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Secant line - Wikipedia
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Available in 34 languages" > <label id="p-lang-btn-label" for="p-lang-btn-checkbox" class="vector-dropdown-label cdx-button cdx-button--fake-button cdx-button--fake-button--enabled cdx-button--weight-quiet cdx-button--action-progressive mw-portlet-lang-heading-34" aria-hidden="true" ><span class="vector-icon mw-ui-icon-language-progressive mw-ui-icon-wikimedia-language-progressive"></span> <span class="vector-dropdown-label-text">34 languages</span> </label> <div class="vector-dropdown-content"> <div class="vector-menu-content"> <ul class="vector-menu-content-list"> <li class="interlanguage-link interwiki-af mw-list-item"><a href="https://af.wikipedia.org/wiki/Snylyn" title="Snylyn – Afrikaans" lang="af" hreflang="af" data-title="Snylyn" data-language-autonym="Afrikaans" data-language-local-name="Afrikaans" class="interlanguage-link-target"><span>Afrikaans</span></a></li><li class="interlanguage-link interwiki-ar mw-list-item"><a href="https://ar.wikipedia.org/wiki/%D9%82%D8%A7%D8%B7%D8%B9_(%D9%85%D9%86%D8%AD%D9%86%D9%89)" title="قاطع (منحنى) – Arabic" lang="ar" hreflang="ar" data-title="قاطع (منحنى)" data-language-autonym="العربية" data-language-local-name="Arabic" class="interlanguage-link-target"><span>العربية</span></a></li><li class="interlanguage-link interwiki-ca mw-list-item"><a href="https://ca.wikipedia.org/wiki/Recta_secant" title="Recta secant – Catalan" lang="ca" hreflang="ca" data-title="Recta secant" data-language-autonym="Català" data-language-local-name="Catalan" class="interlanguage-link-target"><span>Català</span></a></li><li class="interlanguage-link interwiki-cv mw-list-item"><a href="https://cv.wikipedia.org/wiki/%D0%A3%D1%80%D0%BB%D0%B0%D0%B2_(%D0%B3%D0%B5%D0%BE%D0%BC%D0%B5%D1%82%D1%80%D0%B8)" title="Урлав (геометри) – Chuvash" lang="cv" hreflang="cv" data-title="Урлав (геометри)" data-language-autonym="Чӑвашла" data-language-local-name="Chuvash" class="interlanguage-link-target"><span>Чӑвашла</span></a></li><li class="interlanguage-link interwiki-cs mw-list-item"><a href="https://cs.wikipedia.org/wiki/Se%C4%8Dna" title="Sečna – Czech" lang="cs" hreflang="cs" data-title="Sečna" data-language-autonym="Čeština" data-language-local-name="Czech" class="interlanguage-link-target"><span>Čeština</span></a></li><li class="interlanguage-link interwiki-da mw-list-item"><a href="https://da.wikipedia.org/wiki/Sekant" title="Sekant – Danish" lang="da" hreflang="da" data-title="Sekant" data-language-autonym="Dansk" data-language-local-name="Danish" class="interlanguage-link-target"><span>Dansk</span></a></li><li class="interlanguage-link interwiki-de mw-list-item"><a href="https://de.wikipedia.org/wiki/Sekante" title="Sekante – German" lang="de" hreflang="de" data-title="Sekante" data-language-autonym="Deutsch" data-language-local-name="German" class="interlanguage-link-target"><span>Deutsch</span></a></li><li class="interlanguage-link interwiki-es mw-list-item"><a href="https://es.wikipedia.org/wiki/Recta_secante" title="Recta secante – Spanish" lang="es" hreflang="es" data-title="Recta secante" data-language-autonym="Español" data-language-local-name="Spanish" class="interlanguage-link-target"><span>Español</span></a></li><li class="interlanguage-link interwiki-eu mw-list-item"><a href="https://eu.wikipedia.org/wiki/Zuzen_ebakitzaile" title="Zuzen ebakitzaile – Basque" lang="eu" hreflang="eu" data-title="Zuzen ebakitzaile" data-language-autonym="Euskara" data-language-local-name="Basque" class="interlanguage-link-target"><span>Euskara</span></a></li><li class="interlanguage-link interwiki-fa mw-list-item"><a href="https://fa.wikipedia.org/wiki/%D8%AE%D8%B7_%D8%B3%DA%A9%D8%A7%D9%86%D8%AA" title="خط سکانت – Persian" lang="fa" hreflang="fa" data-title="خط سکانت" data-language-autonym="فارسی" data-language-local-name="Persian" class="interlanguage-link-target"><span>فارسی</span></a></li><li class="interlanguage-link interwiki-fr mw-list-item"><a href="https://fr.wikipedia.org/wiki/Droite_s%C3%A9cante" title="Droite sécante – French" lang="fr" hreflang="fr" data-title="Droite sécante" data-language-autonym="Français" data-language-local-name="French" class="interlanguage-link-target"><span>Français</span></a></li><li class="interlanguage-link interwiki-ko mw-list-item"><a href="https://ko.wikipedia.org/wiki/%ED%95%A0%EC%84%A0" title="할선 – Korean" lang="ko" hreflang="ko" data-title="할선" data-language-autonym="한국어" data-language-local-name="Korean" class="interlanguage-link-target"><span>한국어</span></a></li><li class="interlanguage-link interwiki-hi mw-list-item"><a href="https://hi.wikipedia.org/wiki/%E0%A4%9B%E0%A5%87%E0%A4%A6%E0%A4%95_%E0%A4%B0%E0%A5%87%E0%A4%96%E0%A4%BE" title="छेदक रेखा – Hindi" lang="hi" hreflang="hi" data-title="छेदक रेखा" data-language-autonym="हिन्दी" data-language-local-name="Hindi" class="interlanguage-link-target"><span>हिन्दी</span></a></li><li class="interlanguage-link interwiki-id mw-list-item"><a href="https://id.wikipedia.org/wiki/Garis_potong_(geometri)" title="Garis potong (geometri) – Indonesian" lang="id" hreflang="id" data-title="Garis potong (geometri)" data-language-autonym="Bahasa Indonesia" data-language-local-name="Indonesian" class="interlanguage-link-target"><span>Bahasa Indonesia</span></a></li><li class="interlanguage-link interwiki-is mw-list-item"><a href="https://is.wikipedia.org/wiki/Sni%C3%B0ill" title="Sniðill – Icelandic" lang="is" hreflang="is" data-title="Sniðill" data-language-autonym="Íslenska" data-language-local-name="Icelandic" class="interlanguage-link-target"><span>Íslenska</span></a></li><li class="interlanguage-link interwiki-it mw-list-item"><a href="https://it.wikipedia.org/wiki/Secante_(geometria)" title="Secante (geometria) – Italian" lang="it" hreflang="it" data-title="Secante (geometria)" data-language-autonym="Italiano" data-language-local-name="Italian" class="interlanguage-link-target"><span>Italiano</span></a></li><li class="interlanguage-link interwiki-lv mw-list-item"><a href="https://lv.wikipedia.org/wiki/Sekante" title="Sekante – Latvian" lang="lv" hreflang="lv" data-title="Sekante" data-language-autonym="Latviešu" data-language-local-name="Latvian" class="interlanguage-link-target"><span>Latviešu</span></a></li><li class="interlanguage-link interwiki-mk mw-list-item"><a href="https://mk.wikipedia.org/wiki/%D0%A1%D0%B5%D0%BA%D0%B0%D0%BD%D1%82%D0%B0" title="Секанта – Macedonian" lang="mk" hreflang="mk" data-title="Секанта" data-language-autonym="Македонски" data-language-local-name="Macedonian" class="interlanguage-link-target"><span>Македонски</span></a></li><li class="interlanguage-link interwiki-mr mw-list-item"><a href="https://mr.wikipedia.org/wiki/%E0%A4%AD%E0%A5%87%E0%A4%A6%E0%A4%BF%E0%A4%95%E0%A4%BE_(%E0%A4%B0%E0%A5%87%E0%A4%B7%E0%A4%BE)" title="भेदिका (रेषा) – Marathi" lang="mr" hreflang="mr" data-title="भेदिका (रेषा)" data-language-autonym="मराठी" data-language-local-name="Marathi" class="interlanguage-link-target"><span>मराठी</span></a></li><li class="interlanguage-link interwiki-nl mw-list-item"><a href="https://nl.wikipedia.org/wiki/Snijlijn" title="Snijlijn – Dutch" lang="nl" hreflang="nl" data-title="Snijlijn" data-language-autonym="Nederlands" data-language-local-name="Dutch" class="interlanguage-link-target"><span>Nederlands</span></a></li><li class="interlanguage-link interwiki-ja mw-list-item"><a href="https://ja.wikipedia.org/wiki/%E5%89%B2%E7%B7%9A" title="割線 – Japanese" lang="ja" hreflang="ja" data-title="割線" data-language-autonym="日本語" data-language-local-name="Japanese" class="interlanguage-link-target"><span>日本語</span></a></li><li class="interlanguage-link interwiki-no mw-list-item"><a href="https://no.wikipedia.org/wiki/Sekant" title="Sekant – Norwegian Bokmål" lang="nb" hreflang="nb" data-title="Sekant" data-language-autonym="Norsk bokmål" data-language-local-name="Norwegian Bokmål" class="interlanguage-link-target"><span>Norsk bokmål</span></a></li><li class="interlanguage-link interwiki-nn mw-list-item"><a href="https://nn.wikipedia.org/wiki/Sekant" title="Sekant – Norwegian Nynorsk" lang="nn" hreflang="nn" data-title="Sekant" data-language-autonym="Norsk nynorsk" data-language-local-name="Norwegian Nynorsk" class="interlanguage-link-target"><span>Norsk nynorsk</span></a></li><li class="interlanguage-link interwiki-km mw-list-item"><a href="https://km.wikipedia.org/wiki/%E1%9E%94%E1%9E%93%E1%9F%92%E1%9E%91%E1%9E%B6%E1%9E%8F%E1%9F%8B%E1%9E%9F%E1%9F%81%E1%9E%80%E1%9E%84%E1%9F%8B" title="បន្ទាត់សេកង់ – Khmer" lang="km" hreflang="km" data-title="បន្ទាត់សេកង់" data-language-autonym="ភាសាខ្មែរ" data-language-local-name="Khmer" class="interlanguage-link-target"><span>ភាសាខ្មែរ</span></a></li><li class="interlanguage-link interwiki-pl mw-list-item"><a href="https://pl.wikipedia.org/wiki/Sieczna" title="Sieczna – Polish" lang="pl" hreflang="pl" data-title="Sieczna" data-language-autonym="Polski" data-language-local-name="Polish" class="interlanguage-link-target"><span>Polski</span></a></li><li class="interlanguage-link interwiki-pt mw-list-item"><a href="https://pt.wikipedia.org/wiki/Reta_secante" title="Reta secante – Portuguese" lang="pt" hreflang="pt" data-title="Reta secante" data-language-autonym="Português" data-language-local-name="Portuguese" class="interlanguage-link-target"><span>Português</span></a></li><li class="interlanguage-link interwiki-ro mw-list-item"><a href="https://ro.wikipedia.org/wiki/Secant%C4%83" title="Secantă – Romanian" lang="ro" hreflang="ro" data-title="Secantă" data-language-autonym="Română" data-language-local-name="Romanian" class="interlanguage-link-target"><span>Română</span></a></li><li class="interlanguage-link interwiki-ru mw-list-item"><a href="https://ru.wikipedia.org/wiki/%D0%A1%D0%B5%D0%BA%D1%83%D1%89%D0%B0%D1%8F_%D0%BF%D1%80%D1%8F%D0%BC%D0%B0%D1%8F" title="Секущая прямая – Russian" lang="ru" hreflang="ru" data-title="Секущая прямая" data-language-autonym="Русский" data-language-local-name="Russian" class="interlanguage-link-target"><span>Русский</span></a></li><li class="interlanguage-link interwiki-sv mw-list-item"><a href="https://sv.wikipedia.org/wiki/Sekant" title="Sekant – Swedish" lang="sv" hreflang="sv" data-title="Sekant" data-language-autonym="Svenska" data-language-local-name="Swedish" class="interlanguage-link-target"><span>Svenska</span></a></li><li class="interlanguage-link interwiki-tl mw-list-item"><a href="https://tl.wikipedia.org/wiki/Secant" title="Secant – Tagalog" lang="tl" hreflang="tl" data-title="Secant" data-language-autonym="Tagalog" data-language-local-name="Tagalog" class="interlanguage-link-target"><span>Tagalog</span></a></li><li class="interlanguage-link interwiki-ta mw-list-item"><a href="https://ta.wikipedia.org/wiki/%E0%AE%B5%E0%AF%86%E0%AE%9F%E0%AF%8D%E0%AE%9F%E0%AF%81%E0%AE%95%E0%AF%8D%E0%AE%95%E0%AF%8B%E0%AE%9F%E0%AF%81" title="வெட்டுக்கோடு – Tamil" lang="ta" hreflang="ta" data-title="வெட்டுக்கோடு" data-language-autonym="தமிழ்" data-language-local-name="Tamil" class="interlanguage-link-target"><span>தமிழ்</span></a></li><li class="interlanguage-link interwiki-tr mw-list-item"><a href="https://tr.wikipedia.org/wiki/Kesen" title="Kesen – Turkish" lang="tr" hreflang="tr" data-title="Kesen" data-language-autonym="Türkçe" data-language-local-name="Turkish" class="interlanguage-link-target"><span>Türkçe</span></a></li><li class="interlanguage-link interwiki-uk mw-list-item"><a href="https://uk.wikipedia.org/wiki/%D0%A1%D1%96%D1%87%D0%BD%D0%B0" title="Січна – Ukrainian" lang="uk" hreflang="uk" data-title="Січна" data-language-autonym="Українська" data-language-local-name="Ukrainian" class="interlanguage-link-target"><span>Українська</span></a></li><li class="interlanguage-link interwiki-zh mw-list-item"><a href="https://zh.wikipedia.org/wiki/%E5%89%B2%E7%BA%BF" title="割线 – Chinese" lang="zh" hreflang="zh" data-title="割线" data-language-autonym="中文" data-language-local-name="Chinese" class="interlanguage-link-target"><span>中文</span></a></li> </ul> <div class="after-portlet after-portlet-lang"><span class="wb-langlinks-edit wb-langlinks-link"><a href="https://www.wikidata.org/wiki/Special:EntityPage/Q426301#sitelinks-wikipedia" title="Edit interlanguage links" class="wbc-editpage">Edit links</a></span></div> </div> </div> </div> </header> <div class="vector-page-toolbar"> <div class="vector-page-toolbar-container"> <div id="left-navigation"> <nav aria-label="Namespaces"> <div id="p-associated-pages" class="vector-menu vector-menu-tabs mw-portlet mw-portlet-associated-pages" > <div class="vector-menu-content"> <ul class="vector-menu-content-list"> <li id="ca-nstab-main" class="selected vector-tab-noicon mw-list-item"><a href="/wiki/Secant_line" title="View the content page [c]" accesskey="c"><span>Article</span></a></li><li id="ca-talk" class="vector-tab-noicon mw-list-item"><a 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class="mw-content-ltr mw-parser-output" lang="en" dir="ltr"><div class="shortdescription nomobile noexcerpt noprint searchaux" style="display:none">Line that intersects a curve at least twice</div> <style data-mw-deduplicate="TemplateStyles:r1236090951">.mw-parser-output .hatnote{font-style:italic}.mw-parser-output div.hatnote{padding-left:1.6em;margin-bottom:0.5em}.mw-parser-output .hatnote i{font-style:normal}.mw-parser-output .hatnote+link+.hatnote{margin-top:-0.5em}@media print{body.ns-0 .mw-parser-output .hatnote{display:none!important}}</style><div role="note" class="hatnote navigation-not-searchable">For the secant trigonometric function, see <a href="/wiki/Secant_(trigonometry)" class="mw-redirect" title="Secant (trigonometry)">Secant (trigonometry)</a>.</div> <p>In <a href="/wiki/Geometry" title="Geometry">geometry</a>, a <b>secant</b> is a <a href="/wiki/Line_(geometry)" title="Line (geometry)">line</a> that intersects a <a href="/wiki/Curve" title="Curve">curve</a> at a minimum of two distinct <a href="/wiki/Point_(geometry)" title="Point (geometry)">points</a>.<sup id="cite_ref-cag_1-0" class="reference"><a href="#cite_note-cag-1"><span class="cite-bracket">[</span>1<span class="cite-bracket">]</span></a></sup> The word <i>secant</i> comes from the <a href="/wiki/Latin" title="Latin">Latin</a> word <i>secare</i>, meaning <i>to cut</i>.<sup id="cite_ref-2" class="reference"><a href="#cite_note-2"><span class="cite-bracket">[</span>2<span class="cite-bracket">]</span></a></sup> In the case of a <a href="/wiki/Circle" title="Circle">circle</a>, a secant intersects the circle at exactly two points. A <a href="/wiki/Chord_(geometry)" title="Chord (geometry)">chord</a> is the <a href="/wiki/Line_segment" title="Line segment">line segment</a> determined by the two points, that is, the <a href="/wiki/Interval_(mathematics)" title="Interval (mathematics)">interval</a> on the secant whose ends are the two points.<sup id="cite_ref-3" class="reference"><a href="#cite_note-3"><span class="cite-bracket">[</span>3<span class="cite-bracket">]</span></a></sup> </p> <meta property="mw:PageProp/toc" /> <div class="mw-heading mw-heading2"><h2 id="Circles">Circles</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Secant_line&action=edit&section=1" title="Edit section: Circles"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1236090951" /><div role="note" class="hatnote navigation-not-searchable">Further information: <a href="/wiki/Circle#Chord" title="Circle">Circle § Chord</a></div> <figure class="mw-default-size" typeof="mw:File/Thumb"><a href="/wiki/File:CIRCLE_LINES-en.svg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/b/b2/CIRCLE_LINES-en.svg/220px-CIRCLE_LINES-en.svg.png" decoding="async" width="220" height="222" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/b/b2/CIRCLE_LINES-en.svg/330px-CIRCLE_LINES-en.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/b/b2/CIRCLE_LINES-en.svg/440px-CIRCLE_LINES-en.svg.png 2x" data-file-width="612" data-file-height="618" /></a><figcaption>Common lines and line segments on a circle, including a secant</figcaption></figure> <p>A straight line can intersect a circle at zero, one, or two points. A line with intersections at two points is called a <i>secant line</i>, at one point a <i>tangent line</i> and at no points an <i>exterior line</i>. A <i>chord</i> is the line segment that joins two distinct points of a circle. A chord is therefore contained in a unique secant line and each secant line determines a unique chord. </p><p>In rigorous modern treatments of <a href="/wiki/Plane_geometry" class="mw-redirect" title="Plane geometry">plane geometry</a>, results that seem obvious and were assumed (without statement) by <a href="/wiki/Euclid" title="Euclid">Euclid</a> in <a href="/wiki/Euclid%27s_Elements" title="Euclid's Elements">his treatment</a>, are usually proved. </p><p>For example, <i>Theorem (Elementary Circular Continuity)</i>:<sup id="cite_ref-4" class="reference"><a href="#cite_note-4"><span class="cite-bracket">[</span>4<span class="cite-bracket">]</span></a></sup> If <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\mathcal {C}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi class="MJX-tex-caligraphic" mathvariant="script">C</mi> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\mathcal {C}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e7b3edab7022ca9e2976651bc59c489513ee9019" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.239ex; height:2.176ex;" alt="{\displaystyle {\mathcal {C}}}" /></span> is a circle and <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \ell }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>ℓ<!-- ℓ --></mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \ell }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f066e981e530bacc07efc6a10fa82deee985929e" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:0.97ex; height:2.176ex;" alt="{\displaystyle \ell }" /></span> a line that contains a point <span class="texhtml mvar" style="font-style:italic;">A</span> that is inside <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\mathcal {C}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi class="MJX-tex-caligraphic" mathvariant="script">C</mi> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\mathcal {C}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e7b3edab7022ca9e2976651bc59c489513ee9019" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.239ex; height:2.176ex;" alt="{\displaystyle {\mathcal {C}}}" /></span> and a point <span class="texhtml mvar" style="font-style:italic;">B</span> that is outside of <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\mathcal {C}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi class="MJX-tex-caligraphic" mathvariant="script">C</mi> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\mathcal {C}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e7b3edab7022ca9e2976651bc59c489513ee9019" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.239ex; height:2.176ex;" alt="{\displaystyle {\mathcal {C}}}" /></span> then <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \ell }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>ℓ<!-- ℓ --></mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \ell }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f066e981e530bacc07efc6a10fa82deee985929e" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:0.97ex; height:2.176ex;" alt="{\displaystyle \ell }" /></span> is a secant line for <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\mathcal {C}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi class="MJX-tex-caligraphic" mathvariant="script">C</mi> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\mathcal {C}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e7b3edab7022ca9e2976651bc59c489513ee9019" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.239ex; height:2.176ex;" alt="{\displaystyle {\mathcal {C}}}" /></span>. </p><p>In some situations phrasing results in terms of secant lines instead of chords can help to unify statements. As an example of this consider the result:<sup id="cite_ref-5" class="reference"><a href="#cite_note-5"><span class="cite-bracket">[</span>5<span class="cite-bracket">]</span></a></sup> </p> <dl><dd>If two secant lines contain chords <span class="texhtml"><span style="text-decoration:overline;"><i>AB</i></span></span> and <span class="texhtml"><span style="text-decoration:overline;"><i>CD</i></span></span> in a circle and intersect at a point <span class="texhtml mvar" style="font-style:italic;">P</span> that is not on the circle, then the line segment lengths satisfy <span class="texhtml"><i>AP</i>⋅<i>PB</i> = <i>CP</i>⋅<i>PD</i></span>.</dd></dl> <p>If the point <span class="texhtml mvar" style="font-style:italic;">P</span> lies inside the circle this is Euclid III.35, but if the point is outside the circle the result is not contained in the Elements. However, <a href="/wiki/Robert_Simson" title="Robert Simson">Robert Simson</a> following <a href="/wiki/Christopher_Clavius" title="Christopher Clavius">Christopher Clavius</a> demonstrated this result, sometimes called the <a href="/wiki/Intersecting_secants_theorem" title="Intersecting secants theorem">intersecting secants theorem</a>, in their commentaries on Euclid.<sup id="cite_ref-6" class="reference"><a href="#cite_note-6"><span class="cite-bracket">[</span>6<span class="cite-bracket">]</span></a></sup> </p> <div class="mw-heading mw-heading2"><h2 id="Curves">Curves</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Secant_line&action=edit&section=2" title="Edit section: Curves"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>For curves more complicated than simple circles, the possibility that a line that intersects a curve in more than two distinct points arises. Some authors define a secant line to a curve as a line that intersects the curve in two distinct points. This definition leaves open the possibility that the line may have other points of intersection with the curve. When phrased this way the definitions of a secant line for circles and curves are identical and the possibility of additional points of intersection just does not occur for a circle. </p> <div class="mw-heading mw-heading3"><h3 id="Secants_and_tangents">Secants and tangents</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Secant_line&action=edit&section=3" title="Edit section: Secants and tangents"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>Secants may be used to <a href="/wiki/Approximation_theory" title="Approximation theory">approximate</a> the <a href="/wiki/Tangent" title="Tangent">tangent</a> line to a <a href="/wiki/Curve" title="Curve">curve</a>, at some point <span class="texhtml"><i>P</i></span>, if it exists. Define a secant to a curve by two <a href="/wiki/Point_(geometry)" title="Point (geometry)">points</a>, <span class="texhtml"><i>P</i></span> and <span class="texhtml"><i>Q</i></span>, with <span class="texhtml"><i>P</i></span> fixed and <span class="texhtml"><i>Q</i></span> variable. As <span class="texhtml"><i>Q</i></span> approaches <span class="texhtml"><i>P</i></span> along the curve, if the <a href="/wiki/Slope" title="Slope">slope</a> of the secant approaches a <a href="/wiki/Limit_(mathematics)" title="Limit (mathematics)">limit value</a>, then that limit defines the slope of the tangent line at <span class="texhtml"><i>P</i></span>.<sup id="cite_ref-cag_1-1" class="reference"><a href="#cite_note-cag-1"><span class="cite-bracket">[</span>1<span class="cite-bracket">]</span></a></sup> The secant lines <span class="texhtml"><span style="text-decoration:overline;"><i>PQ</i></span></span> are the approximations to the tangent line. In calculus, this idea is the geometric definition of the <a href="/wiki/Derivative" title="Derivative">derivative</a>. </p> <figure class="mw-default-size" typeof="mw:File/Thumb"><a href="/wiki/File:Secanttangent.svg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/3/3f/Secanttangent.svg/220px-Secanttangent.svg.png" decoding="async" width="220" height="220" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/3/3f/Secanttangent.svg/330px-Secanttangent.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/3/3f/Secanttangent.svg/440px-Secanttangent.svg.png 2x" data-file-width="400" data-file-height="400" /></a><figcaption>The tangent line at point <span class="texhtml mvar" style="font-style:italic;">P</span> is a secant line of the curve</figcaption></figure> <p>A tangent line to a curve at a point <span class="texhtml"><i>P</i></span> may be a secant line to that curve if it intersects the curve in at least one point other than <span class="texhtml"><i>P</i></span>. Another way to look at this is to realize that being a tangent line at a point <span class="texhtml"><i>P</i></span> is a <i>local</i> property, depending only on the curve in the immediate neighborhood of <span class="texhtml"><i>P</i></span>, while being a secant line is a <i>global</i> property since the entire domain of the function producing the curve needs to be examined. </p> <div class="mw-heading mw-heading2"><h2 id="Sets_and_n-secants">Sets and <span class="texhtml mvar" style="font-style:italic;">n</span>-secants</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Secant_line&action=edit&section=4" title="Edit section: Sets and n-secants"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>The concept of a secant line can be applied in a more general setting than Euclidean space. Let <span class="texhtml mvar" style="font-style:italic;">K</span> be a finite set of <span class="texhtml mvar" style="font-style:italic;">k</span> points in some geometric setting. A line will be called an <span class="texhtml mvar" style="font-style:italic;">n</span>-secant of <span class="texhtml mvar" style="font-style:italic;">K</span> if it contains exactly <span class="texhtml mvar" style="font-style:italic;">n</span> points of <span class="texhtml mvar" style="font-style:italic;">K</span>.<sup id="cite_ref-7" class="reference"><a href="#cite_note-7"><span class="cite-bracket">[</span>7<span class="cite-bracket">]</span></a></sup> For example, if <span class="texhtml mvar" style="font-style:italic;">K</span> is a set of 50 points arranged on a circle in the Euclidean plane, a line joining two of them would be a 2-secant (or <i>bisecant</i>) and a line passing through only one of them would be a 1-secant (or <i>unisecant</i>). A unisecant in this example need not be a tangent line to the circle. </p><p>This terminology is often used in <a href="/wiki/Incidence_geometry" title="Incidence geometry">incidence geometry</a> and <a href="/wiki/Discrete_geometry" title="Discrete geometry">discrete geometry</a>. For instance, the <a href="/wiki/Sylvester%E2%80%93Gallai_theorem" title="Sylvester–Gallai theorem">Sylvester–Gallai theorem</a> of incidence geometry states that if <span class="texhtml mvar" style="font-style:italic;">n</span> points of Euclidean geometry are not <a href="/wiki/Collinearity" title="Collinearity">collinear</a> then there must exist a 2-secant of them. And the original <a href="/wiki/Orchard-planting_problem" title="Orchard-planting problem">orchard-planting problem</a> of discrete geometry asks for a bound on the number of 3-secants of a finite set of points. </p><p>Finiteness of the set of points is not essential in this definition, as long as each line can intersect the set in only a finite number of points. </p> <div class="mw-heading mw-heading2"><h2 id="See_also">See also</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Secant_line&action=edit&section=5" title="Edit section: See also"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <ul><li><a href="/wiki/Elliptic_curve" title="Elliptic curve">Elliptic curve</a>, a curve for which every secant has a third point of intersection, from which most of a group law may be defined</li> <li><a href="/wiki/Mean_value_theorem" title="Mean value theorem">Mean value theorem</a>, that every secant of the graph of a smooth function has a parallel tangent line</li> <li><a href="/wiki/Quadrisecant" title="Quadrisecant">Quadrisecant</a>, a line that intersects four points of a curve (usually a space curve)</li> <li><a href="/wiki/Secant_plane" title="Secant plane">Secant plane</a>, the three-dimensional equivalent of a secant line</li> <li><a href="/wiki/Secant_variety" title="Secant variety">Secant variety</a>, the union of secant lines and tangent lines to a given projective variety</li></ul> <div class="mw-heading mw-heading2"><h2 id="References">References</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Secant_line&action=edit&section=6" title="Edit section: References"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <style data-mw-deduplicate="TemplateStyles:r1239543626">.mw-parser-output .reflist{margin-bottom:0.5em;list-style-type:decimal}@media screen{.mw-parser-output .reflist{font-size:90%}}.mw-parser-output .reflist .references{font-size:100%;margin-bottom:0;list-style-type:inherit}.mw-parser-output .reflist-columns-2{column-width:30em}.mw-parser-output .reflist-columns-3{column-width:25em}.mw-parser-output .reflist-columns{margin-top:0.3em}.mw-parser-output .reflist-columns ol{margin-top:0}.mw-parser-output .reflist-columns li{page-break-inside:avoid;break-inside:avoid-column}.mw-parser-output .reflist-upper-alpha{list-style-type:upper-alpha}.mw-parser-output .reflist-upper-roman{list-style-type:upper-roman}.mw-parser-output .reflist-lower-alpha{list-style-type:lower-alpha}.mw-parser-output .reflist-lower-greek{list-style-type:lower-greek}.mw-parser-output .reflist-lower-roman{list-style-type:lower-roman}</style><div class="reflist"> <div class="mw-references-wrap"><ol class="references"> <li id="cite_note-cag-1"><span class="mw-cite-backlink">^ <a href="#cite_ref-cag_1-0"><sup><i><b>a</b></i></sup></a> <a href="#cite_ref-cag_1-1"><sup><i><b>b</b></i></sup></a></span> <span class="reference-text"><style data-mw-deduplicate="TemplateStyles:r1238218222">.mw-parser-output cite.citation{font-style:inherit;word-wrap:break-word}.mw-parser-output .citation q{quotes:"\"""\"""'""'"}.mw-parser-output .citation:target{background-color:rgba(0,127,255,0.133)}.mw-parser-output .id-lock-free.id-lock-free a{background:url("//upload.wikimedia.org/wikipedia/commons/6/65/Lock-green.svg")right 0.1em center/9px no-repeat}.mw-parser-output .id-lock-limited.id-lock-limited a,.mw-parser-output .id-lock-registration.id-lock-registration a{background:url("//upload.wikimedia.org/wikipedia/commons/d/d6/Lock-gray-alt-2.svg")right 0.1em center/9px no-repeat}.mw-parser-output .id-lock-subscription.id-lock-subscription a{background:url("//upload.wikimedia.org/wikipedia/commons/a/aa/Lock-red-alt-2.svg")right 0.1em center/9px no-repeat}.mw-parser-output .cs1-ws-icon 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.citation .mw-selflink{font-weight:inherit}@media screen{.mw-parser-output .cs1-format{font-size:95%}html.skin-theme-clientpref-night .mw-parser-output .cs1-maint{color:#18911f}}@media screen and (prefers-color-scheme:dark){html.skin-theme-clientpref-os .mw-parser-output .cs1-maint{color:#18911f}}</style><cite id="CITEREFProtterProtter1988" class="citation cs2"><a href="/wiki/Murray_H._Protter" title="Murray H. Protter">Protter, Murray H.</a>; Protter, Philip E. (1988), <a rel="nofollow" class="external text" href="https://books.google.com/books?id=jTmuOwwGDwoC&pg=PA62"><i>Calculus with Analytic Geometry</i></a>, Jones & Bartlett Learning, p. 62, <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/9780867200935" title="Special:BookSources/9780867200935"><bdi>9780867200935</bdi></a></cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Calculus+with+Analytic+Geometry&rft.pages=62&rft.pub=Jones+%26+Bartlett+Learning&rft.date=1988&rft.isbn=9780867200935&rft.aulast=Protter&rft.aufirst=Murray+H.&rft.au=Protter%2C+Philip+E.&rft_id=https%3A%2F%2Fbooks.google.com%2Fbooks%3Fid%3DjTmuOwwGDwoC%26pg%3DPA62&rfr_id=info%3Asid%2Fen.wikipedia.org%3ASecant+line" class="Z3988"></span>.</span> </li> <li id="cite_note-2"><span class="mw-cite-backlink"><b><a href="#cite_ref-2">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222" /><cite id="CITEREFRedgrove1913" class="citation cs2">Redgrove, Herbert Stanley (1913), <a rel="nofollow" class="external text" href="https://books.google.com/books?id=Nh0yAQAAMAAJ&pg=PA167"><i>Experimental Mensuration: An Elementary Test-book of Inductive Geometry</i></a>, Van Nostrand, p. 167</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Experimental+Mensuration%3A+An+Elementary+Test-book+of+Inductive+Geometry&rft.pages=167&rft.pub=Van+Nostrand&rft.date=1913&rft.aulast=Redgrove&rft.aufirst=Herbert+Stanley&rft_id=https%3A%2F%2Fbooks.google.com%2Fbooks%3Fid%3DNh0yAQAAMAAJ%26pg%3DPA167&rfr_id=info%3Asid%2Fen.wikipedia.org%3ASecant+line" class="Z3988"></span>.</span> </li> <li id="cite_note-3"><span class="mw-cite-backlink"><b><a href="#cite_ref-3">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222" /><cite id="CITEREFGullberg1997" class="citation cs2"><a href="/wiki/Jan_Gullberg" title="Jan Gullberg">Gullberg, Jan</a> (1997), <a rel="nofollow" class="external text" href="https://books.google.com/books?id=E09fBi9StpQC&pg=PA387"><i>Mathematics: From the Birth of Numbers</i></a>, W. W. Norton & Company, p. 387, <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/9780393040029" title="Special:BookSources/9780393040029"><bdi>9780393040029</bdi></a></cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Mathematics%3A+From+the+Birth+of+Numbers&rft.pages=387&rft.pub=W.+W.+Norton+%26+Company&rft.date=1997&rft.isbn=9780393040029&rft.aulast=Gullberg&rft.aufirst=Jan&rft_id=https%3A%2F%2Fbooks.google.com%2Fbooks%3Fid%3DE09fBi9StpQC%26pg%3DPA387&rfr_id=info%3Asid%2Fen.wikipedia.org%3ASecant+line" class="Z3988"></span>.</span> </li> <li id="cite_note-4"><span class="mw-cite-backlink"><b><a href="#cite_ref-4">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222" /><cite id="CITEREFVenema2006" class="citation cs2">Venema, Gerard A. (2006), <i>Foundations of Geometry</i>, Pearson/Prentice-Hall, p. 229, <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-0-13-143700-5" title="Special:BookSources/978-0-13-143700-5"><bdi>978-0-13-143700-5</bdi></a></cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Foundations+of+Geometry&rft.pages=229&rft.pub=Pearson%2FPrentice-Hall&rft.date=2006&rft.isbn=978-0-13-143700-5&rft.aulast=Venema&rft.aufirst=Gerard+A.&rfr_id=info%3Asid%2Fen.wikipedia.org%3ASecant+line" class="Z3988"></span></span> </li> <li id="cite_note-5"><span class="mw-cite-backlink"><b><a href="#cite_ref-5">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222" /><cite id="CITEREFJacobs1974" class="citation cs2">Jacobs, Harold R. (1974), <i>Geometry</i>, W. H. Freeman & Co., p. 482, <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/0-7167-0456-0" title="Special:BookSources/0-7167-0456-0"><bdi>0-7167-0456-0</bdi></a></cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Geometry&rft.pages=482&rft.pub=W.+H.+Freeman+%26+Co.&rft.date=1974&rft.isbn=0-7167-0456-0&rft.aulast=Jacobs&rft.aufirst=Harold+R.&rfr_id=info%3Asid%2Fen.wikipedia.org%3ASecant+line" class="Z3988"></span></span> </li> <li id="cite_note-6"><span class="mw-cite-backlink"><b><a href="#cite_ref-6">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222" /><cite id="CITEREFHeath1956" class="citation cs2"><a href="/wiki/Thomas_Little_Heath" class="mw-redirect" title="Thomas Little Heath">Heath, Thomas L.</a> (1956), <i>The thirteen books of Euclid's Elements (Vol. 2)</i>, Dover, p. 73</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=The+thirteen+books+of+Euclid%27s+Elements+%28Vol.+2%29&rft.pages=73&rft.pub=Dover&rft.date=1956&rft.aulast=Heath&rft.aufirst=Thomas+L.&rfr_id=info%3Asid%2Fen.wikipedia.org%3ASecant+line" class="Z3988"></span></span> </li> <li id="cite_note-7"><span class="mw-cite-backlink"><b><a href="#cite_ref-7">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222" /><cite id="CITEREFHirschfeld1979" class="citation cs2"><a href="/wiki/James_William_Peter_Hirschfeld" title="James William Peter Hirschfeld">Hirschfeld, J. W. P.</a> (1979), <a rel="nofollow" class="external text" href="https://archive.org/details/projectivegeomet0000hirs/page/70"><i>Projective Geometries over Finite Fields</i></a>, Oxford University Press, p. <a rel="nofollow" class="external text" href="https://archive.org/details/projectivegeomet0000hirs/page/70">70</a>, <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/0-19-853526-0" title="Special:BookSources/0-19-853526-0"><bdi>0-19-853526-0</bdi></a></cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Projective+Geometries+over+Finite+Fields&rft.pages=70&rft.pub=Oxford+University+Press&rft.date=1979&rft.isbn=0-19-853526-0&rft.aulast=Hirschfeld&rft.aufirst=J.+W.+P.&rft_id=https%3A%2F%2Farchive.org%2Fdetails%2Fprojectivegeomet0000hirs%2Fpage%2F70&rfr_id=info%3Asid%2Fen.wikipedia.org%3ASecant+line" class="Z3988"></span></span> </li> </ol></div></div> <div class="mw-heading mw-heading2"><h2 id="External_links">External links</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Secant_line&action=edit&section=7" title="Edit section: External links"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <ul><li><span class="citation mathworld" id="Reference-Mathworld-Secant_line"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222" /><cite id="CITEREFWeisstein" class="citation web cs1"><a href="/wiki/Eric_W._Weisstein" title="Eric W. Weisstein">Weisstein, Eric W.</a> <a rel="nofollow" class="external text" href="https://mathworld.wolfram.com/SecantLine.html">"Secant line"</a>. <i><a href="/wiki/MathWorld" title="MathWorld">MathWorld</a></i>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=unknown&rft.jtitle=MathWorld&rft.atitle=Secant+line&rft.au=Weisstein%2C+Eric+W.&rft_id=https%3A%2F%2Fmathworld.wolfram.com%2FSecantLine.html&rfr_id=info%3Asid%2Fen.wikipedia.org%3ASecant+line" class="Z3988"></span></span></li></ul> <div class="navbox-styles"><style data-mw-deduplicate="TemplateStyles:r1129693374">.mw-parser-output .hlist dl,.mw-parser-output .hlist ol,.mw-parser-output .hlist ul{margin:0;padding:0}.mw-parser-output .hlist dd,.mw-parser-output .hlist dt,.mw-parser-output .hlist li{margin:0;display:inline}.mw-parser-output .hlist.inline,.mw-parser-output .hlist.inline dl,.mw-parser-output .hlist.inline ol,.mw-parser-output .hlist.inline ul,.mw-parser-output .hlist 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difference">Finite difference</a></li> <li><a href="/wiki/Free_variables_and_bound_variables" title="Free variables and bound variables">Free variables and bound variables</a></li> <li><a href="/wiki/Graph_of_a_function" title="Graph of a function">Graph of a function</a></li> <li><a href="/wiki/Linear_function" title="Linear function">Linear function</a></li> <li><a href="/wiki/Radian" title="Radian">Radian</a></li> <li><a href="/wiki/Rolle%27s_theorem" title="Rolle's theorem">Rolle's theorem</a></li> <li><a class="mw-selflink selflink">Secant</a></li> <li><a href="/wiki/Slope" title="Slope">Slope</a></li> <li><a href="/wiki/Tangent" title="Tangent">Tangent</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%"><a href="/wiki/Limit_(mathematics)" title="Limit (mathematics)">Limits</a></th><td class="navbox-list-with-group navbox-list navbox-even hlist" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Indeterminate_form" title="Indeterminate form">Indeterminate form</a></li> <li><a href="/wiki/Limit_of_a_function" title="Limit of a function">Limit of a function</a> <ul><li><a href="/wiki/One-sided_limit" title="One-sided limit">One-sided limit</a></li></ul></li> <li><a href="/wiki/Limit_of_a_sequence" title="Limit of a sequence">Limit of a sequence</a></li> <li><a href="/wiki/Order_of_approximation" title="Order of approximation">Order of approximation</a></li> <li><a href="/wiki/(%CE%B5,_%CE%B4)-definition_of_limit" class="mw-redirect" title="(ε, δ)-definition of limit">(ε, δ)-definition of limit</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%"><a href="/wiki/Differential_calculus" title="Differential calculus">Differential calculus</a></th><td class="navbox-list-with-group navbox-list navbox-odd hlist" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Derivative" title="Derivative">Derivative</a></li> <li><a href="/wiki/Second_derivative" title="Second derivative">Second derivative</a></li> <li><a href="/wiki/Partial_derivative" title="Partial derivative">Partial derivative</a></li> <li><a href="/wiki/Differential_(mathematics)" title="Differential (mathematics)">Differential</a></li> <li><a href="/wiki/Differential_operator" title="Differential operator">Differential operator</a></li> <li><a href="/wiki/Mean_value_theorem" title="Mean value theorem">Mean value theorem</a></li> <li><a href="/wiki/Notation_for_differentiation" title="Notation for differentiation">Notation</a> <ul><li><a href="/wiki/Leibniz%27s_notation" title="Leibniz's notation">Leibniz's notation</a></li> <li><a href="/wiki/Newton%27s_notation_for_differentiation" class="mw-redirect" title="Newton's notation for differentiation">Newton's notation</a></li></ul></li> <li><a href="/wiki/Differentiation_rules" title="Differentiation rules">Rules of differentiation</a> <ul><li><a href="/wiki/Linearity_of_differentiation" title="Linearity of differentiation">linearity</a></li> <li><a href="/wiki/Power_rule" title="Power rule">Power</a></li> <li><a href="/wiki/Sum_rule_in_differentiation" class="mw-redirect" title="Sum rule in differentiation">Sum</a></li> <li><a href="/wiki/Chain_rule" title="Chain rule">Chain</a></li> <li><a href="/wiki/L%27H%C3%B4pital%27s_rule" title="L'Hôpital's rule">L'Hôpital's</a></li> <li><a href="/wiki/Product_rule" title="Product rule">Product</a> <ul><li><a href="/wiki/General_Leibniz_rule" title="General Leibniz rule">General Leibniz's rule</a></li></ul></li> <li><a href="/wiki/Quotient_rule" title="Quotient rule">Quotient</a></li></ul></li> <li>Other techniques <ul><li><a href="/wiki/Implicit_differentiation" class="mw-redirect" title="Implicit differentiation">Implicit differentiation</a></li> <li><a href="/wiki/Inverse_functions_and_differentiation" class="mw-redirect" title="Inverse functions and differentiation">Inverse functions and differentiation</a></li> <li><a href="/wiki/Logarithmic_derivative" title="Logarithmic derivative">Logarithmic derivative</a></li> <li><a href="/wiki/Related_rates" title="Related rates">Related rates</a></li></ul></li> <li><a href="/wiki/Stationary_point" title="Stationary point">Stationary points</a> <ul><li><a href="/wiki/First_derivative_test" class="mw-redirect" title="First derivative test">First derivative test</a></li> <li><a href="/wiki/Second_derivative_test" class="mw-redirect" title="Second derivative test">Second derivative test</a></li> <li><a href="/wiki/Extreme_value_theorem" title="Extreme value theorem">Extreme value theorem</a></li> <li><a href="/wiki/Maximum_and_minimum" title="Maximum and minimum">Maximum and minimum</a></li></ul></li> <li>Further applications <ul><li><a href="/wiki/Newton%27s_method" title="Newton's method">Newton's method</a></li> <li><a href="/wiki/Taylor%27s_theorem" title="Taylor's theorem">Taylor's theorem</a></li></ul></li> <li><a href="/wiki/Differential_equation" title="Differential equation">Differential equation</a> <ul><li><a href="/wiki/Ordinary_differential_equation" title="Ordinary differential equation">Ordinary differential equation</a></li> <li><a href="/wiki/Partial_differential_equation" title="Partial differential equation">Partial differential equation</a></li> <li><a href="/wiki/Stochastic_differential_equation" title="Stochastic differential equation">Stochastic differential equation</a></li></ul></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%"><a href="/wiki/Integral_calculus" class="mw-redirect" title="Integral calculus">Integral calculus</a></th><td class="navbox-list-with-group navbox-list navbox-even hlist" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Antiderivative" title="Antiderivative">Antiderivative</a></li> <li><a href="/wiki/Arc_length" title="Arc length">Arc length</a></li> <li><a href="/wiki/Riemann_integral" title="Riemann integral">Riemann integral</a></li> <li><a href="/wiki/Integral#Properties" title="Integral">Basic properties</a></li> <li><a href="/wiki/Constant_of_integration" title="Constant of integration">Constant of integration</a></li> <li><a href="/wiki/Fundamental_theorem_of_calculus" title="Fundamental theorem of calculus">Fundamental theorem of calculus</a> <ul><li><a href="/wiki/Leibniz_integral_rule" title="Leibniz integral rule">Differentiating under the integral sign</a></li></ul></li> <li><a href="/wiki/Integration_by_parts" title="Integration by parts">Integration by parts</a></li> <li><a href="/wiki/Integration_by_substitution" title="Integration by substitution">Integration by substitution</a> <ul><li><a href="/wiki/Trigonometric_substitution" title="Trigonometric substitution">trigonometric</a></li> <li><a href="/wiki/Euler_substitution" title="Euler substitution">Euler</a></li> <li><a href="/wiki/Tangent_half-angle_substitution" title="Tangent half-angle substitution">Tangent half-angle substitution</a></li></ul></li> <li><a href="/wiki/Partial_fractions_in_integration" class="mw-redirect" title="Partial fractions in integration">Partial fractions in integration</a> <ul><li><a href="/wiki/Quadratic_integral" title="Quadratic integral">Quadratic integral</a></li></ul></li> <li><a href="/wiki/Trapezoidal_rule" title="Trapezoidal rule">Trapezoidal rule</a></li> <li>Volumes <ul><li><a href="/wiki/Disc_integration" title="Disc integration">Washer method</a></li> <li><a href="/wiki/Shell_integration" title="Shell integration">Shell method</a></li></ul></li> <li><a href="/wiki/Integral_equation" title="Integral equation">Integral equation</a></li> <li><a href="/wiki/Integro-differential_equation" title="Integro-differential equation">Integro-differential equation</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%"><a href="/wiki/Vector_calculus" title="Vector calculus">Vector calculus</a></th><td class="navbox-list-with-group navbox-list navbox-odd hlist" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li>Derivatives <ul><li><a href="/wiki/Curl_(mathematics)" title="Curl (mathematics)">Curl</a></li> <li><a href="/wiki/Directional_derivative" title="Directional derivative">Directional derivative</a></li> <li><a href="/wiki/Divergence" title="Divergence">Divergence</a></li> <li><a href="/wiki/Gradient" title="Gradient">Gradient</a></li> <li><a href="/wiki/Laplace_operator" title="Laplace operator">Laplacian</a></li></ul></li> <li>Basic theorems <ul><li><a href="/wiki/Fundamental_Theorem_of_Line_Integrals" class="mw-redirect" title="Fundamental Theorem of Line Integrals">Line integrals</a></li> <li><a href="/wiki/Green%27s_theorem" title="Green's theorem">Green's</a></li> <li><a href="/wiki/Stokes%27_theorem" title="Stokes' theorem">Stokes'</a></li> <li><a href="/wiki/Divergence_theorem" title="Divergence theorem">Gauss'</a></li></ul></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%"><a href="/wiki/Multivariable_calculus" title="Multivariable calculus">Multivariable calculus</a></th><td class="navbox-list-with-group navbox-list navbox-even hlist" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Divergence_theorem" title="Divergence theorem">Divergence theorem</a></li> <li><a href="/wiki/Geometric_calculus" title="Geometric calculus">Geometric</a></li> <li><a href="/wiki/Hessian_matrix" title="Hessian matrix">Hessian matrix</a></li> <li><a href="/wiki/Jacobian_matrix_and_determinant" title="Jacobian matrix and determinant">Jacobian matrix and determinant</a></li> <li><a href="/wiki/Lagrange_multiplier" title="Lagrange multiplier">Lagrange multiplier</a></li> <li><a href="/wiki/Line_integral" title="Line integral">Line integral</a></li> <li><a href="/wiki/Matrix_calculus" title="Matrix calculus">Matrix</a></li> <li><a href="/wiki/Multiple_integral" title="Multiple integral">Multiple integral</a></li> <li><a href="/wiki/Partial_derivative" title="Partial derivative">Partial derivative</a></li> <li><a href="/wiki/Surface_integral" title="Surface integral">Surface integral</a></li> <li><a href="/wiki/Volume_integral" title="Volume integral">Volume integral</a></li> <li>Advanced topics <ul><li><a href="/wiki/Differential_form" title="Differential form">Differential forms</a></li> <li><a href="/wiki/Exterior_derivative" title="Exterior derivative">Exterior derivative</a></li> <li><a href="/wiki/Generalized_Stokes%27_theorem" class="mw-redirect" title="Generalized Stokes' theorem">Generalized Stokes' theorem</a></li> <li><a href="/wiki/Tensor_calculus" class="mw-redirect" title="Tensor calculus">Tensor calculus</a></li></ul></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%">Sequences and series</th><td class="navbox-list-with-group navbox-list navbox-odd hlist" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Arithmetico-geometric_sequence" title="Arithmetico-geometric sequence">Arithmetico-geometric sequence</a></li> <li>Types of series <ul><li><a href="/wiki/Alternating_series" title="Alternating series">Alternating</a></li> <li><a href="/wiki/Binomial_series" title="Binomial series">Binomial</a></li> <li><a href="/wiki/Fourier_series" title="Fourier series">Fourier</a></li> <li><a href="/wiki/Geometric_series" title="Geometric series">Geometric</a></li> <li><a href="/wiki/Harmonic_series_(mathematics)" title="Harmonic series (mathematics)">Harmonic</a></li> <li><a href="/wiki/Infinite_series" class="mw-redirect" title="Infinite series">Infinite</a></li> <li><a href="/wiki/Power_series" title="Power series">Power</a> <ul><li><a href="/wiki/Maclaurin_series" class="mw-redirect" title="Maclaurin series">Maclaurin</a></li> <li><a href="/wiki/Taylor_series" title="Taylor series">Taylor</a></li></ul></li> <li><a href="/wiki/Telescoping_series" title="Telescoping series">Telescoping</a></li></ul></li> <li>Tests of convergence <ul><li><a href="/wiki/Abel%27s_test" title="Abel's test">Abel's</a></li> <li><a href="/wiki/Alternating_series_test" title="Alternating series test">Alternating series</a></li> <li><a href="/wiki/Cauchy_condensation_test" title="Cauchy condensation test">Cauchy condensation</a></li> <li><a href="/wiki/Direct_comparison_test" title="Direct comparison test">Direct comparison</a></li> <li><a href="/wiki/Dirichlet%27s_test" title="Dirichlet's test">Dirichlet's</a></li> <li><a href="/wiki/Integral_test_for_convergence" title="Integral test for convergence">Integral</a></li> <li><a href="/wiki/Limit_comparison_test" title="Limit comparison test">Limit comparison</a></li> <li><a href="/wiki/Ratio_test" title="Ratio test">Ratio</a></li> <li><a href="/wiki/Root_test" title="Root test">Root</a></li> <li><a href="/wiki/Term_test" class="mw-redirect" title="Term test">Term</a></li></ul></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%">Special functions<br />and numbers</th><td class="navbox-list-with-group navbox-list navbox-even hlist" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Bernoulli_number" title="Bernoulli number">Bernoulli numbers</a></li> <li><a href="/wiki/E_(mathematical_constant)" title="E (mathematical constant)">e (mathematical constant)</a></li> <li><a href="/wiki/Exponential_function" title="Exponential function">Exponential function</a></li> <li><a href="/wiki/Natural_logarithm" title="Natural logarithm">Natural logarithm</a></li> <li><a href="/wiki/Stirling%27s_approximation" title="Stirling's approximation">Stirling's approximation</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%"><a href="/wiki/History_of_calculus" title="History of calculus">History of calculus</a></th><td class="navbox-list-with-group navbox-list navbox-odd hlist" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Adequality" title="Adequality">Adequality</a></li> <li><a href="/wiki/Brook_Taylor" title="Brook Taylor">Brook Taylor</a></li> <li><a href="/wiki/Colin_Maclaurin" title="Colin Maclaurin">Colin Maclaurin</a></li> <li><a href="/wiki/Generality_of_algebra" title="Generality of algebra">Generality of algebra</a></li> <li><a href="/wiki/Gottfried_Wilhelm_Leibniz" title="Gottfried Wilhelm Leibniz">Gottfried Wilhelm Leibniz</a></li> <li><a href="/wiki/Infinitesimal" title="Infinitesimal">Infinitesimal</a></li> <li><a href="/wiki/Infinitesimal_calculus" class="mw-redirect" title="Infinitesimal calculus">Infinitesimal calculus</a></li> <li><a href="/wiki/Isaac_Newton" title="Isaac Newton">Isaac Newton</a></li> <li><a href="/wiki/Fluxion" title="Fluxion">Fluxion</a></li> <li><a href="/wiki/Law_of_Continuity" class="mw-redirect" title="Law of Continuity">Law of Continuity</a></li> <li><a href="/wiki/Leonhard_Euler" title="Leonhard Euler">Leonhard Euler</a></li> <li><i><a href="/wiki/Method_of_Fluxions" title="Method of Fluxions">Method of Fluxions</a></i></li> <li><i><a href="/wiki/The_Method_of_Mechanical_Theorems" title="The Method of Mechanical Theorems">The Method of Mechanical Theorems</a></i></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%">Lists</th><td class="navbox-list-with-group navbox-list navbox-odd hlist" style="width:100%;padding:0"><div style="padding:0 0.25em"></div><table class="nowraplinks navbox-subgroup" style="border-spacing:0"><tbody><tr><th id="Integrals32" scope="row" class="navbox-group" style="width:1%;text-align:left"><a href="/wiki/Lists_of_integrals" title="Lists of integrals">Integrals</a></th><td class="navbox-list-with-group navbox-list navbox-even" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/List_of_integrals_of_rational_functions" title="List of integrals of rational functions">rational functions</a></li> <li><a href="/wiki/List_of_integrals_of_irrational_functions" title="List of integrals of irrational functions">irrational functions</a></li> <li><a href="/wiki/List_of_integrals_of_exponential_functions" title="List of integrals of exponential functions">exponential functions</a></li> <li><a href="/wiki/List_of_integrals_of_logarithmic_functions" title="List of integrals of logarithmic functions">logarithmic functions</a></li> <li><a href="/wiki/List_of_integrals_of_hyperbolic_functions" title="List of integrals of hyperbolic functions">hyperbolic functions</a> <ul><li><a href="/wiki/List_of_integrals_of_inverse_hyperbolic_functions" title="List of integrals of inverse hyperbolic functions">inverse</a></li></ul></li> <li><a href="/wiki/List_of_integrals_of_trigonometric_functions" title="List of integrals of trigonometric functions">trigonometric functions</a> <ul><li><a href="/wiki/List_of_integrals_of_inverse_trigonometric_functions" title="List of integrals of inverse trigonometric functions">inverse</a></li> <li><a href="/wiki/Integral_of_the_secant_function" title="Integral of the secant function">Secant</a></li> <li><a href="/wiki/Integral_of_secant_cubed" title="Integral of secant cubed">Secant cubed</a></li></ul></li></ul> </div></td></tr><tr><td colspan="2" class="navbox-list navbox-odd" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/List_of_limits" title="List of limits">List of limits</a></li> <li><a href="/wiki/Differentiation_rules" title="Differentiation rules">List of derivatives</a></li></ul> </div></td></tr></tbody></table><div></div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%">Miscellaneous topics</th><td class="navbox-list-with-group navbox-list navbox-even hlist" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li>Complex calculus <ul><li><a href="/wiki/Contour_integral" class="mw-redirect" title="Contour integral">Contour integral</a></li></ul></li> <li>Differential geometry <ul><li><a href="/wiki/Manifold" title="Manifold">Manifold</a></li> <li><a href="/wiki/Curvature" title="Curvature">Curvature</a></li> <li><a href="/wiki/Differential_geometry_of_curves" class="mw-redirect" title="Differential geometry of curves">of curves</a></li> <li><a href="/wiki/Differential_geometry_of_surfaces" title="Differential geometry of surfaces">of surfaces</a></li> <li><a href="/wiki/Tensor" title="Tensor">Tensor</a></li></ul></li> <li><a href="/wiki/Euler%E2%80%93Maclaurin_formula" title="Euler–Maclaurin formula">Euler–Maclaurin formula</a></li> <li><a href="/wiki/Gabriel%27s_horn" title="Gabriel's horn">Gabriel's horn</a></li> <li><a href="/wiki/Integration_Bee" title="Integration Bee">Integration Bee</a></li> <li><a href="/wiki/Proof_that_22/7_exceeds_%CF%80" title="Proof that 22/7 exceeds π">Proof that 22/7 exceeds π</a></li> <li><a href="/wiki/Regiomontanus%27_angle_maximization_problem" title="Regiomontanus' angle maximization problem">Regiomontanus' angle maximization problem</a></li> <li><a href="/wiki/Steinmetz_solid" title="Steinmetz solid">Steinmetz solid</a></li></ul> </div></td></tr></tbody></table></div> <!-- NewPP limit report Parsed by mw‐web.eqiad.main‐8669bc5c8‐hlwks Cached time: 20250318155317 Cache expiry: 2592000 Reduced expiry: false Complications: [vary‐revision‐sha1, show‐toc] CPU time usage: 0.281 seconds Real time usage: 0.426 seconds Preprocessor visited node count: 1684/1000000 Post‐expand include size: 47701/2097152 bytes Template argument size: 1854/2097152 bytes Highest expansion depth: 8/100 Expensive parser function count: 3/500 Unstrip recursion depth: 1/20 Unstrip post‐expand size: 35894/5000000 bytes Lua time usage: 0.172/10.000 seconds Lua memory usage: 5095657/52428800 bytes Number of Wikibase entities loaded: 0/400 --> <!-- Transclusion expansion time report (%,ms,calls,template) 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