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<li id="toc-Basis_in_earlier_work" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Basis_in_earlier_work"> <div class="vector-toc-text"> <span class="vector-toc-numb">1.1</span> <span>Basis in earlier work</span> </div> </a> <ul id="toc-Basis_in_earlier_work-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Transmission_of_the_text" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Transmission_of_the_text"> <div class="vector-toc-text"> <span class="vector-toc-numb">1.2</span> <span>Transmission of the text</span> </div> </a> <ul id="toc-Transmission_of_the_text-sublist" class="vector-toc-list"> </ul> </li> </ul> </li> <li id="toc-Contents" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Contents"> <div class="vector-toc-text"> <span class="vector-toc-numb">2</span> <span>Contents</span> </div> </a> <ul id="toc-Contents-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Euclid's_method_and_style_of_presentation" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Euclid's_method_and_style_of_presentation"> <div class="vector-toc-text"> <span class="vector-toc-numb">3</span> <span>Euclid's method and style of presentation</span> </div> </a> <ul id="toc-Euclid's_method_and_style_of_presentation-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Influence" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Influence"> <div class="vector-toc-text"> <span class="vector-toc-numb">4</span> <span>Influence</span> </div> </a> <button aria-controls="toc-Influence-sublist" class="cdx-button cdx-button--weight-quiet cdx-button--icon-only vector-toc-toggle"> <span class="vector-icon mw-ui-icon-wikimedia-expand"></span> <span>Toggle Influence subsection</span> </button> <ul id="toc-Influence-sublist" class="vector-toc-list"> <li id="toc-In_modern_mathematics" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#In_modern_mathematics"> <div class="vector-toc-text"> <span class="vector-toc-numb">4.1</span> <span>In modern mathematics</span> </div> </a> <ul id="toc-In_modern_mathematics-sublist" class="vector-toc-list"> </ul> </li> </ul> </li> <li id="toc-Criticism" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Criticism"> <div class="vector-toc-text"> <span class="vector-toc-numb">5</span> <span>Criticism</span> </div> </a> <ul id="toc-Criticism-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Apocrypha" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Apocrypha"> <div class="vector-toc-text"> <span class="vector-toc-numb">6</span> <span>Apocrypha</span> </div> </a> <ul id="toc-Apocrypha-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Editions" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Editions"> <div class="vector-toc-text"> <span class="vector-toc-numb">7</span> <span>Editions</span> </div> </a> <button aria-controls="toc-Editions-sublist" class="cdx-button cdx-button--weight-quiet cdx-button--icon-only vector-toc-toggle"> <span class="vector-icon mw-ui-icon-wikimedia-expand"></span> <span>Toggle Editions subsection</span> </button> <ul id="toc-Editions-sublist" class="vector-toc-list"> <li id="toc-Translations" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Translations"> <div class="vector-toc-text"> <span class="vector-toc-numb">7.1</span> <span>Translations</span> </div> </a> <ul id="toc-Translations-sublist" class="vector-toc-list"> <li id="toc-English" class="vector-toc-list-item vector-toc-level-3"> <a class="vector-toc-link" href="#English"> <div class="vector-toc-text"> <span class="vector-toc-numb">7.1.1</span> <span>English</span> </div> </a> <ul id="toc-English-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Other_languages" class="vector-toc-list-item vector-toc-level-3"> <a class="vector-toc-link" href="#Other_languages"> <div class="vector-toc-text"> <span class="vector-toc-numb">7.1.2</span> <span>Other languages</span> </div> </a> <ul id="toc-Other_languages-sublist" class="vector-toc-list"> </ul> </li> </ul> </li> <li id="toc-Book_I_Editions" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Book_I_Editions"> <div class="vector-toc-text"> <span class="vector-toc-numb">7.2</span> <span>Book I Editions</span> </div> </a> <ul id="toc-Book_I_Editions-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Selected_editions_currently_in_print" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Selected_editions_currently_in_print"> <div class="vector-toc-text"> <span class="vector-toc-numb">7.3</span> <span>Selected editions currently in print</span> </div> </a> <ul id="toc-Selected_editions_currently_in_print-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Selected_editions_based_on_Oliver_Byrne's_edition" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Selected_editions_based_on_Oliver_Byrne's_edition"> <div class="vector-toc-text"> <span class="vector-toc-numb">7.4</span> <span>Selected editions based on Oliver Byrne's edition</span> </div> </a> <ul id="toc-Selected_editions_based_on_Oliver_Byrne's_edition-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Free_versions" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Free_versions"> <div class="vector-toc-text"> <span class="vector-toc-numb">7.5</span> <span>Free versions</span> </div> </a> <ul id="toc-Free_versions-sublist" class="vector-toc-list"> </ul> </li> </ul> </li> <li id="toc-See_also" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#See_also"> <div class="vector-toc-text"> <span class="vector-toc-numb">8</span> <span>See also</span> </div> </a> <ul id="toc-See_also-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-References" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#References"> <div class="vector-toc-text"> <span class="vector-toc-numb">9</span> <span>References</span> </div> </a> <button aria-controls="toc-References-sublist" class="cdx-button cdx-button--weight-quiet cdx-button--icon-only vector-toc-toggle"> <span class="vector-icon mw-ui-icon-wikimedia-expand"></span> <span>Toggle References subsection</span> </button> <ul id="toc-References-sublist" class="vector-toc-list"> <li id="toc-Notes" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Notes"> <div class="vector-toc-text"> <span class="vector-toc-numb">9.1</span> <span>Notes</span> </div> </a> <ul id="toc-Notes-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Citations" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Citations"> <div class="vector-toc-text"> <span class="vector-toc-numb">9.2</span> <span>Citations</span> </div> </a> <ul id="toc-Citations-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Sources" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Sources"> <div class="vector-toc-text"> <span class="vector-toc-numb">9.3</span> <span>Sources</span> </div> </a> <ul id="toc-Sources-sublist" class="vector-toc-list"> </ul> </li> </ul> </li> <li id="toc-External_links" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#External_links"> <div class="vector-toc-text"> <span class="vector-toc-numb">10</span> <span>External links</span> </div> </a> <ul id="toc-External_links-sublist" class="vector-toc-list"> </ul> </li> </ul> </div> </div> </nav> </div> </div> <div class="mw-content-container"> <main id="content" class="mw-body"> <header class="mw-body-header vector-page-titlebar"> <nav aria-label="Contents" class="vector-toc-landmark"> <div id="vector-page-titlebar-toc" class="vector-dropdown vector-page-titlebar-toc vector-button-flush-left" title="Table of Contents" > <input type="checkbox" id="vector-page-titlebar-toc-checkbox" role="button" aria-haspopup="true" data-event-name="ui.dropdown-vector-page-titlebar-toc" class="vector-dropdown-checkbox " aria-label="Toggle the table of contents" > <label id="vector-page-titlebar-toc-label" for="vector-page-titlebar-toc-checkbox" class="vector-dropdown-label cdx-button cdx-button--fake-button cdx-button--fake-button--enabled cdx-button--weight-quiet cdx-button--icon-only " aria-hidden="true" ><span class="vector-icon mw-ui-icon-listBullet mw-ui-icon-wikimedia-listBullet"></span> <span class="vector-dropdown-label-text">Toggle the table of contents</span> </label> <div class="vector-dropdown-content"> <div id="vector-page-titlebar-toc-unpinned-container" class="vector-unpinned-container"> </div> </div> </div> </nav> <h1 id="firstHeading" class="firstHeading mw-first-heading">Euclid's <i>Elements</i></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 75 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-75" aria-hidden="true" ><span class="vector-icon mw-ui-icon-language-progressive mw-ui-icon-wikimedia-language-progressive"></span> <span class="vector-dropdown-label-text">75 languages</span> </label> <div class="vector-dropdown-content"> <div class="vector-menu-content"> <ul class="vector-menu-content-list"> <li class="interlanguage-link interwiki-als mw-list-item"><a href="https://als.wikipedia.org/wiki/Euklids_Elemente" title="Euklids Elemente – Alemannic" lang="gsw" hreflang="gsw" data-title="Euklids Elemente" data-language-autonym="Alemannisch" data-language-local-name="Alemannic" class="interlanguage-link-target"><span>Alemannisch</span></a></li><li class="interlanguage-link interwiki-ar mw-list-item"><a href="https://ar.wikipedia.org/wiki/%D8%A7%D9%84%D8%A3%D8%B5%D9%88%D9%84_(%D9%83%D8%AA%D8%A7%D8%A8)" title="الأصول (كتاب) – Arabic" lang="ar" hreflang="ar" data-title="الأصول (كتاب)" data-language-autonym="العربية" data-language-local-name="Arabic" class="interlanguage-link-target"><span>العربية</span></a></li><li class="interlanguage-link interwiki-ast mw-list-item"><a href="https://ast.wikipedia.org/wiki/Elementos_d%27Euclides" title="Elementos d'Euclides – Asturian" lang="ast" hreflang="ast" data-title="Elementos d'Euclides" data-language-autonym="Asturianu" data-language-local-name="Asturian" class="interlanguage-link-target"><span>Asturianu</span></a></li><li class="interlanguage-link interwiki-az mw-list-item"><a href="https://az.wikipedia.org/wiki/Ba%C5%9Flan%C4%9F%C4%B1clar_(Evklid)" title="Başlanğıclar (Evklid) – Azerbaijani" lang="az" hreflang="az" data-title="Başlanğıclar (Evklid)" data-language-autonym="Azərbaycanca" data-language-local-name="Azerbaijani" class="interlanguage-link-target"><span>Azərbaycanca</span></a></li><li class="interlanguage-link interwiki-bn mw-list-item"><a href="https://bn.wikipedia.org/wiki/%E0%A6%AE%E0%A7%8C%E0%A6%B2%E0%A6%BF%E0%A6%95_%E0%A6%89%E0%A6%AA%E0%A6%BE%E0%A6%A6%E0%A6%BE%E0%A6%A8%E0%A6%B8%E0%A6%AE%E0%A7%82%E0%A6%B9_(%E0%A6%87%E0%A6%89%E0%A6%95%E0%A7%8D%E0%A6%B2%E0%A6%BF%E0%A6%A1)" title="মৌলিক উপাদানসমূহ (ইউক্লিড) – Bangla" lang="bn" hreflang="bn" data-title="মৌলিক উপাদানসমূহ (ইউক্লিড)" data-language-autonym="বাংলা" data-language-local-name="Bangla" class="interlanguage-link-target"><span>বাংলা</span></a></li><li class="interlanguage-link interwiki-be mw-list-item"><a href="https://be.wikipedia.org/wiki/%D0%9F%D0%B0%D1%87%D0%B0%D1%82%D0%BA%D1%96_%D0%AD%D1%9E%D0%BA%D0%BB%D1%96%D0%B4%D0%B0" title="Пачаткі Эўкліда – Belarusian" lang="be" hreflang="be" data-title="Пачаткі Эўкліда" data-language-autonym="Беларуская" data-language-local-name="Belarusian" class="interlanguage-link-target"><span>Беларуская</span></a></li><li class="interlanguage-link interwiki-bg mw-list-item"><a href="https://bg.wikipedia.org/wiki/%D0%95%D0%BB%D0%B5%D0%BC%D0%B5%D0%BD%D1%82%D0%B8" title="Елементи – Bulgarian" lang="bg" hreflang="bg" data-title="Елементи" data-language-autonym="Български" data-language-local-name="Bulgarian" class="interlanguage-link-target"><span>Български</span></a></li><li class="interlanguage-link interwiki-ca mw-list-item"><a href="https://ca.wikipedia.org/wiki/Elements_d%27Euclides" title="Elements d'Euclides – Catalan" lang="ca" hreflang="ca" data-title="Elements d'Euclides" data-language-autonym="Català" data-language-local-name="Catalan" class="interlanguage-link-target"><span>Català</span></a></li><li class="interlanguage-link interwiki-cv mw-list-item"><a href="https://cv.wikipedia.org/wiki/%D0%9F%D1%83%C3%A7%D0%BB%D0%B0%D0%BC%C4%83%D1%88%D1%81%D0%B5%D0%BC_(%D0%95%D0%B2%D0%BA%D0%BB%D0%B8%D0%B4)" title="Пуçламăшсем (Евклид) – Chuvash" lang="cv" hreflang="cv" data-title="Пуçламăшсем (Евклид)" data-language-autonym="Чӑвашла" data-language-local-name="Chuvash" class="interlanguage-link-target"><span>Чӑвашла</span></a></li><li class="interlanguage-link interwiki-cs mw-list-item"><a href="https://cs.wikipedia.org/wiki/Eukleidovy_Z%C3%A1klady" title="Eukleidovy Základy – Czech" lang="cs" hreflang="cs" data-title="Eukleidovy Základy" data-language-autonym="Čeština" data-language-local-name="Czech" class="interlanguage-link-target"><span>Čeština</span></a></li><li class="interlanguage-link interwiki-cy mw-list-item"><a href="https://cy.wikipedia.org/wiki/Yr_Elfennau" title="Yr Elfennau – Welsh" lang="cy" hreflang="cy" data-title="Yr Elfennau" data-language-autonym="Cymraeg" data-language-local-name="Welsh" class="interlanguage-link-target"><span>Cymraeg</span></a></li><li class="interlanguage-link interwiki-da mw-list-item"><a href="https://da.wikipedia.org/wiki/Euklids_Elementer" title="Euklids Elementer – Danish" lang="da" hreflang="da" data-title="Euklids Elementer" data-language-autonym="Dansk" data-language-local-name="Danish" class="interlanguage-link-target"><span>Dansk</span></a></li><li class="interlanguage-link interwiki-de mw-list-item"><a href="https://de.wikipedia.org/wiki/Elemente_(Euklid)" title="Elemente (Euklid) – German" lang="de" hreflang="de" data-title="Elemente (Euklid)" data-language-autonym="Deutsch" data-language-local-name="German" class="interlanguage-link-target"><span>Deutsch</span></a></li><li class="interlanguage-link interwiki-et mw-list-item"><a href="https://et.wikipedia.org/wiki/Elemendid_(Eukleides)" title="Elemendid (Eukleides) – Estonian" lang="et" hreflang="et" data-title="Elemendid (Eukleides)" data-language-autonym="Eesti" data-language-local-name="Estonian" class="interlanguage-link-target"><span>Eesti</span></a></li><li class="interlanguage-link interwiki-el mw-list-item"><a href="https://el.wikipedia.org/wiki/%CE%A3%CF%84%CE%BF%CE%B9%CF%87%CE%B5%CE%AF%CE%B1" title="Στοιχεία – Greek" lang="el" hreflang="el" data-title="Στοιχεία" data-language-autonym="Ελληνικά" data-language-local-name="Greek" class="interlanguage-link-target"><span>Ελληνικά</span></a></li><li class="interlanguage-link interwiki-es mw-list-item"><a href="https://es.wikipedia.org/wiki/Elementos_de_Euclides" title="Elementos de Euclides – Spanish" lang="es" hreflang="es" data-title="Elementos de Euclides" data-language-autonym="Español" data-language-local-name="Spanish" class="interlanguage-link-target"><span>Español</span></a></li><li class="interlanguage-link interwiki-eo mw-list-item"><a href="https://eo.wikipedia.org/wiki/Elementoj_de_E%C5%ADklido" title="Elementoj de Eŭklido – Esperanto" lang="eo" hreflang="eo" data-title="Elementoj de Eŭklido" data-language-autonym="Esperanto" data-language-local-name="Esperanto" class="interlanguage-link-target"><span>Esperanto</span></a></li><li class="interlanguage-link interwiki-eu mw-list-item"><a href="https://eu.wikipedia.org/wiki/Euklidesen_Elementuak" title="Euklidesen Elementuak – Basque" lang="eu" hreflang="eu" data-title="Euklidesen Elementuak" data-language-autonym="Euskara" data-language-local-name="Basque" class="interlanguage-link-target"><span>Euskara</span></a></li><li class="interlanguage-link interwiki-fa mw-list-item"><a href="https://fa.wikipedia.org/wiki/%D8%A7%D8%B5%D9%88%D9%84_%D8%A7%D9%82%D9%84%DB%8C%D8%AF%D8%B3" title="اصول اقلیدس – Persian" lang="fa" hreflang="fa" data-title="اصول اقلیدس" data-language-autonym="فارسی" data-language-local-name="Persian" class="interlanguage-link-target"><span>فارسی</span></a></li><li class="interlanguage-link interwiki-fr mw-list-item"><a href="https://fr.wikipedia.org/wiki/%C3%89l%C3%A9ments_(Euclide)" title="Éléments (Euclide) – French" lang="fr" hreflang="fr" data-title="Éléments (Euclide)" data-language-autonym="Français" data-language-local-name="French" class="interlanguage-link-target"><span>Français</span></a></li><li class="interlanguage-link interwiki-ga mw-list-item"><a href="https://ga.wikipedia.org/wiki/C%C3%A9adnithe_an_Chruinne-Th%C3%B3mhais" title="Céadnithe an Chruinne-Thómhais – Irish" lang="ga" hreflang="ga" data-title="Céadnithe an Chruinne-Thómhais" data-language-autonym="Gaeilge" data-language-local-name="Irish" class="interlanguage-link-target"><span>Gaeilge</span></a></li><li class="interlanguage-link interwiki-gl mw-list-item"><a href="https://gl.wikipedia.org/wiki/Elementos_de_Euclides" title="Elementos de Euclides – Galician" lang="gl" hreflang="gl" data-title="Elementos de Euclides" data-language-autonym="Galego" data-language-local-name="Galician" class="interlanguage-link-target"><span>Galego</span></a></li><li class="interlanguage-link interwiki-ko mw-list-item"><a href="https://ko.wikipedia.org/wiki/%EC%97%90%EC%9A%B0%ED%81%B4%EB%A0%88%EC%9D%B4%EB%8D%B0%EC%8A%A4%EC%9D%98_%EC%9B%90%EB%A1%A0" title="에우클레이데스의 원론 – Korean" lang="ko" hreflang="ko" data-title="에우클레이데스의 원론" data-language-autonym="한국어" data-language-local-name="Korean" class="interlanguage-link-target"><span>한국어</span></a></li><li class="interlanguage-link interwiki-hy mw-list-item"><a href="https://hy.wikipedia.org/wiki/%D5%8D%D5%AF%D5%A6%D5%A2%D5%B8%D6%82%D5%B6%D6%84%D5%B6%D5%A5%D6%80_(%D4%B7%D5%BE%D5%AF%D5%AC%D5%AB%D5%A4%D5%A5%D5%BD)" title="Սկզբունքներ (Էվկլիդես) – Armenian" lang="hy" hreflang="hy" data-title="Սկզբունքներ (Էվկլիդես)" data-language-autonym="Հայերեն" data-language-local-name="Armenian" class="interlanguage-link-target"><span>Հայերեն</span></a></li><li class="interlanguage-link interwiki-hi mw-list-item"><a href="https://hi.wikipedia.org/wiki/%E0%A4%8F%E0%A4%B2%E0%A4%BF%E0%A4%AE%E0%A5%87%E0%A4%A8%E0%A5%8D%E0%A4%9F%E0%A5%8D%E0%A4%B8_(%E0%A4%AF%E0%A5%82%E0%A4%95%E0%A5%8D%E0%A4%B2%E0%A4%BF%E0%A4%A1)" title="एलिमेन्ट्स (यूक्लिड) – Hindi" lang="hi" hreflang="hi" data-title="एलिमेन्ट्स (यूक्लिड)" data-language-autonym="हिन्दी" data-language-local-name="Hindi" class="interlanguage-link-target"><span>हिन्दी</span></a></li><li class="interlanguage-link interwiki-hr mw-list-item"><a href="https://hr.wikipedia.org/wiki/Elementi_(Euklid)" title="Elementi (Euklid) – Croatian" lang="hr" hreflang="hr" data-title="Elementi (Euklid)" data-language-autonym="Hrvatski" data-language-local-name="Croatian" class="interlanguage-link-target"><span>Hrvatski</span></a></li><li class="interlanguage-link interwiki-id mw-list-item"><a href="https://id.wikipedia.org/wiki/Elemen_Euklides" title="Elemen Euklides – Indonesian" lang="id" hreflang="id" data-title="Elemen Euklides" data-language-autonym="Bahasa Indonesia" data-language-local-name="Indonesian" class="interlanguage-link-target"><span>Bahasa Indonesia</span></a></li><li class="interlanguage-link interwiki-ia mw-list-item"><a href="https://ia.wikipedia.org/wiki/Elementos" title="Elementos – Interlingua" lang="ia" hreflang="ia" data-title="Elementos" data-language-autonym="Interlingua" data-language-local-name="Interlingua" class="interlanguage-link-target"><span>Interlingua</span></a></li><li class="interlanguage-link interwiki-it mw-list-item"><a href="https://it.wikipedia.org/wiki/Elementi_(Euclide)" title="Elementi (Euclide) – Italian" lang="it" hreflang="it" data-title="Elementi (Euclide)" data-language-autonym="Italiano" data-language-local-name="Italian" class="interlanguage-link-target"><span>Italiano</span></a></li><li class="interlanguage-link interwiki-he mw-list-item"><a href="https://he.wikipedia.org/wiki/%D7%99%D7%A1%D7%95%D7%93%D7%95%D7%AA_(%D7%A1%D7%A4%D7%A8)" title="יסודות (ספר) – Hebrew" lang="he" hreflang="he" data-title="יסודות (ספר)" data-language-autonym="עברית" data-language-local-name="Hebrew" class="interlanguage-link-target"><span>עברית</span></a></li><li class="interlanguage-link interwiki-ka mw-list-item"><a href="https://ka.wikipedia.org/wiki/%E1%83%A1%E1%83%90%E1%83%AC%E1%83%A7%E1%83%98%E1%83%A1%E1%83%94%E1%83%91%E1%83%98" title="საწყისები – Georgian" lang="ka" hreflang="ka" data-title="საწყისები" data-language-autonym="ქართული" data-language-local-name="Georgian" class="interlanguage-link-target"><span>ქართული</span></a></li><li class="interlanguage-link interwiki-kk mw-list-item"><a href="https://kk.wikipedia.org/wiki/%D0%95%D0%B2%D0%BA%D0%BB%D0%B8%D0%B4%D1%82%D1%96%D2%A3_%D0%9D%D0%B5%D0%B3%D1%96%D0%B7%D0%B4%D0%B5%D1%80%D1%96" title="Евклидтің Негіздері – Kazakh" lang="kk" hreflang="kk" data-title="Евклидтің Негіздері" data-language-autonym="Қазақша" data-language-local-name="Kazakh" class="interlanguage-link-target"><span>Қазақша</span></a></li><li class="interlanguage-link interwiki-ky mw-list-item"><a href="https://ky.wikipedia.org/wiki/%D0%95%D0%B2%D0%BA%D0%BB%D0%B8%D0%B4_%C2%AB%D0%BD%D0%B5%D0%B3%D0%B8%D0%B7%D0%B4%D0%B5%D1%80%D0%B8%C2%BB" title="Евклид «негиздери» – Kyrgyz" lang="ky" hreflang="ky" data-title="Евклид «негиздери»" data-language-autonym="Кыргызча" data-language-local-name="Kyrgyz" class="interlanguage-link-target"><span>Кыргызча</span></a></li><li class="interlanguage-link interwiki-la mw-list-item"><a href="https://la.wikipedia.org/wiki/Elementa_(Euclides)" title="Elementa (Euclides) – Latin" lang="la" hreflang="la" data-title="Elementa (Euclides)" data-language-autonym="Latina" data-language-local-name="Latin" class="interlanguage-link-target"><span>Latina</span></a></li><li class="interlanguage-link interwiki-lv mw-list-item"><a href="https://lv.wikipedia.org/wiki/Elementi" title="Elementi – Latvian" lang="lv" hreflang="lv" data-title="Elementi" data-language-autonym="Latviešu" data-language-local-name="Latvian" class="interlanguage-link-target"><span>Latviešu</span></a></li><li class="interlanguage-link interwiki-lt mw-list-item"><a href="https://lt.wikipedia.org/wiki/Pradmenys" title="Pradmenys – Lithuanian" lang="lt" hreflang="lt" data-title="Pradmenys" data-language-autonym="Lietuvių" data-language-local-name="Lithuanian" class="interlanguage-link-target"><span>Lietuvių</span></a></li><li class="interlanguage-link interwiki-lfn mw-list-item"><a href="https://lfn.wikipedia.org/wiki/Elementos_(libro)" title="Elementos (libro) – Lingua Franca Nova" lang="lfn" hreflang="lfn" data-title="Elementos (libro)" data-language-autonym="Lingua Franca Nova" data-language-local-name="Lingua Franca Nova" class="interlanguage-link-target"><span>Lingua Franca Nova</span></a></li><li class="interlanguage-link interwiki-hu mw-list-item"><a href="https://hu.wikipedia.org/wiki/Elemek" title="Elemek – Hungarian" lang="hu" hreflang="hu" data-title="Elemek" data-language-autonym="Magyar" data-language-local-name="Hungarian" class="interlanguage-link-target"><span>Magyar</span></a></li><li class="interlanguage-link interwiki-mk mw-list-item"><a href="https://mk.wikipedia.org/wiki/%D0%95%D0%BB%D0%B5%D0%BC%D0%B5%D0%BD%D1%82%D0%B8_(%D0%95%D0%B2%D0%BA%D0%BB%D0%B8%D0%B4)" title="Елементи (Евклид) – Macedonian" lang="mk" hreflang="mk" data-title="Елементи (Евклид)" data-language-autonym="Македонски" data-language-local-name="Macedonian" class="interlanguage-link-target"><span>Македонски</span></a></li><li class="interlanguage-link interwiki-ml mw-list-item"><a href="https://ml.wikipedia.org/wiki/%E0%B4%8E%E0%B4%B2%E0%B4%BF%E0%B4%AE%E0%B5%86%E0%B4%A8%E0%B5%8D%E0%B4%B1%E0%B5%8D%E0%B4%B8%E0%B5%8D" title="എലിമെന്റ്സ് – Malayalam" lang="ml" hreflang="ml" data-title="എലിമെന്റ്സ്" data-language-autonym="മലയാളം" data-language-local-name="Malayalam" class="interlanguage-link-target"><span>മലയാളം</span></a></li><li class="interlanguage-link interwiki-arz mw-list-item"><a href="https://arz.wikipedia.org/wiki/%D8%B9%D9%86%D8%A7%D8%B5%D8%B1_%D8%A7%D9%88%D9%83%D9%84%D9%8A%D8%AF%D9%8A%D8%B3" title="عناصر اوكليديس – Egyptian Arabic" lang="arz" hreflang="arz" data-title="عناصر اوكليديس" data-language-autonym="مصرى" data-language-local-name="Egyptian Arabic" class="interlanguage-link-target"><span>مصرى</span></a></li><li class="interlanguage-link interwiki-ms mw-list-item"><a href="https://ms.wikipedia.org/wiki/Elemen_(Euclid)" title="Elemen (Euclid) – Malay" lang="ms" hreflang="ms" data-title="Elemen (Euclid)" data-language-autonym="Bahasa Melayu" data-language-local-name="Malay" class="interlanguage-link-target"><span>Bahasa Melayu</span></a></li><li class="interlanguage-link interwiki-mn mw-list-item"><a href="https://mn.wikipedia.org/wiki/%D0%95%D0%B2%D0%BA%D0%BB%D0%B8%D0%B4%D0%B8%D0%B9%D0%BD_%D1%8D%D0%BB%D0%B5%D0%BC%D0%B5%D0%BD%D1%82%D2%AF%D2%AF%D0%B4" title="Евклидийн элементүүд – Mongolian" lang="mn" hreflang="mn" data-title="Евклидийн элементүүд" data-language-autonym="Монгол" data-language-local-name="Mongolian" class="interlanguage-link-target"><span>Монгол</span></a></li><li class="interlanguage-link interwiki-nl mw-list-item"><a href="https://nl.wikipedia.org/wiki/Elementen_(Euclides)" title="Elementen (Euclides) – Dutch" lang="nl" hreflang="nl" data-title="Elementen (Euclides)" data-language-autonym="Nederlands" data-language-local-name="Dutch" class="interlanguage-link-target"><span>Nederlands</span></a></li><li class="interlanguage-link interwiki-ja mw-list-item"><a href="https://ja.wikipedia.org/wiki/%E3%83%A6%E3%83%BC%E3%82%AF%E3%83%AA%E3%83%83%E3%83%89%E5%8E%9F%E8%AB%96" title="ユークリッド原論 – Japanese" lang="ja" hreflang="ja" data-title="ユークリッド原論" data-language-autonym="日本語" data-language-local-name="Japanese" class="interlanguage-link-target"><span>日本語</span></a></li><li class="interlanguage-link interwiki-no mw-list-item"><a href="https://no.wikipedia.org/wiki/Euklids_Elementer" title="Euklids Elementer – Norwegian Bokmål" lang="nb" hreflang="nb" data-title="Euklids Elementer" data-language-autonym="Norsk bokmål" data-language-local-name="Norwegian Bokmål" class="interlanguage-link-target"><span>Norsk bokmål</span></a></li><li class="interlanguage-link interwiki-oc mw-list-item"><a href="https://oc.wikipedia.org/wiki/Elements_d%27Euclides" title="Elements d'Euclides – Occitan" lang="oc" hreflang="oc" data-title="Elements d'Euclides" data-language-autonym="Occitan" data-language-local-name="Occitan" class="interlanguage-link-target"><span>Occitan</span></a></li><li class="interlanguage-link interwiki-uz mw-list-item"><a href="https://uz.wikipedia.org/wiki/Negizlar" title="Negizlar – Uzbek" lang="uz" hreflang="uz" data-title="Negizlar" data-language-autonym="Oʻzbekcha / ўзбекча" data-language-local-name="Uzbek" class="interlanguage-link-target"><span>Oʻzbekcha / ўзбекча</span></a></li><li class="interlanguage-link interwiki-pa mw-list-item"><a href="https://pa.wikipedia.org/wiki/%E0%A8%AF%E0%A9%82%E0%A8%95%E0%A8%B2%E0%A8%BF%E0%A8%A1_%E0%A8%A6%E0%A9%80_%E0%A8%A4%E0%A9%B1%E0%A8%A4" title="ਯੂਕਲਿਡ ਦੀ ਤੱਤ – Punjabi" lang="pa" hreflang="pa" data-title="ਯੂਕਲਿਡ ਦੀ ਤੱਤ" data-language-autonym="ਪੰਜਾਬੀ" data-language-local-name="Punjabi" class="interlanguage-link-target"><span>ਪੰਜਾਬੀ</span></a></li><li class="interlanguage-link interwiki-pl mw-list-item"><a href="https://pl.wikipedia.org/wiki/Elementy_Euklidesa" title="Elementy Euklidesa – Polish" lang="pl" hreflang="pl" data-title="Elementy Euklidesa" data-language-autonym="Polski" data-language-local-name="Polish" class="interlanguage-link-target"><span>Polski</span></a></li><li class="interlanguage-link interwiki-pt mw-list-item"><a href="https://pt.wikipedia.org/wiki/Os_Elementos" title="Os Elementos – Portuguese" lang="pt" hreflang="pt" data-title="Os Elementos" data-language-autonym="Português" data-language-local-name="Portuguese" class="interlanguage-link-target"><span>Português</span></a></li><li class="interlanguage-link interwiki-ro mw-list-item"><a href="https://ro.wikipedia.org/wiki/Elementele" title="Elementele – Romanian" lang="ro" hreflang="ro" data-title="Elementele" data-language-autonym="Română" data-language-local-name="Romanian" class="interlanguage-link-target"><span>Română</span></a></li><li class="interlanguage-link interwiki-ru mw-list-item"><a href="https://ru.wikipedia.org/wiki/%D0%9D%D0%B0%D1%87%D0%B0%D0%BB%D0%B0_(%D0%95%D0%B2%D0%BA%D0%BB%D0%B8%D0%B4)" title="Начала (Евклид) – Russian" lang="ru" hreflang="ru" data-title="Начала (Евклид)" data-language-autonym="Русский" data-language-local-name="Russian" class="interlanguage-link-target"><span>Русский</span></a></li><li class="interlanguage-link interwiki-sco mw-list-item"><a href="https://sco.wikipedia.org/wiki/Euclid%27s_Elements" title="Euclid's Elements – Scots" lang="sco" hreflang="sco" data-title="Euclid's Elements" data-language-autonym="Scots" data-language-local-name="Scots" class="interlanguage-link-target"><span>Scots</span></a></li><li class="interlanguage-link interwiki-sq mw-list-item"><a href="https://sq.wikipedia.org/wiki/Elementet_e_Euklidit" title="Elementet e Euklidit – Albanian" lang="sq" hreflang="sq" data-title="Elementet e Euklidit" data-language-autonym="Shqip" data-language-local-name="Albanian" class="interlanguage-link-target"><span>Shqip</span></a></li><li class="interlanguage-link interwiki-simple mw-list-item"><a href="https://simple.wikipedia.org/wiki/Euclid%27s_Elements" title="Euclid's Elements – Simple English" lang="en-simple" hreflang="en-simple" data-title="Euclid's Elements" data-language-autonym="Simple English" data-language-local-name="Simple English" class="interlanguage-link-target"><span>Simple English</span></a></li><li class="interlanguage-link interwiki-sl mw-list-item"><a href="https://sl.wikipedia.org/wiki/Elementi_(Evklid)" title="Elementi (Evklid) – Slovenian" lang="sl" hreflang="sl" data-title="Elementi (Evklid)" data-language-autonym="Slovenščina" data-language-local-name="Slovenian" class="interlanguage-link-target"><span>Slovenščina</span></a></li><li class="interlanguage-link interwiki-ckb mw-list-item"><a href="https://ckb.wikipedia.org/wiki/%D8%AA%D9%88%D8%AE%D9%85%DB%95%DA%A9%D8%A7%D9%86%DB%8C_%D8%A6%DB%8C%D9%82%D9%84%DB%8C%D8%AF%D8%B3" title="توخمەکانی ئیقلیدس – Central Kurdish" lang="ckb" hreflang="ckb" data-title="توخمەکانی ئیقلیدس" data-language-autonym="کوردی" data-language-local-name="Central Kurdish" class="interlanguage-link-target"><span>کوردی</span></a></li><li class="interlanguage-link interwiki-sr mw-list-item"><a href="https://sr.wikipedia.org/wiki/%D0%95%D1%83%D0%BA%D0%BB%D0%B8%D0%B4%D0%BE%D0%B2%D0%B8_%D0%95%D0%BB%D0%B5%D0%BC%D0%B5%D0%BD%D1%82%D0%B8" title="Еуклидови Елементи – Serbian" lang="sr" hreflang="sr" data-title="Еуклидови Елементи" data-language-autonym="Српски / srpski" data-language-local-name="Serbian" class="interlanguage-link-target"><span>Српски / srpski</span></a></li><li class="interlanguage-link interwiki-sh mw-list-item"><a href="https://sh.wikipedia.org/wiki/Euklidovi_Elementi" title="Euklidovi Elementi – Serbo-Croatian" lang="sh" hreflang="sh" data-title="Euklidovi Elementi" data-language-autonym="Srpskohrvatski / српскохрватски" data-language-local-name="Serbo-Croatian" class="interlanguage-link-target"><span>Srpskohrvatski / српскохрватски</span></a></li><li class="interlanguage-link interwiki-fi mw-list-item"><a href="https://fi.wikipedia.org/wiki/Alkeet" title="Alkeet – Finnish" lang="fi" hreflang="fi" data-title="Alkeet" data-language-autonym="Suomi" data-language-local-name="Finnish" class="interlanguage-link-target"><span>Suomi</span></a></li><li class="interlanguage-link interwiki-sv mw-list-item"><a href="https://sv.wikipedia.org/wiki/Elementa" title="Elementa – Swedish" lang="sv" hreflang="sv" data-title="Elementa" data-language-autonym="Svenska" data-language-local-name="Swedish" class="interlanguage-link-target"><span>Svenska</span></a></li><li class="interlanguage-link interwiki-tl mw-list-item"><a href="https://tl.wikipedia.org/wiki/Mga_Elemento_ni_Euclides" title="Mga Elemento ni Euclides – Tagalog" lang="tl" hreflang="tl" data-title="Mga Elemento ni Euclides" data-language-autonym="Tagalog" data-language-local-name="Tagalog" class="interlanguage-link-target"><span>Tagalog</span></a></li><li class="interlanguage-link interwiki-ta mw-list-item"><a href="https://ta.wikipedia.org/wiki/%E0%AE%AF%E0%AF%82%E0%AE%95%E0%AF%8D%E0%AE%B3%E0%AE%BF%E0%AE%9F%E0%AF%8D%E0%AE%9F%E0%AE%BF%E0%AE%A9%E0%AF%8D_%E0%AE%8E%E0%AE%B2%E0%AE%BF%E0%AE%AE%E0%AF%86%E0%AE%A9%E0%AF%8D%E0%AE%9F%E0%AF%8D%E0%AE%B8%E0%AF%8D" title="யூக்ளிட்டின் எலிமென்ட்ஸ் – Tamil" lang="ta" hreflang="ta" data-title="யூக்ளிட்டின் எலிமென்ட்ஸ்" data-language-autonym="தமிழ்" data-language-local-name="Tamil" class="interlanguage-link-target"><span>தமிழ்</span></a></li><li class="interlanguage-link interwiki-shi mw-list-item"><a href="https://shi.wikipedia.org/wiki/Ifrdisn_n_Uklid" title="Ifrdisn n Uklid – Tachelhit" lang="shi" hreflang="shi" data-title="Ifrdisn n Uklid" data-language-autonym="Taclḥit" data-language-local-name="Tachelhit" class="interlanguage-link-target"><span>Taclḥit</span></a></li><li class="interlanguage-link interwiki-tg mw-list-item"><a href="https://tg.wikipedia.org/wiki/%D0%A3%D1%81%D1%83%D0%BB%D0%B8_%D0%A3%D2%9B%D0%BB%D0%B8%D0%B4%D1%83%D1%81" title="Усули Уқлидус – Tajik" lang="tg" hreflang="tg" data-title="Усули Уқлидус" data-language-autonym="Тоҷикӣ" data-language-local-name="Tajik" class="interlanguage-link-target"><span>Тоҷикӣ</span></a></li><li class="interlanguage-link interwiki-tr mw-list-item"><a href="https://tr.wikipedia.org/wiki/%C3%96klid%27in_Elementleri" title="Öklid'in Elementleri – Turkish" lang="tr" hreflang="tr" data-title="Öklid'in Elementleri" data-language-autonym="Türkçe" data-language-local-name="Turkish" class="interlanguage-link-target"><span>Türkçe</span></a></li><li class="interlanguage-link interwiki-uk mw-list-item"><a href="https://uk.wikipedia.org/wiki/%D0%9D%D0%B0%D1%87%D0%B0%D0%BB%D0%B0_%D0%95%D0%B2%D0%BA%D0%BB%D1%96%D0%B4%D0%B0" title="Начала Евкліда – Ukrainian" lang="uk" hreflang="uk" data-title="Начала Евкліда" data-language-autonym="Українська" data-language-local-name="Ukrainian" class="interlanguage-link-target"><span>Українська</span></a></li><li class="interlanguage-link interwiki-vi mw-list-item"><a href="https://vi.wikipedia.org/wiki/C%C6%A1_s%E1%BB%9F_(Euclid)" title="Cơ sở (Euclid) – Vietnamese" lang="vi" hreflang="vi" data-title="Cơ sở (Euclid)" data-language-autonym="Tiếng Việt" data-language-local-name="Vietnamese" class="interlanguage-link-target"><span>Tiếng Việt</span></a></li><li class="interlanguage-link interwiki-fiu-vro mw-list-item"><a href="https://fiu-vro.wikipedia.org/wiki/Elemendiq" title="Elemendiq – Võro" lang="vro" hreflang="vro" data-title="Elemendiq" data-language-autonym="Võro" data-language-local-name="Võro" class="interlanguage-link-target"><span>Võro</span></a></li><li class="interlanguage-link interwiki-zh-classical mw-list-item"><a href="https://zh-classical.wikipedia.org/wiki/%E5%B9%BE%E4%BD%95%E5%8E%9F%E6%9C%AC" title="幾何原本 – Literary Chinese" lang="lzh" hreflang="lzh" data-title="幾何原本" data-language-autonym="文言" data-language-local-name="Literary Chinese" class="interlanguage-link-target"><span>文言</span></a></li><li class="interlanguage-link interwiki-war mw-list-item"><a href="https://war.wikipedia.org/wiki/Kan_Euclid_Mga_Elemento" title="Kan Euclid Mga Elemento – Waray" lang="war" hreflang="war" data-title="Kan Euclid Mga Elemento" data-language-autonym="Winaray" data-language-local-name="Waray" class="interlanguage-link-target"><span>Winaray</span></a></li><li class="interlanguage-link interwiki-wuu mw-list-item"><a href="https://wuu.wikipedia.org/wiki/%E5%87%A0%E4%BD%95%E5%8E%9F%E6%9C%AC" title="几何原本 – Wu" lang="wuu" hreflang="wuu" data-title="几何原本" data-language-autonym="吴语" 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.navbar{font-size:100%}@media screen{html.skin-theme-clientpref-night .mw-parser-output .infobox-full-data:not(.notheme)>div:not(.notheme)[style]{background:#1f1f23!important;color:#f8f9fa}}@media screen and (prefers-color-scheme:dark){html.skin-theme-clientpref-os .mw-parser-output .infobox-full-data:not(.notheme) div:not(.notheme){background:#1f1f23!important;color:#f8f9fa}}@media(min-width:640px){body.skin--responsive .mw-parser-output .infobox-table{display:table!important}body.skin--responsive .mw-parser-output .infobox-table>caption{display:table-caption!important}body.skin--responsive .mw-parser-output .infobox-table>tbody{display:table-row-group}body.skin--responsive .mw-parser-output .infobox-table tr{display:table-row!important}body.skin--responsive .mw-parser-output .infobox-table th,body.skin--responsive .mw-parser-output .infobox-table td{padding-left:inherit;padding-right:inherit}}</style><table class="infobox vcard"><caption class="infobox-title" style="font-size:125%; font-style:italic; padding-bottom:0.2em;">Elements <span class="Z3988" title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Elements&rft.author=%5B%5BEuclid%5D%5D&rft.date=%3Cabbr+title%3D%22circa%22%3Ec.%3C%2Fabbr%3E+300+BC&rft.pages=13+books"></span></caption><tbody><tr><td colspan="2" class="infobox-image"><span class="mw-default-size" typeof="mw:File/Frameless"><a href="/wiki/File:Title_page_of_Sir_Henry_Billingsley%27s_first_English_version_of_Euclid%27s_Elements,_1570_(560x900).jpg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/c/cf/Title_page_of_Sir_Henry_Billingsley%27s_first_English_version_of_Euclid%27s_Elements%2C_1570_%28560x900%29.jpg/220px-Title_page_of_Sir_Henry_Billingsley%27s_first_English_version_of_Euclid%27s_Elements%2C_1570_%28560x900%29.jpg" decoding="async" width="220" height="337" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/c/cf/Title_page_of_Sir_Henry_Billingsley%27s_first_English_version_of_Euclid%27s_Elements%2C_1570_%28560x900%29.jpg/330px-Title_page_of_Sir_Henry_Billingsley%27s_first_English_version_of_Euclid%27s_Elements%2C_1570_%28560x900%29.jpg 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/c/cf/Title_page_of_Sir_Henry_Billingsley%27s_first_English_version_of_Euclid%27s_Elements%2C_1570_%28560x900%29.jpg/440px-Title_page_of_Sir_Henry_Billingsley%27s_first_English_version_of_Euclid%27s_Elements%2C_1570_%28560x900%29.jpg 2x" data-file-width="1766" data-file-height="2702" /></a></span><div class="infobox-caption">Title page of Sir Henry Billingsley's first English version of Euclid's <i>Elements</i>, 1570. During the <a href="/wiki/Renaissance" title="Renaissance">Renaissance</a>, Euclid was <a href="/wiki/Euclid#Identity_and_historicity" title="Euclid">commonly conflated with</a> the philosopher <a href="/wiki/Euclid_of_Megara" title="Euclid of Megara">Euclid of Megara</a>.</div></td></tr><tr><th scope="row" class="infobox-label">Author</th><td class="infobox-data"><a href="/wiki/Euclid" title="Euclid">Euclid</a></td></tr><tr><th scope="row" class="infobox-label">Language</th><td class="infobox-data"><a href="/wiki/Ancient_Greek" title="Ancient Greek">Ancient Greek</a></td></tr><tr><th scope="row" class="infobox-label">Subject</th><td class="infobox-data"><a href="/wiki/Euclidean_geometry" title="Euclidean geometry">plane</a> and <a href="/wiki/Solid_geometry" title="Solid geometry">solid geometry</a>, <a href="/wiki/Number_theory" title="Number theory">number theory</a>, <a href="/wiki/Commensurability_(mathematics)" title="Commensurability (mathematics)">incommensurable</a> lines</td></tr><tr><th scope="row" class="infobox-label">Genre</th><td class="infobox-data">Mathematics</td></tr><tr><th scope="row" class="infobox-label"><div style="display: inline-block; line-height: 1.2em; padding: .1em 0;">Publication date</div></th><td class="infobox-data"><abbr title="circa">c.</abbr> 300 BC</td></tr><tr><th scope="row" class="infobox-label">Pages</th><td class="infobox-data">13 books</td></tr></tbody></table> <p>The <i><b>Elements</b></i> (<a href="/wiki/Ancient_Greek_language" class="mw-redirect" title="Ancient Greek language">Ancient Greek</a>: <span lang="grc">Στοιχεῖα</span> <span title="Ancient Greek (to 1453)-language romanization"><i lang="grc-Latn">Stoikheîa</i></span>) is a <a href="/wiki/Mathematics" title="Mathematics">mathematical</a> <a href="/wiki/Treatise" title="Treatise">treatise</a> consisting of 13 books attributed to the ancient <a href="/wiki/Greek_mathematics" title="Greek mathematics">Greek mathematician</a> <a href="/wiki/Euclid" title="Euclid">Euclid</a> <abbr title="circa">c.</abbr> 300 BC. It is a collection of definitions, <a href="/wiki/Postulates" class="mw-redirect" title="Postulates">postulates</a>, <a href="/wiki/Propositions" class="mw-redirect" title="Propositions">propositions</a> (<a href="/wiki/Theorem" title="Theorem">theorems</a> and <a href="/wiki/Compass_and_straightedge_constructions" class="mw-redirect" title="Compass and straightedge constructions">constructions</a>), and <a href="/wiki/Mathematical_proof" title="Mathematical proof">mathematical proofs</a> of the propositions. The books cover plane and solid <a href="/wiki/Euclidean_geometry" title="Euclidean geometry">Euclidean geometry</a>, elementary <a href="/wiki/Number_theory" title="Number theory">number theory</a>, and <a href="/wiki/Commensurability_(mathematics)" title="Commensurability (mathematics)">incommensurable</a> lines. <i>Elements</i> is the oldest extant large-scale deductive treatment of mathematics. It has proven instrumental in the development of <a href="/wiki/Logic" title="Logic">logic</a> and modern <a href="/wiki/Science" title="Science">science</a>, and its logical rigor was not <a href="/wiki/Set_theory" title="Set theory">surpassed</a> until the 19th century. </p><p>Euclid's <i>Elements</i> has been referred to as the most successful<sup id="cite_ref-1" class="reference"><a href="#cite_note-1"><span class="cite-bracket">[</span>a<span class="cite-bracket">]</span></a></sup><sup id="cite_ref-2" class="reference"><a href="#cite_note-2"><span class="cite-bracket">[</span>b<span class="cite-bracket">]</span></a></sup> and influential<sup id="cite_ref-3" class="reference"><a href="#cite_note-3"><span class="cite-bracket">[</span>c<span class="cite-bracket">]</span></a></sup> <a href="/wiki/Textbook" title="Textbook">textbook</a> ever written. It was one of the very earliest mathematical works to be printed after the <a href="/wiki/Movable_type" title="Movable type">invention of the printing press</a> and has been estimated to be second only to the <a href="/wiki/Bible" title="Bible">Bible</a> in the number of editions published since the first printing in 1482,<sup id="cite_ref-FOOTNOTEBoyer1991100_4-0" class="reference"><a href="#cite_note-FOOTNOTEBoyer1991100-4"><span class="cite-bracket">[</span>1<span class="cite-bracket">]</span></a></sup> the number reaching well over one thousand.<sup id="cite_ref-5" class="reference"><a href="#cite_note-5"><span class="cite-bracket">[</span>d<span class="cite-bracket">]</span></a></sup> For centuries, when the <a href="/wiki/Quadrivium" title="Quadrivium">quadrivium</a> was included in the curriculum of all university students, knowledge of at least part of Euclid's <i>Elements</i> was required of all students. Not until the 20th century, by which time its content was universally taught through other school textbooks, did it cease to be considered something all educated people had read.<sup class="noprint Inline-Template Template-Fact" style="white-space:nowrap;">[<i><a href="/wiki/Wikipedia:Citation_needed" title="Wikipedia:Citation needed"><span title="Sweeping claim (April 2023)">citation needed</span></a></i>]</sup> </p> <meta property="mw:PageProp/toc" /> <div class="mw-heading mw-heading2"><h2 id="History">History</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Euclid%27s_Elements&action=edit&section=1" title="Edit section: History"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <figure class="mw-default-size" typeof="mw:File/Thumb"><a href="/wiki/File:P._Oxy._I_29.jpg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/8/8d/P._Oxy._I_29.jpg/250px-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/500px-P._Oxy._I_29.jpg 2x" data-file-width="1694" data-file-height="1032" /></a><figcaption>A fragment of Euclid's <i>Elements</i> on part of the <a href="/wiki/Oxyrhynchus_papyri" class="mw-redirect" title="Oxyrhynchus papyri">Oxyrhynchus papyri</a> </figcaption></figure> <figure class="mw-default-size" typeof="mw:File/Thumb"><a href="/wiki/File:Illustrated_Opening._Arabic_Translation_of_Euclid%27s_Elementa_(CBL_Ar_3035,_ff.105b-106a).jpg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/9/96/Illustrated_Opening._Arabic_Translation_of_Euclid%27s_Elementa_%28CBL_Ar_3035%2C_ff.105b-106a%29.jpg/220px-Illustrated_Opening._Arabic_Translation_of_Euclid%27s_Elementa_%28CBL_Ar_3035%2C_ff.105b-106a%29.jpg" decoding="async" width="220" height="153" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/9/96/Illustrated_Opening._Arabic_Translation_of_Euclid%27s_Elementa_%28CBL_Ar_3035%2C_ff.105b-106a%29.jpg/330px-Illustrated_Opening._Arabic_Translation_of_Euclid%27s_Elementa_%28CBL_Ar_3035%2C_ff.105b-106a%29.jpg 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/9/96/Illustrated_Opening._Arabic_Translation_of_Euclid%27s_Elementa_%28CBL_Ar_3035%2C_ff.105b-106a%29.jpg/440px-Illustrated_Opening._Arabic_Translation_of_Euclid%27s_Elementa_%28CBL_Ar_3035%2C_ff.105b-106a%29.jpg 2x" data-file-width="6639" data-file-height="4631" /></a><figcaption>Double-page from the <a href="/wiki/Ishaq_ibn_Hunayn" title="Ishaq ibn Hunayn">Ishaq ibn Hunayn's</a> Arabic translation of the <i>Elements</i>. <a href="/wiki/Iraq" title="Iraq">Iraq</a>, 1270. <a href="/wiki/Chester_Beatty_Library" title="Chester Beatty Library">Chester Beatty Library</a></figcaption></figure> <div class="mw-heading mw-heading3"><h3 id="Basis_in_earlier_work">Basis in earlier work</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Euclid%27s_Elements&action=edit&section=2" title="Edit section: Basis in earlier work"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <figure class="mw-default-size" typeof="mw:File/Thumb"><a href="/wiki/File:Woman_teaching_geometry.jpg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/d/d4/Woman_teaching_geometry.jpg/220px-Woman_teaching_geometry.jpg" decoding="async" width="220" height="243" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/d/d4/Woman_teaching_geometry.jpg/330px-Woman_teaching_geometry.jpg 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/d/d4/Woman_teaching_geometry.jpg/440px-Woman_teaching_geometry.jpg 2x" data-file-width="1039" data-file-height="1148" /></a><figcaption>An illumination from a manuscript based on <a href="/wiki/Adelard_of_Bath" title="Adelard of Bath">Adelard of Bath</a>'s translation of the <i>Elements</i>, <abbr title="circa">c.</abbr> 1309–1316; Adelard's is the oldest surviving translation of the <i>Elements</i> into Latin, done in the 12th-century work and translated from Arabic.<sup id="cite_ref-FOOTNOTERussell2013177_6-0" class="reference"><a href="#cite_note-FOOTNOTERussell2013177-6"><span class="cite-bracket">[</span>2<span class="cite-bracket">]</span></a></sup></figcaption></figure> <p>Scholars believe that the <i>Elements</i> is largely a compilation of propositions based on books by earlier Greek mathematicians.<sup id="cite_ref-FOOTNOTEVan_der_Waerden1975197_7-0" class="reference"><a href="#cite_note-FOOTNOTEVan_der_Waerden1975197-7"><span class="cite-bracket">[</span>3<span class="cite-bracket">]</span></a></sup> </p><p><a href="/wiki/Proclus" title="Proclus">Proclus</a> (412–485 AD), a Greek mathematician who lived around seven centuries after Euclid, wrote in his commentary on the <i>Elements</i>: "Euclid, who put together the <i>Elements</i>, collecting many of <a href="/wiki/Eudoxus_of_Cnidus" title="Eudoxus of Cnidus">Eudoxus</a>' theorems, perfecting many of <a href="/wiki/Theaetetus_(mathematician)" title="Theaetetus (mathematician)">Theaetetus</a>', and also bringing to irrefragable demonstration the things which were only somewhat loosely proved by his predecessors". </p><p><a href="/wiki/Pythagoras" title="Pythagoras">Pythagoras</a> (<abbr title="circa">c.</abbr> 570–495 BC) was probably the source for most of books I and II, <a href="/wiki/Hippocrates_of_Chios" title="Hippocrates of Chios">Hippocrates of Chios</a> (<abbr title="circa">c.</abbr> 470–410 BC, not the better known <a href="/wiki/Hippocrates" title="Hippocrates">Hippocrates of Kos</a>) for book III, and <a href="/wiki/Eudoxus_of_Cnidus" title="Eudoxus of Cnidus">Eudoxus of Cnidus</a> (<abbr title="circa">c.</abbr> 408–355 BC) for book V, while books IV, VI, XI, and XII probably came from other Pythagorean or Athenian mathematicians.<sup id="cite_ref-FOOTNOTEBall1915[httpsarchiveorgdetailsshortaccountofhi00ballrichpage54_p.&nbsp;54]_8-0" class="reference"><a href="#cite_note-FOOTNOTEBall1915[httpsarchiveorgdetailsshortaccountofhi00ballrichpage54_p.&nbsp;54]-8"><span class="cite-bracket">[</span>4<span class="cite-bracket">]</span></a></sup> The <i>Elements</i> may have been based on an earlier textbook by Hippocrates of Chios, who also may have originated the use of letters to refer to figures.<sup id="cite_ref-FOOTNOTEBall1915[httpsarchiveorgdetailsshortaccountofhi00ballrichpage38_p.&nbsp;38]_9-0" class="reference"><a href="#cite_note-FOOTNOTEBall1915[httpsarchiveorgdetailsshortaccountofhi00ballrichpage38_p.&nbsp;38]-9"><span class="cite-bracket">[</span>5<span class="cite-bracket">]</span></a></sup> Other similar works are also reported to have been written by <a href="/wiki/Theudius_of_Magnesia" class="mw-redirect" title="Theudius of Magnesia">Theudius of Magnesia</a>, <a href="/wiki/Leon_(mathematician)" title="Leon (mathematician)">Leon</a>, and Hermotimus of Colophon.<sup id="cite_ref-10" class="reference"><a href="#cite_note-10"><span class="cite-bracket">[</span>6<span class="cite-bracket">]</span></a></sup><sup id="cite_ref-11" class="reference"><a href="#cite_note-11"><span class="cite-bracket">[</span>7<span class="cite-bracket">]</span></a></sup> </p> <div class="mw-heading mw-heading3"><h3 id="Transmission_of_the_text">Transmission of the text</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Euclid%27s_Elements&action=edit&section=3" title="Edit section: Transmission of the text"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>In the 4th century AD, <a href="/wiki/Theon_of_Alexandria" title="Theon of Alexandria">Theon of Alexandria</a> produced an edition of Euclid which was so widely used that it became the only surviving source until <a href="/wiki/Fran%C3%A7ois_Peyrard" title="François Peyrard">François Peyrard</a>'s 1808 discovery at the <a href="/wiki/Vatican_Library" title="Vatican Library">Vatican</a> of a manuscript not derived from Theon's. This manuscript, the <a href="/wiki/Johan_Ludvig_Heiberg_(historian)" title="Johan Ludvig Heiberg (historian)">Heiberg</a> manuscript, is from a Byzantine workshop around 900 and is the basis of modern editions.<sup id="cite_ref-12" class="reference"><a href="#cite_note-12"><span class="cite-bracket">[</span>8<span class="cite-bracket">]</span></a></sup> <a href="/wiki/Papyrus_Oxyrhynchus_29" title="Papyrus Oxyrhynchus 29">Papyrus Oxyrhynchus 29</a> is a tiny fragment of an even older manuscript, but only contains the statement of one proposition. </p><p>Although Euclid was known to <a href="/wiki/Cicero" title="Cicero">Cicero</a>, for instance, no record exists of the text having been translated into Latin prior to <a href="/wiki/Boethius" title="Boethius">Boethius</a> in the fifth or sixth century.<sup id="cite_ref-FOOTNOTERussell2013177_6-1" class="reference"><a href="#cite_note-FOOTNOTERussell2013177-6"><span class="cite-bracket">[</span>2<span class="cite-bracket">]</span></a></sup> The Arabs received the <i>Elements</i> from the Byzantines around 760; this version was translated into <a href="/wiki/Arabic" title="Arabic">Arabic</a> under <a href="/wiki/Harun_al-Rashid" title="Harun al-Rashid">Harun al-Rashid</a> (<abbr title="circa">c.</abbr> 800).<sup id="cite_ref-FOOTNOTERussell2013177_6-2" class="reference"><a href="#cite_note-FOOTNOTERussell2013177-6"><span class="cite-bracket">[</span>2<span class="cite-bracket">]</span></a></sup> The Byzantine scholar <a href="/wiki/Arethas_of_Caesarea" title="Arethas of Caesarea">Arethas</a> commissioned the copying of one of the extant Greek manuscripts of Euclid in the late ninth century.<sup id="cite_ref-FOOTNOTEReynoldsWilson199157_13-0" class="reference"><a href="#cite_note-FOOTNOTEReynoldsWilson199157-13"><span class="cite-bracket">[</span>9<span class="cite-bracket">]</span></a></sup> Although known in Byzantium, the <i>Elements</i> was lost to Western Europe until about 1120, when the English monk <a href="/wiki/Adelard_of_Bath" title="Adelard of Bath">Adelard of Bath</a> translated it into Latin from an Arabic translation.<sup id="cite_ref-15" class="reference"><a href="#cite_note-15"><span class="cite-bracket">[</span>e<span class="cite-bracket">]</span></a></sup> A relatively recent discovery was made of a Greek-to-Latin translation from the 12th century at Palermo, Sicily. The name of the translator is not known other than he was an anonymous medical student from Salerno who was visiting Palermo in order to translate the <a href="/wiki/Almagest" title="Almagest">Almagest</a> to Latin. The Euclid manuscript is extant and quite complete.<sup id="cite_ref-16" class="reference"><a href="#cite_note-16"><span class="cite-bracket">[</span>11<span class="cite-bracket">]</span></a></sup> </p><p>After the translation by Adelard of Bath (known as Adelard I), there was a flurry of translations from Arabic. Notable translators in this period include <a href="/wiki/Herman_of_Carinthia" title="Herman of Carinthia">Herman of Carinthia</a> who wrote an edition around 1140, Robert of Chester (his manuscripts are referred to collectively as Adelard II, written on or before 1251), Johannes de Tinemue,<sup id="cite_ref-17" class="reference"><a href="#cite_note-17"><span class="cite-bracket">[</span>12<span class="cite-bracket">]</span></a></sup> possibly also known as <a href="/wiki/John_of_Tynemouth_(geometer)" title="John of Tynemouth (geometer)">John of Tynemouth</a> (his manuscripts are referred to collectively as Adelard III), late 12th century, and <a href="/wiki/Gerard_of_Cremona" title="Gerard of Cremona">Gerard of Cremona</a> (sometime after 1120 but before 1187). The exact details concerning these translations is still an active area of research.<sup id="cite_ref-FOOTNOTEBusard2005_18-0" class="reference"><a href="#cite_note-FOOTNOTEBusard2005-18"><span class="cite-bracket">[</span>13<span class="cite-bracket">]</span></a></sup><sup class="noprint Inline-Template" style="white-space:nowrap;">[<i><a href="/wiki/Wikipedia:Citing_sources" title="Wikipedia:Citing sources"><span title="This citation requires a reference to the specific page or range of pages in which the material appears. (November 2023)">page needed</span></a></i>]</sup> <a href="/wiki/Campanus_of_Novara" title="Campanus of Novara">Campanus of Novara</a> relied heavily on these Arabic translations to create his edition (sometime before 1260) which ultimately came to dominate Latin editions until the availability of Greek manuscripts in the 16th century. There are more than 100 pre-1482 Campanus manuscripts still available today.<sup id="cite_ref-19" class="reference"><a href="#cite_note-19"><span class="cite-bracket">[</span>14<span class="cite-bracket">]</span></a></sup><sup id="cite_ref-20" class="reference"><a href="#cite_note-20"><span class="cite-bracket">[</span>15<span class="cite-bracket">]</span></a></sup> </p> <figure class="mw-default-size" typeof="mw:File/Thumb"><a href="/wiki/File:Euclid%27s_Elements_1573_Edition.JPG" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/4/49/Euclid%27s_Elements_1573_Edition.JPG/220px-Euclid%27s_Elements_1573_Edition.JPG" decoding="async" width="220" height="165" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/4/49/Euclid%27s_Elements_1573_Edition.JPG/330px-Euclid%27s_Elements_1573_Edition.JPG 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/4/49/Euclid%27s_Elements_1573_Edition.JPG/440px-Euclid%27s_Elements_1573_Edition.JPG 2x" data-file-width="3072" data-file-height="2304" /></a><figcaption>Euclidis – Elementorum libri XV Paris, Hieronymum de Marnef & Guillaume Cavelat, 1573 (second edition after the 1557 ed.); in 8:350, (2)pp. THOMAS–STANFORD, Early Editions of Euclid's <i>Elements</i>, n°32. Mentioned in T.L. Heath's translation. Private collection Hector Zenil.</figcaption></figure><p> The first printed edition appeared in 1482 (based on Campanus's translation),<sup id="cite_ref-FOOTNOTEBusard20051_21-0" class="reference"><a href="#cite_note-FOOTNOTEBusard20051-21"><span class="cite-bracket">[</span>16<span class="cite-bracket">]</span></a></sup> and since then it has been translated into many languages and published in about a thousand different editions. Theon's Greek edition was recovered and <a href="/wiki/List_of_editiones_principes_in_Greek" title="List of editiones principes in Greek">published in 1533</a><sup id="cite_ref-22" class="reference"><a href="#cite_note-22"><span class="cite-bracket">[</span>17<span class="cite-bracket">]</span></a></sup> based on Paris gr. 2343 and Venetus Marcianus 301.<sup id="cite_ref-23" class="reference"><a href="#cite_note-23"><span class="cite-bracket">[</span>18<span class="cite-bracket">]</span></a></sup> In 1570, <a href="/wiki/John_Dee_(mathematician)" class="mw-redirect" title="John Dee (mathematician)">John Dee</a> provided a widely respected "Mathematical Preface", along with copious notes and supplementary material, to the first English edition by <a href="/wiki/Henry_Billingsley" title="Henry Billingsley">Henry Billingsley</a>. </p><p>Copies of the Greek text still exist, some of which can be found in the <a href="/wiki/Vatican_Library" title="Vatican Library">Vatican Library</a> and the <a href="/wiki/Bodleian_Library" title="Bodleian Library">Bodleian Library</a> in Oxford. The manuscripts available are of variable quality, and invariably incomplete. By careful analysis of the translations and originals, hypotheses have been made about the contents of the original text (copies of which are no longer available). </p><p>Ancient texts which refer to the <i>Elements</i> itself, and to other mathematical theories that were current at the time it was written, are also important in this process. Such analyses are conducted by <a href="/wiki/Johan_Ludvig_Heiberg_(historian)" title="Johan Ludvig Heiberg (historian)">J. L. Heiberg</a> and Sir <a href="/wiki/Thomas_Little_Heath" class="mw-redirect" title="Thomas Little Heath">Thomas Little Heath</a> in their editions of the text. </p><p>Also of importance are the <a href="/wiki/Scholia" title="Scholia">scholia</a>, or annotations to the text. These additions, which often distinguished themselves from the main text (depending on the manuscript), gradually accumulated over time as opinions varied upon what was worthy of explanation or further study. </p> <div class="mw-heading mw-heading2"><h2 id="Contents">Contents</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Euclid%27s_Elements&action=edit&section=4" title="Edit section: Contents"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <ul><li>Book 1 contains 5 postulates and 5 common notions, and covers important topics of plane geometry such as the <a href="/wiki/Pythagorean_theorem" title="Pythagorean theorem">Pythagorean theorem</a>, equality of angles and <a href="/wiki/Area" title="Area">areas</a>, parallelism, the sum of the angles in a triangle, and the construction of various geometric figures.</li> <li>Book 2 contains a number of <a href="/wiki/Lemma_(mathematics)" title="Lemma (mathematics)">lemmas</a> concerning the equality of rectangles and squares, sometimes referred to as "<a href="/wiki/Greek_geometric_algebra" class="mw-redirect" title="Greek geometric algebra">geometric algebra</a>", and concludes with a construction of the <a href="/wiki/Golden_ratio" title="Golden ratio">golden ratio</a> and a way of constructing a square equal in area to any rectilineal plane figure.</li> <li>Book 3 deals with circles and their properties: finding the center, <a href="/wiki/Inscribe" class="mw-redirect" title="Inscribe">inscribed</a> angles, <a href="/wiki/Tangent" title="Tangent">tangents</a>, the power of a point, <a href="/wiki/Thales%27_theorem" class="mw-redirect" title="Thales' theorem">Thales' theorem</a>.</li> <li>Book 4 constructs the <a href="/wiki/Incircle" class="mw-redirect" title="Incircle">incircle</a> and <a href="/wiki/Circumcircle" title="Circumcircle">circumcircle</a> of a triangle, as well as <a href="/wiki/Regular_polygon" title="Regular polygon">regular polygons</a> with 4, 5, 6, and 15 sides.</li> <li>Book 5, on proportions of <a href="/wiki/Magnitude_(mathematics)" title="Magnitude (mathematics)">magnitudes</a>, gives the highly sophisticated theory of proportion probably developed by <a href="/wiki/Eudoxus_of_Cnidus" title="Eudoxus of Cnidus">Eudoxus</a>, and proves properties such as "alternation" (if <i>a</i> : <i>b</i> :: <i>c</i> : <i>d</i>, then <i>a</i> : <i>c</i> :: <i>b</i> : <i>d</i>).</li> <li>Book 6 applies proportions to plane geometry, especially the construction and recognition of <a href="/wiki/Similarity_(geometry)" title="Similarity (geometry)">similar</a> figures.</li> <li>Book 7 deals with elementary number theory: <a href="/wiki/Divisibility" class="mw-redirect" title="Divisibility">divisibility</a>, <a href="/wiki/Prime_number" title="Prime number">prime numbers</a> and their relation to <a href="/wiki/Composite_number" title="Composite number">composite numbers</a>, <a href="/wiki/Euclidean_algorithm" title="Euclidean algorithm">Euclid's algorithm</a> for finding the <a href="/wiki/Greatest_common_divisor" title="Greatest common divisor">greatest common divisor</a>, finding the <a href="/wiki/Least_common_multiple" title="Least common multiple">least common multiple</a>.</li> <li>Book 8 deals with the construction and existence of <a href="/wiki/Geometric_progression" title="Geometric progression">geometric sequences</a> of integers.</li> <li>Book 9 applies the results of the preceding two books and gives the <a href="/wiki/Infinitude_of_prime_numbers" class="mw-redirect" title="Infinitude of prime numbers">infinitude of prime numbers</a> and the construction of all even <a href="/wiki/Perfect_number" title="Perfect number">perfect numbers</a>.</li> <li>Book 10 proves the irrationality of the square roots of non-square integers (e.g. <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\sqrt {2}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mn>2</mn> </msqrt> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\sqrt {2}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b4afc1e27d418021bf10898eb44a7f5f315735ff" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:3.098ex; height:3.009ex;" alt="{\displaystyle {\sqrt {2}}}" /></span>) and classifies the square roots of <a href="/wiki/Commensurability_(mathematics)" title="Commensurability (mathematics)">incommensurable</a> lines into thirteen disjoint categories. Euclid here introduces the term "irrational", which has a different meaning than the modern concept of <a href="/wiki/Irrational_number" title="Irrational number">irrational numbers</a>. He also gives a <a href="/wiki/Euclid%27s_formula" class="mw-redirect" title="Euclid's formula">formula</a> to produce <a href="/wiki/Pythagorean_triples" class="mw-redirect" title="Pythagorean triples">Pythagorean triples</a>.<sup id="cite_ref-Joyce_24-0" class="reference"><a href="#cite_note-Joyce-24"><span class="cite-bracket">[</span>19<span class="cite-bracket">]</span></a></sup></li> <li>Book 11 generalizes the results of book 6 to solid figures: perpendicularity, parallelism, volumes and similarity of <a href="/wiki/Parallelepiped" title="Parallelepiped">parallelepipeds</a>.</li> <li>Book 12 studies the volumes of <a href="/wiki/Cone_(geometry)" class="mw-redirect" title="Cone (geometry)">cones</a>, <a href="/wiki/Pyramid_(geometry)" title="Pyramid (geometry)">pyramids</a>, and <a href="/wiki/Cylinder_(geometry)" class="mw-redirect" title="Cylinder (geometry)">cylinders</a> in detail by using the <a href="/wiki/Method_of_exhaustion" title="Method of exhaustion">method of exhaustion</a>, a precursor to <a href="/wiki/Integral" title="Integral">integration</a>, and shows, for example, that the volume of a cone is a third of the volume of the corresponding cylinder. It concludes by showing that the volume of a <a href="/wiki/Sphere" title="Sphere">sphere</a> is proportional to the cube of its radius (in modern language) by approximating its volume by a union of many pyramids.</li> <li>Book 13 constructs the five regular <a href="/wiki/Platonic_solid" title="Platonic solid">Platonic solids</a> inscribed in a sphere and compares the ratios of their edges to the radius of the sphere.</li></ul> <table class="wikitable"> <caption>Summary Contents of Euclid's <i>Elements</i> </caption> <tbody><tr> <th>Book </th> <th>I </th> <th>II </th> <th>III </th> <th>IV </th> <th>V </th> <th>VI </th> <th>VII </th> <th>VIII </th> <th>IX </th> <th>X </th> <th>XI </th> <th>XII </th> <th>XIII </th> <th>Totals </th></tr> <tr> <th>Definitions </th> <td>23</td> <td>2</td> <td>11</td> <td>7</td> <td>18</td> <td>4</td> <td>22</td> <td>–</td> <td>–</td> <td>16</td> <td>28</td> <td>–</td> <td>–</td> <td>131 </td></tr> <tr> <th>Postulates </th> <td>5</td> <td>–</td> <td>–</td> <td>–</td> <td>–</td> <td>–</td> <td>–</td> <td>–</td> <td>–</td> <td>–</td> <td>–</td> <td>–</td> <td>–</td> <td>5 </td></tr> <tr> <th>Common Notions </th> <td>5</td> <td>–</td> <td>–</td> <td>–</td> <td>–</td> <td>–</td> <td>–</td> <td>–</td> <td>–</td> <td>–</td> <td>–</td> <td>–</td> <td>–</td> <td>5 </td></tr> <tr> <th>Propositions </th> <td>48</td> <td>14</td> <td>37</td> <td>16</td> <td>25</td> <td>33</td> <td>39</td> <td>27</td> <td>36</td> <td>115</td> <td>39</td> <td>18</td> <td>18</td> <td>465 </td></tr></tbody></table> <div class="mw-heading mw-heading2"><h2 id="Euclid's_method_and_style_of_presentation"><span id="Euclid.27s_method_and_style_of_presentation"></span>Euclid's method and style of presentation</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Euclid%27s_Elements&action=edit&section=5" title="Edit section: Euclid's method and style of presentation"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <style data-mw-deduplicate="TemplateStyles:r1224211176">.mw-parser-output .quotebox{background-color:#F9F9F9;border:1px solid #aaa;box-sizing:border-box;padding:10px;font-size:88%;max-width:100%}.mw-parser-output .quotebox.floatleft{margin:.5em 1.4em .8em 0}.mw-parser-output .quotebox.floatright{margin:.5em 0 .8em 1.4em}.mw-parser-output .quotebox.centered{overflow:hidden;position:relative;margin:.5em auto .8em auto}.mw-parser-output .quotebox.floatleft span,.mw-parser-output .quotebox.floatright span{font-style:inherit}.mw-parser-output .quotebox>blockquote{margin:0;padding:0;border-left:0;font-family:inherit;font-size:inherit}.mw-parser-output .quotebox-title{text-align:center;font-size:110%;font-weight:bold}.mw-parser-output .quotebox-quote>:first-child{margin-top:0}.mw-parser-output .quotebox-quote:last-child>:last-child{margin-bottom:0}.mw-parser-output .quotebox-quote.quoted:before{font-family:"Times New Roman",serif;font-weight:bold;font-size:large;color:gray;content:" “ ";vertical-align:-45%;line-height:0}.mw-parser-output .quotebox-quote.quoted:after{font-family:"Times New Roman",serif;font-weight:bold;font-size:large;color:gray;content:" ” ";line-height:0}.mw-parser-output .quotebox .left-aligned{text-align:left}.mw-parser-output .quotebox .right-aligned{text-align:right}.mw-parser-output .quotebox .center-aligned{text-align:center}.mw-parser-output .quotebox .quote-title,.mw-parser-output .quotebox .quotebox-quote{display:block}.mw-parser-output .quotebox cite{display:block;font-style:normal}@media screen and (max-width:640px){.mw-parser-output .quotebox{width:100%!important;margin:0 0 .8em!important;float:none!important}}</style><div class="quotebox pullquote floatright" style=";"> <blockquote class="quotebox-quote left-aligned" style=""> <p>• "To draw a straight line from any point to any point."<br /> • "To describe a circle with any center and distance." </p> </blockquote> <div style="padding-bottom: 0; padding-top: 0.5em"><cite class="left-aligned" style="">Euclid, <i>Elements</i>, Book I, Postulates 1 & 3.<sup id="cite_ref-FOOTNOTEHartshorne200018_25-0" class="reference"><a href="#cite_note-FOOTNOTEHartshorne200018-25"><span class="cite-bracket">[</span>20<span class="cite-bracket">]</span></a></sup></cite></div> </div> <figure class="mw-default-size" typeof="mw:File/Thumb"><a href="/wiki/File:HexagonConstructionAni.gif" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/c/cf/HexagonConstructionAni.gif/220px-HexagonConstructionAni.gif" decoding="async" width="220" height="212" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/c/cf/HexagonConstructionAni.gif 1.5x" data-file-width="234" data-file-height="225" /></a><figcaption>An animation showing how Euclid constructed a hexagon (Book IV, Proposition 15). Every two-dimensional figure in the <i>Elements</i> can be constructed using only a compass and straightedge.<sup id="cite_ref-FOOTNOTEHartshorne200018_25-1" class="reference"><a href="#cite_note-FOOTNOTEHartshorne200018-25"><span class="cite-bracket">[</span>20<span class="cite-bracket">]</span></a></sup></figcaption></figure> <figure class="mw-default-size" typeof="mw:File/Thumb"><a href="/wiki/File:Euclid_Vat_ms_no_190_I_prop_47.jpg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/a/af/Euclid_Vat_ms_no_190_I_prop_47.jpg/220px-Euclid_Vat_ms_no_190_I_prop_47.jpg" decoding="async" width="220" height="135" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/a/af/Euclid_Vat_ms_no_190_I_prop_47.jpg/330px-Euclid_Vat_ms_no_190_I_prop_47.jpg 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/a/af/Euclid_Vat_ms_no_190_I_prop_47.jpg/440px-Euclid_Vat_ms_no_190_I_prop_47.jpg 2x" data-file-width="2200" data-file-height="1351" /></a><figcaption>Scan of pages demonstrating <a href="/wiki/Pythagorean_theorem" title="Pythagorean theorem">Pythagorean theorem</a> from manuscript held in the <a href="/wiki/Vatican_Library" title="Vatican Library">Vatican Library</a></figcaption></figure> <p>Euclid's <a href="/wiki/Axiomatic_system#Axiomatic_method" title="Axiomatic system">axiomatic approach</a> and <a href="/wiki/Euclidean_geometry#Methods_of_proof" title="Euclidean geometry">constructive methods</a> were widely influential. </p><p>Many of Euclid's propositions were constructive, demonstrating the existence of some figure by detailing the steps he used to construct the object using a <a href="/wiki/Compass-and-straightedge_construction" class="mw-redirect" title="Compass-and-straightedge construction">compass and straightedge</a>. His constructive approach appears even in his geometry's postulates, as the first and third postulates stating the existence of a line and circle are constructive. Instead of stating that lines and circles exist per his prior definitions, he states that it is possible to 'construct' a line and circle. It also appears that, for him to use a figure in one of his proofs, he needs to construct it in an earlier proposition. For example, he proves the Pythagorean theorem by first inscribing a square on the sides of a right triangle, but only after constructing a square on a given line one proposition earlier.<sup id="cite_ref-FOOTNOTEHartshorne200018–20_26-0" class="reference"><a href="#cite_note-FOOTNOTEHartshorne200018–20-26"><span class="cite-bracket">[</span>21<span class="cite-bracket">]</span></a></sup> </p><p>As was common in ancient mathematical texts, when a proposition needed <a href="/wiki/Mathematical_proof" title="Mathematical proof">proof</a> in several different cases, Euclid often proved only one of them (often the most difficult), leaving the others to the reader. Later editors such as <a href="/wiki/Theon_of_Alexandria" title="Theon of Alexandria">Theon</a> often interpolated their own proofs of these cases. </p><p>Euclid's presentation was limited by the mathematical ideas and notations in common currency in his era, and this causes the treatment to seem awkward to the modern reader in some places. For example, there was no notion of an angle greater than two right angles,<sup id="cite_ref-FOOTNOTEBall1915[httpsarchiveorgdetailsshortaccountofhi00ballrichpage55_p.&nbsp;55]_27-0" class="reference"><a href="#cite_note-FOOTNOTEBall1915[httpsarchiveorgdetailsshortaccountofhi00ballrichpage55_p.&nbsp;55]-27"><span class="cite-bracket">[</span>22<span class="cite-bracket">]</span></a></sup> the number 1 was sometimes treated separately from other positive integers, and as multiplication was treated geometrically he did not use the product of more than 3 different numbers. The geometrical treatment of number theory may have been because the alternative would have been the extremely awkward <a href="/wiki/Greek_numerals" title="Greek numerals">Alexandrian system of numerals</a>.<sup id="cite_ref-FOOTNOTEBall1915<span_class="nowrap">[httpsarchiveorgdetailsshortaccountofhi00ballrichpage54_pp._54],_[httpsarchiveorgdetailsshortaccountofhi00ballrichpage58_58],_[httpsarchiveorgdetailsshortaccountofhi00ballrichpage127_127]</span>_28-0" class="reference"><a href="#cite_note-FOOTNOTEBall1915<span_class="nowrap">[httpsarchiveorgdetailsshortaccountofhi00ballrichpage54_pp._54],_[httpsarchiveorgdetailsshortaccountofhi00ballrichpage58_58],_[httpsarchiveorgdetailsshortaccountofhi00ballrichpage127_127]</span>-28"><span class="cite-bracket">[</span>23<span class="cite-bracket">]</span></a></sup> </p><p>The presentation of each result is given in a stylized form, which, although not invented by Euclid, is recognized as typically classical. It has six different parts: First is the 'enunciation', which states the result in general terms (i.e., the statement of the proposition). Then comes the 'setting-out', which gives the figure and denotes particular geometrical objects by letters. Next comes the 'definition' or 'specification', which restates the enunciation in terms of the particular figure. Then the 'construction' or 'machinery' follows. Here, the original figure is extended to forward the proof. Then, the 'proof' itself follows. Finally, the 'conclusion' connects the proof to the enunciation by stating the specific conclusions drawn in the proof, in the general terms of the enunciation.<sup id="cite_ref-FOOTNOTEHeath1963216_29-0" class="reference"><a href="#cite_note-FOOTNOTEHeath1963216-29"><span class="cite-bracket">[</span>24<span class="cite-bracket">]</span></a></sup> </p><p>No indication is given of the method of reasoning that led to the result, although the <i><a href="/wiki/Data_(Euclid)" class="mw-redirect" title="Data (Euclid)">Data</a></i> does provide instruction about how to approach the types of problems encountered in the first four books of the <i>Elements</i>.<sup id="cite_ref-FOOTNOTEBall1915[httpsarchiveorgdetailsshortaccountofhi00ballrichpage54_p.&nbsp;54]_8-1" class="reference"><a href="#cite_note-FOOTNOTEBall1915[httpsarchiveorgdetailsshortaccountofhi00ballrichpage54_p.&nbsp;54]-8"><span class="cite-bracket">[</span>4<span class="cite-bracket">]</span></a></sup> Some scholars have tried to find fault in Euclid's use of figures in his proofs, accusing him of writing proofs that depended on the specific figures drawn rather than the general underlying logic, especially concerning Proposition II of Book I. However, Euclid's original proof of this proposition, is general, valid, and does not depend on the figure used as an example to illustrate one given configuration.<sup id="cite_ref-FOOTNOTEToussaint199312–23_30-0" class="reference"><a href="#cite_note-FOOTNOTEToussaint199312–23-30"><span class="cite-bracket">[</span>25<span class="cite-bracket">]</span></a></sup> </p> <div class="mw-heading mw-heading2"><h2 id="Influence">Influence</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Euclid%27s_Elements&action=edit&section=6" title="Edit section: Influence"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <figure class="mw-default-size" typeof="mw:File/Thumb"><a href="/wiki/File:Euclid%27s_Elements,_1482.jpg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/d/d4/Euclid%27s_Elements%2C_1482.jpg/220px-Euclid%27s_Elements%2C_1482.jpg" decoding="async" width="220" height="161" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/d/d4/Euclid%27s_Elements%2C_1482.jpg/330px-Euclid%27s_Elements%2C_1482.jpg 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/d/d4/Euclid%27s_Elements%2C_1482.jpg/440px-Euclid%27s_Elements%2C_1482.jpg 2x" data-file-width="1536" data-file-height="1122" /></a><figcaption>A page with marginalia from the first printed edition of <i>Elements</i>, printed by <a href="/wiki/Erhard_Ratdolt" title="Erhard Ratdolt">Erhard Ratdolt</a> in 1482</figcaption></figure> <p>The <i>Elements</i> is still considered a masterpiece in the application of <a href="/wiki/Logic" title="Logic">logic</a> to <a href="/wiki/Mathematics" title="Mathematics">mathematics</a>. In historical context, it has proven enormously influential in many areas of <a href="/wiki/Science" title="Science">science</a>. Scientists <a href="/wiki/Nicolaus_Copernicus" title="Nicolaus Copernicus">Nicolaus Copernicus</a>, <a href="/wiki/Johannes_Kepler" title="Johannes Kepler">Johannes Kepler</a>, <a href="/wiki/Galileo_Galilei" title="Galileo Galilei">Galileo Galilei</a>, <a href="/wiki/Albert_Einstein" title="Albert Einstein">Albert Einstein</a> and Sir <a href="/wiki/Isaac_Newton" title="Isaac Newton">Isaac Newton</a> were all influenced by the <i>Elements</i>, and applied their knowledge of it to their work.<sup id="cite_ref-31" class="reference"><a href="#cite_note-31"><span class="cite-bracket">[</span>26<span class="cite-bracket">]</span></a></sup><sup id="cite_ref-32" class="reference"><a href="#cite_note-32"><span class="cite-bracket">[</span>27<span class="cite-bracket">]</span></a></sup> Mathematicians and philosophers, such as <a href="/wiki/Thomas_Hobbes" title="Thomas Hobbes">Thomas Hobbes</a>, <a href="/wiki/Baruch_Spinoza" title="Baruch Spinoza">Baruch Spinoza</a>, <a href="/wiki/Alfred_North_Whitehead" title="Alfred North Whitehead">Alfred North Whitehead</a>, and <a href="/wiki/Bertrand_Russell" title="Bertrand Russell">Bertrand Russell</a>, have attempted to create their own foundational "Elements" for their respective disciplines, by adopting the axiomatized deductive structures that Euclid's work introduced. </p><p>The austere beauty of Euclidean geometry has been seen by many in western culture as a glimpse of an otherworldly system of perfection and certainty. <a href="/wiki/Abraham_Lincoln" title="Abraham Lincoln">Abraham Lincoln</a> kept a copy of Euclid in his saddlebag, and studied it late at night by lamplight; he related that he said to himself, "You never can make a lawyer if you do not understand what demonstrate means; and I left my situation in <a href="/wiki/Springfield,_Illinois" title="Springfield, Illinois">Springfield</a>, went home to my father's house, and stayed there till I could give any proposition in the six books of Euclid at sight".<sup id="cite_ref-FOOTNOTEKetcham1901_33-0" class="reference"><a href="#cite_note-FOOTNOTEKetcham1901-33"><span class="cite-bracket">[</span>28<span class="cite-bracket">]</span></a></sup><sup id="cite_ref-34" class="reference"><a href="#cite_note-34"><span class="cite-bracket">[</span>29<span class="cite-bracket">]</span></a></sup> <a href="/wiki/Edna_St._Vincent_Millay" title="Edna St. Vincent Millay">Edna St. Vincent Millay</a> wrote in her sonnet "Euclid alone has looked on Beauty bare", "O blinding hour, O holy, terrible day, / When first the shaft into his vision shone / Of light anatomized!". Albert Einstein recalled a copy of the <i>Elements</i> and a magnetic compass as two gifts that had a great influence on him as a boy, referring to the Euclid as the "holy little geometry book".<sup id="cite_ref-35" class="reference"><a href="#cite_note-35"><span class="cite-bracket">[</span>30<span class="cite-bracket">]</span></a></sup><sup id="cite_ref-36" class="reference"><a href="#cite_note-36"><span class="cite-bracket">[</span>31<span class="cite-bracket">]</span></a></sup> </p><p>The success of the <i>Elements</i> is due primarily to its logical presentation of most of the mathematical knowledge available to Euclid. Much of the material is not original to him, although many of the proofs are his. However, Euclid's systematic development of his subject, from a small set of axioms to deep results, and the consistency of his approach throughout the <i>Elements</i>, encouraged its use as a textbook for about 2,000 years. The <i>Elements</i> still influences modern geometry books. Furthermore, its logical, axiomatic approach and rigorous proofs remain the cornerstone of mathematics. </p> <div class="mw-heading mw-heading3"><h3 id="In_modern_mathematics">In modern mathematics</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Euclid%27s_Elements&action=edit&section=7" title="Edit section: In modern mathematics"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>One of the most notable influences of Euclid on modern mathematics is the discussion of the <a href="/wiki/Parallel_postulate" title="Parallel postulate">parallel postulate</a>. In Book I, Euclid lists five postulates, the fifth of which stipulates </p> <style data-mw-deduplicate="TemplateStyles:r1244412712">.mw-parser-output .templatequote{overflow:hidden;margin:1em 0;padding:0 32px}.mw-parser-output .templatequotecite{line-height:1.5em;text-align:left;margin-top:0}@media(min-width:500px){.mw-parser-output .templatequotecite{padding-left:1.6em}}</style><blockquote class="templatequote"><p>If a <a href="/wiki/Line_segment" title="Line segment">line segment</a> intersects two straight <a href="/wiki/Line_(mathematics)" class="mw-redirect" title="Line (mathematics)">lines</a> forming two interior angles on the same side that sum to less than two <a href="/wiki/Right_angle" title="Right angle">right angles</a>, then the two lines, if extended indefinitely, meet on that side on which the angles sum to less than two right angles.</p></blockquote> <figure 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="263" height="66" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/7/78/Noneuclid.svg/500px-Noneuclid.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/7/78/Noneuclid.svg/526px-Noneuclid.svg.png 2x" data-file-width="663" data-file-height="167" /></a><figcaption>The different versions of the parallel postulate result in different geometries.</figcaption></figure> <p>This postulate plagued mathematicians for centuries due to its apparent complexity compared with the other four postulates. Many attempts were made to prove the fifth postulate based on the other four, but they never succeeded. Eventually in 1829, mathematician <a href="/wiki/Nikolai_Lobachevsky" title="Nikolai Lobachevsky">Nikolai Lobachevsky</a> published a description of acute geometry (or <a href="/wiki/Hyperbolic_geometry" title="Hyperbolic geometry">hyperbolic geometry</a>), a geometry which assumed a different form of the parallel postulate. It is in fact possible to create a valid geometry without the fifth postulate entirely, or with different versions of the fifth postulate (<a href="/wiki/Elliptic_geometry" title="Elliptic geometry">elliptic geometry</a>). If one takes the fifth postulate as a given, the result is <a href="/wiki/Euclidean_geometry" title="Euclidean geometry">Euclidean geometry</a>.<sup class="noprint Inline-Template Template-Fact" style="white-space:nowrap;">[<i><a href="/wiki/Wikipedia:Citation_needed" title="Wikipedia:Citation needed"><span title="This claim needs references to reliable sources. (January 2022)">citation needed</span></a></i>]</sup> </p> <div class="mw-heading mw-heading2"><h2 id="Criticism">Criticism</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Euclid%27s_Elements&action=edit&section=8" title="Edit section: Criticism"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>Euclid's <i>Elements</i> contains errors. Some of the foundational theorems are proved using axioms that Euclid did not state explicitly. A few proofs have errors, by relying on assumptions that are intuitive but not explicitly proven. Mathematician and historian <a href="/wiki/W._W._Rouse_Ball" title="W. W. Rouse Ball">W. W. Rouse Ball</a> put the criticisms in perspective, remarking that "the fact that for two thousand years [the <i>Elements</i>] was the usual text-book on the subject raises a strong presumption that it is not unsuitable for that purpose."<sup id="cite_ref-FOOTNOTEBall1915[httpsarchiveorgdetailsshortaccountofhi00ballrichpage55_p.&nbsp;55]_27-1" class="reference"><a href="#cite_note-FOOTNOTEBall1915[httpsarchiveorgdetailsshortaccountofhi00ballrichpage55_p.&nbsp;55]-27"><span class="cite-bracket">[</span>22<span class="cite-bracket">]</span></a></sup> </p><p>Later editors have added Euclid's implicit axiomatic assumptions in their list of formal axioms.<sup id="cite_ref-FOOTNOTEHeath1956a62_37-0" class="reference"><a href="#cite_note-FOOTNOTEHeath1956a62-37"><span class="cite-bracket">[</span>32<span class="cite-bracket">]</span></a></sup> For example, in the first construction of Book 1, Euclid used a premise that was neither postulated nor proved: that two circles with centers at the distance of their radius will intersect in two points.<sup id="cite_ref-FOOTNOTEHeath1956a242_38-0" class="reference"><a href="#cite_note-FOOTNOTEHeath1956a242-38"><span class="cite-bracket">[</span>33<span class="cite-bracket">]</span></a></sup> </p><p>Known errors in Euclid date to at least 1882, when <a href="/wiki/Pasch%27s_axiom" title="Pasch's axiom">Pasch published his missing axiom</a>. </p><p>Early attempts to find all the errors include <a href="/wiki/Hilbert%27s_axioms" title="Hilbert's axioms">Hilbert's geometry axioms</a> and <a href="/wiki/Tarski%27s_axioms" title="Tarski's axioms">Tarski's</a>. In 2017, Michael Beeson et al. used computer <a href="/wiki/Proof_assistant" title="Proof assistant">proof assistants</a> to create a new set of axioms similar to Euclid's and generate proofs that were valid with those axioms.<sup id="cite_ref-39" class="reference"><a href="#cite_note-39"><span class="cite-bracket">[</span>34<span class="cite-bracket">]</span></a></sup> </p><p>Beeson et al. checked only Book I and found these errors: missing axioms, superfluous axioms, gaps in logic (such as failing to prove points were colinear), missing theorems (such as an angle cannot be less than itself), and outright bad proofs. The bad proofs were in Book I, Proof 7 and Book I, Proposition 9. </p> <div class="mw-heading mw-heading2"><h2 id="Apocrypha">Apocrypha</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Euclid%27s_Elements&action=edit&section=9" title="Edit section: Apocrypha"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>It was not uncommon in ancient times to attribute works to celebrated authors that were not written by them. It is by these means that the <a href="/wiki/Apocryphal" class="mw-redirect" title="Apocryphal">apocryphal</a> books XIV and XV of the <i>Elements</i> were sometimes included in the collection.<sup id="cite_ref-FOOTNOTEBoyer1991118–119_40-0" class="reference"><a href="#cite_note-FOOTNOTEBoyer1991118–119-40"><span class="cite-bracket">[</span>35<span class="cite-bracket">]</span></a></sup> The spurious Book XIV was probably written by <a href="/wiki/Hypsicles" title="Hypsicles">Hypsicles</a> on the basis of a treatise by <a href="/wiki/Apollonius_of_Perga" title="Apollonius of Perga">Apollonius</a>. The book continues Euclid's comparison of regular solids inscribed in spheres, with the chief result being that the ratio of the surfaces of the <a href="/wiki/Dodecahedron" title="Dodecahedron">dodecahedron</a> and <a href="/wiki/Icosahedron" title="Icosahedron">icosahedron</a> inscribed in the same sphere is the same as the ratio of their volumes, the ratio being <span class="mwe-math-element"><span class="mwe-math-mathml-display mwe-math-mathml-a11y" style="display: none;"><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\sqrt {\frac {10}{3(5-{\sqrt {5}})}}}={\sqrt {\frac {5+{\sqrt {5}}}{6}}}.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mfrac> <mn>10</mn> <mrow> <mn>3</mn> <mo stretchy="false">(</mo> <mn>5</mn> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mn>5</mn> </msqrt> </mrow> <mo stretchy="false">)</mo> </mrow> </mfrac> </msqrt> </mrow> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mfrac> <mrow> <mn>5</mn> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mn>5</mn> </msqrt> </mrow> </mrow> <mn>6</mn> </mfrac> </msqrt> </mrow> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\sqrt {\frac {10}{3(5-{\sqrt {5}})}}}={\sqrt {\frac {5+{\sqrt {5}}}{6}}}.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/79000800482e38e87a5f85c10f0594f15302dadd" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.505ex; width:27.239ex; height:8.343ex;" alt="{\displaystyle {\sqrt {\frac {10}{3(5-{\sqrt {5}})}}}={\sqrt {\frac {5+{\sqrt {5}}}{6}}}.}" /></span> </p><p>The spurious Book XV was probably written, at least in part, by <a href="/wiki/Isidore_of_Miletus" title="Isidore of Miletus">Isidore of Miletus</a>. This book covers topics such as counting the number of edges and solid angles in the regular solids, and finding the measure of dihedral angles of faces that meet at an edge.<sup id="cite_ref-41" class="reference"><a href="#cite_note-41"><span class="cite-bracket">[</span>f<span class="cite-bracket">]</span></a></sup> </p> <div class="mw-heading mw-heading2"><h2 id="Editions">Editions</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Euclid%27s_Elements&action=edit&section=10" title="Edit section: Editions"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <figure class="mw-default-size" typeof="mw:File/Thumb"><a href="/wiki/File:Ricci_Guangqi_2.jpg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/e/ec/Ricci_Guangqi_2.jpg/250px-Ricci_Guangqi_2.jpg" decoding="async" width="220" height="329" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/e/ec/Ricci_Guangqi_2.jpg/330px-Ricci_Guangqi_2.jpg 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/e/ec/Ricci_Guangqi_2.jpg/500px-Ricci_Guangqi_2.jpg 2x" data-file-width="3338" data-file-height="4988" /></a><figcaption>The <a href="/wiki/Italy" title="Italy">Italian</a> <a href="/wiki/Jesuit" class="mw-redirect" title="Jesuit">Jesuit</a> <a href="/wiki/Matteo_Ricci" title="Matteo Ricci">Matteo Ricci</a> (left) and the Chinese mathematician <a href="/wiki/Xu_Guangqi" title="Xu Guangqi">Xu Guangqi</a> (right) published the first <a href="/wiki/Chinese_language" title="Chinese language">Chinese</a> edition of <i>Euclid's Elements</i> (<span title="Chinese-language romanization"><i lang="zh-Latn">Jīhé yuánběn</i></span> <span title="Chinese-language text"><span lang="zh">幾何原本</span></span>) in 1607.</figcaption></figure> <ul><li>4th century, <a href="/wiki/Theon_of_Alexandria" title="Theon of Alexandria">Theon of Alexandria</a>, 888 AD manuscript extant.</li> <li>9th century, Pre-Theon <a href="/wiki/Fran%C3%A7ois_Peyrard" title="François Peyrard">François Peyrard</a> Vaticanus Graecus 190</li> <li>Many medieval editions, pre 1482</li> <li>1460s, <a href="/wiki/Regiomontanus" title="Regiomontanus">Regiomontanus</a> (incomplete)</li> <li>1482, <a href="/wiki/Erhard_Ratdolt" title="Erhard Ratdolt">Erhard Ratdolt</a> (Venice), <i><a href="/wiki/Editio_princeps" title="Editio princeps">editio princeps</a></i> (in Latin)<sup id="cite_ref-42" class="reference"><a href="#cite_note-42"><span class="cite-bracket">[</span>36<span class="cite-bracket">]</span></a></sup><sup id="cite_ref-43" class="reference"><a href="#cite_note-43"><span class="cite-bracket">[</span>37<span class="cite-bracket">]</span></a></sup></li> <li>1533, <i><a href="/wiki/Editio_princeps" title="Editio princeps">editio princeps</a></i> of the Greek text by <a href="/wiki/Simon_Grynaeus" title="Simon Grynaeus">Simon Grynäus</a><sup id="cite_ref-44" class="reference"><a href="#cite_note-44"><span class="cite-bracket">[</span>38<span class="cite-bracket">]</span></a></sup></li> <li>1557, by Jean Magnien and <a href="/w/index.php?title=Pierre_de_Montdor%C3%A9&action=edit&redlink=1" class="new" title="Pierre de Montdoré (page does not exist)">Pierre de Montdoré</a><span class="noprint" style="font-size:85%; font-style: normal;"> [<a href="https://fr.wikipedia.org/wiki/Pierre_de_Montdor%C3%A9" class="extiw" title="fr:Pierre de Montdoré">fr</a>]</span>, reviewed by Stephanus Gracilis (only propositions, no full proofs, includes original Greek and the Latin translation)</li> <li>1572, <a href="/wiki/Federico_Commandino" title="Federico Commandino">Commandinus</a> Latin edition</li> <li>1574, <a href="/wiki/Christoph_Clavius" class="mw-redirect" title="Christoph Clavius">Christoph Clavius</a></li> <li>1883–1888, <a href="/wiki/Johan_Ludvig_Heiberg_(historian)" title="Johan Ludvig Heiberg (historian)">Johan Ludvig Heiberg</a></li></ul> <div class="mw-heading mw-heading3"><h3 id="Translations">Translations</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Euclid%27s_Elements&action=edit&section=11" title="Edit section: Translations"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <div class="mw-heading mw-heading4"><h4 id="English">English</h4><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Euclid%27s_Elements&action=edit&section=12" title="Edit section: English"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <ol><li>1570, <a href="/wiki/Henry_Billingsley" title="Henry Billingsley">Henry Billingsley</a></li> <li>1651, <a href="/wiki/Thomas_Rudd" title="Thomas Rudd">Thomas Rudd</a></li> <li>1660, <a href="/wiki/Isaac_Barrow" title="Isaac Barrow">Isaac Barrow</a></li> <li>1661, John Leeke and Geo. Serle</li> <li>1685, <a href="/wiki/William_Hallifax" title="William Hallifax">William Hallifax</a></li> <li>1705, <a href="/wiki/Charles_Scarborough" title="Charles Scarborough">Charles Scarborough</a></li> <li>1708, <a href="/wiki/John_Keill" title="John Keill">John Keill</a></li> <li>1714, W. Whiston</li> <li>1756, <a href="/wiki/Robert_Simson" title="Robert Simson">Robert Simson</a></li> <li>1781, 1788 James Williamson</li> <li>1781, William Austin</li> <li>1795, <a href="/wiki/John_Playfair" title="John Playfair">John Playfair</a></li> <li>1826, George Phillips</li> <li>1828, <a href="/wiki/Dionysius_Lardner" title="Dionysius Lardner">Dionysius Lardner</a></li> <li>1833, <a href="/wiki/Thomas_Perronet_Thompson" title="Thomas Perronet Thompson">Thomas Perronet Thompson</a></li> <li>1862, <a href="/wiki/Isaac_Todhunter" title="Isaac Todhunter">Isaac Todhunter</a></li> <li>1908, <a href="/wiki/Thomas_Little_Heath" class="mw-redirect" title="Thomas Little Heath">Thomas Little Heath</a> (revised in 1926) from <a href="/wiki/Johan_Ludvig_Heiberg_(historian)" title="Johan Ludvig Heiberg (historian)">Johan Ludvig Heiberg</a>'s edition</li> <li>1939, <a href="/wiki/R._Catesby_Taliaferro" title="R. Catesby Taliaferro">R. Catesby Taliaferro</a></li></ol> <div class="mw-heading mw-heading4"><h4 id="Other_languages">Other languages</h4><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Euclid%27s_Elements&action=edit&section=13" title="Edit section: Other languages"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <ul><li>1505, <a href="/w/index.php?title=Bartolomeo_Zamberti&action=edit&redlink=1" class="new" title="Bartolomeo Zamberti (page does not exist)">Bartolomeo Zamberti</a><span class="noprint" style="font-size:85%; font-style: normal;"> [<a href="https://de.wikipedia.org/wiki/Bartolomeo_Zamberti" class="extiw" title="de:Bartolomeo Zamberti">de</a>]</span> (Latin)</li> <li>1543, <a href="/wiki/Nicolo_Tartaglia" title="Nicolo Tartaglia">Nicolo Tartaglia</a> (Italian)</li> <li>1557, Jean Magnien and Pierre de Montdoré, reviewed by Stephanus Gracilis (Greek to Latin)</li> <li>1558, <a href="/wiki/Johann_Scheubel" class="mw-redirect" title="Johann Scheubel">Johann Scheubel</a> (German)</li> <li>1562, Jacob Kündig (German)</li> <li>1562, Wilhelm Holtzmann (German)</li> <li>1564–1566, <a href="/w/index.php?title=Pierre_Forcadel&action=edit&redlink=1" class="new" title="Pierre Forcadel (page does not exist)">Pierre Forcadel</a><span class="noprint" style="font-size:85%; font-style: normal;"> [<a href="https://fr.wikipedia.org/wiki/Pierre_Forcadel" class="extiw" title="fr:Pierre Forcadel">fr</a>]</span> de Béziers (French)</li> <li>1572, <a href="/wiki/Federico_Commandino" title="Federico Commandino">Commandinus</a> (Latin)</li> <li>1575, Commandinus (Italian)</li> <li>1576, <a href="/wiki/Rodrigo_de_Zamorano" class="mw-redirect" title="Rodrigo de Zamorano">Rodrigo de Zamorano</a> (Spanish)</li> <li>1594, <a href="/wiki/Typographia_Medicea" class="mw-redirect" title="Typographia Medicea">Typographia Medicea</a> (edition of the Arabic translation of <i>The Recension of Euclid's "Elements"</i><sup id="cite_ref-FOOTNOTENasir_al-Din_al-Tusi1594_45-0" class="reference"><a href="#cite_note-FOOTNOTENasir_al-Din_al-Tusi1594-45"><span class="cite-bracket">[</span>39<span class="cite-bracket">]</span></a></sup>)</li> <li>1604, <a href="/wiki/Jean_Errard" title="Jean Errard">Jean Errard</a> de Bar-le-Duc (French)</li> <li>1606, Jan Pieterszoon Dou (Dutch)</li> <li>1607, <a href="/wiki/Matteo_Ricci" title="Matteo Ricci">Matteo Ricci</a>, <a href="/wiki/Xu_Guangqi" title="Xu Guangqi">Xu Guangqi</a> (Chinese)</li> <li>1613, <a href="/wiki/Pietro_Cataldi" title="Pietro Cataldi">Pietro Cataldi</a> (Italian)</li> <li>1615, <a href="/wiki/Denis_Henrion" title="Denis Henrion">Denis Henrion</a> (French)</li> <li>1617, Frans van Schooten (Dutch)</li> <li>1637, L. Carduchi (Spanish)</li> <li>1639, <a href="/wiki/Pierre_H%C3%A9rigone" title="Pierre Hérigone">Pierre Hérigone</a> (French)</li> <li>1651, Heinrich Hoffmann (German)</li> <li>1663, Domenico Magni (Italian from Latin)</li> <li>1672, <a href="/wiki/Claude_Fran%C3%A7ois_Milliet_Dechales" class="mw-redirect" title="Claude François Milliet Dechales">Claude François Milliet Dechales</a> (French)</li> <li>1680, Vitale Giordano (Italian)</li> <li>1689, Jacob Knesa (Spanish)</li> <li>1690, Vincenzo Viviani (Italian)</li> <li>1694, Ant. Ernst Burkh v. Pirckenstein (German)</li> <li>1695, Claes Jansz Vooght (Dutch)</li> <li>1697, <a href="/w/index.php?title=Samuel_Reyher&action=edit&redlink=1" class="new" title="Samuel Reyher (page does not exist)">Samuel Reyher</a> (German)</li> <li>1702, Hendrik Coets (Dutch)</li> <li>1714, Chr. Schessler (German)</li> <li>1720s, <a href="/wiki/Jagannatha_Samrat" title="Jagannatha Samrat">Jagannatha Samrat</a> (Sanskrit, based on the Arabic translation of Nasir al-Din al-Tusi)<sup id="cite_ref-FOOTNOTESarma1997460–461_46-0" class="reference"><a href="#cite_note-FOOTNOTESarma1997460–461-46"><span class="cite-bracket">[</span>40<span class="cite-bracket">]</span></a></sup></li> <li>1731, Guido Grandi (abbreviation to Italian)</li> <li>1738, Ivan Satarov (Russian from French)</li> <li>1744, <a href="/w/index.php?title=M%C3%A5rten_Str%C3%B6mer&action=edit&redlink=1" class="new" title="Mårten Strömer (page does not exist)">Mårten Strömer</a> (Swedish)</li> <li>1749, Dechales (Italian)</li> <li>1749, Methodios Anthrakitis (Μεθόδιος Ανθρακίτης) (Greek)</li> <li>1745, Ernest Gottlieb Ziegenbalg (Danish)</li> <li>1752, Leonardo Ximenes (Italian)</li> <li>1763, Pibo Steenstra (Dutch)</li> <li>1768, Angelo Brunelli (Portuguese)</li> <li>1773, 1781, J. F. Lorenz (German)</li> <li>1780, <a href="/wiki/Baruch_Schick_of_Shklov" title="Baruch Schick of Shklov">Baruch Schick of Shklov</a> (Hebrew)<sup id="cite_ref-47" class="reference"><a href="#cite_note-47"><span class="cite-bracket">[</span>41<span class="cite-bracket">]</span></a></sup></li> <li>1789, Pr. Suvoroff nad Yos. Nikitin (Russian from Greek)</li> <li>1803, H.C. Linderup (Danish)</li> <li>1804, <a href="/wiki/Fran%C3%A7ois_Peyrard" title="François Peyrard">François Peyrard</a> (French). Peyrard discovered in 1808 the <i>Vaticanus Graecus 190</i>, which enables him to provide a first definitive version in 1814–1818</li> <li>1807, Józef Czech (Polish based on Greek, Latin and English editions)</li> <li>1807, J. K. F. Hauff (German)</li> <li>1818, <a href="/wiki/Vincenzo_Flauti" title="Vincenzo Flauti">Vincenzo Flauti</a> (Italian)</li> <li>1820, Benjamin of Lesbos (Modern Greek)</li> <li>1828, Joh. Josh and Ign. Hoffmann (German)</li> <li>1833, E. S. Unger (German)</li> <li>1836, H. Falk (Swedish)</li> <li>1844, 1845, 1859, P. R. Bråkenhjelm (Swedish)</li> <li>1850, F. A. A. Lundgren (Swedish)</li> <li>1850, H. A. Witt and M. E. Areskong (Swedish)</li> <li>1865, <a href="/wiki/S%C3%A1muel_Brassai" title="Sámuel Brassai">Sámuel Brassai</a> (Hungarian)</li> <li>1873, Masakuni Yamada (Japanese)</li> <li>1880, <a href="/wiki/Mykhailo_Vaschenko-Zakharchenko" class="mw-redirect" title="Mykhailo Vaschenko-Zakharchenko">Vachtchenko-Zakhartchenko</a> (Russian)</li> <li>1897, Thyra Eibe (Danish)</li> <li>1901, Max Simon (German)</li> <li>1907, František Servít (Czech)<sup id="cite_ref-FOOTNOTEServít1907_48-0" class="reference"><a href="#cite_note-FOOTNOTEServít1907-48"><span class="cite-bracket">[</span>42<span class="cite-bracket">]</span></a></sup></li> <li>1953, 1958, 1975, Evangelos Stamatis (Ευάγγελος Σταμάτης) (Modern Greek)</li> <li>1999, Maja Hudoletnjak Grgić (Book I-VI) (Croatian)<sup id="cite_ref-FOOTNOTEEuklid1999_49-0" class="reference"><a href="#cite_note-FOOTNOTEEuklid1999-49"><span class="cite-bracket">[</span>43<span class="cite-bracket">]</span></a></sup></li> <li>2009, Irineu Bicudo (<a href="/wiki/Portuguese_language" title="Portuguese language">Portuguese</a>)</li> <li>2019, Ali Sinan Sertöz (Turkish)<sup id="cite_ref-FOOTNOTESertöz2019_50-0" class="reference"><a href="#cite_note-FOOTNOTESertöz2019-50"><span class="cite-bracket">[</span>44<span class="cite-bracket">]</span></a></sup></li> <li>2022, Ján Čižmár (Slovak)</li></ul> <div class="mw-heading mw-heading3"><h3 id="Book_I_Editions">Book I Editions</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Euclid%27s_Elements&action=edit&section=14" title="Edit section: Book I Editions"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <ul><li>1886, Euclid Book I Hall & Stevens (English)</li> <li>1891,1896, The Harpur Euclid by Edward Langley and Seys Phillips (English)</li> <li>1949, Henry Regnery Company (English)</li></ul> <div class="mw-heading mw-heading3"><h3 id="Selected_editions_currently_in_print">Selected editions currently in print</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Euclid%27s_Elements&action=edit&section=15" title="Edit section: Selected editions currently in print"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <ul><li><i>Euclid's Elements – All thirteen books complete in one volume</i>, Based on Heath's translation, edited by Dana Densmore, et al. Green Lion Press <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><a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/1-888009-18-7" title="Special:BookSources/1-888009-18-7">1-888009-18-7</a>.</li> <li><i>The Elements: Books I–XIII – Complete and Unabridged</i> (2006), Translated by Sir Thomas Heath, Barnes & Noble <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/0-7607-6312-7" title="Special:BookSources/0-7607-6312-7">0-7607-6312-7</a>.</li> <li><i>The Thirteen Books of Euclid's Elements</i>, translation and commentaries by Heath, Thomas L. (1956) in three volumes. Dover Publications. <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/0-486-60088-2" title="Special:BookSources/0-486-60088-2">0-486-60088-2</a> (vol. 1), <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/0-486-60089-0" title="Special:BookSources/0-486-60089-0">0-486-60089-0</a> (vol. 2), <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/0-486-60090-4" title="Special:BookSources/0-486-60090-4">0-486-60090-4</a> (vol. 3)</li> <li><i>Plane Geometry (Euclid's elements Redux) Books I–VI</i>, based on John Casey's translation, edited by Daniel Callahan, <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/978-1977730039" title="Special:BookSources/978-1977730039">978-1977730039</a></li></ul> <div class="mw-heading mw-heading3"><h3 id="Selected_editions_based_on_Oliver_Byrne's_edition"><span id="Selected_editions_based_on_Oliver_Byrne.27s_edition"></span>Selected editions based on Oliver Byrne's edition</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Euclid%27s_Elements&action=edit&section=16" title="Edit section: Selected editions based on Oliver Byrne's edition"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <ul><li><i>The first six books of the Elements of Euclid</i>, edited by Werner Oechslin, Taschen, 2010, <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/3836517752" title="Special:BookSources/3836517752">3836517752</a>, a facsimile of Byrne (1847).</li> <li><i>Oliver Byrne's Elements of Euclid</i>, Art Meets Science, 2022, <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/978-1528770439" title="Special:BookSources/978-1528770439">978-1528770439</a>, a facsimile of Byrne (1847).</li> <li><a rel="nofollow" class="external text" href="https://www.kroneckerwallis.com/product/euclids-elements-completing-oliver-byrnes-work/"><i>Euclid’s Elements: Completing Oliver Byrne's work</i></a>, Kronecker Wallis, 2019, a modern redrawing extended to the rest of the <i>Elements</i>, originally launched on <a href="/wiki/Kickstarter" title="Kickstarter">Kickstarter</a>.</li></ul> <div class="mw-heading mw-heading3"><h3 id="Free_versions">Free versions</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Euclid%27s_Elements&action=edit&section=17" title="Edit section: Free versions"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <ul><li><i>Euclid's Elements Redux, Volume 1</i>, contains books I–III, based on John Casey's translation.<sup id="cite_ref-FOOTNOTECallahanCasey2015_51-0" class="reference"><a href="#cite_note-FOOTNOTECallahanCasey2015-51"><span class="cite-bracket">[</span>45<span class="cite-bracket">]</span></a></sup></li> <li><i>Euclid's Elements Redux, Volume 2</i>, contains books IV–VIII, based on John Casey's translation.<sup id="cite_ref-FOOTNOTECallahanCasey2015_51-1" class="reference"><a href="#cite_note-FOOTNOTECallahanCasey2015-51"><span class="cite-bracket">[</span>45<span class="cite-bracket">]</span></a></sup></li></ul> <div class="mw-heading mw-heading2"><h2 id="See_also">See also</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Euclid%27s_Elements&action=edit&section=18" title="Edit section: See also"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <ul><li><a href="/wiki/Bride%27s_Chair" title="Bride's Chair">Bride's Chair</a></li></ul> <div class="mw-heading mw-heading2"><h2 id="References">References</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Euclid%27s_Elements&action=edit&section=19" title="Edit section: References"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <div class="mw-heading mw-heading3"><h3 id="Notes">Notes</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Euclid%27s_Elements&action=edit&section=20" title="Edit section: Notes"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <style data-mw-deduplicate="TemplateStyles:r1239543626">.mw-parser-output .reflist{margin-bottom:0.5em;list-style-type:decimal}@media screen{.mw-parser-output .reflist{font-size:90%}}.mw-parser-output .reflist .references{font-size:100%;margin-bottom:0;list-style-type:inherit}.mw-parser-output .reflist-columns-2{column-width:30em}.mw-parser-output .reflist-columns-3{column-width:25em}.mw-parser-output .reflist-columns{margin-top:0.3em}.mw-parser-output .reflist-columns ol{margin-top:0}.mw-parser-output .reflist-columns li{page-break-inside:avoid;break-inside:avoid-column}.mw-parser-output .reflist-upper-alpha{list-style-type:upper-alpha}.mw-parser-output .reflist-upper-roman{list-style-type:upper-roman}.mw-parser-output .reflist-lower-alpha{list-style-type:lower-alpha}.mw-parser-output .reflist-lower-greek{list-style-type:lower-greek}.mw-parser-output .reflist-lower-roman{list-style-type:lower-roman}</style><div class="reflist reflist-lower-alpha"> <div class="mw-references-wrap"><ol class="references"> <li id="cite_note-1"><span class="mw-cite-backlink"><b><a href="#cite_ref-1">^</a></b></span> <span class="reference-text"><a href="#CITEREFWilson2006">Wilson 2006</a>, p. 278 states, "Euclid's Elements subsequently became the basis of all mathematical education, not only in the Roman and Byzantine periods, but right down to the mid-20th century, and it could be argued that it is the most successful textbook ever written."</span> </li> <li id="cite_note-2"><span class="mw-cite-backlink"><b><a href="#cite_ref-2">^</a></b></span> <span class="reference-text"><a href="#CITEREFBoyer1991">Boyer 1991</a>, p. 100 notes, "As teachers at the school he called a band of leading scholars, among whom was the author of the most fabulously successful mathematics textbook ever written – the <i>Elements</i> (<i>Stoichia</i>) of Euclid".</span> </li> <li id="cite_note-3"><span class="mw-cite-backlink"><b><a href="#cite_ref-3">^</a></b></span> <span class="reference-text"><a href="#CITEREFBoyer1991">Boyer 1991</a>, p. 119 notes, "The <i>Elements</i> of Euclid not only was the earliest major Greek mathematical work to come down to us, but also the most influential textbook of all times. [...]The first printed versions of the <i>Elements</i> appeared at Venice in 1482, one of the very earliest of mathematical books to be set in type; it has been estimated that since then at least a thousand editions have been published. Perhaps no book other than the Bible can boast so many editions, and certainly no mathematical work has had an influence comparable with that of Euclid's <i>Elements</i>".</span> </li> <li id="cite_note-5"><span class="mw-cite-backlink"><b><a href="#cite_ref-5">^</a></b></span> <span class="reference-text"><a href="#CITEREFBuntJonesBedient1988">Bunt, Jones & Bedient 1988</a>, p. 142 state, "the <i>Elements</i> became known to Western Europe via the Arabs and the Moors. There, the <i>Elements</i> became the foundation of mathematical education. More than 1000 editions of the <i>Elements</i> are known. In all probability, it is, next to the <i>Bible</i>, the most widely spread book in the civilization of the Western world."</span> </li> <li id="cite_note-15"><span class="mw-cite-backlink"><b><a href="#cite_ref-15">^</a></b></span> <span class="reference-text">One older work claims Adelard disguised himself as a Muslim student to obtain a copy in Muslim Córdoba.<sup id="cite_ref-FOOTNOTEBall1915[httpsarchiveorgdetailsshortaccountofhi00ballrichpage165_p.&nbsp;165]_14-0" class="reference"><a href="#cite_note-FOOTNOTEBall1915[httpsarchiveorgdetailsshortaccountofhi00ballrichpage165_p.&nbsp;165]-14"><span class="cite-bracket">[</span>10<span class="cite-bracket">]</span></a></sup> However, more recent biographical work has turned up no clear documentation that Adelard ever went to <a href="/wiki/Al-Andalus" title="Al-Andalus">Muslim-ruled Spain</a>, although he spent time in Norman-ruled Sicily and Crusader-ruled Antioch, both of which had Arabic-speaking populations. Charles Burnett, <i>Adelard of Bath: Conversations with his Nephew</i> (Cambridge, 1999); Charles Burnett, <i>Adelard of Bath</i> (University of London, 1987).</span> </li> <li id="cite_note-41"><span class="mw-cite-backlink"><b><a href="#cite_ref-41">^</a></b></span> <span class="reference-text"><a href="#CITEREFBoyer1991">Boyer 1991</a>, pp. 118–119 writes, "In ancient times it was not uncommon to attribute to a celebrated author works that were not by him; thus, some versions of Euclid's <i>Elements</i> include a fourteenth and even a fifteenth book, both shown by later scholars to be apocryphal. The so-called Book XIV continues Euclid's comparison of the regular solids inscribed in a sphere, the chief results being that the ratio of the surfaces of the dodecahedron and icosahedron inscribed in the same sphere is the same as the ratio of their volumes, the ratio being that of the edge of the cube to the edge of the icosahedron, that is, <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\sqrt {10/[3(5-{\sqrt {5}})]}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mn>10</mn> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <mo stretchy="false">[</mo> <mn>3</mn> <mo stretchy="false">(</mo> <mn>5</mn> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mn>5</mn> </msqrt> </mrow> <mo stretchy="false">)</mo> <mo stretchy="false">]</mo> </msqrt> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\sqrt {10/[3(5-{\sqrt {5}})]}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ee8ac9d7ad5600039dedb60198e06edc23c7b943" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.671ex; width:17.178ex; height:4.843ex;" alt="{\displaystyle {\sqrt {10/[3(5-{\sqrt {5}})]}}}" /></span>. It is thought that this book may have been composed by Hypsicles on the basis of a treatise (now lost) by Apollonius comparing the dodecahedron and icosahedron. [...] The spurious Book XV, which is inferior, is thought to have been (at least in part) the work of Isidore of Miletus (fl. ca. A.D. 532), architect of the cathedral of Holy Wisdom (Hagia Sophia) at Constantinople. This book also deals with the regular solids, counting the number of edges and solid angles in the solids, and finding the measures of the dihedral angles of faces meeting at an edge.</span> </li> </ol></div></div> <div class="mw-heading mw-heading3"><h3 id="Citations">Citations</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Euclid%27s_Elements&action=edit&section=21" title="Edit section: Citations"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1239543626" /><div class="reflist reflist-columns references-column-width" style="column-width: 30em;"> <ol class="references"> <li id="cite_note-FOOTNOTEBoyer1991100-4"><span class="mw-cite-backlink"><b><a href="#cite_ref-FOOTNOTEBoyer1991100_4-0">^</a></b></span> <span class="reference-text"><a href="#CITEREFBoyer1991">Boyer 1991</a>, p. 100.</span> </li> <li id="cite_note-FOOTNOTERussell2013177-6"><span class="mw-cite-backlink">^ <a href="#cite_ref-FOOTNOTERussell2013177_6-0"><sup><i><b>a</b></i></sup></a> <a href="#cite_ref-FOOTNOTERussell2013177_6-1"><sup><i><b>b</b></i></sup></a> <a href="#cite_ref-FOOTNOTERussell2013177_6-2"><sup><i><b>c</b></i></sup></a></span> <span class="reference-text"><a href="#CITEREFRussell2013">Russell 2013</a>, p. 177.</span> </li> <li id="cite_note-FOOTNOTEVan_der_Waerden1975197-7"><span class="mw-cite-backlink"><b><a href="#cite_ref-FOOTNOTEVan_der_Waerden1975197_7-0">^</a></b></span> <span class="reference-text"><a href="#CITEREFVan_der_Waerden1975">Van der Waerden 1975</a>, p. 197.</span> </li> <li id="cite_note-FOOTNOTEBall1915[httpsarchiveorgdetailsshortaccountofhi00ballrichpage54_p.&nbsp;54]-8"><span class="mw-cite-backlink">^ <a href="#cite_ref-FOOTNOTEBall1915[httpsarchiveorgdetailsshortaccountofhi00ballrichpage54_p.&nbsp;54]_8-0"><sup><i><b>a</b></i></sup></a> <a href="#cite_ref-FOOTNOTEBall1915[httpsarchiveorgdetailsshortaccountofhi00ballrichpage54_p.&nbsp;54]_8-1"><sup><i><b>b</b></i></sup></a></span> <span class="reference-text"><a href="#CITEREFBall1915">Ball 1915</a>, <a rel="nofollow" class="external text" href="https://archive.org/details/shortaccountofhi00ballrich/page/54/">p. 54</a>.</span> </li> <li id="cite_note-FOOTNOTEBall1915[httpsarchiveorgdetailsshortaccountofhi00ballrichpage38_p.&nbsp;38]-9"><span class="mw-cite-backlink"><b><a href="#cite_ref-FOOTNOTEBall1915[httpsarchiveorgdetailsshortaccountofhi00ballrichpage38_p.&nbsp;38]_9-0">^</a></b></span> <span class="reference-text"><a href="#CITEREFBall1915">Ball 1915</a>, <a rel="nofollow" class="external text" href="https://archive.org/details/shortaccountofhi00ballrich/page/38/">p. 38</a>.</span> </li> <li id="cite_note-10"><span class="mw-cite-backlink"><b><a href="#cite_ref-10">^</a></b></span> <span class="reference-text">Unguru, S. (1985). <a rel="nofollow" class="external text" href="https://core.ac.uk/download/pdf/82804799.pdf">Digging for Structure into the Elements: Euclid, Hilbert, and Mueller</a>. Historia Mathematica 12, 176</span> </li> <li id="cite_note-11"><span class="mw-cite-backlink"><b><a href="#cite_ref-11">^</a></b></span> <span class="reference-text">Zhmud, L. (1998). <a rel="nofollow" class="external text" href="https://philarchive.org/archive/ZHMPAA">Plato as "Architect of Science"</a>. Phonesis 43, 211</span> </li> <li id="cite_note-12"><span class="mw-cite-backlink"><b><a href="#cite_ref-12">^</a></b></span> <span class="reference-text"><a rel="nofollow" class="external text" href="http://historyofinformation.com/expanded.php?id=2749">The Earliest Surviving Manuscript Closest to Euclid's Original Text (Circa 850)</a>; an <a rel="nofollow" class="external text" href="http://www.ibiblio.org/expo/vatican.exhibit/exhibit/d-mathematics/Greek_math.html">image</a> <a rel="nofollow" class="external text" href="https://web.archive.org/web/20091220042513/http://www.ibiblio.org/expo/vatican.exhibit/exhibit/d-mathematics/Greek_math.html">Archived</a> 2009-12-20 at the <a href="/wiki/Wayback_Machine" title="Wayback Machine">Wayback Machine</a> of one page</span> </li> <li id="cite_note-FOOTNOTEReynoldsWilson199157-13"><span class="mw-cite-backlink"><b><a href="#cite_ref-FOOTNOTEReynoldsWilson199157_13-0">^</a></b></span> <span class="reference-text"><a href="#CITEREFReynoldsWilson1991">Reynolds & Wilson 1991</a>, p. 57.</span> </li> <li id="cite_note-FOOTNOTEBall1915[httpsarchiveorgdetailsshortaccountofhi00ballrichpage165_p.&nbsp;165]-14"><span class="mw-cite-backlink"><b><a href="#cite_ref-FOOTNOTEBall1915[httpsarchiveorgdetailsshortaccountofhi00ballrichpage165_p.&nbsp;165]_14-0">^</a></b></span> <span class="reference-text"><a href="#CITEREFBall1915">Ball 1915</a>, <a rel="nofollow" class="external text" href="https://archive.org/details/shortaccountofhi00ballrich/page/165/">p. 165</a>.</span> </li> <li id="cite_note-16"><span class="mw-cite-backlink"><b><a href="#cite_ref-16">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222" /><cite id="CITEREFMurdoch1967" class="citation journal cs1">Murdoch, John E. (1967). "Euclides Graeco-Latinus: A Hitherto Unknown Medieval Latin Translation of the Elements Made Directly from the Greek". <i>Harvard Studies in Classical Philology</i>. <b>71</b>: <span class="nowrap">249–</span>302. <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%2F310767">10.2307/310767</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/310767">310767</a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=Harvard+Studies+in+Classical+Philology&rft.atitle=Euclides+Graeco-Latinus%3A+A+Hitherto+Unknown+Medieval+Latin+Translation+of+the+Elements+Made+Directly+from+the+Greek&rft.volume=71&rft.pages=%3Cspan+class%3D%22nowrap%22%3E249-%3C%2Fspan%3E302&rft.date=1967&rft_id=info%3Adoi%2F10.2307%2F310767&rft_id=https%3A%2F%2Fwww.jstor.org%2Fstable%2F310767%23id-name%3DJSTOR&rft.aulast=Murdoch&rft.aufirst=John+E.&rfr_id=info%3Asid%2Fen.wikipedia.org%3AEuclid%27s+Elements" class="Z3988"></span></span> </li> <li id="cite_note-17"><span class="mw-cite-backlink"><b><a href="#cite_ref-17">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222" /><cite id="CITEREFKnorr1990" class="citation journal cs1">Knorr, Wilbur R. (1990). <a rel="nofollow" class="external text" href="https://www.jstor.org/stable/4026757">"John of Tynemouth alias John of London: Emerging Portrait of a Singular Medieval Mathematician"</a>. <i>The British Journal for the History of Science</i>. <b>23</b> (3): <span class="nowrap">293–</span>330. <a href="/wiki/Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1017%2FS0007087400044009">10.1017/S0007087400044009</a>. <a href="/wiki/ISSN_(identifier)" class="mw-redirect" title="ISSN (identifier)">ISSN</a> <a rel="nofollow" class="external text" href="https://search.worldcat.org/issn/0007-0874">0007-0874</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/4026757">4026757</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:144172844">144172844</a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=The+British+Journal+for+the+History+of+Science&rft.atitle=John+of+Tynemouth+alias+John+of+London%3A+Emerging+Portrait+of+a+Singular+Medieval+Mathematician&rft.volume=23&rft.issue=3&rft.pages=%3Cspan+class%3D%22nowrap%22%3E293-%3C%2Fspan%3E330&rft.date=1990&rft.issn=0007-0874&rft_id=https%3A%2F%2Fapi.semanticscholar.org%2FCorpusID%3A144172844%23id-name%3DS2CID&rft_id=https%3A%2F%2Fwww.jstor.org%2Fstable%2F4026757%23id-name%3DJSTOR&rft_id=info%3Adoi%2F10.1017%2FS0007087400044009&rft.aulast=Knorr&rft.aufirst=Wilbur+R.&rft_id=https%3A%2F%2Fwww.jstor.org%2Fstable%2F4026757&rfr_id=info%3Asid%2Fen.wikipedia.org%3AEuclid%27s+Elements" class="Z3988"></span></span> </li> <li id="cite_note-FOOTNOTEBusard2005-18"><span class="mw-cite-backlink"><b><a href="#cite_ref-FOOTNOTEBusard2005_18-0">^</a></b></span> <span class="reference-text"><a href="#CITEREFBusard2005">Busard 2005</a>.</span> </li> <li id="cite_note-19"><span class="mw-cite-backlink"><b><a href="#cite_ref-19">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222" /><cite id="CITEREFMenso1989" class="citation book cs1">Menso, Folkerts (1989). <a rel="nofollow" class="external text" href="https://math.berkeley.edu/~wodzicki/160/Euclid_in_Middle_Ages.pdf"><i>Euclid in Medieval Europe</i></a> <span class="cs1-format">(PDF)</span>. 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Retrieved <span class="nowrap">2023-07-28</span></span>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=unknown&rft.jtitle=historyofinformation.com&rft.atitle=The+First+Printed+Edition+of+the+Greek+Text+of+Euclid+is+also+the+First+Edition+to+Include+the+Diagrams+within+the+Text+%3A+History+of+Information&rft_id=https%3A%2F%2Fhistoryofinformation.com%2Fdetail.php%3Fentryid%3D2748&rfr_id=info%3Asid%2Fen.wikipedia.org%3AEuclid%27s+Elements" class="Z3988"></span></span> </li> <li id="cite_note-FOOTNOTENasir_al-Din_al-Tusi1594-45"><span class="mw-cite-backlink"><b><a href="#cite_ref-FOOTNOTENasir_al-Din_al-Tusi1594_45-0">^</a></b></span> <span class="reference-text"><a href="#CITEREFNasir_al-Din_al-Tusi1594">Nasir al-Din al-Tusi 1594</a>.</span> </li> <li id="cite_note-FOOTNOTESarma1997460–461-46"><span class="mw-cite-backlink"><b><a href="#cite_ref-FOOTNOTESarma1997460–461_46-0">^</a></b></span> <span class="reference-text"><a href="#CITEREFSarma1997">Sarma 1997</a>, pp. 460–461.</span> </li> <li id="cite_note-47"><span class="mw-cite-backlink"><b><a href="#cite_ref-47">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222" /><cite class="citation web cs1"><a rel="nofollow" class="external text" href="https://archive.today/20090622080437/http://aleph500.huji.ac.il/nnl/dig/books/bk001139706.html">"JNUL Digitized Book Repository"</a>. <i>huji.ac.il</i>. 22 June 2009. Archived from <a rel="nofollow" class="external text" href="http://aleph500.huji.ac.il/nnl/dig/books/bk001139706.html">the original</a> on 22 June 2009<span class="reference-accessdate">. Retrieved <span class="nowrap">29 April</span> 2018</span>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=unknown&rft.jtitle=huji.ac.il&rft.atitle=JNUL+Digitized+Book+Repository&rft.date=2009-06-22&rft_id=http%3A%2F%2Faleph500.huji.ac.il%2Fnnl%2Fdig%2Fbooks%2Fbk001139706.html&rfr_id=info%3Asid%2Fen.wikipedia.org%3AEuclid%27s+Elements" class="Z3988"></span></span> </li> <li id="cite_note-FOOTNOTEServít1907-48"><span class="mw-cite-backlink"><b><a href="#cite_ref-FOOTNOTEServít1907_48-0">^</a></b></span> <span class="reference-text"><a href="#CITEREFServít1907">Servít 1907</a>.</span> </li> <li id="cite_note-FOOTNOTEEuklid1999-49"><span class="mw-cite-backlink"><b><a href="#cite_ref-FOOTNOTEEuklid1999_49-0">^</a></b></span> <span class="reference-text"><a href="#CITEREFEuklid1999">Euklid 1999</a>.</span> </li> <li id="cite_note-FOOTNOTESertöz2019-50"><span class="mw-cite-backlink"><b><a href="#cite_ref-FOOTNOTESertöz2019_50-0">^</a></b></span> <span class="reference-text"><a href="#CITEREFSertöz2019">Sertöz 2019</a>.</span> </li> <li id="cite_note-FOOTNOTECallahanCasey2015-51"><span class="mw-cite-backlink">^ <a href="#cite_ref-FOOTNOTECallahanCasey2015_51-0"><sup><i><b>a</b></i></sup></a> <a href="#cite_ref-FOOTNOTECallahanCasey2015_51-1"><sup><i><b>b</b></i></sup></a></span> <span class="reference-text"><a href="#CITEREFCallahanCasey2015">Callahan & Casey 2015</a>.</span> </li> </ol></div> <div class="mw-heading mw-heading3"><h3 id="Sources">Sources</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Euclid%27s_Elements&action=edit&section=22" title="Edit section: Sources"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <style data-mw-deduplicate="TemplateStyles:r1239549316">.mw-parser-output .refbegin{margin-bottom:0.5em}.mw-parser-output .refbegin-hanging-indents>ul{margin-left:0}.mw-parser-output .refbegin-hanging-indents>ul>li{margin-left:0;padding-left:3.2em;text-indent:-3.2em}.mw-parser-output .refbegin-hanging-indents ul,.mw-parser-output .refbegin-hanging-indents ul li{list-style:none}@media(max-width:720px){.mw-parser-output .refbegin-hanging-indents>ul>li{padding-left:1.6em;text-indent:-1.6em}}.mw-parser-output .refbegin-columns{margin-top:0.3em}.mw-parser-output .refbegin-columns ul{margin-top:0}.mw-parser-output .refbegin-columns li{page-break-inside:avoid;break-inside:avoid-column}@media screen{.mw-parser-output .refbegin{font-size:90%}}</style><div class="refbegin refbegin-hanging-indents refbegin-columns references-column-width" style="column-width: 30em"> <ul><li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222" /><cite id="CITEREFAlexandersonGreenwalt2012" class="citation cs2"><a href="/wiki/Gerald_L._Alexanderson" title="Gerald L. Alexanderson">Alexanderson, Gerald L.</a>; Greenwalt, William S. (2012), "About the cover: Billingsley's Euclid in English", <i>Bulletin of the American Mathematical Society</i>, New Series, <b>49</b> (1): <span class="nowrap">163–</span>167, <a href="/wiki/Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<span class="id-lock-free" title="Freely accessible"><a rel="nofollow" class="external text" href="https://doi.org/10.1090%2FS0273-0979-2011-01365-9">10.1090/S0273-0979-2011-01365-9</a></span></cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=Bulletin+of+the+American+Mathematical+Society&rft.atitle=About+the+cover%3A+Billingsley%27s+Euclid+in+English&rft.volume=49&rft.issue=1&rft.pages=%3Cspan+class%3D%22nowrap%22%3E163-%3C%2Fspan%3E167&rft.date=2012&rft_id=info%3Adoi%2F10.1090%2FS0273-0979-2011-01365-9&rft.aulast=Alexanderson&rft.aufirst=Gerald+L.&rft.au=Greenwalt%2C+William+S.&rfr_id=info%3Asid%2Fen.wikipedia.org%3AEuclid%27s+Elements" class="Z3988"></span></li> <li><a href="/w/index.php?title=Benno_Artmann&action=edit&redlink=1" class="new" title="Benno Artmann (page does not exist)">Artmann, Benno</a>: <i>Euclid – The Creation of Mathematics.</i> New York, Berlin, Heidelberg: Springer 1999, <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/0-387-98423-2" title="Special:BookSources/0-387-98423-2">0-387-98423-2</a></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222" /><cite id="CITEREFBall1915" class="citation book cs1"><a href="/wiki/W._W._Rouse_Ball" title="W. W. Rouse Ball">Ball, Walter William Rouse</a> (1915) [1st ed. 1888]. <a rel="nofollow" class="external text" href="https://archive.org/details/shortaccountofhi00ballrich/"><i>A Short Account of the History of Mathematics</i></a> (6th ed.). MacMillan.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=A+Short+Account+of+the+History+of+Mathematics&rft.edition=6th&rft.pub=MacMillan&rft.date=1915&rft.aulast=Ball&rft.aufirst=Walter+William+Rouse&rft_id=https%3A%2F%2Farchive.org%2Fdetails%2Fshortaccountofhi00ballrich%2F&rfr_id=info%3Asid%2Fen.wikipedia.org%3AEuclid%27s+Elements" class="Z3988"></span></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222" /><cite id="CITEREFBoyer1991" class="citation book cs1"><a href="/wiki/Carl_Benjamin_Boyer" title="Carl Benjamin Boyer">Boyer, Carl B.</a> (1991). "Euclid of Alexandria". <span class="id-lock-registration" title="Free registration required"><a rel="nofollow" class="external text" href="https://archive.org/details/historyofmathema00boye"><i>A History of Mathematics</i></a></span> (Second ed.). John Wiley & Sons. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/0-471-54397-7" title="Special:BookSources/0-471-54397-7"><bdi>0-471-54397-7</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=bookitem&rft.atitle=Euclid+of+Alexandria&rft.btitle=A+History+of+Mathematics&rft.edition=Second&rft.pub=John+Wiley+%26+Sons&rft.date=1991&rft.isbn=0-471-54397-7&rft.aulast=Boyer&rft.aufirst=Carl+B.&rft_id=https%3A%2F%2Farchive.org%2Fdetails%2Fhistoryofmathema00boye&rfr_id=info%3Asid%2Fen.wikipedia.org%3AEuclid%27s+Elements" class="Z3988"></span></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222" /><cite id="CITEREFBuntJonesBedient1988" class="citation book cs1">Bunt, Lucas Nicolaas Hendrik; Jones, Phillip S.; Bedient, Jack D. (1988). <i>The Historical Roots of Elementary Mathematics</i>. Dover.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=The+Historical+Roots+of+Elementary+Mathematics&rft.pub=Dover&rft.date=1988&rft.aulast=Bunt&rft.aufirst=Lucas+Nicolaas+Hendrik&rft.au=Jones%2C+Phillip+S.&rft.au=Bedient%2C+Jack+D.&rfr_id=info%3Asid%2Fen.wikipedia.org%3AEuclid%27s+Elements" class="Z3988"></span></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222" /><cite id="CITEREFBusard2005" class="citation book cs1">Busard, H.L.L. (2005). "Introduction to the Text". <a rel="nofollow" class="external text" href="https://books.google.com/books?id=tiHUN6jjm6MC&pg=PA1"><i>Campanus of Novara and Euclid's Elements</i></a>. Stuttgart: Franz Steiner Verlag. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-3-515-08645-5" title="Special:BookSources/978-3-515-08645-5"><bdi>978-3-515-08645-5</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=bookitem&rft.atitle=Introduction+to+the+Text&rft.btitle=Campanus+of+Novara+and+Euclid%27s+Elements&rft.place=Stuttgart&rft.pub=Franz+Steiner+Verlag&rft.date=2005&rft.isbn=978-3-515-08645-5&rft.aulast=Busard&rft.aufirst=H.L.L.&rft_id=https%3A%2F%2Fbooks.google.com%2Fbooks%3Fid%3DtiHUN6jjm6MC%26pg%3DPA1&rfr_id=info%3Asid%2Fen.wikipedia.org%3AEuclid%27s+Elements" class="Z3988"></span></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222" /><cite id="CITEREFCallahanCasey2015" class="citation book cs1">Callahan, Daniel; Casey, John (2015). <a rel="nofollow" class="external text" href="http://gutenberg.org/ebooks/21076"><i>Euclid's "Elements" Redux</i></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Euclid%27s+%22Elements%22+Redux&rft.date=2015&rft.aulast=Callahan&rft.aufirst=Daniel&rft.au=Casey%2C+John&rft_id=http%3A%2F%2Fgutenberg.org%2Febooks%2F21076&rfr_id=info%3Asid%2Fen.wikipedia.org%3AEuclid%27s+Elements" class="Z3988"></span></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222" /><cite id="CITEREFDodgsonHagar2009" class="citation book cs1"><a href="/wiki/Lewis_Carroll" title="Lewis Carroll">Dodgson, Charles L.</a>; Hagar, Amit (2009). "Introduction". <a rel="nofollow" class="external text" href="https://books.google.com/books?id=6Uk42tX9P10C"><i>Euclid and His Modern Rivals</i></a>. Cambridge University Press. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-1-108-00100-7" title="Special:BookSources/978-1-108-00100-7"><bdi>978-1-108-00100-7</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=bookitem&rft.atitle=Introduction&rft.btitle=Euclid+and+His+Modern+Rivals&rft.pub=Cambridge+University+Press&rft.date=2009&rft.isbn=978-1-108-00100-7&rft.aulast=Dodgson&rft.aufirst=Charles+L.&rft.au=Hagar%2C+Amit&rft_id=https%3A%2F%2Fbooks.google.com%2Fbooks%3Fid%3D6Uk42tX9P10C&rfr_id=info%3Asid%2Fen.wikipedia.org%3AEuclid%27s+Elements" class="Z3988"></span></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222" /><cite id="CITEREFHartshorne2000" class="citation book cs1">Hartshorne, Robin (2000). <i>Geometry: Euclid and Beyond</i> (2nd ed.). <a href="/wiki/New_York_City" title="New York City">New York, NY</a>: <a href="/wiki/Springer_Publishing" title="Springer Publishing">Springer</a>. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/9780387986500" title="Special:BookSources/9780387986500"><bdi>9780387986500</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Geometry%3A+Euclid+and+Beyond&rft.place=New+York%2C+NY&rft.edition=2nd&rft.pub=Springer&rft.date=2000&rft.isbn=9780387986500&rft.aulast=Hartshorne&rft.aufirst=Robin&rfr_id=info%3Asid%2Fen.wikipedia.org%3AEuclid%27s+Elements" class="Z3988"></span></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222" /><cite id="CITEREFHeath1956a" class="citation book cs1 cs1-prop-long-vol"><a href="/wiki/T._L._Heath" class="mw-redirect" title="T. L. Heath">Heath, Thomas L.</a> (1956a). <i>The Thirteen Books of Euclid's Elements</i>. Vol. 1. Books I and II (2nd ed.). New York: Dover Publications. <a href="/wiki/OL_(identifier)" class="mw-redirect" title="OL (identifier)">OL</a> <a rel="nofollow" class="external text" href="https://openlibrary.org/books/OL22193354M">22193354M</a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=The+Thirteen+Books+of+Euclid%27s+Elements&rft.place=New+York&rft.edition=2nd&rft.pub=Dover+Publications&rft.date=1956&rft_id=https%3A%2F%2Fopenlibrary.org%2Fbooks%2FOL22193354M%23id-name%3DOL&rft.aulast=Heath&rft.aufirst=Thomas+L.&rfr_id=info%3Asid%2Fen.wikipedia.org%3AEuclid%27s+Elements" class="Z3988"></span></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222" /><cite id="CITEREFHeath1956b" class="citation book cs1 cs1-prop-long-vol"><a href="/wiki/T._L._Heath" class="mw-redirect" title="T. L. Heath">Heath, Thomas L.</a> (1956b). <i>The Thirteen Books of Euclid's Elements</i>. Vol. 2. Books III to IX (2nd ed.). New York: Dover Publications. <a href="/wiki/OL_(identifier)" class="mw-redirect" title="OL (identifier)">OL</a> <a rel="nofollow" class="external text" href="https://openlibrary.org/books/OL7650092M">7650092M</a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=The+Thirteen+Books+of+Euclid%27s+Elements&rft.place=New+York&rft.edition=2nd&rft.pub=Dover+Publications&rft.date=1956&rft_id=https%3A%2F%2Fopenlibrary.org%2Fbooks%2FOL7650092M%23id-name%3DOL&rft.aulast=Heath&rft.aufirst=Thomas+L.&rfr_id=info%3Asid%2Fen.wikipedia.org%3AEuclid%27s+Elements" class="Z3988"></span></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222" /><cite id="CITEREFHeath1956c" class="citation book cs1 cs1-prop-long-vol"><a href="/wiki/T._L._Heath" class="mw-redirect" title="T. L. 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New York: Dover Publications. <a href="/wiki/OCLC_(identifier)" class="mw-redirect" title="OCLC (identifier)">OCLC</a> <a rel="nofollow" class="external text" href="https://search.worldcat.org/oclc/929205858">929205858</a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=The+Thirteen+Books+of+Euclid%27s+Elements&rft.place=New+York&rft.edition=2nd&rft.pub=Dover+Publications&rft.date=1956&rft_id=info%3Aoclcnum%2F929205858&rft.aulast=Heath&rft.aufirst=Thomas+L.&rfr_id=info%3Asid%2Fen.wikipedia.org%3AEuclid%27s+Elements" class="Z3988"></span> Heath's authoritative translation plus extensive historical research and detailed commentary throughout the text.</li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222" /><cite id="CITEREFHeath1963" class="citation book cs1"><a href="/wiki/T._L._Heath" class="mw-redirect" title="T. L. Heath">Heath, Thomas L.</a> (1963). <a rel="nofollow" class="external text" href="https://books.google.com/books?id=_HZNr_mGFzQC"><i>A Manual of Greek Mathematics</i></a>. Dover Publications. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-0-486-43231-1" title="Special:BookSources/978-0-486-43231-1"><bdi>978-0-486-43231-1</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=A+Manual+of+Greek+Mathematics&rft.pub=Dover+Publications&rft.date=1963&rft.isbn=978-0-486-43231-1&rft.aulast=Heath&rft.aufirst=Thomas+L.&rft_id=https%3A%2F%2Fbooks.google.com%2Fbooks%3Fid%3D_HZNr_mGFzQC&rfr_id=info%3Asid%2Fen.wikipedia.org%3AEuclid%27s+Elements" class="Z3988"></span></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222" /><cite id="CITEREFKetcham1901" class="citation book cs1">Ketcham, Henry (1901). <a rel="nofollow" class="external text" href="https://archive.org/details/lifeofabrahaml1154ketc/page/n6"><i>The Life of Abraham Lincoln</i></a>. New York: Perkins Book Company.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=The+Life+of+Abraham+Lincoln&rft.place=New+York&rft.pub=Perkins+Book+Company&rft.date=1901&rft.aulast=Ketcham&rft.aufirst=Henry&rft_id=https%3A%2F%2Farchive.org%2Fdetails%2Flifeofabrahaml1154ketc%2Fpage%2Fn6&rfr_id=info%3Asid%2Fen.wikipedia.org%3AEuclid%27s+Elements" class="Z3988"></span></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222" /><cite id="CITEREFNasir_al-Din_al-Tusi1594" class="citation book cs1 cs1-prop-foreign-lang-source"><a href="/wiki/Nasir_al-Din_al-Tusi" title="Nasir al-Din al-Tusi">Nasir al-Din al-Tusi</a> (1594). <a rel="nofollow" class="external text" href="https://www.wdl.org/en/item/10666/"><i>Kitāb taḥrīr uṣūl li-Uqlīdus</i></a> [<i>The Recension of Euclid's "Elements"</i>] (in Arabic).</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Kit%C4%81b+ta%E1%B8%A5r%C4%ABr+u%E1%B9%A3%C5%ABl+li-Uql%C4%ABdus&rft.date=1594&rft.au=Nasir+al-Din+al-Tusi&rft_id=https%3A%2F%2Fwww.wdl.org%2Fen%2Fitem%2F10666%2F&rfr_id=info%3Asid%2Fen.wikipedia.org%3AEuclid%27s+Elements" class="Z3988"></span></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222" /><cite id="CITEREFReynoldsWilson1991" class="citation book cs1">Reynolds, Leighton Durham; Wilson, Nigel Guy (9 May 1991). <a rel="nofollow" class="external text" href="https://books.google.com/books?id=7iSlvWFf1FQC"><i>Scribes and scholars: a guide to the transmission of Greek and Latin literature</i></a> (2nd ed.). Oxford: Clarendon Press. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-0-19-872145-1" title="Special:BookSources/978-0-19-872145-1"><bdi>978-0-19-872145-1</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Scribes+and+scholars%3A+a+guide+to+the+transmission+of+Greek+and+Latin+literature&rft.place=Oxford&rft.edition=2nd&rft.pub=Clarendon+Press&rft.date=1991-05-09&rft.isbn=978-0-19-872145-1&rft.aulast=Reynolds&rft.aufirst=Leighton+Durham&rft.au=Wilson%2C+Nigel+Guy&rft_id=https%3A%2F%2Fbooks.google.com%2Fbooks%3Fid%3D7iSlvWFf1FQC&rfr_id=info%3Asid%2Fen.wikipedia.org%3AEuclid%27s+Elements" class="Z3988"></span></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222" /><cite id="CITEREFRussell2013" class="citation book cs1"><a href="/wiki/Bertrand_Russell" title="Bertrand Russell">Russell, Bertrand</a> (2013). <a rel="nofollow" class="external text" href="https://books.google.com/books?id=Gm_cCZBiOhQC&pg=PA177"><i>History of Western Philosophy: Collectors Edition</i></a>. Routledge. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-1-135-69284-1" title="Special:BookSources/978-1-135-69284-1"><bdi>978-1-135-69284-1</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=History+of+Western+Philosophy%3A+Collectors+Edition&rft.pub=Routledge&rft.date=2013&rft.isbn=978-1-135-69284-1&rft.aulast=Russell&rft.aufirst=Bertrand&rft_id=https%3A%2F%2Fbooks.google.com%2Fbooks%3Fid%3DGm_cCZBiOhQC%26pg%3DPA177&rfr_id=info%3Asid%2Fen.wikipedia.org%3AEuclid%27s+Elements" class="Z3988"></span></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222" /><cite id="CITEREFSarma1997" class="citation book cs1"><a href="/wiki/K._V._Sarma" title="K. V. Sarma">Sarma, K.V.</a> (1997). <a href="/wiki/Helaine_Selin" title="Helaine Selin">Selin, Helaine</a> (ed.). <a rel="nofollow" class="external text" href="https://books.google.com/books?id=yoiXDTXSHi4C&q=rekhaganita&pg=PA460"><i>Encyclopaedia of the history of science, technology, and medicine in non-western cultures</i></a>. Springer. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-0-7923-4066-9" title="Special:BookSources/978-0-7923-4066-9"><bdi>978-0-7923-4066-9</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Encyclopaedia+of+the+history+of+science%2C+technology%2C+and+medicine+in+non-western+cultures&rft.pub=Springer&rft.date=1997&rft.isbn=978-0-7923-4066-9&rft.aulast=Sarma&rft.aufirst=K.V.&rft_id=https%3A%2F%2Fbooks.google.com%2Fbooks%3Fid%3DyoiXDTXSHi4C%26q%3Drekhaganita%26pg%3DPA460&rfr_id=info%3Asid%2Fen.wikipedia.org%3AEuclid%27s+Elements" class="Z3988"></span></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222" /><cite id="CITEREFServít1907" class="citation book cs1 cs1-prop-foreign-lang-source">Servít, František (1907). <a class="external text" href="https://commons.wikimedia.org/wiki/File:Eukleides_Servit.pdf"><i>Eukleidovy Zaklady (Elementa)</i></a> [<i>Euclid's Elements</i>] <span class="cs1-format">(PDF)</span> (in Czech).</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Eukleidovy+Zaklady+%28Elementa%29&rft.date=1907&rft.aulast=Serv%C3%ADt&rft.aufirst=Franti%C5%A1ek&rft_id=https%3A%2F%2Fcommons.wikimedia.org%2Fwiki%2FFile%3AEukleides_Servit.pdf&rfr_id=info%3Asid%2Fen.wikipedia.org%3AEuclid%27s+Elements" class="Z3988"></span></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222" /><cite id="CITEREFSertöz2019" class="citation book cs1 cs1-prop-foreign-lang-source">Sertöz, Ali Sinan (2019). <a rel="nofollow" class="external text" href="https://books.google.com/books?id=X_eIxgEACAAJ"><i>Öklidin Elemanlari: Ciltli</i></a> [<i>Euclid's Elements</i>] (in Turkish). Tübitak. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-605-312-329-3" title="Special:BookSources/978-605-312-329-3"><bdi>978-605-312-329-3</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=%C3%96klidin+Elemanlari%3A+Ciltli&rft.pub=T%C3%BCbitak&rft.date=2019&rft.isbn=978-605-312-329-3&rft.aulast=Sert%C3%B6z&rft.aufirst=Ali+Sinan&rft_id=https%3A%2F%2Fbooks.google.com%2Fbooks%3Fid%3DX_eIxgEACAAJ&rfr_id=info%3Asid%2Fen.wikipedia.org%3AEuclid%27s+Elements" class="Z3988"></span></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222" /><cite id="CITEREFToussaint1993" class="citation journal cs1"><a href="/wiki/Godfried_Toussaint" title="Godfried Toussaint">Toussaint, Godfried</a> (1993). "A new look at euclid's second proposition". <i>The Mathematical Intelligencer</i>. <b>15</b> (3): <span class="nowrap">12–</span>24. <a href="/wiki/Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1007%2FBF03024252">10.1007/BF03024252</a>. <a href="/wiki/ISSN_(identifier)" class="mw-redirect" title="ISSN (identifier)">ISSN</a> <a rel="nofollow" class="external text" href="https://search.worldcat.org/issn/0343-6993">0343-6993</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:26811463">26811463</a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=The+Mathematical+Intelligencer&rft.atitle=A+new+look+at+euclid%27s+second+proposition&rft.volume=15&rft.issue=3&rft.pages=%3Cspan+class%3D%22nowrap%22%3E12-%3C%2Fspan%3E24&rft.date=1993&rft_id=https%3A%2F%2Fapi.semanticscholar.org%2FCorpusID%3A26811463%23id-name%3DS2CID&rft.issn=0343-6993&rft_id=info%3Adoi%2F10.1007%2FBF03024252&rft.aulast=Toussaint&rft.aufirst=Godfried&rfr_id=info%3Asid%2Fen.wikipedia.org%3AEuclid%27s+Elements" class="Z3988"></span></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222" /><cite id="CITEREFVan_der_Waerden1975" class="citation book cs1"><a href="/wiki/Bartel_Leendert_van_der_Waerden" title="Bartel Leendert van der Waerden">Van der Waerden, Bartel Leendert</a> (1975). <a rel="nofollow" class="external text" href="https://books.google.com/books?id=djUPAQAAMAAJ"><i>Science awakening</i></a>. Noordhoff International. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-90-01-93102-5" title="Special:BookSources/978-90-01-93102-5"><bdi>978-90-01-93102-5</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Science+awakening&rft.pub=Noordhoff+International&rft.date=1975&rft.isbn=978-90-01-93102-5&rft.aulast=Van+der+Waerden&rft.aufirst=Bartel+Leendert&rft_id=https%3A%2F%2Fbooks.google.com%2Fbooks%3Fid%3DdjUPAQAAMAAJ&rfr_id=info%3Asid%2Fen.wikipedia.org%3AEuclid%27s+Elements" class="Z3988"></span></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222" /><cite id="CITEREFWilson2006" class="citation book cs1">Wilson, Nigel Guy (2006). <i>Encyclopedia of Ancient Greece</i>. Routledge.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Encyclopedia+of+Ancient+Greece&rft.pub=Routledge&rft.date=2006&rft.aulast=Wilson&rft.aufirst=Nigel+Guy&rfr_id=info%3Asid%2Fen.wikipedia.org%3AEuclid%27s+Elements" class="Z3988"></span></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222" /><cite id="CITEREFEuklid1999" class="citation book cs1">Euklid (1999). <i>Elementi I-VI</i>. Translated by Hudoletnjak Grgić, Maja. KruZak. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/953-96477-6-2" title="Special:BookSources/953-96477-6-2"><bdi>953-96477-6-2</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Elementi+I-VI&rft.pub=KruZak&rft.date=1999&rft.isbn=953-96477-6-2&rft.au=Euklid&rfr_id=info%3Asid%2Fen.wikipedia.org%3AEuclid%27s+Elements" class="Z3988"></span></li></ul> </div> <div class="mw-heading mw-heading2"><h2 id="External_links">External links</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Euclid%27s_Elements&action=edit&section=23" title="Edit section: External links"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <style data-mw-deduplicate="TemplateStyles:r1235681985">.mw-parser-output .side-box{margin:4px 0;box-sizing:border-box;border:1px solid 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href="https://elements.ratherthanpaper.com/"><i>Elements</i> with highlights</a> by ratherthanpaper</li> <li><a rel="nofollow" class="external text" href="https://www2.hf.uio.no/polyglotta/index.php?page=volume&vid=67">Multilingual edition of <i>Elementa</i> in the Bibliotheca Polyglotta</a></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222" /><cite id="CITEREFEuclid1997" class="citation web cs1">Euclid (1997) [c. 300 BC]. <a href="/wiki/David_E._Joyce_(mathematician)" class="mw-redirect" title="David E. Joyce (mathematician)">David E. Joyce</a> (ed.). <a rel="nofollow" class="external text" href="http://aleph0.clarku.edu/~djoyce/java/elements/toc.html">"Elements"</a><span class="reference-accessdate">. Retrieved <span class="nowrap">2006-08-30</span></span>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=unknown&rft.btitle=Elements&rft.date=1997&rft.au=Euclid&rft_id=http%3A%2F%2Faleph0.clarku.edu%2F~djoyce%2Fjava%2Felements%2Ftoc.html&rfr_id=info%3Asid%2Fen.wikipedia.org%3AEuclid%27s+Elements" class="Z3988"></span> In HTML with Java-based interactive figures.</li> <li><a rel="nofollow" class="external text" href="http://farside.ph.utexas.edu/Books/Euclid/Elements.pdf">Richard Fitzpatrick's bilingual edition</a> (freely downloadable PDF, typeset in a two-column format with the original Greek beside a modern English translation; also available in print as <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/979-8589564587" title="Special:BookSources/979-8589564587">979-8589564587</a>)</li> <li><a rel="nofollow" class="external text" href="https://www.perseus.tufts.edu/hopper/text?doc=Euc.+1">Heath's English translation</a> (HTML, without the figures, public domain) (accessed February 4, 2010) <ul><li>Heath's English translation and commentary, with the figures (Google Books): <a rel="nofollow" class="external text" href="https://books.google.com/books?id=UhgPAAAAIAAJ">vol. 1</a>, <a rel="nofollow" class="external text" href="https://books.google.com/books?id=lxkPAAAAIAAJ">vol. 2</a>, <a rel="nofollow" class="external text" href="https://books.google.com/books?id=xhkPAAAAIAAJ">vol. 3</a>, <a rel="nofollow" class="external text" href="https://books.google.com/books?id=KHMDAAAAYAAJ">vol. 3 c. 2</a></li></ul></li> <li><a rel="nofollow" class="external text" href="http://www.math.ubc.ca/~cass/Euclid/byrne.html">Oliver Byrne's 1847 edition</a> (also hosted at <a rel="nofollow" class="external text" href="https://archive.org/details/firstsixbooksofe00eucl">archive.org</a>)– an unusual version by <a href="/wiki/Oliver_Byrne_(mathematician)" title="Oliver Byrne (mathematician)">Oliver Byrne</a> who used color rather than labels such as ABC (scanned page images, public domain)</li> <li><a rel="nofollow" class="external text" href="https://www.c82.net/euclid/#books">Web adapted version of Byrne’s Euclid</a> designed by Nicholas Rougeux</li> <li><a rel="nofollow" class="external text" href="https://www.youtube.com/c/SandyBultena/playlists">Video adaptation</a>, animated and explained by Sandy Bultena, contains books I-VII.</li> <li><a rel="nofollow" class="external text" href="http://gutenberg.org/ebooks/21076">The First Six Books of the <i>Elements</i></a> by John Casey and Euclid scanned by <a href="/wiki/Project_Gutenberg" title="Project Gutenberg">Project Gutenberg</a>.</li> <li><a rel="nofollow" class="external text" href="http://mysite.du.edu/~etuttle/classics/nugreek/contents.htm">Reading Euclid</a> – a course in how to read Euclid in the original Greek, with English translations and commentaries (HTML with figures)</li> <li><a href="/wiki/Thomas_More" title="Thomas More">Sir Thomas More</a>'s <a rel="nofollow" class="external text" href="https://web.archive.org/web/20040807095930/http://www.columbia.edu/acis/textarchive/rare/24.html">manuscript</a></li> <li><a rel="nofollow" class="external text" href="https://web.archive.org/web/20040807101503/http://www.columbia.edu/acis/textarchive/rare/6.html">Latin translation</a> by <a href="/wiki/Aethelhard_of_Bath" class="mw-redirect" title="Aethelhard of Bath">Aethelhard of Bath</a></li> <li><a rel="nofollow" class="external text" href="http://www.physics.ntua.gr/~mourmouras/euclid/index.html">Euclid <i>Elements</i> – The original Greek text</a> Greek HTML</li> <li><a href="/wiki/Clay_Mathematics_Institute" title="Clay Mathematics Institute">Clay Mathematics Institute</a> Historical Archive – <a rel="nofollow" class="external text" href="http://www.claymath.org/euclids-elements">The thirteen books of Euclid's <i>Elements</i></a> copied by Stephen the Clerk for Arethas of Patras, in Constantinople in 888 AD</li> <li><a rel="nofollow" class="external text" href="http://pds.lib.harvard.edu/pds/view/13079270">Kitāb Taḥrīr uṣūl li-Ūqlīdis</a> Arabic translation of the thirteen books of Euclid's <i>Elements</i> by Nasīr al-Dīn al-Ṭūsī. Published by Medici Oriental Press(also, Typographia Medicea). Facsimile hosted by <a rel="nofollow" class="external text" href="http://ocp.hul.harvard.edu/ihp/">Islamic Heritage Project</a>.</li> <li><a rel="nofollow" class="external text" href="https://archive.org/details/euclid-elements-redux_201809">Euclid's <i>Elements</i> Redux</a>, an open textbook based on the <i>Elements</i></li> <li><a rel="nofollow" class="external text" href="https://archive.org/search.php?query=%E5%B9%BE%E4%BD%95%E5%8E%9F%E6%9C%AC">1607 Chinese translations</a> reprinted as part of the <i><a href="/wiki/Complete_Library_of_the_Four_Treasuries" class="mw-redirect" title="Complete Library of the Four Treasuries">Complete Library of the Four Treasuries</a></i>, or <i>Siku Quanshu</i>.</li></ul> <div class="navbox-styles"><style data-mw-deduplicate="TemplateStyles:r1129693374">.mw-parser-output .hlist dl,.mw-parser-output .hlist ol,.mw-parser-output .hlist ul{margin:0;padding:0}.mw-parser-output .hlist dd,.mw-parser-output .hlist dt,.mw-parser-output .hlist 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title="Euclid's Phaenomena">Phaenomena</a></i></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%">Topics</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/List_of_things_named_after_Euclid" title="List of things named after Euclid">Namesakes</a></li> <li><a href="/wiki/Euclidean_geometry" title="Euclidean geometry">Euclidean geometry</a></li> <li><a href="/wiki/Euclidean_algorithm" title="Euclidean algorithm">Euclidean algorithm</a></li> <li><a href="/wiki/Euclid%27s_theorem" title="Euclid's theorem">Euclid's theorem</a></li> <li><a href="/wiki/Euclidean_relation" title="Euclidean relation">Euclidean relation</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%">Scholars</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/Thomas_Heath_(classicist)" title="Thomas Heath (classicist)">Thomas Heath</a></li> <li><a href="/wiki/Johan_Ludvig_Heiberg_(historian)" title="Johan Ludvig Heiberg (historian)">Johan Ludvig Heiberg</a></li> <li><a href="/wiki/Robert_Simson" title="Robert Simson">Robert Simson</a></li> <li><a href="/wiki/Isaac_Todhunter" title="Isaac Todhunter">Isaac Todhunter</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" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Papyrus_Oxyrhynchus_29" title="Papyrus Oxyrhynchus 29">Papyrus Oxyrhynchus 29</a></li></ul> </div></td></tr><tr><td class="navbox-abovebelow" colspan="2"><div> <ul><li><span class="noviewer" typeof="mw:File"><span title="Category"><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" /></span></span> <a href="/wiki/Category:Euclid" title="Category:Euclid">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="Ancient_Greek_mathematics550" style="padding:3px"><table class="nowraplinks mw-collapsible autocollapse navbox-inner" style="border-spacing:0;background:transparent;color:inherit"><tbody><tr><th scope="col" class="navbox-title" colspan="2"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1129693374" /><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:Ancient_Greek_mathematics" title="Template:Ancient Greek mathematics"><abbr title="View this template">v</abbr></a></li><li class="nv-talk"><a href="/wiki/Template_talk:Ancient_Greek_mathematics" title="Template talk:Ancient Greek mathematics"><abbr title="Discuss this template">t</abbr></a></li><li class="nv-edit"><a href="/wiki/Special:EditPage/Template:Ancient_Greek_mathematics" title="Special:EditPage/Template:Ancient Greek mathematics"><abbr title="Edit this template">e</abbr></a></li></ul></div><div id="Ancient_Greek_mathematics550" style="font-size:114%;margin:0 4em"><a href="/wiki/Greek_mathematics" title="Greek mathematics">Ancient Greek mathematics</a></div></th></tr><tr><th scope="row" class="navbox-group" style="width:1%"><a href="/wiki/List_of_Greek_mathematicians" title="List of Greek mathematicians">Mathematicians</a><br /><a href="/wiki/Timeline_of_ancient_Greek_mathematicians" title="Timeline of ancient Greek mathematicians">(timeline)</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/Anaxagoras" title="Anaxagoras">Anaxagoras</a></li> <li><a href="/wiki/Anthemius_of_Tralles" title="Anthemius of Tralles">Anthemius</a></li> <li><a href="/wiki/Archytas" title="Archytas">Archytas</a></li> <li><a href="/wiki/Aristaeus_the_Elder" title="Aristaeus the Elder">Aristaeus the Elder</a></li> <li><a href="/wiki/Aristarchus_of_Samos" title="Aristarchus of Samos">Aristarchus</a></li> <li><a href="/wiki/Aristotle" title="Aristotle">Aristotle</a></li> <li><a href="/wiki/Apollonius_of_Perga" title="Apollonius of Perga">Apollonius</a></li> <li><a href="/wiki/Archimedes" title="Archimedes">Archimedes</a></li> <li><a href="/wiki/Autolycus_of_Pitane" title="Autolycus of Pitane">Autolycus</a></li> <li><a href="/wiki/Bion_of_Abdera" title="Bion of Abdera">Bion</a></li> <li><a href="/wiki/Bryson_of_Heraclea" title="Bryson of Heraclea">Bryson</a></li> <li><a href="/wiki/Callippus" title="Callippus">Callippus</a></li> <li><a href="/wiki/Carpus_of_Antioch" title="Carpus of Antioch">Carpus</a></li> <li><a href="/wiki/Chrysippus" title="Chrysippus">Chrysippus</a></li> <li><a href="/wiki/Cleomedes" title="Cleomedes">Cleomedes</a></li> <li><a href="/wiki/Conon_of_Samos" title="Conon of Samos">Conon</a></li> <li><a href="/wiki/Ctesibius" title="Ctesibius">Ctesibius</a></li> <li><a href="/wiki/Democritus" title="Democritus">Democritus</a></li> <li><a href="/wiki/Dicaearchus" title="Dicaearchus">Dicaearchus</a></li> <li><a href="/wiki/Diocles_(mathematician)" title="Diocles (mathematician)">Diocles</a></li> <li><a href="/wiki/Diophantus" title="Diophantus">Diophantus</a></li> <li><a href="/wiki/Dinostratus" title="Dinostratus">Dinostratus</a></li> <li><a href="/wiki/Dionysodorus" title="Dionysodorus">Dionysodorus</a></li> <li><a href="/wiki/Domninus_of_Larissa" title="Domninus of Larissa">Domninus</a></li> <li><a href="/wiki/Eratosthenes" title="Eratosthenes">Eratosthenes</a></li> <li><a href="/wiki/Eudemus_of_Rhodes" title="Eudemus of Rhodes">Eudemus</a></li> <li><a href="/wiki/Euclid" title="Euclid">Euclid</a></li> <li><a href="/wiki/Eudoxus_of_Cnidus" title="Eudoxus of Cnidus">Eudoxus</a></li> <li><a href="/wiki/Eutocius_of_Ascalon" title="Eutocius of Ascalon">Eutocius</a></li> <li><a href="/wiki/Geminus" title="Geminus">Geminus</a></li> <li><a href="/wiki/Heliodorus_of_Larissa" title="Heliodorus of Larissa">Heliodorus</a></li> <li><a href="/wiki/Hero_of_Alexandria" title="Hero of Alexandria">Heron</a></li> <li><a href="/wiki/Hipparchus" title="Hipparchus">Hipparchus</a></li> <li><a href="/wiki/Hippasus" title="Hippasus">Hippasus</a></li> <li><a href="/wiki/Hippias" title="Hippias">Hippias</a></li> <li><a href="/wiki/Hippocrates_of_Chios" title="Hippocrates of Chios">Hippocrates</a></li> <li><a href="/wiki/Hypatia" title="Hypatia">Hypatia</a></li> <li><a href="/wiki/Hypsicles" title="Hypsicles">Hypsicles</a></li> <li><a href="/wiki/Isidore_of_Miletus" title="Isidore of Miletus">Isidore of Miletus</a></li> <li><a href="/wiki/Leon_(mathematician)" title="Leon (mathematician)">Leon</a></li> <li><a href="/wiki/Marinus_of_Neapolis" title="Marinus of Neapolis">Marinus</a></li> <li><a href="/wiki/Menaechmus" title="Menaechmus">Menaechmus</a></li> <li><a href="/wiki/Menelaus_of_Alexandria" title="Menelaus of Alexandria">Menelaus</a></li> <li><a href="/wiki/Metrodorus_(grammarian)" title="Metrodorus (grammarian)">Metrodorus</a></li> <li><a href="/wiki/Nicomachus" title="Nicomachus">Nicomachus</a></li> <li><a href="/wiki/Nicomedes_(mathematician)" title="Nicomedes (mathematician)">Nicomedes</a></li> <li><a href="/wiki/Nicoteles_of_Cyrene" title="Nicoteles of Cyrene">Nicoteles</a></li> <li><a href="/wiki/Oenopides" title="Oenopides">Oenopides</a></li> <li><a href="/wiki/Pappus_of_Alexandria" title="Pappus of Alexandria">Pappus</a></li> <li><a href="/wiki/Perseus_(geometer)" title="Perseus (geometer)">Perseus</a></li> <li><a href="/wiki/Philolaus" title="Philolaus">Philolaus</a></li> <li><a href="/wiki/Philon" title="Philon">Philon</a></li> <li><a href="/wiki/Philonides_of_Laodicea" title="Philonides of Laodicea">Philonides</a></li> <li><a href="/wiki/Plato" title="Plato">Plato</a></li> <li><a href="/wiki/Porphyry_of_Tyre" title="Porphyry of Tyre">Porphyry of Tyre</a></li> <li><a href="/wiki/Posidonius" title="Posidonius">Posidonius</a></li> <li><a href="/wiki/Proclus" title="Proclus">Proclus</a></li> <li><a href="/wiki/Ptolemy" title="Ptolemy">Ptolemy</a></li> <li><a href="/wiki/Pythagoras" title="Pythagoras">Pythagoras</a></li> <li><a href="/wiki/Serenus_of_Antino%C3%B6polis" title="Serenus of Antinoöpolis">Serenus </a></li> <li><a href="/wiki/Simplicius_of_Cilicia" title="Simplicius of Cilicia">Simplicius</a></li> <li><a href="/wiki/Sosigenes_of_Alexandria" class="mw-redirect" title="Sosigenes of Alexandria">Sosigenes</a></li> <li><a href="/wiki/Sporus_of_Nicaea" title="Sporus of Nicaea">Sporus</a></li> <li><a href="/wiki/Thales_of_Miletus" title="Thales of Miletus">Thales</a></li> <li><a href="/wiki/Theaetetus_(mathematician)" title="Theaetetus (mathematician)">Theaetetus</a></li> <li><a href="/wiki/Theano_(philosopher)" title="Theano (philosopher)">Theano</a></li> <li><a href="/wiki/Theodorus_of_Cyrene" title="Theodorus of Cyrene">Theodorus</a></li> <li><a href="/wiki/Theodosius_of_Bithynia" title="Theodosius of Bithynia">Theodosius</a></li> <li><a href="/wiki/Theon_of_Alexandria" title="Theon of Alexandria">Theon of Alexandria</a></li> <li><a href="/wiki/Theon_of_Smyrna" title="Theon of Smyrna">Theon of Smyrna</a></li> <li><a href="/wiki/Thymaridas" title="Thymaridas">Thymaridas</a></li> <li><a href="/wiki/Xenocrates" title="Xenocrates">Xenocrates</a></li> <li><a href="/wiki/Zeno_of_Elea" title="Zeno of Elea">Zeno of Elea</a></li> <li><a href="/wiki/Zeno_of_Sidon" title="Zeno of Sidon">Zeno of Sidon</a></li> <li><a href="/wiki/Zenodorus_(mathematician)" title="Zenodorus (mathematician)">Zenodorus</a></li></ul> </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><i><a href="/wiki/Almagest" title="Almagest">Almagest</a></i></li> <li><a href="/wiki/Archimedes_Palimpsest" title="Archimedes Palimpsest">Archimedes Palimpsest</a></li> <li><i><a href="/wiki/Arithmetica" title="Arithmetica">Arithmetica</a></i></li> <li><a href="/wiki/Apollonius_of_Perga#Conics" title="Apollonius of Perga"><i>Conics</i> <span style="font-size: 85%;">(Apollonius)</span></a></li> <li><i><a href="/wiki/Catoptrics" title="Catoptrics">Catoptrics</a></i></li> <li><a href="/wiki/Data_(Euclid)" class="mw-redirect" title="Data (Euclid)"><i>Data</i> <span style="font-size: 85%;">(Euclid)</span></a></li> <li><a class="mw-selflink selflink"><i>Elements</i> <span style="font-size: 85%;">(Euclid)</span></a></li> <li><i><a href="/wiki/Measurement_of_a_Circle" title="Measurement of a Circle">Measurement of a Circle</a></i></li> <li><i><a href="/wiki/On_Conoids_and_Spheroids" title="On Conoids and Spheroids">On Conoids and Spheroids</a></i></li> <li><a href="/wiki/On_the_Sizes_and_Distances_(Aristarchus)" title="On the Sizes and Distances (Aristarchus)"><i>On the Sizes and Distances</i> <span style="font-size: 85%;">(Aristarchus)</span></a></li> <li><a href="/wiki/On_Sizes_and_Distances_(Hipparchus)" title="On Sizes and Distances (Hipparchus)"><i>On Sizes and Distances</i> <span style="font-size: 85%;">(Hipparchus)</span></a></li> <li><a href="/wiki/Autolycus_of_Pitane" title="Autolycus of Pitane"><i>On the Moving Sphere</i> <span style="font-size: 85%;">(Autolycus)</span></a></li> <li><a href="/wiki/Euclid%27s_Optics" title="Euclid's Optics"><i>Optics</i> <span style="font-size: 85%;">(Euclid)</span></a></li> <li><i><a href="/wiki/On_Spirals" title="On Spirals">On Spirals</a></i></li> <li><i><a href="/wiki/On_the_Sphere_and_Cylinder" title="On the Sphere and Cylinder">On the Sphere and Cylinder</a></i></li> <li><i><a href="/wiki/Ostomachion" title="Ostomachion">Ostomachion</a></i></li> <li><i><a href="/wiki/Planisphaerium" title="Planisphaerium">Planisphaerium</a></i></li> <li><a href="/wiki/Theodosius%27_Spherics" title="Theodosius' Spherics"><i>Spherics</i> <span style="font-size: 85%;">(Theodosius)</span></a></li> <li><a href="/wiki/Menelaus_of_Alexandria" title="Menelaus of Alexandria"><i>Spherics</i> <span style="font-size: 85%;">(Menelaus)</span></a></li> <li><i><a href="/wiki/The_Quadrature_of_the_Parabola" class="mw-redirect" title="The Quadrature of the Parabola">The Quadrature of the Parabola</a></i></li> <li><i><a href="/wiki/The_Sand_Reckoner" title="The Sand Reckoner">The Sand Reckoner</a></i></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%">Problems</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/Constructible_number" title="Constructible number">Constructible numbers</a> <ul><li><a href="/wiki/Angle_trisection" title="Angle trisection">Angle trisection</a></li> <li><a href="/wiki/Doubling_the_cube" title="Doubling the cube">Doubling the cube</a></li> <li><a href="/wiki/Squaring_the_circle" title="Squaring the circle">Squaring the circle</a></li></ul></li> <li><a href="/wiki/Problem_of_Apollonius" title="Problem of Apollonius">Problem of Apollonius</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%">Concepts<br />and definitions</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/Angle" title="Angle">Angle</a> <ul><li><a href="/wiki/Central_angle" title="Central angle">Central</a></li> <li><a href="/wiki/Inscribed_angle" title="Inscribed angle">Inscribed</a></li></ul></li> <li><a href="/wiki/Axiomatic_system" title="Axiomatic system">Axiomatic system</a> <ul><li><a href="/wiki/Axiom" title="Axiom">Axiom</a></li></ul></li> <li><a href="/wiki/Chord_(geometry)" title="Chord (geometry)">Chord</a></li> <li><a href="/wiki/Circles_of_Apollonius" title="Circles of Apollonius">Circles of Apollonius</a> <ul><li><a href="/wiki/Apollonian_circles" title="Apollonian circles">Apollonian circles</a></li> <li><a href="/wiki/Apollonian_gasket" title="Apollonian gasket">Apollonian gasket</a></li></ul></li> <li><a href="/wiki/Circumscribed_circle" title="Circumscribed circle">Circumscribed circle</a></li> <li><a href="/wiki/Commensurability_(mathematics)" title="Commensurability (mathematics)">Commensurability</a></li> <li><a href="/wiki/Diophantine_equation" title="Diophantine equation">Diophantine equation</a></li> <li><a href="https://en.wikiquote.org/wiki/Doctrine_of_proportion_(mathematics)" class="extiw" title="wikiquote:Doctrine of proportion (mathematics)">Doctrine of proportionality</a></li> <li><a href="/wiki/Euclidean_geometry" title="Euclidean geometry">Euclidean geometry</a></li> <li><a href="/wiki/Golden_ratio" title="Golden ratio">Golden ratio</a></li> <li><a href="/wiki/Greek_numerals" title="Greek numerals">Greek numerals</a></li> <li><a href="/wiki/Incircle_and_excircles_of_a_triangle" class="mw-redirect" title="Incircle and excircles of a triangle">Incircle and excircles of a triangle</a></li> <li><a href="/wiki/Method_of_exhaustion" title="Method of exhaustion">Method of exhaustion</a></li> <li><a href="/wiki/Parallel_postulate" title="Parallel postulate">Parallel postulate</a></li> <li><a href="/wiki/Platonic_solid" title="Platonic solid">Platonic solid</a></li> <li><a href="/wiki/Lune_of_Hippocrates" title="Lune of Hippocrates">Lune of Hippocrates</a></li> <li><a href="/wiki/Quadratrix_of_Hippias" title="Quadratrix of Hippias">Quadratrix of Hippias</a></li> <li><a href="/wiki/Regular_polygon" title="Regular polygon">Regular polygon</a></li> <li><a href="/wiki/Straightedge_and_compass_construction" title="Straightedge and compass construction">Straightedge and compass construction</a></li> <li><a href="/wiki/Triangle_center" title="Triangle center">Triangle center</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%">Results</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%">In <a href="/wiki/Euclid%27s_elements" class="mw-redirect" title="Euclid's elements"><i>Elements</i></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/Angle_bisector_theorem" title="Angle bisector theorem">Angle bisector theorem</a></li> <li><a href="/wiki/Exterior_angle_theorem" title="Exterior angle theorem">Exterior angle theorem</a></li> <li><a href="/wiki/Euclidean_algorithm" title="Euclidean algorithm">Euclidean algorithm</a></li> <li><a href="/wiki/Euclid%27s_theorem" title="Euclid's theorem">Euclid's theorem</a></li> <li><a href="/wiki/Geometric_mean_theorem" title="Geometric mean theorem">Geometric mean theorem</a></li> <li><a href="/wiki/Greek_geometric_algebra" class="mw-redirect" title="Greek geometric algebra">Greek geometric algebra</a></li> <li><a href="/wiki/Hinge_theorem" title="Hinge theorem">Hinge theorem</a></li> <li><a href="/wiki/Inscribed_angle_theorem" class="mw-redirect" title="Inscribed angle theorem">Inscribed angle theorem</a></li> <li><a href="/wiki/Intercept_theorem" title="Intercept theorem">Intercept theorem</a></li> <li><a href="/wiki/Intersecting_chords_theorem" title="Intersecting chords theorem">Intersecting chords theorem</a></li> <li><a href="/wiki/Intersecting_secants_theorem" title="Intersecting secants theorem">Intersecting secants theorem</a></li> <li><a href="/wiki/Law_of_cosines" title="Law of cosines">Law of cosines</a></li> <li><a href="/wiki/Pons_asinorum" title="Pons asinorum">Pons asinorum</a></li> <li><a href="/wiki/Pythagorean_theorem" title="Pythagorean theorem">Pythagorean theorem</a></li> <li><a href="/wiki/Tangent-secant_theorem" class="mw-redirect" title="Tangent-secant theorem">Tangent-secant theorem</a></li> <li><a href="/wiki/Thales%27s_theorem" title="Thales's theorem">Thales's theorem</a></li> <li><a href="/wiki/Theorem_of_the_gnomon" title="Theorem of the gnomon">Theorem of the gnomon</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%"><a href="/wiki/Apollonius_of_Tyana" title="Apollonius of Tyana">Apollonius</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/Apollonius%27s_theorem" title="Apollonius's theorem">Apollonius's theorem</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><a href="/wiki/Aristarchus%27s_inequality" title="Aristarchus's inequality">Aristarchus's inequality</a></li> <li><a href="/wiki/Crossbar_theorem" title="Crossbar theorem">Crossbar theorem</a></li> <li><a href="/wiki/Heron%27s_formula" title="Heron's formula">Heron's formula</a></li> <li><a href="/wiki/Irrational_number" title="Irrational number">Irrational numbers</a></li> <li><a href="/wiki/Law_of_sines" title="Law of sines">Law of sines</a></li> <li><a href="/wiki/Menelaus%27s_theorem" title="Menelaus's theorem">Menelaus's theorem</a></li> <li><a href="/wiki/Pappus%27s_area_theorem" title="Pappus's area theorem">Pappus's area theorem</a></li> <li><a href="/wiki/Diophantus_II.VIII" title="Diophantus II.VIII">Problem II.8 of <i>Arithmetica</i></a></li> <li><a href="/wiki/Ptolemy%27s_inequality" title="Ptolemy's inequality">Ptolemy's inequality</a></li> <li><a href="/wiki/Ptolemy%27s_table_of_chords" title="Ptolemy's table of chords">Ptolemy's table of chords</a></li> <li><a href="/wiki/Ptolemy%27s_theorem" title="Ptolemy's theorem">Ptolemy's theorem</a></li> <li><a href="/wiki/Spiral_of_Theodorus" title="Spiral of Theodorus">Spiral of Theodorus</a></li></ul> </div></td></tr></tbody></table><div></div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%">Centers</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/Cyrene,_Libya" title="Cyrene, Libya">Cyrene</a></li> <li><a href="/wiki/Musaeum" class="mw-redirect" title="Musaeum">Mouseion of Alexandria</a></li> <li><a href="/wiki/Platonic_Academy" title="Platonic Academy">Platonic Academy</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-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><td colspan="2" class="navbox-list navbox-odd" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Ancient_Greek_astronomy" title="Ancient Greek astronomy">Ancient Greek astronomy</a></li> <li><a href="/wiki/Attic_numerals" title="Attic numerals">Attic numerals</a></li> <li><a href="/wiki/Greek_numerals" title="Greek numerals">Greek numerals</a></li> <li><a href="/wiki/Latin_translations_of_the_12th_century" title="Latin translations of the 12th century">Latin translations of the 12th century</a></li> <li><a href="/wiki/Non-Euclidean_geometry" title="Non-Euclidean geometry">Non-Euclidean geometry</a></li> <li><a href="/wiki/Philosophy_of_mathematics" title="Philosophy of mathematics">Philosophy of mathematics</a></li> <li><a href="/wiki/Neusis_construction" title="Neusis construction">Neusis construction</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%">History of</th><td class="navbox-list-with-group navbox-list navbox-even" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><i><a href="/wiki/A_History_of_Greek_Mathematics" title="A History of Greek Mathematics">A History of Greek Mathematics</a></i> <ul><li>by <a href="/wiki/Thomas_Heath_(classicist)" title="Thomas Heath (classicist)">Thomas Heath</a></li></ul></li> <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><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></ul></li> <li><a href="/wiki/History_of_geometry" title="History of geometry">geometry</a> <ul><li><a href="/wiki/Timeline_of_geometry" title="Timeline of geometry">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_mathematics" title="History of mathematics">mathematics</a> <ul><li><a href="/wiki/Timeline_of_mathematics" title="Timeline of mathematics">timeline</a></li></ul></li> <li><a href="/wiki/History_of_numbers" class="mw-redirect" title="History of numbers">numbers</a> <ul><li><a href="/wiki/Prehistoric_counting" class="mw-redirect" title="Prehistoric counting">prehistoric counting</a></li></ul></li> <li><a href="/wiki/History_of_ancient_numeral_systems" title="History of ancient numeral systems">numeral systems</a> <ul><li><a href="/wiki/List_of_numeral_systems" title="List of numeral systems">list</a></li></ul></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%">Other 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/Mathematics_in_the_medieval_Islamic_world" title="Mathematics in the medieval Islamic world">Arabian/Islamic</a></li> <li><a href="/wiki/Babylonian_mathematics" title="Babylonian mathematics">Babylonian</a></li> <li><a href="/wiki/Chinese_mathematics" title="Chinese mathematics">Chinese</a></li> <li><a href="/wiki/Ancient_Egyptian_mathematics" title="Ancient Egyptian mathematics">Egyptian</a></li> <li><a href="/wiki/Mathematics_of_the_Incas" title="Mathematics of the Incas">Incan</a></li> <li><a href="/wiki/Indian_mathematics" title="Indian mathematics">Indian</a></li> <li><a href="/wiki/Japanese_mathematics" title="Japanese mathematics">Japanese</a></li></ul> </div></td></tr></tbody></table><div></div></td></tr><tr><td class="navbox-abovebelow" colspan="2"><div><b><span class="nowrap"><span class="noviewer" typeof="mw:File"><span><img alt="" src="//upload.wikimedia.org/wikipedia/commons/thumb/a/ad/Parthenon_from_west.jpg/20px-Parthenon_from_west.jpg" decoding="async" width="17" height="13" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/a/ad/Parthenon_from_west.jpg/40px-Parthenon_from_west.jpg 1.5x" 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