Jade Mirror of the Four Unknowns
Jade Mirror of the Four Unknowns,[1] Siyuan yujian (simplified Chinese: 四元玉鉴; traditional Chinese: 四元玉鑒), also referred to as Jade Mirror of the Four Origins,[2] is a 1303 mathematical monograph by Yuan dynasty mathematician Zhu Shijie.
[3] Zhu advanced Chinese algebra with this Magnum opus.
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.
The preface of the book describes how Zhu travelled around China for 20 years as a teacher of mathematics.
The four quantities are x, y, z, w can be presented with the following diagram The square of which is: This section deals with Tian yuan shu or problems of one unknown.
; we get: and by method of elimination, we obtain a quadratic equation solution:
Template for solution of problem of three unknowns Zhu Shijie explained the method of elimination in detail.
[5][6][7] 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
; Change back the variables We obtain the hypothenus =5 paces This section deals with simultaneous equations of four unknowns.
Successive elimination of unknowns to get Solve this and obtain 14 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.
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 liLet tian yuan unitary as half of the length, we obtain a 4th order equation solve it and obtain x=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: Solve it to obtain x=180 paces, length =360 paces =one li.
Problem No 5 is the earliest 4th order interpolation formula in the world men summoned :
