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[[File:四元自乘演段图.jpg|thumb|right|300px|Illustrations in Jade Mirror of the Four Unknowns]]
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[[File:Yanghui triangle.gif|thumb|right|300px|Jia Xian triangle]]
 
'''''Jade Mirror of the Four Unknowns''''',<ref>This title was suggested by [[Joseph Dauben]]</ref> '''''Siyuan yujian''''' ({{lang|zh|四元玉鉴}}), also referred to as '''''Jade Mirror of the Four Origins''''',<ref name=Hart>{{cite book|last=Hart|first=Roger|title=Imagined Civilizations China, the West, and Their First Encounter.|year=2013|publisher=Johns Hopkins Univ Pr|location=Baltimore, MD|isbn=1421406063|page=82|url=http://books.google.com/books?id=c9G6Eeh-CMgC&dq=%22Jade+Mirror+of+the+Four+Unknowns%22&source=gbs_navlinks_s}}</ref> is a 1303 mathematical monograph by Yuan dynasty mathematician [[Zhu Shijie]].<ref name=Elman>{{cite book|last=Elman|first=Benjamin A.|title=On their own terms science in China, 1550-1900|year=2005|publisher=Harvard University Press|location=Cambridge, Mass.|isbn=0674036476|page=252|url=http://books.google.com/books?id=qm57OqARqpAC&dq=%22Jade+Mirror+of+the+Four+Unknowns%22&source=gbs_navlinks_s}}</ref> With this masterpiece, Zhu brought Chinese algebra to its highest level.
 
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 [[Southern Song]] dynasty mathematician [[Qin Jiushao]]'s "Ling long kai fang" method published in Shùshū Jiǔzhāng (“[[Mathematical Treatise in Nine Sections]]”) in 1247 (more than 570 years before English mathematician [[William Horner]]'s method using synthetic division). To do this, he makes use of the [[Pascal triangle]], which he labels as the diagram of an ancient method first discovered by [[Jia Xian]] before 1050.
 
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 [[linear equation]]s by reducing the matrix of their coefficients to [[diagonal form]]. His methods predate [[Blaise Pascal]], 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.
 
''Jade Mirror of the Four Unknowns'' consists of four books, with 24 classes and 288 problems, in which 232 problems deal with [[Tian yuan shu]], 36 problems deal with variable of two variables, 13 problems of three variables, and 7 problems of four variables.
 
==Introduction==
[[File:Siyuan1.png|thumb|right|250px|The Square of the Sum of the Four Quantities of a Right Angle Triangle]]
 
The four quantities are ''x'', ''y'', ''z'', ''w''  can be presented with the following diagram
 
:::::::::::::{{v1}}x
::::::::::::y{{v1}} {{Rod0}}太{{v1}}w
:::::::::::::{{v1}}z
 
The square of which is:
 
[[File:Siyuan2.png|center|300px]]
 
[[File:Gouguxian.png|thumb|right|300px|a:"go" base b "gu" vertical  c "Xian" hypothenus]]
 
===The Unitary Nebuls ===
This section deals with [[Tian yuan shu]] or problems of one unknown.
 
:Question:Given the product of ''huangfan'' and ''zhi ji'' equals to 24 paces, and the sum of vertical and hypothenus equals to 9 paces, what is the value of the base ?
:Answer: 3 paces
:Set up  ''unitary tian'' as the base( that is let the base be the unknown quantity ''x'')
 
Since the product of ''huangfang'' and ''zhi ji'' = 24
 
in which
:huangfan is defined as：<math>(a+b-c)</math><ref>Zhu Sijie ''Siyuan yujian'' Science Press p148 2007 ISBN 978-7-03-020112-6</ref>
:''zhi ji''：<math>ab</math>
 
:therefore  <math>(a+b-c)ab=24</math>
:Further, the sum of vertical and hypothenus is
 
:: <math>b+c=9</math>
:Set up the unknown ''unitary tian'' as the vertical
<math>x=a</math>
 
We obtain the following equation
 
::::::::::{{v3}}{{h8}}{{v-8}}{{h8}}    （<math>x^5-9x^4-81x^3+729x^2=3888</math>）
::::::::::::{{Rod0}} 太
:::::::::::{{v7}}{{h2}}{{v9}}
::::::::::::{{v-8}}{{h1}}
:::::::::::::{{v-9}}
::::::::::::::{{v1}}
 
Solve it and obtain  x=3
 
===The Mystery of Two Natures ===
::::::::::::{{v-2}}{{Rod0}}太 Unitary
::::::::::::{{v-1}}{{Rod2}}{{Rod0}}
::::::::::::{{Rod0}}{{Rod2}}{{Rod0}}
::::::::::::{{Rod0}}{{Rod0}}{{v1}}
 
equation： <math>-2y^2-xy^2+2xy+2x^2y+x^3=0</math>;
 
from the given
 
::::::::::::{{Rod2}}{{Rod0}}太
::::::::::::{{v-1}}{{Rod2}}{{Rod0}}
::::::::::::{{Rod0}}{{Rod0}}{{Rod0}}
::::::::::::{{Rod0}}{{Rod0}}{{v1}}
 
equation： <math>2y^2-xy^2+2xy+x^3=0</math>;
 
we get：
 
::::::::::::::太 
::::::::::::::{{v'8}}   
::::::::::::::{{v4}}
::::<math>8x+4x^2=0</math>
 
and
 
:::::::::::::::太
:::::::::::::::{{Rod0}}
:::::::::::::::{{Rod2}}
:::::::::::::::{{v1}}
::::<math>2x^2+x^3=0</math>
 
by method of elimination, we obtain a quadratic equation
 
:::::::::::::::{{v-8}}
:::::::::::::::{{v-2}}
:::::::::::::::{{v1}}
::::<math>x^2-2x-8=0</math>
 
solution:  <math>x=4</math>。
 
===The Evolution of Three Talents ===
Template for solution of problem of three unknowns
 
Zhu Shijie explained the method of elimination in detail. His example has been quoted frequently in scientific literature<ref>Wu Wenjun ''Mechanization of Mathematics (吴文俊 数学机械化 《朱世杰的一个例子》)pp 18-19 Science Press ISBN 7-03-010764-0</ref><ref>Zhu Shijie ''Siyuan yujian'', annotated by Li Zhaohua (朱世杰原著 李兆华校正 《四元玉鉴》)p149-153 Science Press  2007 ISBN 978-7-03-020112-6</ref><ref>J. Hoe  Les Systemes d'Equation Polynomes dans le siyuanyujian[1303],Instude Haute Etudes Chinoise, Paris 1977</ref>。
 
Set up three equations as follows
 
:::::::::::{{v-1}}太{{v-1}} 
::::::::::::{{v1}}
::::::::::{{v-1}}{{Rod0}}{{v-1}}
 
:::::::::::<math>-y-z-y^2 x-x+xyz=0</math> .... I
 
:::::::::::{{v-1}}{{Rod0}}{{v-1}}
::::::::::::{{v1}}
::::::::::::{{v-1}}
 
::::::::::::::<math>-y-z+x-x^2+xz=0</math>.....II
 
:::::::::::{{v1}}{{Rod0}}太{{Rod0}}{{v-1}}
:::::::::::::{{Rod0}}
:::::::::::::{{v1}}
:::::::::::::::<math>y^2-z^2+x^2=0;</math>....III
Elimination of unknown between II and  III
 
by manipulation of exchange of variables
 
We obtain
 
::::::::::::{{v1}}      {{v1}}{{v-2}}太
::::::::::::{{v-1}}{{v1}}{{v-1}} 
::::::::::::{{Rod0}}{{v1}}{{v-2}}
:::::::::::::<math>-x-2x^2+y+y^2+xy-xy^2+x^2y</math> ...IV
and
 
::::::::::::{{v1}}{{v-2}}{{Rod2}}太
::::::::::::{{Rod0}}{{v-2}}{{v4}}{{v-2}} 
::::::::::::{{Rod0}}{{Rod0}}{{v1}}{{v-2}}
 
:::::::::::: <math>-2x-2x^2+2y-2y^2+y^3+4xy-2xy^2+xy^2</math>.... V
 
Elimination of unknown between IV and V we obtain a 3rd order equation
 
<math>x^4-6x^3+4x^2+6x-5=0</math>
 
::::::::::::{{v-5}}
::::::::::::{{v6}}   
::::::::::::{{v4}}
::::::::::::{{v-6}}
::::::::::::{{v1}}
 
Solve to this 3rd order equation to obtain <math>x=5</math> ；
 
Change back the variables
 
We obtain the hypothenus =5 paces
 
===Simultaneous of the Four Elements ====
This section deals with simultaneous equations of four unknowns。
[[File:Sixianghuiyuan.jpg|thumb|right|300px|Equations of four Elements]]
 
: 1：  {{0}} {{0}}        <math>-2y+x+z=0</math>;
: 2：  {{0}} {{0}}        <math>-y^2x+4y+2x-x^2+4z+xz=0</math>;
: 3：  {{0}} {{0}}          <math>x^2+y^2-z^2=0</math>;
: 4：  {{0}} {{0}}          <math>2y-w+2x=0</math>;
 
Successive elimination of unknowns to get
 
::::::::::::{{h6}}{{v'8}}{{h-6}}  <math>4x^2-7x-686=0</math>
:::::::::::::{{v-7}}
:::::::::::::{{v4}}
 
Solve this and obtain 14 paces
 
==Book I==
[[File:SIYUAN YUJIAN PDF-102-102.jpg|thumb|right|200px]]
 
===Problems of Right Angle Triangles and Rectangles===
There are 18 problems in this section.
 
Problem 18
 
Obtain a tenth order  polynomial equation:
 
: <math>16x^10-64x^9+160x^8-384x^7+512x^6-544x^5+456x^4+126x^3+3x^2-4x-177162=0</math>
 
The root of which is ''x'' = 3， multiply by 4, getting 12. That is the final answer。
 
===Problems of Plane Figures===
There are 18 problems in this section
 
===Problems of Piece Goods===
There are 9 problems in this section
 
===Problems on Grain Storage===
There are 6 problems in this section
 
===Problems on Labour===
There are 7 problems in this section
 
===Problmes of Equations for Fractional Roots===
There are 13 problems in this section
 
==Book II==
 
===Mixed Problems===
 
===Containment of Circles and Squares===
 
===Problems on Areas===
 
===Surveying with Right Angle Triangles===
There are eight problems in this section
;Problem 1：
{{quote|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}}
 
Let tian yuan unitary as half of the length, we obtain a 4th order equation
 
<math>x^4+480*x^3-270000*x^2+15552000*x+1866240000=0</math><ref>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-</ref>。
 
solve it and obtain  x=240 paces，hence length =2x= 480 paces=1 li and 120paces。
 
Similarity, let tian yuan unitary(x) equals to half of width
 
we get the equation：
 
<math>x^4+360*x^3-270000*x^2+20736000*x+1866240000=0</math><ref>万有文库第二集 朱世杰撰 罗士琳草 （中） 卷下之五  四一一页</ref>。
 
Solve it to obtain  x=180 paces ，length =360 paces =one li。
 
;Problem 7: Identical to ''The depth of a ravine (using hence-forward cross-bars)'' in [[The Sea Island Mathematical Manual]].
 
;Problem 8: Identical to ''The depth of a transparent pool''  in the [[The Sea Island Mathematical Manual]].
 
===Hay Stacks===
 
===Bundles of Arrows===
 
===Land Measurement===
 
===Summon Men According to Need===
Problem No 5 is the earliest 4th order interpolation formular in the world
 
men summoned :<math>n*a+\frac{1}{2*1}*n*(n-1)*b+\frac{1}{3*2*1}*n*(n-1)*(n-2)*c</math>
 
<math>+\frac{1}{4*3*2*1}n*(n-1)*(n-2)*(n-3)*d</math><ref  name="K2“"><孔国平 440-441</ref>。
 
In whicn
* a=1st order difference
* b=2nd order difference
* c=3rd order difference
* d=4th order difference
 
==Book III==
 
===Fruit pile===
This section contains 20 problems dealing with triangular piles, rectangular piles
 
Problem 1
 
Find the sum of triangular pile
：<math>1+3+6+10+...+</math><math>1 \over 2</math><math>n(n+1)</math>
 
and value of the fruit pile is:
 
<math>v=2+9+24+50+90+147+224+</math>…………<math>1 \over 2</math><math>n(n+1)^2</math>
 
Zhu Shijie use Tian yuan shu to solve this problem by letting x=n
 
and obtained the formular：<math> v=</math><math>1 \over 2*3*4</math><math>(3x+5)*x*(x+1)*(x+2)</math>
 
From given condition <math>v=1320</math>, hence
 
<math>3*x^4+14x^3+21x^2+10x-31680=0</math><ref>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)</ref>
 
Solve it to obtain<math>x=n=9</math>。
 
Therefore
 
<math>v=2+9+24+50+90+147+224+324+450=1320</math>。
 
===Figures within Figure===
 
===Simultaneous Equations===
 
===Equation of two unknowns===
 
===Left and Right===
 
===Equation of Three Unknowns===
 
===Equation of Four Unknowns===
Six problems of four unknowns。
 
Question 2
 
Yield a set of equations in four unknowns：.<ref>Zhu Shijie, Siyuan yujian, annotated by Li Zhaohua , Science Press pp246-249 2007 ISBN 978-7-03-020112-6</ref>
 
:<math>-3*y^2+8*y-8*x+8*z=0</math>
:<math>4*y^2-8*x*y+3*x^2-8*y*z+6*x*z+3*z^2=0</math>
:<math>y^2+x^2-z^2=0</math>
:<math>2*y+4*x+2*z-w=0</math>
 
==References==
{{reflist}}
* Jade Mirror of the Four Unknowns, tr. into English by Professor Chen Zhaixin, Former Head of Mathematics Department, [[Yenching University]] (in 1925),Translated into modern Chinese by Guo Shuchun, Volume I & II, Library of Chinese Classics, Chinese-English, Liaoning Education Press 2006 ISBN 7-5382-6923-1
*Collected Works in the History of Sciencs 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》
*Zhu Shijie Siyuan yujian  Book 1-4, Annotated by Qin Dyasty mathematician Luo Shilin, Commercial Press
*J. Hoe Les Systemes d'Equation Polynomes dans le siyuanyujian[1303],Instude Haute Etudes Chinoise, Paris 1977
 
[[Category:Chinese mathematics]]

Latest revision as of 08:47, 2 August 2014

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