Jade Mirror of the Four Unknowns,Hoe, John (1978) The Jade Mirror of the Four Unknowns - Some Reflections. Math. Chronicle 7, p. 125-156. Siyuan yujian (), also referred to as Jade Mirror of the Four Origins, is a 1303 mathematical monograph by Yuan dynasty mathematician Zhu Shijie.
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 equations by reducing the matrix of their coefficients to diagonal form.
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
The four quantities are x, y, z, w can be presented with the following diagram
x y 太w z
The square of which is:
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 hypotenuse 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：Zhu Sijie Siyuan yujian Science Press p. 148 2007 zhi ji：
therefore Further, the sum of vertical and hypotenuse is
Set up the unknown unitary tian as the vertical
Then use Pythagoras to isolate c and b:
such that we obtain:
Combining everything, we obtain the following equation:
（） 太
We have and ,
so .
Then we get .
Which is .
If we now regroup by power: .
Cleaning up, we finally get: .
Solve it and obtain x=3
The Mystery of Two Natures 太 Unitary
equation: ;
from the given
太
equation: ;
we get:
太
and
太
by method of elimination, we obtain a quadratic equation
solution: .
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.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 Zhu Shijie Siyuan yujian, annotated by Li Zhaohua (朱世杰原著 李兆华校正 《四元玉鉴》) p. 149-153 Science Press, 2007 J. Hoe Les Systèmes d'Equations Polynômes dans le Siyuan Yujian (1303), Paris: Institut des Hautes Etudes Chinoises, 1977
Set up three equations as follows
太
.... I
.....II
太
....III Elimination of unknown between II and III
by manipulation of exchange of variables
We obtain
太
...IV and
太
.... V
Elimination of unknown between IV and V we obtain a 3rd order equation
Solve to this 3rd order equation to obtain ;
Change back the variables
We obtain the hypothenus =5 paces
Simultaneous of the Four Elements This section deals with simultaneous equations of four unknowns.
Successive elimination of unknowns to get
Solve this and obtain 14 paces
Book I
Problems of Right Angle Triangles and Rectangles There are 18 problems in this section.
Problem 18
Obtain a tenth order polynomial equation:
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
Problems 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
Let tian yuan unitary as half of the length, we obtain a 4th order equation
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。
solve it and obtain =240 paces, hence length =2x= 480 paces=1 li and 120 paces.
Similarity, let tian yuan unitary(x) equals to half of width
we get the equation:
万有文库第二集 朱世杰撰 罗士琳草 （中） 卷下之五 四一一页。
Solve it to obtain =180 paces, length =360 paces =one li.
Problem 7 Identical to The depth of a ravine (using hence-forward cross-bars) in Haidao Suanjing.
Problem 8 Identical to The depth of a transparent pool in Haidao Suanjing.
Hay Stacks
Bundles of Arrows
Land Measurement
Summon Men According to Need Problem No 5 is the earliest 4th order interpolation formula in the world
men summoned :孔国平 440-441。
In which =1st order difference =2nd order difference =3rd order difference =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
and value of the fruit pile is:
Zhu Shijie use Tian yuan shu to solve this problem by letting x=n
and obtained the formular
From given condition , hence
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)
Solve it to obtain .
Therefore,
。
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: .Zhu Shijie, Siyuan yujian, annotated by Li Zhaohua, Science Press pp246-249 2007
References
Sources
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 https://www.scribd.com/document/357204551/Siyuan-yujian-2, https://www.scribd.com/document/357204728/Siyuan-yujian-1 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》 Zhu Shijie Siyuan yujian Book 1–4, Annotated by Qin Dynasty mathematician Luo Shilin, Commercial Press J. Hoe, Les systèmes d'équations polynômes dans le Siyuan yujian (1303), Institut des Hautes Études Chinoises, Paris, 1977 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