Jade Mirror of the Four Unknowns, Siyuan yujian, also referred to as Jade Mirror of the Four Origins, is a 1303 mathematical monograph by Yuan dynasty mathematician Zhu Shijie. 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, 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 in 1247. 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. 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
The four quantities are x, y, z, w can be presented with the following diagram The square of which is:
The Unitary Nebuls
This section deals with Tian yuan shu or problems of one unknown. Since the product of huangfang and zhi ji = 24 in which We obtain the following equation Solve it and obtain x=3
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. Set up three equations as follows Elimination of unknown between II and III by manipulation of exchange of variables We obtain and 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
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 solve it and obtain =240 paces, hence length =2x= 480 paces=1 li and 120paces. Similarity, let tian yuan unitary equals to half of width we get the equation: Solve it to obtain =180 paces, length =360 paces =one li. ;Problem 7: Identical toThe depth of a ravine in Haidao Suanjing. ;Problem 8: Identical to The depth of a transparent pool in Haidao Suanjing.
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 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:.
