Geometry | Definition, History, Basics, Branches, & Facts

<!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/3-130"> <link rel="preconnect" href="https://cdn.britannica.com/mendel-resources/3-130"> <link rel="preload" as="script" href="https://www.googletagservices.com/tag/js/gpt.js" /> <link rel="icon" href="/favicon.png" /> <meta name="description" content="Geometry, the branch of mathematics concerned with the shape of individual objects, spatial relationships among various objects, and the properties of surrounding space. It is one of the oldest branches of mathematics, having arisen in response to such practical problems as those found in" /> <meta name="keywords" content="geometry, encyclopedia, encyclopeadia, britannica, article" /> <link rel="canonical" href="https://www.britannica.com/science/geometry" /> <title>Geometry | Definition, History, Basics, Branches, & Facts | Britannica</title> <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","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","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","DEFAULT_S3_REGION": "US_EAST_1","TOPIC_COLLECTION_URL": "/summary","LOGINBOX_URL": "/auth/loginbox","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_LOGIN_URL = "https://cam.britannica.com"; 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/geometry", "amuselabsUrl": "https://cdn3.amuselabs.com", "resourcesPrefixUrl": "https://cdn.britannica.com/mendel-resources/3-130/[url]?v=3.130.14", "date": 20241127, "userInfo": { "type": "ANONYMOUS" ,"currency": "AUUS" ,"country": "SG" ,"state": "XX" ,"timezone": "Asia/Singapore" ,"bcomId": "-1772932940545863509" ,"hasAds": true ,"testVersion": "A" ,"adsTestVersion": "A" ,"consumerId": "" ,"instId": "" ,"consumerUserName": "" ,"instUserName": "" ,"cognito": null }, "tvs":{ "r":[25,25,25,25], "a": [25,25,25,25]}, "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":229851,"currentDate":20241127}, "autocompleteToSearchPage": false,"topicUrl": "https://www.britannica.com/science/geometry" ,"freeTopicReason": "NO_REFERRER" ,"topicId": 229851 ,"template": "DESKTOP" ,"type": "CORE" ,"hasToc": true ,"chatbotApi": "https://www.britannica.com/chat-api" ,"showPreview": false ,"infiniteScrollList": [{"p":10,"t":229851},{"p":3,"t":194901},{"p":1,"t":417456},{"p":2,"t":22548},{"p":4,"t":599686},{"p":1,"t":215500},{"p":2,"t":162938},{"p":1,"t":682092},{"p":27,"t":369194},{"p":23,"t":32876}] ,"sequence": 1 ,"topics": {} }, "GA": {"leg":"A","adLeg":"A","userType":"ANONYMOUS","pageType":"Topic","articleTemplateType":"PAGINATED","gisted":false,"pageNumber":1,"hasSummarizeButton":false,"hasAskButton":false} }; </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/61/67361-050-F48C1CC9/map-millennium-mathematicians-Greco-Roman-Thales-Hypatia-Alexandria.jpg" /> <meta name="twitter:description" content="Geometry, the branch of mathematics concerned with the shape of individual objects, spatial relationships among various objects, and the properties of surrounding space. It is one of the oldest branches of mathematics, having arisen in response to such practical problems as those found in"/> <meta property="og:type" content="ARTICLE"/> <meta property="og:title" content="Geometry | Definition, History, Basics, Branches, & Facts | Britannica"/> <meta property="og:description" content="Geometry, the branch of mathematics concerned with the shape of individual objects, spatial relationships among various objects, and the properties of surrounding space. It is one of the oldest branches of mathematics, having arisen in response to such practical problems as those found in"/> <meta property="og:site_name" content="Encyclopedia Britannica" /> <meta property="og:url" content="https://www.britannica.com/science/geometry"/> <meta property="og:image" content="https://cdn.britannica.com/61/67361-050-F48C1CC9/map-millennium-mathematicians-Greco-Roman-Thales-Hypatia-Alexandria.jpg" /> <meta property="og:image:type" content="image/jpeg" /> <script type="text/javascript" data-type="init opengraph"> Mendel.openGraph = {"type":"ARTICLE","title":"Geometry | Definition, History, Basics, Branches, & Facts","description":"Geometry, the branch of mathematics concerned with the shape of individual objects, spatial relationships among various objects, and the properties of surrounding space. It is one of the oldest branches of mathematics, having arisen in response to such practical problems as those found in","imageUrl":"https://cdn.britannica.com/61/67361-050-F48C1CC9/map-millennium-mathematicians-Greco-Roman-Thales-Hypatia-Alexandria.jpg","imageType":"image/jpeg","pageUrl":"https://www.britannica.com/science/geometry"}</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/3-130/dist/vendor-bundle.css?v=3.130.14" rel="stylesheet" /> <link href="https://cdn.britannica.com/mendel-resources/3-130/dist/mendel-css.css?v=3.130.14" rel="stylesheet" /> <link href="https://cdn.britannica.com/mendel-resources/3-130/dist/topic-page.css?v=3.130.14" rel="stylesheet" /> <script type="text/javascript"> if (self !== top) { top.location = self.location; } // if ('scrollRestoration' in history) { history.scrollRestoration = 'manual'; } </script> <script src="https://cdn.britannica.com/mendel-resources/3-130/js/at.js?v=3.130.14" 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="2024-11-19" /> <script type="application/ld+json"> {"headline":"Geometry | Definition, History, Basics, Branches, & Facts","image":{"url":"https://cdn.britannica.com/61/67361-050-F48C1CC9/map-millennium-mathematicians-Greco-Roman-Thales-Hypatia-Alexandria.jpg","@type":"ImageObject"},"author":[{"name":"J.L. Heilbron","url":"https://www.britannica.com/contributor/JL-Heilbron/4434","@type":"Person"}],"keywords":"geometry","wordcount":9409,"url":"https://www.britannica.com/science/geometry","datePublished":"2001-09-14T00:00:00Z","dateModified":"2024-11-19T00:00:00Z","description":"Geometry, the branch of mathematics concerned with the shape of individual objects, spatial relationships among various objects, and the properties of surrounding space. It is one of the oldest branches of mathematics, having arisen in response to such practical problems as those found in","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="A" class="new-topic topic-desktop first-page-true user-ANONYMOUS user-ads md-desktop leg-ab-ie">  <script>Mendel.config.adProvider='ay';</script> <script async src="https://securepubads.g.doubleclick.net/tag/js/gpt.js"></script> <script> window.googletag = window.googletag || {cmd: []}; googletag.cmd.push(function() { googletag.defineSlot('/15510053/CMP_1x1', [1, 1], 'div-gpt-ad-1709766812090-0').addService(googletag.pubads()); googletag.pubads().enableSingleRequest(); googletag.enableServices(); }); </script> <script async defer src="https://launchpad-wrapper.privacymanager.io/0ccc6fe8-1870-4ad8-b47b-6d029ac116fc/launchpad-liveramp.js"></script> <script async src="https://JRyhoywLYXNLYMAhs.ay.delivery/manager/JRyhoywLYXNLYMAhs" type="text/javascript" referrerpolicy="no-referrer-when-downgrade" ></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":"ProCon","url":"/procon","navbarId":"PROCON"},{"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":"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":"ProCon","url":"/procon"},{"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":"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://subscription.britannica.com/subscribe?partnerCode=BP_Black_Friday_AUUS" class="subscribe-link btn btn-sm btn-outline-white-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://subscription.britannica.com/subscribe?partnerCode=BP_Black_Friday_AUUS" class="subscribe-link btn btn-xs btn-orange-dark 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://subscription.britannica.com/subscribe?partnerCode=BP_Black_Friday_AUUS</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=black-friday-2024" class="subscribe-link btn btn-sm btn-orange py-5" target="_blank"> Subscribe <span class="d-none d-md-inline"> Now</span> </a></DIV></div> <div id="global-nav-react"> <div class="d-none"> <ul> <li><a href="/">Home</a></li> <li><a href="/procon">ProCon</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="/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-PROCON " href="/procon">ProCon</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-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"> <article class="article-content container-lg qa-content px-0 pt-0 pb-40 py-lg-20 content md-expanded" data-topic-id="229851"> <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/geometry">geometry</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/geometry">Introduction</a></div><div class="ml-40 toc-drawer sub-toc-drawer"></div></li><li data-target="#ref217464"><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/geometry#ref217464">Major branches of geometry</a></div><div class="ml-40 toc-drawer sub-toc-drawer"><ul class="list-unstyled" data-level="h2"><li data-target="#ref217465"><a class="w-100 link-gray-900" href="/science/geometry#ref217465">Euclidean geometry</a></li></ul><ul class="list-unstyled" data-level="h2"><li data-target="#ref217466"><a class="w-100 link-gray-900" href="/science/geometry#ref217466">Analytic geometry</a></li></ul><ul class="list-unstyled" data-level="h2"><li data-target="#ref217467"><a class="w-100 link-gray-900" href="/science/geometry#ref217467">Projective geometry</a></li></ul><ul class="list-unstyled" data-level="h2"><li data-target="#ref217468"><a class="w-100 link-gray-900" href="/science/geometry#ref217468">Differential geometry</a></li></ul><ul class="list-unstyled" data-level="h2"><li data-target="#ref217469"><a class="w-100 link-gray-900" href="/science/geometry#ref217469">Non-Euclidean geometries</a></li></ul><ul class="list-unstyled" data-level="h2"><li data-target="#ref217470"><a class="w-100 link-gray-900" href="/science/geometry#ref217470">Topology</a></li></ul></div></li><li data-target="#ref217471"><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/geometry#ref217471">History of geometry</a></div><div class="ml-40 toc-drawer sub-toc-drawer"><ul class="list-unstyled" data-level="h2"><li data-target="#ref217472"><a class="w-100 link-gray-900" href="/science/geometry#ref217472">Ancient geometry: practical and empirical</a><ul class="list-unstyled" data-level="h3"><li data-target="#ref217473"><a class="w-100 link-gray-900" href="/science/geometry#ref217473">Finding the right angle</a></li></ul><ul class="list-unstyled" data-level="h3"><li data-target="#ref217474"><a class="w-100 link-gray-900" href="/science/geometry#ref217474">Locating the inaccessible</a></li></ul><ul class="list-unstyled" data-level="h3"><li data-target="#ref217475"><a class="w-100 link-gray-900" href="/science/geometry#ref217475">Estimating the wealth</a></li></ul></li></ul><ul class="list-unstyled" data-level="h2"><li data-target="#ref217476"><a class="w-100 link-gray-900" href="/science/geometry/Ancient-geometry-abstract-and-applied">Ancient geometry: abstract and applied</a><ul class="list-unstyled" data-level="h3"><li data-target="#ref217477" class="has-children"><a class="w-100 link-gray-900" href="/science/geometry/Ancient-geometry-abstract-and-applied#ref217477">The three classical problems</a><ul class="list-unstyled" data-level="h4"><li data-target="#ref217478"><a class="w-100 link-gray-900" href="/science/geometry/Ancient-geometry-abstract-and-applied#ref217478">Doubling the cube</a></li></ul><ul class="list-unstyled" data-level="h4"><li data-target="#ref217479"><a class="w-100 link-gray-900" href="/science/geometry/Ancient-geometry-abstract-and-applied#ref217479">Trisecting the angle</a></li></ul><ul class="list-unstyled" data-level="h4"><li data-target="#ref217480"><a class="w-100 link-gray-900" href="/science/geometry/Ancient-geometry-abstract-and-applied#ref217480">Squaring the circle</a></li></ul></li></ul><ul class="list-unstyled" data-level="h3"><li data-target="#ref217481"><a class="w-100 link-gray-900" href="/science/geometry/Idealization-and-proof">Idealization and proof</a></li></ul><ul class="list-unstyled" data-level="h3"><li data-target="#ref217482"><a class="w-100 link-gray-900" href="/science/geometry/Idealization-and-proof#ref217482">The Euclidean synthesis</a></li></ul><ul class="list-unstyled" data-level="h3"><li data-target="#ref217483"><a class="w-100 link-gray-900" href="/science/geometry/Idealization-and-proof#ref217483">Gnomonics and the cone</a></li></ul><ul class="list-unstyled" data-level="h3"><li data-target="#ref217484" class="has-children"><a class="w-100 link-gray-900" href="/science/geometry/Astronomy-and-trigonometry">Astronomy and trigonometry</a><ul class="list-unstyled" data-level="h4"><li data-target="#ref217485"><a class="w-100 link-gray-900" href="/science/geometry/Astronomy-and-trigonometry#ref217485">Calculation</a></li></ul><ul class="list-unstyled" data-level="h4"><li data-target="#ref217486"><a class="w-100 link-gray-900" href="/science/geometry/Astronomy-and-trigonometry#ref217486">Epistemology</a></li></ul></li></ul></li></ul><ul class="list-unstyled" data-level="h2"><li data-target="#ref217487"><a class="w-100 link-gray-900" href="/science/geometry/Astronomy-and-trigonometry#ref217487">Ancient geometry: cosmological and metaphysical</a><ul class="list-unstyled" data-level="h3"><li data-target="#ref217488"><a class="w-100 link-gray-900" href="/science/geometry/Astronomy-and-trigonometry#ref217488">Pythagorean numbers and Platonic solids</a></li></ul><ul class="list-unstyled" data-level="h3"><li data-target="#ref217489"><a class="w-100 link-gray-900" href="/science/geometry/Astronomy-and-trigonometry#ref217489">Measuring the Earth and heavens</a></li></ul></li></ul><ul class="list-unstyled" data-level="h2"><li data-target="#ref217490"><a class="w-100 link-gray-900" href="/science/geometry/The-post-classical-period">The post-classical period</a><ul class="list-unstyled" data-level="h3"><li data-target="#ref217491"><a class="w-100 link-gray-900" href="/science/geometry/The-post-classical-period#ref217491">Passage through Islam</a></li></ul><ul class="list-unstyled" data-level="h3"><li data-target="#ref217492"><a class="w-100 link-gray-900" href="/science/geometry/The-post-classical-period#ref217492">Europe rediscovers the classics</a></li></ul><ul class="list-unstyled" data-level="h3"><li data-target="#ref217493"><a class="w-100 link-gray-900" href="/science/geometry/The-post-classical-period#ref217493">Linear perspective</a></li></ul></li></ul><ul class="list-unstyled" data-level="h2"><li data-target="#ref217494"><a class="w-100 link-gray-900" href="/science/geometry/Transformation">Transformation</a><ul class="list-unstyled" data-level="h3"><li data-target="#ref217495" class="has-children"><a class="w-100 link-gray-900" href="/science/geometry/Transformation#ref217495">French circles</a><ul class="list-unstyled" data-level="h4"><li data-target="#ref217496"><a class="w-100 link-gray-900" href="/science/geometry/Transformation#ref217496">Projective geometry</a></li></ul><ul class="list-unstyled" data-level="h4"><li data-target="#ref217497"><a class="w-100 link-gray-900" href="/science/geometry/Cartesian-geometry">Cartesian geometry</a></li></ul></li></ul><ul class="list-unstyled" data-level="h3"><li data-target="#ref217498"><a class="w-100 link-gray-900" href="/science/geometry/Cartesian-geometry#ref217498">Geometrical calculus</a></li></ul><ul class="list-unstyled" data-level="h3"><li data-target="#ref217499"><a class="w-100 link-gray-900" href="/science/geometry/Cartesian-geometry#ref217499">The world system</a></li></ul></li></ul><ul class="list-unstyled" data-level="h2"><li data-target="#ref217500"><a class="w-100 link-gray-900" href="/science/geometry/Relaxation-and-rigour">Relaxation and rigour</a><ul class="list-unstyled" data-level="h3"><li data-target="#ref217501"><a class="w-100 link-gray-900" href="/science/geometry/Relaxation-and-rigour#ref217501">Projection again</a></li></ul><ul class="list-unstyled" data-level="h3"><li data-target="#ref217502"><a class="w-100 link-gray-900" href="/science/geometry/Relaxation-and-rigour#ref217502">Non-Euclidean geometries</a></li></ul><ul class="list-unstyled" data-level="h3"><li data-target="#ref217503"><a class="w-100 link-gray-900" href="/science/geometry/A-grand-synthesis">A grand synthesis</a></li></ul><ul class="list-unstyled" data-level="h3"><li data-target="#ref217504"><a class="w-100 link-gray-900" href="/science/geometry/The-real-world">The real world</a></li></ul></li></ul></div></li></ul> <a class="toc-extra-link link-gray-900" href="https://www.britannica.com/science/geometry/additional-info">References & Edit History</a> <a class="toc-extra-link link-gray-900" href="/facts/geometry">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/geometry/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/61/67361-050-F48C1CC9/map-millennium-mathematicians-Greco-Roman-Thales-Hypatia-Alexandria.jpg" data-href="/media/1/229851/57444" 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/41/66641-004-0A9B7581/comparison-Greek-Chinese-theorem-figure-equivalence-triangles.jpg" data-href="/media/1/229851/57039" class="media-overlay-link d-inline-block mr-5"> <img loading="lazy" src="https://cdn.britannica.com/41/66641-004-0A9B7581/comparison-Greek-Chinese-theorem-figure-equivalence-triangles.jpg" alt="Chinese and Greek geometric theorems" height="50" /> </a> <a href="https://cdn.britannica.com/48/66648-004-0633B972/Greeks-numbers-Polygonal-terms-measurements-dimensions-patterns.jpg" data-href="/media/1/229851/57046" class="media-overlay-link d-inline-block mr-5"> <img loading="lazy" src="https://cdn.britannica.com/48/66648-004-0633B972/Greeks-numbers-Polygonal-terms-measurements-dimensions-patterns.jpg" alt="polygonal numbers" height="50" /> </a> <a href="https://cdn.britannica.com/49/66649-050-22190482/Earth-Sun-measurement-Eratosthenes-distance-Syene-rays.jpg" data-href="/media/1/229851/57047" class="media-overlay-link d-inline-block mr-5"> <img loading="lazy" src="https://cdn.britannica.com/49/66649-004-A8E43489/Earth-Sun-measurement-Eratosthenes-distance-Syene-rays.jpg" alt="Eratosthenes' measurement of the Earth" height="50" /> </a> <a href="https://cdn.britannica.com/66/67366-050-E9EE4BA1/Mathematicians-world-names-Islamic-map-mathematicians-al-Khwarizmi.jpg" data-href="/media/1/229851/57443" class="media-overlay-link d-inline-block mr-5"> <img loading="lazy" src="https://cdn.britannica.com/66/67366-004-DC133B15/Mathematicians-world-names-Islamic-map-mathematicians-al-Khwarizmi.jpg" alt="mathematicians of the Islamic world" height="50" /> </a> <a href="https://cdn.britannica.com/54/66654-004-99D4F14F/figures-area-height-principle-Cavalieri-sections-length.jpg" data-href="/media/1/229851/57052" class="media-overlay-link d-inline-block mr-5"> <img loading="lazy" src="https://cdn.britannica.com/54/66654-004-99D4F14F/figures-area-height-principle-Cavalieri-sections-length.jpg" alt="Cavalieri's principle" height="50" /> </a> <a href="https://cdn.britannica.com/50/66650-004-4FF38AA7/calculus-tangent-method-line-Fermat-approach-curve.jpg" data-href="/media/1/229851/57048" class="media-overlay-link d-inline-block mr-5"> <img loading="lazy" src="https://cdn.britannica.com/50/66650-004-4FF38AA7/calculus-tangent-method-line-Fermat-approach-curve.jpg" alt="Fermat's tangent method" height="50" /> </a> <a href="https://cdn.britannica.com/53/66653-004-0B8D7195/Blaise-Pascal-points-hexagon-section-pairs-line.jpg" data-href="/media/1/229851/57051" class="media-overlay-link d-inline-block mr-5"> <img loading="lazy" src="https://cdn.britannica.com/53/66653-004-0B8D7195/Blaise-Pascal-points-hexagon-section-pairs-line.jpg" alt="Pascal's hexagon" height="50" /> </a> <a href="https://cdn.britannica.com/51/66651-004-D22D2B83/Omar-Khayyam-quadrilateral-Quadrilateral-figure-angles-lines.jpg" data-href="/media/1/229851/57049" class="media-overlay-link d-inline-block mr-5"> <img loading="lazy" src="https://cdn.britannica.com/51/66651-004-D22D2B83/Omar-Khayyam-quadrilateral-Quadrilateral-figure-angles-lines.jpg" alt="quadrilateral of Omar Khayyam" height="50" /> </a> <a href="https://cdn.britannica.com/52/66652-004-DC031B61/pseudosphere-curvature-concavity-surface-entirety-illustration-infinity.jpg" data-href="/media/1/229851/57050" class="media-overlay-link d-inline-block mr-5"> <img loading="lazy" src="https://cdn.britannica.com/52/66652-004-DC031B61/pseudosphere-curvature-concavity-surface-entirety-illustration-infinity.jpg" alt="pseudosphere" 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-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/all-about-math-quiz"> <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/all-about-math-quiz" >All About Math Quiz</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/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 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/7-famous-child-prodigies"> <img loading="lazy" src="https://cdn.britannica.com/38/154338-131-518608F9/some-Wolfgang-Amadeus-Mozart-classical-music-pieces.jpg?w=200&h=200&c=crop" alt="Wolfgang Amadeus Mozart. Mozart rehearsing his 12th Mass with singer and musician. (Austrian composer. (Johann Chrysostom Wolfgang Amadeus Mozart)" 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/7-famous-child-prodigies" >7 Famous Child Prodigies</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-noteworthy-bog-bodies-and-what-they-tell-us"> <img loading="lazy" src="https://cdn.britannica.com/68/178968-131-ADEF94E7/bog-body-Head-Tollund-Man-Human-remains-1950.jpg?w=200&h=200&c=crop" alt="Bog body. Head of Tollund Man. Died at about age 30-40, Dated to about 280 BCE early Iron Age. Found Bjaeldskovdal bog Denmark in 1950 near Elling Woman. Most well preserved bog body to date. Human remains mummified in natural peat bogs. mummy, embalm" 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-noteworthy-bog-bodies-and-what-they-tell-us" >9 Noteworthy Bog Bodies (And What They Tell Us)</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-do-black-holes-really-work"> <img loading="lazy" src="https://cdn.britannica.com/84/193284-131-66EAF092/merging-black-holes-binary-system-each-other-December-26-2015.jpg?w=200&h=200&c=crop" alt="The merger of two black holes and the gravitational waves that ripple outward as the black holes spiral toward each other. The black holes-which represent those detected by LIGO on Dec. 26, 2015-were 14 and 8 times the mass of the sun, until they merged," 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-do-black-holes-really-work" >How Do Black Holes Really Work?</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-santa-claus-a-real-person"> <img loading="lazy" src="https://cdn.britannica.com/92/202092-131-ACFF5B40/Santa-Claus-sleigh.jpg?w=200&h=200&c=crop" alt="Santa Claus flying in his sleigh, christmas, reindeer" 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-santa-claus-a-real-person" >Was Santa Claus a Real Person?</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/7-of-the-worlds-deadliest-shipwrecks"> <img loading="lazy" src="https://cdn.britannica.com/41/182141-131-32BD2459/Lusitania-sinking-U-boat-German-May-1915.jpg?w=200&h=200&c=crop" alt="Lusitania sinking, illustration from The War of the Nations (New York), December 31, 1919. World War I, WWI. This image is one of three on the page. (See source file for the other two.)" 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/7-of-the-worlds-deadliest-shipwrecks" >7 of the World’s Deadliest Shipwrecks</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/where-do-anti-semitic-conspiracy-theories-about-the-rothschild-family-come-from"> <img loading="lazy" src="https://cdn.britannica.com/67/200267-131-4F79E2AC/Mayer-Amschel-Rothschild.jpg?w=200&h=200&c=crop" alt="Mayer Amschel Rothschild (1744-1855), founder of the house of Rothschild; lithograph by Friedrich Lieder, c. 1830." 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/where-do-anti-semitic-conspiracy-theories-about-the-rothschild-family-come-from" >Where Do Anti-Semitic Conspiracy Theories About the Rothschild Family Come From?</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/13-iconic-buildings-to-visit-in-new-york-city"> <img loading="lazy" src="https://cdn.britannica.com/82/156482-131-AEEBFEFC/New-York-York.jpg?w=200&h=200&c=crop" alt="New York city skyline aerial with the Empire State Building, New York City, New York." 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/13-iconic-buildings-to-visit-in-new-york-city" >13 Iconic Buildings to Visit in New York City</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-martin-luther-king-jr-a-republican-or-a-democrat"> <img loading="lazy" src="https://cdn.britannica.com/57/203257-131-48B2BCA1/Martin-Luther-King-Jr-marchers-speech-I-August-28th-1963.jpg?w=200&h=200&c=crop" alt="Dr. Martin Luther King Jr. addresses marchers during his "I Have a Dream" speech at the Lincoln Memorial in Washington. August 28th 1963" 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-martin-luther-king-jr-a-republican-or-a-democrat" >Was Martin Luther King, Jr., a Republican or a Democrat?</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="false"> <script class="page-description-json" type="application/json"> { "url": "/science/geometry", "shareUrl": "https://www.britannica.com/science/geometry", "browserTitle": "Geometry | Definition, History, Basics, Branches, & Facts", "firstTopicPage": true, "topicId":229851 } </script> <div class="reading-channel"> <div class="desktop-header-image module-spacing"> <figure class="md-assembly m-0 mb-20 mb-md-0 card card-borderless print-false" data-assembly-id="13684" data-asm-type="image"> <div class="md-assembly-wrapper card-media " data-type="image"> <a style="min-height: 160px;" 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/1/229851/57444"> <picture> <source media="(min-width: 680px)" srcset="https://cdn.britannica.com/61/67361-050-F48C1CC9/map-millennium-mathematicians-Greco-Roman-Thales-Hypatia-Alexandria.jpg?w=300"> <img src="https://cdn.britannica.com/61/67361-050-F48C1CC9/map-millennium-mathematicians-Greco-Roman-Thales-Hypatia-Alexandria.jpg?w=400&h=300&c=crop" alt="mathematicians of the Greco-Roman world" 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/1/229851/57444">mathematicians of the Greco-Roman world</a> <span><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></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 class="topic-header"> <div class="d-flex align-items-top justify-content-between"> <div class="d-flex flex-column"> <div> <div> <h1>geometry</h1></div> </div> <div class="topic-identifier font-16 font-md-20">mathematics</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-none d-sm-block md-topic-tools qa-action-buttons mb-15" data-topic-id="229851"> <button class="js-tooltip btn btn-sm btn-outline-blue border pr-10 border-2" > <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=229851] .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/229851" 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=229851] .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="J.L. Heilbron" data-title="geometry" data-published-date="19 Nov. 2024" data-url="https://www.britannica.com/science/geometry" > <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=229851] .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="geometry" data-url="https://www.britannica.com/science/geometry" > <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/geometry">https://www.britannica.com/science/geometry</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=229851] .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/229851"> <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://digitalcommons.pittstate.edu/cgi/viewcontent.cgi?article=1496&context=etd" target="_blank" rel="noopener ">Pittsburg State University - Digital Commons - Guide to Geometry</a></li> <li><a class="external" href="https://plato.stanford.edu/entries/epistemology-geometry/" target="_blank" rel="noopener ">Stanford Encyclopedia of Philosophy - Epistemology of Geometry</a></li> <li><a class="external" href="https://www.khanacademy.org/math/geometry/hs-geo-transformations/hs-geo-intro-euclid/v/language-and-notation-of-basic-geometry" target="_blank" rel="noopener ">Khan Academy - Terms and labels in geometry</a></li> <li><a class="external" href="https://www.storyofmathematics.com/geometry/" target="_blank" rel="noopener ">Story of Mathematics - Geometry</a></li> <li><a class="external" href="https://www.mathsisfun.com/geometry/index.html" target="_blank" rel="noopener nofollow">Maths Is Fun - Geometry</a></li> <li><a class="external" href="https://www.gresham.ac.uk/watch-now/4000-years-geometry" target="_blank" rel="noopener nofollow">Gresham College - 4000 Years of Geometry</a></li> <li><a class="external" href="https://fhsu.pressbooks.pub/ecumath/chapter/chapter-14-geometric-measurement-concepts/" target="_blank" rel="noopener ">FHSU Digital Press - Mathematics Methods for Early Childhood - Geometry and Measurement</a></li> <li><a class="external" href="https://uwaterloo.ca/pure-mathematics/about-pure-math/what-is-pure-math/what-is-geometry" target="_blank" rel="noopener ">University of Waterloo - What is Geometry?</a></li> <li><a class="external" href="https://math.libretexts.org/Courses/Fullerton_College/Math_100%3A_Liberal_Arts_Math_(Claassen_and_Ikeda)/02%3A_Geometry" target="_blank" rel="noopener ">Mathematics LibreTexts - Geometry</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/geometry/353174" target="_blank" rel="noopener">geometry - Children's Encyclopedia (Ages 8-11)</a></li> <li><a class="external" href="https://kids.britannica.com/students/article/geometry/274526" target="_blank" rel="noopener">geometry - 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="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 gtm-byline font-12 byline-contributor" href="/contributor/JL-Heilbron/4434" > <div class="editor-title font-16 font-weight-bold">J.L. Heilbron</div> <div class="editor-description font-12 font-serif mt-5 clamp-description text-black">Senior Research Fellow at the University of Oxford, England. Author of <i>Geometry Civilized</i> and <i>The Sun in the Church</i> among 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"> J.L. Heilbron</span></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 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="2024-11-19T00:00:00CST" >Nov 19, 2024</time> •</span> <a class="byline-edit-history" href="https://www.britannica.com/science/geometry/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/Blaise-Pascal" topicid="445406">Blaise Pascal</a></dd> <dd><a href="/biography/Euclid-Greek-mathematician" topicid="194880">Euclid</a></dd> <dd><a href="/biography/Ptolemy" topicid="482098">Ptolemy</a></dd> <dd><a href="/biography/Pierre-de-Fermat" topicid="204668">Pierre de Fermat</a></dd> <dd><a href="/biography/Bernhard-Riemann" topicid="503201">Bernhard Riemann</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/topology" topicid="599686">topology</a></dd> <dd><a href="/science/Euclidean-geometry" topicid="194901">Euclidean geometry</a></dd> <dd><a href="/science/analytic-geometry" topicid="22548">analytic geometry</a></dd> <dd><a href="/science/differential-geometry" topicid="162938">differential geometry</a></dd> <dd><a href="/science/projective-geometry" topicid="478486">projective geometry</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>On the Web: </dt> <dd><a href="https://digitalcommons.pittstate.edu/cgi/viewcontent.cgi?article=1496&context=etd" target="_blank">Pittsburg State University - Digital Commons - Guide to Geometry</a> (Nov. 19, 2024)</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/geometry"> See all related content </a> </div> </div> </div> </div><span class="marker before-article"></span><section data-level="1" id="ref1"><span class="marker PREMOD1 mod-inline"></span><p class="topic-paragraph"><strong><span id="ref726375"></span>geometry</strong>, the branch of <a href="https://www.britannica.com/science/mathematics" class="md-crosslink " data-show-preview="true">mathematics</a> concerned with the shape of individual objects, <a class="md-dictionary-link md-dictionary-tt-off eb" data-term="spatial" href="https://www.britannica.com/dictionary/spatial" data-type="EB">spatial</a> relationships among various objects, and the properties of surrounding <a href="https://www.britannica.com/science/space-physics-and-metaphysics" class="md-crosslink autoxref " data-show-preview="true">space</a>. It is one of the oldest branches of mathematics, having arisen in response to such practical problems as those found in <a href="https://www.britannica.com/technology/surveying" class="md-crosslink autoxref " data-show-preview="true">surveying</a>, and its name is derived from <a href="https://www.britannica.com/topic/Greek-language" class="md-crosslink " data-show-preview="true">Greek</a> words meaning “Earth measurement.” Eventually it was realized that geometry need not be limited to the study of flat surfaces (plane geometry) and rigid three-dimensional objects (solid geometry) but that even the most abstract thoughts and images might be represented and developed in geometric terms.</p><span class="marker MOD1 mod-inline"></span><span class="marker PREMOD2 mod-inline"></span><p class="topic-paragraph">This article begins with a brief guidepost to the major branches of geometry and then proceeds to an extensive historical treatment. For information on specific branches of geometry, <em>see</em> <a href="https://www.britannica.com/science/Euclidean-geometry" class="md-crosslink " data-show-preview="true">Euclidean geometry</a>, <a href="https://www.britannica.com/science/analytic-geometry" class="md-crosslink " data-show-preview="true">analytic geometry</a>, <a href="https://www.britannica.com/science/projective-geometry" class="md-crosslink " data-show-preview="true">projective geometry</a>, <a href="https://www.britannica.com/science/differential-geometry" class="md-crosslink " data-show-preview="true">differential geometry</a>, <a href="https://www.britannica.com/science/non-Euclidean-geometry" class="md-crosslink " data-show-preview="true">non-Euclidean geometries</a>, and <a href="https://www.britannica.com/science/topology" class="md-crosslink " data-show-preview="true">topology</a>.</p><span class="marker MOD2 mod-inline"></span></section> <span class="marker h2"></span><section data-level="1" id="ref217464"> <h2 class="h1">Major branches of geometry</h2> <section data-level="2" id="ref217465"> <h2 class="h2"><span id="ref726376"></span><a href="https://www.britannica.com/science/Euclidean-geometry" class="md-crosslink " data-show-preview="true">Euclidean geometry</a></h2> <span class="marker PREMOD3 mod-inline"></span><p class="topic-paragraph">In several ancient <a class="md-dictionary-link md-dictionary-tt-off mw" data-term="cultures" href="https://www.merriam-webster.com/dictionary/cultures" data-type="MW">cultures</a> there developed a form of geometry suited to the relationships between lengths, areas, and volumes of physical objects. This geometry was codified in Euclid’s <em>Elements</em> about 300 <span class="text-smallcaps">bce</span> on the basis of 10 axioms, or postulates, from which several hundred theorems were proved by deductive logic. The <em>Elements</em> epitomized the axiomatic-deductive method for many centuries.</p><span class="marker MOD3 mod-inline"></span> </section> <section data-level="2" id="ref217466"> <h2 class="h2"><span id="ref726377"></span><a href="https://www.britannica.com/science/analytic-geometry" class="md-crosslink " data-show-preview="true">Analytic geometry</a></h2> <span class="marker PREMOD4 mod-inline"></span><p class="topic-paragraph"><a class="md-dictionary-link md-dictionary-tt-off mw" data-term="Analytic" href="https://www.merriam-webster.com/dictionary/Analytic" data-type="MW">Analytic</a> geometry was initiated by the French mathematician <a href="https://www.britannica.com/biography/Rene-Descartes" class="md-crosslink " data-show-preview="true">René Descartes</a> (1596–1650), who introduced rectangular <a href="https://www.britannica.com/science/coordinate-system" class="md-crosslink autoxref " data-show-preview="true">coordinates</a> to locate points and to enable lines and curves to be represented with algebraic equations. Algebraic geometry is a modern extension of the subject to multidimensional and non-Euclidean spaces.</p><span class="marker MOD4 mod-inline"></span> </section> <section data-level="2" id="ref217467"> <h2 class="h2"><span id="ref726378"></span><a href="https://www.britannica.com/science/projective-geometry" class="md-crosslink " data-show-preview="true">Projective geometry</a></h2> <span class="marker PREMOD5 mod-inline"></span><p class="topic-paragraph">Projective geometry originated with the French mathematician <a href="https://www.britannica.com/biography/Girard-Desargues" class="md-crosslink " data-show-preview="true">Girard Desargues</a> (1591–1661) to deal with those properties of geometric figures that are not altered by projecting their image, or “shadow,” onto another <a href="https://www.britannica.com/science/surface-geometry" class="md-crosslink autoxref " data-show-preview="true">surface</a>.</p><a class="link-module shadow-sm d-block qa-quiz-module" href="/quiz/all-about-math-quiz" data-link-module-iframe-link=""> <img loading="lazy" src="https://cdn.britannica.com/86/94086-131-0BAE374D/Equations-blackboard.jpg" alt="Equations written on blackboard" class="rounded-sm mr-15" width="70" /> <div class="line-clamp clamp-5"> <div class="module-title bg-green">Britannica Quiz</div> <div class="font-weight-semi-bold mt-5">All About Math Quiz</div> </div> </a><span class="marker MOD5 mod-inline"></span> </section> <section data-level="2" id="ref217468"> <h2 class="h2"><span id="ref726379"></span><a href="https://www.britannica.com/science/differential-geometry" class="md-crosslink " data-show-preview="true">Differential geometry</a></h2> <span class="marker PREMOD6 mod-inline"></span><p class="topic-paragraph">The German mathematician <a href="https://www.britannica.com/biography/Carl-Friedrich-Gauss" class="md-crosslink " data-show-preview="true">Carl Friedrich Gauss</a> (1777–1855), in connection with practical problems of surveying and geodesy, initiated the field of differential geometry. Using <a href="https://www.britannica.com/science/differential-calculus" class="md-crosslink autoxref " data-show-preview="true">differential calculus</a>, he characterized the <a class="md-dictionary-link md-dictionary-tt-off mw" data-term="intrinsic" href="https://www.merriam-webster.com/dictionary/intrinsic" data-type="MW">intrinsic</a> properties of curves and surfaces. For instance, he showed that the intrinsic <a href="https://www.britannica.com/science/curvature" class="md-crosslink autoxref " data-show-preview="true">curvature</a> of a <a href="https://www.britannica.com/science/cylinder-mathematics" class="md-crosslink autoxref " data-show-preview="true">cylinder</a> is the same as that of a plane, as can be seen by cutting a cylinder along its axis and flattening, but not the same as that of a <a href="https://www.britannica.com/science/sphere" class="md-crosslink autoxref " data-show-preview="true">sphere</a>, which cannot be flattened without distortion.</p><span class="marker MOD6 mod-inline"></span> </section> <section data-level="2" id="ref217469"> <h2 class="h2"><span id="ref726380"></span><a href="https://www.britannica.com/science/non-Euclidean-geometry" class="md-crosslink " data-show-preview="true">Non-Euclidean geometries</a></h2> <span class="marker PREMOD7 mod-inline"></span><p class="topic-paragraph">Beginning in the 19th century, various mathematicians substituted <a class="md-dictionary-link md-dictionary-tt-off mw" data-term="alternatives" href="https://www.merriam-webster.com/dictionary/alternatives" data-type="MW">alternatives</a> to Euclid’s <a href="https://www.britannica.com/science/parallel-postulate" class="md-crosslink autoxref " data-show-preview="true">parallel postulate</a>, which, in its modern form, reads, “given a <a href="https://www.britannica.com/science/line-mathematics" class="md-crosslink autoxref " data-show-preview="true">line</a> and a point not on the line, it is possible to draw exactly one line through the given point parallel to the line.” They hoped to show that the alternatives were logically impossible. Instead, they discovered that consistent non-Euclidean geometries exist.</p><div class="module-spacing"> <DIV class="marketing-INLINE_SUBSCRIPTION marketing-content" data-marketing-id="INLINE_SUBSCRIPTION"><style> .student-promo-banner-wrapper { container-type: inline-size; margin-bottom: 15px; } @container (min-width: 475px) { .student-promo-banner { flex-direction: row; } .student-promo-banner-img-wrapper { margin-bottom: 0; margin-right: 10px; justify-content: flex-start; } .student-promo-banner-text-wrapper { text-align: left; margin-bottom: 0px; margin-left: 10px; } .student-promo-banner-button-wrapper { margin-right: 0; } }</style> <div class="student-promo-banner-wrapper"> <div class="student-promo-banner d-flex flex-column align-items-center bg-blue rounded p-20"> <div class="student-promo-banner-img-wrapper mb-20 mr-0 d-flex justify-content-center"> <img class="rounded" style="max-width: 100px; min-width: 80px" src="https://cdn.britannica.com/marketing/BlueThistle.webp" /> </div> <div class="student-promo-banner-text-wrapper ml-0 mb-10 text-center text-white"> <div class="h2 mb-10">Get Unlimited Access</div> <div class="h4 font-weight-semi-bold">Try Britannica Premium for free and discover more.</div> </div> <div class="student-promo-banner-button-wrapper d-flex justify-content-center align-items-center ml-auto mr-auto"> <a class="btn btn-m btn-orange" href="https://premium.britannica.com/premium-membership/?utm_source=premium&utm_medium=inline-cta&utm_campaign=black-friday-2024">Subscribe</a> </div> </div> </div> </DIV></div><span class="marker MOD7 mod-inline"></span> </section> <section data-level="2" id="ref217470"> <h2 class="h2">Topology</h2> <span class="marker PREMOD8 mod-inline"></span><p class="topic-paragraph">Topology, the youngest and most sophisticated branch of geometry, focuses on the properties of geometric objects that remain unchanged upon continuous deformation—shrinking, stretching, and folding, but not tearing. The continuous development of topology dates from 1911, when the Dutch mathematician <a href="https://www.britannica.com/biography/Luitzen-Egbertus-Jan-Brouwer" class="md-crosslink " data-show-preview="true">L.E.J. Brouwer</a> (1881–1966) introduced methods generally applicable to the topic.</p><span class="marker MOD8 mod-inline"></span> </section> </section> <span class="marker h3"></span><section data-level="1" id="ref217471"> <h2 class="h1">History of geometry</h2> <span class="marker PREMOD9 mod-inline"></span><p class="topic-paragraph">The earliest known unambiguous examples of written records—dating from Egypt and Mesopotamia about 3100 <span class="text-smallcaps">bce</span>—demonstrate that ancient peoples had already begun to devise mathematical rules and techniques useful for surveying land areas, constructing buildings, and measuring storage containers. Beginning about the 6th century <span class="text-smallcaps">bce</span>, the Greeks gathered and extended this practical knowledge and from it generalized the abstract subject now known as geometry, from the combination of the Greek words <em>geo</em> (“Earth”) and <em>metron</em> (“measure”) for the measurement of the Earth.</p><span class="marker MOD9 mod-inline"></span> <span class="marker PREMOD10 mod-inline"></span><p class="topic-paragraph">In addition to describing some of the achievements of the ancient Greeks, notably Euclid’s logical development of geometry in the <em>Elements</em>, this article examines some applications of geometry to <a href="https://www.britannica.com/science/astronomy" class="md-crosslink autoxref " data-show-preview="true">astronomy</a>, cartography, and painting from classical Greece through <a class="md-dictionary-link md-dictionary-tt-off mw" data-term="medieval" href="https://www.merriam-webster.com/dictionary/medieval" data-type="MW">medieval</a> Islam and Renaissance Europe. It concludes with a brief discussion of extensions to non-Euclidean and multidimensional geometries in the modern age.</p><span class="marker MOD10 mod-inline"></span> <section data-level="2" id="ref217472"> <h2 class="h2">Ancient geometry: practical and empirical</h2> <span class="marker PREMOD11 mod-inline"></span><p class="topic-paragraph">The origin of geometry lies in the concerns of everyday life. The traditional account, preserved in Herodotus’s <em>History</em> (5th century <span class="text-smallcaps">bce</span>), credits the Egyptians with inventing surveying in order to reestablish property values after the annual flood of the Nile. Similarly, eagerness to know the volumes of solid figures derived from the need to evaluate tribute, store oil and grain, and build dams and pyramids. Even the three <a class="md-dictionary-link md-dictionary-tt-off mw" data-term="abstruse" href="https://www.merriam-webster.com/dictionary/abstruse" data-type="MW">abstruse</a> geometrical problems of ancient times—to double a <a href="https://www.britannica.com/science/cube-mathematics" class="md-crosslink autoxref " data-show-preview="true">cube</a>, trisect an angle, and <a href="https://www.britannica.com/science/square-mathematics" class="md-crosslink autoxref " data-show-preview="true">square</a> a <a href="https://www.britannica.com/science/circle-mathematics" class="md-crosslink autoxref " data-show-preview="true">circle</a>, all of which will be discussed later—probably arose from practical matters, from religious ritual, timekeeping, and <a href="https://www.britannica.com/technology/construction" class="md-crosslink autoxref " data-show-preview="true">construction</a>, respectively, in pre-Greek societies of the Mediterranean. And the main subject of later Greek geometry, the theory of <a href="https://www.britannica.com/science/conic-section" class="md-crosslink " data-show-preview="true">conic sections</a>, owed its general importance, and perhaps also its origin, to its application to optics and astronomy.</p><span class="marker MOD11 mod-inline"></span> <span class="marker PREMOD12 mod-inline"></span><p class="topic-paragraph">While many ancient individuals, known and unknown, contributed to the subject, none equaled the impact of Euclid and his <em><span id="ref726398"></span><a href="https://www.britannica.com/topic/Elements-by-Euclid" class="md-crosslink " data-show-preview="true">Elements</a></em> of geometry, a book now 2,300 years old and the object of as much painful and <a class="md-dictionary-link md-dictionary-tt-off eb" data-term="painstaking" href="https://www.britannica.com/dictionary/painstaking" data-type="EB">painstaking</a> study as the Bible. Much less is known about <span id="ref726397"></span><a href="https://www.britannica.com/biography/Euclid-Greek-mathematician" class="md-crosslink " data-show-preview="true">Euclid</a>, however, than about Moses. In fact, the only thing known with a fair degree of confidence is that Euclid taught at the <a href="https://www.britannica.com/topic/Library-of-Alexandria" class="md-crosslink " data-show-preview="true">Library of Alexandria</a> during the reign of <a href="https://www.britannica.com/biography/Ptolemy" class="md-crosslink autoxref " data-show-preview="true">Ptolemy</a> I (323–285/283 <span class="text-smallcaps">bce</span>). Euclid wrote not only on geometry but also on astronomy and optics and perhaps also on mechanics and music. Only the <em>Elements</em>, which was extensively copied and translated, has survived intact.</p><span class="marker MOD12 mod-inline"></span> <span class="marker PREMOD13 mod-inline"></span><p class="topic-paragraph">Euclid’s <em>Elements</em> was so complete and clearly written that it literally obliterated the work of his predecessors. What is known about Greek geometry before him comes primarily from bits quoted by Plato and Aristotle and by later mathematicians and commentators. Among other <a class="md-dictionary-link md-dictionary-tt-off mw" data-term="precious" href="https://www.merriam-webster.com/dictionary/precious" data-type="MW">precious</a> items they preserved are some results and the general approach of <a href="https://www.britannica.com/biography/Pythagoras" class="md-crosslink " data-show-preview="true">Pythagoras</a> (<em>c.</em> 580–<em>c.</em> 500 <span class="text-smallcaps">bce</span>) and his followers. The <span id="ref726399"></span><a href="https://www.britannica.com/science/Pythagoreanism" class="md-crosslink " data-show-preview="true">Pythagoreans</a> convinced themselves that all things are, or owe their relationships to, numbers. The doctrine gave mathematics supreme importance in the investigation and understanding of the world. Plato developed a similar view, and philosophers influenced by Pythagoras or Plato often wrote ecstatically about geometry as the key to the interpretation of the <a href="https://www.britannica.com/science/universe" class="md-crosslink autoxref " data-show-preview="true">universe</a>. Thus ancient geometry gained an association with the <a class="md-dictionary-link md-dictionary-tt-off mw" data-term="sublime" href="https://www.merriam-webster.com/dictionary/sublime" data-type="MW">sublime</a> to complement its earthy origins and its reputation as the exemplar of precise reasoning.</p><span class="marker MOD13 mod-inline"></span> <section data-level="3" id="ref217473"> <h2 class="h3">Finding the <span id="ref726435"></span>right angle</h2> <span class="marker PREMOD14 mod-inline"></span><p class="topic-paragraph">Ancient builders and surveyors needed to be able to construct right angles in the field on demand. The method employed by the Egyptians earned them the name “rope pullers” in Greece, apparently because they employed a <span id="ref726436"></span><a href="https://www.britannica.com/technology/rope" class="md-crosslink " data-show-preview="true">rope</a> for laying out their construction guidelines. One way that they could have employed a rope to construct right triangles was to mark a looped rope with knots so that, when held at the knots and pulled tight, the rope must form a right <a href="https://www.britannica.com/science/triangle-mathematics" class="md-crosslink autoxref " data-show-preview="true">triangle</a>. The simplest way to perform the trick is to take a rope that is 12 units long, make a knot 3 units from one end and another 5 units from the other end, and then knot the ends together to form a loop. However, the Egyptian <a class="md-dictionary-link md-dictionary-tt-off eb" data-term="scribes" href="https://www.britannica.com/dictionary/scribes" data-type="EB">scribes</a> have not left us instructions about these procedures, much less any hint that they knew how to generalize them to obtain the <a href="https://www.britannica.com/topic/Euclids-Windmill-1688351" class="md-crosslink " data-show-preview="true">Pythagorean theorem</a>: the square on the line opposite the right angle equals the sum of the squares on the other two sides. Similarly, the Vedic scriptures of ancient India contain sections called <em>sulvasutra</em>s, or “rules of the rope,” for the exact positioning of sacrificial altars. The required right angles were made by ropes marked to give the triads (3, 4, 5) and (5, 12, 13).</p><span class="marker MOD14 mod-inline"></span> <span class="marker PREMOD15 mod-inline"></span><p class="topic-paragraph">In Babylonian clay tablets (<em>c.</em> 1700–1500 <span class="text-smallcaps">bce</span>) modern historians have discovered problems whose solutions indicate that the Pythagorean theorem and some special <a class="md-dictionary-link md-dictionary-tt-off eb" data-term="triads" href="https://www.britannica.com/dictionary/triads" data-type="EB">triads</a> were known more than a thousand years before Euclid. A right triangle made at random, however, is very unlikely to have all its sides measurable by the same unit—that is, every side a whole-number multiple of some common unit of measurement. This fact, which came as a shock when discovered by the Pythagoreans, gave rise to the concept and theory of <a href="https://www.britannica.com/topic/Incommensurables-1688515" class="md-crosslink " data-show-preview="true">incommensurability</a>.</p><span class="marker MOD15 mod-inline"></span> </section> <section data-level="3" id="ref217474"> <h2 class="h3">Locating the inaccessible</h2> <span class="marker PREMOD16 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="57039" data-asm-type="image"><div class="md-assembly-wrapper card-media " data-type="image"><a href="https://cdn.britannica.com/41/66641-004-0A9B7581/comparison-Greek-Chinese-theorem-figure-equivalence-triangles.jpg" class="gtm-assembly-link position-relative d-flex align-items-center justify-content-center media-overlay-link card-media" data-href="/media/1/229851/57039"><picture><source media="(min-width: 680px)" srcset="https://cdn.britannica.com/41/66641-004-0A9B7581/comparison-Greek-Chinese-theorem-figure-equivalence-triangles.jpg"><img src="https://cdn.britannica.com/41/66641-004-0A9B7581/comparison-Greek-Chinese-theorem-figure-equivalence-triangles.jpg?w=300" alt="Chinese and Greek geometric theorems" data-width="360" data-height="386" 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/41/66641-004-0A9B7581/comparison-Greek-Chinese-theorem-figure-equivalence-triangles.jpg" data-href="/media/1/229851/57039">Chinese and Greek geometric theorems</a><span>A comparison of a Chinese and a Greek geometric theoremThe figure illustrates the equivalence of the Chinese complementary rectangles theorem and the Greek similar triangles theorem.</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">By ancient tradition, <span id="ref726455"></span><a href="https://www.britannica.com/biography/Thales-of-Miletus" class="md-crosslink " data-show-preview="true">Thales of Miletus</a>, who lived before Pythagoras in the 6th century <span class="text-smallcaps">bce</span>, invented a way to measure inaccessible heights, such as the Egyptian pyramids. Although none of his writings survives, Thales may well have known about a Babylonian observation that for similar triangles (triangles having the same shape but not necessarily the same size) the length of each corresponding side is increased (or decreased) by the same multiple. The ancient Chinese arrived at measures of inaccessible heights and distances by another route, using “complementary” rectangles, as seen in the next <span class="link-blue media-overlay-link asmref" data-href="/media/1/229851/57039">figure</span>, which can be shown to give results equivalent to those of the Greek method involving triangles.</p><span class="marker MOD16 mod-inline"></span> </section> <section data-level="3" id="ref217475"> <h2 class="h3">Estimating the wealth</h2> <span class="marker PREMOD17 mod-inline"></span><p class="topic-paragraph">A Babylonian cuneiform tablet written some 3,500 years ago treats problems about <a class="md-dictionary-link md-dictionary-tt-off eb" data-term="dams" href="https://www.britannica.com/dictionary/dams" data-type="EB">dams</a>, wells, water clocks, and excavations. It also has an exercise on circular enclosures with an implied value of π = 3. The contractor for King Solomon’s swimming pool, who made a pond 10 cubits across and 30 cubits around (1 Kings 7:23), used the same value. However, the Hebrews should have taken their π from the Egyptians before crossing the <a href="https://www.britannica.com/place/Red-Sea" class="md-crosslink autoxref " data-show-preview="true">Red Sea</a>, for the <span id="ref726456"></span><a href="https://www.britannica.com/topic/Rhind-papyrus" class="md-crosslink " data-show-preview="true">Rhind papyrus</a> (<em>c.</em> 2000 <span class="text-smallcaps">bce</span>; our principal source for ancient Egyptian mathematics) implies π = 3.1605.</p><div class="one-good-fact-module"> </div><span class="marker MOD17 mod-inline"></span> <span class="marker PREMOD18 mod-inline"></span><p class="topic-paragraph">Knowledge of the area of a circle was of practical value to the officials who kept track of the pharaoh’s tribute as well as to the builders of altars and swimming pools. <span id="ref726457"></span><a href="https://www.britannica.com/biography/Ahmes" class="md-crosslink ">Ahmes</a>, the scribe who copied and <a class="md-dictionary-link md-dictionary-tt-off mw" data-term="annotated" href="https://www.merriam-webster.com/dictionary/annotated" data-type="MW">annotated</a> the Rhind papyrus (<em>c.</em> 1650 <span class="text-smallcaps">bce</span>), has much to say about cylindrical granaries and pyramids, whole and truncated. He could calculate their volumes, and, as appears from his taking the Egyptian <em>seked</em>, the horizontal distance associated with a vertical rise of one cubit, as the defining quantity for the pyramid’s slope, he knew something about similar triangles.</p><span class="marker MOD18 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":1,"pagesTotal":10,"pageId":229851,"pageLength":1629,"initialLoad":true,"lastPageOfScroll":false} </script> <script class="marketing-content-info" type="application/json"> [] </script> <script src="https://cdn.britannica.com/mendel-resources/3-130/js/libs/jquery-3.5.0.min.js?v=3.130.14"></script> <script type="text/javascript" data-type="Init Mendel Code Splitting"> (function() { $.ajax({ dataType: 'script', cache: true, url: 'https://cdn.britannica.com/mendel-resources/3-130/dist/topic-page.js?v=3.130.14' }); })(); </script> <script class="analytics-metadata" type="application/json"> {"leg":"A","adLeg":"A","userType":"ANONYMOUS","pageType":"Topic","pageSubtype":null,"articleTemplateType":"PAGINATED","gisted":false,"pageNumber":1,"hasSummarizeButton":false,"hasAskButton":false} </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

Geometry | Definition, History, Basics, Branches, & Facts | Britannica