving these are 2/3 the surface area and volume of a cylinder circumscribing the sphere.<a href="#cite_note-68">[68]</a> <figure class="mw-default-size mw-halign-right" typeof="mw:File/Thumb"><a href="/wiki/File:Conic_sections_2.png" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/4/48/Conic_sections_2.png/280px-Conic_sections_2.png" decoding="async" width="280" height="156" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/4/48/Conic_sections_2.png/420px-Conic_sections_2.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/4/48/Conic_sections_2.png/560px-Conic_sections_2.png 2x" data-file-width="1080" data-file-height="600" /></a><figcaption><a href="/wiki/Apollonius_of_Perga" title="Apollonius of Perga">Apollonius of Perga</a> made significant advances in the study of <a href="/wiki/Conic_sections" class="mw-redirect" title="Conic sections">conic sections</a>.</figcaption></figure> <a href="/wiki/Apollonius_of_Perga" title="Apollonius of Perga">Apollonius of Perga</a> (<abbr title="circa">c.</abbr> 262–190 BC) made significant advances to the study of <a href="/wiki/Conic_sections" class="mw-redirect" title="Conic sections">conic sections</a>, showing that one can obtain all three varieties of conic section by varying the angle of the plane that cuts a double-napped cone.<a href="#cite_note-69">[69]</a> He also coined the terminology in use today for conic sections, namely <a href="/wiki/Parabola" title="Parabola">parabola</a> ("place beside" or "comparison"), "ellipse" ("deficiency"), and "hyperbola" ("a throw beyond").<a href="#cite_note-70">[70]</a> His work Conics is one of the best known and preserved mathematical works from antiquity, and in it he derives many theorems concerning conic sections that would prove invaluable to later mathematicians and astronomers studying planetary motion, such as Isaac Newton.<a href="#cite_note-71">[71]</a> While neither Apollonius nor any other Greek mathematicians made the leap to coordinate geometry, Apollonius' treatment of curves is in some ways similar to the modern treatment, and some of his work seems to anticipate the development of analytical geometry by Descartes some 1800 years later.<a href="#cite_note-72">[72]</a> Around the same time, <a href="/wiki/Eratosthenes_of_Cyrene" class="mw-redirect" title="Eratosthenes of Cyrene">Eratosthenes of Cyrene</a> (<abbr title="circa">c.</abbr> 276–194 BC) devised the <a href="/wiki/Sieve_of_Eratosthenes" title="Sieve of Eratosthenes">Sieve of Eratosthenes</a> for finding <a href="/wiki/Prime_numbers" class="mw-redirect" title="Prime numbers">prime numbers</a>.<a href="#cite_note-73">[73]</a> The 3rd century BC is generally regarded as the "Golden Age" of Greek mathematics, with advances in pure mathematics henceforth in relative decline.<a href="#cite_note-autogenerated3-74">[74]</a> Nevertheless, in the centuries that followed significant advances were made in applied mathematics, most notably <a href="/wiki/Trigonometry" title="Trigonometry">trigonometry</a>, largely to address the needs of astronomers.<a href="#cite_note-autogenerated3-74">[74]</a> <a href="/wiki/Hipparchus_of_Nicaea" class="mw-redirect" title="Hipparchus of Nicaea">Hipparchus of Nicaea</a> (<abbr title="circa">c.</abbr> 190–120 BC) is considered the founder of trigonometry for compiling the first known trigonometric table, and to him is also due the systematic use of the 360 degree circle.<a href="#cite_note-75">[75]</a> <a href="/wiki/Heron_of_Alexandria" class="mw-redirect" title="Heron of Alexandria">Heron of Alexandria</a> (<abbr title="circa">c.</abbr> 10–70 AD) is credited with <a href="/wiki/Heron%27s_formula" title="Heron's formula">Heron's formula</a> for finding the area of a scalene triangle and with being the first to recognize the possibility of negative numbers possessing square roots.<a href="#cite_note-76">[76]</a> <a href="/wiki/Menelaus_of_Alexandria" title="Menelaus of Alexandria">Menelaus of Alexandria</a> (<abbr title="circa">c.</abbr> 100 AD) pioneered <a href="/wiki/Spherical_trigonometry" title="Spherical trigonometry">spherical trigonometry</a> through <a href="/wiki/Menelaus%27_theorem" class="mw-redirect" title="Menelaus' theorem">Menelaus' theorem</a>.<a href="#cite_note-77">[77]</a> The most complete and influential trigonometric work of antiquity is the <a href="/wiki/Almagest" title="Almagest">Almagest</a> of <a href="/wiki/Claudius_Ptolemy" class="mw-redirect" title="Claudius Ptolemy">Ptolemy</a> (<abbr title="circa">c.</abbr> AD 90–168), a landmark astronomical treatise whose trigonometric tables would be used by astronomers for the next thousand years.<a href="#cite_note-78">[78]</a> Ptolemy is also credited with <a href="/wiki/Ptolemy%27s_theorem" title="Ptolemy's theorem">Ptolemy's theorem</a> for deriving trigonometric quantities, and the most accurate value of π outside of China until the medieval period, 3.1416.<a href="#cite_note-79">[79]</a> <figure class="mw-default-size mw-halign-right" typeof="mw:File/Thumb"><a href="/wiki/File:Diophantus-cover.png" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/3/3b/Diophantus-cover.png/170px-Diophantus-cover.png" decoding="async" width="170" height="266" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/3/3b/Diophantus-cover.png/255px-Diophantus-cover.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/3/3b/Diophantus-cover.png/340px-Diophantus-cover.png 2x" data-file-width="828" data-file-height="1295" /></a><figcaption>Title page of the 1621 edition of Diophantus' Arithmetica, translated into <a href="/wiki/Latin" title="Latin">Latin</a> by <a href="/wiki/Claude_Gaspard_Bachet_de_M%C3%A9ziriac" class="mw-redirect" title="Claude Gaspard Bachet de Méziriac">Claude Gaspard Bachet de Méziriac</a>.</figcaption></figure> Following a period of stagnation after Ptolemy, the period between 250 and 350 AD is sometimes referred to as the "Silver Age" of Greek mathematics.<a href="#cite_note-80">[80]</a> During this period, <a href="/wiki/Diophantus" title="Diophantus">Diophantus</a> made significant advances in algebra, particularly <a href="/wiki/Indeterminate_equation" title="Indeterminate equation">indeterminate analysis</a>, which is also known as "Diophantine analysis".<a href="#cite_note-81">[81]</a> The study of <a href="/wiki/Diophantine_equations" class="mw-redirect" title="Diophantine equations">Diophantine equations</a> and <a href="/wiki/Diophantine_approximations" class="mw-redirect" title="Diophantine approximations">Diophantine approximations</a> is a significant area of research to this day. His main work was the Arithmetica, a collection of 150 algebraic problems dealing with exact solutions to determinate and <a href="/wiki/Indeterminate_equation" title="Indeterminate equation">indeterminate equations</a>.<a href="#cite_note-autogenerated1-82">[82]</a> The Arithmetica had a significant influence on later mathematicians, such as <a href="/wiki/Pierre_de_Fermat" title="Pierre de Fermat">Pierre de Fermat</a>, who arrived at his famous <a href="/wiki/Fermat%27s_Last_Theorem" title="Fermat's Last Theorem">Last Theorem</a> after trying to generalize a problem he had read in the Arithmetica (that of dividing a square into two squares).<a href="#cite_note-83">[83]</a> Diophantus also made significant advances in notation, the Arithmetica being the first instance of algebraic symbolism and syncopation.<a href="#cite_note-autogenerated1-82">[82]</a> <figure class="mw-default-size mw-halign-left" typeof="mw:File/Thumb"><a href="/wiki/File:Hagia_Sophia_Mars_2013.jpg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/2/22/Hagia_Sophia_Mars_2013.jpg/220px-Hagia_Sophia_Mars_2013.jpg" decoding="async" width="220" height="147" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/2/22/Hagia_Sophia_Mars_2013.jpg/330px-Hagia_Sophia_Mars_2013.jpg 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/2/22/Hagia_Sophia_Mars_2013.jpg/440px-Hagia_Sophia_Mars_2013.jpg 2x" data-file-width="5514" data-file-height="3681" /></a><figcaption>The <a href="/wiki/Hagia_Sophia" title="Hagia Sophia">Hagia Sophia</a> was designed by mathematicians <a href="/wiki/Anthemius_of_Tralles" title="Anthemius of Tralles">Anthemius of Tralles</a> and <a href="/wiki/Isidore_of_Miletus" title="Isidore of Miletus">Isidore of Miletus</a>.</figcaption></figure> Among the last great Greek mathematicians is <a href="/wiki/Pappus_of_Alexandria" title="Pappus of Alexandria">Pappus of Alexandria</a> (4th century AD). He is known for his <a href="/wiki/Pappus%27s_hexagon_theorem" title="Pappus's hexagon theorem">hexagon theorem</a> and <a href="/wiki/Pappus%27s_centroid_theorem" title="Pappus's centroid theorem">centroid theorem</a>, as well as the <a href="/wiki/Pappus_configuration" title="Pappus configuration">Pappus configuration</a> and <a href="/wiki/Pappus_graph" title="Pappus graph">Pappus graph</a>. His Collection is a major source of knowledge on Greek mathematics as most of it has survived.<a href="#cite_note-84">[84]</a> Pappus is considered the last major innovator in Greek mathematics, with subsequent work consisting mostly of commentaries on earlier work. The first woman mathematician recorded by history was <a href="/wiki/Hypatia" title="Hypatia">Hypatia</a> of Alexandria (AD 350–415). She succeeded her father (<a href="/wiki/Theon_of_Alexandria" title="Theon of Alexandria">Theon of Alexandria</a>) as Librarian at the Great Library[<a href="/wiki/Wikipedia:Citation_needed" title="Wikipedia:Citation needed">citation needed</a>] and wrote many works on applied mathematics. Because of a political dispute, the <a href="/wiki/Christianity_in_the_Roman_Empire" class="mw-redirect" title="Christianity in the Roman Empire">Christian community</a> in Alexandria had her stripped publicly and executed.<a href="#cite_note-85">[85]</a> Her death is sometimes taken as the end of the era of the Alexandrian Greek mathematics, although work did continue in Athens for another century with figures such as <a href="/wiki/Proclus" title="Proclus">Proclus</a>, <a href="/wiki/Simplicius_of_Cilicia" title="Simplicius of Cilicia">Simplicius</a> and <a href="/wiki/Eutocius" class="mw-redirect" title="Eutocius">Eutocius</a>.<a href="#cite_note-86">[86]</a> Although Proclus and Simplicius were more philosophers than mathematicians, their commentaries on earlier works are valuable sources on Greek mathematics. The closure of the neo-Platonic <a href="/wiki/Platonic_Academy" title="Platonic Academy">Academy of Athens</a> by the emperor <a href="/wiki/Justinian" class="mw-redirect" title="Justinian">Justinian</a> in 529 AD is traditionally held as marking the end of the era of Greek mathematics, although the Greek tradition continued unbroken in the <a href="/wiki/Byzantine_empire" class="mw-redirect" title="Byzantine empire">Byzantine empire</a> with mathematicians such as <a href="/wiki/Anthemius_of_Tralles" title="Anthemius of Tralles">Anthemius of Tralles</a> and <a href="/wiki/Isidore_of_Miletus" title="Isidore of Miletus">Isidore of Miletus</a>, the architects of the <a href="/wiki/Hagia_Sophia" title="Hagia Sophia">Hagia Sophia</a>.<a href="#cite_note-87">[87]</a> Nevertheless, Byzantine mathematics consisted mostly of commentaries, with little in the way of innovation, and the centers of mathematical innovation were to be found elsewhere by this time.<a href="#cite_note-88">[88]</a> <div class="mw-heading mw-heading2"><h2 id="Roman">Roman</h2>[<a href="/w/index.php?title=History_of_mathematics&action=edit&section=5" title="Edit section: Roman">edit</a>]</div> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1236090951"><div role="note" class="hatnote navigation-not-searchable">Further information: <a href="/wiki/Roman_abacus" title="Roman abacus">Roman abacus</a> and <a href="/wiki/Roman_numerals" title="Roman numerals">Roman numerals</a></div> <figure class="mw-default-size" typeof="mw:File/Thumb"><a href="/wiki/File:Aquinqum_BHM_IMG_0662_land_surveyor_equipment.jpg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/c/cd/Aquinqum_BHM_IMG_0662_land_surveyor_equipment.jpg/220px-Aquinqum_BHM_IMG_0662_land_surveyor_equipment.jpg" decoding="async" width="220" height="221" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/c/cd/Aquinqum_BHM_IMG_0662_land_surveyor_equipment.jpg/330px-Aquinqum_BHM_IMG_0662_land_surveyor_equipment.jpg 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/c/cd/Aquinqum_BHM_IMG_0662_land_surveyor_equipment.jpg/440px-Aquinqum_BHM_IMG_0662_land_surveyor_equipment.jpg 2x" data-file-width="3122" data-file-height="3134" /></a><figcaption>Equipment used by an <a href="/wiki/Ancient_Rome" title="Ancient Rome">ancient Roman</a> land <a href="/wiki/Surveyor" class="mw-redirect" title="Surveyor">surveyor</a> (<a href="/wiki/Gromatici" title="Gromatici">gromatici</a>), found at the site of <a href="/wiki/Aquincum" title="Aquincum">Aquincum</a>, modern <a href="/wiki/Budapest" title="Budapest">Budapest</a>, <a href="/wiki/Hungary" title="Hungary">Hungary</a></figcaption></figure> Although <a href="/wiki/Greeks" title="Greeks">ethnic Greek</a> mathematicians continued under the rule of the late <a href="/wiki/Roman_Republic" title="Roman Republic">Roman Republic</a> and subsequent <a href="/wiki/Roman_Empire" title="Roman Empire">Roman Empire</a>, there were no noteworthy <a href="/wiki/Latins_(Italic_tribe)" title="Latins (Italic tribe)">native Latin</a> mathematicians in comparison.<a href="#cite_note-89">[89]</a><a href="#cite_note-90">[90]</a> <a href="/wiki/Ancient_Rome" title="Ancient Rome">Ancient Romans</a> such as <a href="/wiki/Cicero" title="Cicero">Cicero</a> (106–43 BC), an influential Roman statesman who studied mathematics in Greece, believed that Roman <a href="/wiki/Surveyor" class="mw-redirect" title="Surveyor">surveyors</a> and <a href="/wiki/Mental_calculator" title="Mental calculator">calculators</a> were far more interested in <a href="/wiki/Applied_mathematics" title="Applied mathematics">applied mathematics</a> than the <a href="/wiki/Theoretical_mathematics" class="mw-redirect" title="Theoretical mathematics">theoretical mathematics</a> and geometry that were prized by the Greeks.<a href="#cite_note-91">[91]</a> It is unclear if the Romans first derived <a href="/wiki/Roman_numerals" title="Roman numerals">their numerical system</a> directly from <a href="/wiki/Greek_numerals" title="Greek numerals">the Greek precedent</a> or from <a href="/wiki/Etruscan_numerals" title="Etruscan numerals">Etruscan numerals</a> used by the <a href="/wiki/Etruscan_civilization" title="Etruscan civilization">Etruscan civilization</a> centered in what is now <a href="/wiki/Tuscany" title="Tuscany">Tuscany</a>, <a href="/wiki/Central_Italy" title="Central Italy">central Italy</a>.<a href="#cite_note-92">[92]</a> Using calculation, Romans were adept at both instigating and detecting financial <a href="/wiki/Fraud" title="Fraud">fraud</a>, as well as <a href="/wiki/List_of_Roman_taxes" title="List of Roman taxes">managing taxes</a> for the <a href="/wiki/Treasury" title="Treasury">treasury</a>.<a href="#cite_note-93">[93]</a> <a href="/wiki/Siculus_Flaccus" title="Siculus Flaccus">Siculus Flaccus</a>, one of the Roman <a href="/wiki/Gromatici" title="Gromatici">gromatici</a> (i.e. land surveyor), wrote the Categories of Fields, which aided Roman surveyors in measuring the <a href="/wiki/Surface_area" title="Surface area">surface areas</a> of allotted lands and territories.<a href="#cite_note-94">[94]</a> Aside from managing trade and taxes, the Romans also regularly applied mathematics to solve problems in <a href="/wiki/Roman_engineering" class="mw-redirect" title="Roman engineering">engineering</a>, including the erection of <a href="/wiki/Roman_architecture" class="mw-redirect" title="Roman architecture">architecture</a> such as <a href="/wiki/Roman_bridge" title="Roman bridge">bridges</a>, <a href="/wiki/Roman_roads" title="Roman roads">road-building</a>, and <a href="/wiki/Roman_military_engineering" title="Roman military engineering">preparation for military campaigns</a>.<a href="#cite_note-95">[95]</a> <a href="/wiki/Roman_art" title="Roman art">Arts and crafts</a> such as <a href="/wiki/Roman_mosaic" title="Roman mosaic">Roman mosaics</a>, inspired by previous <a href="/wiki/Mosaics_of_Delos" title="Mosaics of Delos">Greek designs</a>, created illusionist geometric patterns and rich, detailed scenes that required precise measurements for each <a href="/wiki/Tessera" title="Tessera">tessera</a> tile, the <a href="/wiki/Opus_tessellatum" title="Opus tessellatum">opus tessellatum</a> pieces on average measuring eight millimeters square and the finer <a href="/wiki/Opus_vermiculatum" title="Opus vermiculatum">opus vermiculatum</a> pieces having an average surface of four millimeters square.<a href="#cite_note-96">[96]</a><a href="#cite_note-97">[97]</a> The creation of the <a href="/wiki/Roman_calendar" title="Roman calendar">Roman calendar</a> also necessitated basic mathematics. The first calendar allegedly dates back to 8th century BC during the <a href="/wiki/Roman_Kingdom" title="Roman Kingdom">Roman Kingdom</a> and included 356 days plus a <a href="/wiki/Leap_year" title="Leap year">leap year</a> every other year.<a href="#cite_note-98">[98]</a> In contrast, the <a href="/wiki/Lunar_calendar" title="Lunar calendar">lunar calendar</a> of the Republican era contained 355 days, roughly ten-and-one-fourth days shorter than the <a href="/wiki/Solar_year" class="mw-redirect" title="Solar year">solar year</a>, a discrepancy that was solved by adding an extra month into the calendar after the 23rd of February.<a href="#cite_note-99">[99]</a> This calendar was supplanted by the <a href="/wiki/Julian_calendar" title="Julian calendar">Julian calendar</a>, a <a href="/wiki/Solar_calendar" title="Solar calendar">solar calendar</a> organized by <a href="/wiki/Julius_Caesar" title="Julius Caesar">Julius Caesar</a> (100–44 BC) and devised by <a href="/wiki/Sosigenes_of_Alexandria" class="mw-redirect" title="Sosigenes of Alexandria">Sosigenes of Alexandria</a> to include a <a href="/wiki/Leap_day" class="mw-redirect" title="Leap day">leap day</a> every four years in a 365-day cycle.<a href="#cite_note-100">[100]</a> This calendar, which contained an error of 11 minutes and 14 seconds, was later corrected by the <a href="/wiki/Gregorian_calendar" title="Gregorian calendar">Gregorian calendar</a> organized by <a href="/wiki/Pope_Gregory_XIII" title="Pope Gregory XIII">Pope Gregory XIII</a> (<abbr title="reigned">r.</abbr> 1572–1585), virtually the same solar calendar used in modern times as the international standard calendar.<a href="#cite_note-101">[101]</a> At roughly the same time, <a href="/wiki/Science_and_technology_of_the_Han_dynasty" title="Science and technology of the Han dynasty">the Han Chinese</a> and the Romans both invented the wheeled <a href="/wiki/Odometer" title="Odometer">odometer</a> device for measuring <a href="/wiki/Distance" title="Distance">distances</a> traveled, the Roman model first described by the Roman civil engineer and architect <a href="/wiki/Vitruvius" title="Vitruvius">Vitruvius</a> (<abbr title="circa">c.</abbr> 80 BC – c. 15 BC).<a href="#cite_note-102">[102]</a> The device was used at least until the reign of emperor <a href="/wiki/Commodus" title="Commodus">Commodus</a> (<abbr title="reigned">r.</abbr> 177 – 192 AD), but its design seems to have been lost until experiments were made during the 15th century in Western Europe.<a href="#cite_note-103">[103]</a> Perhaps relying on similar gear-work and <a href="/wiki/Roman_technology" class="mw-redirect" title="Roman technology">technology</a> found in the <a href="/wiki/Antikythera_mechanism" title="Antikythera mechanism">Antikythera mechanism</a>, the odometer of Vitruvius featured chariot wheels measuring 4 feet (1.2 m) in diameter turning four-hundred times in one <a href="/wiki/Roman_mile" class="mw-redirect" title="Roman mile">Roman mile</a> (roughly 4590 ft/1400 m). With each revolution, a pin-and-axle device engaged a 400-tooth <a href="/wiki/Cogwheel" class="mw-redirect" title="Cogwheel">cogwheel</a> that turned a second gear responsible for dropping pebbles into a box, each pebble representing one mile traversed.<a href="#cite_note-104">[104]</a> <div class="mw-heading mw-heading2"><h2 id="Chinese">Chinese</h2>[<a href="/w/index.php?title=History_of_mathematics&action=edit&section=6" title="Edit section: Chinese">edit</a>]</div> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1236090951"><div role="note" class="hatnote navigation-not-searchable">Main article: <a href="/wiki/Chinese_mathematics" title="Chinese mathematics">Chinese mathematics</a></div> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1236090951"><div role="note" class="hatnote navigation-not-searchable">Further information: <a href="/wiki/Book_on_Numbers_and_Computation" title="Book on Numbers and Computation">Book on Numbers and Computation</a></div> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1236090951"><div role="note" class="hatnote navigation-not-searchable">See also: <a href="/wiki/History_of_science#Chinese_mathematics" title="History of science">History of science § Chinese mathematics</a></div> <figure class="mw-default-size mw-halign-right" typeof="mw:File/Thumb"><a href="/wiki/File:Qinghuajian,_Suan_Biao.jpg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/2/24/Qinghuajian%2C_Suan_Biao.jpg/170px-Qinghuajian%2C_Suan_Biao.jpg" decoding="async" width="170" height="233" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/2/24/Qinghuajian%2C_Suan_Biao.jpg/255px-Qinghuajian%2C_Suan_Biao.jpg 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/2/24/Qinghuajian%2C_Suan_Biao.jpg/340px-Qinghuajian%2C_Suan_Biao.jpg 2x" data-file-width="1528" data-file-height="2096" /></a><figcaption>The <a href="/wiki/Tsinghua_Bamboo_Slips" title="Tsinghua Bamboo Slips">Tsinghua Bamboo Slips</a>, containing the world's earliest <a href="/wiki/Decimal" title="Decimal">decimal</a> multiplication table, dated 305 BC during the <a href="/wiki/Warring_States" class="mw-redirect" title="Warring States">Warring States</a> period</figcaption></figure> An analysis of early Chinese mathematics has demonstrated its unique development compared to other parts of the world, leading scholars to assume an entirely independent development.<a href="#cite_note-105">[105]</a> The oldest extant mathematical text from China is the <a href="/wiki/Zhoubi_Suanjing" title="Zhoubi Suanjing">Zhoubi Suanjing</a> (周髀算經), variously dated to between 1200 BC and 100 BC, though a date of about 300 BC during the <a href="/wiki/Warring_States_Period" class="mw-redirect" title="Warring States Period">Warring States Period</a> appears reasonable.<a href="#cite_note-Boyer_1991_loc=China_and_India_p._196-106">[106]</a> However, the <a href="/wiki/Tsinghua_Bamboo_Slips" title="Tsinghua Bamboo Slips">Tsinghua Bamboo Slips</a>, containing the earliest known <a href="/wiki/Decimal" title="Decimal">decimal</a> <a href="/wiki/Multiplication_table" title="Multiplication table">multiplication table</a> (although ancient Babylonians had ones with a base of 60), is dated around 305 BC and is perhaps the oldest surviving mathematical text of China.<a href="#cite_note-Nature-47">[47]</a> <figure class="mw-default-size mw-halign-left" typeof="mw:File/Thumb"><a href="/wiki/File:Chounumerals.svg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/9/9d/Chounumerals.svg/220px-Chounumerals.svg.png" decoding="async" width="220" height="73" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/9/9d/Chounumerals.svg/330px-Chounumerals.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/9/9d/Chounumerals.svg/440px-Chounumerals.svg.png 2x" data-file-width="900" data-file-height="300" /></a><figcaption><a href="/wiki/Counting_rod_numerals" class="mw-redirect" title="Counting rod numerals">Counting rod numerals</a></figcaption></figure> Of particular note is the use in Chinese mathematics of a decimal positional notation system, the so-called "rod numerals" in which distinct ciphers were used for numbers between 1 and 10, and additional ciphers for powers of ten.<a href="#cite_note-107">[107]</a> Thus, the number 123 would be written using the symbol for "1", followed by the symbol for "100", then the symbol for "2" followed by the symbol for "10", followed by the symbol for "3". This was the most advanced number system in the world at the time, apparently in use several centuries before the common era and well before the development of the Indian numeral system.<a href="#cite_note-108">[108]</a> <a href="/wiki/Counting_rods" title="Counting rods">Rod numerals</a> allowed the representation of numbers as large as desired and allowed calculations to be carried out on the <a href="/wiki/Suanpan" title="Suanpan">suan pan</a>, or Chinese abacus. The date of the invention of the suan pan is not certain, but the earliest written mention dates from AD 190, in <a href="/wiki/Xu_Yue_(mathematician)" title="Xu Yue (mathematician)">Xu Yue</a>'s Supplementary Notes on the Art of Figures. The oldest extant work on geometry in China comes from the philosophical <a href="/wiki/Mohism" title="Mohism">Mohist</a> canon <abbr title="circa">c.</abbr> 330 BC, compiled by the followers of <a href="/wiki/Mozi" title="Mozi">Mozi</a> (470–390 BC). The Mo Jing described various aspects of many fields associated with physical science, and provided a small number of geometrical theorems as well.<a href="#cite_note-109">[109]</a> It also defined the concepts of <a href="/wiki/Circumference" title="Circumference">circumference</a>, <a href="/wiki/Diameter" title="Diameter">diameter</a>, <a href="/wiki/Radius" title="Radius">radius</a>, and <a href="/wiki/Volume" title="Volume">volume</a>.<a href="#cite_note-110">[110]</a> <figure class="mw-default-size mw-halign-right" typeof="mw:File/Thumb"><a href="/wiki/File:%E4%B9%9D%E7%AB%A0%E7%AE%97%E8%A1%93.gif" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/8/88/%E4%B9%9D%E7%AB%A0%E7%AE%97%E8%A1%93.gif/170px-%E4%B9%9D%E7%AB%A0%E7%AE%97%E8%A1%93.gif" decoding="async" width="170" height="222" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/8/88/%E4%B9%9D%E7%AB%A0%E7%AE%97%E8%A1%93.gif/255px-%E4%B9%9D%E7%AB%A0%E7%AE%97%E8%A1%93.gif 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/8/88/%E4%B9%9D%E7%AB%A0%E7%AE%97%E8%A1%93.gif/340px-%E4%B9%9D%E7%AB%A0%E7%AE%97%E8%A1%93.gif 2x" data-file-width="419" data-file-height="546" /></a><figcaption><a href="/wiki/The_Nine_Chapters_on_the_Mathematical_Art" title="The Nine Chapters on the Mathematical Art">The Nine Chapters on the Mathematical Art</a>, one of the earliest surviving mathematical texts from <a href="/wiki/China" title="China">China</a> (2nd century AD).</figcaption></figure> In 212 BC, the Emperor <a href="/wiki/Qin_Shi_Huang" title="Qin Shi Huang">Qin Shi Huang</a> commanded all books in the <a href="/wiki/Qin_Empire" class="mw-redirect" title="Qin Empire">Qin Empire</a> other than officially sanctioned ones be burned. This decree was not universally obeyed, but as a consequence of this order little is known about ancient Chinese mathematics before this date. After the <a href="/wiki/Burning_of_books_and_burying_of_scholars" title="Burning of books and burying of scholars">book burning</a> of 212 BC, the <a href="/wiki/Han_dynasty" title="Han dynasty">Han dynasty</a> (202 BC–220 AD) produced works of mathematics which presumably expanded on works that are now lost. The most important of these is <a href="/wiki/The_Nine_Chapters_on_the_Mathematical_Art" title="The Nine Chapters on the Mathematical Art">The Nine Chapters on the Mathematical Art</a>, the full title of which appeared by AD 179, but existed in part under other titles beforehand. It consists of 246 word problems involving agriculture, business, employment of geometry to figure height spans and dimension ratios for <a href="/wiki/Chinese_pagoda" class="mw-redirect" title="Chinese pagoda">Chinese pagoda</a> towers, engineering, <a href="/wiki/Surveying" title="Surveying">surveying</a>, and includes material on <a href="/wiki/Right_triangle" title="Right triangle">right triangles</a>.<a href="#cite_note-Boyer_1991_loc=China_and_India_p._196-106">[106]</a> It created mathematical proof for the <a href="/wiki/Pythagorean_theorem" title="Pythagorean theorem">Pythagorean theorem</a>,<a href="#cite_note-111">[111]</a> and a mathematical formula for <a href="/wiki/Gaussian_elimination" title="Gaussian elimination">Gaussian elimination</a>.<a href="#cite_note-112">[112]</a> The treatise also provides values of <a href="/wiki/Pi" title="Pi">π</a>,<a href="#cite_note-Boyer_1991_loc=China_and_India_p._196-106">[106]</a> which Chinese mathematicians originally approximated as 3 until <a href="/wiki/Liu_Xin_(scholar)" title="Liu Xin (scholar)">Liu Xin</a> (d. 23 AD) provided a figure of 3.1457 and subsequently <a href="/wiki/Zhang_Heng" title="Zhang Heng">Zhang Heng</a> (78–139) approximated pi as 3.1724,<a href="#cite_note-113">[113]</a> as well as 3.162 by taking the <a href="/wiki/Square_root" title="Square root">square root</a> of 10.<a href="#cite_note-114">[114]</a><a href="#cite_note-115">[115]</a> <a href="/wiki/Liu_Hui" title="Liu Hui">Liu Hui</a> commented on the Nine Chapters in the 3rd century AD and <a href="/wiki/Liu_Hui%27s_%CF%80_algorithm" title="Liu Hui's π algorithm">gave a value of π</a> accurate to 5 decimal places (i.e. 3.14159).<a href="#cite_note-Boyer_1991_loc=China_and_India_p._202-116">[116]</a><a href="#cite_note-117">[117]</a> Though more of a matter of computational stamina than theoretical insight, in the 5th century AD <a href="/wiki/Zu_Chongzhi" title="Zu Chongzhi">Zu Chongzhi</a> computed <a href="/wiki/Mil%C3%BC" title="Milü">the value of π</a> to seven decimal places (between 3.1415926 and 3.1415927), which remained the most accurate value of π for almost the next 1000 years.<a href="#cite_note-Boyer_1991_loc=China_and_India_p._202-116">[116]</a><a href="#cite_note-118">[118]</a> He also established a method which would later be called <a href="/wiki/Cavalieri%27s_principle" title="Cavalieri's principle">Cavalieri's principle</a> to find the volume of a <a href="/wiki/Sphere" title="Sphere">sphere</a>.<a href="#cite_note-119">[119]</a> The high-water mark of Chinese mathematics occurred in the 13th century during the latter half of the <a href="/wiki/Song_dynasty" title="Song dynasty">Song dynasty</a> (960–1279), with the development of Chinese algebra. The most important text from that period is the <a href="/wiki/Jade_Mirror_of_the_Four_Unknowns" title="Jade Mirror of the Four Unknowns">Precious Mirror of the Four Elements</a> by <a href="/wiki/Zhu_Shijie" title="Zhu Shijie">Zhu Shijie</a> (1249–1314), dealing with the solution of simultaneous higher order algebraic equations using a method similar to <a href="/wiki/Horner%27s_method" title="Horner's method">Horner's method</a>.<a href="#cite_note-Boyer_1991_loc=China_and_India_p._202-116">[116]</a> The Precious Mirror also contains a diagram of <a href="/wiki/Pascal%27s_triangle" title="Pascal's triangle">Pascal's triangle</a> with coefficients of binomial expansions through the eighth power, though both appear in Chinese works as early as 1100.<a href="#cite_note-Boyer_1991_loc=China_and_India_p._205-120">[120]</a> The Chinese also made use of the complex combinatorial diagram known as the <a href="/wiki/Magic_square" title="Magic square">magic square</a> and <a href="/wiki/Magic_circle_(mathematics)" title="Magic circle (mathematics)">magic circles</a>, described in ancient times and perfected by <a href="/wiki/Yang_Hui" title="Yang Hui">Yang Hui</a> (AD 1238–1298).<a href="#cite_note-Boyer_1991_loc=China_and_India_p._205-120">[120]</a> Even after European mathematics began to flourish during the <a href="/wiki/Renaissance" title="Renaissance">Renaissance</a>, European and Chinese mathematics were separate traditions, with significant Chinese mathematical output in decline from the 13th century onwards. <a href="/wiki/Jesuit" class="mw-redirect" title="Jesuit">Jesuit</a> missionaries such as <a href="/wiki/Matteo_Ricci" title="Matteo Ricci">Matteo Ricci</a> carried mathematical ideas back and forth between the two cultures from the 16th to 18th centuries, though at this point far more mathematical ideas were entering China than leaving.<a href="#cite_note-Boyer_1991_loc=China_and_India_p._205-120">[120]</a> <a href="/wiki/Japanese_mathematics" title="Japanese mathematics">Japanese mathematics</a>, <a href="/wiki/Korean_numerals" title="Korean numerals">Korean mathematics</a>, and <a href="/wiki/Vietnamese_numerals" title="Vietnamese numerals">Vietnamese mathematics</a> are traditionally viewed as stemming from Chinese mathematics and belonging to the <a href="/wiki/Confucian" class="mw-redirect" title="Confucian">Confucian</a>-based <a href="/wiki/East_Asian_cultural_sphere" class="mw-redirect" title="East Asian cultural sphere">East Asian cultural sphere</a>.<a href="#cite_note-121">[121]</a> Korean and Japanese mathematics were heavily influenced by the algebraic works produced during China's Song dynasty, whereas Vietnamese mathematics was heavily indebted to popular works of China's <a href="/wiki/Ming_dynasty" title="Ming dynasty">Ming dynasty</a> (1368–1644).<a href="#cite_note-122">[122]</a> For instance, although Vietnamese mathematical treatises were written in either <a href="/wiki/Chinese_characters" title="Chinese characters">Chinese</a> or the native Vietnamese <a href="/wiki/Ch%E1%BB%AF_N%C3%B4m" title="Chữ Nôm">Chữ Nôm</a> script, all of them followed the Chinese format of presenting a collection of problems with <a href="/wiki/Algorithm" title="Algorithm">algorithms</a> for solving them, followed by numerical answers.<a href="#cite_note-123">[123]</a> Mathematics in Vietnam and Korea were mostly associated with the professional court bureaucracy of <a href="/wiki/History_of_astronomy" title="History of astronomy">mathematicians and astronomers</a>, whereas in Japan it was more prevalent in the realm of <a href="/wiki/Private_school" title="Private school">private schools</a>.<a href="#cite_note-124">[124]</a> <div class="mw-heading mw-heading2"><h2 id="Japan">Japan</h2>[<a href="/w/index.php?title=History_of_mathematics&action=edit&section=7" title="Edit section: Japan">edit</a>]</div> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1236090951"><div role="note" class="hatnote navigation-not-searchable">Further information: <a href="/wiki/Sangaku" title="Sangaku">Sangaku</a> and <a href="/wiki/Seki_Takakazu" title="Seki Takakazu">Seki Takakazu</a></div> <figure class="mw-default-size" typeof="mw:File/Thumb"><a href="/wiki/File:Sangaku_at_Enmanji.jpg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/5/5f/Sangaku_at_Enmanji.jpg/220px-Sangaku_at_Enmanji.jpg" decoding="async" width="220" height="128" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/5/5f/Sangaku_at_Enmanji.jpg/330px-Sangaku_at_Enmanji.jpg 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/5/5f/Sangaku_at_Enmanji.jpg/440px-Sangaku_at_Enmanji.jpg 2x" data-file-width="2450" data-file-height="1431" /></a><figcaption>Sangaku dedicated at Enmanji Temple in Nara city, Japan.</figcaption></figure> The mathematics that developed in <a href="/wiki/Japan" title="Japan">Japan</a> during the <a href="/wiki/Edo_period" title="Edo period">Edo period</a> (1603-1887) is independent of Western mathematics; To this period belongs the mathematician <a href="/wiki/Seki_Takakazu" title="Seki Takakazu">Seki Takakazu</a>, of great influence, for example, in the development of <a href="/wiki/Wasan" title="Wasan">wasan</a> (traditional Japanese mathematics), and whose discoveries (in areas such as <a href="/wiki/Integral_calculus" class="mw-redirect" title="Integral calculus">integral calculus</a>), are almost simultaneous with contemporary European mathematicians such as <a href="/wiki/Gottfried_Leibniz" class="mw-redirect" title="Gottfried Leibniz">Gottfried Leibniz</a>. Japanese mathematics of this period is inspired by Chinese mathematics and is oriented towards essentially geometric problems. On wooden tablets called sangaku, "geometric enigmas" are proposed and solved; That's where, for example, <a href="/wiki/Soddy%27s_hexlet" title="Soddy's hexlet">Soddy's hexlet</a> theorem comes from. <div class="mw-heading mw-heading2"><h2 id="Indian">Indian</h2>[<a href="/w/index.php?title=History_of_mathematics&action=edit&section=8" title="Edit section: Indian">edit</a>]</div> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1236090951"><div role="note" class="hatnote navigation-not-searchable">Main article: <a href="/wiki/Indian_mathematics" title="Indian mathematics">Indian mathematics</a></div> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1236090951"><div role="note" class="hatnote navigation-not-searchable">Further information: <a href="/wiki/History_of_science#Indian_mathematics" title="History of science">History of science § Indian mathematics</a></div> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1236090951"><div role="note" class="hatnote navigation-not-searchable">See also: <a href="/wiki/History_of_the_Hindu%E2%80%93Arabic_numeral_system" title="History of the Hindu–Arabic numeral system">History of the Hindu–Arabic numeral system</a></div> <figure class="mw-default-size mw-halign-right" typeof="mw:File/Thumb"><a href="/wiki/File:Bakhshali_numerals_2.jpg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/5/58/Bakhshali_numerals_2.jpg/330px-Bakhshali_numerals_2.jpg" decoding="async" width="330" height="50" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/5/58/Bakhshali_numerals_2.jpg/495px-Bakhshali_numerals_2.jpg 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/5/58/Bakhshali_numerals_2.jpg/660px-Bakhshali_numerals_2.jpg 2x" data-file-width="1958" data-file-height="295" /></a><figcaption>The numerals used in the <a href="/wiki/Bakhshali_manuscript" title="Bakhshali manuscript">Bakhshali manuscript</a>, dated between the 2nd century BC and the 2nd century AD.</figcaption></figure> <style data-mw-deduplicate="TemplateStyles:r1237032888/mw-parser-output/.tmulti">.mw-parser-output .tmulti .multiimageinner{display:flex;flex-direction:column}.mw-parser-output .tmulti .trow{display:flex;flex-direction:row;clear:left;flex-wrap:wrap;width:100%;box-sizing:border-box}.mw-parser-output .tmulti .tsingle{margin:1px;float:left}.mw-parser-output .tmulti .theader{clear:both;font-weight:bold;text-align:center;align-self:center;background-color:transparent;width:100%}.mw-parser-output .tmulti .thumbcaption{background-color:transparent}.mw-parser-output .tmulti .text-align-left{text-align:left}.mw-parser-output .tmulti .text-align-right{text-align:right}.mw-parser-output .tmulti .text-align-center{text-align:center}@media all and (max-width:720px){.mw-parser-output .tmulti .thumbinner{width:100%!important;box-sizing:border-box;max-width:none!important;align-items:center}.mw-parser-output .tmulti .trow{justify-content:center}.mw-parser-output .tmulti .tsingle{float:none!important;max-width:100%!important;box-sizing:border-box;text-align:center}.mw-parser-output .tmulti .tsingle .thumbcaption{text-align:left}.mw-parser-output .tmulti .trow>.thumbcaption{text-align:center}}@media screen{html.skin-theme-clientpref-night .mw-parser-output .tmulti .multiimageinner img{background-color:white}}@media screen and (prefers-color-scheme:dark){html.skin-theme-clientpref-os .mw-parser-output .tmulti .multiimageinner img{background-color:white}}</style><div class="thumb tmulti tright"><div class="thumbinner multiimageinner" style="width:334px;max-width:334px"><div class="trow"><div class="tsingle" style="width:332px;max-width:332px"><div class="thumbimage"><a href="/wiki/File:1911_sketch_of_numerals_script_history_ancient_India,_mathematical_symbols_shapes.jpg" class="mw-file-description"><img alt="Numerals evolution in India" src="//upload.wikimedia.org/wikipedia/commons/thumb/9/96/1911_sketch_of_numerals_script_history_ancient_India%2C_mathematical_symbols_shapes.jpg/330px-1911_sketch_of_numerals_script_history_ancient_India%2C_mathematical_symbols_shapes.jpg" decoding="async" width="330" height="168" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/9/96/1911_sketch_of_numerals_script_history_ancient_India%2C_mathematical_symbols_shapes.jpg/495px-1911_sketch_of_numerals_script_history_ancient_India%2C_mathematical_symbols_shapes.jpg 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/9/96/1911_sketch_of_numerals_script_history_ancient_India%2C_mathematical_symbols_shapes.jpg/660px-1911_sketch_of_numerals_script_history_ancient_India%2C_mathematical_symbols_shapes.jpg 2x" data-file-width="1208" data-file-height="614" /></a></div><div class="thumbcaption">Indian numerals in stone and copper inscriptions<a href="#cite_note-britnanaghat-125">[125]</a></div></div></div><div class="trow"><div class="tsingle" style="width:332px;max-width:332px"><div class="thumbimage"><a href="/wiki/File:Indian_numerals_100AD.svg" class="mw-file-description"><img alt="Brahmi numerals" src="//upload.wikimedia.org/wikipedia/commons/thumb/f/fb/Indian_numerals_100AD.svg/330px-Indian_numerals_100AD.svg.png" decoding="async" width="330" height="69" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/f/fb/Indian_numerals_100AD.svg/495px-Indian_numerals_100AD.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/f/fb/Indian_numerals_100AD.svg/660px-Indian_numerals_100AD.svg.png 2x" data-file-width="343" data-file-height="72" /></a></div><div class="thumbcaption">Ancient Brahmi numerals in a part of India</div></div></div></div></div> The earliest civilization on the Indian subcontinent is the <a href="/wiki/Indus_Valley_civilization" class="mw-redirect" title="Indus Valley civilization">Indus Valley civilization</a> (mature second phase: 2600 to 1900 BC) that flourished in the <a href="/wiki/Indus_river" class="mw-redirect" title="Indus river">Indus river</a> basin. Their cities were laid out with geometric regularity, but no known mathematical documents survive from this civilization.<a href="#cite_note-126">[126]</a> The oldest extant mathematical records from India are the <a href="/wiki/Sulba_Sutras" class="mw-redirect" title="Sulba Sutras">Sulba Sutras</a> (dated variously between the 8th century BC and the 2nd century AD),<a href="#cite_note-Boyer_1991_loc=China_and_India_p._207-127">[127]</a> appendices to religious texts which give simple rules for constructing altars of various shapes, such as squares, rectangles, parallelograms, and others.<a href="#cite_note-128">[128]</a> As with Egypt, the preoccupation with temple functions points to an origin of mathematics in religious ritual.<a href="#cite_note-Boyer_1991_loc=China_and_India_p._207-127">[127]</a> The Sulba Sutras give methods for constructing a <a href="/wiki/Squaring_the_circle" title="Squaring the circle">circle with approximately the same area as a given square</a>, which imply several different approximations of the value of π.<a href="#cite_note-129">[129]</a><a href="#cite_note-Indian_sulbasutras-130">[130]</a><a href="#cite_note-131">[a]</a> In addition, they compute the <a href="/wiki/Square_root" title="Square root">square root</a> of 2 to several decimal places, list Pythagorean triples, and give a statement of the <a href="/wiki/Pythagorean_theorem" title="Pythagorean theorem">Pythagorean theorem</a>.<a href="#cite_note-Indian_sulbasutras-130">[130]</a> All of these results are present in Babylonian mathematics, indicating Mesopotamian influence.<a href="#cite_note-Boyer_1991_loc=China_and_India_p._207-127">[127]</a> It is not known to what extent the Sulba Sutras influenced later Indian mathematicians. As in China, there is a lack of continuity in Indian mathematics; significant advances are separated by long periods of inactivity.<a href="#cite_note-Boyer_1991_loc=China_and_India_p._207-127">[127]</a> <a href="/wiki/P%C4%81%E1%B9%87ini" title="Pāṇini">Pāṇini</a> (c. 5th century BC) formulated the rules for <a href="/wiki/Sanskrit_grammar" title="Sanskrit grammar">Sanskrit grammar</a>.<a href="#cite_note-132">[131]</a> His notation was similar to modern mathematical notation, and used metarules, <a href="/wiki/Transformation_(geometry)" class="mw-redirect" title="Transformation (geometry)">transformations</a>, and <a href="/wiki/Recursion" title="Recursion">recursion</a>.<a href="#cite_note-133">[132]</a> <a href="/wiki/Pingala" title="Pingala">Pingala</a> (roughly 3rd–1st centuries BC) in his treatise of <a href="/wiki/Prosody_(poetry)" class="mw-redirect" title="Prosody (poetry)">prosody</a> uses a device corresponding to a <a href="/wiki/Binary_numeral_system" class="mw-redirect" title="Binary numeral system">binary numeral system</a>.<a href="#cite_note-134">[133]</a><a href="#cite_note-135">[134]</a> His discussion of the <a href="/wiki/Combinatorics" title="Combinatorics">combinatorics</a> of <a href="/wiki/Metre_(music)" title="Metre (music)">meters</a> corresponds to an elementary version of the <a href="/wiki/Binomial_theorem" title="Binomial theorem">binomial theorem</a>. Pingala's work also contains the basic ideas of <a href="/wiki/Fibonacci_number" class="mw-redirect" title="Fibonacci number">Fibonacci numbers</a> (called mātrāmeru).<a href="#cite_note-136">[135]</a> The next significant mathematical documents from India after the Sulba Sutras are the Siddhantas, astronomical treatises from the 4th and 5th centuries AD (<a href="/wiki/Gupta_period" class="mw-redirect" title="Gupta period">Gupta period</a>) showing strong Hellenistic influence.<a href="#cite_note-137">[136]</a> They are significant in that they contain the first instance of trigonometric relations based on the half-chord, as is the case in modern trigonometry, rather than the full chord, as was the case in Ptolemaic trigonometry.<a href="#cite_note-autogenerated2-138">[137]</a> Through a series of translation errors, the words "sine" and "cosine" derive from the Sanskrit "jiya" and "kojiya".<a href="#cite_note-autogenerated2-138">[137]</a> <figure class="mw-default-size mw-halign-left" typeof="mw:File/Thumb"><a href="/wiki/File:Yuktibhasa.svg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/3/3e/Yuktibhasa.svg/170px-Yuktibhasa.svg.png" decoding="async" width="170" height="252" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/3/3e/Yuktibhasa.svg/255px-Yuktibhasa.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/3/3e/Yuktibhasa.svg/340px-Yuktibhasa.svg.png 2x" data-file-width="263" data-file-height="390" /></a><figcaption>Explanation of the <a href="/wiki/Law_of_sines" title="Law of sines">sine rule</a> in <a href="/wiki/Yuktibh%C4%81%E1%B9%A3%C4%81" title="Yuktibhāṣā">Yuktibhāṣā</a></figcaption></figure> Around 500 AD, <a href="/wiki/Aryabhata" title="Aryabhata">Aryabhata</a> wrote the <a href="/wiki/Aryabhatiya" title="Aryabhatiya">Aryabhatiya</a>, a slim volume, written in verse, intended to supplement the rules of calculation used in astronomy and mathematical mensuration, though with no feeling for logic or deductive methodology.<a href="#cite_note-139">[138]</a> It is in the Aryabhatiya that the decimal place-value system first appears. Several centuries later, the <a href="/wiki/Islamic_mathematics" class="mw-redirect" title="Islamic mathematics">Muslim mathematician</a> <a href="/wiki/Abu_Rayhan_Biruni" class="mw-redirect" title="Abu Rayhan Biruni">Abu Rayhan Biruni</a> described the Aryabhatiya as a "mix of common pebbles and costly crystals".<a href="#cite_note-140">[139]</a> In the 7th century, <a href="/wiki/Brahmagupta" title="Brahmagupta">Brahmagupta</a> identified the <a href="/wiki/Brahmagupta_theorem" title="Brahmagupta theorem">Brahmagupta theorem</a>, <a href="/wiki/Brahmagupta%27s_identity" title="Brahmagupta's identity">Brahmagupta's identity</a> and <a href="/wiki/Brahmagupta%27s_formula" title="Brahmagupta's formula">Brahmagupta's formula</a>, and for the first time, in <a href="/wiki/Brahmasphutasiddhanta" class="mw-redirect" title="Brahmasphutasiddhanta">Brahma-sphuta-siddhanta</a>, he lucidly explained the use of <a href="/wiki/0_(number)" class="mw-redirect" title="0 (number)">zero</a> as both a placeholder and <a href="/wiki/Decimal_digit" class="mw-redirect" title="Decimal digit">decimal digit</a>, and explained the <a href="/wiki/Hindu%E2%80%93Arabic_numeral_system" title="Hindu–Arabic numeral system">Hindu–Arabic numeral system</a>.<a href="#cite_note-Boyer_Siddhanta-141">[140]</a> It was from a translation of this Indian text on mathematics (c. 770) that Islamic mathematicians were introduced to this numeral system, which they adapted as <a href="/wiki/Arabic_numerals" title="Arabic numerals">Arabic numerals</a>. Islamic scholars carried knowledge of this number system to Europe by the 12th century, and it has now displaced all older number systems throughout the world. Various symbol sets are used to represent numbers in the Hindu–Arabic numeral system, all of which evolved from the <a href="/wiki/Brahmi_numeral" class="mw-redirect" title="Brahmi numeral">Brahmi numerals</a>. Each of the roughly dozen major scripts of India has its own numeral glyphs. In the 10th century, <a href="/wiki/Halayudha" title="Halayudha">Halayudha</a>'s commentary on <a href="/wiki/Pingala" title="Pingala">Pingala</a>'s work contains a study of the <a href="/wiki/Fibonacci_sequence" title="Fibonacci sequence">Fibonacci sequence</a> and <a href="/wiki/Pascal%27s_triangle" title="Pascal's triangle">Pascal's triangle</a>, and describes the formation of a <a href="/wiki/Matrix_(mathematics)" title="Matrix (mathematics)">matrix</a>.[<a href="/wiki/Wikipedia:Citation_needed" title="Wikipedia:Citation needed">citation needed</a>] In the 12th century, <a href="/wiki/Bh%C4%81skara_II" title="Bhāskara II">Bhāskara II</a>,<a href="#cite_note-142">[141]</a> who lived in southern India, wrote extensively on all then known branches of mathematics. His work contains mathematical objects equivalent or approximately equivalent to infinitesimals, <a href="/wiki/Mean_value_theorem" title="Mean value theorem">the mean value theorem</a> and the derivative of the sine function although he did not develop the notion of a derivative.<a href="#cite_note-143">[142]</a><a href="#cite_note-144">[143]</a> In the 14th century, <a href="/wiki/Narayana_Pandita_(mathematician)" title="Narayana Pandita (mathematician)">Narayana Pandita</a> completed his <a href="/wiki/Ganita_Kaumudi" title="Ganita Kaumudi">Ganita Kaumudi</a>.<a href="#cite_note-145">[144]</a> Also in the 14th century, <a href="/wiki/Madhava_of_Sangamagrama" title="Madhava of Sangamagrama">Madhava of Sangamagrama</a>, the founder of the <a href="/wiki/Kerala_School_of_Astronomy_and_Mathematics" class="mw-redirect" title="Kerala School of Astronomy and Mathematics">Kerala School of Mathematics</a>, found the <a href="/wiki/Leibniz_formula_for_pi" class="mw-redirect" title="Leibniz formula for pi">Madhava–Leibniz series</a> and obtained from it a <a href="/wiki/Approximations_of_%CF%80#Middle_Ages" title="Approximations of π">transformed series</a>, whose first 21 terms he used to compute the value of π as 3.14159265359. Madhava also found <a href="/wiki/Gregory%27s_series" class="mw-redirect" title="Gregory's series">the Madhava-Gregory series</a> to determine the arctangent, the Madhava-Newton <a href="/wiki/Power_series" title="Power series">power series</a> to determine sine and cosine and <a href="/wiki/Taylor_series" title="Taylor series">the Taylor approximation</a> for sine and cosine functions.<a href="#cite_note-146">[145]</a> In the 16th century, <a href="/wiki/Jyesthadeva" class="mw-redirect" title="Jyesthadeva">Jyesthadeva</a> consolidated many of the Kerala School's developments and theorems in the Yukti-bhāṣā.<a href="#cite_note-rajujournal-147">[146]</a><a href="#cite_note-148">[147]</a> It has been argued that certain ideas of calculus like infinite series and taylor series of some trigonometry functions, were transmitted to Europe in the 16th century<a href="#cite_note-:2-6">[6]</a> via <a href="/wiki/Jesuit" class="mw-redirect" title="Jesuit">Jesuit</a> missionaries and traders who were active around the ancient port of <a href="/wiki/Muziris" title="Muziris">Muziris</a> at the time and, as a result, directly influenced later European developments in analysis and calculus.<a href="#cite_note-almeida-149">[148]</a> However, other scholars argue that the Kerala School did not formulate a systematic theory of <a href="/wiki/Derivative" title="Derivative">differentiation</a> and <a href="/wiki/Integral" title="Integral">integration</a>, and that there is not any direct evidence of their results being transmitted outside Kerala.<a href="#cite_note-150">[149]</a><a href="#cite_note-151">[150]</a><a href="#cite_note-152">[151]</a><a href="#cite_note-153">[152]</a> <div class="mw-heading mw-heading2"><h2 id="Islamic_empires">Islamic empires</h2>[<a href="/w/index.php?title=History_of_mathematics&action=edit&section=9" title="Edit section: Islamic empires">edit</a>]</div> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1236090951"><div role="note" class="hatnote navigation-not-searchable">Main article: <a href="/wiki/Mathematics_in_medieval_Islam" class="mw-redirect" title="Mathematics in medieval Islam">Mathematics in medieval Islam</a></div> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1236090951"><div role="note" class="hatnote navigation-not-searchable">See also: <a href="/wiki/History_of_the_Hindu%E2%80%93Arabic_numeral_system" title="History of the Hindu–Arabic numeral system">History of the Hindu–Arabic numeral system</a></div> <figure class="mw-default-size" typeof="mw:File/Thumb"><a href="/wiki/File:Image-Al-Kit%C4%81b_al-mu%E1%B8%ABta%E1%B9%A3ar_f%C4%AB_%E1%B8%A5is%C4%81b_al-%C4%9Fabr_wa-l-muq%C4%81bala.jpg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/2/23/Image-Al-Kit%C4%81b_al-mu%E1%B8%ABta%E1%B9%A3ar_f%C4%AB_%E1%B8%A5is%C4%81b_al-%C4%9Fabr_wa-l-muq%C4%81bala.jpg/220px-Image-Al-Kit%C4%81b_al-mu%E1%B8%ABta%E1%B9%A3ar_f%C4%AB_%E1%B8%A5is%C4%81b_al-%C4%9Fabr_wa-l-muq%C4%81bala.jpg" decoding="async" width="220" height="348" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/2/23/Image-Al-Kit%C4%81b_al-mu%E1%B8%ABta%E1%B9%A3ar_f%C4%AB_%E1%B8%A5is%C4%81b_al-%C4%9Fabr_wa-l-muq%C4%81bala.jpg 1.5x" data-file-width="240" data-file-height="380" /></a><figcaption>Page from <a href="/wiki/The_Compendious_Book_on_Calculation_by_Completion_and_Balancing" class="mw-redirect" title="The Compendious Book on Calculation by Completion and Balancing">The Compendious Book on Calculation by Completion and Balancing</a> by <a href="/wiki/Muhammad_ibn_M%C5%ABs%C4%81_al-Khw%C4%81rizm%C4%AB" class="mw-redirect" title="Muhammad ibn Mūsā al-Khwārizmī">Muhammad ibn Mūsā al-Khwārizmī</a> (c. AD 820)</figcaption></figure> The <a href="/wiki/Caliphate" title="Caliphate">Islamic Empire</a> established across the <a href="/wiki/Middle_East" title="Middle East">Middle East</a>, <a href="/wiki/Central_Asia" title="Central Asia">Central Asia</a>, <a href="/wiki/North_Africa" title="North Africa">North Africa</a>, <a href="/wiki/Iberian_Peninsula" title="Iberian Peninsula">Iberia</a>, and in parts of <a href="/wiki/History_of_India" title="History of India">India</a> in the 8th century made significant contributions towards mathematics. Although most Islamic texts on mathematics were written in <a href="/wiki/Arabic_language" class="mw-redirect" title="Arabic language">Arabic</a>, they were not all written by <a href="/wiki/Arab" class="mw-redirect" title="Arab">Arabs</a>, since much like the status of Greek in the Hellenistic world, Arabic was used as the written language of non-Arab scholars throughout the Islamic world at the time.<a href="#cite_note-154">[153]</a> In the 9th century, the Persian mathematician <a href="/wiki/Mu%E1%B8%A5ammad_ibn_M%C5%ABs%C4%81_al-Khw%C4%81rizm%C4%AB" class="mw-redirect" title="Muḥammad ibn Mūsā al-Khwārizmī">Muḥammad ibn Mūsā al-Khwārizmī</a> wrote an important book on the <a href="/wiki/Hindu%E2%80%93Arabic_numerals" class="mw-redirect" title="Hindu–Arabic numerals">Hindu–Arabic numerals</a> and one on methods for solving equations. His book On the Calculation with Hindu Numerals, written about 825, along with the work of <a href="/wiki/Al-Kindi" title="Al-Kindi">Al-Kindi</a>, were instrumental in spreading <a href="/wiki/Indian_mathematics" title="Indian mathematics">Indian mathematics</a> and <a href="/wiki/Hindu%E2%80%93Arabic_numeral_system" title="Hindu–Arabic numeral system">Indian numerals</a> to the West. The word <a href="/wiki/Algorithm" title="Algorithm">algorithm</a> is derived from the Latinization of his name, Algoritmi, and the word algebra from the title of one of his works, <a href="/wiki/The_Compendious_Book_on_Calculation_by_Completion_and_Balancing" class="mw-redirect" title="The Compendious Book on Calculation by Completion and Balancing">Al-Kitāb al-mukhtaṣar fī hīsāb al-ğabr wa’l-muqābala</a> (The Compendious Book on Calculation by Completion and Balancing). He gave an exhaustive explanation for the algebraic solution of quadratic equations with positive roots,<a href="#cite_note-155">[154]</a> and he was the first to teach algebra in an <a href="/wiki/Elementary_algebra" title="Elementary algebra">elementary form</a> and for its own sake.<a href="#cite_note-156">[155]</a> He also discussed the fundamental method of "<a href="/wiki/Reduction_(mathematics)" title="Reduction (mathematics)">reduction</a>" and "balancing", referring to the transposition of subtracted terms to the other side of an equation, that is, the cancellation of like terms on opposite sides of the equation. This is the operation which al-Khwārizmī originally described as al-jabr.<a href="#cite_note-Boyer-229-157">[156]</a> His algebra was also no longer concerned "with a series of problems to be resolved, but an <a href="/wiki/Expository_writing" class="mw-redirect" title="Expository writing">exposition</a> which starts with primitive terms in which the combinations must give all possible prototypes for equations, which henceforward explicitly constitute the true object of study." He also studied an equation for its own sake and "in a generic manner, insofar as it does not simply emerge in the course of solving a problem, but is specifically called on to define an infinite class of problems."<a href="#cite_note-Rashed-Armstrong-158">[157]</a> In Egypt, <a href="/wiki/Abu_Kamil" title="Abu Kamil">Abu Kamil</a> extended algebra to the set of <a href="/wiki/Irrational_numbers" class="mw-redirect" title="Irrational numbers">irrational numbers</a>, accepting square roots and fourth roots as solutions and coefficients to quadratic equations. He also developed techniques used to solve three non-linear simultaneous equations with three unknown variables. One unique feature of his works was trying to find all the possible solutions to some of his problems, including one where he found 2676 solutions.<a href="#cite_note-HSTM-159">[158]</a> His works formed an important foundation for the development of algebra and influenced later mathematicians, such as al-Karaji and Fibonacci. Further developments in algebra were made by <a href="/wiki/Al-Karaji" title="Al-Karaji">Al-Karaji</a> in his treatise al-Fakhri, where he extends the methodology to incorporate integer powers and integer roots of unknown quantities. Something close to a <a href="/wiki/Mathematical_proof" title="Mathematical proof">proof</a> by <a href="/wiki/Mathematical_induction" title="Mathematical induction">mathematical induction</a> appears in a book written by Al-Karaji around 1000 AD, who used it to prove the <a href="/wiki/Binomial_theorem" title="Binomial theorem">binomial theorem</a>, <a href="/wiki/Pascal%27s_triangle" title="Pascal's triangle">Pascal's triangle</a>, and the sum of integral <a href="/wiki/Cube_(algebra)" title="Cube (algebra)">cubes</a>.<a href="#cite_note-160">[159]</a> The <a href="/wiki/Historian" title="Historian">historian</a> of mathematics, F. Woepcke,<a href="#cite_note-161">[160]</a> praised Al-Karaji for being "the first who introduced the <a href="/wiki/Theory" title="Theory">theory</a> of <a href="/wiki/Algebra" title="Algebra">algebraic</a> <a href="/wiki/Calculus" title="Calculus">calculus</a>." Also in the 10th century, <a href="/wiki/Abul_Wafa" class="mw-redirect" title="Abul Wafa">Abul Wafa</a> translated the works of <a href="/wiki/Diophantus" title="Diophantus">Diophantus</a> into Arabic. <a href="/wiki/Ibn_al-Haytham" title="Ibn al-Haytham">Ibn al-Haytham</a> was the first mathematician to derive the formula for the sum of the fourth powers, using a method that is readily generalizable for determining the general formula for the sum of any integral powers. He performed an integration in order to find the volume of a <a href="/wiki/Paraboloid" title="Paraboloid">paraboloid</a>, and was able to generalize his result for the integrals of <a href="/wiki/Polynomial" title="Polynomial">polynomials</a> up to the <a href="/wiki/Quartic_polynomial" class="mw-redirect" title="Quartic polynomial">fourth degree</a>. He thus came close to finding a general formula for the integrals of polynomials, but he was not concerned with any polynomials higher than the fourth degree.<a href="#cite_note-Katz-162">[161]</a> In the late 11th century, <a href="/wiki/Omar_Khayyam" title="Omar Khayyam">Omar Khayyam</a> wrote Discussions of the Difficulties in Euclid, a book about what he perceived as flaws in <a href="/wiki/Euclid%27s_Elements" title="Euclid's Elements">Euclid's Elements</a>, especially the <a href="/wiki/Parallel_postulate" title="Parallel postulate">parallel postulate</a>. He was also the first to find the general geometric solution to <a href="/wiki/Cubic_equation" title="Cubic equation">cubic equations</a>. He was also very influential in <a href="/wiki/Calendar_reform" title="Calendar reform">calendar reform</a>.<a href="#cite_note-163">[162]</a> In the 13th century, <a href="/wiki/Nasir_al-Din_Tusi" class="mw-redirect" title="Nasir al-Din Tusi">Nasir al-Din Tusi</a> (Nasireddin) made advances in <a href="/wiki/Spherical_trigonometry" title="Spherical trigonometry">spherical trigonometry</a>. He also wrote influential work on Euclid's <a href="/wiki/Parallel_postulate" title="Parallel postulate">parallel postulate</a>. In the 15th century, <a href="/wiki/Ghiyath_al-Kashi" class="mw-redirect" title="Ghiyath al-Kashi">Ghiyath al-Kashi</a> computed the value of π to the 16th decimal place. Kashi also had an algorithm for calculating nth roots, which was a special case of the methods given many centuries later by <a href="/wiki/Paolo_Ruffini_(mathematician)" class="mw-redirect" title="Paolo Ruffini (mathematician)">Ruffini</a> and <a href="/wiki/William_George_Horner" title="William George Horner">Horner</a>. Other achievements of Muslim mathematicians during this period include the addition of the <a href="/wiki/Decimal_point" class="mw-redirect" title="Decimal point">decimal point</a> notation to the <a href="/wiki/Arabic_numerals" title="Arabic numerals">Arabic numerals</a>, the discovery of all the modern <a href="/wiki/Trigonometric_function" class="mw-redirect" title="Trigonometric function">trigonometric functions</a> besides the sine, <a href="/wiki/Al-Kindi" title="Al-Kindi">al-Kindi</a>'s introduction of <a href="/wiki/Cryptanalysis" title="Cryptanalysis">cryptanalysis</a> and <a href="/wiki/Frequency_analysis" title="Frequency analysis">frequency analysis</a>, the development of <a href="/wiki/Analytic_geometry" title="Analytic geometry">analytic geometry</a> by <a href="/wiki/Ibn_al-Haytham" title="Ibn al-Haytham">Ibn al-Haytham</a>, the beginning of <a href="/wiki/Algebraic_geometry" title="Algebraic geometry">algebraic geometry</a> by <a href="/wiki/Omar_Khayyam" title="Omar Khayyam">Omar Khayyam</a> and the development of an <a href="/wiki/Mathematical_notation" title="Mathematical notation">algebraic notation</a> by <a href="/wiki/Ab%C5%AB_al-Hasan_ibn_Al%C4%AB_al-Qalas%C4%81d%C4%AB" class="mw-redirect" title="Abū al-Hasan ibn Alī al-Qalasādī">al-Qalasādī</a>.<a href="#cite_note-Qalasadi-164">[163]</a> During the time of the <a href="/wiki/Ottoman_Empire" title="Ottoman Empire">Ottoman Empire</a> and <a href="/wiki/Safavid_Empire" class="mw-redirect" title="Safavid Empire">Safavid Empire</a> from the 15th century, the development of Islamic mathematics became stagnant. <div class="mw-heading mw-heading2"><h2 id="Maya">Maya</h2>[<a href="/w/index.php?title=History_of_mathematics&action=edit&section=10" title="Edit section: Maya">edit</a>]</div> <figure class="mw-default-size" typeof="mw:File/Thumb"><a href="/wiki/File:Maya_Hieroglyphs_Fig_40.jpg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/2/25/Maya_Hieroglyphs_Fig_40.jpg/220px-Maya_Hieroglyphs_Fig_40.jpg" decoding="async" width="220" height="217" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/2/25/Maya_Hieroglyphs_Fig_40.jpg/330px-Maya_Hieroglyphs_Fig_40.jpg 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/2/25/Maya_Hieroglyphs_Fig_40.jpg/440px-Maya_Hieroglyphs_Fig_40.jpg 2x" data-file-width="1398" data-file-height="1379" /></a><figcaption>The <a href="/wiki/Maya_numerals" title="Maya numerals">Maya numerals</a> for numbers 1 through 19, written in the <a href="/wiki/Maya_script" title="Maya script">Maya script</a></figcaption></figure> In the <a href="/wiki/Pre-Columbian_Americas" class="mw-redirect" title="Pre-Columbian Americas">Pre-Columbian Americas</a>, the <a href="/wiki/Maya_civilization" title="Maya civilization">Maya civilization</a> that flourished in <a href="/wiki/Mexico" title="Mexico">Mexico</a> and <a href="/wiki/Central_America" title="Central America">Central America</a> during the 1st millennium AD developed a unique tradition of mathematics that, due to its geographic isolation, was entirely independent of existing European, Egyptian, and Asian mathematics.<a href="#cite_note-Goodman_2016_p121-165">[164]</a> <a href="/wiki/Maya_numerals" title="Maya numerals">Maya numerals</a> used a <a href="/wiki/Radix" title="Radix">base</a> of twenty, the <a href="/wiki/Vigesimal" title="Vigesimal">vigesimal</a> system, instead of a base of ten that forms the basis of the <a href="/wiki/Decimal" title="Decimal">decimal</a> system used by most modern cultures.<a href="#cite_note-Goodman_2016_p121-165">[164]</a> The Maya used mathematics to create the <a href="/wiki/Maya_calendar" title="Maya calendar">Maya calendar</a> as well as to predict astronomical phenomena in their native <a href="/wiki/Maya_astronomy" title="Maya astronomy">Maya astronomy</a>.<a href="#cite_note-Goodman_2016_p121-165">[164]</a> While the concept of <a href="/wiki/Zero" class="mw-redirect" title="Zero">zero</a> had to be inferred in the mathematics of many contemporary cultures, the Maya developed a standard symbol for it.<a href="#cite_note-Goodman_2016_p121-165">[164]</a> <div class="mw-heading mw-heading2"><h2 id="Medieval_European">Medieval European</h2>[<a href="/w/index.php?title=History_of_mathematics&action=edit&section=11" title="Edit section: Medieval European">edit</a>]</div> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1236090951"><div role="note" class="hatnote navigation-not-searchable">Further information: <a href="/wiki/List_of_medieval_European_scientists" title="List of medieval European scientists">List of medieval European scientists</a> and <a href="/wiki/European_science_in_the_Middle_Ages" title="European science in the Middle Ages">European science in the Middle Ages</a></div> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1236090951"><div role="note" class="hatnote navigation-not-searchable">See also: <a href="/wiki/Latin_translations_of_the_12th_century" title="Latin translations of the 12th century">Latin translations of the 12th century</a></div> Medieval European interest in mathematics was driven by concerns quite different from those of modern mathematicians. One driving element was the belief that mathematics provided the key to understanding the created order of nature, frequently justified by <a href="/wiki/Plato" title="Plato">Plato</a>'s <a href="/wiki/Timaeus_(dialogue)" title="Timaeus (dialogue)">Timaeus</a> and the biblical passage (in the <a href="/wiki/Book_of_Wisdom" title="Book of Wisdom">Book of Wisdom</a>) that God had ordered all things in measure, and number, and weight.<a href="#cite_note-166">[165]</a> <a href="/wiki/Boethius" title="Boethius">Boethius</a> provided a place for mathematics in the curriculum in the 6th century when he coined the term <a href="/wiki/Quadrivium" title="Quadrivium">quadrivium</a> to describe the study of arithmetic, geometry, astronomy, and music. He wrote De institutione arithmetica, a free translation from the Greek of <a href="/wiki/Nicomachus" title="Nicomachus">Nicomachus</a>'s Introduction to Arithmetic; De institutione musica, also derived from Greek sources; and a series of excerpts from Euclid's Elements. His works were theoretical, rather than practical, and were the basis of mathematical study until the recovery of Greek and Arabic mathematical works.<a href="#cite_note-167">[166]</a><a href="#cite_note-168">[167]</a> In the 12th century, European scholars traveled to Spain and Sicily <a href="/wiki/Latin_translations_of_the_12th_century" title="Latin translations of the 12th century">seeking scientific Arabic texts</a>, including <a href="/wiki/Al-Khw%C4%81rizm%C4%AB" class="mw-redirect" title="Al-Khwārizmī">al-Khwārizmī</a>'s <a href="/wiki/The_Compendious_Book_on_Calculation_by_Completion_and_Balancing" class="mw-redirect" title="The Compendious Book on Calculation by Completion and Balancing">The Compendious Book on Calculation by Completion and Balancing</a>, translated into Latin by <a href="/wiki/Robert_of_Chester" title="Robert of Chester">Robert of Chester</a>, and the complete text of Euclid's Elements, translated in various versions by <a href="/wiki/Adelard_of_Bath" title="Adelard of Bath">Adelard of Bath</a>, <a href="/wiki/Herman_of_Carinthia" title="Herman of Carinthia">Herman of Carinthia</a>, and <a href="/wiki/Gerard_of_Cremona" title="Gerard of Cremona">Gerard of Cremona</a>.<a href="#cite_note-169">[168]</a><a href="#cite_note-170">[169]</a> These and other new sources sparked a renewal of mathematics. Leonardo of Pisa, now known as <a href="/wiki/Fibonacci" title="Fibonacci">Fibonacci</a>, serendipitously learned about the <a href="/wiki/Hindu%E2%80%93Arabic_numerals" class="mw-redirect" title="Hindu–Arabic numerals">Hindu–Arabic numerals</a> on a trip to what is now <a href="/wiki/B%C3%A9ja%C3%AFa" title="Béjaïa">Béjaïa</a>, <a href="/wiki/Algeria" title="Algeria">Algeria</a> with his merchant father. (Europe was still using <a href="/wiki/Roman_numerals" title="Roman numerals">Roman numerals</a>.) There, he observed a system of <a href="/wiki/Arithmetic" title="Arithmetic">arithmetic</a> (specifically <a href="/wiki/Algorism" title="Algorism">algorism</a>) which due to the <a href="/wiki/Positional_notation" title="Positional notation">positional notation</a> of Hindu–Arabic numerals was much more efficient and greatly facilitated commerce. Leonardo wrote <a href="/wiki/Liber_Abaci" title="Liber Abaci">Liber Abaci</a> in 1202 (updated in 1254) introducing the technique to Europe and beginning a long period of popularizing it. The book also brought to Europe what is now known as the <a href="/wiki/Fibonacci_sequence" title="Fibonacci sequence">Fibonacci sequence</a> (known to Indian mathematicians for hundreds of years before that)<a href="#cite_note-171">[170]</a> which Fibonacci used as an unremarkable example. The 14th century saw the development of new mathematical concepts to investigate a wide range of problems.<a href="#cite_note-172">[171]</a> One important contribution was development of mathematics of local motion. <a href="/wiki/Thomas_Bradwardine" title="Thomas Bradwardine">Thomas Bradwardine</a> proposed that speed (V) increases in arithmetic proportion as the ratio of force (F) to resistance (R) increases in geometric proportion. Bradwardine expressed this by a series of specific examples, but although the logarithm had not yet been conceived, we can express his conclusion anachronistically by writing: V = log (F/R).<a href="#cite_note-173">[172]</a> Bradwardine's analysis is an example of transferring a mathematical technique used by <a href="/wiki/Al-Kindi" title="Al-Kindi">al-Kindi</a> and <a href="/wiki/Arnald_of_Villanova" class="mw-redirect" title="Arnald of Villanova">Arnald of Villanova</a> to quantify the nature of compound medicines to a different physical problem.<a href="#cite_note-174">[173]</a> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1237032888/mw-parser-output/.tmulti"><div class="thumb tmulti tright"><div class="thumbinner multiimageinner" style="width:319px;max-width:319px"><div class="trow"><div class="tsingle" style="width:158px;max-width:158px"><div class="thumbimage"><a href="/wiki/File:Oresme.jpg" class="mw-file-description"><img alt="" src="//upload.wikimedia.org/wikipedia/commons/thumb/3/35/Oresme.jpg/156px-Oresme.jpg" decoding="async" width="156" height="154" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/3/35/Oresme.jpg/234px-Oresme.jpg 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/3/35/Oresme.jpg/312px-Oresme.jpg 2x" data-file-width="3787" data-file-height="3742" /></a></div><div class="thumbcaption"><a href="/wiki/Nicole_Oresme" title="Nicole Oresme">Nicole Oresme</a> (1323–1382), shown in this contemporary <a href="/wiki/Illuminated_manuscript" title="Illuminated manuscript">illuminated manuscript</a> with an <a href="/wiki/Armillary_sphere" title="Armillary sphere">armillary sphere</a> in the foreground, was the first to offer a mathematical proof for the <a href="/wiki/Divergent_series" title="Divergent series">divergence</a> of the <a href="/wiki/Harmonic_series_(mathematics)" title="Harmonic series (mathematics)">harmonic series</a>.<a href="#cite_note-175">[174]</a></div></div><div class="tsingle" style="width:157px;max-width:157px"><div class="thumbimage"><a href="/wiki/File:Ries.PNG" class="mw-file-description"><img alt="" src="//upload.wikimedia.org/wikipedia/commons/thumb/1/1f/Ries.PNG/155px-Ries.PNG" decoding="async" width="155" height="154" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/1/1f/Ries.PNG/233px-Ries.PNG 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/1/1f/Ries.PNG/310px-Ries.PNG 2x" data-file-width="440" data-file-height="438" /></a></div><div class="thumbcaption"><a href="/wiki/Adam_Ries" title="Adam Ries">Adam Ries</a> is known as the "father of modern calculating" because of his decisive contribution to the recognition that <a href="/wiki/Roman_numerals" title="Roman numerals">Roman numerals</a> are unpractical and to their replacement by the considerably more practical <a href="/wiki/Arabic_numerals" title="Arabic numerals">Arabic numerals</a>.<a href="#cite_note-176">[175]</a></div></div></div></div></div> One of the 14th-century <a href="/wiki/Oxford_Calculators" title="Oxford Calculators">Oxford Calculators</a>, <a href="/wiki/William_Heytesbury" class="mw-redirect" title="William Heytesbury">William Heytesbury</a>, lacking <a href="/wiki/Differential_calculus" title="Differential calculus">differential calculus</a> and the concept of <a href="/wiki/Limit_of_a_function" title="Limit of a function">limits</a>, proposed to measure instantaneous speed "by the path that would be described by [a body] if... it were moved uniformly at the same degree of speed with which it is moved in that given instant".<a href="#cite_note-177">[176]</a> Heytesbury and others mathematically determined the distance covered by a body undergoing uniformly accelerated motion (today solved by integration), stating that "a moving body uniformly acquiring or losing that increment [of speed] will traverse in some given time a [distance] completely equal to that which it would traverse if it were moving continuously through the same time with the mean degree [of speed]".<a href="#cite_note-178">[177]</a> <a href="/wiki/Nicole_Oresme" title="Nicole Oresme">Nicole Oresme</a> at the <a href="/wiki/University_of_Paris" title="University of Paris">University of Paris</a> and the Italian <a href="/wiki/Giovanni_di_Casali" title="Giovanni di Casali">Giovanni di Casali</a> independently provided graphical demonstrations of this relationship, asserting that the area under the line depicting the constant acceleration, represented the total distance traveled.<a href="#cite_note-179">[178]</a> In a later mathematical commentary on Euclid's Elements, Oresme made a more detailed general analysis in which he demonstrated that a body will acquire in each successive increment of time an increment of any quality that increases as the odd numbers. Since Euclid had demonstrated the sum of the odd numbers are the square numbers, the total quality acquired by the body increases as the square of the time.<a href="#cite_note-180">[179]</a> <div class="mw-heading mw-heading2"><h2 id="Renaissance">Renaissance</h2>[<a href="/w/index.php?title=History_of_mathematics&action=edit&section=12" title="Edit section: Renaissance">edit</a>]</div> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1236090951"><div role="note" class="hatnote navigation-not-searchable">Further information: <a href="/wiki/Mathematics_and_art" title="Mathematics and art">Mathematics and art</a></div> During the <a href="/wiki/Renaissance" title="Renaissance">Renaissance</a>, the development of mathematics and of <a href="/wiki/Accounting" title="Accounting">accounting</a> were intertwined.<a href="#cite_note-181">[180]</a> While there is no direct relationship between algebra and accounting, the teaching of the subjects and the books published often intended for the children of merchants who were sent to reckoning schools (in <a href="/wiki/Flanders" title="Flanders">Flanders</a> and <a href="/wiki/Germany" title="Germany">Germany</a>) or <a href="/wiki/Abacus_school" title="Abacus school">abacus schools</a> (known as abbaco in Italy), where they learned the skills useful for trade and commerce. There is probably no need for algebra in performing <a href="/wiki/Bookkeeping" title="Bookkeeping">bookkeeping</a> operations, but for complex bartering operations or the calculation of <a href="/wiki/Compound_interest" title="Compound interest">compound interest</a>, a basic knowledge of arithmetic was mandatory and knowledge of algebra was very useful. <a href="/wiki/Piero_della_Francesca" title="Piero della Francesca">Piero della Francesca</a> (c. 1415–1492) wrote books on <a href="/wiki/Solid_geometry" title="Solid geometry">solid geometry</a> and <a href="/wiki/Perspective_(graphical)" title="Perspective (graphical)">linear perspective</a>, including <a href="/wiki/De_Prospectiva_Pingendi" class="mw-redirect" title="De Prospectiva Pingendi">De Prospectiva Pingendi</a> (On Perspective for Painting), Trattato d’Abaco (Abacus Treatise), and <a href="/wiki/De_quinque_corporibus_regularibus" title="De quinque corporibus regularibus">De quinque corporibus regularibus</a> (On the Five Regular Solids).<a href="#cite_note-182">[181]</a><a href="#cite_note-183">[182]</a><a href="#cite_note-184">[183]</a> <figure class="mw-default-size mw-halign-left" typeof="mw:File/Thumb"><a href="/wiki/File:Pacioli.jpg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/2/2a/Pacioli.jpg/220px-Pacioli.jpg" decoding="async" width="220" height="183" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/2/2a/Pacioli.jpg/330px-Pacioli.jpg 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/2/2a/Pacioli.jpg/440px-Pacioli.jpg 2x" data-file-width="1500" data-file-height="1250" /></a><figcaption><a href="/wiki/Portrait_of_Luca_Pacioli" title="Portrait of Luca Pacioli">Portrait of Luca Pacioli</a>, a painting traditionally attributed to <a href="/wiki/Jacopo_de%27_Barbari" title="Jacopo de' Barbari">Jacopo de' Barbari</a>, 1495, (<a href="/wiki/Museo_di_Capodimonte" title="Museo di Capodimonte">Museo di Capodimonte</a>).</figcaption></figure> <a href="/wiki/Luca_Pacioli" title="Luca Pacioli">Luca Pacioli</a>'s <a href="/wiki/Summa_de_arithmetica" title="Summa de arithmetica">Summa de Arithmetica, Geometria, Proportioni et Proportionalità</a> (Italian: "Review of <a href="/wiki/Arithmetic" title="Arithmetic">Arithmetic</a>, <a href="/wiki/Geometry" title="Geometry">Geometry</a>, <a href="/wiki/Ratio" title="Ratio">Ratio</a> and <a href="/wiki/Proportionality_(mathematics)" title="Proportionality (mathematics)">Proportion</a>") was first printed and published in <a href="/wiki/Venice" title="Venice">Venice</a> in 1494. It included a 27-page treatise on bookkeeping, "Particularis de Computis et Scripturis" (Italian: "Details of Calculation and Recording"). It was written primarily for, and sold mainly to, merchants who used the book as a reference text, as a source of pleasure from the <a href="/wiki/Mathematical_puzzles" class="mw-redirect" title="Mathematical puzzles">mathematical puzzles</a> it contained, and to aid the education of their sons.<a href="#cite_note-185">[184]</a> In Summa Arithmetica, Pacioli introduced symbols for <a href="/wiki/Plus_and_minus" class="mw-redirect" title="Plus and minus">plus and minus</a> for the first time in a printed book, symbols that became standard notation in Italian Renaissance mathematics. Summa Arithmetica was also the first known book printed in Italy to contain algebra. Pacioli obtained many of his ideas from Piero Della Francesca whom he plagiarized. In Italy, during the first half of the 16th century, <a href="/wiki/Scipione_del_Ferro" title="Scipione del Ferro">Scipione del Ferro</a> and <a href="/wiki/Niccol%C3%B2_Fontana_Tartaglia" class="mw-redirect" title="Niccolò Fontana Tartaglia">Niccolò Fontana Tartaglia</a> discovered solutions for <a href="/wiki/Cubic_equation" title="Cubic equation">cubic equations</a>. <a href="/wiki/Gerolamo_Cardano" title="Gerolamo Cardano">Gerolamo Cardano</a> published them in his 1545 book <a href="/wiki/Ars_Magna_(Gerolamo_Cardano)" class="mw-redirect" title="Ars Magna (Gerolamo Cardano)">Ars Magna</a>, together with a solution for the <a href="/wiki/Quartic_equation" title="Quartic equation">quartic equations</a>, discovered by his student <a href="/wiki/Lodovico_Ferrari" title="Lodovico Ferrari">Lodovico Ferrari</a>. In 1572 <a href="/wiki/Rafael_Bombelli" title="Rafael Bombelli">Rafael Bombelli</a> published his L'Algebra in which he showed how to deal with the <a href="/wiki/Imaginary_number" title="Imaginary number">imaginary quantities</a> that could appear in Cardano's formula for solving cubic equations. <a href="/wiki/Simon_Stevin" title="Simon Stevin">Simon Stevin</a>'s <a href="/wiki/De_Thiende" title="De Thiende">De Thiende</a> ('the art of tenths'), first published in Dutch in 1585, contained the first systematic treatment of <a href="/wiki/Decimal_notation" class="mw-redirect" title="Decimal notation">decimal notation</a> in Europe, which influenced all later work on the <a href="/wiki/Real_number_system" class="mw-redirect" title="Real number system">real number system</a>.<a href="#cite_note-186">[185]</a><a href="#cite_note-GS35-187">[186]</a> Driven by the demands of navigation and the growing need for accurate maps of large areas, <a href="/wiki/Trigonometry" title="Trigonometry">trigonometry</a> grew to be a major branch of mathematics. <a href="/wiki/Bartholomaeus_Pitiscus" title="Bartholomaeus Pitiscus">Bartholomaeus Pitiscus</a> was the first to use the word, publishing his Trigonometria in 1595. Regiomontanus's table of sines and cosines was published in 1533.<a href="#cite_note-188">[187]</a> During the Renaissance the desire of artists to represent the natural world realistically, together with the rediscovered philosophy of the Greeks, led artists to study mathematics. They were also the engineers and architects of that time, and so had need of mathematics in any case. The art of painting in perspective, and the developments in geometry that were involved, were studied intensely.<a href="#cite_note-Kline-189">[188]</a> <div class="mw-heading mw-heading2"><h2 id="Mathematics_during_the_Scientific_Revolution">Mathematics during the Scientific Revolution</h2>[<a href="/w/index.php?title=History_of_mathematics&action=edit&section=13" title="Edit section: Mathematics during the Scientific Revolution">edit</a>]</div> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1236090951"><div role="note" class="hatnote navigation-not-searchable">See also: <a href="/wiki/Scientific_Revolution" title="Scientific Revolution">Scientific Revolution</a></div> <div class="mw-heading mw-heading3"><h3 id="17th_century">17th century</h3>[<a href="/w/index.php?title=History_of_mathematics&action=edit&section=14" title="Edit section: 17th century">edit</a>]</div> <figure class="mw-default-size mw-halign-left" typeof="mw:File/Thumb"><a href="/wiki/File:Gottfried_Wilhelm_Leibniz,_Bernhard_Christoph_Francke.jpg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/c/ce/Gottfried_Wilhelm_Leibniz%2C_Bernhard_Christoph_Francke.jpg/170px-Gottfried_Wilhelm_Leibniz%2C_Bernhard_Christoph_Francke.jpg" decoding="async" width="170" height="210" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/c/ce/Gottfried_Wilhelm_Leibniz%2C_Bernhard_Christoph_Francke.jpg/255px-Gottfried_Wilhelm_Leibniz%2C_Bernhard_Christoph_Francke.jpg 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/c/ce/Gottfried_Wilhelm_Leibniz%2C_Bernhard_Christoph_Francke.jpg/340px-Gottfried_Wilhelm_Leibniz%2C_Bernhard_Christoph_Francke.jpg 2x" data-file-width="4486" data-file-height="5538" /></a><figcaption><a href="/wiki/Gottfried_Wilhelm_Leibniz" title="Gottfried Wilhelm Leibniz">Gottfried Wilhelm Leibniz</a></figcaption></figure> The 17th century saw an unprecedented increase of mathematical and scientific ideas across Europe. <a href="/wiki/Galileo" class="mw-redirect" title="Galileo">Galileo</a> observed the moons of Jupiter in orbit about that planet, using a telescope based <a href="/wiki/Hans_Lipperhey" title="Hans Lipperhey">Hans Lipperhey</a>'s. <a href="/wiki/Tycho_Brahe" title="Tycho Brahe">Tycho Brahe</a> had gathered a large quantity of mathematical data describing the positions of the planets in the sky. By his position as Brahe's assistant, <a href="/wiki/Johannes_Kepler" title="Johannes Kepler">Johannes Kepler</a> was first exposed to and seriously interacted with the topic of planetary motion. Kepler's calculations were made simpler by the contemporaneous invention of <a href="/wiki/Logarithm" title="Logarithm">logarithms</a> by <a href="/wiki/John_Napier" title="John Napier">John Napier</a> and <a href="/wiki/Jost_B%C3%BCrgi" title="Jost Bürgi">Jost Bürgi</a>. Kepler succeeded in formulating mathematical laws of planetary motion.<a href="#cite_note-190">[189]</a> The <a href="/wiki/Analytic_geometry" title="Analytic geometry">analytic geometry</a> developed by <a href="/wiki/Ren%C3%A9_Descartes" title="René Descartes">René Descartes</a> (1596–1650) allowed those orbits to be plotted on a graph, in <a href="/wiki/Cartesian_coordinates" class="mw-redirect" title="Cartesian coordinates">Cartesian coordinates</a>. Building on earlier work by many predecessors, <a href="/wiki/Isaac_Newton" title="Isaac Newton">Isaac Newton</a> discovered the laws of physics that explain <a href="/wiki/Kepler%27s_Laws" class="mw-redirect" title="Kepler's Laws">Kepler's Laws</a>, and brought together the concepts now known as <a href="/wiki/Calculus" title="Calculus">calculus</a>. Independently, <a href="/wiki/Gottfried_Wilhelm_Leibniz" title="Gottfried Wilhelm Leibniz">Gottfried Wilhelm Leibniz</a>, developed calculus and much of the calculus notation still in use today. He also refined the <a href="/wiki/Binary_number" title="Binary number">binary number</a> system, which is the foundation of nearly all digital (<a href="/wiki/Scientific_calculator" title="Scientific calculator">electronic</a>, <a href="/wiki/Solid-state_electronics" title="Solid-state electronics">solid-state</a>, <a href="/wiki/Logic_gate" title="Logic gate">discrete logic</a>) <a href="/wiki/Computer" title="Computer">computers</a>, including the <a href="/wiki/Von_Neumann_architecture" title="Von Neumann architecture">Von Neumann architecture</a>, which is the standard design paradigm, or "<a href="/wiki/Computer_architecture" title="Computer architecture">computer architecture</a>", followed from the second half of the 20th century, and into the 21st. Leibniz has been called the "founder of computer science".<a href="#cite_note-191">[190]</a> Science and mathematics had become an international endeavor, which would soon spread over the entire world.<a href="#cite_note-192">[191]</a> In addition to the application of mathematics to the studies of the heavens, <a href="/wiki/Applied_mathematics" title="Applied mathematics">applied mathematics</a> began to expand into new areas, with the correspondence of <a href="/wiki/Pierre_de_Fermat" title="Pierre de Fermat">Pierre de Fermat</a> and <a href="/wiki/Blaise_Pascal" title="Blaise Pascal">Blaise Pascal</a>. Pascal and Fermat set the groundwork for the investigations of <a href="/wiki/Probability_theory" title="Probability theory">probability theory</a> and the corresponding rules of <a href="/wiki/Combinatorics" title="Combinatorics">combinatorics</a> in their discussions over a game of <a href="/wiki/Gambling" title="Gambling">gambling</a>. Pascal, with his <a href="/wiki/Pascal%27s_Wager" class="mw-redirect" title="Pascal's Wager">wager</a>, attempted to use the newly developing probability theory to argue for a life devoted to religion, on the grounds that even if the probability of success was small, the rewards were infinite. In some sense, this foreshadowed the development of <a href="/wiki/Utility_theory" class="mw-redirect" title="Utility theory">utility theory</a> in the 18th and 19th centuries. <div class="mw-heading mw-heading3"><h3 id="18th_century">18th century</h3>[<a href="/w/index.php?title=History_of_mathematics&action=edit&section=15" title="Edit section: 18th century">edit</a>]</div> <figure class="mw-default-size mw-halign-right" typeof="mw:File/Thumb"><a href="/wiki/File:Leonhard_Euler_-_Jakob_Emanuel_Handmann_(Kunstmuseum_Basel).jpg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/f/f9/Leonhard_Euler_-_Jakob_Emanuel_Handmann_%28Kunstmuseum_Basel%29.jpg/170px-Leonhard_Euler_-_Jakob_Emanuel_Handmann_%28Kunstmuseum_Basel%29.jpg" decoding="async" width="170" height="220" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/f/f9/Leonhard_Euler_-_Jakob_Emanuel_Handmann_%28Kunstmuseum_Basel%29.jpg/255px-Leonhard_Euler_-_Jakob_Emanuel_Handmann_%28Kunstmuseum_Basel%29.jpg 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/f/f9/Leonhard_Euler_-_Jakob_Emanuel_Handmann_%28Kunstmuseum_Basel%29.jpg/340px-Leonhard_Euler_-_Jakob_Emanuel_Handmann_%28Kunstmuseum_Basel%29.jpg 2x" data-file-width="4672" data-file-height="6040" /></a><figcaption><a href="/wiki/Leonhard_Euler" title="Leonhard Euler">Leonhard Euler</a></figcaption></figure> The most influential mathematician of the 18th century was arguably <a href="/wiki/Leonhard_Euler" title="Leonhard Euler">Leonhard Euler</a> (1707–1783). His contributions range from founding the study of <a href="/wiki/Graph_theory" title="Graph theory">graph theory</a> with the <a href="/wiki/Seven_Bridges_of_K%C3%B6nigsberg" title="Seven Bridges of Königsberg">Seven Bridges of Königsberg</a> problem to standardizing many modern mathematical terms and notations. For example, he named the square root of minus 1 with the symbol <a href="/wiki/Imaginary_unit" title="Imaginary unit">i</a>, and he popularized the use of the Greek letter <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \pi }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>π</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \pi }</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/9be4ba0bb8df3af72e90a0535fabcc17431e540a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.332ex; height:1.676ex;" alt="{\displaystyle \pi }"> to stand for the ratio of a circle's circumference to its diameter. He made numerous contributions to the study of topology, graph theory, calculus, combinatorics, and complex analysis, as evidenced by the multitude of theorems and notations named for him. Other important European mathematicians of the 18th century included <a href="/wiki/Joseph_Louis_Lagrange" class="mw-redirect" title="Joseph Louis Lagrange">Joseph Louis Lagrange</a>, who did pioneering work in number theory, algebra, differential calculus, and the calculus of variations, and <a href="/wiki/Pierre-Simon_Laplace" title="Pierre-Simon Laplace">Pierre-Simon Laplace</a>, who, in the age of <a href="/wiki/Napoleon" title="Napoleon">Napoleon</a>, did important work on the foundations of <a href="/wiki/Celestial_mechanics" title="Celestial mechanics">celestial mechanics</a> and on <a href="/wiki/Statistics" title="Statistics">statistics</a>. <div class="mw-heading mw-heading2"><h2 id="Modern">Modern</h2>[<a href="/w/index.php?title=History_of_mathematics&action=edit&section=16" title="Edit section: Modern">edit</a>]</div> <style data-mw-deduplicate="TemplateStyles:r1251242444">.mw-parser-output .ambox{border:1px solid #a2a9b1;border-left:10px solid #36c;background-color:#fbfbfb;box-sizing:border-box}.mw-parser-output .ambox+link+.ambox,.mw-parser-output .ambox+link+style+.ambox,.mw-parser-output .ambox+link+link+.ambox,.mw-parser-output .ambox+.mw-empty-elt+link+.ambox,.mw-parser-output .ambox+.mw-empty-elt+link+style+.ambox,.mw-parser-output .ambox+.mw-empty-elt+link+link+.ambox{margin-top:-1px}html body.mediawiki .mw-parser-output .ambox.mbox-small-left{margin:4px 1em 4px 0;overflow:hidden;width:238px;border-collapse:collapse;font-size:88%;line-height:1.25em}.mw-parser-output .ambox-speedy{border-left:10px solid #b32424;background-color:#fee7e6}.mw-parser-output .ambox-delete{border-left:10px solid #b32424}.mw-parser-output .ambox-content{border-left:10px solid #f28500}.mw-parser-output .ambox-style{border-left:10px solid #fc3}.mw-parser-output .ambox-move{border-left:10px solid #9932cc}.mw-parser-output .ambox-protection{border-left:10px solid #a2a9b1}.mw-parser-output .ambox .mbox-text{border:none;padding:0.25em 0.5em;width:100%}.mw-parser-output .ambox .mbox-image{border:none;padding:2px 0 2px 0.5em;text-align:center}.mw-parser-output .ambox .mbox-imageright{border:none;padding:2px 0.5em 2px 0;text-align:center}.mw-parser-output .ambox .mbox-empty-cell{border:none;padding:0;width:1px}.mw-parser-output .ambox .mbox-image-div{width:52px}@media(min-width:720px){.mw-parser-output .ambox{margin:0 10%}}@media print{body.ns-0 .mw-parser-output .ambox{display:none!important}}</style><table class="box-More_citations_needed_section plainlinks metadata ambox ambox-content ambox-Refimprove" role="presentation"><tbody><tr><td class="mbox-image"><div class="mbox-image-div"><a href="/wiki/File:Question_book-new.svg" class="mw-file-description"><img alt="" src="//upload.wikimedia.org/wikipedia/en/thumb/9/99/Question_book-new.svg/50px-Question_book-new.svg.png" decoding="async" width="50" height="39" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/en/thumb/9/99/Question_book-new.svg/75px-Question_book-new.svg.png 1.5x, //upload.wikimedia.org/wikipedia/en/thumb/9/99/Question_book-new.svg/100px-Question_book-new.svg.png 2x" data-file-width="512" data-file-height="399" /></a></div></td><td class="mbox-text"><div class="mbox-text-span">This section needs additional citations for <a href="/wiki/Wikipedia:Verifiability" title="Wikipedia:Verifiability">verification</a>. Please help <a href="/wiki/Special:EditPage/History_of_mathematics" title="Special:EditPage/History of mathematics">improve this article</a> by <a href="/wiki/Help:Referencing_for_beginners" title="Help:Referencing for beginners">adding citations to reliable sources</a> in this section. Unsourced material may be challenged and removed. Find sources: <a rel="nofollow" class="external text" href="https://www.google.com/search?as_eq=wikipedia&q=%22History+of+mathematics%22">"History of mathematics"</a> – <a rel="nofollow" class="external text" href="https://www.google.com/search?tbm=nws&q=%22History+of+mathematics%22+-wikipedia&tbs=ar:1">news</a> · <a rel="nofollow" class="external text" href="https://www.google.com/search?&q=%22History+of+mathematics%22&tbs=bkt:s&tbm=bks">newspapers</a> · <a rel="nofollow" class="external text" href="https://www.google.com/search?tbs=bks:1&q=%22History+of+mathematics%22+-wikipedia">books</a> · <a rel="nofollow" class="external text" href="https://scholar.google.com/scholar?q=%22History+of+mathematics%22">scholar</a> · <a rel="nofollow" class="external text" href="https://www.jstor.org/action/doBasicSearch?Query=%22History+of+mathematics%22&acc=on&wc=on">JSTOR</a> (April 2021) (<a href="/wiki/Help:Maintenance_template_removal" title="Help:Maintenance template removal">Learn how and when to remove this message</a>)</div></td></tr></tbody></table> <div class="mw-heading mw-heading3"><h3 id="19th_century">19th century</h3>[<a href="/w/index.php?title=History_of_mathematics&action=edit&section=17" title="Edit section: 19th century">edit</a>]</div> <figure class="mw-default-size mw-halign-right" typeof="mw:File/Thumb"><a href="/wiki/File:Carl_Friedrich_Gauss.jpg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/9/9b/Carl_Friedrich_Gauss.jpg/170px-Carl_Friedrich_Gauss.jpg" decoding="async" width="170" height="219" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/9/9b/Carl_Friedrich_Gauss.jpg/255px-Carl_Friedrich_Gauss.jpg 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/9/9b/Carl_Friedrich_Gauss.jpg/340px-Carl_Friedrich_Gauss.jpg 2x" data-file-width="917" data-file-height="1180" /></a><figcaption><a href="/wiki/Carl_Friedrich_Gauss" title="Carl Friedrich Gauss">Carl Friedrich Gauss</a></figcaption></figure> Throughout the 19th century mathematics became increasingly abstract.<a href="#cite_note-193">[192]</a> <a href="/wiki/Carl_Friedrich_Gauss" title="Carl Friedrich Gauss">Carl Friedrich Gauss</a> (1777–1855) epitomizes this trend.[<a href="/wiki/Wikipedia:Citation_needed" title="Wikipedia:Citation needed">citation needed</a>] He did revolutionary work on <a href="/wiki/Function_(mathematics)" title="Function (mathematics)">functions</a> of <a href="/wiki/Complex_variable" class="mw-redirect" title="Complex variable">complex variables</a>, in <a href="/wiki/Geometry" title="Geometry">geometry</a>, and on the convergence of <a href="/wiki/Series_(mathematics)" title="Series (mathematics)">series</a>, leaving aside his many contributions to science. He also gave the first satisfactory proofs of the <a href="/wiki/Fundamental_theorem_of_algebra" title="Fundamental theorem of algebra">fundamental theorem of algebra</a> and of the <a href="/wiki/Quadratic_reciprocity_law" class="mw-redirect" title="Quadratic reciprocity law">quadratic reciprocity law</a>.[<a href="/wiki/Wikipedia:Citation_needed" title="Wikipedia:Citation needed">citation needed</a>] <figure class="mw-default-size mw-halign-left" typeof="mw:File/Thumb"><a href="/wiki/File:Noneuclid.svg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/7/78/Noneuclid.svg/330px-Noneuclid.svg.png" decoding="async" width="330" height="83" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/7/78/Noneuclid.svg/495px-Noneuclid.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/7/78/Noneuclid.svg/660px-Noneuclid.svg.png 2x" data-file-width="663" data-file-height="167" /></a><figcaption>Behavior of lines with a common perpendicular in each of the three types of geometry</figcaption></figure> This century saw the development of the two forms of <a href="/wiki/Non-Euclidean_geometry" title="Non-Euclidean geometry">non-Euclidean geometry</a>, where the <a href="/wiki/Parallel_postulate" title="Parallel postulate">parallel postulate</a> of Euclidean geometry no longer holds. The Russian mathematician <a href="/wiki/Nikolai_Ivanovich_Lobachevsky" class="mw-redirect" title="Nikolai Ivanovich Lobachevsky">Nikolai Ivanovich Lobachevsky</a> and his rival, the Hungarian mathematician <a href="/wiki/J%C3%A1nos_Bolyai" title="János Bolyai">János Bolyai</a>, independently defined and studied <a href="/wiki/Hyperbolic_geometry" title="Hyperbolic geometry">hyperbolic geometry</a>, where uniqueness of parallels no longer holds. In this geometry the sum of angles in a triangle add up to less than 180°. <a href="/wiki/Elliptic_geometry" title="Elliptic geometry">Elliptic geometry</a> was developed later in the 19th century by the German mathematician <a href="/wiki/Bernhard_Riemann" title="Bernhard Riemann">Bernhard Riemann</a>; here no parallel can be found and the angles in a triangle add up to more than 180°. Riemann also developed <a href="/wiki/Riemannian_geometry" title="Riemannian geometry">Riemannian geometry</a>, which unifies and vastly generalizes the three types of geometry, and he defined the concept of a <a href="/wiki/Manifold" title="Manifold">manifold</a>, which generalizes the ideas of <a href="/wiki/Curve" title="Curve">curves</a> and <a href="/wiki/Surface_(topology)" title="Surface (topology)">surfaces</a>, and set the mathematical foundations for the <a href="/wiki/General_relativity" title="General relativity">theory of general relativity</a>.<a href="#cite_note-194">[193]</a> The 19th century saw the beginning of a great deal of <a href="/wiki/Abstract_algebra" title="Abstract algebra">abstract algebra</a>. <a href="/wiki/Hermann_Grassmann" title="Hermann Grassmann">Hermann Grassmann</a> in Germany gave a first version of <a href="/wiki/Vector_space" title="Vector space">vector spaces</a>, <a href="/wiki/William_Rowan_Hamilton" title="William Rowan Hamilton">William Rowan Hamilton</a> in Ireland developed <a href="/wiki/Noncommutative_algebra" class="mw-redirect" title="Noncommutative algebra">noncommutative algebra</a>.[<a href="/wiki/Wikipedia:Citation_needed" title="Wikipedia:Citation needed">citation needed</a>] The British mathematician <a href="/wiki/George_Boole" title="George Boole">George Boole</a> devised an algebra that soon evolved into what is now called <a href="/wiki/Boolean_algebra" title="Boolean algebra">Boolean algebra</a>, in which the only numbers were 0 and 1. Boolean algebra is the starting point of <a href="/wiki/Mathematical_logic" title="Mathematical logic">mathematical logic</a> and has important applications in <a href="/wiki/Electrical_engineering" title="Electrical engineering">electrical engineering</a> and <a href="/wiki/Computer_science" title="Computer science">computer science</a>.[<a href="/wiki/Wikipedia:Citation_needed" title="Wikipedia:Citation needed">citation needed</a>] <a href="/wiki/Augustin-Louis_Cauchy" title="Augustin-Louis Cauchy">Augustin-Louis Cauchy</a>, <a href="/wiki/Bernhard_Riemann" title="Bernhard Riemann">Bernhard Riemann</a>, and <a href="/wiki/Karl_Weierstrass" title="Karl Weierstrass">Karl Weierstrass</a> reformulated the calculus in a more rigorous fashion.[<a href="/wiki/Wikipedia:Citation_needed" title="Wikipedia:Citation needed">citation needed</a>] Also, for the first time, the limits of mathematics were explored. <a href="/wiki/Niels_Henrik_Abel" title="Niels Henrik Abel">Niels Henrik Abel</a>, a Norwegian, and <a href="/wiki/%C3%89variste_Galois" title="Évariste Galois">Évariste Galois</a>, a Frenchman, proved that there is no general algebraic method for solving polynomial equations of degree greater than four (<a href="/wiki/Abel%E2%80%93Ruffini_theorem" title="Abel–Ruffini theorem">Abel–Ruffini theorem</a>).<a href="#cite_note-195">[194]</a> Other 19th-century mathematicians used this in their proofs that straight edge and compass alone are not sufficient to <a href="/wiki/Trisect_an_arbitrary_angle" class="mw-redirect" title="Trisect an arbitrary angle">trisect an arbitrary angle</a>, to construct the side of a cube twice the volume of a given cube, <a href="/wiki/Squaring_the_circle" title="Squaring the circle">nor to construct a square equal in area to a given circle</a>.[<a href="/wiki/Wikipedia:Citation_needed" title="Wikipedia:Citation needed">citation needed</a>] Mathematicians had vainly attempted to solve all of these problems since the time of the ancient Greeks.[<a href="/wiki/Wikipedia:Citation_needed" title="Wikipedia:Citation needed">citation needed</a>] On the other hand, the limitation of three <a href="/wiki/Dimension" title="Dimension">dimensions</a> in geometry was surpassed in the 19th century through considerations of <a href="/wiki/Parameter_space" title="Parameter space">parameter space</a> and <a href="/wiki/Hypercomplex_number" title="Hypercomplex number">hypercomplex numbers</a>.[<a href="/wiki/Wikipedia:Citation_needed" title="Wikipedia:Citation needed">citation needed</a>] Abel and Galois's investigations into the solutions of various polynomial equations laid the groundwork for further developments of <a href="/wiki/Group_theory" title="Group theory">group theory</a>, and the associated fields of <a href="/wiki/Abstract_algebra" title="Abstract algebra">abstract algebra</a>. In the 20th century physicists and other scientists have seen group theory as the ideal way to study <a href="/wiki/Symmetry" title="Symmetry">symmetry</a>.[<a href="/wiki/Wikipedia:Citation_needed" title="Wikipedia:Citation needed">citation needed</a>] In the later 19th century, <a href="/wiki/Georg_Cantor" title="Georg Cantor">Georg Cantor</a> established the first foundations of <a href="/wiki/Set_theory" title="Set theory">set theory</a>, which enabled the rigorous treatment of the notion of infinity and has become the common language of nearly all mathematics. Cantor's set theory, and the rise of <a href="/wiki/Mathematical_logic" title="Mathematical logic">mathematical logic</a> in the hands of <a href="/wiki/Peano" class="mw-redirect" title="Peano">Peano</a>, <a href="/wiki/L.E.J._Brouwer" class="mw-redirect" title="L.E.J. Brouwer">L.E.J. Brouwer</a>, <a href="/wiki/David_Hilbert" title="David Hilbert">David Hilbert</a>, <a href="/wiki/Bertrand_Russell" title="Bertrand Russell">Bertrand Russell</a>, and <a href="/wiki/A.N._Whitehead" class="mw-redirect" title="A.N. Whitehead">A.N. Whitehead</a>, initiated a long running debate on the <a href="/wiki/Foundations_of_mathematics" title="Foundations of mathematics">foundations of mathematics</a>.[<a href="/wiki/Wikipedia:Citation_needed" title="Wikipedia:Citation needed">citation needed</a>] The 19th century saw the founding of a number of national mathematical societies: the <a href="/wiki/London_Mathematical_Society" title="London Mathematical Society">London Mathematical Society</a> in 1865,<a href="#cite_note-196">[195]</a> the <a href="/wiki/Soci%C3%A9t%C3%A9_Math%C3%A9matique_de_France" class="mw-redirect" title="Société Mathématique de France">Société Mathématique de France</a> in 1872,<a href="#cite_note-197">[196]</a> the <a href="/wiki/Circolo_Matematico_di_Palermo" title="Circolo Matematico di Palermo">Circolo Matematico di Palermo</a> in 1884,<a href="#cite_note-198">[197]</a><a href="#cite_note-199">[198]</a> the <a href="/wiki/Edinburgh_Mathematical_Society" title="Edinburgh Mathematical Society">Edinburgh Mathematical Society</a> in 1883,<a href="#cite_note-200">[199]</a> and the <a href="/wiki/American_Mathematical_Society" title="American Mathematical Society">American Mathematical Society</a> in 1888.<a href="#cite_note-201">[200]</a> The first international, special-interest society, the <a href="/wiki/Quaternion_Society" title="Quaternion Society">Quaternion Society</a>, was formed in 1899, in the context of a <a href="/wiki/Hyperbolic_quaternion#Historical_review" title="Hyperbolic quaternion">vector controversy</a>.<a href="#cite_note-202">[201]</a> In 1897, <a href="/wiki/Kurt_Hensel" title="Kurt Hensel">Kurt Hensel</a> introduced <a href="/wiki/P-adic_number" title="P-adic number">p-adic numbers</a>.<a href="#cite_note-203">[202]</a> <div class="mw-heading mw-heading3"><h3 id="20th_century">20th century</h3>[<a href="/w/index.php?title=History_of_mathematics&action=edit&section=18" title="Edit section: 20th century">edit</a>]</div> The 20th century saw mathematics become a major profession. By the end of the century, thousands of new Ph.D.s in mathematics were being awarded every year, and jobs were available in both teaching and industry.<a href="#cite_note-204">[203]</a> An effort to catalogue the areas and applications of mathematics was undertaken in <a href="/wiki/Klein%27s_encyclopedia" class="mw-redirect" title="Klein's encyclopedia">Klein's encyclopedia</a>.<a href="#cite_note-205">[204]</a> In a 1900 speech to the <a href="/wiki/International_Congress_of_Mathematicians" title="International Congress of Mathematicians">International Congress of Mathematicians</a>, <a href="/wiki/David_Hilbert" title="David Hilbert">David Hilbert</a> set out a list of <a href="/wiki/Hilbert%27s_problems" title="Hilbert's problems">23 unsolved problems in mathematics</a>.<a href="#cite_note-206">[205]</a> These problems, spanning many areas of mathematics, formed a central focus for much of 20th-century mathematics. Today, 10 have been solved, 7 are partially solved, and 2 are still open. The remaining 4 are too loosely formulated to be stated as solved or not.[<a href="/wiki/Wikipedia:Citation_needed" title="Wikipedia:Citation needed">citation needed</a>] <figure class="mw-default-size mw-halign-left" typeof="mw:File/Thumb"><a href="/wiki/File:Four_Colour_Map_Example.svg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/8/8a/Four_Colour_Map_Example.svg/170px-Four_Colour_Map_Example.svg.png" decoding="async" width="170" height="227" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/8/8a/Four_Colour_Map_Example.svg/255px-Four_Colour_Map_Example.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/8/8a/Four_Colour_Map_Example.svg/340px-Four_Colour_Map_Example.svg.png 2x" data-file-width="300" data-file-height="400" /></a><figcaption>A map illustrating the <a href="/wiki/Four_Color_Theorem" class="mw-redirect" title="Four Color Theorem">Four Color Theorem</a></figcaption></figure> Notable historical conjectures were finally proven. In 1976, <a href="/wiki/Wolfgang_Haken" title="Wolfgang Haken">Wolfgang Haken</a> and <a href="/wiki/Kenneth_Appel" title="Kenneth Appel">Kenneth Appel</a> proved the <a href="/wiki/Four_color_theorem" title="Four color theorem">four color theorem</a>, controversial at the time for the use of a computer to do so.<a href="#cite_note-207">[206]</a> <a href="/wiki/Andrew_Wiles" title="Andrew Wiles">Andrew Wiles</a>, building on the work of others, proved <a href="/wiki/Fermat%27s_Last_Theorem" title="Fermat's Last Theorem">Fermat's Last Theorem</a> in 1995.<a href="#cite_note-208">[207]</a> <a href="/wiki/Paul_Cohen_(mathematician)" class="mw-redirect" title="Paul Cohen (mathematician)">Paul Cohen</a> and <a href="/wiki/Kurt_G%C3%B6del" title="Kurt Gödel">Kurt Gödel</a> proved that the <a href="/wiki/Continuum_hypothesis" title="Continuum hypothesis">continuum hypothesis</a> is <a href="/wiki/Logical_independence" class="mw-redirect" title="Logical independence">independent</a> of (could neither be proved nor disproved from) the <a href="/wiki/ZFC" class="mw-redirect" title="ZFC">standard axioms of set theory</a>.<a href="#cite_note-209">[208]</a> In 1998, <a href="/wiki/Thomas_Callister_Hales" title="Thomas Callister Hales">Thomas Callister Hales</a> proved the <a href="/wiki/Kepler_conjecture" title="Kepler conjecture">Kepler conjecture</a>, also using a computer.<a href="#cite_note-210">[209]</a> Mathematical collaborations of unprecedented size and scope took place. An example is the <a href="/wiki/Classification_of_finite_simple_groups" title="Classification of finite simple groups">classification of finite simple groups</a> (also called the "enormous theorem"), whose proof between 1955 and 2004 required 500-odd journal articles by about 100 authors, and filling tens of thousands of pages.<a href="#cite_note-211">[210]</a> A group of French mathematicians, including <a href="/wiki/Jean_Dieudonn%C3%A9" title="Jean Dieudonné">Jean Dieudonné</a> and <a href="/wiki/Andr%C3%A9_Weil" title="André Weil">André Weil</a>, publishing under the <a href="/wiki/Pseudonym" title="Pseudonym">pseudonym</a> "<a href="/wiki/Nicolas_Bourbaki" title="Nicolas Bourbaki">Nicolas Bourbaki</a>", attempted to exposit all of known mathematics as a coherent rigorous whole. The resulting several dozen volumes has had a controversial influence on mathematical education.<a href="#cite_note-212">[211]</a> <figure class="mw-default-size" typeof="mw:File/Thumb"><a href="/wiki/File:Relativistic_precession.svg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/2/28/Relativistic_precession.svg/220px-Relativistic_precession.svg.png" decoding="async" width="220" height="204" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/2/28/Relativistic_precession.svg/330px-Relativistic_precession.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/2/28/Relativistic_precession.svg/440px-Relativistic_precession.svg.png 2x" data-file-width="420" data-file-height="390" /></a><figcaption>Newtonian (red) vs. Einsteinian orbit (blue) of a lone planet orbiting a star, with <a href="/wiki/General_relativity#Orbital_effects_and_the_relativity_of_direction" title="General relativity">relativistic precession of apsides</a></figcaption></figure> <a href="/wiki/Differential_geometry" title="Differential geometry">Differential geometry</a> came into its own when <a href="/wiki/Albert_Einstein" title="Albert Einstein">Albert Einstein</a> used it in <a href="/wiki/General_relativity" title="General relativity">general relativity</a>.[<a href="/wiki/Wikipedia:Citation_needed" title="Wikipedia:Citation needed">citation needed</a>] Entirely new areas of mathematics such as <a href="/wiki/Mathematical_logic" title="Mathematical logic">mathematical logic</a>, <a href="/wiki/Topology" title="Topology">topology</a>, and <a href="/wiki/John_von_Neumann" title="John von Neumann">John von Neumann</a>'s <a href="/wiki/Game_theory" title="Game theory">game theory</a> changed the kinds of questions that could be answered by mathematical methods.[<a href="/wiki/Wikipedia:Citation_needed" title="Wikipedia:Citation needed">citation needed</a>] All kinds of <a href="/wiki/Mathematical_structure" title="Mathematical structure">structures</a> were abstracted using axioms and given names like <a href="/wiki/Metric_space" title="Metric space">metric spaces</a>, <a href="/wiki/Topological_space" title="Topological space">topological spaces</a> etc.[<a href="/wiki/Wikipedia:Citation_needed" title="Wikipedia:Citation needed">citation needed</a>] As mathematicians do, the concept of an abstract structure was itself abstracted and led to <a href="/wiki/Category_theory" title="Category theory">category theory</a>.[<a href="/wiki/Wikipedia:Citation_needed" title="Wikipedia:Citation needed">citation needed</a>] <a href="/wiki/Grothendieck" class="mw-redirect" title="Grothendieck">Grothendieck</a> and <a href="/wiki/Jean-Pierre_Serre" title="Jean-Pierre Serre">Serre</a> recast <a href="/wiki/Algebraic_geometry" title="Algebraic geometry">algebraic geometry</a> using <a href="/wiki/Sheaf_(mathematics)" title="Sheaf (mathematics)">sheaf theory</a>.[<a href="/wiki/Wikipedia:Citation_needed" title="Wikipedia:Citation needed">citation needed</a>] Large advances were made in the qualitative study of <a href="/wiki/Dynamical_systems_theory" title="Dynamical systems theory">dynamical systems</a> that <a href="/wiki/Henri_Poincar%C3%A9" title="Henri Poincaré">Poincaré</a> had begun in the 1890s.[<a href="/wiki/Wikipedia:Citation_needed" title="Wikipedia:Citation needed">citation needed</a>] <a href="/wiki/Measure_theory" class="mw-redirect" title="Measure theory">Measure theory</a> was developed in the late 19th and early 20th centuries. Applications of measures include the <a href="/wiki/Lebesgue_integral" title="Lebesgue integral">Lebesgue integral</a>, <a href="/wiki/Kolmogorov" class="mw-redirect" title="Kolmogorov">Kolmogorov</a>'s axiomatisation of <a href="/wiki/Probability_theory" title="Probability theory">probability theory</a>, and <a href="/wiki/Ergodic_theory" title="Ergodic theory">ergodic theory</a>.[<a href="/wiki/Wikipedia:Citation_needed" title="Wikipedia:Citation needed">citation needed</a>] <a href="/wiki/Knot_theory" title="Knot theory">Knot theory</a> greatly expanded.[<a href="/wiki/Wikipedia:Citation_needed" title="Wikipedia:Citation needed">citation needed</a>] <a href="/wiki/Quantum_mechanics" title="Quantum mechanics">Quantum mechanics</a> led to the development of <a href="/wiki/Functional_analysis" title="Functional analysis">functional analysis</a>,[<a href="/wiki/Wikipedia:Citation_needed" title="Wikipedia:Citation needed">citation needed</a>] a branch of mathematics that was greatly developed by <a href="/wiki/Stefan_Banach" title="Stefan Banach">Stefan Banach</a> and his collaborators who formed the <a href="/wiki/Lw%C3%B3w_School_of_Mathematics" title="Lwów School of Mathematics">Lwów School of Mathematics</a>.<a href="#cite_note-213">[212]</a> Other new areas include <a href="/wiki/Laurent_Schwartz" title="Laurent Schwartz">Laurent Schwartz</a>'s <a href="/wiki/Distribution_(mathematics)" title="Distribution (mathematics)">distribution theory</a>, <a href="/wiki/Fixed-point_theorem" title="Fixed-point theorem">fixed point theory</a>, <a href="/wiki/Singularity_theory" title="Singularity theory">singularity theory</a> and <a href="/wiki/Ren%C3%A9_Thom" title="René Thom">René Thom</a>'s <a href="/wiki/Catastrophe_theory" title="Catastrophe theory">catastrophe theory</a>, <a href="/wiki/Model_theory" title="Model theory">model theory</a>, and <a href="/wiki/Benoit_Mandelbrot" title="Benoit Mandelbrot">Mandelbrot</a>'s <a href="/wiki/Fractals" class="mw-redirect" title="Fractals">fractals</a>.[<a href="/wiki/Wikipedia:Citation_needed" title="Wikipedia:Citation needed">citation needed</a>] <a href="/wiki/Lie_theory" title="Lie theory">Lie theory</a> with its <a href="/wiki/Lie_group" title="Lie group">Lie groups</a> and <a href="/wiki/Lie_algebra" title="Lie algebra">Lie algebras</a> became one of the major areas of study.<a href="#cite_note-214">[213]</a> <a href="/wiki/Non-standard_analysis" class="mw-redirect" title="Non-standard analysis">Non-standard analysis</a>, introduced by <a href="/wiki/Abraham_Robinson" title="Abraham Robinson">Abraham Robinson</a>, rehabilitated the <a href="/wiki/Infinitesimal" title="Infinitesimal">infinitesimal</a> approach to calculus, which had fallen into disrepute in favour of the theory of <a href="/wiki/Limit_of_a_function" title="Limit of a function">limits</a>, by extending the field of real numbers to the <a href="/wiki/Hyperreal_number" title="Hyperreal number">Hyperreal numbers</a> which include infinitesimal and infinite quantities.[<a href="/wiki/Wikipedia:Citation_needed" title="Wikipedia:Citation needed">citation needed</a>] An even larger number system, the <a href="/wiki/Surreal_number" title="Surreal number">surreal numbers</a> were discovered by <a href="/wiki/John_Horton_Conway" title="John Horton Conway">John Horton Conway</a> in connection with <a href="/wiki/Combinatorial_game" class="mw-redirect" title="Combinatorial game">combinatorial games</a>.[<a href="/wiki/Wikipedia:Citation_needed" title="Wikipedia:Citation needed">citation needed</a>] The development and continual improvement of <a href="/wiki/Computer" title="Computer">computers</a>, at first mechanical analog machines and then digital electronic machines, allowed <a href="/wiki/Private_industry" class="mw-redirect" title="Private industry">industry</a> to deal with larger and larger amounts of data to facilitate mass production and distribution and communication, and new areas of mathematics were developed to deal with this: <a href="/wiki/Alan_Turing" title="Alan Turing">Alan Turing</a>'s <a href="/wiki/Computability_theory" title="Computability theory">computability theory</a>; <a href="/wiki/Computational_complexity_theory" title="Computational complexity theory">complexity theory</a>; <a href="/wiki/Derrick_Henry_Lehmer" class="mw-redirect" title="Derrick Henry Lehmer">Derrick Henry Lehmer</a>'s use of <a href="/wiki/ENIAC" title="ENIAC">ENIAC</a> to further number theory and the <a href="/wiki/Lucas%E2%80%93Lehmer_primality_test" title="Lucas–Lehmer primality test">Lucas–Lehmer primality test</a>; <a href="/wiki/R%C3%B3zsa_P%C3%A9ter" title="Rózsa Péter">Rózsa Péter</a>'s <a href="/wiki/Recursive_function_theory" class="mw-redirect" title="Recursive function theory">recursive function theory</a>; <a href="/wiki/Claude_Shannon" title="Claude Shannon">Claude Shannon</a>'s <a href="/wiki/Information_theory" title="Information theory">information theory</a>; <a href="/wiki/Signal_processing" title="Signal processing">signal processing</a>; <a href="/wiki/Data_analysis" title="Data analysis">data analysis</a>; <a href="/wiki/Mathematical_optimization" title="Mathematical optimization">optimization</a> and other areas of <a href="/wiki/Operations_research" title="Operations research">operations research</a>.[<a href="/wiki/Wikipedia:Citation_needed" title="Wikipedia:Citation needed">citation needed</a>] In the preceding centuries much mathematical focus was on calculus and continuous functions, but the rise of computing and communication networks led to an increasing importance of <a href="/wiki/Discrete_mathematics" title="Discrete mathematics">discrete</a> concepts and the expansion of <a href="/wiki/Combinatorics" title="Combinatorics">combinatorics</a> including <a href="/wiki/Graph_theory" title="Graph theory">graph theory</a>. The speed and data processing abilities of computers also enabled the handling of mathematical problems that were too time-consuming to deal with by pencil and paper calculations, leading to areas such as <a href="/wiki/Numerical_analysis" title="Numerical analysis">numerical analysis</a> and <a href="/wiki/Symbolic_computation" class="mw-redirect" title="Symbolic computation">symbolic computation</a>.[<a href="/wiki/Wikipedia:Citation_needed" title="Wikipedia:Citation needed">citation needed</a>] Some of the most important methods and <a href="/wiki/Algorithm" title="Algorithm">algorithms</a> of the 20th century are: the <a href="/wiki/Simplex_algorithm" title="Simplex algorithm">simplex algorithm</a>, the <a href="/wiki/Fast_Fourier_transform" title="Fast Fourier transform">fast Fourier transform</a>, <a href="/wiki/Error-correcting_code" class="mw-redirect" title="Error-correcting code">error-correcting codes</a>, the <a href="/wiki/Kalman_filter" title="Kalman filter">Kalman filter</a> from <a href="/wiki/Control_theory" title="Control theory">control theory</a> and the <a href="/wiki/RSA_algorithm" class="mw-redirect" title="RSA algorithm">RSA algorithm</a> of <a href="/wiki/Public-key_cryptography" title="Public-key cryptography">public-key cryptography</a>.[<a href="/wiki/Wikipedia:Citation_needed" title="Wikipedia:Citation needed">citation needed</a>] At the same time, deep insights were made about the limitations to mathematics. In 1929 and 1930, it was proved[<a href="/wiki/Wikipedia:Manual_of_Style/Words_to_watch#Unsupported_attributions" title="Wikipedia:Manual of Style/Words to watch">by whom?</a>] the truth or falsity of all statements formulated about the <a href="/wiki/Natural_number" title="Natural number">natural numbers</a> plus either addition or multiplication (but not both), was <a href="/wiki/Decidability_(logic)" title="Decidability (logic)">decidable</a>, i.e. could be determined by some algorithm.[<a href="/wiki/Wikipedia:Citation_needed" title="Wikipedia:Citation needed">citation needed</a>] In 1931, <a href="/wiki/Kurt_G%C3%B6del" title="Kurt Gödel">Kurt Gödel</a> found that this was not the case for the natural numbers plus both addition and multiplication; this system, known as <a href="/wiki/Peano_arithmetic" class="mw-redirect" title="Peano arithmetic">Peano arithmetic</a>, was in fact <a href="/wiki/Incompleteness_theorem" class="mw-redirect" title="Incompleteness theorem">incomplete</a>. (Peano arithmetic is adequate for a good deal of <a href="/wiki/Number_theory" title="Number theory">number theory</a>, including the notion of <a href="/wiki/Prime_number" title="Prime number">prime number</a>.) A consequence of Gödel's two <a href="/wiki/Incompleteness_theorem" class="mw-redirect" title="Incompleteness theorem">incompleteness theorems</a> is that in any mathematical system that includes Peano arithmetic (including all of <a href="/wiki/Mathematical_analysis" title="Mathematical analysis">analysis</a> and geometry), truth necessarily outruns proof, i.e. there are true statements that <a href="/wiki/Incompleteness_theorem" class="mw-redirect" title="Incompleteness theorem">cannot be proved</a> within the system. Hence mathematics cannot be reduced to mathematical logic, and <a href="/wiki/David_Hilbert" title="David Hilbert">David Hilbert</a>'s dream of making all of mathematics complete and consistent needed to be reformulated.[<a href="/wiki/Wikipedia:Citation_needed" title="Wikipedia:Citation needed">citation needed</a>] <figure class="mw-default-size mw-halign-right" typeof="mw:File/Thumb"><a href="/wiki/File:GammaAbsSmallPlot.png" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/c/cc/GammaAbsSmallPlot.png/220px-GammaAbsSmallPlot.png" decoding="async" width="220" height="179" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/c/cc/GammaAbsSmallPlot.png/330px-GammaAbsSmallPlot.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/c/cc/GammaAbsSmallPlot.png/440px-GammaAbsSmallPlot.png 2x" data-file-width="700" data-file-height="570" /></a><figcaption>The <a href="/wiki/Absolute_value" title="Absolute value">absolute value</a> of the Gamma function on the complex plane</figcaption></figure> One of the more colorful figures in 20th-century mathematics was <a href="/wiki/Srinivasa_Aiyangar_Ramanujan" class="mw-redirect" title="Srinivasa Aiyangar Ramanujan">Srinivasa Aiyangar Ramanujan</a> (1887–1920), an Indian <a href="/wiki/Autodidact" class="mw-redirect" title="Autodidact">autodidact</a><a href="#cite_note-:3-215">[214]</a> who conjectured or proved over 3000 theorems[<a href="/wiki/Wikipedia:Citation_needed" title="Wikipedia:Citation needed">citation needed</a>], including properties of <a href="/wiki/Highly_composite_number" title="Highly composite number">highly composite numbers</a>,<a href="#cite_note-216">[215]</a> the <a href="/wiki/Partition_function_(number_theory)" title="Partition function (number theory)">partition function</a><a href="#cite_note-:3-215">[214]</a> and its <a href="/wiki/Asymptotics" class="mw-redirect" title="Asymptotics">asymptotics</a>,<a href="#cite_note-217">[216]</a> and <a href="/wiki/Ramanujan_theta_function" title="Ramanujan theta function">mock theta functions</a>.<a href="#cite_note-:3-215">[214]</a> He also made major investigations in the areas of <a href="/wiki/Gamma_function" title="Gamma function">gamma functions</a>,<a href="#cite_note-218">[217]</a><a href="#cite_note-219">[218]</a> <a href="/wiki/Modular_form" title="Modular form">modular forms</a>,<a href="#cite_note-:3-215">[214]</a> <a href="/wiki/Divergent_series" title="Divergent series">divergent series</a>,<a href="#cite_note-:3-215">[214]</a> <a href="/wiki/General_hypergeometric_function" title="General hypergeometric function">hypergeometric series</a><a href="#cite_note-:3-215">[214]</a> and prime number theory.<a href="#cite_note-:3-215">[214]</a> <a href="/wiki/Paul_Erd%C5%91s" title="Paul Erdős">Paul Erdős</a> published more papers than any other mathematician in history,<a href="#cite_note-220">[219]</a> working with hundreds of collaborators. Mathematicians have a game equivalent to the <a href="/wiki/Kevin_Bacon_Game" class="mw-redirect" title="Kevin Bacon Game">Kevin Bacon Game</a>, which leads to the <a href="/wiki/Erd%C5%91s_number" title="Erdős number">Erdős number</a> of a mathematician. This describes the "collaborative distance" between a person and Erdős, as measured by joint authorship of mathematical papers.<a href="#cite_note-221">[220]</a><a href="#cite_note-222">[221]</a> <a href="/wiki/Emmy_Noether" title="Emmy Noether">Emmy Noether</a> has been described by many as the most important woman in the history of mathematics.<a href="#cite_note-223">[222]</a> She studied the theories of <a href="/wiki/Ring_(mathematics)" title="Ring (mathematics)">rings</a>, <a href="/wiki/Field_(mathematics)" title="Field (mathematics)">fields</a>, and <a href="/wiki/Algebra_over_a_field" title="Algebra over a field">algebras</a>.<a href="#cite_note-224">[223]</a> As in most areas of study, the explosion of knowledge in the scientific age has led to specialization: by the end of the century, there were hundreds of specialized areas in mathematics, and the <a href="/wiki/Mathematics_Subject_Classification" title="Mathematics Subject Classification">Mathematics Subject Classification</a> was dozens of pages long.<a href="#cite_note-225">[224]</a> More and more <a href="/wiki/Mathematical_journal" class="mw-redirect" title="Mathematical journal">mathematical journals</a> were published and, by the end of the century, the development of the <a href="/wiki/World_Wide_Web" title="World Wide Web">World Wide Web</a> led to online publishing.[<a href="/wiki/Wikipedia:Citation_needed" title="Wikipedia:Citation needed">citation needed</a>] <div class="mw-heading mw-heading3"><h3 id="21st_century">21st century</h3>[<a href="/w/index.php?title=History_of_mathematics&action=edit&section=19" title="Edit section: 21st century">edit</a>]</div> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1236090951"><div role="note" class="hatnote navigation-not-searchable">See also: <a href="/wiki/List_of_unsolved_problems_in_mathematics#Problems_solved_since_1995" title="List of unsolved problems in mathematics">List of unsolved problems in mathematics § Problems solved since 1995</a></div> In 2000, the <a href="/wiki/Clay_Mathematics_Institute" title="Clay Mathematics Institute">Clay Mathematics Institute</a> announced the seven <a href="/wiki/Millennium_Prize_Problems" title="Millennium Prize Problems">Millennium Prize Problems</a>.<a href="#cite_note-226">[225]</a> In 2003 the <a href="/wiki/Poincar%C3%A9_conjecture" title="Poincaré conjecture">Poincaré conjecture</a> was solved by <a href="/wiki/Grigori_Perelman" title="Grigori Perelman">Grigori Perelman</a> (who declined to accept an award, as he was critical of the mathematics establishment).<a href="#cite_note-227">[226]</a> Most mathematical journals now have online versions as well as print versions, and many online-only journals are launched.<a href="#cite_note-228">[227]</a><a href="#cite_note-229">[228]</a> There is an increasing drive toward <a href="/wiki/Open_access_(publishing)" class="mw-redirect" title="Open access (publishing)">open access publishing</a>, first made popular by <a href="/wiki/ArXiv" title="ArXiv">arXiv</a>.[<a href="/wiki/Wikipedia:Citation_needed" title="Wikipedia:Citation needed">citation needed</a>] <div class="mw-heading mw-heading2"><h2 id="Future">Future</h2>[<a href="/w/index.php?title=History_of_mathematics&action=edit&section=20" title="Edit section: Future">edit</a>]</div> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1236090951"><div role="note" class="hatnote navigation-not-searchable">Main article: <a href="/wiki/Future_of_mathematics" title="Future of mathematics">Future of mathematics</a></div> There are many observable trends in mathematics, the most notable being that the subject is growing ever larger as computers are ever more important and powerful; the volume of data being produced by science and industry, facilitated by computers, continues expanding exponentially. As a result, there is a corresponding growth in the demand for mathematics to help process and understand this <a href="/wiki/Big_data" title="Big data">big data</a>.<a href="#cite_note-230">[229]</a> Math science careers are also expected to continue to grow, with the US <a href="/wiki/Bureau_of_Labor_Statistics" title="Bureau of Labor Statistics">Bureau of Labor Statistics</a> estimating (in 2018) that "employment of mathematical science occupations is projected to grow 27.9 percent from 2016 to 2026."<a href="#cite_note-231">[230]</a> <div class="mw-heading mw-heading2"><h2 id="See_also">See also</h2>[<a href="/w/index.php?title=History_of_mathematics&action=edit&section=21" title="Edit section: See also">edit</a>]</div> <style data-mw-deduplicate="TemplateStyles:r1259569809">.mw-parser-output .portalbox{padding:0;margin:0.5em 0;display:table;box-sizing:border-box;max-width:175px;list-style:none}.mw-parser-output .portalborder{border:1px solid 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src="//upload.wikimedia.org/wikipedia/commons/thumb/3/3e/Nuvola_apps_edu_mathematics_blue-p.svg/28px-Nuvola_apps_edu_mathematics_blue-p.svg.png" decoding="async" width="28" height="28" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/3/3e/Nuvola_apps_edu_mathematics_blue-p.svg/42px-Nuvola_apps_edu_mathematics_blue-p.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/3/3e/Nuvola_apps_edu_mathematics_blue-p.svg/56px-Nuvola_apps_edu_mathematics_blue-p.svg.png 2x" data-file-width="128" data-file-height="128" /></a><a href="/wiki/Portal:Mathematics" title="Portal:Mathematics">Mathematics portal</a></li></ul><style data-mw-deduplicate="TemplateStyles:r1184024115">.mw-parser-output .div-col{margin-top:0.3em;column-width:30em}.mw-parser-output .div-col-small{font-size:90%}.mw-parser-output .div-col-rules{column-rule:1px solid #aaa}.mw-parser-output .div-col dl,.mw-parser-output .div-col ol,.mw-parser-output .div-col 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 class="div-col" style="column-width: 20em;"> <ul><li><a href="/wiki/Archives_of_American_Mathematics" title="Archives of American Mathematics">Archives of American Mathematics</a></li> <li><a href="/wiki/Ethnomathematics" title="Ethnomathematics">Ethnomathematics</a></li> <li><a href="/wiki/History_of_algebra" title="History of algebra">History of algebra</a></li> <li><a href="/wiki/History_of_arithmetic" class="mw-redirect" title="History of arithmetic">History of arithmetic</a></li> <li><a href="/wiki/History_of_calculus" title="History of calculus">History of calculus</a></li> <li><a href="/wiki/History_of_combinatorics" title="History of combinatorics">History of combinatorics</a></li> <li><a href="/wiki/History_of_the_function_concept" title="History of the function concept">History of the function concept</a></li> <li><a href="/wiki/History_of_geometry" title="History of geometry">History of geometry</a></li> <li><a href="/wiki/History_of_group_theory" title="History of group theory">History of group theory</a></li> <li><a href="/wiki/History_of_logic" title="History of logic">History of logic</a></li> <li><a href="/wiki/History_of_mathematicians" class="mw-redirect" title="History of mathematicians">History of mathematicians</a></li> <li><a href="/wiki/History_of_mathematical_notation" title="History of mathematical notation">History of mathematical notation</a></li> <li><a href="/wiki/History_of_measurement" title="History of measurement">History of measurement</a></li> <li><a href="/wiki/History_of_numbers" class="mw-redirect" title="History of numbers">History of numbers</a> <ul><li><a href="/wiki/History_of_ancient_numeral_systems" title="History of ancient numeral systems">History of ancient numeral systems</a></li> <li><a href="/wiki/Prehistoric_counting" title="Prehistoric counting">Prehistoric counting</a></li></ul></li> <li><a href="/wiki/History_of_number_theory" class="mw-redirect" title="History of number theory">History of number theory</a></li> <li><a href="/wiki/History_of_statistics" title="History of statistics">History of statistics</a></li> <li><a href="/wiki/History_of_trigonometry" title="History of trigonometry">History of trigonometry</a></li> <li><a href="/wiki/History_of_writing_numbers" class="mw-redirect" title="History of writing numbers">History of writing numbers</a></li> <li><a href="/wiki/Kenneth_O._May_Prize" title="Kenneth O. May Prize">Kenneth O. May Prize</a></li> <li><a href="/wiki/List_of_important_publications_in_mathematics" title="List of important publications in mathematics">List of important publications in mathematics</a></li> <li><a href="/wiki/Lists_of_mathematicians" title="Lists of mathematicians">Lists of mathematicians</a></li> <li><a href="/wiki/List_of_mathematics_history_topics" title="List of mathematics history topics">List of mathematics history topics</a></li> <li><a href="/wiki/Mathematical_folklore" title="Mathematical folklore">Mathematical folklore</a></li> <li><a href="/wiki/Timeline_of_mathematics" title="Timeline of mathematics">Timeline of mathematics</a></li></ul> </div> <div class="mw-heading mw-heading2"><h2 id="Notes">Notes</h2>[<a href="/w/index.php?title=History_of_mathematics&action=edit&section=22" title="Edit section: Notes">edit</a>]</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 reflist-lower-alpha"> <div class="mw-references-wrap"><ol class="references"> <li id="cite_note-131"><a href="#cite_ref-131">^</a> The approximate values for π are 4 x (13/15)2 (3.0044...), 25/8 (3.125), 900/289 (3.11418685...), 1156/361 (3.202216...), and 339/108 (3.1389) </li> </ol></div></div> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1239543626"><div class="reflist"> <div class="mw-references-wrap mw-references-columns"><ol class="references"> <li id="cite_note-Boyer_1991_loc=Euclid_of_Alexandria_p._119-1">^ <a href="#cite_ref-Boyer_1991_loc=Euclid_of_Alexandria_p._119_1-0">a</a> <a href="#cite_ref-Boyer_1991_loc=Euclid_of_Alexandria_p._119_1-1">b</a> (<a href="#CITEREFBoyer1991">Boyer 1991</a>, "Euclid of Alexandria" p. 119) </li> <li id="cite_note-:0-2"><a href="#cite_ref-:0_2-0">^</a> Friberg, J. (1981). "Methods and traditions of Babylonian mathematics. Plimpton 322, Pythagorean triples, and the Babylonian triangle parameter equations", Historia Mathematica, 8, pp. 277–318. </li> <li id="cite_note-:1-3"><a href="#cite_ref-:1_3-0">^</a> <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="CITEREFNeugebauer1969" class="citation book cs1"><a href="/wiki/Otto_E._Neugebauer" title="Otto E. Neugebauer">Neugebauer, Otto</a> (1969) [1957]. <a rel="nofollow" class="external text" href="https://books.google.com/books?id=JVhTtVA2zr8C">The Exact Sciences in Antiquity</a>. Vol. 9 (2 ed.). <a href="/wiki/Dover_Publications" title="Dover Publications">Dover Publications</a>. pp. 1–191. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-0-486-22332-2" title="Special:BookSources/978-0-486-22332-2"><bdi>978-0-486-22332-2</bdi></a>. <a href="/wiki/PMID_(identifier)" class="mw-redirect" title="PMID (identifier)">PMID</a> <a rel="nofollow" class="external text" href="https://pubmed.ncbi.nlm.nih.gov/14884919">14884919</a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=The+Exact+Sciences+in+Antiquity&rft.pages=1-191&rft.edition=2&rft.pub=Dover+Publications&rft.date=1969&rft_id=info%3Apmid%2F14884919&rft.isbn=978-0-486-22332-2&rft.aulast=Neugebauer&rft.aufirst=Otto&rft_id=https%3A%2F%2Fbooks.google.com%2Fbooks%3Fid%3DJVhTtVA2zr8C&rfr_id=info%3Asid%2Fen.wikipedia.org%3AHistory+of+mathematics" class="Z3988"> <code class="cs1-code">{{<a href="/wiki/Template:Cite_book" title="Template:Cite book">cite book</a>}}</code>: <code class="cs1-code">|journal=</code> ignored (<a href="/wiki/Help:CS1_errors#periodical_ignored" title="Help:CS1 errors">help</a>) Chap. IV "Egyptian Mathematics and Astronomy", pp. 71–96. </li> <li id="cite_note-4"><a href="#cite_ref-4">^</a> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFTurnbull1931" class="citation journal cs1">Turnbull (1931). "A Manual of Greek Mathematics". Nature. 128 (3235): 5. <a href="/wiki/Bibcode_(identifier)" class="mw-redirect" title="Bibcode (identifier)">Bibcode</a>:<a rel="nofollow" class="external text" href="https://ui.adsabs.harvard.edu/abs/1931Natur.128..739T">1931Natur.128..739T</a>. <a href="/wiki/Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1038%2F128739a0">10.1038/128739a0</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:3994109">3994109</a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=Nature&rft.atitle=A+Manual+of+Greek+Mathematics&rft.volume=128&rft.issue=3235&rft.pages=5&rft.date=1931&rft_id=https%3A%2F%2Fapi.semanticscholar.org%2FCorpusID%3A3994109%23id-name%3DS2CID&rft_id=info%3Adoi%2F10.1038%2F128739a0&rft_id=info%3Abibcode%2F1931Natur.128..739T&rft.au=Turnbull&rfr_id=info%3Asid%2Fen.wikipedia.org%3AHistory+of+mathematics" class="Z3988"> </li> <li id="cite_note-5"><a href="#cite_ref-5">^</a> Heath, Thomas L. (1963). A Manual of Greek Mathematics, Dover, p. 1: "In the case of mathematics, it is the Greek contribution which it is most essential to know, for it was the Greeks who first made mathematics a science." </li> <li id="cite_note-:2-6">^ <a href="#cite_ref-:2_6-0">a</a> <a href="#cite_ref-:2_6-1">b</a> Joseph, George Gheverghese (1991). The Crest of the Peacock: Non-European Roots of Mathematics. Penguin Books, London, pp. 140–48. </li> <li id="cite_note-7"><a href="#cite_ref-7">^</a> Ifrah, Georges (1986). Universalgeschichte der Zahlen. Campus, Frankfurt/New York, pp. 428–37. </li> <li id="cite_note-8"><a href="#cite_ref-8">^</a> Kaplan, Robert (1999). The Nothing That Is: A Natural History of Zero. Allen Lane/The Penguin Press, London. </li> <li id="cite_note-9"><a href="#cite_ref-9">^</a> "The ingenious method of expressing every possible number using a set of ten symbols (each symbol having a place value and an absolute value) emerged in India. The idea seems so simple nowadays that its significance and profound importance is no longer appreciated. Its simplicity lies in the way it facilitated calculation and placed arithmetic foremost amongst useful inventions. the importance of this invention is more readily appreciated when one considers that it was beyond the two greatest men of Antiquity, Archimedes and Apollonius." – Pierre Simon Laplace <a rel="nofollow" class="external free" href="http://www-history.mcs.st-and.ac.uk/HistTopics/Indian_numerals.html">http://www-history.mcs.st-and.ac.uk/HistTopics/Indian_numerals.html</a> </li> <li id="cite_note-10"><a href="#cite_ref-10">^</a> <a href="/wiki/Adolf_Yushkevich" class="mw-redirect" title="Adolf Yushkevich">Juschkewitsch, A. P.</a> (1964). Geschichte der Mathematik im Mittelalter. Teubner, Leipzig. </li> <li id="cite_note-11"><a href="#cite_ref-11">^</a> Eves, Howard (1990). History of Mathematics, 6th Edition, "After Pappus, Greek mathematics ceased to be a living study, ..." p. 185; "The Athenian school struggled on against growing opposition from Christians until the latter finally, in A.D. 529, obtained a decree from Emperor Justinian that closed the doors of the school forever." p. 186; "The period starting with the fall of the Roman Empire, in the middle of the fifth century, and extending into the eleventh century is known in Europe as the Dark Ages... Schooling became almost nonexistent." p. 258. </li> <li id="cite_note-Boyer_1991_loc=Origins_p._3-12">^ <a href="#cite_ref-Boyer_1991_loc=Origins_p._3_12-0">a</a> <a href="#cite_ref-Boyer_1991_loc=Origins_p._3_12-1">b</a> (<a href="#CITEREFBoyer1991">Boyer 1991</a>, "Origins" p. 3) </li> <li id="cite_note-Diaspora-13"><a href="#cite_ref-Diaspora_13-0">^</a> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFWilliams2005" class="citation web cs1">Williams, Scott W. (2005). <a rel="nofollow" class="external text" href="http://www.math.buffalo.edu/mad/Ancient-Africa/lebombo.html">"The Oldest Mathematical Object is in Swaziland"</a>. Mathematicians of the African Diaspora. SUNY Buffalo mathematics department. Retrieved 2006-05-06.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=unknown&rft.jtitle=Mathematicians+of+the+African+Diaspora&rft.atitle=The+Oldest+Mathematical+Object+is+in+Swaziland&rft.date=2005&rft.aulast=Williams&rft.aufirst=Scott+W.&rft_id=http%3A%2F%2Fwww.math.buffalo.edu%2Fmad%2FAncient-Africa%2Flebombo.html&rfr_id=info%3Asid%2Fen.wikipedia.org%3AHistory+of+mathematics" class="Z3988"> </li> <li id="cite_note-Marshack-14"><a href="#cite_ref-Marshack_14-0">^</a> Marshack, Alexander (1991). The Roots of Civilization, Colonial Hill, Mount Kisco, NY. </li> <li id="cite_note-15"><a href="#cite_ref-15">^</a> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFRudman2007" class="citation book cs1">Rudman, Peter Strom (2007). <a rel="nofollow" class="external text" href="https://archive.org/details/howmathematicsha0000rudm/page/64">How Mathematics Happened: The First 50,000 Years</a>. Prometheus Books. p. <a rel="nofollow" class="external text" href="https://archive.org/details/howmathematicsha0000rudm/page/64">64</a>. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-1-59102-477-4" title="Special:BookSources/978-1-59102-477-4"><bdi>978-1-59102-477-4</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=How+Mathematics+Happened%3A+The+First+50%2C000+Years&rft.pages=64&rft.pub=Prometheus+Books&rft.date=2007&rft.isbn=978-1-59102-477-4&rft.aulast=Rudman&rft.aufirst=Peter+Strom&rft_id=https%3A%2F%2Farchive.org%2Fdetails%2Fhowmathematicsha0000rudm%2Fpage%2F64&rfr_id=info%3Asid%2Fen.wikipedia.org%3AHistory+of+mathematics" class="Z3988"> </li> <li id="cite_note-16"><a href="#cite_ref-16">^</a> Marshack, A. (1972). The Roots of Civilization: the Cognitive Beginning of Man's First Art, Symbol and Notation. New York: McGraw-Hill. </li> <li id="cite_note-17"><a href="#cite_ref-17">^</a> Thom, Alexander; Archie Thom (1988). "The metrology and geometry of Megalithic Man", pp. 132–51 in Ruggles, C. L. N. (ed.), Records in Stone: Papers in memory of Alexander Thom. Cambridge University Press. <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/0-521-33381-4" title="Special:BookSources/0-521-33381-4">0-521-33381-4</a>. </li> <li id="cite_note-18"><a href="#cite_ref-18">^</a> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFDamerow1996" class="citation book cs1">Damerow, Peter (1996). <a rel="nofollow" class="external text" href="https://books.google.com/books?id=c4yBmjnY1JIC&pg=PA199">"The Development of Arithmetical Thinking: On the Role of Calculating Aids in Ancient Egyptian & Babylonian Arithmetic"</a>. Abstraction & Representation: Essays on the Cultural Evolution of Thinking (Boston Studies in the Philosophy & History of Science). Springer. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/0792338162" title="Special:BookSources/0792338162"><bdi>0792338162</bdi></a>. Retrieved 2019-08-17.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=bookitem&rft.atitle=The+Development+of+Arithmetical+Thinking%3A+On+the+Role+of+Calculating+Aids+in+Ancient+Egyptian+%26+Babylonian+Arithmetic&rft.btitle=Abstraction+%26+Representation%3A+Essays+on+the+Cultural+Evolution+of+Thinking+%28Boston+Studies+in+the+Philosophy+%26+History+of+Science%29&rft.pub=Springer&rft.date=1996&rft.isbn=0792338162&rft.aulast=Damerow&rft.aufirst=Peter&rft_id=https%3A%2F%2Fbooks.google.com%2Fbooks%3Fid%3Dc4yBmjnY1JIC%26pg%3DPA199&rfr_id=info%3Asid%2Fen.wikipedia.org%3AHistory+of+mathematics" class="Z3988"> </li> <li id="cite_note-19"><a href="#cite_ref-19">^</a> (<a href="#CITEREFBoyer1991">Boyer 1991</a>, "Mesopotamia" p. 24) </li> <li id="cite_note-Boyer_1991_loc=Mesopotamia_p._26-20">^ <a href="#cite_ref-Boyer_1991_loc=Mesopotamia_p._26_20-0">a</a> <a href="#cite_ref-Boyer_1991_loc=Mesopotamia_p._26_20-1">b</a> <a href="#cite_ref-Boyer_1991_loc=Mesopotamia_p._26_20-2">c</a> <a href="#cite_ref-Boyer_1991_loc=Mesopotamia_p._26_20-3">d</a> (<a href="#CITEREFBoyer1991">Boyer 1991</a>, "Mesopotamia" p. 26) </li> <li id="cite_note-Boyer_1991_loc=Mesopotamia_p._25-21">^ <a href="#cite_ref-Boyer_1991_loc=Mesopotamia_p._25_21-0">a</a> <a href="#cite_ref-Boyer_1991_loc=Mesopotamia_p._25_21-1">b</a> <a href="#cite_ref-Boyer_1991_loc=Mesopotamia_p._25_21-2">c</a> (<a href="#CITEREFBoyer1991">Boyer 1991</a>, "Mesopotamia" p. 25) </li> <li id="cite_note-Boyer_1991_loc=Mesopotamia_p._41-22">^ <a href="#cite_ref-Boyer_1991_loc=Mesopotamia_p._41_22-0">a</a> <a href="#cite_ref-Boyer_1991_loc=Mesopotamia_p._41_22-1">b</a> (<a href="#CITEREFBoyer1991">Boyer 1991</a>, "Mesopotamia" p. 41) </li> <li id="cite_note-23"><a href="#cite_ref-23">^</a> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFSharlach2006" class="citation cs2">Sharlach, Tonia (2006), <a rel="nofollow" class="external text" href="https://dx.doi.org/10.4324/9780203096604.ch15">"Calendars and Counting"</a>, The Sumerian World, Routledge, pp. 307–308, <a href="/wiki/Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.4324%2F9780203096604.ch15">10.4324/9780203096604.ch15</a>, <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-0-203-09660-4" title="Special:BookSources/978-0-203-09660-4"><bdi>978-0-203-09660-4</bdi></a>, retrieved 2023-07-07</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=The+Sumerian+World&rft.atitle=Calendars+and+Counting&rft.pages=307-308&rft.date=2006&rft_id=info%3Adoi%2F10.4324%2F9780203096604.ch15&rft.isbn=978-0-203-09660-4&rft.aulast=Sharlach&rft.aufirst=Tonia&rft_id=http%3A%2F%2Fdx.doi.org%2F10.4324%2F9780203096604.ch15&rfr_id=info%3Asid%2Fen.wikipedia.org%3AHistory+of+mathematics" class="Z3988"> </li> <li id="cite_note-24"><a href="#cite_ref-24">^</a> Melville, Duncan J. 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(2007). Microcontroller programming : the microchip PIC. Boca Raton, Florida: CRC Press. p. 37. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-0-8493-7189-9" title="Special:BookSources/978-0-8493-7189-9"><bdi>978-0-8493-7189-9</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Microcontroller+programming+%3A+the+microchip+PIC&rft.place=Boca+Raton%2C+Florida&rft.pages=37&rft.pub=CRC+Press&rft.date=2007&rft.isbn=978-0-8493-7189-9&rft.aulast=Sanchez&rft.aufirst=Julio&rft.au=Canton%2C+Maria+P.&rfr_id=info%3Asid%2Fen.wikipedia.org%3AHistory+of+mathematics" class="Z3988"> </li> <li id="cite_note-135"><a href="#cite_ref-135">^</a> Anglin, W. S. and J. Lambek (1995). The Heritage of Thales, Springer, <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/0-387-94544-X" title="Special:BookSources/0-387-94544-X">0-387-94544-X</a> </li> <li id="cite_note-136"><a href="#cite_ref-136">^</a> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFHall2008" class="citation journal cs1">Hall, Rachel W. (2008). <a rel="nofollow" class="external text" href="http://people.sju.edu/~rhall/mathforpoets.pdf">"Math for poets and drummers"</a> (PDF). Math Horizons. 15 (3): 10–11. <a href="/wiki/Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1080%2F10724117.2008.11974752">10.1080/10724117.2008.11974752</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:3637061">3637061</a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=Math+Horizons&rft.atitle=Math+for+poets+and+drummers&rft.volume=15&rft.issue=3&rft.pages=10-11&rft.date=2008&rft_id=info%3Adoi%2F10.1080%2F10724117.2008.11974752&rft_id=https%3A%2F%2Fapi.semanticscholar.org%2FCorpusID%3A3637061%23id-name%3DS2CID&rft.aulast=Hall&rft.aufirst=Rachel+W.&rft_id=http%3A%2F%2Fpeople.sju.edu%2F~rhall%2Fmathforpoets.pdf&rfr_id=info%3Asid%2Fen.wikipedia.org%3AHistory+of+mathematics" class="Z3988"> </li> <li id="cite_note-137"><a href="#cite_ref-137">^</a> (<a href="#CITEREFBoyer1991">Boyer 1991</a>, "China and India" p. 208) </li> <li id="cite_note-autogenerated2-138">^ <a href="#cite_ref-autogenerated2_138-0">a</a> <a href="#cite_ref-autogenerated2_138-1">b</a> (<a href="#CITEREFBoyer1991">Boyer 1991</a>, "China and India" p. 209) </li> <li id="cite_note-139"><a href="#cite_ref-139">^</a> (<a href="#CITEREFBoyer1991">Boyer 1991</a>, "China and India" p. 210) </li> <li id="cite_note-140"><a href="#cite_ref-140">^</a> (<a href="#CITEREFBoyer1991">Boyer 1991</a>, "China and India" p. 211) </li> <li id="cite_note-Boyer_Siddhanta-141"><a href="#cite_ref-Boyer_Siddhanta_141-0">^</a> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite class="citation book cs1"><a href="/wiki/Carl_Benjamin_Boyer" title="Carl Benjamin Boyer">Boyer</a> (1991). "The Arabic Hegemony". <a rel="nofollow" class="external text" href="https://archive.org/details/historyofmathema00boye">History of Mathematics</a>. Wiley. p. <a rel="nofollow" class="external text" href="https://archive.org/details/historyofmathema00boye/page/226">226</a>. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/9780471543978" title="Special:BookSources/9780471543978"><bdi>9780471543978</bdi></a>. <q>By 766 we learn that an astronomical-mathematical work, known to the Arabs as the Sindhind, was brought to Baghdad from India. It is generally thought that this was the Brahmasphuta Siddhanta, although it may have been the Surya Siddhanata. A few years later, perhaps about 775, this Siddhanata was translated into Arabic, and it was not long afterwards (ca. 780) that Ptolemy's astrological <a href="/wiki/Tetrabiblos" title="Tetrabiblos">Tetrabiblos</a> was translated into Arabic from the Greek.</q></cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=bookitem&rft.atitle=The+Arabic+Hegemony&rft.btitle=History+of+Mathematics&rft.pages=226&rft.pub=Wiley&rft.date=1991&rft.isbn=9780471543978&rft.au=Boyer&rft_id=https%3A%2F%2Farchive.org%2Fdetails%2Fhistoryofmathema00boye&rfr_id=info%3Asid%2Fen.wikipedia.org%3AHistory+of+mathematics" class="Z3988"> </li> <li id="cite_note-142"><a href="#cite_ref-142">^</a> Plofker 2009 182–207 </li> <li id="cite_note-143"><a href="#cite_ref-143">^</a> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFCooke1997" class="citation book cs1">Cooke, Roger (1997). <a rel="nofollow" class="external text" href="https://archive.org/details/historyofmathema0000cook/page/213">"The Mathematics of the Hindus"</a>. The History of Mathematics: A Brief Course. Wiley-Interscience. pp. <a rel="nofollow" class="external text" href="https://archive.org/details/historyofmathema0000cook/page/213">213–215</a>. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/0-471-18082-3" title="Special:BookSources/0-471-18082-3"><bdi>0-471-18082-3</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=bookitem&rft.atitle=The+Mathematics+of+the+Hindus&rft.btitle=The+History+of+Mathematics%3A+A+Brief+Course&rft.pages=213-215&rft.pub=Wiley-Interscience&rft.date=1997&rft.isbn=0-471-18082-3&rft.aulast=Cooke&rft.aufirst=Roger&rft_id=https%3A%2F%2Farchive.org%2Fdetails%2Fhistoryofmathema0000cook%2Fpage%2F213&rfr_id=info%3Asid%2Fen.wikipedia.org%3AHistory+of+mathematics" class="Z3988"> </li> <li id="cite_note-144"><a href="#cite_ref-144">^</a> Plofker 2009 pp. 197–98; George Gheverghese Joseph, The Crest of the Peacock: Non-European Roots of Mathematics, Penguin Books, London, 1991 pp. 298–300; Takao Hayashi, "Indian Mathematics", pp. 118–30 in Companion History of the History and Philosophy of the Mathematical Sciences, ed. I. Grattan. Guinness, Johns Hopkins University Press, Baltimore and London, 1994, p. 126. </li> <li id="cite_note-145"><a href="#cite_ref-145">^</a> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite class="citation web cs1"><a rel="nofollow" class="external text" href="https://mathshistory.st-andrews.ac.uk/Biographies/Narayana/">"Narayana - Biography"</a>. Maths History. Retrieved 2022-10-03.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=unknown&rft.jtitle=Maths+History&rft.atitle=Narayana+-+Biography&rft_id=https%3A%2F%2Fmathshistory.st-andrews.ac.uk%2FBiographies%2FNarayana%2F&rfr_id=info%3Asid%2Fen.wikipedia.org%3AHistory+of+mathematics" class="Z3988"> </li> <li id="cite_note-146"><a href="#cite_ref-146">^</a> Plofker 2009 pp. 217–53. </li> <li id="cite_note-rajujournal-147"><a href="#cite_ref-rajujournal_147-0">^</a> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFRaju,_C._K.2001" class="citation journal cs1">Raju, C. K. (2001). <a rel="nofollow" class="external text" href="http://ckraju.net/papers/Hawaii.pdf">"Computers, mathematics education, and the alternative epistemology of the calculus in the Yuktibhāṣā"</a> (PDF). Philosophy East & West. 51 (3): 325–362. <a href="/wiki/Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1353%2Fpew.2001.0045">10.1353/pew.2001.0045</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:170341845">170341845</a>. Retrieved 2020-02-11.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=Philosophy+East+%26+West&rft.atitle=Computers%2C+mathematics+education%2C+and+the+alternative+epistemology+of+the+calculus+in+the+Yuktibh%C4%81%E1%B9%A3%C4%81&rft.volume=51&rft.issue=3&rft.pages=325-362&rft.date=2001&rft_id=info%3Adoi%2F10.1353%2Fpew.2001.0045&rft_id=https%3A%2F%2Fapi.semanticscholar.org%2FCorpusID%3A170341845%23id-name%3DS2CID&rft.au=Raju%2C+C.+K.&rft_id=http%3A%2F%2Fckraju.net%2Fpapers%2FHawaii.pdf&rfr_id=info%3Asid%2Fen.wikipedia.org%3AHistory+of+mathematics" class="Z3988"> </li> <li id="cite_note-148"><a href="#cite_ref-148">^</a> Divakaran, P. P. (2007). "The first textbook of calculus: Yukti-bhāṣā", Journal of Indian Philosophy 35, pp. 417–33. </li> <li id="cite_note-almeida-149"><a href="#cite_ref-almeida_149-0">^</a> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFAlmeida,_D._F.;_J._K._John_and_A._Zadorozhnyy2001" class="citation journal cs1">Almeida, D. F.; J. K. John and A. Zadorozhnyy (2001). "Keralese mathematics: its possible transmission to Europe and the consequential educational implications". Journal of Natural Geometry. 20 (1): 77–104.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=Journal+of+Natural+Geometry&rft.atitle=Keralese+mathematics%3A+its+possible+transmission+to+Europe+and+the+consequential+educational+implications&rft.volume=20&rft.issue=1&rft.pages=77-104&rft.date=2001&rft.au=Almeida%2C+D.+F.%3B+J.+K.+John+and+A.+Zadorozhnyy&rfr_id=info%3Asid%2Fen.wikipedia.org%3AHistory+of+mathematics" class="Z3988"><code class="cs1-code">{{<a href="/wiki/Template:Cite_journal" title="Template:Cite journal">cite journal</a>}}</code>: CS1 maint: multiple names: authors list (<a href="/wiki/Category:CS1_maint:_multiple_names:_authors_list" title="Category:CS1 maint: multiple names: authors list">link</a>) </li> <li id="cite_note-150"><a href="#cite_ref-150">^</a> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFPingree1992" class="citation journal cs1"><a href="/wiki/David_Pingree" title="David Pingree">Pingree, David</a> (December 1992). "Hellenophilia versus the History of Science". Isis. 83 (4): 554–563. <a href="/wiki/Bibcode_(identifier)" class="mw-redirect" title="Bibcode (identifier)">Bibcode</a>:<a rel="nofollow" class="external text" href="https://ui.adsabs.harvard.edu/abs/1992Isis...83..554P">1992Isis...83..554P</a>. <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%2F356288">10.1086/356288</a>. <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/234257">234257</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:68570164">68570164</a>. <q>One example I can give you relates to the Indian Mādhava's demonstration, in about 1400 A.D., of the infinite power series of trigonometrical functions using geometrical and algebraic arguments. When this was first described in English by <a href="/wiki/C._M._Whish" title="C. M. Whish">Charles Whish</a>, in the 1830s, it was heralded as the Indians' discovery of the calculus. This claim and Mādhava's achievements were ignored by Western historians, presumably at first because they could not admit that an Indian discovered the calculus, but later because no one read anymore the Transactions of the Royal Asiatic Society, in which Whish's article was published. The matter resurfaced in the 1950s, and now we have the Sanskrit texts properly edited, and we understand the clever way that Mādhava derived the series without the calculus; but many historians still find it impossible to conceive of the problem and its solution in terms of anything other than the calculus and proclaim that the calculus is what Mādhava found. In this case the elegance and brilliance of Mādhava's mathematics are being distorted as they are buried under the current mathematical solution to a problem to which he discovered an alternate and powerful solution.</q></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=Hellenophilia+versus+the+History+of+Science&rft.volume=83&rft.issue=4&rft.pages=554-563&rft.date=1992-12&rft_id=info%3Adoi%2F10.1086%2F356288&rft_id=https%3A%2F%2Fapi.semanticscholar.org%2FCorpusID%3A68570164%23id-name%3DS2CID&rft_id=https%3A%2F%2Fwww.jstor.org%2Fstable%2F234257%23id-name%3DJSTOR&rft_id=info%3Abibcode%2F1992Isis...83..554P&rft.aulast=Pingree&rft.aufirst=David&rfr_id=info%3Asid%2Fen.wikipedia.org%3AHistory+of+mathematics" class="Z3988"> </li> <li id="cite_note-151"><a href="#cite_ref-151">^</a> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFBressoud2002" class="citation journal cs1"><a href="/wiki/David_Bressoud" title="David Bressoud">Bressoud, David</a> (2002). "Was Calculus Invented in India?". College Mathematics Journal. 33 (1): 2–13. <a href="/wiki/Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.2307%2F1558972">10.2307/1558972</a>. <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/1558972">1558972</a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=College+Mathematics+Journal&rft.atitle=Was+Calculus+Invented+in+India%3F&rft.volume=33&rft.issue=1&rft.pages=2-13&rft.date=2002&rft_id=info%3Adoi%2F10.2307%2F1558972&rft_id=https%3A%2F%2Fwww.jstor.org%2Fstable%2F1558972%23id-name%3DJSTOR&rft.aulast=Bressoud&rft.aufirst=David&rfr_id=info%3Asid%2Fen.wikipedia.org%3AHistory+of+mathematics" class="Z3988"> </li> <li id="cite_note-152"><a href="#cite_ref-152">^</a> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFPlofker2001" class="citation journal cs1"><a href="/wiki/Kim_Plofker" title="Kim Plofker">Plofker, Kim</a> (November 2001). <a rel="nofollow" class="external text" href="https://doi.org/10.1006%2Fhmat.2001.2331">"The 'Error' in the Indian "Taylor Series Approximation" to the Sine"</a>. Historia Mathematica. 28 (4): 293. <a href="/wiki/Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1006%2Fhmat.2001.2331">10.1006/hmat.2001.2331</a>. <q>It is not unusual to encounter in discussions of Indian mathematics such assertions as that 'the concept of differentiation was understood [in India] from the time of Manjula (... in the 10th century)' [Joseph 1991, 300], or that 'we may consider Madhava to have been the founder of mathematical analysis' (Joseph 1991, 293), or that Bhaskara II may claim to be 'the precursor of Newton and Leibniz in the discovery of the principle of the differential calculus' (Bag 1979, 294).... The points of resemblance, particularly between early European calculus and the Keralese work on power series, have even inspired suggestions of a possible transmission of mathematical ideas from the Malabar coast in or after the 15th century to the Latin scholarly world (e.g., in (Bag 1979, 285))... It should be borne in mind, however, that such an emphasis on the similarity of Sanskrit (or Malayalam) and Latin mathematics risks diminishing our ability fully to see and comprehend the former. To speak of the Indian 'discovery of the principle of the differential calculus' somewhat obscures the fact that Indian techniques for expressing changes in the Sine by means of the Cosine or vice versa, as in the examples we have seen, remained within that specific trigonometric context. The differential 'principle' was not generalized to arbitrary functions – in fact, the explicit notion of an arbitrary function, not to mention that of its derivative or an algorithm for taking the derivative, is irrelevant here</q></cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=Historia+Mathematica&rft.atitle=The+%27Error%27+in+the+Indian+%22Taylor+Series+Approximation%22+to+the+Sine&rft.volume=28&rft.issue=4&rft.pages=293&rft.date=2001-11&rft_id=info%3Adoi%2F10.1006%2Fhmat.2001.2331&rft.aulast=Plofker&rft.aufirst=Kim&rft_id=https%3A%2F%2Fdoi.org%2F10.1006%252Fhmat.2001.2331&rfr_id=info%3Asid%2Fen.wikipedia.org%3AHistory+of+mathematics" class="Z3988"> </li> <li id="cite_note-153"><a href="#cite_ref-153">^</a> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFKatz1995" class="citation journal cs1">Katz, Victor J. (June 1995). <a rel="nofollow" class="external text" href="http://www2.kenyon.edu/Depts/Math/Aydin/Teach/Fall12/128/CalcIslamIndia.pdf">"Ideas of Calculus in Islam and India"</a> (PDF). Mathematics Magazine. 68 (3): 163–74. <a href="/wiki/Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.2307%2F2691411">10.2307/2691411</a>. <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/2691411">2691411</a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=Mathematics+Magazine&rft.atitle=Ideas+of+Calculus+in+Islam+and+India&rft.volume=68&rft.issue=3&rft.pages=163-74&rft.date=1995-06&rft_id=info%3Adoi%2F10.2307%2F2691411&rft_id=https%3A%2F%2Fwww.jstor.org%2Fstable%2F2691411%23id-name%3DJSTOR&rft.aulast=Katz&rft.aufirst=Victor+J.&rft_id=http%3A%2F%2Fwww2.kenyon.edu%2FDepts%2FMath%2FAydin%2FTeach%2FFall12%2F128%2FCalcIslamIndia.pdf&rfr_id=info%3Asid%2Fen.wikipedia.org%3AHistory+of+mathematics" class="Z3988"> </li> <li id="cite_note-154"><a href="#cite_ref-154">^</a> Abdel Haleem, Muhammad A. S. "The Semitic Languages", <a rel="nofollow" class="external free" href="https://doi.org/10.1515/9783110251586.811">https://doi.org/10.1515/9783110251586.811</a>, "Arabic became the language of scholarship in science and philosophy in the 9th century when the ‘translation movement’ saw concerted work on translations of Greek, Indian, Persian and Chinese, medical, philosophical and scientific texts", p. 811. </li> <li id="cite_note-155"><a href="#cite_ref-155">^</a> (<a href="#CITEREFBoyer1991">Boyer 1991</a>, "The Arabic Hegemony" p. 230) "The six cases of equations given above exhaust all possibilities for linear and quadratic equations having positive root. So systematic and exhaustive was al-Khwārizmī's exposition that his readers must have had little difficulty in mastering the solutions." </li> <li id="cite_note-156"><a href="#cite_ref-156">^</a> Gandz and Saloman (1936). "The sources of Khwarizmi's algebra", Osiris i, pp. 263–77: "In a sense, Khwarizmi is more entitled to be called "the father of algebra" than Diophantus because Khwarizmi is the first to teach algebra in an elementary form and for its own sake, Diophantus is primarily concerned with the theory of numbers". </li> <li id="cite_note-Boyer-229-157"><a href="#cite_ref-Boyer-229_157-0">^</a> (<a href="#CITEREFBoyer1991">Boyer 1991</a>, "The Arabic Hegemony" p. 229) "It is not certain just what the terms al-jabr and muqabalah mean, but the usual interpretation is similar to that implied in the translation above. The word al-jabr presumably meant something like "restoration" or "completion" and seems to refer to the transposition of subtracted terms to the other side of an equation; the word muqabalah is said to refer to "reduction" or "balancing" – that is, the cancellation of like terms on opposite sides of the equation." </li> <li id="cite_note-Rashed-Armstrong-158"><a href="#cite_ref-Rashed-Armstrong_158-0">^</a> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFRashedArmstrong1994" class="citation book cs1">Rashed, R.; Armstrong, Angela (1994). 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Indian Institute of Social Reform & Research International Journal of Research.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=Indian+Institute+of+Social+Reform+%26+Research+International+Journal+of+Research&rft.atitle=Mathematics+for+All+and+Forever&rft.date=2015&rft.aulast=Alam&rft.aufirst=S&rft_id=http%3A%2F%2Fwww.iisrr.in%2Fmainsite%2Fwp-content%2Fuploads%2F2015%2F01%2FIISRR-IJR-1-Mathematics-for-All-...-Syed-Samsul-Alam.pdf&rfr_id=info%3Asid%2Fen.wikipedia.org%3AHistory+of+mathematics" class="Z3988"> </li> <li id="cite_note-Qalasadi-164"><a href="#cite_ref-Qalasadi_164-0">^</a> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFO'ConnorRobertson" class="citation cs2">O'Connor, John J.; <a href="/wiki/Edmund_F._Robertson" title="Edmund F. 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"The De Institutione Arithmetica and the De Institutione Musica", pp. 135–54 in <a href="/wiki/Margaret_Gibson_(historian)" title="Margaret Gibson (historian)">Margaret Gibson</a>, ed., Boethius: His Life, Thought, and Influence, (Oxford: Basil Blackwell). </li> <li id="cite_note-168"><a href="#cite_ref-168">^</a> Folkerts, Menso (1970). "Boethius" Geometrie II, Wiesbaden: Franz Steiner Verlag. </li> <li id="cite_note-169"><a href="#cite_ref-169">^</a> <a href="/wiki/Marie-Th%C3%A9r%C3%A8se_d%27Alverny" title="Marie-Thérèse d'Alverny">Marie-Thérèse d'Alverny</a>, "Translations and Translators", pp. 421–62 in Robert L. Benson and Giles Constable, Renaissance and Renewal in the Twelfth Century, (Cambridge: Harvard University Press, 1982). </li> <li id="cite_note-170"><a href="#cite_ref-170">^</a> Beaujouan, Guy. "The Transformation of the Quadrivium", pp. 463–87 in Robert L. Benson and Giles Constable, Renaissance and Renewal in the Twelfth Century. Cambridge: Harvard University Press, 1982. </li> <li id="cite_note-171"><a href="#cite_ref-171">^</a> Singh, Parmanand (1985). "The So-called Fibonacci numbers in ancient and medieval India", Historia Mathematica, 12 (3): 229–44, doi:10.1016/0315-0860(85)90021-7 </li> <li id="cite_note-172"><a href="#cite_ref-172">^</a> Grant, Edward and John E. Murdoch, eds. (1987). Mathematics and Its Applications to Science and Natural Philosophy in the Middle Ages. Cambridge: Cambridge University Press. <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/0-521-32260-X" title="Special:BookSources/0-521-32260-X">0-521-32260-X</a>. </li> <li id="cite_note-173"><a href="#cite_ref-173">^</a> Clagett, Marshall (1961). The Science of Mechanics in the Middle Ages. Madison: University of Wisconsin Press, pp. 421–40. </li> <li id="cite_note-174"><a href="#cite_ref-174">^</a> Murdoch, John E. (1969). "Mathesis in Philosophiam Scholasticam Introducta: The Rise and Development of the Application of Mathematics in Fourteenth Century Philosophy and Theology", in Arts libéraux et philosophie au Moyen Âge (Montréal: Institut d'Études Médiévales), pp. 224–27. </li> <li id="cite_note-175"><a href="#cite_ref-175">^</a> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFPickover2009" class="citation cs2"><a href="/wiki/Clifford_A._Pickover" title="Clifford A. Pickover">Pickover, Clifford A.</a> (2009), <a rel="nofollow" class="external text" href="https://books.google.com/books?id=JrslMKTgSZwC&pg=PA104">The Math Book: From Pythagoras to the 57th Dimension, 250 Milestones in the History of Mathematics</a>, Sterling Publishing Company, Inc., p. 104, <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-1-4027-5796-9" title="Special:BookSources/978-1-4027-5796-9"><bdi>978-1-4027-5796-9</bdi></a>, <q>Nicole Oresme ... was the first to prove the divergence of the harmonic series (c. 1350). His results were lost for several centuries, and the result was proved again by Italian mathematician <a href="/wiki/Pietro_Mengoli" title="Pietro Mengoli">Pietro Mengoli</a> in 1647 and by Swiss mathematician <a href="/wiki/Johann_Bernoulli" title="Johann Bernoulli">Johann Bernoulli</a> in 1687.</q></cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=The+Math+Book%3A+From+Pythagoras+to+the+57th+Dimension%2C+250+Milestones+in+the+History+of+Mathematics&rft.pages=104&rft.pub=Sterling+Publishing+Company%2C+Inc.&rft.date=2009&rft.isbn=978-1-4027-5796-9&rft.aulast=Pickover&rft.aufirst=Clifford+A.&rft_id=https%3A%2F%2Fbooks.google.com%2Fbooks%3Fid%3DJrslMKTgSZwC%26pg%3DPA104&rfr_id=info%3Asid%2Fen.wikipedia.org%3AHistory+of+mathematics" class="Z3988"> </li> <li id="cite_note-176"><a href="#cite_ref-176">^</a> extent.<a rel="nofollow" class="external free" href="https://www.scientificlib.com/en/Mathematics/Biographies/AdamRies.html#:~:text=Adam%20Ries%20is%20generally%20considered%20to%20be%20the,more%20structured%20Arabic%20numerals%20to%20a%20large%20extent">https://www.scientificlib.com/en/Mathematics/Biographies/AdamRies.html#:~:text=Adam%20Ries%20is%20generally%20considered%20to%20be%20the,more%20structured%20Arabic%20numerals%20to%20a%20large%20extent</a>. </li> <li id="cite_note-177"><a href="#cite_ref-177">^</a> Clagett, Marshall (1961). 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Madison: University of Wisconsin Press, 1968. </li> <li id="cite_note-181"><a href="#cite_ref-181">^</a> Heeffer, Albrecht: On the curious historical coincidence of algebra and double-entry bookkeeping, Foundations of the Formal Sciences, <a href="/wiki/Ghent_University" title="Ghent University">Ghent University</a>, November 2009, p. 7 <a rel="nofollow" class="external autonumber" href="http://logica.ugent.be/albrecht/thesis/FOTFS2008-Heeffer.pdf">[2]</a> </li> <li id="cite_note-182"><a href="#cite_ref-182">^</a> della Francesca, Piero (1942). De Prospectiva Pingendi, ed. G. Nicco Fasola, 2 vols., Florence. </li> <li id="cite_note-183"><a href="#cite_ref-183">^</a> della Francesca, Piero. Trattato d'Abaco, ed. G. Arrighi, Pisa (1970). </li> <li id="cite_note-184"><a href="#cite_ref-184">^</a> della Francesca, Piero (1916). L'opera "De corporibus regularibus" di Pietro Franceschi detto della Francesca usurpata da Fra Luca Pacioli, ed. G. 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(2004), Pi: A Source Book, New York: Springer, <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-0-387-20571-7" title="Special:BookSources/978-0-387-20571-7"><bdi>978-0-387-20571-7</bdi></a></cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Pi%3A+A+Source+Book&rft.place=New+York&rft.pub=Springer&rft.date=2004&rft.isbn=978-0-387-20571-7&rft.aulast=Berggren&rft.aufirst=Lennart&rft.au=Borwein%2C+Jonathan+M.&rft.au=Borwein%2C+Peter+B.&rfr_id=info%3Asid%2Fen.wikipedia.org%3AHistory+of+mathematics" class="Z3988"></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFBoyer1991" class="citation cs2"><a href="/wiki/Carl_Benjamin_Boyer" title="Carl Benjamin Boyer">Boyer, C.B.</a> (1991) [1989], <a rel="nofollow" class="external text" href="https://archive.org/details/historyofmathema00boye">A History of Mathematics</a> (2nd ed.), New York: Wiley, <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-0-471-54397-8" title="Special:BookSources/978-0-471-54397-8"><bdi>978-0-471-54397-8</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+History+of+Mathematics&rft.place=New+York&rft.edition=2nd&rft.pub=Wiley&rft.date=1991&rft.isbn=978-0-471-54397-8&rft.aulast=Boyer&rft.aufirst=C.B.&rft_id=https%3A%2F%2Farchive.org%2Fdetails%2Fhistoryofmathema00boye&rfr_id=info%3Asid%2Fen.wikipedia.org%3AHistory+of+mathematics" class="Z3988"></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFCuomo2001" class="citation cs2">Cuomo, Serafina (2001), Ancient Mathematics, London: Routledge, <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-0-415-16495-5" title="Special:BookSources/978-0-415-16495-5"><bdi>978-0-415-16495-5</bdi></a></cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Ancient+Mathematics&rft.place=London&rft.pub=Routledge&rft.date=2001&rft.isbn=978-0-415-16495-5&rft.aulast=Cuomo&rft.aufirst=Serafina&rfr_id=info%3Asid%2Fen.wikipedia.org%3AHistory+of+mathematics" class="Z3988"></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFGoodman2016" class="citation cs2">Goodman, Michael, K.J. (2016), An introduction of the Early Development of Mathematics, Hoboken: Wiley, <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-1-119-10497-1" title="Special:BookSources/978-1-119-10497-1"><bdi>978-1-119-10497-1</bdi></a></cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=An+introduction+of+the+Early+Development+of+Mathematics&rft.place=Hoboken&rft.pub=Wiley&rft.date=2016&rft.isbn=978-1-119-10497-1&rft.aulast=Goodman&rft.aufirst=Michael%2C+K.J.&rfr_id=info%3Asid%2Fen.wikipedia.org%3AHistory+of+mathematics" class="Z3988"><code class="cs1-code">{{<a href="/wiki/Template:Citation" title="Template:Citation">citation</a>}}</code>: CS1 maint: multiple names: authors list (<a href="/wiki/Category:CS1_maint:_multiple_names:_authors_list" title="Category:CS1 maint: multiple names: authors list">link</a>)</li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFGullberg1997" class="citation cs2">Gullberg, Jan (1997), <a rel="nofollow" class="external text" href="https://archive.org/details/mathematicsfromb1997gull">Mathematics: From the Birth of Numbers</a>, New York: W.W. Norton and Company, <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-0-393-04002-9" title="Special:BookSources/978-0-393-04002-9"><bdi>978-0-393-04002-9</bdi></a></cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Mathematics%3A+From+the+Birth+of+Numbers&rft.place=New+York&rft.pub=W.W.+Norton+and+Company&rft.date=1997&rft.isbn=978-0-393-04002-9&rft.aulast=Gullberg&rft.aufirst=Jan&rft_id=https%3A%2F%2Farchive.org%2Fdetails%2Fmathematicsfromb1997gull&rfr_id=info%3Asid%2Fen.wikipedia.org%3AHistory+of+mathematics" class="Z3988"></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFJoyce1979" class="citation cs2">Joyce, Hetty (July 1979), "Form, Function and Technique in the Pavements of Delos and Pompeii", American Journal of Archaeology, 83 (3): 253–63, <a href="/wiki/Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.2307%2F505056">10.2307/505056</a>, <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/505056">505056</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:191394716">191394716</a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=American+Journal+of+Archaeology&rft.atitle=Form%2C+Function+and+Technique+in+the+Pavements+of+Delos+and+Pompeii&rft.volume=83&rft.issue=3&rft.pages=253-63&rft.date=1979-07&rft_id=https%3A%2F%2Fapi.semanticscholar.org%2FCorpusID%3A191394716%23id-name%3DS2CID&rft_id=https%3A%2F%2Fwww.jstor.org%2Fstable%2F505056%23id-name%3DJSTOR&rft_id=info%3Adoi%2F10.2307%2F505056&rft.aulast=Joyce&rft.aufirst=Hetty&rfr_id=info%3Asid%2Fen.wikipedia.org%3AHistory+of+mathematics" class="Z3988"></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFKatz1998" class="citation cs2">Katz, Victor J. (1998), <a rel="nofollow" class="external text" href="https://archive.org/details/historyofmathema00katz">A History of Mathematics: An Introduction</a> (2nd ed.), <a href="/wiki/Addison-Wesley" title="Addison-Wesley">Addison-Wesley</a>, <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-0-321-01618-8" title="Special:BookSources/978-0-321-01618-8"><bdi>978-0-321-01618-8</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+History+of+Mathematics%3A+An+Introduction&rft.edition=2nd&rft.pub=Addison-Wesley&rft.date=1998&rft.isbn=978-0-321-01618-8&rft.aulast=Katz&rft.aufirst=Victor+J.&rft_id=https%3A%2F%2Farchive.org%2Fdetails%2Fhistoryofmathema00katz&rfr_id=info%3Asid%2Fen.wikipedia.org%3AHistory+of+mathematics" class="Z3988"></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFKatz2007" class="citation cs2">Katz, Victor J. (2007), The Mathematics of Egypt, Mesopotamia, China, India, and Islam: A Sourcebook, Princeton, NJ: Princeton University Press, <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-0-691-11485-9" title="Special:BookSources/978-0-691-11485-9"><bdi>978-0-691-11485-9</bdi></a></cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=The+Mathematics+of+Egypt%2C+Mesopotamia%2C+China%2C+India%2C+and+Islam%3A+A+Sourcebook&rft.place=Princeton%2C+NJ&rft.pub=Princeton+University+Press&rft.date=2007&rft.isbn=978-0-691-11485-9&rft.aulast=Katz&rft.aufirst=Victor+J.&rfr_id=info%3Asid%2Fen.wikipedia.org%3AHistory+of+mathematics" class="Z3988"></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFNeedhamWang1995" class="citation cs2"><a href="/wiki/Joseph_Needham" title="Joseph Needham">Needham, Joseph</a>; <a href="/wiki/Wang_Ling_(historian)" title="Wang Ling (historian)">Wang, Ling</a> (1995) [1959], Science and Civilization in China: Mathematics and the Sciences of the Heavens and the Earth, vol. 3, Cambridge: Cambridge University Press, <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-0-521-05801-8" title="Special:BookSources/978-0-521-05801-8"><bdi>978-0-521-05801-8</bdi></a></cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Science+and+Civilization+in+China%3A+Mathematics+and+the+Sciences+of+the+Heavens+and+the+Earth&rft.place=Cambridge&rft.pub=Cambridge+University+Press&rft.date=1995&rft.isbn=978-0-521-05801-8&rft.aulast=Needham&rft.aufirst=Joseph&rft.au=Wang%2C+Ling&rfr_id=info%3Asid%2Fen.wikipedia.org%3AHistory+of+mathematics" class="Z3988"></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFNeedhamWang2000" class="citation cs2">Needham, Joseph; Wang, Ling (2000) [1965], Science and Civilization in China: Physics and Physical Technology: Mechanical Engineering, vol. 4 (reprint ed.), Cambridge: Cambridge University Press, <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-0-521-05803-2" title="Special:BookSources/978-0-521-05803-2"><bdi>978-0-521-05803-2</bdi></a></cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Science+and+Civilization+in+China%3A+Physics+and+Physical+Technology%3A+Mechanical+Engineering&rft.place=Cambridge&rft.edition=reprint&rft.pub=Cambridge+University+Press&rft.date=2000&rft.isbn=978-0-521-05803-2&rft.aulast=Needham&rft.aufirst=Joseph&rft.au=Wang%2C+Ling&rfr_id=info%3Asid%2Fen.wikipedia.org%3AHistory+of+mathematics" class="Z3988"></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFSleeswyk1981" class="citation cs2">Sleeswyk, Andre (October 1981), "Vitruvius' odometer", Scientific American, 252 (4): 188–200, <a href="/wiki/Bibcode_(identifier)" class="mw-redirect" title="Bibcode (identifier)">Bibcode</a>:<a rel="nofollow" class="external text" href="https://ui.adsabs.harvard.edu/abs/1981SciAm.245d.188S">1981SciAm.245d.188S</a>, <a href="/wiki/Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1038%2Fscientificamerican1081-188">10.1038/scientificamerican1081-188</a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=Scientific+American&rft.atitle=Vitruvius%27+odometer&rft.volume=252&rft.issue=4&rft.pages=188-200&rft.date=1981-10&rft_id=info%3Adoi%2F10.1038%2Fscientificamerican1081-188&rft_id=info%3Abibcode%2F1981SciAm.245d.188S&rft.aulast=Sleeswyk&rft.aufirst=Andre&rfr_id=info%3Asid%2Fen.wikipedia.org%3AHistory+of+mathematics" class="Z3988"></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFStraffin1998" class="citation cs2">Straffin, Philip D. (1998), "Liu Hui and the First Golden Age of Chinese Mathematics", Mathematics Magazine, 71 (3): 163–81, <a href="/wiki/Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1080%2F0025570X.1998.11996627">10.1080/0025570X.1998.11996627</a></cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=Mathematics+Magazine&rft.atitle=Liu+Hui+and+the+First+Golden+Age+of+Chinese+Mathematics&rft.volume=71&rft.issue=3&rft.pages=163-81&rft.date=1998&rft_id=info%3Adoi%2F10.1080%2F0025570X.1998.11996627&rft.aulast=Straffin&rft.aufirst=Philip+D.&rfr_id=info%3Asid%2Fen.wikipedia.org%3AHistory+of+mathematics" class="Z3988"></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFTang2005" class="citation cs2">Tang, Birgit (2005), <a rel="nofollow" class="external text" href="https://books.google.com/books?id=nw5eupvkvfEC">Delos, Carthage, Ampurias: the Housing of Three Mediterranean Trading Centres</a>, Rome: L'Erma di Bretschneider (Accademia di Danimarca), <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-88-8265-305-7" title="Special:BookSources/978-88-8265-305-7"><bdi>978-88-8265-305-7</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Delos%2C+Carthage%2C+Ampurias%3A+the+Housing+of+Three+Mediterranean+Trading+Centres&rft.place=Rome&rft.pub=L%27Erma+di+Bretschneider+%28Accademia+di+Danimarca%29&rft.date=2005&rft.isbn=978-88-8265-305-7&rft.aulast=Tang&rft.aufirst=Birgit&rft_id=https%3A%2F%2Fbooks.google.com%2Fbooks%3Fid%3Dnw5eupvkvfEC&rfr_id=info%3Asid%2Fen.wikipedia.org%3AHistory+of+mathematics" class="Z3988"></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFVolkov2009" class="citation cs2">Volkov, Alexei (2009), "Mathematics and Mathematics Education in Traditional Vietnam", in Robson, Eleanor; Stedall, Jacqueline (eds.), The Oxford Handbook of the History of Mathematics, Oxford: Oxford University Press, pp. 153–76, <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-0-19-921312-2" title="Special:BookSources/978-0-19-921312-2"><bdi>978-0-19-921312-2</bdi></a></cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=bookitem&rft.atitle=Mathematics+and+Mathematics+Education+in+Traditional+Vietnam&rft.btitle=The+Oxford+Handbook+of+the+History+of+Mathematics&rft.place=Oxford&rft.pages=153-76&rft.pub=Oxford+University+Press&rft.date=2009&rft.isbn=978-0-19-921312-2&rft.aulast=Volkov&rft.aufirst=Alexei&rfr_id=info%3Asid%2Fen.wikipedia.org%3AHistory+of+mathematics" class="Z3988"></li></ul> <div class="mw-heading mw-heading2"><h2 id="Further_reading">Further reading</h2>[<a href="/w/index.php?title=History_of_mathematics&action=edit&section=24" title="Edit section: Further reading">edit</a>]</div> <div class="mw-heading mw-heading3"><h3 id="General">General</h3>[<a href="/w/index.php?title=History_of_mathematics&action=edit&section=25" title="Edit section: General">edit</a>]</div> <ul><li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFAaboe1964" class="citation book cs1"><a href="/wiki/Asger_Aaboe" title="Asger Aaboe">Aaboe, Asger</a> (1964). Episodes from the Early History of Mathematics. New York: Random House.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Episodes+from+the+Early+History+of+Mathematics&rft.place=New+York&rft.pub=Random+House&rft.date=1964&rft.aulast=Aaboe&rft.aufirst=Asger&rfr_id=info%3Asid%2Fen.wikipedia.org%3AHistory+of+mathematics" class="Z3988"></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFBell1937" class="citation book cs1"><a href="/wiki/Eric_Temple_Bell" title="Eric Temple Bell">Bell, E. T.</a> (1937). <a rel="nofollow" class="external text" href="https://archive.org/details/menofmathematics0041bell">Men of Mathematics</a>. Simon and Schuster.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Men+of+Mathematics&rft.pub=Simon+and+Schuster&rft.date=1937&rft.aulast=Bell&rft.aufirst=E.+T.&rft_id=https%3A%2F%2Farchive.org%2Fdetails%2Fmenofmathematics0041bell&rfr_id=info%3Asid%2Fen.wikipedia.org%3AHistory+of+mathematics" class="Z3988"></li> <li><a href="/w/index.php?title=David_M._Burton&action=edit&redlink=1" class="new" title="David M. Burton (page does not exist)">Burton, David M.</a> (1997). The History of Mathematics: An Introduction. McGraw Hill.</li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFGrattan-Guinness2003" class="citation book cs1"><a href="/wiki/Ivor_Grattan-Guinness" title="Ivor Grattan-Guinness">Grattan-Guinness, Ivor</a> (2003). Companion Encyclopedia of the History and Philosophy of the Mathematical Sciences. The Johns Hopkins University Press. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-0-8018-7397-3" title="Special:BookSources/978-0-8018-7397-3"><bdi>978-0-8018-7397-3</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Companion+Encyclopedia+of+the+History+and+Philosophy+of+the+Mathematical+Sciences&rft.pub=The+Johns+Hopkins+University+Press&rft.date=2003&rft.isbn=978-0-8018-7397-3&rft.aulast=Grattan-Guinness&rft.aufirst=Ivor&rfr_id=info%3Asid%2Fen.wikipedia.org%3AHistory+of+mathematics" class="Z3988"></li> <li><a href="/wiki/Morris_Kline" title="Morris Kline">Kline, Morris</a>. Mathematical Thought from Ancient to Modern Times.</li> <li><a href="/wiki/Dirk_Jan_Struik" title="Dirk Jan Struik">Struik, D. J.</a> (1987). A Concise History of Mathematics, fourth revised edition. Dover Publications, New York.</li></ul> <div class="mw-heading mw-heading3"><h3 id="Books_on_a_specific_period">Books on a specific period</h3>[<a href="/w/index.php?title=History_of_mathematics&action=edit&section=26" title="Edit section: Books on a specific period">edit</a>]</div> <ul><li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFGillings1972" class="citation book cs1"><a href="/w/index.php?title=Richard_J._Gillings&action=edit&redlink=1" class="new" title="Richard J. Gillings (page does not exist)">Gillings, Richard J.</a> (1972). Mathematics in the Time of the Pharaohs. Cambridge, MA: MIT Press.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Mathematics+in+the+Time+of+the+Pharaohs&rft.place=Cambridge%2C+MA&rft.pub=MIT+Press&rft.date=1972&rft.aulast=Gillings&rft.aufirst=Richard+J.&rfr_id=info%3Asid%2Fen.wikipedia.org%3AHistory+of+mathematics" class="Z3988"></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFHeath1921" class="citation book cs1"><a href="/wiki/Thomas_Little_Heath" class="mw-redirect" title="Thomas Little Heath">Heath, Thomas Little</a> (1921). <a href="/wiki/A_History_of_Greek_Mathematics" title="A History of Greek Mathematics">A History of Greek Mathematics</a>. Oxford, Claredon Press.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=A+History+of+Greek+Mathematics&rft.pub=Oxford%2C+Claredon+Press&rft.date=1921&rft.aulast=Heath&rft.aufirst=Thomas+Little&rfr_id=info%3Asid%2Fen.wikipedia.org%3AHistory+of+mathematics" class="Z3988"></li> <li><a href="/wiki/Bartel_Leendert_van_der_Waerden" title="Bartel Leendert van der Waerden">van der Waerden, B. L.</a> (1983). Geometry and Algebra in Ancient Civilizations, Springer, <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/0-387-12159-5" title="Special:BookSources/0-387-12159-5">0-387-12159-5</a>.</li></ul> <div class="mw-heading mw-heading3"><h3 id="Books_on_a_specific_topic">Books on a specific topic</h3>[<a href="/w/index.php?title=History_of_mathematics&action=edit&section=27" title="Edit section: Books on a specific topic">edit</a>]</div> <ul><li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFCorry2015" class="citation cs2">Corry, Leo (2015), A Brief History of Numbers, Oxford University Press, <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-0198702597" title="Special:BookSources/978-0198702597"><bdi>978-0198702597</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+Brief+History+of+Numbers&rft.pub=Oxford+University+Press&rft.date=2015&rft.isbn=978-0198702597&rft.aulast=Corry&rft.aufirst=Leo&rfr_id=info%3Asid%2Fen.wikipedia.org%3AHistory+of+mathematics" class="Z3988"></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFHoffman1998" class="citation book cs1"><a href="/wiki/Paul_Hoffman_(science_writer)" title="Paul Hoffman (science writer)">Hoffman, Paul</a> (1998). The Man Who Loved Only Numbers: The Story of Paul Erdős and the Search for Mathematical Truth. Hyperion. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/0-7868-6362-5" title="Special:BookSources/0-7868-6362-5"><bdi>0-7868-6362-5</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=The+Man+Who+Loved+Only+Numbers%3A+The+Story+of+Paul+Erd%C5%91s+and+the+Search+for+Mathematical+Truth&rft.pub=Hyperion&rft.date=1998&rft.isbn=0-7868-6362-5&rft.aulast=Hoffman&rft.aufirst=Paul&rfr_id=info%3Asid%2Fen.wikipedia.org%3AHistory+of+mathematics" class="Z3988"></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFMenninger1969" class="citation book cs1"><a href="/wiki/Karl_Menninger" title="Karl Menninger">Menninger, Karl W.</a> (1969). Number Words and Number Symbols: A Cultural History of Numbers. MIT Press. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-0-262-13040-0" title="Special:BookSources/978-0-262-13040-0"><bdi>978-0-262-13040-0</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Number+Words+and+Number+Symbols%3A+A+Cultural+History+of+Numbers&rft.pub=MIT+Press&rft.date=1969&rft.isbn=978-0-262-13040-0&rft.aulast=Menninger&rft.aufirst=Karl+W.&rfr_id=info%3Asid%2Fen.wikipedia.org%3AHistory+of+mathematics" class="Z3988"></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFStigler1990" class="citation book cs1"><a href="/wiki/Stephen_Stigler" title="Stephen Stigler">Stigler, Stephen M.</a> (1990). The History of Statistics: The Measurement of Uncertainty before 1900. Belknap Press. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-0-674-40341-3" title="Special:BookSources/978-0-674-40341-3"><bdi>978-0-674-40341-3</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=The+History+of+Statistics%3A+The+Measurement+of+Uncertainty+before+1900&rft.pub=Belknap+Press&rft.date=1990&rft.isbn=978-0-674-40341-3&rft.aulast=Stigler&rft.aufirst=Stephen+M.&rfr_id=info%3Asid%2Fen.wikipedia.org%3AHistory+of+mathematics" class="Z3988"></li></ul> <div class="mw-heading mw-heading2"><h2 id="External_links">External links</h2>[<a href="/w/index.php?title=History_of_mathematics&action=edit&section=28" title="Edit section: External links">edit</a>]</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"><img alt="" src="//upload.wikimedia.org/wikipedia/commons/thumb/f/fa/Wikiquote-logo.svg/34px-Wikiquote-logo.svg.png" decoding="async" width="34" height="40" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/f/fa/Wikiquote-logo.svg/51px-Wikiquote-logo.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/f/fa/Wikiquote-logo.svg/68px-Wikiquote-logo.svg.png 2x" data-file-width="300" data-file-height="355" /></div> <div class="side-box-text plainlist">Wikiquote has quotations related to <a href="https://en.wikiquote.org/wiki/Special:Search/History_of_mathematics" class="extiw" title="q:Special:Search/History of mathematics">History of mathematics</a>.</div></div> </div> <div class="mw-heading mw-heading3"><h3 id="Documentaries">Documentaries</h3>[<a href="/w/index.php?title=History_of_mathematics&action=edit&section=29" title="Edit section: Documentaries">edit</a>]</div> <ul><li><a href="/wiki/BBC" title="BBC">BBC</a> (2008). <a href="/wiki/The_Story_of_Maths" title="The Story of Maths">The Story of Maths</a>.</li> <li><a rel="nofollow" class="external text" href="https://www.bbc.co.uk/programmes/p003k9hq">Renaissance Mathematics</a>, BBC Radio 4 discussion with Robert Kaplan, Jim Bennett & Jackie Stedall (In Our Time, Jun 2, 2005)</li></ul> <div class="mw-heading mw-heading3"><h3 id="Educational_material">Educational material</h3>[<a href="/w/index.php?title=History_of_mathematics&action=edit&section=30" title="Edit section: Educational material">edit</a>]</div> <ul><li><a rel="nofollow" class="external text" href="http://www-history.mcs.st-andrews.ac.uk/">MacTutor History of Mathematics archive</a> (John J. O'Connor and Edmund F. Robertson; University of St Andrews, Scotland). An award-winning website containing detailed biographies on many historical and contemporary mathematicians, as well as information on notable curves and various topics in the history of mathematics.</li> <li><a rel="nofollow" class="external text" href="http://aleph0.clarku.edu/~djoyce/mathhist/">History of Mathematics Home Page</a> (David E. Joyce; Clark University). Articles on various topics in the history of mathematics with an extensive bibliography.</li> <li><a rel="nofollow" class="external text" href="http://www.maths.tcd.ie/pub/HistMath/">The History of Mathematics</a> (David R. Wilkins; Trinity College, Dublin). Collections of material on the mathematics between the 17th and 19th century.</li> <li><a rel="nofollow" class="external text" href="http://jeff560.tripod.com/mathword.html">Earliest Known Uses of Some of the Words of Mathematics</a> (Jeff Miller). Contains information on the earliest known uses of terms used in mathematics.</li> <li><a rel="nofollow" class="external text" href="http://jeff560.tripod.com/mathsym.html">Earliest Uses of Various Mathematical Symbols</a> (Jeff Miller). Contains information on the history of mathematical notations.</li> <li><a rel="nofollow" class="external text" href="http://www.economics.soton.ac.uk/staff/aldrich/Mathematical%20Words.htm">Mathematical Words: Origins and Sources</a> (John Aldrich, University of Southampton) Discusses the origins of the modern mathematical word stock.</li> <li><a rel="nofollow" class="external text" href="http://www.agnesscott.edu/lriddle/women/women.htm">Biographies of Women Mathematicians</a> (Larry Riddle; Agnes Scott College).</li> <li><a rel="nofollow" class="external text" href="http://www.math.buffalo.edu/mad/">Mathematicians of the African Diaspora</a> (Scott W. Williams; University at Buffalo).</li> <li><a rel="nofollow" class="external text" href="http://fredrickey.info/hm/mini/MinicourseDocuments-09.pdf">Notes for MAA minicourse: teaching a course in the history of mathematics. (2009)</a> (<a href="/wiki/V._Frederick_Rickey" title="V. Frederick Rickey">V. Frederick Rickey</a> & <a href="/wiki/Victor_J._Katz" title="Victor J. Katz">Victor J. Katz</a>).</li> <li><a rel="nofollow" class="external text" href="https://www.history-of-physics.com/2017/08/ancient-rome-odometer-of-vitruv.html#:~:text=What%20Was%20the%20Odometer%20of%20Vitruvius.%20The%20Odometer,wheel%20which%20was%20manually%20moved%20along%20by%20hand">Ancient Rome: The Odometer Of Vitruv</a>. Pictorial (moving) re-construction of Vitusius' Roman ododmeter.</li></ul> <div class="mw-heading mw-heading3"><h3 id="Bibliographies">Bibliographies</h3>[<a href="/w/index.php?title=History_of_mathematics&action=edit&section=31" title="Edit section: Bibliographies">edit</a>]</div> <ul><li><a rel="nofollow" class="external text" href="http://mathematics.library.cornell.edu/additional/Collected-Works-of-Mathematicians">A Bibliography of Collected Works and Correspondence of Mathematicians</a> <a rel="nofollow" class="external text" href="https://web.archive.org/web/20070317034718/http://astech.library.cornell.edu/ast/math/find/Collected-Works-of-Mathematicians.cfm">archive dated 2007/3/17</a> (Steven W. Rockey; Cornell University Library).</li></ul> <div class="mw-heading mw-heading3"><h3 id="Organizations">Organizations</h3>[<a href="/w/index.php?title=History_of_mathematics&action=edit&section=32" title="Edit section: Organizations">edit</a>]</div> <ul><li><a rel="nofollow" class="external text" href="http://www.unizar.es/ichm/">International Commission for the History of Mathematics</a></li></ul> <div class="mw-heading mw-heading3"><h3 id="Journals">Journals</h3>[<a href="/w/index.php?title=History_of_mathematics&action=edit&section=33" title="Edit section: Journals">edit</a>]</div> <ul><li><a href="/wiki/Historia_Mathematica" title="Historia Mathematica">Historia Mathematica</a></li> <li><a rel="nofollow" class="external text" href="http://www.maa.org/press/periodicals/convergence">Convergence</a> <a rel="nofollow" class="external text" href="https://web.archive.org/web/20200908223859/https://www.maa.org/press/periodicals/convergence">Archived</a> 2020-09-08 at the <a href="/wiki/Wayback_Machine" title="Wayback Machine">Wayback Machine</a>, the <a href="/wiki/Mathematical_Association_of_America" title="Mathematical Association of America">Mathematical Association of America</a>'s online Math History Magazine</li> <li><a rel="nofollow" class="external text" href="http://archives.math.utk.edu/topics/history.html">History of Mathematics</a> <a rel="nofollow" class="external text" href="https://web.archive.org/web/20061004065105/http://archives.math.utk.edu/topics/history.html">Archived</a> 2006-10-04 at the <a href="/wiki/Wayback_Machine" title="Wayback Machine">Wayback Machine</a> Math Archives (University of Tennessee, Knoxville)</li> <li><a rel="nofollow" class="external text" href="http://mathforum.org/library/topics/history/">History/Biography</a> The Math Forum (Drexel University)</li> <li><a rel="nofollow" class="external text" href="https://web.archive.org/web/20020716102307/http://www.otterbein.edu/resources/library/libpages/subject/mathhis.htm">History of Mathematics</a> (Courtright Memorial Library).</li> <li><a rel="nofollow" class="external text" href="http://homepages.bw.edu/~dcalvis/history.html">History of Mathematics Web Sites</a> <a rel="nofollow" class="external text" href="https://web.archive.org/web/20090525100502/http://homepages.bw.edu/~dcalvis/history.html">Archived</a> 2009-05-25 at the <a href="/wiki/Wayback_Machine" title="Wayback Machine">Wayback Machine</a> (David Calvis; Baldwin-Wallace College)</li> <li><a rel="nofollow" class="external text" href="https://web.archive.org/web/20030219004407/http://webpages.ull.es/users/jbarrios/hm/">Historia de las Matemáticas</a> (Universidad de La La guna)</li> <li><a rel="nofollow" class="external text" href="http://www.mat.uc.pt/~jaimecs/indexhm.html">História da Matemática</a> (Universidade de Coimbra)</li> <li><a rel="nofollow" class="external text" href="https://web.archive.org/web/20110707053917/http://math.illinoisstate.edu/marshall">Using History in Math Class</a></li> <li><a rel="nofollow" class="external text" href="http://mathres.kevius.com/history.html">Mathematical Resources: History of Mathematics</a> (Bruno Kevius)</li> <li><a rel="nofollow" class="external text" href="https://web.archive.org/web/20080615051823/http://www.dm.unipi.it/~tucci/index.html">History of Mathematics</a> (Roberta Tucci)</li></ul> <div class="navbox-styles"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1129693374"><style data-mw-deduplicate="TemplateStyles:r1236075235">.mw-parser-output .navbox{box-sizing:border-box;border:1px solid #a2a9b1;width:100%;clear:both;font-size:88%;text-align:center;padding:1px;margin:1em auto 0}.mw-parser-output .navbox .navbox{margin-top:0}.mw-parser-output .navbox+.navbox,.mw-parser-output .navbox+.navbox-styles+.navbox{margin-top:-1px}.mw-parser-output .navbox-inner,.mw-parser-output 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.navbox-abovebelow{background-color:#e6e6ff}.mw-parser-output .navbox-even{background-color:#f7f7f7}.mw-parser-output .navbox-odd{background-color:transparent}.mw-parser-output .navbox .hlist td dl,.mw-parser-output .navbox .hlist td ol,.mw-parser-output .navbox .hlist td ul,.mw-parser-output .navbox td.hlist dl,.mw-parser-output .navbox td.hlist ol,.mw-parser-output .navbox td.hlist ul{padding:0.125em 0}.mw-parser-output .navbox .navbar{display:block;font-size:100%}.mw-parser-output .navbox-title .navbar{float:left;text-align:left;margin-right:0.5em}body.skin--responsive .mw-parser-output .navbox-image img{max-width:none!important}@media print{body.ns-0 .mw-parser-output .navbox{display:none!important}}</style></div><div role="navigation" class="navbox" aria-labelledby="Major_mathematics_areas" style="padding:3px"><table class="nowraplinks hlist 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"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1239400231"><div class="navbar plainlinks hlist navbar-mini"><ul><li class="nv-view"><a href="/wiki/Template:Areas_of_mathematics" title="Template:Areas of mathematics"><abbr title="View this template">v</abbr></a></li><li class="nv-talk"><a href="/wiki/Template_talk:Areas_of_mathematics" title="Template talk:Areas of mathematics"><abbr title="Discuss this template">t</abbr></a></li><li class="nv-edit"><a href="/wiki/Special:EditPage/Template:Areas_of_mathematics" title="Special:EditPage/Template:Areas of mathematics"><abbr title="Edit this template">e</abbr></a></li></ul></div><div id="Major_mathematics_areas" style="font-size:114%;margin:0 4em">Major <a href="/wiki/Mathematics" title="Mathematics">mathematics</a> areas</div></th></tr><tr><td class="navbox-abovebelow" colspan="2"><div> <ul><li><a class="mw-selflink selflink">History</a> <ul><li><a href="/wiki/Timeline_of_mathematics" title="Timeline of mathematics">Timeline</a></li> <li><a href="/wiki/Future_of_mathematics" title="Future of mathematics">Future</a></li></ul></li> <li><a href="/wiki/Lists_of_mathematics_topics" title="Lists of mathematics topics">Lists</a></li> <li><a href="/wiki/Glossary_of_mathematical_symbols" title="Glossary of mathematical symbols">Glossary</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%"><a href="/wiki/Foundations_of_mathematics" title="Foundations of mathematics">Foundations</a></th><td class="navbox-list-with-group navbox-list navbox-odd" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Category_theory" title="Category theory">Category theory</a></li> <li><a href="/wiki/Information_theory" title="Information theory">Information theory</a></li> <li><a href="/wiki/Mathematical_logic" title="Mathematical logic">Mathematical logic</a></li> <li><a href="/wiki/Philosophy_of_mathematics" title="Philosophy of mathematics">Philosophy of mathematics</a></li> <li><a href="/wiki/Set_theory" title="Set theory">Set theory</a></li> <li><a href="/wiki/Type_theory" title="Type theory">Type theory</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%"><a href="/wiki/Algebra" title="Algebra">Algebra</a></th><td class="navbox-list-with-group navbox-list navbox-even" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Abstract_algebra" title="Abstract algebra">Abstract</a></li> <li><a href="/wiki/Commutative_algebra" title="Commutative algebra">Commutative</a></li> <li><a href="/wiki/Elementary_algebra" title="Elementary algebra">Elementary</a></li> <li><a href="/wiki/Group_theory" title="Group theory">Group theory</a></li> <li><a href="/wiki/Linear_algebra" title="Linear algebra">Linear</a></li> <li><a href="/wiki/Multilinear_algebra" title="Multilinear algebra">Multilinear</a></li> <li><a href="/wiki/Universal_algebra" title="Universal algebra">Universal</a></li> <li><a href="/wiki/Homological_algebra" title="Homological algebra">Homological</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%"><a href="/wiki/Mathematical_analysis" title="Mathematical analysis">Analysis</a></th><td class="navbox-list-with-group navbox-list navbox-odd" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Calculus" title="Calculus">Calculus</a></li> <li><a href="/wiki/Real_analysis" title="Real analysis">Real analysis</a></li> <li><a href="/wiki/Complex_analysis" title="Complex analysis">Complex analysis</a></li> <li><a href="/wiki/Hypercomplex_analysis" title="Hypercomplex analysis">Hypercomplex analysis</a></li> <li><a href="/wiki/Differential_equation" title="Differential equation">Differential equations</a></li> <li><a href="/wiki/Functional_analysis" title="Functional analysis">Functional analysis</a></li> <li><a href="/wiki/Harmonic_analysis" title="Harmonic analysis">Harmonic analysis</a></li> <li><a href="/wiki/Measure_(mathematics)" title="Measure (mathematics)">Measure theory</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%"><a href="/wiki/Discrete_mathematics" title="Discrete mathematics">Discrete</a></th><td class="navbox-list-with-group navbox-list navbox-even" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Combinatorics" title="Combinatorics">Combinatorics</a></li> <li><a href="/wiki/Graph_theory" title="Graph theory">Graph theory</a></li> <li><a href="/wiki/Order_theory" title="Order theory">Order theory</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%"><a href="/wiki/Geometry" title="Geometry">Geometry</a></th><td class="navbox-list-with-group navbox-list navbox-odd" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Algebraic_geometry" title="Algebraic geometry">Algebraic</a></li> <li><a href="/wiki/Analytic_geometry" title="Analytic geometry">Analytic</a></li> <li><a href="/wiki/Arithmetic_geometry" title="Arithmetic geometry">Arithmetic</a></li> <li><a href="/wiki/Differential_geometry" title="Differential geometry">Differential</a></li> <li><a href="/wiki/Discrete_geometry" title="Discrete geometry">Discrete</a></li> <li><a href="/wiki/Euclidean_geometry" title="Euclidean geometry">Euclidean</a></li> <li><a href="/wiki/Finite_geometry" title="Finite geometry">Finite</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%"><a href="/wiki/Number_theory" title="Number theory">Number theory</a></th><td class="navbox-list-with-group navbox-list navbox-even" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Arithmetic" title="Arithmetic">Arithmetic</a></li> <li><a href="/wiki/Algebraic_number_theory" title="Algebraic number theory">Algebraic number theory</a></li> <li><a href="/wiki/Analytic_number_theory" title="Analytic number theory">Analytic number theory</a></li> <li><a href="/wiki/Diophantine_geometry" title="Diophantine geometry">Diophantine geometry</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%"><a href="/wiki/Topology" title="Topology">Topology</a></th><td class="navbox-list-with-group navbox-list navbox-odd" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/General_topology" title="General topology">General</a></li> <li><a href="/wiki/Algebraic_topology" title="Algebraic topology">Algebraic</a></li> <li><a href="/wiki/Differential_topology" title="Differential topology">Differential</a></li> <li><a href="/wiki/Geometric_topology" title="Geometric topology">Geometric</a></li> <li><a href="/wiki/Homotopy_theory" title="Homotopy theory">Homotopy theory</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%"><a href="/wiki/Applied_mathematics" title="Applied mathematics">Applied</a></th><td class="navbox-list-with-group navbox-list navbox-even" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Engineering_mathematics" title="Engineering mathematics">Engineering mathematics</a></li> <li><a href="/wiki/Mathematical_and_theoretical_biology" title="Mathematical and theoretical biology">Mathematical biology</a></li> <li><a href="/wiki/Mathematical_chemistry" title="Mathematical chemistry">Mathematical chemistry</a></li> <li><a href="/wiki/Mathematical_economics" title="Mathematical economics">Mathematical economics</a></li> <li><a href="/wiki/Mathematical_finance" title="Mathematical finance">Mathematical finance</a></li> <li><a href="/wiki/Mathematical_physics" title="Mathematical physics">Mathematical physics</a></li> <li><a href="/wiki/Mathematical_psychology" title="Mathematical psychology">Mathematical psychology</a></li> <li><a href="/wiki/Mathematical_sociology" title="Mathematical sociology">Mathematical sociology</a></li> <li><a href="/wiki/Mathematical_statistics" title="Mathematical statistics">Mathematical statistics</a></li> <li><a href="/wiki/Probability_theory" title="Probability theory">Probability</a></li> <li><a href="/wiki/Statistics" title="Statistics">Statistics</a></li> <li><a href="/wiki/Systems_science" title="Systems science">Systems science</a> <ul><li><a href="/wiki/Control_theory" title="Control theory">Control theory</a></li> <li><a href="/wiki/Game_theory" title="Game theory">Game theory</a></li> <li><a href="/wiki/Operations_research" title="Operations research">Operations research</a></li></ul></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%"><a href="/wiki/Computational_mathematics" title="Computational mathematics">Computational</a></th><td class="navbox-list-with-group navbox-list navbox-odd" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Computer_science" title="Computer science">Computer science</a></li> <li><a href="/wiki/Theory_of_computation" title="Theory of computation">Theory of computation</a></li> <li><a href="/wiki/Computational_complexity_theory" title="Computational complexity theory">Computational complexity theory</a></li> <li><a href="/wiki/Numerical_analysis" title="Numerical analysis">Numerical analysis</a></li> <li><a href="/wiki/Mathematical_optimization" title="Mathematical optimization">Optimization</a></li> <li><a href="/wiki/Computer_algebra" title="Computer algebra">Computer algebra</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%"><a href="/wiki/Lists_of_mathematics_topics" title="Lists of mathematics topics">Related topics</a></th><td class="navbox-list-with-group navbox-list navbox-even" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Mathematicians" class="mw-redirect" title="Mathematicians">Mathematicians</a> <ul><li><a href="/wiki/List_of_mathematicians" class="mw-redirect" title="List of mathematicians">lists</a></li></ul></li> <li><a href="/wiki/Informal_mathematics" title="Informal mathematics">Informal mathematics</a></li> <li><a href="/wiki/List_of_films_about_mathematicians" title="List of films about mathematicians">Films about mathematicians</a></li> <li><a href="/wiki/Recreational_mathematics" title="Recreational mathematics">Recreational mathematics</a></li> <li><a href="/wiki/Mathematics_and_art" title="Mathematics and art">Mathematics and art</a></li> <li><a href="/wiki/Mathematics_education" title="Mathematics education">Mathematics education</a></li></ul> </div></td></tr><tr><td class="navbox-abovebelow" colspan="2"><div> <ul><li><a href="/wiki/File:Nuvola_apps_edu_mathematics_blue-p.svg" class="mw-file-description"><img alt="icon" src="//upload.wikimedia.org/wikipedia/commons/thumb/3/3e/Nuvola_apps_edu_mathematics_blue-p.svg/16px-Nuvola_apps_edu_mathematics_blue-p.svg.png" decoding="async" width="16" height="16" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/3/3e/Nuvola_apps_edu_mathematics_blue-p.svg/24px-Nuvola_apps_edu_mathematics_blue-p.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/3/3e/Nuvola_apps_edu_mathematics_blue-p.svg/32px-Nuvola_apps_edu_mathematics_blue-p.svg.png 2x" data-file-width="128" data-file-height="128" /></a> <a href="/wiki/Portal:Mathematics" title="Portal:Mathematics">Mathematics portal</a></li> <li><img alt="" src="//upload.wikimedia.org/wikipedia/en/thumb/9/96/Symbol_category_class.svg/16px-Symbol_category_class.svg.png" decoding="async" width="16" height="16" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/en/thumb/9/96/Symbol_category_class.svg/23px-Symbol_category_class.svg.png 1.5x, //upload.wikimedia.org/wikipedia/en/thumb/9/96/Symbol_category_class.svg/31px-Symbol_category_class.svg.png 2x" data-file-width="180" data-file-height="185" /> <a href="/wiki/Category:Fields_of_mathematics" title="Category:Fields of mathematics">Category</a></li> <li><img alt="" src="//upload.wikimedia.org/wikipedia/en/thumb/4/4a/Commons-logo.svg/12px-Commons-logo.svg.png" decoding="async" width="12" height="16" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/en/thumb/4/4a/Commons-logo.svg/18px-Commons-logo.svg.png 1.5x, //upload.wikimedia.org/wikipedia/en/thumb/4/4a/Commons-logo.svg/24px-Commons-logo.svg.png 2x" data-file-width="1024" data-file-height="1376" /> <a href="https://commons.wikimedia.org/wiki/Category:Mathematics" class="extiw" title="commons:Category:Mathematics">Commons</a></li> <li><img alt="" src="//upload.wikimedia.org/wikipedia/commons/thumb/3/37/People_icon.svg/16px-People_icon.svg.png" decoding="async" width="16" height="16" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/3/37/People_icon.svg/24px-People_icon.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/3/37/People_icon.svg/32px-People_icon.svg.png 2x" data-file-width="100" data-file-height="100" /> <a href="/wiki/Wikipedia:WikiProject_Mathematics" title="Wikipedia:WikiProject Mathematics">WikiProject</a></li></ul> </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"></div><div role="navigation" class="navbox" aria-labelledby="Indian_mathematics" style="padding:3px"><table class="nowraplinks mw-collapsible mw-collapsed 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"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1239400231"><div class="navbar plainlinks hlist navbar-mini"><ul><li class="nv-view"><a href="/wiki/Template:Indian_mathematics" title="Template:Indian mathematics"><abbr title="View this template">v</abbr></a></li><li class="nv-talk"><a href="/wiki/Template_talk:Indian_mathematics" title="Template talk:Indian mathematics"><abbr title="Discuss this template">t</abbr></a></li><li class="nv-edit"><a href="/wiki/Special:EditPage/Template:Indian_mathematics" title="Special:EditPage/Template:Indian mathematics"><abbr title="Edit this template">e</abbr></a></li></ul></div><div id="Indian_mathematics" style="font-size:114%;margin:0 4em"><a href="/wiki/Indian_mathematics" title="Indian mathematics">Indian mathematics</a></div></th></tr><tr><th scope="row" class="navbox-group" style="width:1%"><a href="/wiki/List_of_Indian_mathematicians" title="List of Indian mathematicians">Mathematicians</a></th><td class="navbox-list-with-group navbox-list navbox-odd hlist" style="width:100%;padding:0"><div style="padding:0 0.25em"></div><table class="nowraplinks navbox-subgroup" style="border-spacing:0"><tbody><tr><th scope="row" class="navbox-group" style="width:1%">Ancient</th><td class="navbox-list-with-group navbox-list navbox-odd" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Apastamba_Dharmasutra" title="Apastamba Dharmasutra">Apastamba</a></li> <li><a href="/wiki/Baudhayana_sutras" title="Baudhayana sutras">Baudhayana</a></li> <li><a href="/wiki/K%C4%81ty%C4%81yana" title="Kātyāyana">Katyayana</a></li> <li><a href="/wiki/Manava" title="Manava">Manava</a></li> <li><a href="/wiki/P%C4%81%E1%B9%87ini" title="Pāṇini">Pāṇini</a></li> <li><a href="/wiki/Pingala" title="Pingala">Pingala</a></li> <li><a href="/wiki/Yajnavalkya" title="Yajnavalkya">Yajnavalkya</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%">Classical</th><td class="navbox-list-with-group navbox-list navbox-even" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Aryabhata" title="Aryabhata">Āryabhaṭa I</a></li> <li><a href="/wiki/Aryabhata_II" title="Aryabhata II">Āryabhaṭa II</a></li> <li><a href="/wiki/Bh%C4%81skara_I" title="Bhāskara I">Bhāskara I</a></li> <li><a href="/wiki/Bh%C4%81skara_II" title="Bhāskara II">Bhāskara II</a></li> <li><a href="/wiki/Melpathur_Narayana_Bhattathiri" title="Melpathur Narayana Bhattathiri">Melpathur Narayana Bhattathiri</a></li> <li><a href="/wiki/Brahmadeva" title="Brahmadeva">Brahmadeva</a></li> <li><a href="/wiki/Brahmagupta" title="Brahmagupta">Brahmagupta</a></li> <li><a href="/wiki/Govindasv%C4%81mi" title="Govindasvāmi">Govindasvāmi</a></li> <li><a href="/wiki/Halayudha" title="Halayudha">Halayudha</a></li> <li><a href="/wiki/Jye%E1%B9%A3%E1%B9%ADhadeva" title="Jyeṣṭhadeva">Jyeṣṭhadeva</a></li> <li><a href="/wiki/Kamalakara" title="Kamalakara">Kamalakara</a></li> <li><a href="/wiki/Madhava_of_Sangamagrama" title="Madhava of Sangamagrama">Mādhava of Saṅgamagrāma</a></li> <li><a href="/wiki/Mah%C4%81v%C4%ABra_(mathematician)" title="Mahāvīra (mathematician)">Mahāvīra</a></li> <li><a href="/wiki/Mahendra_S%C5%ABri" title="Mahendra Sūri">Mahendra Sūri</a></li> <li><a href="/wiki/Munishvara" title="Munishvara">Munishvara</a></li> <li><a href="/wiki/Narayana_Pandit" class="mw-redirect" title="Narayana Pandit">Narayana</a></li> <li><a href="/wiki/Parameshvara" class="mw-redirect" title="Parameshvara">Parameshvara</a></li> <li><a href="/wiki/Achyuta_Pisharati" class="mw-redirect" title="Achyuta Pisharati">Achyuta Pisharati</a></li> <li><a href="/wiki/Jagannatha_Samrat" title="Jagannatha Samrat">Jagannatha Samrat</a></li> <li><a href="/wiki/Nilakantha_Somayaji" title="Nilakantha Somayaji">Nilakantha Somayaji</a></li> <li><a href="/wiki/%C5%9Ar%C4%ABpati" title="Śrīpati">Śrīpati</a></li> <li><a href="/wiki/Sridhara" title="Sridhara">Sridhara</a></li> <li><a href="/wiki/Gangesha_Upadhyaya" class="mw-redirect" title="Gangesha Upadhyaya">Gangesha Upadhyaya</a></li> <li><a href="/wiki/Var%C4%81hamihira" title="Varāhamihira">Varāhamihira</a></li> <li><a href="/wiki/Sankara_Variar" title="Sankara Variar">Sankara Variar</a></li> <li><a href="/wiki/Virasena" title="Virasena">Virasena</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%">Modern</th><td class="navbox-list-with-group navbox-list navbox-odd" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Srinivasa_Ramanujan" title="Srinivasa Ramanujan">Srinivasa Ramanujan</a></li> <li><a href="/wiki/Satyendra_Nath_Bose" title="Satyendra Nath Bose">Satyendra Nath Bose</a></li> <li><a href="/wiki/P.C._Mahalanobis" class="mw-redirect" title="P.C. Mahalanobis">P.C. Mahalanobis</a></li> <li><a href="/wiki/Subrahmanyan_Chandrasekhar" title="Subrahmanyan Chandrasekhar">Subrahmanyan Chandrasekhar</a></li> <li><a href="/wiki/C.R._Rao" class="mw-redirect" title="C.R. Rao">C.R. Rao</a></li> <li><a href="/wiki/Veeravalli_S._Varadarajan" title="Veeravalli S. Varadarajan">Veeravalli S. Varadarajan</a></li> <li><a href="/wiki/S._R._Srinivasa_Varadhan" title="S. R. Srinivasa Varadhan">S. R. Srinivasa Varadhan</a></li> <li><a href="/wiki/K._R._Parthasarathy_(probabilist)" title="K. R. Parthasarathy (probabilist)">K. R. Parthasarathy (probabilist)</a></li> <li><a href="/wiki/M._S._Narasimhan" title="M. S. Narasimhan">M. S. Narasimhan</a></li> <li><a href="/wiki/C._S._Seshadri" title="C. S. Seshadri">C. S. Seshadri</a></li> <li><a href="/wiki/Harish-Chandra" title="Harish-Chandra">Harish-Chandra</a></li> <li><a href="/wiki/Subbayya_Sivasankaranarayana_Pillai" title="Subbayya Sivasankaranarayana Pillai">Subbayya Sivasankaranarayana Pillai</a></li> <li><a href="/wiki/Tilak_Raj_Prabhakar" title="Tilak Raj Prabhakar">Tilak Raj Prabhakar</a></li> <li><a href="/wiki/Manjul_Bhargava" title="Manjul Bhargava">Manjul Bhargava</a></li> <li><a href="/wiki/Akshay_Venkatesh" title="Akshay Venkatesh">Akshay Venkatesh</a></li> <li><a href="/wiki/Ravi_Vakil" title="Ravi Vakil">Ravi Vakil</a></li> <li><a href="/wiki/Kannan_Soundararajan" title="Kannan Soundararajan">Kannan Soundararajan</a></li> <li><a href="/wiki/Template:SSBPST_recipients_in_Mathematical_Science" title="Template:SSBPST recipients in Mathematical Science">Shanti Swarup Bhatnagar Prize recipients in Mathematical Science</a></li></ul> </div></td></tr></tbody></table><div></div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%">Treatises</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/Aryabhatiya" title="Aryabhatiya">Aryabhatiya</a></li> <li><a href="/wiki/Bakhshali_manuscript" title="Bakhshali manuscript">Bakhshali manuscript</a></li> <li><a href="/wiki/Bijaganita" title="Bijaganita">Bijaganita</a></li> <li><a href="/wiki/Br%C4%81hmasphu%E1%B9%ADasiddh%C4%81nta" title="Brāhmasphuṭasiddhānta">Brāhmasphuṭasiddhānta</a></li> <li><a href="/wiki/Ganita_Kaumudi" title="Ganita Kaumudi">Ganita Kaumudi</a></li> <li><a href="/wiki/Kanakkusaram" title="Kanakkusaram">Kanakkusaram</a></li> <li><a href="/wiki/Karanapaddhati" title="Karanapaddhati">Karanapaddhati</a></li> <li><a href="/wiki/L%C4%ABl%C4%81vat%C4%AB" title="Līlāvatī">Līlāvatī</a></li> <li><a href="/wiki/Lokavibhaga" title="Lokavibhaga">Lokavibhaga</a></li> <li><a href="/wiki/P%C4%81t%C4%ABga%E1%B9%87ita" title="Pātīgaṇita">Pātīgaṇita</a></li> <li><a href="/wiki/Paulisa_Siddhanta" title="Paulisa Siddhanta">Paulisa Siddhanta</a></li> <li><a href="/wiki/Var%C4%81hamihira#Pancha-Siddhantika" title="Varāhamihira">Paitamaha Siddhanta</a></li> <li><a href="/wiki/Romaka_Siddhanta" title="Romaka Siddhanta">Romaka Siddhanta</a></li> <li><a href="/wiki/Sadratnamala" title="Sadratnamala">Sadratnamala</a></li> <li><a href="/wiki/Siddh%C4%81nta_Shiromani" title="Siddhānta Shiromani">Siddhānta Shiromani</a></li> <li><a href="/wiki/Shulba_Sutras" title="Shulba Sutras">Śulba Sūtras</a></li> <li><a href="/wiki/Surya_Siddhanta" title="Surya Siddhanta">Surya Siddhanta</a></li> <li><a href="/wiki/Tantrasamgraha" title="Tantrasamgraha">Tantrasamgraha</a></li> <li><a href="/wiki/Vasishtha_Siddhanta" title="Vasishtha Siddhanta">Vasishtha Siddhanta</a></li> <li><a href="/wiki/Venvaroha" title="Venvaroha">Veṇvāroha</a></li> <li><a href="/wiki/Yuktibh%C4%81%E1%B9%A3%C4%81" title="Yuktibhāṣā">Yuktibhāṣā</a></li> <li><a href="/wiki/Yavanajataka" title="Yavanajataka">Yavanajataka</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%">Pioneering innovations</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/Brahmi_numerals" title="Brahmi numerals">Brahmi numerals</a></li> <li><a href="/wiki/Hindu%E2%80%93Arabic_numeral_system" title="Hindu–Arabic numeral system">Hindu–Arabic numeral system</a></li> <li>Symbol for <a href="/wiki/0" title="0">zero (0)</a></li> <li><a href="/wiki/Madhava_series" title="Madhava series">Infinite series expansions for the trigonometric functions</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%">Centres</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/Kerala_school_of_astronomy_and_mathematics" title="Kerala school of astronomy and mathematics">Kerala school of astronomy and mathematics</a></li> <li><a href="/wiki/Jantar_Mantar" title="Jantar Mantar">Jantar Mantar</a> (<a href="/wiki/Jantar_Mantar,_Jaipur" title="Jantar Mantar, Jaipur">Jaipur</a>, <a href="/wiki/Jantar_Mantar,_New_Delhi" title="Jantar Mantar, New Delhi">New Delhi</a>, <a href="/wiki/Jantar_Mantar,_Ujjain" title="Jantar Mantar, Ujjain">Ujjain</a>, <a href="/wiki/Jantar_Mantar,_Varanasi" title="Jantar Mantar, Varanasi">Varanasi</a>)</li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%">Historians of mathematics</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/Bapudeva_Sastri" title="Bapudeva Sastri">Bapudeva Sastri</a> (1821–1900)</li> <li><a href="/w/index.php?title=Shankar_Balakrishna_Dikshit&action=edit&redlink=1" class="new" title="Shankar Balakrishna Dikshit (page does not exist)">Shankar Balakrishna Dikshit</a> (1853–1898)</li> <li><a href="/wiki/Sudhakara_Dvivedi" title="Sudhakara Dvivedi">Sudhakara Dvivedi</a> (1855–1910)</li> <li><a href="/w/index.php?title=M._Rangacarya&action=edit&redlink=1" class="new" title="M. Rangacarya (page does not exist)">M. Rangacarya</a> (1861–1916)</li> <li><a href="/wiki/P._C._Sengupta" class="mw-redirect" title="P. C. Sengupta">P. C. Sengupta</a> (1876–1962)</li> <li><a href="/wiki/B._B._Datta" class="mw-redirect" title="B. B. Datta">B. B. Datta</a> (1888–1958)</li> <li><a href="/wiki/Takao_Hayashi" title="Takao Hayashi">T. Hayashi</a></li> <li><a href="/w/index.php?title=A._A._Krishnaswamy_Ayyangar&action=edit&redlink=1" class="new" title="A. A. Krishnaswamy Ayyangar (page does not exist)">A. A. Krishnaswamy Ayyangar</a> (1892– 1953)</li> <li><a href="/wiki/Avadhesh_Narayan_Singh" title="Avadhesh Narayan Singh">A. N. Singh</a> (1901–1954)</li> <li><a href="/wiki/C._T._Rajagopal" class="mw-redirect" title="C. T. Rajagopal">C. T. Rajagopal</a> (1903–1978)</li> <li><a href="/wiki/T._A._Saraswati_Amma" class="mw-redirect" title="T. A. Saraswati Amma">T. A. Saraswati Amma</a> (1918–2000)</li> <li><a href="/w/index.php?title=S._N._Sen&action=edit&redlink=1" class="new" title="S. N. Sen (page does not exist)">S. N. Sen</a> (1918–1992)</li> <li><a href="/wiki/K._S._Shukla" class="mw-redirect" title="K. S. Shukla">K. S. Shukla</a> (1918–2007)</li> <li><a href="/wiki/K._V._Sarma" title="K. V. Sarma">K. V. Sarma</a> (1919–2005)</li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%">Translators</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/Walter_Eugene_Clark" title="Walter Eugene Clark">Walter Eugene Clark</a></li> <li><a href="/wiki/Henry_Thomas_Colebrooke" title="Henry Thomas Colebrooke">Henry Thomas Colebrooke </a></li> <li><a href="/wiki/David_Pingree" title="David Pingree">David Pingree</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%">Other regions</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/Babylonian_mathematics" title="Babylonian mathematics">Babylon</a></li> <li><a href="/wiki/Chinese_mathematics" title="Chinese mathematics">China</a></li> <li><a href="/wiki/Greek_mathematics" title="Greek mathematics">Greece</a></li> <li><a href="/wiki/Mathematics_in_the_medieval_Islamic_world" title="Mathematics in the medieval Islamic world">Islamic mathematics</a></li> <li><a class="mw-selflink-fragment" href="#Medieval_European_mathematics">Europe</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%">Modern institutions</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/Indian_Statistical_Institute" title="Indian Statistical Institute">Indian Statistical Institute</a></li> <li><a href="/wiki/Bhaskaracharya_Pratishthana" title="Bhaskaracharya Pratishthana">Bhaskaracharya Pratishthana</a></li> <li><a href="/wiki/Chennai_Mathematical_Institute" title="Chennai Mathematical Institute">Chennai Mathematical Institute</a></li> <li><a href="/wiki/Institute_of_Mathematical_Sciences,_Chennai" title="Institute of Mathematical Sciences, Chennai">Institute of Mathematical Sciences</a></li> <li><a href="/wiki/Indian_Institute_of_Science" title="Indian Institute of Science">Indian Institute of Science</a></li> <li><a href="/wiki/Harish-Chandra_Research_Institute" title="Harish-Chandra Research Institute">Harish-Chandra Research Institute</a></li> <li><a href="/wiki/Homi_Bhabha_Centre_for_Science_Education" title="Homi Bhabha Centre for Science Education">Homi Bhabha Centre for Science Education</a></li> <li><a href="/wiki/Ramanujan_Institute_for_Advanced_Study_in_Mathematics" title="Ramanujan Institute for Advanced Study in Mathematics">Ramanujan Institute for Advanced Study in Mathematics</a></li> <li><a href="/wiki/Tata_Institute_of_Fundamental_Research" title="Tata Institute of Fundamental Research">TIFR</a></li></ul> </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"></div><div role="navigation" class="navbox" aria-labelledby="Mathematics_in_the_medieval_Islamic_world" 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="3"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1129693374"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1239400231"><div class="navbar plainlinks hlist navbar-mini"><ul><li class="nv-view"><a href="/wiki/Template:Islamic_mathematics" title="Template:Islamic mathematics"><abbr title="View this template">v</abbr></a></li><li class="nv-talk"><a href="/wiki/Template_talk:Islamic_mathematics" title="Template talk:Islamic mathematics"><abbr title="Discuss this template">t</abbr></a></li><li class="nv-edit"><a href="/wiki/Special:EditPage/Template:Islamic_mathematics" title="Special:EditPage/Template:Islamic mathematics"><abbr title="Edit this template">e</abbr></a></li></ul></div><div id="Mathematics_in_the_medieval_Islamic_world" style="font-size:114%;margin:0 4em"><a href="/wiki/Mathematics_in_the_medieval_Islamic_world" title="Mathematics in the medieval Islamic world">Mathematics in the medieval Islamic world</a></div></th></tr><tr><th scope="row" class="navbox-group" style="width:1%"><a href="/wiki/Category:Mathematicians_of_the_medieval_Islamic_world" title="Category:Mathematicians of the medieval Islamic world">Mathematicians</a></th><td class="navbox-list-with-group navbox-list navbox-odd hlist" style="width:100%;padding:0"><div style="padding:0 0.25em"></div><table class="nowraplinks navbox-subgroup" style="border-spacing:0"><tbody><tr><th scope="row" class="navbox-group" style="width:1%">9th century</th><td class="navbox-list-with-group navbox-list navbox-odd" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/%27Abd_al-Ham%C4%ABd_ibn_Turk" class="mw-redirect" title="'Abd al-Hamīd ibn Turk">'Abd al-Hamīd ibn Turk</a></li> <li><a href="/wiki/Sanad_ibn_Ali" title="Sanad ibn Ali">Sanad ibn Ali</a></li> <li><a href="/wiki/Al-Abb%C4%81s_ibn_Said_al-Jawhar%C4%AB" title="Al-Abbās ibn Said al-Jawharī">al-Jawharī</a></li> <li><a href="/wiki/Al-%E1%B8%A4ajj%C4%81j_ibn_Y%C5%ABsuf_ibn_Ma%E1%B9%ADar" title="Al-Ḥajjāj ibn Yūsuf ibn Maṭar">Al-Ḥajjāj ibn Yūsuf</a></li> <li><a href="/wiki/Al-Kindi" title="Al-Kindi">Al-Kindi</a></li> <li><a href="/wiki/Qusta_ibn_Luqa" title="Qusta ibn Luqa">Qusta ibn Luqa</a></li> <li><a href="/wiki/Al-Mahani" title="Al-Mahani">Al-Mahani</a></li> <li><a href="/wiki/Abu_Hanifa_Dinawari" title="Abu Hanifa Dinawari">al-Dinawari</a></li> <li><a href="/wiki/Ban%C5%AB_M%C5%ABs%C4%81_brothers" title="Banū Mūsā brothers">Banū Mūsā brothers</a></li> <li><a href="/wiki/Hunayn_ibn_Ishaq" title="Hunayn ibn Ishaq">Hunayn ibn Ishaq</a></li> <li><a href="/wiki/Al-Khwarizmi" title="Al-Khwarizmi">Al-Khwarizmi</a></li> <li><a href="/wiki/Yusuf_al-Khuri" title="Yusuf al-Khuri">Yusuf al-Khuri</a></li> <li><a href="/wiki/Ishaq_ibn_Hunayn" title="Ishaq ibn Hunayn">Ishaq ibn Hunayn</a></li> <li><a href="/wiki/Na%27im_ibn_Musa" title="Na'im ibn Musa">Na'im ibn Musa</a></li> <li><a href="/wiki/Th%C4%81bit_ibn_Qurra" title="Thābit ibn Qurra">Thābit ibn Qurra</a></li> <li><a href="/wiki/Habash_al-Hasib_al-Marwazi" class="mw-redirect" title="Habash al-Hasib al-Marwazi">al-Marwazi</a></li> <li><a href="/wiki/Abu_Said_Gorgani" title="Abu Said Gorgani">Abu Said Gorgani</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%">10th century</th><td class="navbox-list-with-group navbox-list navbox-even" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Abu_al-Wafa%27_Buzjani" class="mw-redirect" title="Abu al-Wafa' Buzjani">Abu al-Wafa</a></li> <li><a href="/wiki/Abu_Ja%27far_al-Khazin" title="Abu Ja'far al-Khazin">al-Khazin</a></li> <li><a href="/wiki/Al-Qabisi" title="Al-Qabisi">Al-Qabisi</a></li> <li><a href="/wiki/Abu_Kamil" title="Abu Kamil">Abu Kamil</a></li> <li><a href="/wiki/Ahmad_ibn_Yusuf" title="Ahmad ibn Yusuf">Ahmad ibn Yusuf</a></li> <li><a href="/wiki/A%E1%B9%A3-%E1%B9%A2aidan%C4%81n%C4%AB" title="Aṣ-Ṣaidanānī">Aṣ-Ṣaidanānī</a></li> <li><a href="/wiki/Sin%C4%81n_ibn_al-Fat%E1%B8%A5" title="Sinān ibn al-Fatḥ">Sinān ibn al-Fatḥ</a></li> <li><a href="/wiki/Abu-Mahmud_Khojandi" class="mw-redirect" title="Abu-Mahmud Khojandi">al-Khojandi</a></li> <li><a href="/wiki/Al-Nayrizi" title="Al-Nayrizi">Al-Nayrizi</a></li> <li><a href="/wiki/Al-Saghani" class="mw-redirect" title="Al-Saghani">Al-Saghani</a></li> <li><a href="/wiki/Brethren_of_Purity" title="Brethren of Purity">Brethren of Purity</a></li> <li><a href="/wiki/Ibn_Sahl_(mathematician)" title="Ibn Sahl (mathematician)">Ibn Sahl</a></li> <li><a href="/wiki/Ibn_Yunus" title="Ibn Yunus">Ibn Yunus</a></li> <li><a href="/wiki/Abu%27l-Hasan_al-Uqlidisi" title="Abu'l-Hasan al-Uqlidisi">al-Uqlidisi</a></li> <li><a href="/wiki/Al-Battani" title="Al-Battani">Al-Battani</a></li> <li><a href="/wiki/Sinan_ibn_Thabit" title="Sinan ibn Thabit">Sinan ibn Thabit</a></li> <li><a href="/wiki/Ibrahim_ibn_Sinan" title="Ibrahim ibn Sinan">Ibrahim ibn Sinan</a></li> <li><a href="/wiki/Al-Isfahani" title="Al-Isfahani">Al-Isfahani</a></li> <li><a href="/wiki/Nazif_ibn_Yumn" title="Nazif ibn Yumn">Nazif ibn Yumn</a></li> <li><a href="/wiki/Abu_Sahl_al-Quhi" title="Abu Sahl al-Quhi">al-Qūhī</a></li> <li><a href="/wiki/Abu_al-Jud" title="Abu al-Jud">Abu al-Jud</a></li> <li><a href="/wiki/Al-Sijzi" title="Al-Sijzi">Al-Sijzi</a></li> <li><a href="/wiki/Al-Karaji" title="Al-Karaji">Al-Karaji</a></li> <li><a href="/wiki/Maslama_al-Majriti" title="Maslama al-Majriti">al-Majriti</a></li> <li><a href="/wiki/Mohammed_ibn_Abdun_al-Jabali" title="Mohammed ibn Abdun al-Jabali">al-Jabali</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%">11th century</th><td class="navbox-list-with-group navbox-list navbox-odd" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Abu_Nasr_Mansur" title="Abu Nasr Mansur">Abu Nasr Mansur</a></li> <li><a href="/wiki/Ibn_al-Haytham" title="Ibn al-Haytham">Alhazen</a></li> <li><a href="/wiki/Kushyar_Gilani" title="Kushyar Gilani">Kushyar Gilani</a></li> <li><a href="/wiki/Al-Biruni" title="Al-Biruni">Al-Biruni</a></li> <li><a href="/wiki/Ibn_al-Samh" title="Ibn al-Samh">Ibn al-Samh</a></li> <li><a href="/wiki/Abu_Mansur_al-Baghdadi" title="Abu Mansur al-Baghdadi">Abu Mansur al-Baghdadi</a></li> <li><a href="/wiki/Avicenna" title="Avicenna">Avicenna</a></li> <li><a href="/wiki/Ibn_Mu%27adh_al-Jayyani" title="Ibn Mu'adh al-Jayyani">al-Jayyānī</a></li> <li><a href="/wiki/Al%C4%AB_ibn_Ahmad_al-Nasaw%C4%AB" class="mw-redirect" title="Alī ibn Ahmad al-Nasawī"> al-Nasawī</a></li> <li><a href="/wiki/Ab%C5%AB_Is%E1%B8%A5%C4%81q_Ibr%C4%81h%C4%ABm_al-Zarq%C4%81l%C4%AB" class="mw-redirect" title="Abū Isḥāq Ibrāhīm al-Zarqālī">al-Zarqālī</a></li> <li><a href="/wiki/Yusuf_al-Mu%27taman_ibn_Hud" title="Yusuf al-Mu'taman ibn Hud"> ibn Hud</a></li> <li><a href="/wiki/Al-Isfizari" title="Al-Isfizari">Al-Isfizari</a></li> <li><a href="/wiki/Omar_Khayyam" title="Omar Khayyam">Omar Khayyam</a></li> <li><a href="/wiki/Muhammad_al-Baghdadi" title="Muhammad al-Baghdadi">Muhammad al-Baghdadi</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%">12th century</th><td class="navbox-list-with-group navbox-list navbox-even" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Jabir_ibn_Aflah" title="Jabir ibn Aflah">Jabir ibn Aflah</a></li> <li><a href="/wiki/Al-Kharaq%C4%AB" title="Al-Kharaqī">Al-Kharaqī</a></li> <li><a href="/wiki/Al-Khazini" title="Al-Khazini">Al-Khazini</a></li> <li><a href="/wiki/Al-Samawal_al-Maghribi" title="Al-Samawal al-Maghribi">Al-Samawal al-Maghribi</a></li> <li><a href="/wiki/Abu_Bakr_al-Hassar" title="Abu Bakr al-Hassar">al-Hassar</a></li> <li><a href="/wiki/Sharaf_al-Din_al-Tusi" title="Sharaf al-Din al-Tusi">Sharaf al-Din al-Tusi</a></li> <li><a href="/wiki/Ibn_al-Yasamin" title="Ibn al-Yasamin">Ibn al-Yasamin</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%">13th century</th><td class="navbox-list-with-group navbox-list navbox-odd" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Ibn_al%E2%80%90Ha%27im_al%E2%80%90Ishbili" class="mw-redirect" title="Ibn al‐Ha'im al‐Ishbili">Ibn al‐Ha'im al‐Ishbili</a></li> <li><a href="/wiki/Ahmad_al-Buni" title="Ahmad al-Buni">Ahmad al-Buni</a></li> <li><a href="/wiki/Ahmad_ibn_Munim_al-Abdari" title="Ahmad ibn Munim al-Abdari">Ibn Munim</a></li> <li><a href="/wiki/Alam_al-Din_al-Hanafi" title="Alam al-Din al-Hanafi">Alam al-Din al-Hanafi</a></li> <li><a href="/wiki/Ibn_Adlan" title="Ibn Adlan">Ibn Adlan</a></li> <li><a href="/wiki/Mu%27ayyad_al-Din_al-Urdi" title="Mu'ayyad al-Din al-Urdi">al-Urdi</a></li> <li><a href="/wiki/Nasir_al-Din_al-Tusi" title="Nasir al-Din al-Tusi">Nasir al-Din al-Tusi</a></li> <li><a href="/wiki/Athir_al-Din_al-Abhari" title="Athir al-Din al-Abhari">al-Abhari</a></li> <li><a href="/wiki/Muhyi_al-Din_al-Maghribi" title="Muhyi al-Din al-Maghribi">Muhyi al-Din al-Maghribi</a></li> <li><a href="/wiki/Abu_Ali_al-Hasan_al-Marrakushi" title="Abu Ali al-Hasan al-Marrakushi">al-Hasan al-Marrakushi</a></li> <li><a href="/wiki/Qutb_al-Din_al-Shirazi" title="Qutb al-Din al-Shirazi">Qutb al-Din al-Shirazi</a></li> <li><a href="/wiki/Shams_al-Din_al-Samarqandi" title="Shams al-Din al-Samarqandi">Shams al-Din al-Samarqandi</a></li> <li><a href="/wiki/Ibn_al-Banna%27_al-Marrakushi" title="Ibn al-Banna' al-Marrakushi">Ibn al-Banna'</a></li> <li><a href="/wiki/Kam%C4%81l_al-D%C4%ABn_al-F%C4%81ris%C4%AB" title="Kamāl al-Dīn al-Fārisī">Kamāl al-Dīn al-Fārisī</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%">14th century</th><td class="navbox-list-with-group navbox-list navbox-even" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Nizam_al-Din_al-Nisapuri" class="mw-redirect" title="Nizam al-Din al-Nisapuri">Nizam al-Din al-Nisapuri</a></li> <li><a href="/wiki/Ibn_al-Shatir" title="Ibn al-Shatir">Ibn al-Shatir</a></li> <li><a href="/wiki/Ibn_al-Durayhim" title="Ibn al-Durayhim">Ibn al-Durayhim</a></li> <li><a href="/wiki/Shams_al-Din_Abu_Abd_Allah_al-Khalili" title="Shams al-Din Abu Abd Allah al-Khalili">Al-Khalili</a></li> <li><a href="/wiki/Ya%27ish_ibn_Ibrahim_al-Umawi" title="Ya'ish ibn Ibrahim al-Umawi">al-Umawi</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%">15th century</th><td class="navbox-list-with-group navbox-list navbox-odd" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Ibn_al-Majdi" title="Ibn al-Majdi">Ibn al-Majdi</a></li> <li><a href="/wiki/Q%C4%81%E1%B8%8D%C4%AB_Z%C4%81da_al-R%C5%ABm%C4%AB" title="Qāḍī Zāda al-Rūmī">al-Rūmī</a></li> <li><a href="/wiki/Jamshid_al-Kashi" title="Jamshid al-Kashi">al-Kāshī</a></li> <li><a href="/wiki/Ulugh_Beg" title="Ulugh Beg">Ulugh Beg</a></li> <li><a href="/wiki/Ali_Qushji" title="Ali Qushji">Ali Qushji</a></li> <li><a href="/wiki/%27Abd_al-%27Aziz_al-Wafa%27i" title="'Abd al-'Aziz al-Wafa'i">al-Wafa'i</a></li> <li><a href="/wiki/Abu%27l-Hasan_ibn_Ali_al-Qalasadi" title="Abu'l-Hasan ibn Ali al-Qalasadi">al-Qalaṣādī</a></li> <li><a href="/wiki/Sibt_al-Maridini" title="Sibt al-Maridini">Sibt al-Maridini</a></li> <li><a href="/wiki/Ibn_Ghazi_al-Miknasi" title="Ibn Ghazi al-Miknasi">Ibn Ghazi al-Miknasi</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%">16th century</th><td class="navbox-list-with-group navbox-list navbox-even" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Al-Birjandi" title="Al-Birjandi">Al-Birjandi</a></li> <li><a href="/wiki/Muhammad_Baqir_Yazdi" title="Muhammad Baqir Yazdi">Muhammad Baqir Yazdi</a></li> <li><a href="/wiki/Taqi_ad-Din_Muhammad_ibn_Ma%27ruf" title="Taqi ad-Din Muhammad ibn Ma'ruf">Taqi ad-Din</a></li> <li><a href="/wiki/Ibn_Hamza_al-Maghribi" title="Ibn Hamza al-Maghribi">Ibn Hamza al-Maghribi</a></li></ul> </div></td></tr></tbody></table><div></div></td><td class="noviewer navbox-image" rowspan="7" style="width:1px;padding:0 0 0 2px"><div><a href="/wiki/File:Gravure_originale_du_compas_parfait_par_Ab%C5%AB_Sahl_al-Q%C5%ABh%C4%AB.jpg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/9/91/Gravure_originale_du_compas_parfait_par_Ab%C5%AB_Sahl_al-Q%C5%ABh%C4%AB.jpg/80px-Gravure_originale_du_compas_parfait_par_Ab%C5%AB_Sahl_al-Q%C5%ABh%C4%AB.jpg" decoding="async" width="80" height="107" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/9/91/Gravure_originale_du_compas_parfait_par_Ab%C5%AB_Sahl_al-Q%C5%ABh%C4%AB.jpg/120px-Gravure_originale_du_compas_parfait_par_Ab%C5%AB_Sahl_al-Q%C5%ABh%C4%AB.jpg 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/9/91/Gravure_originale_du_compas_parfait_par_Ab%C5%AB_Sahl_al-Q%C5%ABh%C4%AB.jpg/160px-Gravure_originale_du_compas_parfait_par_Ab%C5%AB_Sahl_al-Q%C5%ABh%C4%AB.jpg 2x" data-file-width="458" data-file-height="612" /></a></div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%"><a href="/wiki/Category:Mathematical_works_of_the_medieval_Islamic_world" title="Category:Mathematical works of the medieval Islamic world">Mathematical works</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/The_Compendious_Book_on_Calculation_by_Completion_and_Balancing" class="mw-redirect" title="The Compendious Book on Calculation by Completion and Balancing">The Compendious Book on Calculation by Completion and Balancing</a></li> <li><a href="/wiki/De_Gradibus" title="De Gradibus">De Gradibus</a></li> <li><a href="/wiki/Principles_of_Hindu_Reckoning" title="Principles of Hindu Reckoning">Principles of Hindu Reckoning</a></li> <li><a href="/wiki/Book_of_Optics" title="Book of Optics">Book of Optics</a></li> <li><a href="/wiki/The_Book_of_Healing" title="The Book of Healing">The Book of Healing</a></li> <li><a href="/wiki/Book_on_the_Measurement_of_Plane_and_Spherical_Figures" title="Book on the Measurement of Plane and Spherical Figures">Book on the Measurement of Plane and Spherical Figures</a></li> <li><a href="/wiki/Encyclopedia_of_the_Brethren_of_Purity" title="Encyclopedia of the Brethren of Purity">Encyclopedia of the Brethren of Purity</a></li> <li><a href="/wiki/Toledan_Tables" title="Toledan Tables">Toledan Tables</a></li> <li><a href="/wiki/Tabula_Rogeriana" title="Tabula Rogeriana">Tabula Rogeriana</a></li> <li><a href="/wiki/Zij" title="Zij">Zij</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%">Concepts</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/Alhazen%27s_problem" title="Alhazen's problem">Alhazen's problem</a></li> <li><a href="/wiki/Islamic_geometric_patterns" title="Islamic geometric patterns">Islamic geometric patterns</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%">Centers</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/Al-Azhar_University" title="Al-Azhar University">Al-Azhar University</a></li> <li><a href="/wiki/Al-Mustansiriya_University" title="Al-Mustansiriya University">Al-Mustansiriya University</a></li> <li><a href="/wiki/House_of_Knowledge" title="House of Knowledge">House of Knowledge</a></li> <li><a href="/wiki/House_of_Wisdom" title="House of Wisdom">House of Wisdom</a></li> <li><a href="/wiki/Constantinople_observatory_of_Taqi_ad-Din" title="Constantinople observatory of Taqi ad-Din">Constantinople observatory of Taqi ad-Din</a></li> <li><a href="/wiki/Madrasa" title="Madrasa">Madrasa</a></li> <li><a href="/wiki/Maragheh_observatory" title="Maragheh observatory">Maragheh observatory</a></li> <li><a href="/wiki/University_of_al-Qarawiyyin" title="University of al-Qarawiyyin">University of al-Qarawiyyin</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%">Influences</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/Babylonian_mathematics" title="Babylonian mathematics">Babylonian mathematics</a></li> <li><a href="/wiki/Greek_mathematics" title="Greek mathematics">Greek mathematics</a></li> <li><a href="/wiki/Indian_mathematics" title="Indian mathematics">Indian mathematics</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%">Influenced</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/Byzantine_science" title="Byzantine science">Byzantine mathematics</a></li> <li><a class="mw-selflink-fragment" href="#Medieval_European_mathematics">European mathematics</a></li> <li><a href="/wiki/Indian_mathematics" title="Indian mathematics">Indian mathematics</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%">Related</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/Hindu%E2%80%93Arabic_numeral_system" title="Hindu–Arabic numeral system">Hindu–Arabic numeral system</a></li> <li><a href="/wiki/Arabic_numerals" title="Arabic numerals">Arabic numerals</a> (<a href="/wiki/Eastern_Arabic_numerals" title="Eastern Arabic numerals">Eastern Arabic numerals</a>, <a href="/wiki/Arabic_numerals" title="Arabic numerals">Western Arabic numerals</a>)</li> <li><a href="/wiki/Trigonometric_functions" title="Trigonometric functions">Trigonometric functions</a></li> <li><a href="/wiki/History_of_trigonometry" title="History of trigonometry">History of trigonometry</a></li> <li><a href="/wiki/History_of_algebra" title="History of algebra">History of algebra</a></li></ul> </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"></div><div role="navigation" class="navbox" aria-labelledby="History_of_science" style="padding:3px"><table class="nowraplinks hlist mw-collapsible autocollapse navbox-inner" style="border-spacing:0;background:transparent;color:inherit"><tbody><tr><th scope="col" class="navbox-title" colspan="3"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1129693374"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1239400231"><div class="navbar plainlinks hlist navbar-mini"><ul><li class="nv-view"><a href="/wiki/Template:History_of_science" title="Template:History of science"><abbr title="View this template">v</abbr></a></li><li class="nv-talk"><a href="/wiki/Template_talk:History_of_science" class="mw-redirect" title="Template talk:History of science"><abbr title="Discuss this template">t</abbr></a></li><li class="nv-edit"><a href="/wiki/Special:EditPage/Template:History_of_science" title="Special:EditPage/Template:History of science"><abbr title="Edit this template">e</abbr></a></li></ul></div><div id="History_of_science" style="font-size:114%;margin:0 4em"><a href="/wiki/History_of_science" title="History of science">History of science</a></div></th></tr><tr><th scope="row" class="navbox-group" style="width:1%">Background</th><td class="navbox-list-with-group navbox-list navbox-odd" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Sociology_of_the_history_of_science" title="Sociology of the history of science">Theories and sociology</a></li> <li><a href="/wiki/Historiography_of_science" title="Historiography of science">Historiography</a></li> <li><a href="/wiki/History_of_pseudoscience" title="History of pseudoscience">Pseudoscience</a></li> <li><a href="/wiki/History_and_philosophy_of_science" title="History and philosophy of science">History and philosophy of science</a></li></ul> </div></td><td class="noviewer navbox-image" rowspan="8" style="width:1px;padding:0 0 0 2px"><div><a href="/wiki/File:Johannes-kepler-tabulae-rudolphinae-google-arts-culture.jpg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/e/e5/Johannes-kepler-tabulae-rudolphinae-google-arts-culture.jpg/80px-Johannes-kepler-tabulae-rudolphinae-google-arts-culture.jpg" decoding="async" width="80" height="118" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/e/e5/Johannes-kepler-tabulae-rudolphinae-google-arts-culture.jpg/120px-Johannes-kepler-tabulae-rudolphinae-google-arts-culture.jpg 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/e/e5/Johannes-kepler-tabulae-rudolphinae-google-arts-culture.jpg/160px-Johannes-kepler-tabulae-rudolphinae-google-arts-culture.jpg 2x" data-file-width="3992" data-file-height="5880" /></a></div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%">By era</th><td class="navbox-list-with-group navbox-list navbox-even" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Science_in_the_ancient_world" title="Science in the ancient world">Ancient world</a></li> <li><a href="/wiki/Science_in_classical_antiquity" title="Science in classical antiquity">Classical Antiquity</a></li> <li><a href="/wiki/European_science_in_the_Middle_Ages" title="European science in the Middle Ages">Medieval European</a></li> <li><a href="/wiki/History_of_science_in_the_Renaissance" class="mw-redirect" title="History of science in the Renaissance">Renaissance</a></li> <li><a href="/wiki/Scientific_Revolution" title="Scientific Revolution">Scientific Revolution</a></li> <li><a href="/wiki/Science_in_the_Age_of_Enlightenment" title="Science in the Age of Enlightenment">Age of Enlightenment</a></li> <li><a href="/wiki/Romanticism_in_science" title="Romanticism in science">Romanticism</a></li> <li><a href="/wiki/19th_century_in_science" title="19th century in science">19th century in science</a></li> <li><a href="/wiki/20th_century_in_science" title="20th century in science">20th century in science</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%">By culture</th><td class="navbox-list-with-group navbox-list navbox-odd" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/History_of_science_and_technology_in_Africa" title="History of science and technology in Africa">African</a></li> <li><a href="/wiki/History_of_science_and_technology_in_Argentina" title="History of science and technology in Argentina">Argentine</a></li> <li><a href="/wiki/History_of_science_and_technology_in_Brazil" class="mw-redirect" title="History of science and technology in Brazil">Brazilian</a></li> <li><a href="/wiki/Byzantine_science" title="Byzantine science">Byzantine</a></li> <li><a href="/wiki/History_of_science_and_technology_in_France" class="mw-redirect" title="History of science and technology in France">French</a></li> <li><a href="/wiki/History_of_science_and_technology_in_China" title="History of science and technology in China">Chinese</a></li> <li><a href="/wiki/History_of_science_and_technology_in_the_Indian_subcontinent" class="mw-redirect" title="History of science and technology in the Indian subcontinent">Indian</a></li> <li><a href="/wiki/Science_in_the_medieval_Islamic_world" title="Science in the medieval Islamic world">Medieval Islamic</a></li> <li><a href="/wiki/History_of_science_and_technology_in_Japan" title="History of science and technology in Japan">Japanese</a></li> <li><a href="/wiki/History_of_science_and_technology_in_Korea" title="History of science and technology in Korea">Korean</a></li> <li><a href="/wiki/History_of_science_and_technology_in_Mexico" title="History of science and technology in Mexico">Mexican</a></li> <li><a href="/wiki/History_of_science_and_technology_in_Russia" class="mw-redirect" title="History of science and technology in Russia">Russian</a></li> <li><a href="/wiki/History_of_science_and_technology_in_Spain" title="History of science and technology in Spain">Spanish</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%"><a href="/wiki/History_of_natural_science" class="mw-redirect" title="History of natural science">Natural sciences</a></th><td class="navbox-list-with-group navbox-list navbox-even" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/History_of_astronomy" title="History of astronomy">Astronomy</a></li> <li><a href="/wiki/History_of_biology" title="History of biology">Biology</a></li> <li><a href="/wiki/History_of_chemistry" title="History of chemistry">Chemistry</a></li> <li><a href="/wiki/Outline_of_Earth_sciences#History_of_Earth_science" title="Outline of Earth sciences">Earth science</a></li> <li><a href="/wiki/History_of_physics" title="History of physics">Physics</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%"><a class="mw-selflink selflink">Mathematics</a></th><td class="navbox-list-with-group navbox-list navbox-odd" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/History_of_algebra" title="History of algebra">Algebra</a></li> <li><a href="/wiki/History_of_calculus" title="History of calculus">Calculus</a></li> <li><a href="/wiki/History_of_combinatorics" title="History of combinatorics">Combinatorics</a></li> <li><a href="/wiki/History_of_geometry" title="History of geometry">Geometry</a></li> <li><a href="/wiki/History_of_logic" title="History of logic">Logic</a></li> <li><a href="/wiki/History_of_probability" title="History of probability">Probability</a></li> <li><a href="/wiki/History_of_statistics" title="History of statistics">Statistics</a></li> <li><a href="/wiki/History_of_trigonometry" title="History of trigonometry">Trigonometry</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%"><a href="/wiki/History_of_the_social_sciences" title="History of the social sciences">Social sciences</a></th><td class="navbox-list-with-group navbox-list navbox-even" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/History_of_anthropology" title="History of anthropology">Anthropology</a></li> <li><a href="/wiki/History_of_archaeology" title="History of archaeology">Archaeology</a></li> <li><a href="/wiki/History_of_economic_thought" title="History of economic thought">Economics</a></li> <li><a href="/wiki/History" title="History">History</a></li> <li><a href="/wiki/History_of_political_science" title="History of political science">Political science</a></li> <li><a href="/wiki/History_of_psychology" title="History of psychology">Psychology</a></li> <li><a href="/wiki/History_of_sociology" title="History of sociology">Sociology</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%"><a href="/wiki/History_of_technology" title="History of technology">Technology</a></th><td class="navbox-list-with-group navbox-list navbox-odd" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/History_of_agricultural_science" title="History of agricultural science">Agricultural science</a></li> <li><a href="/wiki/History_of_computer_science" title="History of computer science">Computer science</a></li> <li><a href="/wiki/History_of_materials_science" title="History of materials science">Materials science</a></li> <li><a href="/wiki/History_of_engineering" title="History of engineering">Engineering</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%"><a href="/wiki/History_of_medicine" title="History of medicine">Medicine</a></th><td class="navbox-list-with-group navbox-list navbox-even" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/History_of_medicine" title="History of medicine">Human medicine</a></li> <li><a href="/wiki/History_of_veterinary_medicine" class="mw-redirect" title="History of veterinary medicine">Veterinary medicine</a></li> <li><a href="/wiki/History_of_anatomy" title="History of anatomy">Anatomy</a></li> <li><a href="/wiki/History_of_neuroscience" title="History of neuroscience">Neuroscience</a></li> <li><a href="/wiki/History_of_neurology_and_neurosurgery" title="History of neurology and neurosurgery">Neurology and neurosurgery </a></li> <li><a href="/wiki/History_of_nutrition" class="mw-redirect" title="History of nutrition">Nutrition</a></li> <li><a href="/wiki/History_of_pathology" title="History of pathology">Pathology</a></li> <li><a href="/wiki/History_of_pharmacy" title="History of pharmacy">Pharmacy</a></li></ul> </div></td></tr><tr><td class="navbox-abovebelow hlist" colspan="3" style="margin-right:0.5em; padding:0.1em 0 0.4em;line-height:1.7em;"><div> <ul><li><img alt="" src="//upload.wikimedia.org/wikipedia/en/thumb/d/db/Symbol_list_class.svg/16px-Symbol_list_class.svg.png" decoding="async" width="16" height="16" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/en/thumb/d/db/Symbol_list_class.svg/23px-Symbol_list_class.svg.png 1.5x, //upload.wikimedia.org/wikipedia/en/thumb/d/db/Symbol_list_class.svg/31px-Symbol_list_class.svg.png 2x" data-file-width="180" data-file-height="185" /> <a href="/wiki/List_of_timelines#Science" title="List of timelines">Timelines</a></li> <li><a href="/wiki/File:Symbol_portal_class.svg" class="mw-file-description" title="Portal"><img alt="" src="//upload.wikimedia.org/wikipedia/en/thumb/e/e2/Symbol_portal_class.svg/16px-Symbol_portal_class.svg.png" decoding="async" width="16" height="16" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/en/thumb/e/e2/Symbol_portal_class.svg/23px-Symbol_portal_class.svg.png 1.5x, //upload.wikimedia.org/wikipedia/en/thumb/e/e2/Symbol_portal_class.svg/31px-Symbol_portal_class.svg.png 2x" data-file-width="180" data-file-height="185" /></a> <a href="/wiki/Portal:History_of_science" title="Portal:History of science">Portal</a></li> <li><img alt="" src="//upload.wikimedia.org/wikipedia/en/thumb/9/96/Symbol_category_class.svg/16px-Symbol_category_class.svg.png" decoding="async" width="16" height="16" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/en/thumb/9/96/Symbol_category_class.svg/23px-Symbol_category_class.svg.png 1.5x, //upload.wikimedia.org/wikipedia/en/thumb/9/96/Symbol_category_class.svg/31px-Symbol_category_class.svg.png 2x" data-file-width="180" data-file-height="185" /> <a href="/wiki/Category:History_of_science" title="Category:History of science">Category</a></li></ul> </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"></div><div role="navigation" class="navbox" aria-labelledby="History_of_mathematics_(timeline)" style="padding:3px"><table class="nowraplinks hlist mw-collapsible mw-collapsed 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"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1239400231"><div class="navbar plainlinks hlist navbar-mini"><ul><li class="nv-view"><a href="/wiki/Template:History_of_mathematics" title="Template:History of mathematics"><abbr title="View this template">v</abbr></a></li><li class="nv-talk"><a href="/wiki/Template_talk:History_of_mathematics" title="Template talk:History of mathematics"><abbr title="Discuss this template">t</abbr></a></li><li class="nv-edit"><a href="/wiki/Special:EditPage/Template:History_of_mathematics" title="Special:EditPage/Template:History of mathematics"><abbr title="Edit this template">e</abbr></a></li></ul></div><div id="History_of_mathematics_(timeline)" style="font-size:114%;margin:0 4em"><a class="mw-selflink selflink">History of mathematics</a> (<a href="/wiki/Timeline_of_mathematics" title="Timeline of mathematics">timeline</a>)</div></th></tr><tr><th scope="row" class="navbox-group" style="width:1%">By topic</th><td class="navbox-list-with-group navbox-list navbox-odd" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/History_of_algebra" title="History of algebra">Algebra</a> <ul><li><a href="/wiki/Timeline_of_algebra" title="Timeline of algebra">timeline</a></li></ul></li> <li>Algorithms <ul><li><a href="/wiki/Timeline_of_algorithms" title="Timeline of algorithms">timeline</a></li></ul></li> <li><a href="/wiki/History_of_arithmetic" class="mw-redirect" title="History of arithmetic">Arithmetic</a> <ul><li><a href="/wiki/Timeline_of_numerals_and_arithmetic" title="Timeline of numerals and arithmetic">timeline</a></li></ul></li> <li><a href="/wiki/History_of_calculus" title="History of calculus">Calculus</a> <ul><li><a href="/wiki/Timeline_of_calculus_and_mathematical_analysis" title="Timeline of calculus and mathematical analysis">timeline</a></li> <li><a href="/wiki/History_of_Grandi%27s_series" title="History of Grandi's series">Grandi's series</a></li></ul></li> <li>Category theory <ul><li><a href="/wiki/Timeline_of_category_theory_and_related_mathematics" title="Timeline of category theory and related mathematics">timeline</a></li> <li><a href="/wiki/History_of_topos_theory" title="History of topos theory">Topos theory</a></li></ul></li> <li><a href="/wiki/History_of_combinatorics" title="History of combinatorics">Combinatorics</a></li> <li><a href="/wiki/History_of_the_function_concept" title="History of the function concept">Functions</a> <ul><li><a href="/wiki/History_of_logarithms" title="History of logarithms">Logarithms</a></li></ul></li> <li><a href="/wiki/History_of_geometry" title="History of geometry">Geometry</a> <ul><li><a href="/wiki/History_of_trigonometry" title="History of trigonometry">Trigonometry</a></li> <li><a href="/wiki/Timeline_of_geometry" title="Timeline of geometry">timeline</a></li></ul></li> <li><a href="/wiki/History_of_group_theory" title="History of group theory">Group theory</a></li> <li><a href="/wiki/History_of_information_theory" title="History of information theory">Information theory</a> <ul><li><a href="/wiki/Timeline_of_information_theory" title="Timeline of information theory">timeline</a></li></ul></li> <li><a href="/wiki/History_of_logic" title="History of logic">Logic</a> <ul><li><a href="/wiki/Timeline_of_mathematical_logic" title="Timeline of mathematical logic">timeline</a></li></ul></li> <li><a href="/wiki/History_of_mathematical_notation" title="History of mathematical notation">Math notation</a></li> <li>Number theory <ul><li><a href="/wiki/Timeline_of_number_theory" title="Timeline of number theory">timeline</a></li></ul></li> <li><a href="/wiki/History_of_statistics" title="History of statistics">Statistics</a> <ul><li><a href="/wiki/Timeline_of_probability_and_statistics" title="Timeline of probability and statistics">timeline</a></li> <li><a href="/wiki/History_of_probability" title="History of probability">Probability</a></li></ul></li> <li>Topology <ul><li><a href="/wiki/History_of_manifolds_and_varieties" title="History of manifolds and varieties">Manifolds</a> <ul><li><a href="/wiki/Timeline_of_manifolds" title="Timeline of manifolds">timeline</a></li></ul></li> <li><a href="/wiki/History_of_the_separation_axioms" title="History of the separation axioms">Separation axioms</a></li></ul></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%">Numeral systems</th><td class="navbox-list-with-group navbox-list navbox-even" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Prehistoric_counting" title="Prehistoric counting">Prehistoric</a></li> <li><a href="/wiki/History_of_ancient_numeral_systems" title="History of ancient numeral systems">Ancient</a></li> <li><a href="/wiki/History_of_the_Hindu%E2%80%93Arabic_numeral_system" title="History of the Hindu–Arabic numeral system">Hindu-Arabic</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%">By ancient cultures</th><td class="navbox-list-with-group navbox-list navbox-odd" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Babylonian_mathematics" title="Babylonian mathematics">Mesopotamia</a></li> <li><a href="/wiki/Ancient_Egyptian_mathematics" title="Ancient Egyptian mathematics">Ancient Egypt</a></li> <li><a href="/wiki/Greek_mathematics" title="Greek mathematics">Ancient Greece</a></li> <li><a href="/wiki/Chinese_mathematics" title="Chinese mathematics">China</a></li> <li><a href="/wiki/Indian_mathematics" title="Indian mathematics">India</a></li> <li><a href="/wiki/Mathematics_in_the_medieval_Islamic_world" title="Mathematics in the medieval Islamic world">Medieval Islamic world</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%">Controversies</th><td class="navbox-list-with-group navbox-list navbox-even" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Brouwer%E2%80%93Hilbert_controversy" title="Brouwer–Hilbert controversy">Brouwer–Hilbert</a></li> <li><a href="/wiki/Controversy_over_Cantor%27s_theory" title="Controversy over Cantor's theory">Over Cantor's theory</a></li> <li><a href="/wiki/Leibniz%E2%80%93Newton_calculus_controversy" title="Leibniz–Newton calculus controversy">Leibniz–Newton</a></li> <li><a href="/wiki/Hobbes%E2%80%93Wallis_controversy" title="Hobbes–Wallis controversy">Hobbes–Wallis</a></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>Women in mathematics <ul><li><a href="/wiki/Timeline_of_women_in_mathematics" title="Timeline of women in mathematics">timeline</a></li></ul></li> <li><a href="/wiki/Approximations_of_%CF%80" title="Approximations of π">Approximations of π</a> <ul><li><a href="/wiki/Chronology_of_computation_of_%CF%80" title="Chronology of computation of π">timeline</a></li></ul></li> <li><a href="/wiki/Future_of_mathematics" title="Future of mathematics">Future of mathematics</a></li></ul> </div></td></tr><tr><td class="navbox-abovebelow" colspan="2"><div> <ul><li><img alt="" src="//upload.wikimedia.org/wikipedia/en/thumb/9/96/Symbol_category_class.svg/16px-Symbol_category_class.svg.png" decoding="async" width="16" height="16" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/en/thumb/9/96/Symbol_category_class.svg/23px-Symbol_category_class.svg.png 1.5x, //upload.wikimedia.org/wikipedia/en/thumb/9/96/Symbol_category_class.svg/31px-Symbol_category_class.svg.png 2x" data-file-width="180" data-file-height="185" /> <a href="/wiki/Category:History_of_mathematics" 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