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<li id="toc-The_Unitary_Nebuls" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#The_Unitary_Nebuls"> <div class="vector-toc-text"> <span class="vector-toc-numb">1.1</span> <span>The Unitary Nebuls</span> </div> </a> <ul id="toc-The_Unitary_Nebuls-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-The_Mystery_of_Two_Natures" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#The_Mystery_of_Two_Natures"> <div class="vector-toc-text"> <span class="vector-toc-numb">1.2</span> <span>The Mystery of Two Natures</span> </div> </a> <ul id="toc-The_Mystery_of_Two_Natures-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-The_Evolution_of_Three_Talents" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#The_Evolution_of_Three_Talents"> <div class="vector-toc-text"> <span class="vector-toc-numb">1.3</span> <span>The Evolution of Three Talents</span> </div> </a> <ul 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id="toc-Problems_of_Right_Angle_Triangles_and_Rectangles" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Problems_of_Right_Angle_Triangles_and_Rectangles"> <div class="vector-toc-text"> <span class="vector-toc-numb">2.1</span> <span>Problems of Right Angle Triangles and Rectangles</span> </div> </a> <ul id="toc-Problems_of_Right_Angle_Triangles_and_Rectangles-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Problems_of_Plane_Figures" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Problems_of_Plane_Figures"> <div class="vector-toc-text"> <span class="vector-toc-numb">2.2</span> <span>Problems of Plane Figures</span> </div> </a> <ul id="toc-Problems_of_Plane_Figures-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Problems_of_Piece_Goods" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Problems_of_Piece_Goods"> <div class="vector-toc-text"> <span class="vector-toc-numb">2.3</span> <span>Problems of Piece Goods</span> </div> </a> <ul id="toc-Problems_of_Piece_Goods-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Problems_on_Grain_Storage" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Problems_on_Grain_Storage"> <div class="vector-toc-text"> <span class="vector-toc-numb">2.4</span> <span>Problems on Grain Storage</span> </div> </a> <ul id="toc-Problems_on_Grain_Storage-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Problems_on_Labour" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Problems_on_Labour"> <div class="vector-toc-text"> <span class="vector-toc-numb">2.5</span> <span>Problems on Labour</span> </div> </a> <ul id="toc-Problems_on_Labour-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Problems_of_Equations_for_Fractional_Roots" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Problems_of_Equations_for_Fractional_Roots"> <div class="vector-toc-text"> <span class="vector-toc-numb">2.6</span> <span>Problems of Equations for Fractional Roots</span> </div> </a> <ul id="toc-Problems_of_Equations_for_Fractional_Roots-sublist" class="vector-toc-list"> </ul> </li> </ul> </li> <li id="toc-Book_II" class="vector-toc-list-item vector-toc-level-1"> <a class="vector-toc-link" href="#Book_II"> <div class="vector-toc-text"> <span class="vector-toc-numb">3</span> <span>Book II</span> </div> </a> <button aria-controls="toc-Book_II-sublist" class="cdx-button cdx-button--weight-quiet cdx-button--icon-only vector-toc-toggle"> <span class="vector-icon mw-ui-icon-wikimedia-expand"></span> <span>Toggle Book II subsection</span> </button> <ul id="toc-Book_II-sublist" class="vector-toc-list"> <li id="toc-Mixed_Problems" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Mixed_Problems"> <div class="vector-toc-text"> <span class="vector-toc-numb">3.1</span> <span>Mixed Problems</span> </div> </a> <ul id="toc-Mixed_Problems-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Containment_of_Circles_and_Squares" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Containment_of_Circles_and_Squares"> <div class="vector-toc-text"> <span class="vector-toc-numb">3.2</span> <span>Containment of Circles and Squares</span> </div> </a> <ul id="toc-Containment_of_Circles_and_Squares-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Problems_on_Areas" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Problems_on_Areas"> <div class="vector-toc-text"> <span class="vector-toc-numb">3.3</span> <span>Problems on Areas</span> </div> </a> <ul id="toc-Problems_on_Areas-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Surveying_with_Right_Angle_Triangles" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Surveying_with_Right_Angle_Triangles"> <div class="vector-toc-text"> <span class="vector-toc-numb">3.4</span> <span>Surveying with Right Angle Triangles</span> </div> </a> <ul id="toc-Surveying_with_Right_Angle_Triangles-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Hay_Stacks" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Hay_Stacks"> <div class="vector-toc-text"> <span class="vector-toc-numb">3.5</span> <span>Hay Stacks</span> </div> </a> <ul id="toc-Hay_Stacks-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Bundles_of_Arrows" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Bundles_of_Arrows"> <div class="vector-toc-text"> <span class="vector-toc-numb">3.6</span> <span>Bundles of Arrows</span> </div> </a> <ul id="toc-Bundles_of_Arrows-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Land_Measurement" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Land_Measurement"> <div class="vector-toc-text"> <span class="vector-toc-numb">3.7</span> <span>Land Measurement</span> </div> </a> <ul id="toc-Land_Measurement-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Summon_Men_According_to_Need" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Summon_Men_According_to_Need"> <div class="vector-toc-text"> <span class="vector-toc-numb">3.8</span> <span>Summon Men According to Need</span> </div> </a> <ul id="toc-Summon_Men_According_to_Need-sublist" class="vector-toc-list"> </ul> </li> </ul> </li> <li id="toc-Book_III" class="vector-toc-list-item vector-toc-level-1"> <a class="vector-toc-link" href="#Book_III"> <div class="vector-toc-text"> <span class="vector-toc-numb">4</span> <span>Book III</span> </div> </a> <button aria-controls="toc-Book_III-sublist" class="cdx-button cdx-button--weight-quiet cdx-button--icon-only vector-toc-toggle"> <span class="vector-icon mw-ui-icon-wikimedia-expand"></span> <span>Toggle Book III subsection</span> </button> <ul id="toc-Book_III-sublist" class="vector-toc-list"> <li id="toc-Fruit_pile" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Fruit_pile"> <div class="vector-toc-text"> <span class="vector-toc-numb">4.1</span> <span>Fruit pile</span> </div> </a> <ul id="toc-Fruit_pile-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Figures_within_Figure" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Figures_within_Figure"> <div class="vector-toc-text"> <span class="vector-toc-numb">4.2</span> <span>Figures within Figure</span> </div> </a> <ul id="toc-Figures_within_Figure-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Simultaneous_Equations" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Simultaneous_Equations"> <div class="vector-toc-text"> <span class="vector-toc-numb">4.3</span> <span>Simultaneous Equations</span> </div> </a> <ul id="toc-Simultaneous_Equations-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Equation_of_two_unknowns" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Equation_of_two_unknowns"> <div class="vector-toc-text"> <span class="vector-toc-numb">4.4</span> <span>Equation of two unknowns</span> </div> </a> <ul id="toc-Equation_of_two_unknowns-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Left_and_Right" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Left_and_Right"> <div class="vector-toc-text"> <span class="vector-toc-numb">4.5</span> <span>Left and Right</span> </div> </a> <ul id="toc-Left_and_Right-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Equation_of_Three_Unknowns" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Equation_of_Three_Unknowns"> <div class="vector-toc-text"> <span class="vector-toc-numb">4.6</span> <span>Equation of Three Unknowns</span> </div> </a> <ul id="toc-Equation_of_Three_Unknowns-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Equation_of_Four_Unknowns" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Equation_of_Four_Unknowns"> <div class="vector-toc-text"> <span class="vector-toc-numb">4.7</span> <span>Equation of Four Unknowns</span> </div> </a> <ul id="toc-Equation_of_Four_Unknowns-sublist" class="vector-toc-list"> </ul> </li> </ul> </li> <li id="toc-References" class="vector-toc-list-item vector-toc-level-1"> <a class="vector-toc-link" href="#References"> <div class="vector-toc-text"> <span class="vector-toc-numb">5</span> <span>References</span> </div> </a> <ul id="toc-References-sublist" class="vector-toc-list"> </ul> </li> </ul> </div> </div> </nav> </div> </div> <div class="mw-content-container"> <main id="content" class="mw-body"> <header class="mw-body-header vector-page-titlebar"> 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dir="ltr"><div class="shortdescription nomobile noexcerpt noprint searchaux" style="display:none">1303 mathematical monograph by Zhu Shijie</div> <p class="mw-empty-elt"> </p> <figure class="mw-halign-right" typeof="mw:File/Thumb"><a href="/wiki/File:%E5%9B%9B%E5%85%83%E8%87%AA%E4%B9%98%E6%BC%94%E6%AE%B5%E5%9B%BE.jpg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/9/9b/%E5%9B%9B%E5%85%83%E8%87%AA%E4%B9%98%E6%BC%94%E6%AE%B5%E5%9B%BE.jpg/300px-%E5%9B%9B%E5%85%83%E8%87%AA%E4%B9%98%E6%BC%94%E6%AE%B5%E5%9B%BE.jpg" decoding="async" width="300" height="200" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/9/9b/%E5%9B%9B%E5%85%83%E8%87%AA%E4%B9%98%E6%BC%94%E6%AE%B5%E5%9B%BE.jpg/450px-%E5%9B%9B%E5%85%83%E8%87%AA%E4%B9%98%E6%BC%94%E6%AE%B5%E5%9B%BE.jpg 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/9/9b/%E5%9B%9B%E5%85%83%E8%87%AA%E4%B9%98%E6%BC%94%E6%AE%B5%E5%9B%BE.jpg/600px-%E5%9B%9B%E5%85%83%E8%87%AA%E4%B9%98%E6%BC%94%E6%AE%B5%E5%9B%BE.jpg 2x" data-file-width="3432" data-file-height="2285" /></a><figcaption>Illustrations in Jade Mirror of the Four Unknowns</figcaption></figure> <figure class="mw-halign-right" typeof="mw:File/Thumb"><a href="/wiki/File:Yanghui_triangle.gif" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/e/ea/Yanghui_triangle.gif/300px-Yanghui_triangle.gif" decoding="async" width="300" height="467" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/e/ea/Yanghui_triangle.gif/450px-Yanghui_triangle.gif 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/e/ea/Yanghui_triangle.gif/600px-Yanghui_triangle.gif 2x" data-file-width="704" data-file-height="1095" /></a><figcaption>Jia Xian triangle</figcaption></figure> <p><i><b>Jade Mirror of the Four Unknowns</b></i>,<sup id="cite_ref-1" class="reference"><a href="#cite_note-1"><span class="cite-bracket">[</span>1<span class="cite-bracket">]</span></a></sup> <i><b>Siyuan yujian</b></i> (<a href="/wiki/Simplified_Chinese_characters" title="Simplified Chinese characters">simplified Chinese</a>: <span lang="zh-Hans">四元玉鉴</span>; <a href="/wiki/Traditional_Chinese_characters" title="Traditional Chinese characters">traditional Chinese</a>: <span lang="zh-Hant">四元玉鑒</span>), also referred to as <i><b>Jade Mirror of the Four Origins</b></i>,<sup id="cite_ref-Hart_2-0" class="reference"><a href="#cite_note-Hart-2"><span class="cite-bracket">[</span>2<span class="cite-bracket">]</span></a></sup> is a 1303 mathematical monograph by Yuan dynasty mathematician <a href="/wiki/Zhu_Shijie" title="Zhu Shijie">Zhu Shijie</a>.<sup id="cite_ref-Elman_3-0" class="reference"><a href="#cite_note-Elman-3"><span class="cite-bracket">[</span>3<span class="cite-bracket">]</span></a></sup> Zhu advanced Chinese algebra with this <a href="/wiki/Magnum_opus" class="mw-redirect" title="Magnum opus">Magnum opus</a>. </p><p>The book consists of an introduction and three books, with a total of 288 problems. The first four problems in the introduction illustrate his method of the four unknowns. He showed how to convert a problem stated verbally into a system of polynomial equations (up to the 14th order), by using up to four unknowns: 天 Heaven, 地 Earth, 人 Man, 物 Matter, and then how to reduce the system to a single polynomial equation in one unknown by successive elimination of unknowns. He then solved the high-order equation by <a href="/wiki/Southern_Song_dynasty" class="mw-redirect" title="Southern Song dynasty">Southern Song dynasty</a> mathematician <a href="/wiki/Qin_Jiushao" title="Qin Jiushao">Qin Jiushao</a>'s "Ling long kai fang" method published in Shùshū Jiǔzhāng (“<a href="/wiki/Mathematical_Treatise_in_Nine_Sections" title="Mathematical Treatise in Nine Sections">Mathematical Treatise in Nine Sections</a>”) in 1247 (more than 570 years before English mathematician <a href="/wiki/William_George_Horner" title="William George Horner">William Horner</a>'s method using synthetic division). To do this, he makes use of the <a href="/wiki/Pascal_triangle" class="mw-redirect" title="Pascal triangle">Pascal triangle</a>, which he labels as the diagram of an ancient method first discovered by <a href="/wiki/Jia_Xian" title="Jia Xian">Jia Xian</a> before 1050. </p><p>Zhu also solved square and cube roots problems by solving quadratic and cubic equations, and added to the understanding of series and progressions, classifying them according to the coefficients of the Pascal triangle. He also showed how to solve systems of <a href="/wiki/Linear_equation" title="Linear equation">linear equations</a> by reducing the matrix of their coefficients to <a href="/wiki/Diagonal_form" title="Diagonal form">diagonal form</a>. His methods predate <a href="/wiki/Blaise_Pascal" title="Blaise Pascal">Blaise Pascal</a>, William Horner, and modern matrix methods by many centuries. The preface of the book describes how Zhu travelled around China for 20 years as a teacher of mathematics. </p><p><i>Jade Mirror of the Four Unknowns</i> consists of four books, with 24 classes and 288 problems, in which 232 problems deal with <a href="/wiki/Tian_yuan_shu" title="Tian yuan shu">Tian yuan shu</a>, 36 problems deal with variable of two variables, 13 problems of three variables, and 7 problems of four variables. </p> <meta property="mw:PageProp/toc" /> <div class="mw-heading mw-heading2"><h2 id="Introduction">Introduction</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Jade_Mirror_of_the_Four_Unknowns&action=edit&section=1" title="Edit section: Introduction"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <figure class="mw-halign-right" typeof="mw:File/Thumb"><a href="/wiki/File:Siyuan1.png" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/a/ad/Siyuan1.png/250px-Siyuan1.png" decoding="async" width="250" height="235" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/a/ad/Siyuan1.png/375px-Siyuan1.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/a/ad/Siyuan1.png/500px-Siyuan1.png 2x" data-file-width="568" data-file-height="534" /></a><figcaption>The Square of the Sum of the Four Quantities of a Right Angle Triangle</figcaption></figure> <p>The four quantities are <i>x</i>, <i>y</i>, <i>z</i>, <i>w</i> can be presented with the following diagram </p> <dl><dd><dl><dd><dl><dd><dl><dd><dl><dd><dl><dd><dl><dd><dl><dd><dl><dd><dl><dd><dl><dd><dl><dd><dl><dd><span class="mw-default-size" typeof="mw:File"><a href="/wiki/File:Counting_rod_v1.png" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/1/19/Counting_rod_v1.png" decoding="async" width="7" height="29" class="mw-file-element" data-file-width="7" data-file-height="29" /></a></span>x</dd></dl></dd> <dd>y<span class="mw-default-size" typeof="mw:File"><a href="/wiki/File:Counting_rod_v1.png" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/1/19/Counting_rod_v1.png" decoding="async" width="7" height="29" class="mw-file-element" data-file-width="7" data-file-height="29" /></a></span> <span class="mw-default-size" typeof="mw:File"><a href="/wiki/File:Counting_rod_0.png" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/5/50/Counting_rod_0.png" decoding="async" width="23" height="23" class="mw-file-element" data-file-width="23" data-file-height="23" /></a></span>太<span class="mw-default-size" typeof="mw:File"><a href="/wiki/File:Counting_rod_v1.png" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/1/19/Counting_rod_v1.png" decoding="async" width="7" height="29" class="mw-file-element" data-file-width="7" data-file-height="29" /></a></span>w <dl><dd><span class="mw-default-size" typeof="mw:File"><a href="/wiki/File:Counting_rod_v1.png" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/1/19/Counting_rod_v1.png" decoding="async" width="7" height="29" class="mw-file-element" data-file-width="7" data-file-height="29" /></a></span>z</dd></dl></dd></dl></dd></dl></dd></dl></dd></dl></dd></dl></dd></dl></dd></dl></dd></dl></dd></dl></dd></dl></dd></dl></dd></dl> <p>The square of which is: </p> <figure class="mw-halign-center" typeof="mw:File"><a href="/wiki/File:Siyuan2.png" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/8/8c/Siyuan2.png/300px-Siyuan2.png" decoding="async" width="300" height="249" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/8/8c/Siyuan2.png/450px-Siyuan2.png 1.5x, //upload.wikimedia.org/wikipedia/commons/8/8c/Siyuan2.png 2x" data-file-width="538" data-file-height="447" /></a><figcaption></figcaption></figure> <figure class="mw-halign-right" typeof="mw:File/Thumb"><a href="/wiki/File:Gouguxian.png" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/0/0e/Gouguxian.png/300px-Gouguxian.png" decoding="async" width="300" height="249" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/0/0e/Gouguxian.png/450px-Gouguxian.png 1.5x, //upload.wikimedia.org/wikipedia/commons/0/0e/Gouguxian.png 2x" data-file-width="538" data-file-height="447" /></a><figcaption>a:"go" base b "gu" vertical c "Xian" hypothenus</figcaption></figure> <div class="mw-heading mw-heading3"><h3 id="The_Unitary_Nebuls">The Unitary Nebuls</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Jade_Mirror_of_the_Four_Unknowns&action=edit&section=2" title="Edit section: The Unitary Nebuls"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>This section deals with <a href="/wiki/Tian_yuan_shu" title="Tian yuan shu">Tian yuan shu</a> or problems of one unknown. </p> <dl><dd>Question: Given the product of <i>huangfan</i> and <i>zhi ji</i> equals to 24 paces, and the sum of vertical and hypotenuse equals to 9 paces, what is the value of the base?</dd> <dd>Answer: 3 paces</dd> <dd>Set up <i>unitary tian</i> as the base( that is let the base be the unknown quantity <i>x</i>)</dd></dl> <p>Since the product of <i>huangfang</i> and <i>zhi ji</i> = 24 </p><p>in which </p> <dl><dd>huangfan is defined as：<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 (a+b-c)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">(</mo> <mi>a</mi> <mo>+</mo> <mi>b</mi> <mo>−</mo> <mi>c</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle (a+b-c)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/824193e1cee47bc8b74afbf105778b90f138f0ee" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:10.724ex; height:2.843ex;" alt="{\displaystyle (a+b-c)}"></span><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></dd> <dd><i>zhi ji</i>：<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 ab}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>a</mi> <mi>b</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle ab}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/49337c5cf256196e2292f7047cb5da68c24ca95d" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.227ex; height:2.176ex;" alt="{\displaystyle ab}"></span></dd></dl> <dl><dd>therefore <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 (a+b-c)ab=24}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">(</mo> <mi>a</mi> <mo>+</mo> <mi>b</mi> <mo>−</mo> <mi>c</mi> <mo stretchy="false">)</mo> <mi>a</mi> <mi>b</mi> <mo>=</mo> <mn>24</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle (a+b-c)ab=24}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/1de32a4cee47171bc1a333fe4ab776b67e070b04" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:18.375ex; height:2.843ex;" alt="{\displaystyle (a+b-c)ab=24}"></span></dd> <dd>Further, the sum of vertical and hypotenuse is</dd></dl> <dl><dd><dl><dd><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 b+c=9}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>b</mi> <mo>+</mo> <mi>c</mi> <mo>=</mo> <mn>9</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle b+c=9}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e11afa6f2137d44dfdeb13e9a3e2463a8745c389" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:9.106ex; height:2.343ex;" alt="{\displaystyle b+c=9}"></span></dd></dl></dd> <dd>Set up the unknown <i>unitary tian</i> as the vertical</dd></dl> <p><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 x=a}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>x</mi> <mo>=</mo> <mi>a</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle x=a}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/aaae23950e96a955ab5b07015a168fd931d4d82b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:5.658ex; height:1.676ex;" alt="{\displaystyle x=a}"></span> </p><p>We obtain the following equation </p> <dl><dd><dl><dd><dl><dd><dl><dd><dl><dd><dl><dd><dl><dd><dl><dd><dl><dd><dl><dd><span class="mw-default-size" typeof="mw:File"><a href="/wiki/File:Counting_rod_v3.png" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/b/b5/Counting_rod_v3.png" decoding="async" width="19" height="29" class="mw-file-element" data-file-width="19" data-file-height="29" /></a></span><span class="mw-default-size" typeof="mw:File"><a href="/wiki/File:Counting_rod_h8.png" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/e/e9/Counting_rod_h8.png" decoding="async" width="29" height="43" class="mw-file-element" data-file-width="29" data-file-height="43" /></a></span><span class="mw-default-size" typeof="mw:File"><a href="/wiki/File:Counting_rod_v-8.png" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/7/72/Counting_rod_v-8.png" decoding="async" width="29" height="32" class="mw-file-element" data-file-width="29" data-file-height="32" /></a></span><span class="mw-default-size" typeof="mw:File"><a href="/wiki/File:Counting_rod_h8.png" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/e/e9/Counting_rod_h8.png" decoding="async" width="29" height="43" class="mw-file-element" data-file-width="29" data-file-height="43" /></a></span> （<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 x^{5}-9x^{4}-81x^{3}+729x^{2}=3888}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>5</mn> </mrow> </msup> <mo>−</mo> <mn>9</mn> <msup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>4</mn> </mrow> </msup> <mo>−</mo> <mn>81</mn> <msup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msup> <mo>+</mo> <mn>729</mn> <msup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>=</mo> <mn>3888</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle x^{5}-9x^{4}-81x^{3}+729x^{2}=3888}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a6a0bf29ac599d659ea11438244aa6a0e2fc43d9" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:32.78ex; height:2.843ex;" alt="{\displaystyle x^{5}-9x^{4}-81x^{3}+729x^{2}=3888}"></span>） <dl><dd><dl><dd><span class="mw-default-size" typeof="mw:File"><a href="/wiki/File:Counting_rod_0.png" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/5/50/Counting_rod_0.png" decoding="async" width="23" height="23" class="mw-file-element" data-file-width="23" data-file-height="23" /></a></span> 太</dd></dl></dd> <dd><span class="mw-default-size" typeof="mw:File"><a href="/wiki/File:Counting_rod_v7.png" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/d/d1/Counting_rod_v7.png" decoding="async" width="29" height="32" class="mw-file-element" data-file-width="29" data-file-height="32" /></a></span><span class="mw-default-size" typeof="mw:File"><a href="/wiki/File:Counting_rod_h2.png" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/5/51/Counting_rod_h2.png" decoding="async" width="29" height="13" class="mw-file-element" data-file-width="29" data-file-height="13" /></a></span><span class="mw-default-size" typeof="mw:File"><a href="/wiki/File:Counting_rod_v9.png" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/7/7a/Counting_rod_v9.png" decoding="async" width="29" height="32" class="mw-file-element" data-file-width="29" data-file-height="32" /></a></span> <dl><dd><span class="mw-default-size" typeof="mw:File"><a href="/wiki/File:Counting_rod_v-8.png" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/7/72/Counting_rod_v-8.png" decoding="async" width="29" height="32" class="mw-file-element" data-file-width="29" data-file-height="32" /></a></span><span class="mw-default-size" typeof="mw:File"><a href="/wiki/File:Counting_rod_h1.png" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/1/17/Counting_rod_h1.png" decoding="async" width="29" height="7" class="mw-file-element" data-file-width="29" data-file-height="7" /></a></span> <dl><dd><span class="mw-default-size" typeof="mw:File"><a href="/wiki/File:Counting_rod_v-9.png" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/0/0d/Counting_rod_v-9.png" decoding="async" width="31" height="32" class="mw-file-element" data-file-width="31" data-file-height="32" /></a></span> <dl><dd><span class="mw-default-size" typeof="mw:File"><a href="/wiki/File:Counting_rod_v1.png" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/1/19/Counting_rod_v1.png" decoding="async" width="7" height="29" class="mw-file-element" data-file-width="7" data-file-height="29" /></a></span></dd></dl></dd></dl></dd></dl></dd></dl></dd></dl></dd></dl></dd></dl></dd></dl></dd></dl></dd></dl></dd></dl></dd></dl></dd></dl></dd></dl> <p>Solve it and obtain x=3 </p> <div class="mw-heading mw-heading3"><h3 id="The_Mystery_of_Two_Natures">The Mystery of Two Natures</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Jade_Mirror_of_the_Four_Unknowns&action=edit&section=3" title="Edit section: The Mystery of Two Natures"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <dl><dd><dl><dd><dl><dd><dl><dd><dl><dd><dl><dd><dl><dd><dl><dd><dl><dd><dl><dd><dl><dd><dl><dd><span class="mw-default-size" typeof="mw:File"><a href="/wiki/File:Counting_rod_v-2.png" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/6/64/Counting_rod_v-2.png" decoding="async" width="23" height="29" class="mw-file-element" data-file-width="23" data-file-height="29" /></a></span><span class="mw-default-size" typeof="mw:File"><a href="/wiki/File:Counting_rod_0.png" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/5/50/Counting_rod_0.png" decoding="async" width="23" height="23" class="mw-file-element" data-file-width="23" data-file-height="23" /></a></span>太 Unitary</dd> <dd><span class="mw-default-size" typeof="mw:File"><a href="/wiki/File:Counting_rod_v-1.png" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/6/6f/Counting_rod_v-1.png" decoding="async" width="23" height="29" class="mw-file-element" data-file-width="23" data-file-height="29" /></a></span><span class="mw-default-size" typeof="mw:File"><a href="/wiki/File:Counting_rod_v2.png" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/f/ff/Counting_rod_v2.png" decoding="async" width="13" height="29" class="mw-file-element" data-file-width="13" data-file-height="29" /></a></span><span class="mw-default-size" typeof="mw:File"><a href="/wiki/File:Counting_rod_0.png" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/5/50/Counting_rod_0.png" decoding="async" width="23" height="23" class="mw-file-element" data-file-width="23" data-file-height="23" /></a></span></dd> <dd><span class="mw-default-size" typeof="mw:File"><a href="/wiki/File:Counting_rod_0.png" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/5/50/Counting_rod_0.png" decoding="async" width="23" height="23" class="mw-file-element" data-file-width="23" data-file-height="23" /></a></span><span class="mw-default-size" typeof="mw:File"><a href="/wiki/File:Counting_rod_v2.png" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/f/ff/Counting_rod_v2.png" decoding="async" width="13" height="29" class="mw-file-element" data-file-width="13" data-file-height="29" /></a></span><span class="mw-default-size" typeof="mw:File"><a href="/wiki/File:Counting_rod_0.png" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/5/50/Counting_rod_0.png" decoding="async" width="23" height="23" class="mw-file-element" data-file-width="23" data-file-height="23" /></a></span></dd> <dd><span class="mw-default-size" typeof="mw:File"><a href="/wiki/File:Counting_rod_0.png" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/5/50/Counting_rod_0.png" decoding="async" width="23" height="23" class="mw-file-element" data-file-width="23" data-file-height="23" /></a></span><span class="mw-default-size" typeof="mw:File"><a href="/wiki/File:Counting_rod_0.png" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/5/50/Counting_rod_0.png" decoding="async" width="23" height="23" class="mw-file-element" data-file-width="23" data-file-height="23" /></a></span><span class="mw-default-size" typeof="mw:File"><a href="/wiki/File:Counting_rod_v1.png" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/1/19/Counting_rod_v1.png" decoding="async" width="7" height="29" class="mw-file-element" data-file-width="7" data-file-height="29" /></a></span></dd></dl></dd></dl></dd></dl></dd></dl></dd></dl></dd></dl></dd></dl></dd></dl></dd></dl></dd></dl></dd></dl></dd></dl> <p>equation: <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 -2y^{2}-xy^{2}+2xy+2x^{2}y+x^{3}=0}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo>−</mo> <mn>2</mn> <msup> <mi>y</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>−</mo> <mi>x</mi> <msup> <mi>y</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>+</mo> <mn>2</mn> <mi>x</mi> <mi>y</mi> <mo>+</mo> <mn>2</mn> <msup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mi>y</mi> <mo>+</mo> <msup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msup> <mo>=</mo> <mn>0</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle -2y^{2}-xy^{2}+2xy+2x^{2}y+x^{3}=0}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f66c09fe60354a501135354d63d4e5fb4edf0676" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:35.086ex; height:3.009ex;" alt="{\displaystyle -2y^{2}-xy^{2}+2xy+2x^{2}y+x^{3}=0}"></span>; </p><p>from the given </p> <dl><dd><dl><dd><dl><dd><dl><dd><dl><dd><dl><dd><dl><dd><dl><dd><dl><dd><dl><dd><dl><dd><dl><dd><span class="mw-default-size" typeof="mw:File"><a href="/wiki/File:Counting_rod_v2.png" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/f/ff/Counting_rod_v2.png" decoding="async" width="13" height="29" class="mw-file-element" data-file-width="13" data-file-height="29" /></a></span><span class="mw-default-size" typeof="mw:File"><a href="/wiki/File:Counting_rod_0.png" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/5/50/Counting_rod_0.png" decoding="async" width="23" height="23" class="mw-file-element" data-file-width="23" data-file-height="23" /></a></span>太</dd> <dd><span class="mw-default-size" typeof="mw:File"><a href="/wiki/File:Counting_rod_v-1.png" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/6/6f/Counting_rod_v-1.png" decoding="async" width="23" height="29" class="mw-file-element" data-file-width="23" data-file-height="29" /></a></span><span class="mw-default-size" typeof="mw:File"><a href="/wiki/File:Counting_rod_v2.png" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/f/ff/Counting_rod_v2.png" decoding="async" width="13" height="29" class="mw-file-element" data-file-width="13" data-file-height="29" /></a></span><span class="mw-default-size" typeof="mw:File"><a href="/wiki/File:Counting_rod_0.png" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/5/50/Counting_rod_0.png" decoding="async" width="23" height="23" class="mw-file-element" data-file-width="23" data-file-height="23" /></a></span></dd> <dd><span class="mw-default-size" typeof="mw:File"><a href="/wiki/File:Counting_rod_0.png" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/5/50/Counting_rod_0.png" decoding="async" width="23" height="23" class="mw-file-element" data-file-width="23" data-file-height="23" /></a></span><span class="mw-default-size" typeof="mw:File"><a href="/wiki/File:Counting_rod_0.png" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/5/50/Counting_rod_0.png" decoding="async" width="23" height="23" class="mw-file-element" data-file-width="23" data-file-height="23" /></a></span><span class="mw-default-size" typeof="mw:File"><a href="/wiki/File:Counting_rod_0.png" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/5/50/Counting_rod_0.png" decoding="async" width="23" height="23" class="mw-file-element" data-file-width="23" data-file-height="23" /></a></span></dd> <dd><span class="mw-default-size" typeof="mw:File"><a href="/wiki/File:Counting_rod_0.png" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/5/50/Counting_rod_0.png" decoding="async" width="23" height="23" class="mw-file-element" data-file-width="23" data-file-height="23" /></a></span><span class="mw-default-size" typeof="mw:File"><a href="/wiki/File:Counting_rod_0.png" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/5/50/Counting_rod_0.png" decoding="async" width="23" height="23" class="mw-file-element" data-file-width="23" data-file-height="23" /></a></span><span class="mw-default-size" typeof="mw:File"><a href="/wiki/File:Counting_rod_v1.png" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/1/19/Counting_rod_v1.png" decoding="async" width="7" height="29" class="mw-file-element" data-file-width="7" data-file-height="29" /></a></span></dd></dl></dd></dl></dd></dl></dd></dl></dd></dl></dd></dl></dd></dl></dd></dl></dd></dl></dd></dl></dd></dl></dd></dl> <p>equation: <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 2y^{2}-xy^{2}+2xy+x^{3}=0}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mn>2</mn> <msup> <mi>y</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>−</mo> <mi>x</mi> <msup> <mi>y</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>+</mo> <mn>2</mn> <mi>x</mi> <mi>y</mi> <mo>+</mo> <msup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msup> <mo>=</mo> <mn>0</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 2y^{2}-xy^{2}+2xy+x^{3}=0}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/42c49c665a5904f8ef004976a070766123b605a3" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:25.735ex; height:3.009ex;" alt="{\displaystyle 2y^{2}-xy^{2}+2xy+x^{3}=0}"></span>; </p><p>we get: </p> <dl><dd><dl><dd><dl><dd><dl><dd><dl><dd><dl><dd><dl><dd><dl><dd><dl><dd><dl><dd><dl><dd><dl><dd><dl><dd><dl><dd>太</dd> <dd><span class="mw-default-size" typeof="mw:File"><a href="/wiki/File:Counting_rod_v8.png" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/6/64/Counting_rod_v8.png" decoding="async" width="29" height="32" class="mw-file-element" data-file-width="29" data-file-height="32" /></a></span></dd> <dd><span class="mw-default-size" typeof="mw:File"><a href="/wiki/File:Counting_rod_v4.png" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/6/67/Counting_rod_v4.png" decoding="async" width="25" height="29" class="mw-file-element" data-file-width="25" data-file-height="29" /></a></span></dd></dl></dd></dl></dd></dl></dd></dl></dd></dl></dd></dl></dd></dl></dd></dl></dd></dl></dd></dl></dd> <dd><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 8x+4x^{2}=0}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mn>8</mn> <mi>x</mi> <mo>+</mo> <mn>4</mn> <msup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>=</mo> <mn>0</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 8x+4x^{2}=0}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/fd7429885081f30cd40485d2aa5e46672ebf4159" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:13.14ex; height:2.843ex;" alt="{\displaystyle 8x+4x^{2}=0}"></span></dd></dl></dd></dl></dd></dl></dd></dl> <p>and </p> <dl><dd><dl><dd><dl><dd><dl><dd><dl><dd><dl><dd><dl><dd><dl><dd><dl><dd><dl><dd><dl><dd><dl><dd><dl><dd><dl><dd><dl><dd>太</dd> <dd><span class="mw-default-size" typeof="mw:File"><a href="/wiki/File:Counting_rod_0.png" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/5/50/Counting_rod_0.png" decoding="async" width="23" height="23" class="mw-file-element" data-file-width="23" data-file-height="23" /></a></span></dd> <dd><span class="mw-default-size" typeof="mw:File"><a href="/wiki/File:Counting_rod_v2.png" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/f/ff/Counting_rod_v2.png" decoding="async" width="13" height="29" class="mw-file-element" data-file-width="13" data-file-height="29" /></a></span></dd> <dd><span class="mw-default-size" typeof="mw:File"><a href="/wiki/File:Counting_rod_v1.png" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/1/19/Counting_rod_v1.png" decoding="async" width="7" height="29" class="mw-file-element" data-file-width="7" data-file-height="29" /></a></span></dd></dl></dd></dl></dd></dl></dd></dl></dd></dl></dd></dl></dd></dl></dd></dl></dd></dl></dd></dl></dd></dl></dd> <dd><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 2x^{2}+x^{3}=0}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mn>2</mn> <msup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>+</mo> <msup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msup> <mo>=</mo> <mn>0</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 2x^{2}+x^{3}=0}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/39fe60006a9e7569a9cbaeaf5df42ca31399b86e" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:13.032ex; height:2.843ex;" alt="{\displaystyle 2x^{2}+x^{3}=0}"></span></dd></dl></dd></dl></dd></dl></dd></dl> <p>by method of elimination, we obtain a quadratic equation </p> <dl><dd><dl><dd><dl><dd><dl><dd><dl><dd><dl><dd><dl><dd><dl><dd><dl><dd><dl><dd><dl><dd><dl><dd><dl><dd><dl><dd><dl><dd><span class="mw-default-size" typeof="mw:File"><a href="/wiki/File:Counting_rod_v-8.png" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/7/72/Counting_rod_v-8.png" decoding="async" width="29" height="32" class="mw-file-element" data-file-width="29" data-file-height="32" /></a></span></dd> <dd><span class="mw-default-size" typeof="mw:File"><a href="/wiki/File:Counting_rod_v-2.png" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/6/64/Counting_rod_v-2.png" decoding="async" width="23" height="29" class="mw-file-element" data-file-width="23" data-file-height="29" /></a></span></dd> <dd><span class="mw-default-size" typeof="mw:File"><a href="/wiki/File:Counting_rod_v1.png" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/1/19/Counting_rod_v1.png" decoding="async" width="7" height="29" class="mw-file-element" data-file-width="7" data-file-height="29" /></a></span></dd></dl></dd></dl></dd></dl></dd></dl></dd></dl></dd></dl></dd></dl></dd></dl></dd></dl></dd></dl></dd></dl></dd> <dd><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 x^{2}-2x-8=0}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>−</mo> <mn>2</mn> <mi>x</mi> <mo>−</mo> <mn>8</mn> <mo>=</mo> <mn>0</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle x^{2}-2x-8=0}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/3d32db3de25dfd7963a3911bd0d2c3d71ab1ee15" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:15.98ex; height:2.843ex;" alt="{\displaystyle x^{2}-2x-8=0}"></span></dd></dl></dd></dl></dd></dl></dd></dl> <p>solution: <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 x=4}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>x</mi> <mo>=</mo> <mn>4</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle x=4}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/73bfdd8100b6a4ce07d011900560f102e3965064" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:5.591ex; height:2.176ex;" alt="{\displaystyle x=4}"></span>. </p> <div class="mw-heading mw-heading3"><h3 id="The_Evolution_of_Three_Talents">The Evolution of Three Talents</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Jade_Mirror_of_the_Four_Unknowns&action=edit&section=4" title="Edit section: The Evolution of Three Talents"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>Template for solution of problem of three unknowns </p><p>Zhu Shijie explained the method of elimination in detail. His example has been quoted frequently in scientific literature.<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><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><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> </p><p>Set up three equations as follows </p> <dl><dd><dl><dd><dl><dd><dl><dd><dl><dd><dl><dd><dl><dd><dl><dd><dl><dd><dl><dd><dl><dd><span class="mw-default-size" typeof="mw:File"><a href="/wiki/File:Counting_rod_v-1.png" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/6/6f/Counting_rod_v-1.png" decoding="async" width="23" height="29" class="mw-file-element" data-file-width="23" data-file-height="29" /></a></span>太<span class="mw-default-size" typeof="mw:File"><a href="/wiki/File:Counting_rod_v-1.png" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/6/6f/Counting_rod_v-1.png" decoding="async" width="23" height="29" class="mw-file-element" data-file-width="23" data-file-height="29" /></a></span> <dl><dd><span class="mw-default-size" typeof="mw:File"><a href="/wiki/File:Counting_rod_v1.png" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/1/19/Counting_rod_v1.png" decoding="async" width="7" height="29" class="mw-file-element" data-file-width="7" data-file-height="29" /></a></span></dd></dl></dd></dl></dd> <dd><span class="mw-default-size" typeof="mw:File"><a href="/wiki/File:Counting_rod_v-1.png" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/6/6f/Counting_rod_v-1.png" decoding="async" width="23" height="29" class="mw-file-element" data-file-width="23" data-file-height="29" /></a></span><span class="mw-default-size" typeof="mw:File"><a href="/wiki/File:Counting_rod_0.png" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/5/50/Counting_rod_0.png" decoding="async" width="23" height="23" class="mw-file-element" data-file-width="23" data-file-height="23" /></a></span><span class="mw-default-size" typeof="mw:File"><a href="/wiki/File:Counting_rod_v-1.png" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/6/6f/Counting_rod_v-1.png" decoding="async" width="23" height="29" class="mw-file-element" data-file-width="23" data-file-height="29" /></a></span></dd></dl></dd></dl></dd></dl></dd></dl></dd></dl></dd></dl></dd></dl></dd></dl></dd></dl></dd></dl> <dl><dd><dl><dd><dl><dd><dl><dd><dl><dd><dl><dd><dl><dd><dl><dd><dl><dd><dl><dd><dl><dd><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 -y-z-y^{2}x-x+xyz=0}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo>−</mo> <mi>y</mi> <mo>−</mo> <mi>z</mi> <mo>−</mo> <msup> <mi>y</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mi>x</mi> <mo>−</mo> <mi>x</mi> <mo>+</mo> <mi>x</mi> <mi>y</mi> <mi>z</mi> <mo>=</mo> <mn>0</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle -y-z-y^{2}x-x+xyz=0}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/1912537d85522a70b576934b5200c95ce856b1c0" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:28.122ex; height:3.009ex;" alt="{\displaystyle -y-z-y^{2}x-x+xyz=0}"></span> .... I</dd></dl></dd></dl></dd></dl></dd></dl></dd></dl></dd></dl></dd></dl></dd></dl></dd></dl></dd></dl></dd></dl> <dl><dd><dl><dd><dl><dd><dl><dd><dl><dd><dl><dd><dl><dd><dl><dd><dl><dd><dl><dd><dl><dd><span class="mw-default-size" typeof="mw:File"><a href="/wiki/File:Counting_rod_v-1.png" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/6/6f/Counting_rod_v-1.png" decoding="async" width="23" height="29" class="mw-file-element" data-file-width="23" data-file-height="29" /></a></span><span class="mw-default-size" typeof="mw:File"><a href="/wiki/File:Counting_rod_0.png" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/5/50/Counting_rod_0.png" decoding="async" width="23" height="23" class="mw-file-element" data-file-width="23" data-file-height="23" /></a></span><span class="mw-default-size" typeof="mw:File"><a href="/wiki/File:Counting_rod_v-1.png" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/6/6f/Counting_rod_v-1.png" decoding="async" width="23" height="29" class="mw-file-element" data-file-width="23" data-file-height="29" /></a></span> <dl><dd><span class="mw-default-size" typeof="mw:File"><a href="/wiki/File:Counting_rod_v1.png" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/1/19/Counting_rod_v1.png" decoding="async" width="7" height="29" class="mw-file-element" data-file-width="7" data-file-height="29" /></a></span></dd> <dd><span class="mw-default-size" typeof="mw:File"><a href="/wiki/File:Counting_rod_v-1.png" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/6/6f/Counting_rod_v-1.png" decoding="async" width="23" height="29" class="mw-file-element" data-file-width="23" data-file-height="29" /></a></span></dd></dl></dd></dl></dd></dl></dd></dl></dd></dl></dd></dl></dd></dl></dd></dl></dd></dl></dd></dl></dd></dl></dd></dl> <dl><dd><dl><dd><dl><dd><dl><dd><dl><dd><dl><dd><dl><dd><dl><dd><dl><dd><dl><dd><dl><dd><dl><dd><dl><dd><dl><dd><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 -y-z+x-x^{2}+xz=0}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo>−</mo> <mi>y</mi> <mo>−</mo> <mi>z</mi> <mo>+</mo> <mi>x</mi> <mo>−</mo> <msup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>+</mo> <mi>x</mi> <mi>z</mi> <mo>=</mo> <mn>0</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle -y-z+x-x^{2}+xz=0}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/83c108d816b3f496245b936f47b0083984c449c5" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:25.806ex; height:3.009ex;" alt="{\displaystyle -y-z+x-x^{2}+xz=0}"></span>.....II</dd></dl></dd></dl></dd></dl></dd></dl></dd></dl></dd></dl></dd></dl></dd></dl></dd></dl></dd></dl></dd></dl></dd></dl></dd></dl></dd></dl> <dl><dd><dl><dd><dl><dd><dl><dd><dl><dd><dl><dd><dl><dd><dl><dd><dl><dd><dl><dd><dl><dd><span class="mw-default-size" typeof="mw:File"><a href="/wiki/File:Counting_rod_v1.png" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/1/19/Counting_rod_v1.png" decoding="async" width="7" height="29" class="mw-file-element" data-file-width="7" data-file-height="29" /></a></span><span class="mw-default-size" typeof="mw:File"><a href="/wiki/File:Counting_rod_0.png" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/5/50/Counting_rod_0.png" decoding="async" width="23" height="23" class="mw-file-element" data-file-width="23" data-file-height="23" /></a></span>太<span class="mw-default-size" typeof="mw:File"><a href="/wiki/File:Counting_rod_0.png" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/5/50/Counting_rod_0.png" decoding="async" width="23" height="23" class="mw-file-element" data-file-width="23" data-file-height="23" /></a></span><span class="mw-default-size" typeof="mw:File"><a href="/wiki/File:Counting_rod_v-1.png" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/6/6f/Counting_rod_v-1.png" decoding="async" width="23" height="29" class="mw-file-element" data-file-width="23" data-file-height="29" /></a></span> <dl><dd><dl><dd><span class="mw-default-size" typeof="mw:File"><a href="/wiki/File:Counting_rod_0.png" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/5/50/Counting_rod_0.png" decoding="async" width="23" height="23" class="mw-file-element" data-file-width="23" data-file-height="23" /></a></span></dd> <dd><span class="mw-default-size" typeof="mw:File"><a href="/wiki/File:Counting_rod_v1.png" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/1/19/Counting_rod_v1.png" decoding="async" width="7" height="29" class="mw-file-element" data-file-width="7" data-file-height="29" /></a></span> <dl><dd><dl><dd><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 y^{2}-z^{2}+x^{2}=0;}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi>y</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>−</mo> <msup> <mi>z</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>+</mo> <msup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>=</mo> <mn>0</mn> <mo>;</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle y^{2}-z^{2}+x^{2}=0;}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/5d495086417b7c68a1f3b3053363126f9f47f631" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:17.332ex; height:3.009ex;" alt="{\displaystyle y^{2}-z^{2}+x^{2}=0;}"></span>....III</dd></dl></dd></dl></dd></dl></dd></dl></dd></dl></dd></dl></dd></dl></dd></dl></dd></dl></dd></dl></dd></dl></dd></dl></dd></dl></dd></dl></dd></dl> <p>Elimination of unknown between II and III </p><p>by manipulation of exchange of variables </p><p>We obtain </p> <dl><dd><dl><dd><dl><dd><dl><dd><dl><dd><dl><dd><dl><dd><dl><dd><dl><dd><dl><dd><dl><dd><dl><dd><span class="mw-default-size" typeof="mw:File"><a href="/wiki/File:Counting_rod_v1.png" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/1/19/Counting_rod_v1.png" decoding="async" width="7" height="29" class="mw-file-element" data-file-width="7" data-file-height="29" /></a></span> <span class="mw-default-size" typeof="mw:File"><a href="/wiki/File:Counting_rod_v1.png" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/1/19/Counting_rod_v1.png" decoding="async" width="7" height="29" class="mw-file-element" data-file-width="7" data-file-height="29" /></a></span><span class="mw-default-size" typeof="mw:File"><a href="/wiki/File:Counting_rod_v-2.png" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/6/64/Counting_rod_v-2.png" decoding="async" width="23" height="29" class="mw-file-element" data-file-width="23" data-file-height="29" /></a></span>太</dd> <dd><span class="mw-default-size" typeof="mw:File"><a href="/wiki/File:Counting_rod_v-1.png" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/6/6f/Counting_rod_v-1.png" decoding="async" width="23" height="29" class="mw-file-element" data-file-width="23" data-file-height="29" /></a></span><span class="mw-default-size" typeof="mw:File"><a href="/wiki/File:Counting_rod_v1.png" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/1/19/Counting_rod_v1.png" decoding="async" width="7" height="29" class="mw-file-element" data-file-width="7" data-file-height="29" /></a></span><span class="mw-default-size" typeof="mw:File"><a href="/wiki/File:Counting_rod_v-1.png" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/6/6f/Counting_rod_v-1.png" decoding="async" width="23" height="29" class="mw-file-element" data-file-width="23" data-file-height="29" /></a></span></dd> <dd><span class="mw-default-size" typeof="mw:File"><a href="/wiki/File:Counting_rod_0.png" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/5/50/Counting_rod_0.png" decoding="async" width="23" height="23" class="mw-file-element" data-file-width="23" data-file-height="23" /></a></span><span class="mw-default-size" typeof="mw:File"><a href="/wiki/File:Counting_rod_v1.png" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/1/19/Counting_rod_v1.png" decoding="async" width="7" height="29" class="mw-file-element" data-file-width="7" data-file-height="29" /></a></span><span class="mw-default-size" typeof="mw:File"><a href="/wiki/File:Counting_rod_v-2.png" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/6/64/Counting_rod_v-2.png" decoding="async" width="23" height="29" class="mw-file-element" data-file-width="23" data-file-height="29" /></a></span> <dl><dd><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 -x-2x^{2}+y+y^{2}+xy-xy^{2}+x^{2}y}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo>−</mo> <mi>x</mi> <mo>−</mo> <mn>2</mn> <msup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>+</mo> <mi>y</mi> <mo>+</mo> <msup> <mi>y</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>+</mo> <mi>x</mi> <mi>y</mi> <mo>−</mo> <mi>x</mi> <msup> <mi>y</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>+</mo> <msup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mi>y</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle -x-2x^{2}+y+y^{2}+xy-xy^{2}+x^{2}y}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4a1c61edf4a8d3b26fb651ab9719c279dbe2426f" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:36.666ex; height:3.009ex;" alt="{\displaystyle -x-2x^{2}+y+y^{2}+xy-xy^{2}+x^{2}y}"></span> ...IV</dd></dl></dd></dl></dd></dl></dd></dl></dd></dl></dd></dl></dd></dl></dd></dl></dd></dl></dd></dl></dd></dl></dd></dl></dd></dl> <p>and </p> <dl><dd><dl><dd><dl><dd><dl><dd><dl><dd><dl><dd><dl><dd><dl><dd><dl><dd><dl><dd><dl><dd><dl><dd><span class="mw-default-size" typeof="mw:File"><a href="/wiki/File:Counting_rod_v1.png" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/1/19/Counting_rod_v1.png" decoding="async" width="7" height="29" class="mw-file-element" data-file-width="7" data-file-height="29" /></a></span><span class="mw-default-size" typeof="mw:File"><a href="/wiki/File:Counting_rod_v-2.png" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/6/64/Counting_rod_v-2.png" decoding="async" width="23" height="29" class="mw-file-element" data-file-width="23" data-file-height="29" /></a></span><span class="mw-default-size" typeof="mw:File"><a href="/wiki/File:Counting_rod_v2.png" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/f/ff/Counting_rod_v2.png" decoding="async" width="13" height="29" class="mw-file-element" data-file-width="13" data-file-height="29" /></a></span>太</dd> <dd><span class="mw-default-size" typeof="mw:File"><a href="/wiki/File:Counting_rod_0.png" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/5/50/Counting_rod_0.png" decoding="async" width="23" height="23" class="mw-file-element" data-file-width="23" data-file-height="23" /></a></span><span class="mw-default-size" typeof="mw:File"><a href="/wiki/File:Counting_rod_v-2.png" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/6/64/Counting_rod_v-2.png" decoding="async" width="23" height="29" class="mw-file-element" data-file-width="23" data-file-height="29" /></a></span><span class="mw-default-size" typeof="mw:File"><a href="/wiki/File:Counting_rod_v4.png" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/6/67/Counting_rod_v4.png" decoding="async" width="25" height="29" class="mw-file-element" data-file-width="25" data-file-height="29" /></a></span><span class="mw-default-size" typeof="mw:File"><a href="/wiki/File:Counting_rod_v-2.png" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/6/64/Counting_rod_v-2.png" decoding="async" width="23" height="29" class="mw-file-element" data-file-width="23" data-file-height="29" /></a></span></dd> <dd><span class="mw-default-size" typeof="mw:File"><a href="/wiki/File:Counting_rod_0.png" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/5/50/Counting_rod_0.png" decoding="async" width="23" height="23" class="mw-file-element" data-file-width="23" data-file-height="23" /></a></span><span class="mw-default-size" typeof="mw:File"><a href="/wiki/File:Counting_rod_0.png" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/5/50/Counting_rod_0.png" decoding="async" width="23" height="23" class="mw-file-element" data-file-width="23" data-file-height="23" /></a></span><span class="mw-default-size" typeof="mw:File"><a href="/wiki/File:Counting_rod_v1.png" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/1/19/Counting_rod_v1.png" decoding="async" width="7" height="29" class="mw-file-element" data-file-width="7" data-file-height="29" /></a></span><span class="mw-default-size" typeof="mw:File"><a href="/wiki/File:Counting_rod_v-2.png" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/6/64/Counting_rod_v-2.png" decoding="async" width="23" height="29" class="mw-file-element" data-file-width="23" data-file-height="29" /></a></span></dd></dl></dd></dl></dd></dl></dd></dl></dd></dl></dd></dl></dd></dl></dd></dl></dd></dl></dd></dl></dd></dl></dd></dl> <dl><dd><dl><dd><dl><dd><dl><dd><dl><dd><dl><dd><dl><dd><dl><dd><dl><dd><dl><dd><dl><dd><dl><dd><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 -2x-2x^{2}+2y-2y^{2}+y^{3}+4xy-2xy^{2}+xy^{2}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo>−</mo> <mn>2</mn> <mi>x</mi> <mo>−</mo> <mn>2</mn> <msup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>+</mo> <mn>2</mn> <mi>y</mi> <mo>−</mo> <mn>2</mn> <msup> <mi>y</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>+</mo> <msup> <mi>y</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msup> <mo>+</mo> <mn>4</mn> <mi>x</mi> <mi>y</mi> <mo>−</mo> <mn>2</mn> <mi>x</mi> <msup> <mi>y</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>+</mo> <mi>x</mi> <msup> <mi>y</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle -2x-2x^{2}+2y-2y^{2}+y^{3}+4xy-2xy^{2}+xy^{2}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/59a2e29bde9c7ba6d7f749e781c39be99a09bff9" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:47.538ex; height:3.009ex;" alt="{\displaystyle -2x-2x^{2}+2y-2y^{2}+y^{3}+4xy-2xy^{2}+xy^{2}}"></span>.... V</dd></dl></dd></dl></dd></dl></dd></dl></dd></dl></dd></dl></dd></dl></dd></dl></dd></dl></dd></dl></dd></dl></dd></dl> <p>Elimination of unknown between IV and V we obtain a 3rd order equation </p><p><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 x^{4}-6x^{3}+4x^{2}+6x-5=0}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>4</mn> </mrow> </msup> <mo>−</mo> <mn>6</mn> <msup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msup> <mo>+</mo> <mn>4</mn> <msup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>+</mo> <mn>6</mn> <mi>x</mi> <mo>−</mo> <mn>5</mn> <mo>=</mo> <mn>0</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle x^{4}-6x^{3}+4x^{2}+6x-5=0}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/aee50a9e7759258a3e9e355e4bb1b3c9c5c3517d" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:28.754ex; height:2.843ex;" alt="{\displaystyle x^{4}-6x^{3}+4x^{2}+6x-5=0}"></span> </p> <dl><dd><dl><dd><dl><dd><dl><dd><dl><dd><dl><dd><dl><dd><dl><dd><dl><dd><dl><dd><dl><dd><dl><dd><span class="mw-default-size" typeof="mw:File"><a href="/wiki/File:Counting_rod_v-5.png" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/6/66/Counting_rod_v-5.png" decoding="async" width="35" height="29" class="mw-file-element" data-file-width="35" data-file-height="29" /></a></span></dd> <dd><span class="mw-default-size" typeof="mw:File"><a href="/wiki/File:Counting_rod_v6.png" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/1/19/Counting_rod_v6.png" decoding="async" width="29" height="32" class="mw-file-element" data-file-width="29" data-file-height="32" /></a></span></dd> <dd><span class="mw-default-size" typeof="mw:File"><a href="/wiki/File:Counting_rod_v4.png" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/6/67/Counting_rod_v4.png" decoding="async" width="25" height="29" class="mw-file-element" data-file-width="25" data-file-height="29" /></a></span></dd> <dd><span class="mw-default-size" typeof="mw:File"><a href="/wiki/File:Counting_rod_v-6.png" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/1/11/Counting_rod_v-6.png" decoding="async" width="29" height="32" class="mw-file-element" data-file-width="29" data-file-height="32" /></a></span></dd> <dd><span class="mw-default-size" typeof="mw:File"><a href="/wiki/File:Counting_rod_v1.png" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/1/19/Counting_rod_v1.png" decoding="async" width="7" height="29" class="mw-file-element" data-file-width="7" data-file-height="29" /></a></span></dd></dl></dd></dl></dd></dl></dd></dl></dd></dl></dd></dl></dd></dl></dd></dl></dd></dl></dd></dl></dd></dl></dd></dl> <p>Solve to this 3rd order equation to obtain <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 x=5}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>x</mi> <mo>=</mo> <mn>5</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle x=5}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e0f8441cf157c3f0ed6b88edd716956517c9d66c" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:5.591ex; height:2.176ex;" alt="{\displaystyle x=5}"></span>; </p><p>Change back the variables </p><p>We obtain the hypothenus =5 paces </p> <div class="mw-heading mw-heading3"><h3 id="Simultaneous_of_the_Four_Elements">Simultaneous of the Four Elements</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Jade_Mirror_of_the_Four_Unknowns&action=edit&section=5" title="Edit section: Simultaneous of the Four Elements"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>This section deals with simultaneous equations of four unknowns. </p> <figure class="mw-halign-right" typeof="mw:File/Thumb"><a href="/wiki/File:Sixianghuiyuan.jpg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/4/44/Sixianghuiyuan.jpg/300px-Sixianghuiyuan.jpg" decoding="async" width="300" height="222" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/4/44/Sixianghuiyuan.jpg/450px-Sixianghuiyuan.jpg 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/4/44/Sixianghuiyuan.jpg/600px-Sixianghuiyuan.jpg 2x" data-file-width="2000" data-file-height="1480" /></a><figcaption>Equations of four Elements</figcaption></figure> <dl><dd><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 {\begin{cases}-2y+x+z=0\\-y^{2}x+4y+2x-x^{2}+4z+xz=0\\x^{2}+y^{2}-z^{2}=0\\2y-w+2x=0\end{cases}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow> <mo>{</mo> <mtable columnalign="left left" rowspacing=".2em" columnspacing="1em" displaystyle="false"> <mtr> <mtd> <mo>−</mo> <mn>2</mn> <mi>y</mi> <mo>+</mo> <mi>x</mi> <mo>+</mo> <mi>z</mi> <mo>=</mo> <mn>0</mn> </mtd> </mtr> <mtr> <mtd> <mo>−</mo> <msup> <mi>y</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mi>x</mi> <mo>+</mo> <mn>4</mn> <mi>y</mi> <mo>+</mo> <mn>2</mn> <mi>x</mi> <mo>−</mo> <msup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>+</mo> <mn>4</mn> <mi>z</mi> <mo>+</mo> <mi>x</mi> <mi>z</mi> <mo>=</mo> <mn>0</mn> </mtd> </mtr> <mtr> <mtd> <msup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>+</mo> <msup> <mi>y</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>−</mo> <msup> <mi>z</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>=</mo> <mn>0</mn> </mtd> </mtr> <mtr> <mtd> <mn>2</mn> <mi>y</mi> <mo>−</mo> <mi>w</mi> <mo>+</mo> <mn>2</mn> <mi>x</mi> <mo>=</mo> <mn>0</mn> </mtd> </mtr> </mtable> <mo fence="true" stretchy="true" symmetric="true"></mo> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\begin{cases}-2y+x+z=0\\-y^{2}x+4y+2x-x^{2}+4z+xz=0\\x^{2}+y^{2}-z^{2}=0\\2y-w+2x=0\end{cases}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/059c8406577a7875269188ff1a2876b1dd4f4397" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -5.171ex; width:38.495ex; height:11.509ex;" alt="{\displaystyle {\begin{cases}-2y+x+z=0\\-y^{2}x+4y+2x-x^{2}+4z+xz=0\\x^{2}+y^{2}-z^{2}=0\\2y-w+2x=0\end{cases}}}"></span></dd></dl> <p>Successive elimination of unknowns to get </p> <dl><dd><dl><dd><dl><dd><dl><dd><dl><dd><dl><dd><dl><dd><dl><dd><dl><dd><dl><dd><dl><dd><dl><dd><span class="mw-default-size" typeof="mw:File"><a href="/wiki/File:Counting_rod_h6.png" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/c/c1/Counting_rod_h6.png" decoding="async" width="29" height="31" class="mw-file-element" data-file-width="29" data-file-height="31" /></a></span><span class="mw-default-size" typeof="mw:File"><a href="/wiki/File:Counting_rod_v8.png" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/6/64/Counting_rod_v8.png" decoding="async" width="29" height="32" class="mw-file-element" data-file-width="29" data-file-height="32" /></a></span><span class="mw-default-size" typeof="mw:File"><a href="/wiki/File:Counting_rod_h-6.png" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/a/a2/Counting_rod_h-6.png" decoding="async" width="40" height="40" class="mw-file-element" data-file-width="40" data-file-height="40" /></a></span> <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 4x^{2}-7x-686=0}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mn>4</mn> <msup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>−</mo> <mn>7</mn> <mi>x</mi> <mo>−</mo> <mn>686</mn> <mo>=</mo> <mn>0</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 4x^{2}-7x-686=0}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b5a86b6ce4934f2f7062c03d352b16afefe61b00" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:19.468ex; height:2.843ex;" alt="{\displaystyle 4x^{2}-7x-686=0}"></span> <dl><dd><span class="mw-default-size" typeof="mw:File"><a href="/wiki/File:Counting_rod_v-7.png" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/0/04/Counting_rod_v-7.png" decoding="async" width="29" height="32" class="mw-file-element" data-file-width="29" data-file-height="32" /></a></span></dd> <dd><span class="mw-default-size" typeof="mw:File"><a href="/wiki/File:Counting_rod_v4.png" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/6/67/Counting_rod_v4.png" decoding="async" width="25" height="29" class="mw-file-element" data-file-width="25" data-file-height="29" /></a></span></dd></dl></dd></dl></dd></dl></dd></dl></dd></dl></dd></dl></dd></dl></dd></dl></dd></dl></dd></dl></dd></dl></dd></dl></dd></dl> <p>Solve this and obtain 14 paces </p> <div class="mw-heading mw-heading2"><h2 id="Book_I">Book I</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Jade_Mirror_of_the_Four_Unknowns&action=edit&section=6" title="Edit section: Book I"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <figure class="mw-halign-right" typeof="mw:File/Thumb"><a href="/wiki/File:SIYUAN_YUJIAN_PDF-102-102.jpg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/a/a5/SIYUAN_YUJIAN_PDF-102-102.jpg/200px-SIYUAN_YUJIAN_PDF-102-102.jpg" decoding="async" width="200" height="316" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/a/a5/SIYUAN_YUJIAN_PDF-102-102.jpg/300px-SIYUAN_YUJIAN_PDF-102-102.jpg 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/a/a5/SIYUAN_YUJIAN_PDF-102-102.jpg/400px-SIYUAN_YUJIAN_PDF-102-102.jpg 2x" data-file-width="919" data-file-height="1450" /></a><figcaption></figcaption></figure> <div class="mw-heading mw-heading3"><h3 id="Problems_of_Right_Angle_Triangles_and_Rectangles">Problems of Right Angle Triangles and Rectangles</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Jade_Mirror_of_the_Four_Unknowns&action=edit&section=7" title="Edit section: Problems of Right Angle Triangles and Rectangles"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>There are 18 problems in this section. </p><p>Problem 18 </p><p>Obtain a tenth order polynomial equation: </p> <dl><dd><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 16x^{10}-64x^{9}+160x^{8}-384x^{7}+512x^{6}-544x^{5}+456x^{4}+126x^{3}+3x^{2}-4x-177162=0}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mn>16</mn> <msup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>10</mn> </mrow> </msup> <mo>−</mo> <mn>64</mn> <msup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>9</mn> </mrow> </msup> <mo>+</mo> <mn>160</mn> <msup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>8</mn> </mrow> </msup> <mo>−</mo> <mn>384</mn> <msup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>7</mn> </mrow> </msup> <mo>+</mo> <mn>512</mn> <msup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>6</mn> </mrow> </msup> <mo>−</mo> <mn>544</mn> <msup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>5</mn> </mrow> </msup> <mo>+</mo> <mn>456</mn> <msup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>4</mn> </mrow> </msup> <mo>+</mo> <mn>126</mn> <msup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msup> <mo>+</mo> <mn>3</mn> <msup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>−</mo> <mn>4</mn> <mi>x</mi> <mo>−</mo> <mn>177162</mn> <mo>=</mo> <mn>0</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 16x^{10}-64x^{9}+160x^{8}-384x^{7}+512x^{6}-544x^{5}+456x^{4}+126x^{3}+3x^{2}-4x-177162=0}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/bc41602654071f7f1a9c916c9c7fea764171d6ac" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:91.146ex; height:2.843ex;" alt="{\displaystyle 16x^{10}-64x^{9}+160x^{8}-384x^{7}+512x^{6}-544x^{5}+456x^{4}+126x^{3}+3x^{2}-4x-177162=0}"></span></dd></dl> <p>The root of which is <i>x</i> = 3, multiply by 4, getting 12. That is the final answer. </p> <div class="mw-heading mw-heading3"><h3 id="Problems_of_Plane_Figures">Problems of Plane Figures</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Jade_Mirror_of_the_Four_Unknowns&action=edit&section=8" title="Edit section: Problems of Plane Figures"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>There are 18 problems in this section </p> <div class="mw-heading mw-heading3"><h3 id="Problems_of_Piece_Goods">Problems of Piece Goods</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Jade_Mirror_of_the_Four_Unknowns&action=edit&section=9" title="Edit section: Problems of Piece Goods"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>There are 9 problems in this section </p> <div class="mw-heading mw-heading3"><h3 id="Problems_on_Grain_Storage">Problems on Grain Storage</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Jade_Mirror_of_the_Four_Unknowns&action=edit&section=10" title="Edit section: Problems on Grain Storage"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>There are 6 problems in this section </p> <div class="mw-heading mw-heading3"><h3 id="Problems_on_Labour">Problems on Labour</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Jade_Mirror_of_the_Four_Unknowns&action=edit&section=11" title="Edit section: Problems on Labour"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>There are 7 problems in this section </p> <div class="mw-heading mw-heading3"><h3 id="Problems_of_Equations_for_Fractional_Roots">Problems of Equations for Fractional Roots</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Jade_Mirror_of_the_Four_Unknowns&action=edit&section=12" title="Edit section: Problems of Equations for Fractional Roots"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>There are 13 problems in this section </p> <div class="mw-heading mw-heading2"><h2 id="Book_II">Book II</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Jade_Mirror_of_the_Four_Unknowns&action=edit&section=13" title="Edit section: Book II"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <div class="mw-heading mw-heading3"><h3 id="Mixed_Problems">Mixed Problems</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Jade_Mirror_of_the_Four_Unknowns&action=edit&section=14" title="Edit section: Mixed Problems"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <div class="mw-heading mw-heading3"><h3 id="Containment_of_Circles_and_Squares">Containment of Circles and Squares</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Jade_Mirror_of_the_Four_Unknowns&action=edit&section=15" title="Edit section: Containment of Circles and Squares"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <div class="mw-heading mw-heading3"><h3 id="Problems_on_Areas">Problems on Areas</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Jade_Mirror_of_the_Four_Unknowns&action=edit&section=16" title="Edit section: Problems on Areas"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <div class="mw-heading mw-heading3"><h3 id="Surveying_with_Right_Angle_Triangles">Surveying with Right Angle Triangles</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Jade_Mirror_of_the_Four_Unknowns&action=edit&section=17" title="Edit section: Surveying with Right Angle Triangles"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>There are eight problems in this section </p> <dl><dt>Problem 1</dt> <dd></dd></dl> <style data-mw-deduplicate="TemplateStyles:r1244412712">.mw-parser-output .templatequote{overflow:hidden;margin:1em 0;padding:0 32px}.mw-parser-output .templatequotecite{line-height:1.5em;text-align:left;margin-top:0}@media(min-width:500px){.mw-parser-output .templatequotecite{padding-left:1.6em}}</style><blockquote class="templatequote"><p>Question: There is a rectangular town of unknown dimension which has one gate on each side. There is a pagoda located at 240 paces from the south gate. A man walking 180 paces from the west gate can see the pagoda, he then walks towards the south-east corner for 240 paces and reaches the pagoda; what is the length and width of the rectangular town? Answer: 120 paces in length and width one li</p></blockquote> <p>Let tian yuan unitary as half of the length, we obtain a 4th order equation </p> <dl><dd><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 x^{4}+480x^{3}-270000x^{2}+15552000x+1866240000=0}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>4</mn> </mrow> </msup> <mo>+</mo> <mn>480</mn> <msup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msup> <mo>−</mo> <mn>270000</mn> <msup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>+</mo> <mn>15552000</mn> <mi>x</mi> <mo>+</mo> <mn>1866240000</mn> <mo>=</mo> <mn>0</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle x^{4}+480x^{3}-270000x^{2}+15552000x+1866240000=0}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/04cfad37d7d79016639e700e13c9ef53652161cb" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:55.49ex; height:2.843ex;" alt="{\displaystyle x^{4}+480x^{3}-270000x^{2}+15552000x+1866240000=0}"></span><sup id="cite_ref-8" class="reference"><a href="#cite_note-8"><span class="cite-bracket">[</span>8<span class="cite-bracket">]</span></a></sup></dd></dl> <p>solve it and obtain <span class="texhtml mvar" style="font-style:italic;">x</span>=240 paces, hence length =2x= 480 paces=1 li and 120 paces. </p><p>Similarity, let tian yuan unitary(x) equals to half of width </p><p>we get the equation: </p> <dl><dd><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 x^{4}+360x^{3}-270000x^{2}+20736000x+1866240000=0}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>4</mn> </mrow> </msup> <mo>+</mo> <mn>360</mn> <msup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msup> <mo>−</mo> <mn>270000</mn> <msup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>+</mo> <mn>20736000</mn> <mi>x</mi> <mo>+</mo> <mn>1866240000</mn> <mo>=</mo> <mn>0</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle x^{4}+360x^{3}-270000x^{2}+20736000x+1866240000=0}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c154c47fb023c7537cd41656a88cc8d00e0ad037" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:55.49ex; height:2.843ex;" alt="{\displaystyle x^{4}+360x^{3}-270000x^{2}+20736000x+1866240000=0}"></span><sup id="cite_ref-9" class="reference"><a href="#cite_note-9"><span class="cite-bracket">[</span>9<span class="cite-bracket">]</span></a></sup></dd></dl> <p>Solve it to obtain <span class="texhtml mvar" style="font-style:italic;">x</span>=180 paces, length =360 paces =one li. </p> <dl><dt>Problem 7</dt> <dd>Identical to <i>The depth of a ravine (using hence-forward cross-bars)</i> in <a href="/wiki/Haidao_Suanjing" title="Haidao Suanjing">Haidao Suanjing</a>.</dd></dl> <dl><dt>Problem 8</dt> <dd>Identical to <i>The depth of a transparent pool</i> in <a href="/wiki/Haidao_Suanjing" title="Haidao Suanjing">Haidao Suanjing</a>.</dd></dl> <div class="mw-heading mw-heading3"><h3 id="Hay_Stacks">Hay Stacks</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Jade_Mirror_of_the_Four_Unknowns&action=edit&section=18" title="Edit section: Hay Stacks"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <div class="mw-heading mw-heading3"><h3 id="Bundles_of_Arrows">Bundles of Arrows</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Jade_Mirror_of_the_Four_Unknowns&action=edit&section=19" title="Edit section: Bundles of Arrows"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <div class="mw-heading mw-heading3"><h3 id="Land_Measurement">Land Measurement</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Jade_Mirror_of_the_Four_Unknowns&action=edit&section=20" title="Edit section: Land Measurement"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <div class="mw-heading mw-heading3"><h3 id="Summon_Men_According_to_Need">Summon Men According to Need</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Jade_Mirror_of_the_Four_Unknowns&action=edit&section=21" title="Edit section: Summon Men According to Need"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>Problem No 5 is the earliest 4th order interpolation formula in the world </p><p>men summoned :<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 na+{\tfrac {1}{2!}}n(n-1)b+{\tfrac {1}{3!}}n(n-1)(n-2)c+{\tfrac {1}{4!}}n(n-1)(n-2)(n-3)d}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>n</mi> <mi>a</mi> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mfrac> <mn>1</mn> <mrow> <mn>2</mn> <mo>!</mo> </mrow> </mfrac> </mstyle> </mrow> <mi>n</mi> <mo stretchy="false">(</mo> <mi>n</mi> <mo>−</mo> <mn>1</mn> <mo stretchy="false">)</mo> <mi>b</mi> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mfrac> <mn>1</mn> <mrow> <mn>3</mn> <mo>!</mo> </mrow> </mfrac> </mstyle> </mrow> <mi>n</mi> <mo stretchy="false">(</mo> <mi>n</mi> <mo>−</mo> <mn>1</mn> <mo stretchy="false">)</mo> <mo stretchy="false">(</mo> <mi>n</mi> <mo>−</mo> <mn>2</mn> <mo stretchy="false">)</mo> <mi>c</mi> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mfrac> <mn>1</mn> <mrow> <mn>4</mn> <mo>!</mo> </mrow> </mfrac> </mstyle> </mrow> <mi>n</mi> <mo stretchy="false">(</mo> <mi>n</mi> <mo>−</mo> <mn>1</mn> <mo stretchy="false">)</mo> <mo stretchy="false">(</mo> <mi>n</mi> <mo>−</mo> <mn>2</mn> <mo stretchy="false">)</mo> <mo stretchy="false">(</mo> <mi>n</mi> <mo>−</mo> <mn>3</mn> <mo stretchy="false">)</mo> <mi>d</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle na+{\tfrac {1}{2!}}n(n-1)b+{\tfrac {1}{3!}}n(n-1)(n-2)c+{\tfrac {1}{4!}}n(n-1)(n-2)(n-3)d}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/affb66af3e3999c53866beae1fe882f89d9f46b9" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.338ex; width:68.138ex; height:3.676ex;" alt="{\displaystyle na+{\tfrac {1}{2!}}n(n-1)b+{\tfrac {1}{3!}}n(n-1)(n-2)c+{\tfrac {1}{4!}}n(n-1)(n-2)(n-3)d}"></span><sup id="cite_ref-K2_10-0" class="reference"><a href="#cite_note-K2-10"><span class="cite-bracket">[</span>10<span class="cite-bracket">]</span></a></sup> </p><p>In which </p> <ul><li><span class="texhtml mvar" style="font-style:italic;">a</span>=1st order difference</li> <li><span class="texhtml mvar" style="font-style:italic;">b</span>=2nd order difference</li> <li><span class="texhtml mvar" style="font-style:italic;">c</span>=3rd order difference</li> <li><span class="texhtml mvar" style="font-style:italic;">d</span>=4th order difference</li></ul> <div class="mw-heading mw-heading2"><h2 id="Book_III">Book III</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Jade_Mirror_of_the_Four_Unknowns&action=edit&section=22" title="Edit section: Book III"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <div class="mw-heading mw-heading3"><h3 id="Fruit_pile">Fruit pile</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Jade_Mirror_of_the_Four_Unknowns&action=edit&section=23" title="Edit section: Fruit pile"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>This section contains 20 problems dealing with triangular piles, rectangular piles </p><p>Problem 1 </p><p>Find the sum of triangular pile </p> <dl><dd><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 1+3+6+10+...+{\frac {1}{2}}n(n+1)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mn>1</mn> <mo>+</mo> <mn>3</mn> <mo>+</mo> <mn>6</mn> <mo>+</mo> <mn>10</mn> <mo>+</mo> <mo>.</mo> <mo>.</mo> <mo>.</mo> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mn>2</mn> </mfrac> </mrow> <mi>n</mi> <mo stretchy="false">(</mo> <mi>n</mi> <mo>+</mo> <mn>1</mn> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 1+3+6+10+...+{\frac {1}{2}}n(n+1)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f2be318838069dede4d2553f3102b4163e9b1d76" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:31.652ex; height:5.176ex;" alt="{\displaystyle 1+3+6+10+...+{\frac {1}{2}}n(n+1)}"></span></dd></dl> <p>and value of the fruit pile is: </p> <dl><dd><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 v=2+9+24+50+90+147+224+\cdots +{\frac {1}{2}}n(n+1)^{2}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>v</mi> <mo>=</mo> <mn>2</mn> <mo>+</mo> <mn>9</mn> <mo>+</mo> <mn>24</mn> <mo>+</mo> <mn>50</mn> <mo>+</mo> <mn>90</mn> <mo>+</mo> <mn>147</mn> <mo>+</mo> <mn>224</mn> <mo>+</mo> <mo>⋯</mo> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mn>2</mn> </mfrac> </mrow> <mi>n</mi> <mo stretchy="false">(</mo> <mi>n</mi> <mo>+</mo> <mn>1</mn> <msup> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle v=2+9+24+50+90+147+224+\cdots +{\frac {1}{2}}n(n+1)^{2}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a0a818068fcc1bd5e186be5c52d8e30769d0151d" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:57.601ex; height:5.176ex;" alt="{\displaystyle v=2+9+24+50+90+147+224+\cdots +{\frac {1}{2}}n(n+1)^{2}}"></span></dd></dl> <p>Zhu Shijie use Tian yuan shu to solve this problem by letting x=n </p><p>and obtained the formular </p> <dl><dd><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 v={\frac {1}{2\cdot 3\cdot 4}}(3x+5)x(x+1)(x+2)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>v</mi> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mrow> <mn>2</mn> <mo>⋅</mo> <mn>3</mn> <mo>⋅</mo> <mn>4</mn> </mrow> </mfrac> </mrow> <mo stretchy="false">(</mo> <mn>3</mn> <mi>x</mi> <mo>+</mo> <mn>5</mn> <mo stretchy="false">)</mo> <mi>x</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo>+</mo> <mn>1</mn> <mo stretchy="false">)</mo> <mo stretchy="false">(</mo> <mi>x</mi> <mo>+</mo> <mn>2</mn> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle v={\frac {1}{2\cdot 3\cdot 4}}(3x+5)x(x+1)(x+2)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4504350c92655079f864c24b38026a3160161980" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.005ex; width:35.825ex; height:5.343ex;" alt="{\displaystyle v={\frac {1}{2\cdot 3\cdot 4}}(3x+5)x(x+1)(x+2)}"></span></dd></dl> <p>From given condition <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 v=1320}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>v</mi> <mo>=</mo> <mn>1320</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle v=1320}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/eca77e8d0f270d43bfe30f5df31fb25b0e95e829" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:8.876ex; height:2.176ex;" alt="{\displaystyle v=1320}"></span>, hence </p> <dl><dd><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 3x^{4}+14x^{3}+21x^{2}+10x-31680=0}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mn>3</mn> <msup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>4</mn> </mrow> </msup> <mo>+</mo> <mn>14</mn> <msup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msup> <mo>+</mo> <mn>21</mn> <msup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>+</mo> <mn>10</mn> <mi>x</mi> <mo>−</mo> <mn>31680</mn> <mo>=</mo> <mn>0</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 3x^{4}+14x^{3}+21x^{2}+10x-31680=0}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/16121d198d6054ff67caf8d0141ce379bf8688bd" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:38.053ex; height:2.843ex;" alt="{\displaystyle 3x^{4}+14x^{3}+21x^{2}+10x-31680=0}"></span><sup id="cite_ref-11" class="reference"><a href="#cite_note-11"><span class="cite-bracket">[</span>11<span class="cite-bracket">]</span></a></sup></dd></dl> <p>Solve it to obtain <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 x=n=9}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>x</mi> <mo>=</mo> <mi>n</mi> <mo>=</mo> <mn>9</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle x=n=9}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d80634ef74d5bbd8deee6a8ec8e4b0cea7d85ba9" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:10.084ex; height:2.176ex;" alt="{\displaystyle x=n=9}"></span>. </p><p>Therefore, </p> <dl><dd><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 v=2+9+24+50+90+147+224+324+450=1320}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>v</mi> <mo>=</mo> <mn>2</mn> <mo>+</mo> <mn>9</mn> <mo>+</mo> <mn>24</mn> <mo>+</mo> <mn>50</mn> <mo>+</mo> <mn>90</mn> <mo>+</mo> <mn>147</mn> <mo>+</mo> <mn>224</mn> <mo>+</mo> <mn>324</mn> <mo>+</mo> <mn>450</mn> <mo>=</mo> <mn>1320</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle v=2+9+24+50+90+147+224+324+450=1320}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c51f97b2c84e4bc07e60ea95097ca34972dad3a5" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:57.947ex; height:2.343ex;" alt="{\displaystyle v=2+9+24+50+90+147+224+324+450=1320}"></span>。</dd></dl> <div class="mw-heading mw-heading3"><h3 id="Figures_within_Figure">Figures within Figure</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Jade_Mirror_of_the_Four_Unknowns&action=edit&section=24" title="Edit section: Figures within Figure"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <div class="mw-heading mw-heading3"><h3 id="Simultaneous_Equations">Simultaneous Equations</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Jade_Mirror_of_the_Four_Unknowns&action=edit&section=25" title="Edit section: Simultaneous Equations"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <div class="mw-heading mw-heading3"><h3 id="Equation_of_two_unknowns">Equation of two unknowns</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Jade_Mirror_of_the_Four_Unknowns&action=edit&section=26" title="Edit section: Equation of two unknowns"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <div class="mw-heading mw-heading3"><h3 id="Left_and_Right">Left and Right</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Jade_Mirror_of_the_Four_Unknowns&action=edit&section=27" title="Edit section: Left and Right"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <div class="mw-heading mw-heading3"><h3 id="Equation_of_Three_Unknowns">Equation of Three Unknowns</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Jade_Mirror_of_the_Four_Unknowns&action=edit&section=28" title="Edit section: Equation of Three Unknowns"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <div class="mw-heading mw-heading3"><h3 id="Equation_of_Four_Unknowns">Equation of Four Unknowns</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Jade_Mirror_of_the_Four_Unknowns&action=edit&section=29" title="Edit section: Equation of Four Unknowns"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>Six problems of four unknowns. </p><p>Question 2 </p><p>Yield a set of equations in four unknowns: .<sup id="cite_ref-12" class="reference"><a href="#cite_note-12"><span class="cite-bracket">[</span>12<span class="cite-bracket">]</span></a></sup> </p> <dl><dd><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 {\begin{cases}-3y^{2}+8y-8x+8z=0\\4y^{2}-8xy+3x^{2}-8yz+6xz+3z^{2}=0\\y^{2}+x^{2}-z^{2}=0\\2y+4x+2z-w=0\end{cases}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow> <mo>{</mo> <mtable columnalign="left left" rowspacing=".2em" columnspacing="1em" displaystyle="false"> <mtr> <mtd> <mo>−</mo> <mn>3</mn> <msup> <mi>y</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>+</mo> <mn>8</mn> <mi>y</mi> <mo>−</mo> <mn>8</mn> <mi>x</mi> <mo>+</mo> <mn>8</mn> <mi>z</mi> <mo>=</mo> <mn>0</mn> </mtd> </mtr> <mtr> <mtd> <mn>4</mn> <msup> <mi>y</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>−</mo> <mn>8</mn> <mi>x</mi> <mi>y</mi> <mo>+</mo> <mn>3</mn> <msup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>−</mo> <mn>8</mn> <mi>y</mi> <mi>z</mi> <mo>+</mo> <mn>6</mn> <mi>x</mi> <mi>z</mi> <mo>+</mo> <mn>3</mn> <msup> <mi>z</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>=</mo> <mn>0</mn> </mtd> </mtr> <mtr> <mtd> <msup> <mi>y</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>+</mo> <msup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>−</mo> <msup> <mi>z</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>=</mo> <mn>0</mn> </mtd> </mtr> <mtr> <mtd> <mn>2</mn> <mi>y</mi> <mo>+</mo> <mn>4</mn> <mi>x</mi> <mo>+</mo> <mn>2</mn> <mi>z</mi> <mo>−</mo> <mi>w</mi> <mo>=</mo> <mn>0</mn> </mtd> </mtr> </mtable> <mo fence="true" stretchy="true" symmetric="true"></mo> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\begin{cases}-3y^{2}+8y-8x+8z=0\\4y^{2}-8xy+3x^{2}-8yz+6xz+3z^{2}=0\\y^{2}+x^{2}-z^{2}=0\\2y+4x+2z-w=0\end{cases}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/fd4d6e92439e9c04fa3469352f42a3c77d636bdc" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -5.171ex; width:42.145ex; height:11.509ex;" alt="{\displaystyle {\begin{cases}-3y^{2}+8y-8x+8z=0\\4y^{2}-8xy+3x^{2}-8yz+6xz+3z^{2}=0\\y^{2}+x^{2}-z^{2}=0\\2y+4x+2z-w=0\end{cases}}}"></span></dd></dl> <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=Jade_Mirror_of_the_Four_Unknowns&action=edit&section=30" 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 mw-references-columns"><ol class="references"> <li id="cite_note-1"><span class="mw-cite-backlink"><b><a href="#cite_ref-1">^</a></b></span> <span class="reference-text">Hoe, John (1978) The Jade Mirror of the Four Unknowns - Some Reflections. Math. Chronicle 7, p. 125-156.</span> </li> <li id="cite_note-Hart-2"><span class="mw-cite-backlink"><b><a href="#cite_ref-Hart_2-0">^</a></b></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 a{background:url("//upload.wikimedia.org/wikipedia/commons/4/4c/Wikisource-logo.svg")right 0.1em center/12px no-repeat}body:not(.skin-timeless):not(.skin-minerva) .mw-parser-output .id-lock-free a,body:not(.skin-timeless):not(.skin-minerva) .mw-parser-output .id-lock-limited a,body:not(.skin-timeless):not(.skin-minerva) .mw-parser-output .id-lock-registration a,body:not(.skin-timeless):not(.skin-minerva) .mw-parser-output .id-lock-subscription a,body:not(.skin-timeless):not(.skin-minerva) .mw-parser-output .cs1-ws-icon a{background-size:contain;padding:0 1em 0 0}.mw-parser-output .cs1-code{color:inherit;background:inherit;border:none;padding:inherit}.mw-parser-output .cs1-hidden-error{display:none;color:var(--color-error,#d33)}.mw-parser-output .cs1-visible-error{color:var(--color-error,#d33)}.mw-parser-output .cs1-maint{display:none;color:#085;margin-left:0.3em}.mw-parser-output .cs1-kern-left{padding-left:0.2em}.mw-parser-output .cs1-kern-right{padding-right:0.2em}.mw-parser-output .citation .mw-selflink{font-weight:inherit}@media screen{.mw-parser-output .cs1-format{font-size:95%}html.skin-theme-clientpref-night .mw-parser-output .cs1-maint{color:#18911f}}@media screen and (prefers-color-scheme:dark){html.skin-theme-clientpref-os .mw-parser-output .cs1-maint{color:#18911f}}</style><cite id="CITEREFHart2013" class="citation book cs1">Hart, Roger (2013). <a rel="nofollow" class="external text" href="https://books.google.com/books?id=c9G6Eeh-CMgC&q=%22Jade+Mirror+of+the+Four+Unknowns%22"><i>Imagined Civilizations China, the West, and Their First Encounter</i></a>. Baltimore, MD: Johns Hopkins Univ Pr. p. 82. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-1421406060" title="Special:BookSources/978-1421406060"><bdi>978-1421406060</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Imagined+Civilizations+China%2C+the+West%2C+and+Their+First+Encounter.&rft.place=Baltimore%2C+MD&rft.pages=82&rft.pub=Johns+Hopkins+Univ+Pr&rft.date=2013&rft.isbn=978-1421406060&rft.aulast=Hart&rft.aufirst=Roger&rft_id=https%3A%2F%2Fbooks.google.com%2Fbooks%3Fid%3Dc9G6Eeh-CMgC%26q%3D%2522Jade%2BMirror%2Bof%2Bthe%2BFour%2BUnknowns%2522&rfr_id=info%3Asid%2Fen.wikipedia.org%3AJade+Mirror+of+the+Four+Unknowns" class="Z3988"></span></span> </li> <li id="cite_note-Elman-3"><span class="mw-cite-backlink"><b><a href="#cite_ref-Elman_3-0">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFElman2005" class="citation book cs1">Elman, Benjamin A. (2005). <a rel="nofollow" class="external text" href="https://books.google.com/books?id=qm57OqARqpAC&q=%22Jade+Mirror+of+the+Four+Unknowns%22"><i>On their own terms science in China, 1550-1900</i></a>. Cambridge, Mass.: Harvard University Press. p. 252. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/0674036476" title="Special:BookSources/0674036476"><bdi>0674036476</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=On+their+own+terms+science+in+China%2C+1550-1900&rft.place=Cambridge%2C+Mass.&rft.pages=252&rft.pub=Harvard+University+Press&rft.date=2005&rft.isbn=0674036476&rft.aulast=Elman&rft.aufirst=Benjamin+A.&rft_id=https%3A%2F%2Fbooks.google.com%2Fbooks%3Fid%3Dqm57OqARqpAC%26q%3D%2522Jade%2BMirror%2Bof%2Bthe%2BFour%2BUnknowns%2522&rfr_id=info%3Asid%2Fen.wikipedia.org%3AJade+Mirror+of+the+Four+Unknowns" 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">Zhu Sijie <i>Siyuan yujian</i> Science Press p. 148 2007 <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-7-03-020112-6" title="Special:BookSources/978-7-03-020112-6">978-7-03-020112-6</a></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">Wu Wenjun <i>Mechanization of Mathematics (吴文俊数学机械化《朱世杰的一个例子》) pp. 18-19 Science Press <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/7-03-010764-0" title="Special:BookSources/7-03-010764-0">7-03-010764-0</a></i></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">Zhu Shijie <i>Siyuan yujian</i>, annotated by Li Zhaohua (朱世杰原著李兆华校正《四元玉鉴》) p. 149-153 Science Press, 2007 <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-7-03-020112-6" title="Special:BookSources/978-7-03-020112-6">978-7-03-020112-6</a></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">J. Hoe Les Systèmes d'Equations Polynômes dans le Siyuan Yujian (1303), Paris: Institut des Hautes Etudes Chinoises, 1977</span> </li> <li id="cite_note-8"><span class="mw-cite-backlink"><b><a href="#cite_ref-8">^</a></b></span> <span class="reference-text">万有文库第二集朱世杰撰罗士琳草（中）卷下之五四一0-四一一。</span> </li> <li id="cite_note-9"><span class="mw-cite-backlink"><b><a href="#cite_ref-9">^</a></b></span> <span class="reference-text">万有文库第二集朱世杰撰罗士琳草（中）卷下之五四一一页。</span> </li> <li id="cite_note-K2-10"><span class="mw-cite-backlink"><b><a href="#cite_ref-K2_10-0">^</a></b></span> <span class="reference-text">孔国平 440-441。</span> </li> <li id="cite_note-11"><span class="mw-cite-backlink"><b><a href="#cite_ref-11">^</a></b></span> <span class="reference-text">Zhu Shijie Siyuan yujian, with Luo Shilin's procedures. (万有文库第二集朱世杰撰罗士琳草（中）卷下之一六四六-六四八)</span> </li> <li id="cite_note-12"><span class="mw-cite-backlink"><b><a href="#cite_ref-12">^</a></b></span> <span class="reference-text">Zhu Shijie, Siyuan yujian, annotated by Li Zhaohua, Science Press pp246-249 2007 <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-7-03-020112-6" title="Special:BookSources/978-7-03-020112-6">978-7-03-020112-6</a></span> </li> </ol></div></div> <p><b>Sources</b> </p> <style data-mw-deduplicate="TemplateStyles:r1239549316">.mw-parser-output .refbegin{margin-bottom:0.5em}.mw-parser-output .refbegin-hanging-indents>ul{margin-left:0}.mw-parser-output .refbegin-hanging-indents>ul>li{margin-left:0;padding-left:3.2em;text-indent:-3.2em}.mw-parser-output .refbegin-hanging-indents ul,.mw-parser-output .refbegin-hanging-indents ul li{list-style:none}@media(max-width:720px){.mw-parser-output .refbegin-hanging-indents>ul>li{padding-left:1.6em;text-indent:-1.6em}}.mw-parser-output .refbegin-columns{margin-top:0.3em}.mw-parser-output .refbegin-columns ul{margin-top:0}.mw-parser-output .refbegin-columns li{page-break-inside:avoid;break-inside:avoid-column}@media screen{.mw-parser-output .refbegin{font-size:90%}}</style><div class="refbegin" style=""> <ul><li>Jade Mirror of the Four Unknowns, tr. into English by Professor Chen Zhaixin, Former Head of Mathematics Department, <a href="/wiki/Yenching_University" title="Yenching University">Yenching University</a> (in 1925), Translated into modern Chinese by Guo Shuchun, Volume I & II, Library of Chinese Classics, Chinese-English, Liaoning Education Press 2006 <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/7-5382-6923-1" title="Special:BookSources/7-5382-6923-1">7-5382-6923-1</a> <a rel="nofollow" class="external free" href="https://www.scribd.com/document/357204551/Siyuan-yujian-2">https://www.scribd.com/document/357204551/Siyuan-yujian-2</a>, <a rel="nofollow" class="external free" href="https://www.scribd.com/document/357204728/Siyuan-yujian-1">https://www.scribd.com/document/357204728/Siyuan-yujian-1</a></li> <li>Collected Works in the History of Sciences by Li Yan and Qian Baocong, Volume 1 《李俨钱宝琮科学史全集》第一卷钱宝琮《中国算学史上编》</li> <li>Zhu Shijie Siyuan yujian Book 1–4, Annotated by Qin Dynasty mathematician Luo Shilin, Commercial Press</li> <li>J. Hoe, Les systèmes d'équations polynômes dans le Siyuan yujian (1303), Institut des Hautes Études Chinoises, Paris, 1977</li> <li>J. Hoe, A study of the fourteenth-century manual on polynomial equations "The jade mirror of the four unknowns" by Zhu Shijie, Mingming Bookroom, P.O. Box 29-316, Christchurch, New Zealand, 2007</li></ul> </div>    </div><noscript><img src="https://login.wikimedia.org/wiki/Special:CentralAutoLogin/start?type=1x1" alt="" width="1" height="1" style="border: none; position: absolute;"></noscript> <div class="printfooter" data-nosnippet="">Retrieved from "<a dir="ltr" href="https://en.wikipedia.org/w/index.php?title=Jade_Mirror_of_the_Four_Unknowns&oldid=1245564490">https://en.wikipedia.org/w/index.php?title=Jade_Mirror_of_the_Four_Unknowns&oldid=1245564490</a>"</div></div> <div id="catlinks" class="catlinks" data-mw="interface"><div id="mw-normal-catlinks" class="mw-normal-catlinks"><a href="/wiki/Help:Category" title="Help:Category">Categories</a>: <ul><li><a href="/wiki/Category:Chinese_mathematics" title="Category:Chinese mathematics">Chinese mathematics</a></li><li><a href="/wiki/Category:1303_books" title="Category:1303 books">1303 books</a></li><li><a href="/wiki/Category:Mathematics_books" title="Category:Mathematics books">Mathematics books</a></li></ul></div><div id="mw-hidden-catlinks" class="mw-hidden-catlinks mw-hidden-cats-hidden">Hidden categories: <ul><li><a href="/wiki/Category:Articles_with_short_description" title="Category:Articles with short description">Articles with short description</a></li><li><a href="/wiki/Category:Short_description_is_different_from_Wikidata" title="Category:Short description is different from Wikidata">Short description is different from Wikidata</a></li><li><a href="/wiki/Category:Articles_containing_simplified_Chinese-language_text" title="Category:Articles containing simplified Chinese-language text">Articles containing simplified Chinese-language text</a></li><li><a href="/wiki/Category:Articles_containing_traditional_Chinese-language_text" title="Category:Articles containing traditional Chinese-language text">Articles containing traditional Chinese-language text</a></li></ul></div></div> </div> </main> </div> <div class="mw-footer-container"> <footer id="footer" class="mw-footer" > <ul id="footer-info"> <li id="footer-info-lastmod"> This page was last edited on 13 September 2024, at 18:47<span class="anonymous-show"> (UTC)</span>.</li> <li id="footer-info-copyright">Text is available under the <a href="/wiki/Wikipedia:Text_of_the_Creative_Commons_Attribution-ShareAlike_4.0_International_License" title="Wikipedia:Text of the Creative Commons Attribution-ShareAlike 4.0 International License">Creative Commons Attribution-ShareAlike 4.0 License</a>; additional terms may apply. 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