Mathematics - Egyptian, Assessment, History

<!doctype html> <html lang="en" class="topic-desktop ui-ie7 ui-ie"> <head prefix="og: https://ogp.me/ns# fb: https://ogp.me/ns/fb#"> <meta charset="utf-8"> <meta http-equiv="Content-Type" content="text/html; charset=UTF-8" /> <meta name="viewport" content="width=device-width, initial-scale=1.0" /> <link rel="dns-prefetch" href="https://cdn.britannica.com/mendel-resources/3-133"> <link rel="preconnect" href="https://cdn.britannica.com/mendel-resources/3-133"> <link rel="preload" as="script" href="https://www.googletagservices.com/tag/js/gpt.js" /> <link rel="icon" href="/favicon.png" /> <meta name="description" content="Mathematics - Egyptian, Assessment, History: The papyri thus bear witness to a mathematical tradition closely tied to the practical accounting and surveying activities of the scribes. Occasionally, the scribes loosened up a bit: one problem (Rhind papyrus, problem 79), for example, seeks the total from seven houses, seven cats per house, seven mice per cat, seven ears of wheat per mouse, and seven hekat of grain per ear (result: 19,607). Certainly the scribe’s interest in progressions (for which he appears to have a rule) goes beyond practical considerations. Other than this, however, Egyptian mathematics falls firmly within the range of practice. Even allowing for the" /> <meta name="keywords" content="mathematics, encyclopedia, encyclopeadia, britannica, article" /> <link rel="canonical" href="https://www.britannica.com/science/mathematics/Assessment-of-Egyptian-mathematics" /> <title>Mathematics - Egyptian, Assessment, History | Britannica</title>  <script> !function(){"use strict";function e(e){const t=e.match(/((?=([a-z0-9._!#$%+^&*()[\]<>-]+))\2@[a-z0-9._-]+\.[a-z0-9._-]+)/gi);return t?t[0]:""}function t(t){return e(a(t.toLowerCase()))}function a(e){return e.replace(/\s/g,"")}async function n(e){const t={sha256Hash:"",sha1Hash:""};if(!("msCrypto"in window)&&"https:"===location.protocol&&"crypto"in window&&"TextEncoder"in window){const a=(new TextEncoder).encode(e),[n,c]=await Promise.all([s("SHA-256",a),s("SHA-1",a)]);t.sha256Hash=n,t.sha1Hash=c}return t}async function s(e,t){const a=await crypto.subtle.digest(e,t);return Array.from(new Uint8Array(a)).map(e=>("00"+e.toString(16)).slice(-2)).join("")}function c(e){let t=!0;return Object.keys(e).forEach(a=>{0===e[a].length&&(t=!1)}),t}function i(e,t,a){e.splice(t,1);const n="?"+e.join("&")+a.hash;history.replaceState(null,"",n)}var o={checkEmail:e,validateEmail:t,trimInput:a,hashEmail:n,hasHashes:c,removeEmailAndReplaceHistory:i,detectEmails:async function(){const e=new URL(window.location.href),a=Array.from(e.searchParams.entries()).map(e=>`=`);let s,o;const r=["adt_eih","sh_kit"];if(a.forEach((e,t)=>{const a=decodeURIComponent(e),[n,c]=a.split("=");if("adt_ei"===n&&(s={value:c,index:t,emsrc:"url"}),r.includes(n)){o={value:c,index:t,emsrc:"sh_kit"===n?"urlhck":"urlh"}}}),s)t(s.value)&&n(s.value).then(e=>{if(c(e)){const t={value:e,created:Date.now()};localStorage.setItem("adt_ei",JSON.stringify(t)),localStorage.setItem("adt_emsrc",s.emsrc)}});else if(o){const e={value:{sha256Hash:o.value,sha1Hash:""},created:Date.now()};localStorage.setItem("adt_ei",JSON.stringify(e)),localStorage.setItem("adt_emsrc",o.emsrc)}s&&i(a,s.index,e),o&&i(a,o.index,e)},cb:"adthrive"};const{detectEmails:r,cb:l}=o;r()}(); </script> <script type="text/javascript" data-type="Init Mendel"> window.$UI = {}; window.Constants = {"LICENSE_URL": "/bps/license","DEFAULT_TEST_VERSION": "A","DEFAULT_STATE": "XX","QUIZ_URL": "/quiz","SPOTLIGHT_BROWSE_URL": "/stories/spotlight","CONTENT_TYPE_TEXT": "text/plain;charset=UTF-8","TOPIC_FACTS_DATA_URL": "/facts","QUIZ_BROWSE_IMAGE_QUIZZES": "images","TOPIC_MEDIA_PATH": "/images-videos","USER_PROFILE_URL": "/user","DEBUG_URL": "/debug","ONE_GOOD_FACT_URL": "/one-good-fact","ERROR_404_URL": "/error404","PROCON_CITED_IN_THE_NEWS_URL": "/procon/ProCon-in-the-News","PROCON_URL": "/procon","TOPIC_PAGE_CONTENT_AJAX_URL": "/topic-content/page","INFINITE_SCROLL_PREFIX_URL": "/scroll","TOPIC_TOP_QUESTION_BROWSE_URL": "/questions","CC_USD": "USD","domain": "britannica.com","PROCON_EDITOR_ID": "12941390","SURVEY_URL": "/survey","CATEGORY_BROWSE_URL": "/browse","STORY_BROWSE_URL": "/stories","COUNTRY_US": "US","OPEN_MEDIA_OVERLAY_PARAMETER": "/media","NEWSLETTER_SUBSCRIPTION_URL": "/newsletter-subscription","MAINTENANCE_ERROR_URL": "https://maintenance.eb.com","IMARS_EDITOR_ID": "12365882","PROFILE_EB_EDITOR_URL": "/editor","WEB_INF_RESOURCES_PATH": "WEB-INF/resources","AI_ABOUT_PAGE_URL": "/about-britannica-ai","TOPIC_ADDITIONAL_INFO_PATH": "/additional-info","SUDOKU_GAME_URL": "/games/sudoku","CC_INR": "INR","ARTICLE_PRINT_URL": "/print/article","FIRST_EDITION_URL": "/subscriber/firstedition","WW1_PORTAL_URL": "/discover/World-War-I","MENDEL_COOKIE": "__mendel","topicUrlClasses": "[topic, animal, art, biography, event, place, plant, science, sports, technology, procon, money]","DEMYSTIFIED_BROWSE_URL": "/stories/demystified","LIST_BROWSE_URL": "/list/browse","PROFILE_EXPERT_URL": "/contributor","ASSEMBLY_IMAGE_URL": "/image/assembly","DAY_IN_HISTORY_URL": "/on-this-day","DEFAULT_CURRENCY": "USD","CONTENT_TYPE_XML": "text/xml;charset=UTF-8","PORTAL_FINANCE_BROWSE_URL_PREFIX": "/money/browse","MONEY_IMARS_CATEGORY": "13000","AJAX_PREFIX_URL": "/ajax","TOPIC_BROWSE_URL": "/topic-browse","MARKETING_CONTENT": "/marketing-content","ENV_RUNTIME": "runtime","GALLERY_URL": "/gallery","topicUrlClassesList": "topic|animal|art|biography|event|place|plant|science|sports|technology|procon","CONTENT_TYPE_HTML": "text/html;charset=UTF-8","ENV_LOCAL": "override","MEDIA_OVERLAY_URL": "/media-overlay","CHATBOT_PAGE_URL": "/chatbot","NEWSLETTER_PAGE_URL": "/newsletters","ENV_DEV": "development","MEDIA_URL": "/media","TOPIC_TOP_QUESTION_URL": "/question","PORTAL_FINANCE_URL_PREFIX": "/money","PODCASTS_URL": "/podcasts","STAND_ALONE_VIDEO_URL": "/video","MORE_ON_THIS_DAY_URL": "/more-on-this-day","TOPIC_QUOTES_URL": "/quotes","SEARCH_PAGE_URL": "/search","PROCON_CLASS": "PROCON","KUSTOM_MENDEL_APPLICATION_ID": "1","TOPIC_CONTENT_AJAX_URL": "/topic-content/topic","ENV_BRANCH": "branch","ERROR_URL": "/error","MAIN_VERSION": "mainVersion","TOPIC_COLLECTION_URL": "/summary","LOGINBOX_URL": "/auth/loginbox","PROCON_DEBATE_TOPICS_URL": "/procon/Debate-Topics","ONE_GOOD_FACT_BROWSE_URL": "/one-good-fact/all-good-facts","QUIZ_BROWSE_URL": "/quiz/browse","BIO_BROWSE_URL": "/browse/biographies","LIST_URL": "/list","TIGHTROPE_QUIZ_URL": "/quiz/tightrope","ALPHA_BROWSE_URL": "/sitemap","CONTENT_TYPE_JSON": "application/json","DICTIONARY_URL": "/dictionary","COBRAND_IMAGE_URL": "/image/cobrand","PROCON_IN_THE_NEWS_URL": "/procon/pro-and-con-issues-in-the-news","PROCON_BROWSE_URL": "/procon","QUIZ_BROWSE_VOCAB_QUIZZES": "vocabulary-quizzes","SUBMISSION_URL": "/submission","EB_LOG_OUT": "/auth2/logout","ENV_PRODUCTION": "production","EXPLORE_PORTAL_URL": "/explore","TOPIC_AJAX_URL": "/ajax/topic","TOPIC_SUMMARY_BROWSE_URL": "/summaries","WTFACT_BROWSE_URL": "/stories/wtfact","VIDEO_CHANNEL_URL": "/videos","GALLERY_BROWSE_URL": "/gallery/browse","CACHE_URL": "/cache","PROCON_ABOUT_URL": "/procon/About-ProCon","COMPANION_BROWSE_URL": "/stories/companion","MEDIA_FOLDER": "/eb-media","SHOW_ALL_CONTRIBUTORS": "/additional-info#contributors","BRITANNICA_EDITORS_ID": "4419","ENV_CACHE_DISABLED": "mendelCache","CALCULATORS_BROWSE_URL": "/calculators","STORY_URL": "/story","DEFAULT_COUNTRY": "US","NAVBAR_URL": "/ajax/navbar","EB_LOGIN_URL": "/auth/eb-login","NEW_ARTICLES_URL": "/new-articles",}; window.CDN = "https://cdn.britannica.com"; window.CAM_SETTINGS_URL = "https://cam.britannica.com/settings"; window.CAM_LOGIN_URL = "https://cam.britannica.com/login"; window.CAM_SIGN_UP_URL = "https://cam.britannica.com/registration" window.Mendel = { "config" : { "domain": "britannica.com", "page": "Topic", "videoPlayerId": "UyMCoK2v", "sharedUrl": "https://www.britannica.com/science/mathematics/Assessment-of-Egyptian-mathematics", "amuselabsUrl": "https://cdn3.amuselabs.com", "resourcesPrefixUrl": "https://cdn.britannica.com/mendel-resources/3-133/[url]?v=3.133.9", "date": 20250216, "userInfo": { "type": "ANONYMOUS" ,"currency": "AUUS" ,"country": "SG" ,"state": "XX" ,"timezone": "Asia/Singapore" ,"bcomId": "1485315548685719051" ,"hasAds": true ,"testVersion": "C" ,"adsTestVersion": "C" ,"consumerId": "" ,"instId": "" ,"consumerUserName": "" ,"instUserName": "" ,"cognito": null }, "tvs":{ "r":[25,25,25,25], "a": [25,25,45,5]}, "isLoggedInAsUser": false, "isPhone": false, "isDesktop": true, "logoutUrl": "/auth2/logout", "selfServiceUrl": "https://myaccount.britannica.com", "cdnUrl": "https://cdn.britannica.com", "chatbotApi": "https://www.britannica.com/chat-api", "fetchOffset": 800, "mendelCookieName": "__mendel", "mendelCookie": {"surveyShown":false,"visitedTopicId":369194,"currentDate":20250216}, "autocompleteToSearchPage": false,"topicUrl": "https://www.britannica.com/science/mathematics", "freeTopicReason": "NO_REFERRER", "topicId": 369194, "template": "DESKTOP", "type": "CORE", "hasToc": true, "chatbotApi": "https://www.britannica.com/chat-api", "showPreview": false, }, "GA": {"leg":"C","adLeg":"C","userType":"ANONYMOUS","pageType":"Topic","articleTemplateType":"PAGINATED","gisted":false,"pageNumber":5,"hasSummarizeButton":false,"hasAskButton":true} }; </script> <meta property="fb:app_id" content="1887621861548296"/ <meta name="twitter:card" content="summary_large_image" /> <meta name="twitter:site" content="@britannica" /> <meta name="twitter:image" content="https://cdn.britannica.com/54/2354-050-B47B3735/Babylonian-tablet.jpg" /> <meta name="twitter:description" content="Mathematics - Egyptian, Assessment, History: The papyri thus bear witness to a mathematical tradition closely tied to the practical accounting and surveying activities of the scribes. Occasionally, the scribes loosened up a bit: one problem (Rhind papyrus, problem 79), for example, seeks the total from seven houses, seven cats per house, seven mice per cat, seven ears of wheat per mouse, and seven hekat of grain per ear (result: 19,607). Certainly the scribe’s interest in progressions (for which he appears to have a rule) goes beyond practical considerations. Other than this, however, Egyptian mathematics falls firmly within the range of practice. Even allowing for the"/> <meta property="og:type" content="ARTICLE"/> <meta property="og:title" content="Mathematics - Egyptian, Assessment, History | Britannica"/> <meta property="og:description" content="Mathematics - Egyptian, Assessment, History: The papyri thus bear witness to a mathematical tradition closely tied to the practical accounting and surveying activities of the scribes. Occasionally, the scribes loosened up a bit: one problem (Rhind papyrus, problem 79), for example, seeks the total from seven houses, seven cats per house, seven mice per cat, seven ears of wheat per mouse, and seven hekat of grain per ear (result: 19,607). Certainly the scribe’s interest in progressions (for which he appears to have a rule) goes beyond practical considerations. Other than this, however, Egyptian mathematics falls firmly within the range of practice. Even allowing for the"/> <meta property="og:site_name" content="Encyclopedia Britannica" /> <meta property="og:url" content="https://www.britannica.com/science/mathematics"/> <meta property="og:image" content="https://cdn.britannica.com/54/2354-050-B47B3735/Babylonian-tablet.jpg" /> <meta property="og:image:type" content="image/jpeg" /> <script type="text/javascript" data-type="init opengraph"> Mendel.openGraph = {"type":"ARTICLE","title":"Mathematics - Egyptian, Assessment, History","description":"Mathematics - Egyptian, Assessment, History: The papyri thus bear witness to a mathematical tradition closely tied to the practical accounting and surveying activities of the scribes. Occasionally, the scribes loosened up a bit: one problem (Rhind papyrus, problem 79), for example, seeks the total from seven houses, seven cats per house, seven mice per cat, seven ears of wheat per mouse, and seven hekat of grain per ear (result: 19,607). Certainly the scribe’s interest in progressions (for which he appears to have a rule) goes beyond practical considerations. Other than this, however, Egyptian mathematics falls firmly within the range of practice. Even allowing for the","imageUrl":"https://cdn.britannica.com/54/2354-050-B47B3735/Babylonian-tablet.jpg","imageType":"image/jpeg","pageUrl":"https://www.britannica.com/science/mathematics"}</script> <link rel="preconnect" href="https://fonts.googleapis.com/"> <link rel="dns-prefetch" href="https://fonts.googleapis.com/" > <link rel="stylesheet" href="https://fonts.googleapis.com/icon?family=Material+Icons"> <link href="https://cdn.britannica.com/mendel-resources/3-133/dist/vendor-bundle.css?v=3.133.9" rel="stylesheet" /> <link href="https://cdn.britannica.com/mendel-resources/3-133/dist/mendel-css.css?v=3.133.9" rel="stylesheet" /> <link href="https://cdn.britannica.com/mendel-resources/3-133/dist/topic-page.css?v=3.133.9" rel="stylesheet" /> <script type="text/javascript"> if (self !== top) { top.location = self.location; } </script> <script src="https://cdn.britannica.com/mendel-resources/3-133/js/at.js?v=3.133.9" async ></script> <script> dataLayer = []; </script> <script type="text/javascript">(function(w,d,s,l,i){w[l]=w[l]||[];w[l].push({'gtm.start': new Date().getTime(),event:'gtm.js'});var f=d.getElementsByTagName(s)[0], j=d.createElement(s),dl=l!='dataLayer'?'&l='+l:'';j.async=true;j.src= '//www.googletagmanager.com/gtm.js?id='+i+dl;f.parentNode.insertBefore(j,f); })(window,document,'script','dataLayer','GTM-5W6NC8'); </script> <meta name="last-modified" content="2025-02-14" /> <script type="application/ld+json"> {"headline":"Mathematics | Definition, History, & Importance","image":{"url":"https://cdn.britannica.com/54/2354-050-B47B3735/Babylonian-tablet.jpg","@type":"ImageObject"},"author":[{"name":"Jeremy John Gray","url":"https://www.britannica.com/contributor/Jeremy-John-Gray/3547","@type":"Person"},{"name":"John L. Berggren","url":"https://www.britannica.com/contributor/John-L-Berggren/3484","@type":"Person"}],"keywords":"mathematics","wordcount":41449,"url":"https://www.britannica.com/science/mathematics","datePublished":"1999-07-26T00:00:00Z","dateModified":"2025-02-14T00:00:00Z","description":"Mathematics, the science of structure, order, and relation that has evolved from counting, measuring, and describing the shapes of objects. Mathematics has been an indispensable adjunct to the physical sciences and technology and has assumed a similar role in the life sciences.","publisher":{"name":"Encyclopedia Britannica","@type":"Organization","logo":{"url":"https://corporate.britannica.com/wp-content/themes/eb-corporate/_img/logo.png","@type":"ImageObject"}},"@context":"https://schema.org","@type":"article"} </script> </head> <body data-leg="C" class="new-topic topic-desktop first-page-false user-ANONYMOUS user-ads md-desktop leg-c b-ie">  <script>Mendel.config.adProvider='cafemedia';</script> <script data-no-optimize="1" data-cfasync="false"> (function(w, d) { w.adthrive = w.adthrive || {}; w.adthrive.cmd = w. adthrive.cmd || []; w.adthrive.plugin = 'adthrive-ads-manual'; w.adthrive.host = 'ads.adthrive.com';var s = d.createElement('script'); s.async = true; s.referrerpolicy='no-referrer-when-downgrade'; s.src = 'https://' + w.adthrive.host + '/sites/61575e5c934c481d714b3ca9/ads.min.js?referrer=' + w.encodeURIComponent(w.location.href) + '&cb=' + (Math.floor(Math.random() * 100) + 1); var n = d.getElementsByTagName('script')[0]; n.parentNode.insertBefore(s, n); })(window, document); </script> <div class="ie-warning d-flex align-items-center align-self-center justify-content-center site-alert bg-orange"> <div> You are using an <strong>outdated</strong> browser. Please <a class="text-white text-underscore" href="https://browsehappy.com/">upgrade your browser</a> to improve your experience and security. </div> </div> <script id="json-navbar-info" type="application/json"> {"topSectionLinks":[{"title":"Ask the Chatbot","url":"/chatbot","navbarId":"CHATBOT"},{"title":"Games & Quizzes","url":"/quiz/browse","navbarId":"QUIZZES"},{"title":"History & Society","url":"/History-Society","navbarId":"HISTORY"},{"title":"Science & Tech","url":"/Science-Tech","selected":true,"navbarId":"SCIENCE"},{"title":"Biographies","url":"/Biographies","navbarId":"BIOS"},{"title":"Animals & Nature","url":"/Animals-Nature","navbarId":"ANIMALS"},{"title":"Geography & Travel","url":"/Geography-Travel","navbarId":"GEOGRAPHY"},{"title":"Arts & Culture","url":"/Arts-Culture","navbarId":"ART"},{"title":"ProCon","url":"/procon","navbarId":"PROCON"},{"title":"Money","url":"/money","navbarId":"MONEY"},{"title":"Videos","url":"/videos","navbarId":"VIDEOS"}],"selectedSuperCategory":{"id":6,"title":"Science & Tech","url":"Science-Tech","description":"Explore science and technology; astronomy; biology; chemistry; earth science; mathematics; physics; technology, agriculture, cars, computers, engineering, industry, inventions, communication","keywords":"astronomy; biology; chemistry; earth science; mathematics; physics; technology, agriculture, cars, computers, engineering, industry, inventions, communication","classId":"SCIENCE","sortOrder":2},"selectedNavbarLink":{"title":"Science & Tech","url":"/Science-Tech","selected":true,"navbarId":"SCIENCE"}} </script> <script id="json-hamburger-menu" type="application/json"> {"britannicaMenu1":[{"title":"Home","url":"/"},{"title":"History & Society","url":"/History-Society"},{"title":"Science & Tech","url":"/Science-Tech"},{"title":"Biographies","url":"/Biographies"},{"title":"Animals & Nature","url":"/Animals-Nature"},{"title":"Geography & Travel","url":"/Geography-Travel"},{"title":"Arts & Culture","url":"/Arts-Culture"},{"title":"ProCon","url":"/procon"},{"title":"Money","url":"/money"}],"britannicaMenu2":[{"title":"Games & Quizzes","url":"/quiz/browse"},{"title":"Videos","url":"/videos"},{"title":"On This Day","url":"/on-this-day"},{"title":"One Good Fact","url":"/one-good-fact"},{"title":"Dictionary","url":"/dictionary"},{"title":"New Articles","url":"/new-articles"}],"browseByCategory":[{"title":{"id":5,"title":"History & Society","url":"/History-Society"},"links":[{"title":"Lifestyles & Social Issues","url":"/browse/Lifestyles-Social-Issues"},{"title":"Philosophy & Religion","url":"/browse/Philosophy-Religion"},{"title":"Politics, Law & Government","url":"/browse/Politics-Law-Government"},{"title":"World History","url":"/browse/World-History"}]},{"title":{"id":6,"title":"Science & Tech","url":"/Science-Tech"},"links":[{"title":"Health & Medicine","url":"/browse/Health-Medicine"},{"title":"Science","url":"/browse/Science"},{"title":"Technology","url":"/browse/Technology"}]},{"title":{"id":3,"title":"Biographies","url":"/Biographies"},"links":[{"title":"Browse Biographies","url":"/browse/biographies"}]},{"title":{"id":1,"title":"Animals & Nature","url":"/Animals-Nature"},"links":[{"title":"Birds, Reptiles & Other Vertebrates","url":"/browse/Birds-Reptiles-Vertebrates"},{"title":"Bugs, Mollusks & Other Invertebrates","url":"/browse/Bugs-Mollusks-Invertebrates"},{"title":"Environment","url":"/browse/Environment"},{"title":"Fossils & Geologic Time","url":"/browse/Fossil-Geologic-Time"},{"title":"Mammals","url":"/browse/Mammals"},{"title":"Plants","url":"/browse/Plants"}]},{"title":{"id":4,"title":"Geography & Travel","url":"/Geography-Travel"},"links":[{"title":"Geography & Travel","url":"/browse/Geography-Travel"}]},{"title":{"id":2,"title":"Arts & Culture","url":"/Arts-Culture"},"links":[{"title":"Entertainment & Pop Culture","url":"/browse/Entertainment-Pop-Culture"},{"title":"Literature","url":"/browse/Literature"},{"title":"Sports & Recreation","url":"/browse/Sports-Recreation"},{"title":"Visual Arts","url":"/browse/Visual-Arts"}]}],"browseByFeature":[{"title":"Companions","url":"/stories/companion"},{"title":"Demystified","url":"/stories/demystified"},{"title":"Image Galleries","url":"/gallery/browse"},{"title":"Lists","url":"/list/browse"},{"title":"Podcasts","url":"/podcasts"},{"title":"Spotlight","url":"/stories/spotlight"},{"title":"Summaries","url":"/summary"},{"title":"The Forum","url":"/stories/the-forum"},{"title":"Top Questions","url":"/question"},{"title":"#WTFact","url":"/stories/wtfact"}],"moreFromBritannica":[{"title":"Britannica Kids","url":"https://kids.britannica.com/","newTab":true}],"menuType":"DEFAULT"} </script> <header id="header" class="bg-navy-dark"> <div class="global-nav-top-bar"> <div class="grid gx-0 h-100 justify-content-between align-items-center container-lg mx-auto p-0 position-relative"> <div class="d-flex align-items-center"> <button class="d-flex align-items-center justify-self-start js-toggle js-toggle-hamburger btn btn-link link-white btn-sm rounded-0 p-10"> <div class="hamburger-tooltip"> <em class="material-icons d-inline-block font-24" id="nav-toggle" data-icon="menu"></em> </div> <em class="material-icons d-inline-block font-24 global-nav-search-icon" id="nav-search-icon" data-icon="search" ></em> </button> <a href="/" class="d-flex align-items-center justify-content-center ml-10"> <img loading="lazy" src="https://cdn.britannica.com/mendel/eb-logo/MendelNewThistleLogo.png" alt="Encyclopedia Britannica" class="global-nav-logo global-nav-logo-left" /> </a> <div class="global-nav-top-search-bar global-nav-top-search-container global-nav-search-container" id="global-nav-top-search-bar"> <form method="get" action="/search" id="global-nav-search" class="md-search-form m-0 global-nav-search-bar-small"> <div class="search-box position-relative col-100"> <label class="sr-only" for="global-nav-search-query">Search Britannica</label> <input name="query" id="global-nav-search-query" placeholder="Search Britannica..." class="form-control form-control-lg rounded-lg font-16 search-query pl-20 pr-70 shadow-sm" maxlength="200" autocomplete="off" aria-label="Search Britannica" /> <button class="search-reset-btn btn btn-link px-10 position-absolute top-0 h-100 d-none" type="reset"> <em class="material-icons" data-icon="close"></em> </button> <button class="search-submit btn btn-link text-blue px-10 position-absolute top-0 right-0 h-100" type="submit" disabled> <span class="sr-only">Click here to search</span> <em class="material-icons search-icon" data-icon="search"></em> </button> </div> </form> </div> </div> <a href="/" class="d-flex align-items-center justify-content-center"> <img loading="lazy" src="https://cdn.britannica.com/mendel/eb-logo/MendelNewThistleLogo.png" alt="Encyclopedia Britannica" class="global-nav-center global-nav-logo non-homepage-logo" /> </a> <form method="get" action="/search" id="global-nav-search" class="md-search-form m-0 global-nav-search-bar-small global-nav-center search global-nav-center-search-container"> <div class="search-box position-relative col-100"> <label class="sr-only" for="global-nav-search-query">Search Britannica</label> <input name="query" id="global-nav-search-query" placeholder="Search Britannica..." class="form-control form-control-lg rounded-lg font-16 search-query pl-20 pr-70 shadow-sm" maxlength="200" autocomplete="off" aria-label="Search Britannica" /> <button class="search-reset-btn btn btn-link px-10 position-absolute top-0 h-100 d-none" type="reset"> <em class="material-icons" data-icon="close"></em> </button> <button class="search-submit btn btn-link text-blue px-10 position-absolute top-0 right-0 h-100" type="submit" disabled> <span class="sr-only">Click here to search</span> <em class="material-icons search-icon" data-icon="search"></em> </button> </div> </form> <div class="col-35 col-sm-auto text-right order-3 mr-lg-15 align-items-center d-flex justify-content-end"> <div class="d-none d-md-inline-block"> <SPAN class="marketing-HEADER_SUBSCRIPTION_DESKTOP2 marketing-content" data-marketing-id="HEADER_SUBSCRIPTION_DESKTOP2"><a href="https://premium.britannica.com/premium-membership/?utm_source=premium&utm_medium=global-nav&utm_campaign=evergreen-cap" class="subscribe-link btn btn-sm btn-orange py-5 mr-10" target="_blank" rel="noopener"> SUBSCRIBE </a></SPAN></div> <div class="d-inline-block d-md-none mr-5 mr-sm-10"> <SPAN class="marketing-HEADER_SUBSCRIPTION_MOBILE marketing-content" data-marketing-id="HEADER_SUBSCRIPTION_MOBILE"><a href="https://premium.britannica.com/premium-membership/?utm_source=premium&utm_medium=global-nav-mobile&utm_campaign=evergreen" class="subscribe-link btn btn-xs btn-orange p-5" target="_blank" rel="noopener"> SUBSCRIBE </a></SPAN></div> <button class="js-toggle-user-dropdown js-toggle btn btn-sm btn-link link-white rounded-0 px-md-15 pl-5 pr-5"> <span class="d-none d-md-inline-block mr-5">Login</span> <em class="material-icons d-inline-block d-md-none font-16 font-sm-20" data-icon="account_circle"></em> <div class="d-none dropdown-menu-subscription-link">https://premium.britannica.com/premium-membership/?utm_source=premium&utm_medium=nav-login-box&utm_campaign=evergreen</div> <em class="material-icons inactive-icon d-inline-block font-18" data-icon="keyboard_arrow_down"></em> <em class="material-icons active-icon d-inline-block font-18" data-icon="keyboard_arrow_up"></em> </button> </div> </div> </div> <div class="d-none hamburger-menu-subscription-link"><DIV class="marketing-HAMBURGER_MENU_CTA marketing-content" data-marketing-id="HAMBURGER_MENU_CTA"><a href="https://premium.britannica.com/premium-membership/?utm_source=premium&utm_medium=hamburger-menu&utm_campaign=evergreen" class="subscribe-link btn btn-sm btn-orange py-5" target="_blank"> SUBSCRIBE </a></DIV></div> <div id="global-nav-react"> <div class="d-none"> <ul> <li><a href="/">Home</a></li> <li><a href="/History-Society">History & Society</a></li> <li><a href="/Science-Tech">Science & Tech</a></li> <li><a href="/Biographies">Biographies</a></li> <li><a href="/Animals-Nature">Animals & Nature</a></li> <li><a href="/Geography-Travel">Geography & Travel</a></li> <li><a href="/Arts-Culture">Arts & Culture</a></li> <li><a href="/procon">ProCon</a></li> <li><a href="/money">Money</a></li> </ul> <ul> <li><a href="/quiz/browse">Games & Quizzes</a></li> <li><a href="/videos">Videos</a></li> <li><a href="/on-this-day">On This Day</a></li> <li><a href="/one-good-fact">One Good Fact</a></li> <li><a href="/dictionary">Dictionary</a></li> <li><a href="/new-articles">New Articles</a></li> </ul> <a href="/History-Society">History & Society</a> <ul> <li><a href="/browse/Lifestyles-Social-Issues">Lifestyles & Social Issues</a></li> <li><a href="/browse/Philosophy-Religion">Philosophy & Religion</a></li> <li><a href="/browse/Politics-Law-Government">Politics, Law & Government</a></li> <li><a href="/browse/World-History">World History</a></li> </ul> <a href="/Science-Tech">Science & Tech</a> <ul> <li><a href="/browse/Health-Medicine">Health & Medicine</a></li> <li><a href="/browse/Science">Science</a></li> <li><a href="/browse/Technology">Technology</a></li> </ul> <a href="/Biographies">Biographies</a> <ul> <li><a href="/browse/biographies">Browse Biographies</a></li> </ul> <a href="/Animals-Nature">Animals & Nature</a> <ul> <li><a href="/browse/Birds-Reptiles-Vertebrates">Birds, Reptiles & Other Vertebrates</a></li> <li><a href="/browse/Bugs-Mollusks-Invertebrates">Bugs, Mollusks & Other Invertebrates</a></li> <li><a href="/browse/Environment">Environment</a></li> <li><a href="/browse/Fossil-Geologic-Time">Fossils & Geologic Time</a></li> <li><a href="/browse/Mammals">Mammals</a></li> <li><a href="/browse/Plants">Plants</a></li> </ul> <a href="/Geography-Travel">Geography & Travel</a> <ul> <li><a href="/browse/Geography-Travel">Geography & Travel</a></li> </ul> <a href="/Arts-Culture">Arts & Culture</a> <ul> <li><a href="/browse/Entertainment-Pop-Culture">Entertainment & Pop Culture</a></li> <li><a href="/browse/Literature">Literature</a></li> <li><a href="/browse/Sports-Recreation">Sports & Recreation</a></li> <li><a href="/browse/Visual-Arts">Visual Arts</a></li> </ul> <ul> <li><a href="/stories/companion">Companions</a></li> <li><a href="/stories/demystified">Demystified</a></li> <li><a href="/gallery/browse">Image Galleries</a></li> <li><a href="/list/browse">Lists</a></li> <li><a href="/podcasts">Podcasts</a></li> <li><a href="/stories/spotlight">Spotlight</a></li> <li><a href="/summary">Summaries</a></li> <li><a href="/stories/the-forum">The Forum</a></li> <li><a href="/question">Top Questions</a></li> <li><a href="/stories/wtfact">#WTFact</a></li> </ul> <ul> <li><a href="https://kids.britannica.com/">Britannica Kids</a></li> </ul> </div> </div> </header> <div class="bg-navy-dark"> <div class="container-lg p-0 d-flex justify-content-center global-nav-categories-bar overflow-hidden"> <div class="slider js-slider position-relative d-inline-flex align-items-center mw-100 global-nav-slider category-snap-slider"> <div class="slider-container js-slider-container overflow-hidden d-flex font-14 overflow-hidden text-nowrap mx-5"> <a class="nav-bar-category mx-5 category-link-CHATBOT " href="/chatbot">Ask the Chatbot</a> <a class="nav-bar-category mx-5 category-link-QUIZZES " href="/quiz/browse">Games & Quizzes</a> <a class="nav-bar-category mx-5 category-link-HISTORY " href="/History-Society">History & Society</a> <a class="nav-bar-category mx-5 category-link-SCIENCE selected selected" href="/Science-Tech">Science & Tech</a> <a class="nav-bar-category mx-5 category-link-BIOS " href="/Biographies">Biographies</a> <a class="nav-bar-category mx-5 category-link-ANIMALS " href="/Animals-Nature">Animals & Nature</a> <a class="nav-bar-category mx-5 category-link-GEOGRAPHY " href="/Geography-Travel">Geography & Travel</a> <a class="nav-bar-category mx-5 category-link-ART " href="/Arts-Culture">Arts & Culture</a> <a class="nav-bar-category mx-5 category-link-PROCON " href="/procon">ProCon</a> <a class="nav-bar-category mx-5 category-link-MONEY " href="/money">Money</a> <a class="nav-bar-category mx-5 category-link-VIDEOS " href="/videos">Videos</a> </div> <button disabled class="prev-button js-prev-button position-absolute btn btn-circle shadow btn-blue " aria-label="Previous"> <span class="material-icons md-24" data-icon="keyboard_arrow_left"></span> </button> <button disabled class="next-button js-next-button position-absolute btn btn-circle shadow btn-blue " aria-label="Next"> <span class="material-icons md-24" data-icon="keyboard_arrow_right"></span> </button> </div> </div> </div> <main> <div class="md-page-wrapper"> <div id="content" class="md-content"> <div class="md-article-container template-desktop infinite-pagination"> <div class="infinite-scroll-container article last"> <script> Object.assign( window.Mendel.config, { "infiniteScrollList": [{"p":27,"t":369194},{"p":11,"t":14885},{"p":1,"t":89161},{"p":10,"t":229851},{"p":4,"t":422325},{"p":10,"t":564172},{"p":4,"t":599686},{"p":8,"t":536159},{"p":1,"t":213751},{"p":1,"t":2228673}], "sequence": 5, "topics": {} }); </script> <article class="article-content container-lg qa-content px-0 pt-0 pb-40 py-lg-20 content md-expanded" data-topic-id="369194"> <div class="grid gx-0"> <div class="col-auto"> <div class="topic-left-rail md-article-drawer position-relative d-flex border-right-sm border-left-sm open"> <div class="drawer d-flex flex-column open"> <div class="left-rail-section-content"> <div class="topic-left-rail-header text-truncate bg-gray-50 position-relative text-right d-flex align-items-center"> <div class="tlr-title px-20 py-15 text-left"> <em class="material-icons text-gray-400 d-lg-none" data-icon="toc"></em> <a class="font-serif font-weight-bold text-black link-blue" href="https://www.britannica.com/science/mathematics">mathematics</a> </div> <button aria-label="Close" class="js-sections-close-button btn-link btn-sm btn d-lg-none position-absolute top-0 p-10 right-0" > <em class="material-icons font-26" data-icon="close"></em> </button> </div> <div class="section-content pl-10 pr-20 pl-sm-50 pr-sm-60 pl-lg-5 pr-lg-10 pt-10 pt-lg-0 bg-gray-50 clear-catfish-ad"> <div class="toc mb-20"> <div class="font-serif font-14 font-weight-bold mx-15 mb-15 mt-20"> Table of Contents </div> <ul class="list-unstyled my-0" data-level="h1"><li data-target="#ref1"><div class="pl-25"><a class="link-gray-900 w-100" href="/science/mathematics">Introduction</a></div><div class="ml-40 toc-drawer sub-toc-drawer"></div></li><li data-target="#ref253521"><div class="d-flex align-items-center"><div class="ml-25"></div><a class="w-100 link-gray-900" href="/science/mathematics/Ancient-mathematical-sources">Ancient mathematical sources</a></div><div class="ml-40 toc-drawer sub-toc-drawer"></div></li><li data-target="#ref65969"><div class="d-flex align-items-center"><button class="h1-link-drawer-button btn btn-xs btn-circle d-flex rounded" type="button" aria-label="Toggle Heading"><em class="material-icons font-18" data-icon="keyboard_arrow_right"></em></button><a class="w-100 link-gray-900" href="/science/mathematics/Ancient-mathematical-sources#ref65969">Mathematics in ancient Mesopotamia</a></div><div class="ml-40 toc-drawer sub-toc-drawer"><ul class="list-unstyled" data-level="h2"><li data-target="#ref65970"><a class="w-100 link-gray-900" href="/science/mathematics/Ancient-mathematical-sources#ref65970">The numeral system and arithmetic operations</a></li></ul><ul class="list-unstyled" data-level="h2"><li data-target="#ref65971"><a class="w-100 link-gray-900" href="/science/mathematics/Geometric-and-algebraic-problems">Geometric and algebraic problems</a></li></ul><ul class="list-unstyled" data-level="h2"><li data-target="#ref65972"><a class="w-100 link-gray-900" href="/science/mathematics/Geometric-and-algebraic-problems#ref65972">Mathematical astronomy</a></li></ul></div></li><li data-target="#ref65973"><div class="d-flex align-items-center"><button class="h1-link-drawer-button btn btn-xs btn-circle d-flex rounded" type="button" aria-label="Toggle Heading"><em class="material-icons font-18" data-icon="keyboard_arrow_right"></em></button><a class="w-100 link-gray-900" href="/science/mathematics/Mathematics-in-ancient-Egypt">Mathematics in ancient Egypt</a></div><div class="ml-40 toc-drawer sub-toc-drawer"><ul class="list-unstyled" data-level="h2"><li data-target="#ref65974"><a class="w-100 link-gray-900" href="/science/mathematics/Mathematics-in-ancient-Egypt#ref65974">The numeral system and arithmetic operations</a></li></ul><ul class="list-unstyled" data-level="h2"><li data-target="#ref65975"><a class="w-100 link-gray-900" href="/science/mathematics/Mathematics-in-ancient-Egypt#ref65975">Geometry</a></li></ul><ul class="list-unstyled" data-level="h2"><li data-target="#ref65976"><a class="w-100 link-gray-900" href="/science/mathematics/Assessment-of-Egyptian-mathematics">Assessment of Egyptian mathematics</a></li></ul></div></li><li data-target="#ref65977"><div class="d-flex align-items-center"><button class="h1-link-drawer-button btn btn-xs btn-circle d-flex rounded" type="button" aria-label="Toggle Heading"><em class="material-icons font-18" data-icon="keyboard_arrow_right"></em></button><a class="w-100 link-gray-900" href="/science/mathematics/Assessment-of-Egyptian-mathematics#ref65977">Greek mathematics</a></div><div class="ml-40 toc-drawer sub-toc-drawer"><ul class="list-unstyled" data-level="h2"><li data-target="#ref65978"><a class="w-100 link-gray-900" href="/science/mathematics/Assessment-of-Egyptian-mathematics#ref65978">The development of pure mathematics</a><ul class="list-unstyled" data-level="h3"><li data-target="#ref65979"><a class="w-100 link-gray-900" href="/science/mathematics/Assessment-of-Egyptian-mathematics#ref65979">The pre-Euclidean period</a></li></ul><ul class="list-unstyled" data-level="h3"><li data-target="#ref65980"><a class="w-100 link-gray-900" href="/science/mathematics/Assessment-of-Egyptian-mathematics#ref65980">The <em>Elements</em></a></li></ul><ul class="list-unstyled" data-level="h3"><li data-target="#ref65981"><a class="w-100 link-gray-900" href="/science/mathematics/The-three-classical-problems">The three classical problems</a></li></ul></li></ul><ul class="list-unstyled" data-level="h2"><li data-target="#ref65982"><a class="w-100 link-gray-900" href="/science/mathematics/The-three-classical-problems#ref65982">Geometry in the 3rd century <span class="text-smallcaps">bce</span></a><ul class="list-unstyled" data-level="h3"><li data-target="#ref65983"><a class="w-100 link-gray-900" href="/science/mathematics/The-three-classical-problems#ref65983">Archimedes</a></li></ul><ul class="list-unstyled" data-level="h3"><li data-target="#ref65984"><a class="w-100 link-gray-900" href="/science/mathematics/Apollonius">Apollonius</a></li></ul><ul class="list-unstyled" data-level="h3"><li data-target="#ref65985"><a class="w-100 link-gray-900" href="/science/mathematics/Applied-geometry">Applied geometry</a></li></ul></li></ul><ul class="list-unstyled" data-level="h2"><li data-target="#ref65986"><a class="w-100 link-gray-900" href="/science/mathematics/Applied-geometry#ref65986">Later trends in geometry and arithmetic</a><ul class="list-unstyled" data-level="h3"><li data-target="#ref65987"><a class="w-100 link-gray-900" href="/science/mathematics/Applied-geometry#ref65987">Greek trigonometry and mensuration</a></li></ul><ul class="list-unstyled" data-level="h3"><li data-target="#ref65988"><a class="w-100 link-gray-900" href="/science/mathematics/Number-theory">Number theory</a></li></ul><ul class="list-unstyled" data-level="h3"><li data-target="#ref65989"><a class="w-100 link-gray-900" href="/science/mathematics/Number-theory#ref65989">Survival and influence of Greek mathematics</a></li></ul></li></ul></div></li><li data-target="#ref65990"><div class="d-flex align-items-center"><button class="h1-link-drawer-button btn btn-xs btn-circle d-flex rounded" type="button" aria-label="Toggle Heading"><em class="material-icons font-18" data-icon="keyboard_arrow_right"></em></button><a class="w-100 link-gray-900" href="/science/mathematics/Mathematics-in-the-Islamic-world-8th-15th-century">Mathematics in the Islamic world (8th–15th century)</a></div><div class="ml-40 toc-drawer sub-toc-drawer"><ul class="list-unstyled" data-level="h2"><li data-target="#ref65991"><a class="w-100 link-gray-900" href="/science/mathematics/Mathematics-in-the-Islamic-world-8th-15th-century#ref65991">Origins</a></li></ul><ul class="list-unstyled" data-level="h2"><li data-target="#ref65992"><a class="w-100 link-gray-900" href="/science/mathematics/Mathematics-in-the-Islamic-world-8th-15th-century#ref65992">Mathematics in the 9th century</a></li></ul><ul class="list-unstyled" data-level="h2"><li data-target="#ref65993"><a class="w-100 link-gray-900" href="/science/mathematics/Mathematics-in-the-Islamic-world-8th-15th-century#ref65993">Mathematics in the 10th century</a></li></ul><ul class="list-unstyled" data-level="h2"><li data-target="#ref65994"><a class="w-100 link-gray-900" href="/science/mathematics/Omar-Khayyam">Omar Khayyam</a></li></ul><ul class="list-unstyled" data-level="h2"><li data-target="#ref65995"><a class="w-100 link-gray-900" href="/science/mathematics/Omar-Khayyam#ref65995">Islamic mathematics to the 15th century</a></li></ul></div></li><li data-target="#ref65996"><div class="d-flex align-items-center"><button class="h1-link-drawer-button btn btn-xs btn-circle d-flex rounded" type="button" aria-label="Toggle Heading"><em class="material-icons font-18" data-icon="keyboard_arrow_right"></em></button><a class="w-100 link-gray-900" href="/science/mathematics/Omar-Khayyam#ref65996">European mathematics during the Middle Ages and Renaissance</a></div><div class="ml-40 toc-drawer sub-toc-drawer"><ul class="list-unstyled" data-level="h2"><li data-target="#ref65997"><a class="w-100 link-gray-900" href="/science/mathematics/The-transmission-of-Greek-and-Arabic-learning">The transmission of Greek and Arabic learning</a></li></ul><ul class="list-unstyled" data-level="h2"><li data-target="#ref65998"><a class="w-100 link-gray-900" href="/science/mathematics/The-transmission-of-Greek-and-Arabic-learning#ref65998">The universities</a></li></ul><ul class="list-unstyled" data-level="h2"><li data-target="#ref65999"><a class="w-100 link-gray-900" href="/science/mathematics/The-transmission-of-Greek-and-Arabic-learning#ref65999">The Renaissance</a></li></ul></div></li><li data-target="#ref66000"><div class="d-flex align-items-center"><button class="h1-link-drawer-button btn btn-xs btn-circle d-flex rounded" type="button" aria-label="Toggle Heading"><em class="material-icons font-18" data-icon="keyboard_arrow_right"></em></button><a class="w-100 link-gray-900" href="/science/mathematics/Mathematics-in-the-17th-and-18th-centuries">Mathematics in the 17th and 18th centuries</a></div><div class="ml-40 toc-drawer sub-toc-drawer"><ul class="list-unstyled" data-level="h2"><li data-target="#ref66001"><a class="w-100 link-gray-900" href="/science/mathematics/Mathematics-in-the-17th-and-18th-centuries#ref66001">The 17th century</a><ul class="list-unstyled" data-level="h3"><li data-target="#ref66002"><a class="w-100 link-gray-900" href="/science/mathematics/Mathematics-in-the-17th-and-18th-centuries#ref66002">Institutional background</a></li></ul><ul class="list-unstyled" data-level="h3"><li data-target="#ref66003"><a class="w-100 link-gray-900" href="/science/mathematics/Mathematics-in-the-17th-and-18th-centuries#ref66003">Numerical calculation</a></li></ul><ul class="list-unstyled" data-level="h3"><li data-target="#ref66004"><a class="w-100 link-gray-900" href="/science/mathematics/Analytic-geometry">Analytic geometry</a></li></ul><ul class="list-unstyled" data-level="h3"><li data-target="#ref66005" class="has-children"><a class="w-100 link-gray-900" href="/science/mathematics/The-calculus">The calculus</a><ul class="list-unstyled" data-level="h4"><li data-target="#ref66006"><a class="w-100 link-gray-900" href="/science/mathematics/The-calculus#ref66006">The precalculus period</a></li></ul><ul class="list-unstyled" data-level="h4"><li data-target="#ref66007"><a class="w-100 link-gray-900" href="/science/mathematics/Newton-and-Leibniz">Newton and Leibniz</a></li></ul></li></ul></li></ul><ul class="list-unstyled" data-level="h2"><li data-target="#ref66008"><a class="w-100 link-gray-900" href="/science/mathematics/Newton-and-Leibniz#ref66008">The 18th century</a><ul class="list-unstyled" data-level="h3"><li data-target="#ref66009"><a class="w-100 link-gray-900" href="/science/mathematics/Newton-and-Leibniz#ref66009">Institutional background</a></li></ul><ul class="list-unstyled" data-level="h3"><li data-target="#ref66010"><a class="w-100 link-gray-900" href="/science/mathematics/Analysis-and-mechanics">Analysis and mechanics</a></li></ul><ul class="list-unstyled" data-level="h3"><li data-target="#ref66011"><a class="w-100 link-gray-900" href="/science/mathematics/Analysis-and-mechanics#ref66011">History of analysis</a></li></ul></li></ul><ul class="list-unstyled" data-level="h2"><li data-target="#ref66012"><a class="w-100 link-gray-900" href="/science/mathematics/Analysis-and-mechanics#ref66012">Other developments</a><ul class="list-unstyled" data-level="h3"><li data-target="#ref66013"><a class="w-100 link-gray-900" href="/science/mathematics/Theory-of-equations">Theory of equations</a></li></ul><ul class="list-unstyled" data-level="h3"><li data-target="#ref66014"><a class="w-100 link-gray-900" href="/science/mathematics/Theory-of-equations#ref66014">Foundations of geometry</a></li></ul></li></ul></div></li><li data-target="#ref335980"><div class="d-flex align-items-center"><button class="h1-link-drawer-button btn btn-xs btn-circle d-flex rounded" type="button" aria-label="Toggle Heading"><em class="material-icons font-18" data-icon="keyboard_arrow_right"></em></button><a class="w-100 link-gray-900" href="/science/mathematics/Mathematics-in-the-19th-century">Mathematics in the 19th century</a></div><div class="ml-40 toc-drawer sub-toc-drawer"><ul class="list-unstyled" data-level="h2"><li data-target="#ref337058"><a class="w-100 link-gray-900" href="/science/mathematics/Mathematics-in-the-19th-century#ref337058">Projective geometry</a></li></ul><ul class="list-unstyled" data-level="h2"><li data-target="#ref337059"><a class="w-100 link-gray-900" href="/science/mathematics/Mathematics-in-the-19th-century#ref337059">Making the calculus rigorous</a></li></ul><ul class="list-unstyled" data-level="h2"><li data-target="#ref337060"><a class="w-100 link-gray-900" href="/science/mathematics/Fourier-series">Fourier series</a></li></ul><ul class="list-unstyled" data-level="h2"><li data-target="#ref337061"><a class="w-100 link-gray-900" href="/science/mathematics/Fourier-series#ref337061">Elliptic functions</a></li></ul><ul class="list-unstyled" data-level="h2"><li data-target="#ref337062"><a class="w-100 link-gray-900" href="/science/mathematics/Fourier-series#ref337062">The theory of numbers</a></li></ul><ul class="list-unstyled" data-level="h2"><li data-target="#ref337063"><a class="w-100 link-gray-900" href="/science/mathematics/The-theory-of-equations">The theory of equations</a></li></ul><ul class="list-unstyled" data-level="h2"><li data-target="#ref337064"><a class="w-100 link-gray-900" href="/science/mathematics/The-theory-of-equations#ref337064">Gauss</a></li></ul><ul class="list-unstyled" data-level="h2"><li data-target="#ref337065"><a class="w-100 link-gray-900" href="/science/mathematics/The-theory-of-equations#ref337065">Non-Euclidean geometry</a></li></ul><ul class="list-unstyled" data-level="h2"><li data-target="#ref337066"><a class="w-100 link-gray-900" href="/science/mathematics/Riemann">Riemann</a></li></ul><ul class="list-unstyled" data-level="h2"><li data-target="#ref337067"><a class="w-100 link-gray-900" href="/science/mathematics/Riemann#ref337067">Riemann’s influence</a></li></ul><ul class="list-unstyled" data-level="h2"><li data-target="#ref337068"><a class="w-100 link-gray-900" href="/science/mathematics/Differential-equations">Differential equations</a></li></ul><ul class="list-unstyled" data-level="h2"><li data-target="#ref337069"><a class="w-100 link-gray-900" href="/science/mathematics/Differential-equations#ref337069">Linear algebra</a></li></ul><ul class="list-unstyled" data-level="h2"><li data-target="#ref337070"><a class="w-100 link-gray-900" href="/science/mathematics/Differential-equations#ref337070">The foundations of geometry</a></li></ul><ul class="list-unstyled" data-level="h2"><li data-target="#ref337071"><a class="w-100 link-gray-900" href="/science/mathematics/Differential-equations#ref337071">The foundations of mathematics</a></li></ul></div></li><li data-target="#ref337072"><div class="d-flex align-items-center"><button class="h1-link-drawer-button btn btn-xs btn-circle d-flex rounded" type="button" aria-label="Toggle Heading"><em class="material-icons font-18" data-icon="keyboard_arrow_right"></em></button><a class="w-100 link-gray-900" href="/science/mathematics/Mathematics-in-the-20th-and-21st-centuries">Mathematics in the 20th and 21st centuries</a></div><div class="ml-40 toc-drawer sub-toc-drawer"><ul class="list-unstyled" data-level="h2"><li data-target="#ref66030"><a class="w-100 link-gray-900" href="/science/mathematics/Mathematics-in-the-20th-and-21st-centuries#ref66030">Cantor</a></li></ul><ul class="list-unstyled" data-level="h2"><li data-target="#ref66031"><a class="w-100 link-gray-900" href="/science/mathematics/Mathematics-in-the-20th-and-21st-centuries#ref66031">Mathematical physics</a></li></ul><ul class="list-unstyled" data-level="h2"><li data-target="#ref66032"><a class="w-100 link-gray-900" href="/science/mathematics/Algebraic-topology">Algebraic topology</a></li></ul><ul class="list-unstyled" data-level="h2"><li data-target="#ref66033"><a class="w-100 link-gray-900" href="/science/mathematics/Algebraic-topology#ref66033">Developments in pure mathematics</a></li></ul><ul class="list-unstyled" data-level="h2"><li data-target="#ref66034"><a class="w-100 link-gray-900" href="/science/mathematics/Mathematical-physics-and-the-theory-of-groups">Mathematical physics and the theory of groups</a></li></ul><ul class="list-unstyled" data-level="h2"><li data-target="#ref335981"><a class="w-100 link-gray-900" href="/science/mathematics/Probabilistic-mathematics">Probabilistic mathematics</a></li></ul></div></li></ul> <a class="toc-extra-link link-gray-900" href="https://www.britannica.com/science/mathematics/additional-info">References & Edit History</a> <a class="toc-extra-link link-gray-900" href="/facts/mathematics">Quick Facts & Related Topics</a> </div> <div class="tlr-media-slider pb-10 mb-30"> <a class="section-header link-gray-900 font-serif font-14 font-weight-bold mb-10 mx-10" href="https://www.britannica.com/science/mathematics/images-videos">Images & Videos</a> <div class="slider js-slider position-relative d-inline-flex align-items-center mw-100 "> <div class="slider-container js-slider-container overflow-hidden d-flex overflow-hidden text-nowrap ml-15"> <a href="https://cdn.britannica.com/54/2354-050-B47B3735/Babylonian-tablet.jpg" data-href="/media/1/369194/2213" class="media-overlay-link d-inline-block mr-5"> <img loading="lazy" src="https://cdn.britannica.com/54/2354-004-969C8844/Babylonian-tablet.jpg" alt="Babylonian mathematical tablet" height="50" /> </a> <a href="https://cdn.britannica.com/92/74892-050-8F76B0AA/Egyptians-right-left-system-power-symbols.jpg" data-href="/media/1/369194/68823" class="media-overlay-link d-inline-block mr-5"> <img loading="lazy" src="https://cdn.britannica.com/92/74892-004-4EFCA3ED/Egyptians-right-left-system-power-symbols.jpg" alt="Ancient Egyptian numerals" height="50" /> </a> <a href="https://cdn.britannica.com/93/74893-050-AF385FBC/Egyptian-numerals.jpg" data-href="/media/1/369194/68835" class="media-overlay-link d-inline-block mr-5"> <img loading="lazy" src="https://cdn.britannica.com/93/74893-004-93F0C348/Egyptian-numerals.jpg" alt="Egyptian hieratic numerals" height="50" /> </a> <a href="https://cdn.britannica.com/02/69702-050-1C1B19A6/reciprocal-seked-Egyptian-run-ratio-Egyptians-definition.jpg" data-href="/media/1/369194/59963" class="media-overlay-link d-inline-block mr-5"> <img loading="lazy" src="https://cdn.britannica.com/02/69702-004-599357AC/reciprocal-seked-Egyptian-run-ratio-Egyptians-definition.jpg" alt="Egyptian seked" height="50" /> </a> <a href="https://cdn.britannica.com/61/67361-050-F48C1CC9/map-millennium-mathematicians-Greco-Roman-Thales-Hypatia-Alexandria.jpg" data-href="/media/1/369194/57444" class="media-overlay-link d-inline-block mr-5"> <img loading="lazy" src="https://cdn.britannica.com/61/67361-004-46CBD13A/map-millennium-mathematicians-Greco-Roman-Thales-Hypatia-Alexandria.jpg" alt="mathematicians of the Greco-Roman world" height="50" /> </a> <a href="https://cdn.britannica.com/06/80206-004-F6463F0F/cube-Menaechmus-volume-solution-problem-two-curves.jpg" data-href="/media/1/369194/90066" class="media-overlay-link d-inline-block mr-5"> <img loading="lazy" src="https://cdn.britannica.com/06/80206-004-F6463F0F/cube-Menaechmus-volume-solution-problem-two-curves.jpg" alt="doubling the volume of a cube" height="50" /> </a> <a href="https://cdn.britannica.com/11/66811-050-C6924436/circumscribing-cylinder-Sphere-volume-sphere-surface-area.jpg" data-href="/media/1/369194/57320" class="media-overlay-link d-inline-block mr-5"> <img loading="lazy" src="https://cdn.britannica.com/11/66811-004-96E09D0B/circumscribing-cylinder-Sphere-volume-sphere-surface-area.jpg" alt="sphere with circumscribing cylinder" height="50" /> </a> <a href="https://cdn.britannica.com/40/840-004-543CF3D4/sections-plane-figure-cone-families-ellipse-parabola.jpg" data-href="/media/1/369194/1743" class="media-overlay-link d-inline-block mr-5"> <img loading="lazy" src="https://cdn.britannica.com/40/840-004-543CF3D4/sections-plane-figure-cone-families-ellipse-parabola.jpg" alt="conic sections" height="50" /> </a> <a href="https://cdn.britannica.com/08/80208-004-0CEE43E7/curve-Conchoid-lines-P-distance-line-points.jpg" data-href="/media/1/369194/74591" class="media-overlay-link d-inline-block mr-5"> <img loading="lazy" src="https://cdn.britannica.com/08/80208-004-0CEE43E7/curve-Conchoid-lines-P-distance-line-points.jpg" alt="conchoid curve" height="50" /> </a> <a href="https://cdn.britannica.com/09/80209-004-F51212B2/Nicomedes-angle-conchoid-Angle-trisection-curve-side.jpg" data-href="/media/1/369194/90801" class="media-overlay-link d-inline-block mr-5"> <img loading="lazy" src="https://cdn.britannica.com/09/80209-004-F51212B2/Nicomedes-angle-conchoid-Angle-trisection-curve-side.jpg" alt="angle trisection using a conchoid" height="50" /> </a> </div> <button disabled class="prev-button js-prev-button position-absolute btn btn-circle shadow btn-blue " aria-label="Previous"> <span class="material-icons md-24" data-icon="keyboard_arrow_left"></span> </button> <button disabled class="next-button js-next-button position-absolute btn btn-circle shadow btn-blue " aria-label="Next"> <span class="material-icons md-24" data-icon="keyboard_arrow_right"></span> </button> </div> </div> <div class="mb-30 tlr-student-links"> <div class="text-gray-900 p-5 font-serif font-14 font-weight-bold mx-10 mb-10"> For Students </div> <div class="imagelink-with-image-on-the-side card card-horizontal tlr-img-with-side-link ml-15 link-gray-900 mb-10" > <div class="position-relative card-media" style="flex: 0;"> <a class="ilf-image position-relative" href="/summary/mathematics"> <img loading="lazy" src="https://cdn.britannica.com/mendel-resources/3-133/images/shared/default3.png?v=3.133.9" class="default " height="200" width="200"/> </a> </div> <div class="card-body ilf-content"> <a class="font-weight-semi-bold d-block mb-5 font-16 ilf-title" href="/summary/mathematics" >mathematics summary</a> </div> </div> </div> <div class="mb-30 tlr-related-quizzes"> <div class="text-gray-900 p-5 font-serif font-14 font-weight-bold mx-10 mb-10"> Quizzes </div> <div class="imagelink-with-image-on-the-side card card-horizontal tlr-img-with-side-link ml-15 link-gray-900 mb-10" > <div class="position-relative card-media" style="flex: 0;"> <a class="ilf-image position-relative" href="/quiz/Numbers-and-mathematics"> <img loading="lazy" src="https://cdn.britannica.com/86/94086-131-0BAE374D/Equations-blackboard.jpg?w=200&h=200&c=crop" alt="Equations written on blackboard" width="200" height="200" /> </a> </div> <div class="card-body ilf-content"> <a class="font-weight-semi-bold d-block mb-5 font-16 ilf-title" href="/quiz/Numbers-and-mathematics" >Numbers and Mathematics</a> </div> </div> <div class="imagelink-with-image-on-the-side card card-horizontal tlr-img-with-side-link ml-15 link-gray-900 mb-10" > <div class="position-relative card-media" style="flex: 0;"> <a class="ilf-image position-relative" href="/quiz/fun-facts-of-measurement-math"> <img loading="lazy" src="https://cdn.britannica.com/68/166068-131-03976F7B/barometer-Barometer-Technology-measurement-readout-pressure-mathematics.jpg?w=200&h=200&c=crop" alt="barometer. Antique Barometer with readout. Technology measurement, mathematics, measure atmospheric pressure" width="200" height="200" /> </a> </div> <div class="card-body ilf-content"> <a class="font-weight-semi-bold d-block mb-5 font-16 ilf-title" href="/quiz/fun-facts-of-measurement-math" >Fun Facts of Measurement & Math</a> </div> </div> <div class="imagelink-with-image-on-the-side card card-horizontal tlr-img-with-side-link ml-15 link-gray-900 mb-10" > <div class="position-relative card-media" style="flex: 0;"> <a class="ilf-image position-relative" href="/quiz/define-it-math-terms"> <img loading="lazy" src="https://cdn.britannica.com/01/115001-131-7278E518/Enrico-Fermi-Italian-problem-physics-1950.jpg?w=200&h=200&c=crop" alt="Italian-born physicist Dr. Enrico Fermi draws a diagram at a blackboard with mathematical equations. circa 1950." width="200" height="200" /> </a> </div> <div class="card-body ilf-content"> <a class="font-weight-semi-bold d-block mb-5 font-16 ilf-title" href="/quiz/define-it-math-terms" >Define It: Math Terms</a> </div> </div> </div> <div class="mb-30 tlr-read-next"> <div class="text-gray-900 p-5 font-serif font-14 font-weight-bold mx-10 mb-10"> Read Next </div> <div class="imagelink-with-image-on-the-side card card-horizontal tlr-img-with-side-link ml-15 link-gray-900 mb-10" > <div class="position-relative card-media" style="flex: 0;"> <a class="ilf-image position-relative" href="/story/unusual-counting-systems"> <img loading="lazy" src="https://cdn.britannica.com/41/191041-131-7ECE668A/bunch-numbers.jpg?w=200&h=200&c=crop" alt="bunch of numbers" width="200" height="200" /> </a> </div> <div class="card-body ilf-content"> <a class="font-weight-semi-bold d-block mb-5 font-16 ilf-title" href="/story/unusual-counting-systems" >Unusual Counting Systems</a> </div> </div> <div class="imagelink-with-image-on-the-side card card-horizontal tlr-img-with-side-link ml-15 link-gray-900 mb-10" > <div class="position-relative card-media" style="flex: 0;"> <a class="ilf-image position-relative" href="/story/is-zero-an-even-or-an-odd-number"> <img loading="lazy" src="https://cdn.britannica.com/49/191949-131-3E2AC277/balloon.jpg?w=200&h=200&c=crop" alt="number zero, 0 balloon" width="200" height="200" /> </a> </div> <div class="card-body ilf-content"> <a class="font-weight-semi-bold d-block mb-5 font-16 ilf-title" href="/story/is-zero-an-even-or-an-odd-number" >Is Zero an Even or an Odd Number?</a> </div> </div> <div class="imagelink-with-image-on-the-side card card-horizontal tlr-img-with-side-link ml-15 link-gray-900 mb-10" > <div class="position-relative card-media" style="flex: 0;"> <a class="ilf-image position-relative" href="/story/how-fast-is-the-universe-expanding"> <img loading="lazy" src="https://cdn.britannica.com/27/238127-131-9A610712/galaxy-cluster-Abell-2744.jpg?w=200&h=200&c=crop" alt="Galaxy clusters like Abell 2744 can act as a natural cosmic lens, magnifying light from more distant, background objects through gravity. NASA's James Webb Space Telescope may be able to detect light from the first stars in the universe if they are gravitationally lensed by such clusters. (astronomy, space exploration, galaxies)" width="200" height="200" /> </a> </div> <div class="card-body ilf-content"> <a class="font-weight-semi-bold d-block mb-5 font-16 ilf-title" href="/story/how-fast-is-the-universe-expanding" >How Fast Is the Universe Expanding?</a> </div> </div> </div> <div class="mb-30 tlr-discover"> <div class="text-gray-900 p-5 font-serif font-14 font-weight-bold mx-10 mb-10"> Discover </div> <div class="imagelink-with-image-on-the-side card card-horizontal tlr-img-with-side-link ml-15 link-gray-900 mb-10" > <div class="position-relative card-media" style="flex: 0;"> <a class="ilf-image position-relative" href="/list/9-of-the-worlds-deadliest-snakes"> <img loading="lazy" src="https://cdn.britannica.com/68/152568-131-E6B869A4/King-cobra-world-snake.jpg?w=200&h=200&c=crop" alt="King Cobra snake in Malaysia. (reptile)" width="200" height="200" /> </a> </div> <div class="card-body ilf-content"> <a class="font-weight-semi-bold d-block mb-5 font-16 ilf-title" href="/list/9-of-the-worlds-deadliest-snakes" >9 of the World’s Deadliest Snakes</a> </div> </div> <div class="imagelink-with-image-on-the-side card card-horizontal tlr-img-with-side-link ml-15 link-gray-900 mb-10" > <div class="position-relative card-media" style="flex: 0;"> <a class="ilf-image position-relative" href="/list/secret-service-code-names-of-11-us-presidents"> <img loading="lazy" src="https://cdn.britannica.com/45/189145-131-45FF672E/Secret-Service-Agent-Earpiece.jpg?w=200&h=200&c=crop" alt="Secret Service Agent Listens To Earpiece" width="200" height="200" /> </a> </div> <div class="card-body ilf-content"> <a class="font-weight-semi-bold d-block mb-5 font-16 ilf-title" href="/list/secret-service-code-names-of-11-us-presidents" >Secret Service Code Names of 11 U.S. Presidents</a> </div> </div> <div class="imagelink-with-image-on-the-side card card-horizontal tlr-img-with-side-link ml-15 link-gray-900 mb-10" > <div class="position-relative card-media" style="flex: 0;"> <a class="ilf-image position-relative" href="/list/new-seven-wonders-of-the-world"> <img loading="lazy" src="https://cdn.britannica.com/36/162636-131-E4AA93A0/Colosseum-Rome-Italy.jpg?w=200&h=200&c=crop" alt="The Colosseum, Rome, Italy. Giant amphitheatre built in Rome under the Flavian emperors. (ancient architecture; architectural ruins)" width="200" height="200" /> </a> </div> <div class="card-body ilf-content"> <a class="font-weight-semi-bold d-block mb-5 font-16 ilf-title" href="/list/new-seven-wonders-of-the-world" >New Seven Wonders of the World</a> </div> </div> <div class="imagelink-with-image-on-the-side card card-horizontal tlr-img-with-side-link ml-15 link-gray-900 mb-10" > <div class="position-relative card-media" style="flex: 0;"> <a class="ilf-image position-relative" href="/story/why-do-some-people-call-football-soccer"> <img loading="lazy" src="https://cdn.britannica.com/51/190751-131-B431C216/soccer-ball-goal.jpg?w=200&h=200&c=crop" alt="soccer ball in the goal" width="200" height="200" /> </a> </div> <div class="card-body ilf-content"> <a class="font-weight-semi-bold d-block mb-5 font-16 ilf-title" href="/story/why-do-some-people-call-football-soccer" >Why Do Some People Call Football “Soccer”?</a> </div> </div> <div class="imagelink-with-image-on-the-side card card-horizontal tlr-img-with-side-link ml-15 link-gray-900 mb-10" > <div class="position-relative card-media" style="flex: 0;"> <a class="ilf-image position-relative" href="/story/was-jesse-owens-snubbed-by-adolf-hitler-at-the-berlin-olympics"> <img loading="lazy" src="https://cdn.britannica.com/51/189351-131-104BA669/Jesse-Owens-Olympic-Games-1936.jpg?w=200&h=200&c=crop" alt="Berlin, 1936 - Jesse Owens of the USA in action in the mens 200m at the Summer Olympic Games. Owens won a total of four gold medals." width="200" height="200" /> </a> </div> <div class="card-body ilf-content"> <a class="font-weight-semi-bold d-block mb-5 font-16 ilf-title" href="/story/was-jesse-owens-snubbed-by-adolf-hitler-at-the-berlin-olympics" >Was Jesse Owens Snubbed by Adolf Hitler at the Berlin Olympics?</a> </div> </div> <div class="imagelink-with-image-on-the-side card card-horizontal tlr-img-with-side-link ml-15 link-gray-900 mb-10" > <div class="position-relative card-media" style="flex: 0;"> <a class="ilf-image position-relative" href="/list/12-greek-gods-and-goddesses"> <img loading="lazy" src="https://cdn.britannica.com/23/176123-131-94DCF6F8/Aphrodite.jpg?w=200&h=200&c=crop" alt="Aphrodite. Greek mythology. Sculpture. Aphrodite is the Greek goddess of love and beauty." width="200" height="200" /> </a> </div> <div class="card-body ilf-content"> <a class="font-weight-semi-bold d-block mb-5 font-16 ilf-title" href="/list/12-greek-gods-and-goddesses" >12 Greek Gods and Goddesses</a> </div> </div> <div class="imagelink-with-image-on-the-side card card-horizontal tlr-img-with-side-link ml-15 link-gray-900 mb-10" > <div class="position-relative card-media" style="flex: 0;"> <a class="ilf-image position-relative" href="/story/from-sport-to-spectacle-the-history-of-the-super-bowl"> <img loading="lazy" src="https://cdn.britannica.com/94/188294-050-BE037BD2/analysis-game-results-Super-Bowls-teams-wins-MVP-NFL-football-infographic.jpg?w=200&h=200&c=crop" alt="Data analysis of the Super Bowl's "winningest" teams, game locations, MVP winners by position, teams with the most game losses. football, sports, infographic" width="200" height="200" /> </a> </div> <div class="card-body ilf-content"> <a class="font-weight-semi-bold d-block mb-5 font-16 ilf-title" href="/story/from-sport-to-spectacle-the-history-of-the-super-bowl" >From Sport to Spectacle: The History of the Super Bowl</a> </div> </div> </div> </div> </div> </div> <button class="drawerToggle btn position-sticky border btn-xs btn-white btn-circle rounded-sm d-none d-lg-flex " type="button" aria-label="Toggle Drawer"> <em class="material-icons font-18 text-blue" data-icon="keyboard_arrow_left"></em> </button> </div> </div> <div class="col"> <div class="h-100 ml-0 pr-lg-0 "> <div class="h-100 grid gx-0 gx-lg-20"> <div class="h-100 col-sm"> <div class="h-100 infinite-pagination-container d-flex flex-column position-relative"> <div class="position-absolute top-0 h-100 w-100"> <div class="toc-sticky-header d-none d-lg-none bg-gray-50 px-10 px-sm-30 position-sticky w-100 "> <div class="toc-sticky-header-inner-container align-items-center d-flex mx-auto h-100 w-100"> <button class="d-flex d-lg-none btn btn-sm btn-white text-blue border-2 border-gray-100 gtm-mobile-toc-header-button js-sections-button d-lg-none p-10"> <em class="material-icons my-n5 md-icon" data-icon="toc"></em> Contents </button> <div class="header-ai-ask-button-placeholder"></div> <div class="header-ai-summarize-button-placeholder"></div> </div> </div> </div> <div class="grey-box w-100 grey-box-top"> <div class="grey-box-content mx-auto w-100"> <script type="application/ld+json"> { "@context" : "https://schema.org", "@type" : "BreadcrumbList", "itemListElement" : [ { "@type" : "ListItem", "position" : 1, "item" : { "@id" : "https://www.britannica.com/browse/Science", "name": "Science" } } , { "@type" : "ListItem", "position" : 2, "item" : { "@id" : "https://www.britannica.com/browse/Mathematics", "name": "Mathematics" } } ] } </script> <nav class="breadcrumb mt-20"> <span class="breadcrumb-item "> <a class="link-gray-600" href="/browse/Science">Science</a> </span> <span class="breadcrumb-item "> <a class="link-gray-600" href="/browse/Mathematics">Mathematics</a> </span> </nav> <div class="page2ref-true topic-content topic-type-REGULAR" data-student-article="true"> <script class="page-description-json" type="application/json"> { "url": "/science/mathematics/Assessment-of-Egyptian-mathematics", "shareUrl": "https://www.britannica.com/science/mathematics", "browserTitle": "Mathematics - Egyptian, Assessment, History", "firstTopicPage": false, "topicId":369194 } </script> <div class="reading-channel"> <div class="topic-header"> <div class="d-flex align-items-top justify-content-between"> <div class="d-flex flex-column"> <div> <div> <h2 class="h2">Assessment of Egyptian mathematics</h2></div> <div class="in-container"> <em class="material-icons in-arrow" data-icon="subdirectory_arrow_right"></em> <span class="in-bc">in</span><a class="in-link" href="https://www.britannica.com/science/mathematics">mathematics</a> <span class="in-bc">in</span><a class="in-link" href="/science/mathematics/Mathematics-in-ancient-Egypt">Mathematics in ancient Egypt</a> </div> </div> </div> </div> <div class="d-none d-sm-flex flex-row"> <div class="mr-10 mb-15"> <button class="ai-ask-button btn border-2 btn-sm js-inline-ai-ask-button btn-outline-red-400 border-red-400"> Ask the Chatbot a Question </button> </div> <div class="d-block md-topic-tools qa-action-buttons mb-15" data-topic-id="369194"> <button class="js-tooltip btn btn-sm btn-outline-blue border pr-10 border-2 text-nowrap" > <em class="material-icons md-icon ml-n10 my-n5 mr-5" data-icon="more_vert"></em> More Actions </button> <div class="md-more-popover popover popover-sm p-0 font-14 z-1"> <div> <button class="js-print-modal-button js-modal gtm-topic-tool btn btn-sm btn-link gtm-topic-tool font-weight-bold btn-link" data-modal="[data-topic-id=369194] .md-print-modal" > <em class="material-icons mr-5 ml-n10 my-n5 md-icon" data-icon="print"></em> Print </button> <div class="md-print-modal size-lg d-none"> <div class="md-modal-body"> <div class="h2 font-serif d-flex align-items-center pb-15 border-bottom"> <em class="material-icons text-blue mr-10">print</em> Print </div> <div class="mt-20 mb-10"> Please select which sections you would like to print: </div> <form action="/print/article/369194" method="post" target="_blank" rel="noopener"> <div class="print-box-items"> <ul class="list-unstyled"> <li><label><input class="mr-10" type="checkbox" name="sequence[]" value="0">Table Of Contents</label></li> </ul> </div> <input type="submit" class="btn btn-blue md-disabled" value="Print" /> </form> </div> </div> </div> <div> <button class="js-modal qa-cite-modal-button btn btn-sm btn-link gtm-topic-tool font-weight-bold btn-link" data-modal="[data-topic-id=369194] .md-cite-modal"> <em class="material-icons mr-5 ml-n10 my-n5 md-icon" data-icon="verified"></em> Cite </button> <div class="md-cite-modal size-lg d-none"> <div class="md-modal-body"> <div class="h2 font-serif d-flex align-items-center pb-15 border-bottom mb-15"> <em class="material-icons text-blue mr-10">verified</em>Cite </div> <div class="font-serif"> While every effort has been made to follow citation style rules, there may be some discrepancies. Please refer to the appropriate style manual or other sources if you have any questions. </div> <div class="label mt-20 mb-10">Select Citation Style</div> <select class="js-citation-format-select form-select"> <option selected value="mla">MLA</option> <option value="apa">APA</option> <option value="chicago">Chicago Manual of Style</option> </select> <div class="citation font-serif border rounded p-15 mt-20" data-authors="Gray, Jeremy John, Berggren, John L., Folkerts, Menso, Fraser, Craig G., Knorr, Wilbur R." data-authors-initial="Gray, J.J., Berggren, J.L., Folkerts, M., Fraser, C.G., Knorr, W.R." data-title="mathematics" data-published-date="14 Feb. 2025" data-url="https://www.britannica.com/science/mathematics" > <div class="citation-text"></div> </div> <button class="js-copy-citation-button mt-20 btn btn-xs btn-outline-blue border shadow-sm pr-10" > <em class="material-icons md-icon ml-n10 my-n5 mr-5" data-icon="file_copy"></em> <span class="js-citation-status-text">Copy Citation</span> </button> </div> </div> </div> <div> <button class="js-share-modal-button js-modal btn btn-sm btn-link gtm-topic-tool font-weight-bold btn-link" data-modal="[data-topic-id=369194] .md-share-modal"> <em class="material-icons mr-5 ml-n10 my-n5 md-icon" data-icon="share"></em> Share </button> <div class="md-share-modal size-lg d-none qa-share-modal"> <div class="md-modal-body"> <div class="h2 font-serif d-flex align-items-center pb-15 border-bottom"> <em class="material-icons text-blue mr-10" data-icon="share"></em> Share </div> <div class="label my-20">Share to social media</div> <div class="md-social-toolbar-circle d-flex align-items-start inverted" data-value="share" title="mathematics" data-url="https://www.britannica.com/science/mathematics" > <a class="social-icon facebook justify-content-center d-flex align-items-center align-self-center" data-provider="facebook" href="https://www.facebook.com/BRITANNICA/" target="_blank" rel="noopener"><span>Facebook</span></a> <a class="social-icon x justify-content-center d-flex align-items-center align-self-center" data-provider="x" href="https://x.com/britannica" target="_blank" rel="noopener"><span>X</span></a> </div> <div class="label pt-20 mt-20 mb-5 border-top">URL</div> <a class="font-serif text-truncate d-inline-block" href="https://www.britannica.com/science/mathematics">https://www.britannica.com/science/mathematics</a> </div> </div> </div> <div> <button class="js-feedback-modal-button js-modal btn btn-sm btn-link gtm-topic-tool font-weight-bold btn-link" data-modal=".md-feedback-modal"> <em class="material-icons mr-5 ml-n10 my-n5 md-icon" data-icon="message"></em> Feedback </button> </div> <div> <button class="qa-external-website-modal-button js-modal btn btn-sm btn-link gtm-topic-tool font-weight-bold btn-link" data-modal="[data-topic-id=369194] .md-websites-modal"> <em class="material-icons md-icon ml-n10 mr-5" data-icon="link"></em> External Websites </button> </div> </div> <div class="md-feedback-modal size-lg d-none"> <div class="md-modal-body"> <div class="h2 font-serif pb-15 border-bottom"> Feedback </div> <form method="post" action="/submission/feedback/369194"> <div class="my-20"> Corrections? Updates? Omissions? Let us know if you have suggestions to improve this article (requires login). </div> <div class="type-menu"> <label for="feedback-type" class="label mb-10">Feedback Type</label> <select id="feedback-type" class="form-select mb-30" name="feedbackTypeId" required> <option value="" selected="selected">Select a type (Required)</option> <option value="1">Factual Correction</option> <option value="2">Spelling/Grammar Correction</option> <option value="3">Link Correction</option> <option value="4">Additional Information</option> <option value="5">Other</option> </select> </div> <label for="feedback" class="label mb-10">Your Feedback</label> <textarea id="feedback" class="form-control mb-30" name="feedback" maxlength="3000" rows="7" required></textarea> <button class="btn btn-blue" type="submit">Submit Feedback</button> </form> <div class="success-messaging d-none mt-30"> <div class="title">Thank you for your feedback</div> <p>Our editors will review what you’ve submitted and determine whether to revise the article.</p> </div> </div> </div> <div class="md-websites-modal size-lg d-none"> <div class="md-modal-body"> <div class="h2 font-serif pb-15 border-bottom font-weight-bold"> External Websites </div> <div class="pb-20"> <ul class="list-unstyled mt-20 lh-lg"> <li><a class="external" href="https://www.livescience.com/38936-mathematics.html" target="_blank" rel="noopener ">LiveScience - What is Mathematics?</a></li> <li><a class="external" href="https://socialsci.libretexts.org/Courses/Arapahoe_Community_College/Introduction_to_Curriculum_for_Early_Childhood_Education/07%3A_Mathematics" target="_blank" rel="noopener ">Social Sci LibreTexts - Mathematics</a></li> <li><a class="external" href="https://abelprize.no/" target="_blank" rel="noopener nofollow">Official Site of The Abel Prize</a></li> <li><a class="external" href="https://illinois.pbslearningmedia.org/resource/nvmm-math-mathmusic/ancient-math-music/" target="_blank" rel="noopener ">PBS LearningMedia - Ancient Math & Music</a></li> <li><a class="external" href="https://mathweb.ucsd.edu/~harel/What%20Is%20Mathematics.pdf" target="_blank" rel="noopener ">University of California, San Diego - What is Mathematics?</a></li> </ul> </div> <div class="md-websites-ebk-title">Britannica Websites</div> <div class="md-websites-ebk-subtitle">Articles from Britannica Encyclopedias for elementary and high school students.</div> <ul class="list-unstyled bps-topic-web-sites lh-lg"> <li><a class="external" href="https://kids.britannica.com/kids/article/mathematics/353443" target="_blank" rel="noopener">mathematics - Children's Encyclopedia (Ages 8-11)</a></li> <li><a class="external" href="https://kids.britannica.com/students/article/mathematics/275734" target="_blank" rel="noopener">mathematics - Student Encyclopedia (Ages 11 and up)</a></li> </ul> </div> </div> </div> </div> <div class="toc-header-marker"></div> <button class="ai-ask-button btn border-2 js-header-ai-ask-button d-none btn-sm btn-outline-red-400 border-red-400 mr-0 mr-lg-10 ml-5 ml-sm-10 ml-lg-0 p-10"> Ask the Chatbot a Question </button> <div class="caption alternate-titles">Also known as: math</div> <div class="md-byline module-spacing "> <div class="font-serif font-12"> <span class="written-by text-gray-700"> Written by </span> <div class="editor-popover popover p-0"> <a class="d-block p-20 qa-editor-popup gtm-byline font-12 byline-contributor" href="/contributor/Jeremy-John-Gray/3547" > <div class="editor-title font-16 font-weight-bold">Jeremy John Gray</div> <div class="editor-description font-12 font-serif mt-5 clamp-description text-black">Emeritus Professor, School of Mathematics and Statistics, Open University. Author of <em>Plato's Ghost</em>; <em>Henri Poincaré: A Scientific Biography</em>; <em>Ideas of Space</em>; and others.</div> </a> <div data-popper-arrow></div> </div> <span class="btn btn-link editor-link p-0 qa-byline-link gtm-byline font-12 byline-contributor text-decoration-underline"> Jeremy John Gray</span>, <div class="editor-popover popover p-0"> <a class="d-block p-20 qa-editor-popup gtm-byline font-12 byline-contributor" href="/contributor/John-L-Berggren/3484" > <div class="editor-title font-16 font-weight-bold">John L. Berggren</div> <div class="editor-description font-12 font-serif mt-5 clamp-description text-black">Professor of Mathematics, Simon Fraser University, Burnaby, British Columbia. Author of <i>Episodes in the Mathematics of Medieval Islam.</i></div> </a> <div data-popper-arrow></div> </div> <span class="btn btn-link editor-link p-0 qa-byline-link gtm-byline font-12 byline-contributor text-decoration-underline"> John L. Berggren</span><span class="text-gray-700 mx-5">•</span><a class="see-all border-gray-700 gtm-byline" rel="nofollow" href="https://www.britannica.com/science/mathematics/additional-info#contributors">All</a> </div> <div class="font-serif font-12 text-gray-700"> <span class="qa-fact-checked-by">Fact-checked by</span> <div class="editor-popover popover p-0"> <a class="d-block p-20 qa-editor-popup font-12" href="/editor/The-Editors-of-Encyclopaedia-Britannica/4419" > <div class="editor-title font-16 font-weight-bold">The Editors of Encyclopaedia Britannica</div> <div class="editor-description font-12 font-serif mt-5 text-black">Encyclopaedia Britannica's editors oversee subject areas in which they have extensive knowledge, whether from years of experience gained by working on that content or via study for an advanced degree. They write new content and verify and edit content received from contributors.</div> </a> <div data-popper-arrow></div> </div> <span class="btn btn-link editor-link p-0 qa-byline-link font-12 "> The Editors of Encyclopaedia Britannica</span></div> <div class="last-updated font-12 font-serif"> <span class="text-gray-700"> Last Updated: <time datetime="2025-02-14T00:00:00CST" >Feb 14, 2025</time> •</span> <a class="byline-edit-history" href="https://www.britannica.com/science/mathematics/additional-info#history" rel="nofollow">Article History</a> </div></div> </div> <button class="d-flex d-lg-none btn btn-outline-blue border rounded-sm shadow-sm mobile-toc-button gtm-mobile-toc-inline-button d-none d-sm-block js-sections-inline-button module-spacing btn d-lg-none"> <em class="material-icons mr-5 ml-n10 my-n5 md-icon" data-icon="toc"></em> Table of Contents </button> <div class="d-flex d-sm-none flex-row"> <button class="d-flex d-lg-none btn btn-outline-blue border rounded-sm shadow-sm mobile-toc-button gtm-mobile-toc-inline-button js-sections-inline-button module-spacing"> <em class="material-icons mr-5 ml-n10 my-n5 md-icon" data-icon="toc"></em> Table of Contents </button> <button class="ai-ask-button btn border-2 ai-ask-button btn border-2 module-spacing btn-sm js-inline-ai-ask-button btn-outline-red-400 border-red-400 p-10 ml-5"> Ask the Chatbot a Question </button> </div> <div class="js-qf-module qf-module px-40 px-sm-20 py-15 mx-auto module-spacing font-14 bg-gray-50 rounded"> <div class="facts-list mt-10"> <div class=""> <div class="js-fact mb-10 line-clamp clamp-3"> <dl> <dt>Key People: </dt> <dd><a href="/biography/Mary-Cartwright" topicid="2252584">Mary Cartwright</a></dd> <dd><a href="/biography/Gladys-West" topicid="2188733">Gladys West</a></dd> <dd><a href="/biography/Isaac-Newton" topicid="413189">Isaac Newton</a></dd> <dd><a href="/biography/Galileo-Galilei" topicid="224058">Galileo</a></dd> <dd><a href="/biography/Bertrand-Russell" topicid="513124">Bertrand Russell</a></dd> </dl> <button class="js-more-btn d-none btn btn-unstyled font-12 bg-gray-50" aria-label="Toggle more/less fact data"> <em class="js-content link-blue">(Show more)</em> </button> </div> </div> <div class=""> <div class="js-fact mb-10 line-clamp clamp-3"> <dl> <dt>Related Topics: </dt> <dd><a href="/science/analysis-mathematics" topicid="22486">analysis</a></dd> <dd><a href="/science/probability-theory" topicid="477530">probability theory</a></dd> <dd><a href="/topic/cryptology" topicid="145058">cryptology</a></dd> <dd><a href="/science/computer-science" topicid="130675">computer science</a></dd> <dd><a href="/topic/percentage" topicid="2124609">percentage</a></dd> </dl> <button class="js-more-btn d-none btn btn-unstyled font-12 bg-gray-50" aria-label="Toggle more/less fact data"> <em class="js-content link-blue">(Show more)</em> </button> </div> <div class="text-center"> <a class="btn btn-sm btn-link p-0" href="/facts/mathematics"> See all related content </a> </div> </div> </div> </div><div class="bg-gray-50 p-15 rounded module-spacing recent-news d-flex flex-column float-false"> <div> <h2 class="font-weight-bold font-14 m-0 d-inline"> News <span class="text-gray-600">•</span> </h2> <div class="recent-news-item first-recent-news-item d-inline"> <a class="font-14 gtm-ap-news-link" href="https://www.thehindu.com/news/cities/Coimbatore/math-teacher-arrested-for-sexually-harassing-students/article69213832.ece" rel="nofollow">Math teacher arrested in Dharmapuri for sexually harassing students</a> <span class="font-14 text-gray-600"> <span>•</span> Feb. 13, 2025, 2:39 AM ET (The Hindu) <button class="btn btn-link d-inline p-0 font-12 js-toggle-recent-news"> <span class="text-gray-500">...</span><span>(Show more)</span> </button> </span> </div> </div> <div class="rest-of-recent-news-items"> <div class="recent-news-item mt-5"> <a class="font-14 gtm-ap-news-link" href="https://indianexpress.com/article/education/jee-main-2025-session-1-day-3-btech-question-paper-analysis-students-experts-feedback-topics-asked-9794456/" rel="nofollow">JEE Main 2025 Day 3 Exam Analysis: ‘Math challenging, Physics easiest among all,’ Here’s what experts say</a> <span class="font-14 text-gray-600"> <span>•</span> Jan. 24, 2025, 5:07 PM ET (The Indian Express) </span> </div> <div class="recent-news-item mt-5"> <a class="font-14 gtm-ap-news-link" href="https://www.thehindu.com/news/cities/chennai/sri-ramakrishna-math-wins-spirit-of-mylapore-award-2025/article69131213.ece" rel="nofollow">Sri Ramakrishna Math wins Spirit of Mylapore Award 2025</a> <span class="font-14 text-gray-600"> <span>•</span> Jan. 23, 2025, 4:37 AM ET (The Hindu) </span> </div> <button class="js-toggle-recent-news d-flex btn btn-unstyled font-14 pr-10 rounded-sm mt-10" aria-label="Toggle additional news items"> Show less <em class="material-icons" data-icon="expand_less"></em> </button> </div> </div><span class="marker before-article"></span><section data-level="1"><span class="marker before-article"></span><section data-level="2" id="ref65976">  <span class="marker PREMOD1 mod-inline"></span><p class="topic-paragraph">The papyri thus bear witness to a mathematical tradition closely tied to the practical accounting and surveying activities of the scribes. Occasionally, the scribes loosened up a bit: one problem (Rhind papyrus, problem 79), for example, seeks the total from seven houses, seven cats per house, seven mice per cat, seven ears of wheat per mouse, and seven <em>hekat</em> of grain per ear (result: 19,607). Certainly the scribe’s interest in progressions (for which he appears to have a rule) goes beyond practical <a class="md-dictionary-link md-dictionary-tt-off eb" data-term="considerations" href="https://www.britannica.com/dictionary/considerations" data-type="EB">considerations</a>. Other than this, however, Egyptian mathematics falls firmly within the range of practice.</p><span class="marker MOD1 mod-inline"></span> <span class="marker PREMOD2 mod-inline"></span><p class="topic-paragraph">Even allowing for the scantiness of the documentation that survives, the Egyptian achievement in mathematics must be viewed as modest. Its most striking features are competence and <a href="https://www.britannica.com/science/continuity" class="md-crosslink autoxref " data-show-preview="true">continuity</a>. The scribes managed to work out the basic arithmetic and geometry necessary for their official duties as civil managers, and their methods persisted with little evident change for at least a millennium, perhaps two. Indeed, when Egypt came under Greek domination in the <span id="ref536060"></span><a href="https://www.britannica.com/event/Hellenistic-Age" class="md-crosslink " data-show-preview="true">Hellenistic</a> period (from the 3rd century <span class="text-smallcaps">bce</span> onward), the older school methods continued. Quite remarkably, the older unit-fraction methods are still prominent in Egyptian school papyri written in the <a href="https://www.britannica.com/topic/demotic-script" class="md-crosslink " data-show-preview="true">demotic</a> (Egyptian) and Greek languages as late as the 7th century <span class="text-smallcaps">ce</span>, for example.</p><span class="marker MOD2 mod-inline"></span> <span class="marker PREMOD3 mod-inline"></span><p class="topic-paragraph">To the extent that Egyptian mathematics left a <a class="md-dictionary-link md-dictionary-tt-off mw" data-term="legacy" href="https://www.merriam-webster.com/dictionary/legacy" data-type="MW">legacy</a> at all, it was through its impact on the emerging Greek mathematical tradition between the 6th and 4th centuries <span class="text-smallcaps">bce</span>. Because the documentation from this period is limited, the manner and significance of the influence can only be conjectured. But the report about Thales measuring the height of pyramids is only one of several such accounts of Greek <a class="md-dictionary-link md-dictionary-tt-off mw" data-term="intellectuals" href="https://www.merriam-webster.com/dictionary/intellectuals" data-type="MW">intellectuals</a> learning from Egyptians; <a href="https://www.britannica.com/biography/Herodotus-Greek-historian" class="md-crosslink " data-show-preview="true">Herodotus</a> and <a href="https://www.britannica.com/biography/Plato" class="md-crosslink " data-show-preview="true">Plato</a> describe with approval Egyptian practices in the teaching and application of mathematics. This literary evidence has historical support, since the Greeks maintained continuous trade and military operations in Egypt from the 7th century <span class="text-smallcaps">bce</span> onward. It is thus plausible that basic precedents for the Greeks’ earliest mathematical efforts—how they dealt with fractional parts or measured areas and volumes, or their use of ratios in connection with similar figures—came from the learning of the ancient Egyptian scribes.</p><span class="marker MOD3 mod-inline"></span> </section> </section> <span class="marker h5"></span><section data-level="1" id="ref65977"> <h2 class="h1"><span id="ref536061"></span><a href="https://www.britannica.com/place/ancient-Greece" class="md-crosslink " data-show-preview="true">Greek</a> mathematics</h2> <section data-level="2" id="ref65978"> <h2 class="h2">The development of pure mathematics</h2> <section data-level="3" id="ref65979"> <h2 class="h3">The pre-Euclidean period</h2> <span class="marker PREMOD4 mod-inline"></span><div class="assemblies"><div class="w-100"><figure class="md-assembly m-0 mb-md-0 card card-borderless print-false" data-assembly-id="57444" data-asm-type="image"><div class="md-assembly-wrapper card-media" data-type="image"><a href="https://cdn.britannica.com/61/67361-050-F48C1CC9/map-millennium-mathematicians-Greco-Roman-Thales-Hypatia-Alexandria.jpg" class="gtm-assembly-link position-relative d-flex align-items-center justify-content-center media-overlay-link card-media" data-href="/media/1/369194/57444"><picture><source media="(min-width: 680px)" srcset="https://cdn.britannica.com/61/67361-050-F48C1CC9/map-millennium-mathematicians-Greco-Roman-Thales-Hypatia-Alexandria.jpg"><img src="https://cdn.britannica.com/61/67361-050-F48C1CC9/map-millennium-mathematicians-Greco-Roman-Thales-Hypatia-Alexandria.jpg?w=300" alt="mathematicians of the Greco-Roman world" data-width="1600" data-height="887" loading="eager"></picture><button class="magnifying-glass btn btn-circle position-absolute shadow btn-white top-10 right-10" aria-label="Zoom in"><em class="material-icons link-blue" data-icon="zoom_in"></em></button></a></div><figcaption class="card-body"><div class="md-assembly-caption text-muted font-14 font-serif line-clamp"><span><a class="gtm-assembly-link md-assembly-title font-weight-bold d-inline font-sans-serif mr-5 media-overlay-link" href="https://cdn.britannica.com/61/67361-050-F48C1CC9/map-millennium-mathematicians-Greco-Roman-Thales-Hypatia-Alexandria.jpg" data-href="/media/1/369194/57444">mathematicians of the Greco-Roman world</a><span>This map spans a millennium of prominent Greco-Roman mathematicians, from Thales of Miletus (c. 600 <span class="text-smallcaps">bce</span>) to Hypatia of Alexandria (c. 400 <span class="text-smallcaps">ce</span>).</span><button class="js-more-btn d-none btn btn-unstyled font-12 bg-white js-content" aria-label="Toggle more/less fact data"><span class="link-blue">(more)</span></button></span></div></figcaption></figure></div></div><p class="topic-paragraph">The Greeks divided the field of mathematics into arithmetic (the study of “multitude,” or discrete quantity) and geometry (that of “magnitude,” or continuous quantity) and considered both to have originated in practical activities. <span id="ref536062"></span><a href="https://www.britannica.com/biography/Proclus" class="md-crosslink " data-show-preview="true">Proclus</a>, in his <em><span id="ref536063"></span>Commentary on Euclid</em>, observes that geometry—literally, “measurement of land”—first arose in surveying practices among the ancient Egyptians, for the flooding of the Nile compelled them each year to redefine the boundaries of properties. Similarly, arithmetic started with the commerce and trade of <a href="https://www.britannica.com/place/Phoenicia" class="md-crosslink " data-show-preview="true">Phoenician</a> merchants. Although Proclus wrote quite late in the ancient period (in the 5th century <span class="text-smallcaps">ce</span>), his account drew upon views proposed much earlier—by <span id="ref536064"></span><a href="https://www.britannica.com/biography/Herodotus-Greek-historian" class="md-crosslink " data-show-preview="true">Herodotus</a> (mid-5th century <span class="text-smallcaps">bce</span>), for example, and by <a href="https://www.britannica.com/biography/Eudemus-of-Rhodes" class="md-crosslink " data-show-preview="true">Eudemus</a>, a <a class="md-dictionary-link md-dictionary-tt-off mw" data-term="disciple" href="https://www.merriam-webster.com/dictionary/disciple" data-type="MW">disciple</a> of <a href="https://www.britannica.com/biography/Aristotle" class="md-crosslink " data-show-preview="true">Aristotle</a> (late 4th century <span class="text-smallcaps">bce</span>).</p><span class="marker MOD4 mod-inline"></span> <span class="marker PREMOD5 mod-inline"></span><p class="topic-paragraph">However plausible, this view is difficult to check, for there is only meagre evidence of practical mathematics from the early Greek period (roughly, the 8th through the 4th century <span class="text-smallcaps">bce</span>). Inscriptions on stone, for example, reveal use of a <a href="https://www.britannica.com/science/numeral-system" class="md-crosslink autoxref " data-show-preview="true">numeral system</a> the same in principle as the familiar <a href="https://www.britannica.com/topic/Roman-numeral" class="md-crosslink " data-show-preview="true">Roman numerals</a>. Herodotus seems to have known of the <span id="ref536065"></span><a href="https://www.britannica.com/technology/abacus-calculating-device" class="md-crosslink " data-show-preview="true">abacus</a> as an aid for computation by both Greeks and Egyptians, and about a dozen stone specimens of Greek abaci survive from the 5th and 4th centuries <span class="text-smallcaps">bce</span>. In the surveying of new cities in the Greek colonies of the 6th and 5th centuries, there was regular use of a standard length of 70 <em>plethra</em> (one <em><span id="ref536066"></span>plethron</em> equals 100 feet) as the diagonal of a square of side 50 <em>plethra</em>; in fact, the actual diagonal of the square is 50<span class="md-root"><span class="ada-visuallyhidden">Square root of</span><span class="root-symbol">√</span><span class="root-content">2</span></span> <em>plethra</em>, so this was equivalent to using 7/5 (or 1.4) as an estimate for <span class="md-root"><span class="ada-visuallyhidden">Square root of</span><span class="root-symbol">√</span><span class="root-content">2</span></span>, which is now known to equal 1.414…. In the 6th century <span class="text-smallcaps">bce</span> the engineer <span id="ref536067"></span>Eupalinus of Megara directed an aqueduct through a mountain on the island of Samos, and historians still debate how he did it. In a further indication of the practical aspects of early Greek mathematics, <span id="ref536068"></span><a href="https://www.britannica.com/biography/Plato" class="md-crosslink " data-show-preview="true">Plato</a> describes in his <em>Laws</em> how the Egyptians drilled their children in practical problems in arithmetic and geometry; he clearly considered this a model for the Greeks to imitate.</p><a class="link-module shadow-sm d-block qa-read-more-module" href="/story/is-zero-an-even-or-an-odd-number" data-link-module-iframe-link=""> <img loading="lazy" src="https://cdn.britannica.com/49/191949-118-B1262DEA/balloon.jpg" alt="number zero, 0 balloon" class="rounded-sm mr-15" width="70" /> <div class="line-clamp clamp-5"> <div class="module-title bg-navy-dark">More From Britannica</div> <div class="font-weight-semi-bold mt-5">Is Zero an Even or an Odd Number?</div> </div> </a><span class="marker MOD5 mod-inline"></span> <span class="marker PREMOD6 mod-inline"></span><p class="topic-paragraph">Such hints about the nature of early Greek practical mathematics are confirmed in later sources—for example, in the arithmetic problems in papyrus texts from Ptolemaic Egypt (from the 3rd century <span class="text-smallcaps">bce</span> onward) and the geometric manuals by <a href="https://www.britannica.com/biography/Heron-of-Alexandria" class="md-crosslink " data-show-preview="true">Heron of Alexandria</a> (1st century <span class="text-smallcaps">ce</span>). In its basic manner this Greek tradition was much like the earlier traditions in Egypt and Mesopotamia. Indeed, it is likely that the Greeks borrowed from such older sources to some extent.</p><span class="marker MOD6 mod-inline"></span> <span class="marker PREMOD7 mod-inline"></span><p class="topic-paragraph">What was distinctive of the Greeks’ contribution to mathematics—and what in effect made them the creators of “mathematics,” as the term is usually understood—was its development as a theoretical <a class="md-dictionary-link md-dictionary-tt-off mw" data-term="discipline" href="https://www.merriam-webster.com/dictionary/discipline" data-type="MW">discipline</a>. This means two things: mathematical statements are general, and they are confirmed by <span id="ref536069"></span><a href="https://www.britannica.com/topic/proof-logic" class="md-crosslink " data-show-preview="true">proof</a>. For example, the Mesopotamians had procedures for finding whole numbers <em>a</em>, <em>b</em>, and <em>c</em> for which <em>a</em><sup>2</sup> + <em>b</em><sup>2</sup> = <em>c</em><sup>2</sup> (e.g., 3, 4, 5; 5, 12, 13; or 119, 120, 169). From the Greeks came a proof of a general rule for finding all such sets of numbers (now called <span id="ref536070"></span><a href="https://www.britannica.com/science/Pythagorean-triple" class="md-crosslink ">Pythagorean triples</a>): if one takes two whole numbers <em>p</em> and <em>q</em>, both being even or both odd and such that <em>p</em><em>q</em> is a square number, then <em>a</em> = (<em>p</em> − <em>q</em>)/2, <em>b</em> = <span class="md-root"><span class="ada-visuallyhidden">Square root of</span><span class="root-symbol">√</span><span class="root-content"><em>p</em><em>q</em></span></span>, and <em>c</em> = (<em>p</em>+ <em>q</em>)/2. As Euclid proves in Book X of the <em><span id="ref536071"></span><a href="https://www.britannica.com/topic/Elements-by-Euclid" class="md-crosslink " data-show-preview="true">Elements</a></em>, numbers of this form satisfy the relation for Pythagorean triples. Further, the Mesopotamians appear to have understood that sets of such numbers <em>a</em>, <em>b</em>, and <em>c</em> form the sides of right triangles, but the Greeks proved this result (Euclid, in fact, proves it twice: in <em>Elements</em>, Book I, proposition 47, and in a more general form in <em>Elements</em>, Book VI, proposition 31), and these proofs occur in the <a class="md-dictionary-link md-dictionary-tt-off mw" data-term="context" href="https://www.merriam-webster.com/dictionary/context" data-type="MW">context</a> of a systematic presentation of the properties of plane geometric figures.</p><span class="marker MOD7 mod-inline"></span> <span class="marker PREMOD8 mod-inline"></span><p class="topic-paragraph">The <em>Elements</em>, composed by <span id="ref536072"></span><a href="https://www.britannica.com/biography/Euclid-Greek-mathematician" class="md-crosslink " data-show-preview="true">Euclid</a> of Alexandria about 300 <span class="text-smallcaps">bce</span>, was the pivotal contribution to theoretical geometry, but the transition from practical to theoretical mathematics had occurred much earlier, sometime in the 5th century <span class="text-smallcaps">bce</span>. Initiated by men like <a href="https://www.britannica.com/biography/Pythagoras" class="md-crosslink " data-show-preview="true">Pythagoras of Samos</a> (late 6th century) and <a href="https://www.britannica.com/biography/Hippocrates-of-Chios" class="md-crosslink " data-show-preview="true">Hippocrates of Chios</a> (late 5th century), the theoretical form of geometry was advanced by others, most prominently the Pythagorean <a href="https://www.britannica.com/biography/Archytas-of-Tarentum" class="md-crosslink " data-show-preview="true">Archytas of Tarentum</a>, <a href="https://www.britannica.com/biography/Theaetetus" class="md-crosslink " data-show-preview="true">Theaetetus of Athens</a>, and <span id="ref536073"></span><a href="https://www.britannica.com/biography/Eudoxus-of-Cnidus" class="md-crosslink " data-show-preview="true">Eudoxus of Cnidus</a> (4th century). Because the actual writings of these men do not survive, knowledge about their work depends on remarks made by later writers. While even this limited evidence reveals how heavily Euclid depended on them, it does not <a href="https://www.britannica.com/topic/set-mathematics-and-logic" class="md-crosslink autoxref " data-show-preview="true">set</a> out clearly the motives behind their studies.</p><span class="marker MOD8 mod-inline"></span> <span class="marker PREMOD9 mod-inline"></span><p class="topic-paragraph">It is thus a matter of debate how and why this theoretical transition took place. A frequently cited factor is the discovery of <span id="ref536074"></span><a href="https://www.britannica.com/science/irrational-number" class="md-crosslink " data-show-preview="true">irrational numbers</a>. The early <span id="ref536075"></span><a href="https://www.britannica.com/science/Pythagoreanism" class="md-crosslink " data-show-preview="true">Pythagoreans</a> held that “all things are number.” This might be taken to mean that any geometric <a href="https://www.britannica.com/science/measure-mathematics" class="md-crosslink autoxref " data-show-preview="true">measure</a> can be associated with some number (that is, some <a href="https://www.britannica.com/science/natural-number" class="md-crosslink autoxref " data-show-preview="true">whole number</a> or fraction; in modern terminology, rational number), for in Greek usage the term for number, <em>arithmos</em>, refers exclusively to whole numbers or, in some <a class="md-dictionary-link md-dictionary-tt-off mw" data-term="contexts" href="https://www.merriam-webster.com/dictionary/contexts" data-type="MW">contexts</a>, to ordinary fractions. This assumption is common enough in practice, as when the length of a given <a href="https://www.britannica.com/science/line-mathematics" class="md-crosslink autoxref " data-show-preview="true">line</a> is said to be so many feet plus a fractional part. However, it breaks down for the lines that form the side and diagonal of the <span id="ref536076"></span><a href="https://www.britannica.com/science/square-mathematics" class="md-crosslink " data-show-preview="true">square</a>. (For example, if it is supposed that the <a href="https://www.britannica.com/science/ratio" class="md-crosslink autoxref " data-show-preview="true">ratio</a> between the side and diagonal may be expressed as the ratio of two whole numbers, it can be shown that both of these numbers must be even. This is impossible, since every <a href="https://www.britannica.com/science/fraction" class="md-crosslink autoxref " data-show-preview="true">fraction</a> may be expressed as a ratio of two whole numbers having no common factors.) Geometrically, this means that there is no length that could serve as a unit of measure of both the side and diagonal; that is, the side and diagonal cannot each equal the same length multiplied by (different) whole numbers. Accordingly, the Greeks called such pairs of lengths “<span id="ref536077"></span><a href="https://www.britannica.com/topic/Incommensurables-1688515" class="md-crosslink " data-show-preview="true">incommensurable</a>.” (In modern <a class="md-dictionary-link md-dictionary-tt-off eb" data-term="terminology" href="https://www.britannica.com/dictionary/terminology" data-type="EB">terminology</a>, unlike that of the Greeks, the term “number” is applied to such quantities as <span class="md-root"><span class="ada-visuallyhidden">Square root of</span><span class="root-symbol">√</span><span class="root-content">2</span></span>, but they are called irrational.)</p><span class="marker MOD9 mod-inline"></span> <span class="marker PREMOD10 mod-inline"></span><p class="topic-paragraph">This result was already well known at the time of Plato and may well have been discovered within the school of Pythagoras in the 5th century <span class="text-smallcaps">bce</span>, as some late authorities like <a href="https://www.britannica.com/biography/Pappus-of-Alexandria" class="md-crosslink " data-show-preview="true">Pappus of Alexandria</a> (4th century <span class="text-smallcaps">ce</span>) maintain. In any case, by 400 <span class="text-smallcaps">bce</span> it was known that lines corresponding to <span class="md-root"><span class="ada-visuallyhidden">Square root of</span><span class="root-symbol">√</span><span class="root-content">3</span></span>, <span class="md-root"><span class="ada-visuallyhidden">Square root of</span><span class="root-symbol">√</span><span class="root-content">5</span></span>, and other square roots are incommensurable with a fixed unit length. The more general result, the geometric equivalent of the theorem that <span class="md-root"><span class="ada-visuallyhidden">Square root of</span><span class="root-symbol">√</span><span class="root-content"><em>p</em></span></span> is irrational whenever <em>p</em> is not a rational square number, is associated with Plato’s friend <span id="ref536078"></span><a href="https://www.britannica.com/biography/Theaetetus" class="md-crosslink " data-show-preview="true">Theaetetus</a>. Both Theaetetus and Eudoxus contributed to the further study of irrationals, and their followers collected the results into a substantial theory, as represented by the 115 propositions of Book X of the <em>Elements</em>.</p><span class="marker MOD10 mod-inline"></span> <span class="marker PREMOD11 mod-inline"></span><p class="topic-paragraph">The discovery of irrationals must have affected the very nature of early mathematical research, for it made clear that arithmetic was insufficient for the purposes of geometry, despite the assumptions made in practical work. Further, once such seemingly obvious assumptions as the commensurability of all lines turned out to be in fact false, then in principle all mathematical assumptions were rendered suspect. At the least it became necessary to justify carefully all claims made about mathematics. Even more basically, it became necessary to establish what a reasoning has to be like to qualify as a proof. Apparently, Hippocrates of Chios, in the 5th century <span class="text-smallcaps">bce</span>, and others soon after him had already begun the work of organizing geometric results into a systematic form in textbooks called “elements” (meaning “fundamental results” of geometry). These were to serve as sources for Euclid in his <a class="md-dictionary-link md-dictionary-tt-off mw" data-term="comprehensive" href="https://www.merriam-webster.com/dictionary/comprehensive" data-type="MW">comprehensive</a> textbook a century later.</p><span class="marker MOD11 mod-inline"></span> <span class="marker PREMOD12 mod-inline"></span><p class="topic-paragraph">The early mathematicians were not an isolated <a href="https://www.britannica.com/science/group-mathematics" class="md-crosslink autoxref " data-show-preview="true">group</a> but part of a larger, intensely competitive <a class="md-dictionary-link md-dictionary-tt-off mw" data-term="intellectual" href="https://www.merriam-webster.com/dictionary/intellectual" data-type="MW">intellectual</a> <a class="md-dictionary-link md-dictionary-tt-off mw" data-term="environment" href="https://www.merriam-webster.com/dictionary/environment" data-type="MW">environment</a> of <span id="ref536079"></span><a href="https://www.britannica.com/topic/pre-Socratic-philosophy" class="md-crosslink " data-show-preview="true">pre-Socratic</a> thinkers in Ionia and Italy, as well as <a href="https://www.britannica.com/topic/Sophist-philosophy" class="md-crosslink " data-show-preview="true">Sophists</a> at Athens. By insisting that only permanent things could have real existence, the philosopher <span id="ref536080"></span><a href="https://www.britannica.com/biography/Parmenides-Greek-philosopher" class="md-crosslink " data-show-preview="true">Parmenides</a> (5th century <span class="text-smallcaps">bce</span>) called into question the most basic claims about knowledge itself. In contrast, <span id="ref536081"></span><a href="https://www.britannica.com/biography/Heraclitus" class="md-crosslink " data-show-preview="true">Heracleitus</a> (c. 500 <span class="text-smallcaps">bce</span>) maintained that all permanence is an <a class="md-dictionary-link md-dictionary-tt-off mw" data-term="illusion" href="https://www.merriam-webster.com/dictionary/illusion" data-type="MW">illusion</a>, for the things that are perceived arise through a subtle balance of opposing tensions. What is meant by “knowledge” and “proof” thus came into debate.</p><span class="marker MOD12 mod-inline"></span> <span class="marker PREMOD13 mod-inline"></span><p class="topic-paragraph">Mathematical issues were often drawn into these debates. For some, like the Pythagoreans (and, later, Plato), the certainty of mathematics was held as a model for reasoning in other areas, like politics and <a class="md-dictionary-link md-dictionary-tt-off mw" data-term="ethics" href="https://www.merriam-webster.com/dictionary/ethics" data-type="MW">ethics</a>. But for others mathematics seemed prone to contradiction. <a href="https://www.britannica.com/biography/Zeno-of-Elea" class="md-crosslink " data-show-preview="true">Zeno of Elea</a> (5th century <span class="text-smallcaps">bce</span>) posed <span id="ref536082"></span><a href="https://www.britannica.com/topic/paradoxes-of-Zeno" class="md-crosslink " data-show-preview="true">paradoxes</a> about quantity and <a href="https://www.britannica.com/science/motion-mechanics" class="md-crosslink autoxref " data-show-preview="true">motion</a>. In one such <a class="md-dictionary-link md-dictionary-tt-off mw" data-term="paradox" href="https://www.merriam-webster.com/dictionary/paradox" data-type="MW">paradox</a> it is assumed that a line can be bisected again and again without limit; if the division ultimately results in a set of points of zero length, then even infinitely many of them sum up only to zero, but, if it results in tiny line segments, then their sum will be <a class="md-dictionary-link md-dictionary-tt-off mw" data-term="infinite" href="https://www.merriam-webster.com/dictionary/infinite" data-type="MW">infinite</a>. In effect, the length of the given line must be both zero and infinite. In the 5th century <span class="text-smallcaps">bce</span> a solution of such <a class="md-dictionary-link md-dictionary-tt-off mw" data-term="paradoxes" href="https://www.merriam-webster.com/dictionary/paradoxes" data-type="MW">paradoxes</a> was attempted by <span id="ref536084"></span><a href="https://www.britannica.com/biography/Democritus" class="md-crosslink " data-show-preview="true">Democritus</a> and the <span id="ref536083"></span><a href="https://www.britannica.com/topic/atomism" class="md-crosslink " data-show-preview="true">atomists</a>, philosophers who held that all material bodies are ultimately made up of invisibly small “atoms” (the Greek word <em>atomon</em> means “indivisible”). But in <span id="ref536085"></span><a href="https://www.britannica.com/science/geometry" class="md-crosslink " data-show-preview="true">geometry</a> such a view came into conflict with the existence of incommensurable lines, since the atoms would become the measuring units of all lines, even incommensurable ones. Democritus and the Sophist <a href="https://www.britannica.com/biography/Protagoras-Greek-philosopher" class="md-crosslink " data-show-preview="true">Protagoras</a> puzzled over whether the <a href="https://www.britannica.com/science/tangent-mathematical-function" class="md-crosslink autoxref " data-show-preview="true">tangent</a> to a <a href="https://www.britannica.com/science/circle-mathematics" class="md-crosslink autoxref " data-show-preview="true">circle</a> meets it at a point or a line. The Sophists <a href="https://www.britannica.com/biography/Antiphon-Greek-writer-and-statesman" class="md-crosslink " data-show-preview="true">Antiphon</a> and Bryson (both 5th century <span class="text-smallcaps">bce</span>) considered how to compare the circle to polygons inscribed in it.</p><span class="marker MOD13 mod-inline"></span> <span class="marker PREMOD14 mod-inline"></span><p class="topic-paragraph">The pre-Socratics thus revealed difficulties in specific assumptions about the infinitely many and the infinitely small and about the relation of geometry to physical reality, as well as in more general <a class="md-dictionary-link md-dictionary-tt-off mw" data-term="conceptions" href="https://www.merriam-webster.com/dictionary/conceptions" data-type="MW">conceptions</a> like “existence” and “proof.” Philosophical questions such as these need not have affected the technical researches of mathematicians, but they did make them aware of difficulties that could bear on fundamental matters and so made them the more cautious in defining their subject matter.</p><span class="marker MOD14 mod-inline"></span> <span class="marker PREMOD15 mod-inline"></span><p class="topic-paragraph">Any such review of the possible effects of factors such as these is purely conjectural, since the sources are fragmentary and never make explicit how the mathematicians responded to the issues that were raised. But it is the particular concern over fundamental assumptions and proofs that distinguishes Greek mathematics from the earlier traditions. <a class="md-dictionary-link md-dictionary-tt-off eb" data-term="Plausible" href="https://www.britannica.com/dictionary/Plausible" data-type="EB">Plausible</a> factors behind this concern can be identified in the special circumstances of the early Greek tradition—its technical discoveries and its cultural environment—even if it is not possible to describe in detail how these changes took place.</p><span class="marker MOD15 mod-inline"></span> </section> <section data-level="3" id="ref65980"> <h2 class="h3">The <em>Elements</em></h2> <span class="marker PREMOD16 mod-inline"></span><p class="topic-paragraph">The principal source for reconstructing pre-Euclidean mathematics is Euclid’s <em>Elements</em>, for the major part of its contents can be traced back to research from the 4th century <span class="text-smallcaps">bce</span> and in some cases even earlier. The first four books present constructions and proofs of plane <span id="ref536086"></span>geometric figures: Book I deals with the <a href="https://www.britannica.com/science/congruence" class="md-crosslink autoxref " data-show-preview="true">congruence</a> of triangles, the properties of parallel lines, and the area relations of triangles and parallelograms; Book II establishes equalities relating to squares, rectangles, and triangles; Book III covers basic properties of circles; and Book IV sets out constructions of polygons in circles. Much of the content of Books I–III was already familiar to Hippocrates, and the material of Book IV can be associated with the Pythagoreans, so that this portion of the <em>Elements</em> has roots in 5th-century research. It is known, however, that questions about <span id="ref536087"></span><a href="https://www.britannica.com/science/parallel-postulate" class="md-crosslink " data-show-preview="true">parallels</a> were debated in Aristotle’s school (c. 350 <span class="text-smallcaps">bce</span>), and so it may be assumed that efforts to prove results—such as the theorem stating that for any given line and given point, there always exists a <a class="md-dictionary-link md-dictionary-tt-off eb" data-term="unique" href="https://www.britannica.com/dictionary/unique" data-type="EB">unique</a> line through that point and parallel to the line—were tried and failed. Thus, the decision to found the theory of parallels on a postulate, as in Book I of the <em>Elements</em>, must have been a relatively recent development in Euclid’s time. (The postulate would later become the subject of much study, and in modern times it led to the discovery of the so-called <a href="https://www.britannica.com/science/non-Euclidean-geometry" class="md-crosslink " data-show-preview="true">non-Euclidean geometries</a>.)</p><span class="marker MOD16 mod-inline"></span> <span class="marker PREMOD17 mod-inline"></span><p class="topic-paragraph">Book V sets out a general theory of proportion—that is, a theory that does not require any restriction to commensurable magnitudes. This general theory derives from Eudoxus. On the basis of the theory, Book VI describes the properties of <span id="ref536088"></span><a href="https://www.britannica.com/science/similarity-mathematics" class="md-crosslink ">similar</a> plane rectilinear figures and so generalizes the congruence theory of Book I. It appears that the technique of similar figures was already known in the 5th century <span class="text-smallcaps">bce</span>, even though a fully valid justification could not have been given before Eudoxus worked out his theory of proportion.</p><span class="marker MOD17 mod-inline"></span> <span class="marker PREMOD18 mod-inline"></span><p class="topic-paragraph">Books VII–IX deal with what the Greeks called “<span id="ref536089"></span><a href="https://www.britannica.com/science/arithmetic" class="md-crosslink " data-show-preview="true">arithmetic</a>,” the theory of whole numbers. It includes the properties of numerical proportions, greatest common divisors, least common multiples, and relative primes (Book VII); propositions on numerical progressions and square and <a href="https://www.britannica.com/science/cube-mathematics" class="md-crosslink autoxref " data-show-preview="true">cube</a> numbers (Book VIII); and special results, like unique factorization into primes, the existence of an unlimited number of primes, and the formation of “<span id="ref536090"></span><a href="https://www.britannica.com/science/perfect-number" class="md-crosslink " data-show-preview="true">perfect numbers</a>”—that is, those numbers that equal the sum of their proper divisors (Book IX). In some form Book VII stems from Theaetetus and Book VIII from Archytas.</p><span class="marker MOD18 mod-inline"></span> <span class="marker PREMOD19 mod-inline"></span><p class="topic-paragraph">Book X presents a theory of irrational lines and derives from the work of Theaetetus and Eudoxus. The remaining books treat the geometry of solids. Book XI sets out results on solid figures <a class="md-dictionary-link md-dictionary-tt-off mw" data-term="analogous" href="https://www.merriam-webster.com/dictionary/analogous" data-type="MW">analogous</a> to those for planes in Books I and VI; Book XII proves theorems on the ratios of circles, the ratios of spheres, and the volumes of pyramids and cones; Book XIII shows how to inscribe the five regular solids—known as the <a href="https://www.britannica.com/science/Platonic-solid" class="md-crosslink " data-show-preview="true">Platonic solids</a>—in a given <a href="https://www.britannica.com/science/sphere" class="md-crosslink autoxref " data-show-preview="true">sphere</a> (compare the constructions of plane figures in Book IV). The <a href="https://www.britannica.com/technology/measurement" class="md-crosslink autoxref " data-show-preview="true">measurement</a> of curved figures in Book XII is inferred from that of <span id="ref536091"></span>rectilinear figures; for a particular curved figure, a sequence of rectilinear figures is considered in which succeeding figures in the sequence become continually closer to the curved figure; the particular method used by Euclid <a class="md-dictionary-link md-dictionary-tt-off eb" data-term="derives" href="https://www.britannica.com/dictionary/derives" data-type="EB">derives</a> from Eudoxus. The solid constructions in Book XIII derive from Theaetetus.</p><span class="marker MOD19 mod-inline"></span> <span class="marker PREMOD20 mod-inline"></span><p class="topic-paragraph">In sum the <em>Elements</em> gathered together the whole field of elementary geometry and arithmetic that had developed in the two centuries before Euclid. Doubtless, Euclid must be credited with particular aspects of this work, certainly with its editing as a comprehensive whole. But it is not possible to identify for certain even a single one of its results as having been his discovery. Other, more advanced fields, though not touched on in the <em>Elements</em>, were already being vigorously studied in Euclid’s time, in some cases by Euclid himself. For these fields his textbook, true to its name, provides the <a class="md-dictionary-link md-dictionary-tt-off eb" data-term="appropriate" href="https://www.britannica.com/dictionary/appropriate" data-type="EB">appropriate</a> “elementary” introduction.</p><span class="marker MOD20 mod-inline"></span> <span class="marker PREMOD21 mod-inline"></span><p class="topic-paragraph">One such field is the study of geometric <span id="ref536092"></span>constructions. Euclid, like geometers in the generation before him, divided mathematical propositions into two kinds: “theorems” and “<span id="ref536094"></span><a href="https://www.britannica.com/science/problem" class="md-crosslink ">problems</a>.” A <span id="ref536093"></span><a href="https://www.britannica.com/topic/theorem" class="md-crosslink " data-show-preview="true">theorem</a> makes the claim that all terms of a certain description have a specified property; a problem seeks the construction of a term that is to have a specified property. In the <em>Elements</em> all the problems are constructible on the basis of three stated <a class="md-dictionary-link md-dictionary-tt-off eb" data-term="postulates" href="https://www.britannica.com/dictionary/postulates" data-type="EB">postulates</a>: that a line can be constructed by joining two given points, that a given line segment can be extended in a line indefinitely, and that a circle can be constructed with a given point as centre and a given line segment as radius. These postulates in effect restricted the constructions to the use of the so-called <span id="ref536095"></span>Euclidean tools—i.e., a compass and a straightedge or unmarked ruler.</p><span class="marker MOD21 mod-inline"></span> </section> </section></section><span class="marker end-of-content"></span><span class="marker after-article"></span></div> <div id="chatbot-root"></div> </div> </div> </div> <div class="ai-dialog-placeholder"></div> </div> </div> <aside class="col-md-da-320"></aside> </div> </div> </div> </div> </article> </div> </div> </div> </div> </main> <div id="md-footer"></div> <noscript><iframe src="//www.googletagmanager.com/ns.html?id=GTM-5W6NC8" height="0" width="0" style="display:none;visibility:hidden"></iframe></noscript> <script type="text/javascript" id="_informizely_script_tag"> var IzWidget = IzWidget || {}; (function (d) { var scriptElement = d.createElement('script'); scriptElement.type = 'text/javascript'; scriptElement.async = true; scriptElement.src = "https://insitez.blob.core.windows.net/site/f780f33e-a610-4ac2-af81-3eb184037547.js"; var node = d.getElementById('_informizely_script_tag'); node.parentNode.insertBefore(scriptElement, node); } )(document); </script>  <script> window.ap3c = window.ap3c || {}; var ap3c = window.ap3c; ap3c.cmd = ap3c.cmd || []; ap3c.cmd.push(function() { ap3c.init('ZO4siT4cLwnykPnzZWJtd3Byb2Q', 'https://engage.email.britannica.com/'); ap3c.track({v: 0}); }); ap3c.activity = function(act) { ap3c.act = (ap3c.act || []); ap3c.act.push(act); }; var s, t; s = document.createElement('script'); s.type = 'text/javascript'; s.src = "https://engage.email.britannica.com/app.js"; t = document.getElementsByTagName('script')[0]; t.parentNode.insertBefore(s, t); </script> <script class="marketing-page-info" type="application/json"> {"pageType":"Topic","templateName":"DESKTOP","pageNumber":5,"pagesTotal":27,"pageId":369194,"pageLength":2992,"initialLoad":true,"lastPageOfScroll":false} </script> <script class="marketing-content-info" type="application/json"> [] </script> <script src="https://cdn.britannica.com/mendel-resources/3-133/js/libs/jquery-3.5.0.min.js?v=3.133.9"></script> <script type="text/javascript" data-type="Init Mendel Code Splitting"> (function() { $.ajax({ dataType: 'script', cache: true, url: 'https://cdn.britannica.com/mendel-resources/3-133/dist/topic-page.js?v=3.133.9' }); })(); </script> <script class="analytics-metadata" type="application/json"> {"leg":"C","adLeg":"C","userType":"ANONYMOUS","pageType":"Topic","pageSubtype":null,"articleTemplateType":"PAGINATED","gisted":false,"pageNumber":5,"hasSummarizeButton":false,"hasAskButton":true} </script> <script type="text/javascript"> EBStat={accountId:-1,hostnameOverride:'webstats.eb.com',domain:'www.britannica.com', json:''}; </script> <script type="text/javascript"> ( function() { $.ajax( { dataType: 'script', cache: true, url: '//www.britannica.com/webstats/mendelstats.js?v=1' } ) .done( function() { try {writeStat(null,EBStat);} catch(err){} } ); })(); </script> <div id="bc-fixed-dialogue"></div> </body> </html>

CINXE.COM

Mathematics - Egyptian, Assessment, History | Britannica