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 edit

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 hypothenuse 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：${\displaystyle (a+b-c)}$ ^[4]

zhi ji：${\displaystyle ab}$ 

therefore ${\displaystyle (a+b-c)ab=24}$ 

Further, the sum of vertical and hypothenuse is

${\displaystyle b+c=9}$ 

Set up the unknown unitary tian as the vertical

${\displaystyle x=a}$ 

We obtain the following equation

     （${\displaystyle x^{5}-9x^{4}-81x^{3}+729x^{2}=3888}$ ）

  太

   

  

 

 

Solve it and obtain x=3

The Mystery of Two Natures edit

  太 Unitary

   

   

   

equation: ${\displaystyle -2y^{2}-xy^{2}+2xy+2x^{2}y+x^{3}=0}$ ;

from the given

  太

   

   

   

equation: ${\displaystyle 2y^{2}-xy^{2}+2xy+x^{3}=0}$ ;

we get:

太

 

 

${\displaystyle 8x+4x^{2}=0}$ 

and

太

 

 

 

${\displaystyle 2x^{2}+x^{3}=0}$ 

by method of elimination, we obtain a quadratic equation

 

 

 

${\displaystyle x^{2}-2x-8=0}$ 

solution: ${\displaystyle x=4}$ .

The Evolution of Three Talents edit

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.^[5][6][7]

Set up three equations as follows

 太 

 

   

${\displaystyle -y-z-y^{2}x-x+xyz=0}$  .... I

${\displaystyle -y-z+x-x^{2}+xz=0}$ .....II

  太  

 

 

${\displaystyle y^{2}-z^{2}+x^{2}=0;}$ ....III

Elimination of unknown between II and III

by manipulation of exchange of variables

We obtain

    太

   

   

${\displaystyle -x-2x^{2}+y+y^{2}+xy-xy^{2}+x^{2}y}$  ...IV

and

   太

    

    

${\displaystyle -2x-2x^{2}+2y-2y^{2}+y^{3}+4xy-2xy^{2}+xy^{2}}$ .... V

Elimination of unknown between IV and V we obtain a 3rd order equation

${\displaystyle x^{4}-6x^{3}+4x^{2}+6x-5=0}$ 

Solve to this 3rd order equation to obtain ${\displaystyle x=5}$ ;

Change back the variables

We obtain the hypothenus =5 paces

Simultaneous of the Four Elements edit

This section deals with simultaneous equations of four unknowns.

${\displaystyle {\begin{cases}-2y+x+z=0\\-y^{2}x+4y+2x-x^{2}+4z+xz=0\\x^{2}+y^{2}-z^{2}=0\\2y-w+2x=0\end{cases}}}$ 

Successive elimination of unknowns to get

    ${\displaystyle 4x^{2}-7x-686=0}$ 

 

 

Solve this and obtain 14 paces

Problems of Right Angle Triangles and Rectangles edit

There are 18 problems in this section.

Problem 18

Obtain a tenth order polynomial equation:

${\displaystyle 16x^{10}-64x^{9}+160x^{8}-384x^{7}+512x^{6}-544x^{5}+456x^{4}+126x^{3}+3x^{2}-4x-177162=0}$ 

The root of which is x = 3, multiply by 4, getting 12. That is the final answer.

Problems of Plane Figures edit

There are 18 problems in this section

Problems of Piece Goods edit

There are 9 problems in this section

Problems on Grain Storage edit

There are 6 problems in this section

Problems on Labour edit

There are 7 problems in this section

Problems of Equations for Fractional Roots edit

There are 13 problems in this section

Mixed Problems edit

Containment of Circles and Squares edit

Problems on Areas edit

Surveying with Right Angle Triangles edit

There are eight problems in this section

Problem 1

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

${\displaystyle x^{4}+480x^{3}-270000x^{2}+15552000x+1866240000=0}$ ^[8]

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:

${\displaystyle x^{4}+360x^{3}-270000x^{2}+20736000x+1866240000=0}$ ^[9]

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 Haidao Suanjing.

Problem 8

Identical to The depth of a transparent pool in Haidao Suanjing.

Hay Stacks edit

Bundles of Arrows edit

Land Measurement edit

Summon Men According to Need edit

Problem No 5 is the earliest 4th order interpolation formula in the world

men summoned :${\displaystyle na+{\tfrac {1}{2!}}n(n-1)b+{\tfrac {1}{3!}}n(n-1)(n-2)c+{\tfrac {1}{4!}}n(n-1)(n-2)(n-3)d}$ ^[10]

In which

• a=1st order difference
• b=2nd order difference
• c=3rd order difference
• d=4th order difference

Fruit pile edit

This section contains 20 problems dealing with triangular piles, rectangular piles

Problem 1

Find the sum of triangular pile

${\displaystyle 1+3+6+10+...+{\frac {1}{2}}n(n+1)}$ 

and value of the fruit pile is:

${\displaystyle v=2+9+24+50+90+147+224+\cdots +{\frac {1}{2}}n(n+1)^{2}}$ 

Zhu Shijie use Tian yuan shu to solve this problem by letting x=n

and obtained the formular

${\displaystyle v={\frac {1}{2\cdot 3\cdot 4}}(3x+5)x(x+1)(x+2)}$ 

From given condition ${\displaystyle v=1320}$ , hence

${\displaystyle 3x^{4}+14x^{3}+21x^{2}+10x-31680=0}$ ^[11]

Solve it to obtain ${\displaystyle x=n=9}$ .

Therefore,

${\displaystyle v=2+9+24+50+90+147+224+324+450=1320}$ 。

Figures within Figure edit

Simultaneous Equations edit

Equation of two unknowns edit

Left and Right edit

Equation of Three Unknowns edit

Equation of Four Unknowns edit

Six problems of four unknowns.

Question 2

Yield a set of equations in four unknowns: .^[12]

${\displaystyle {\begin{cases}-3y^{2}+8y-8x+8z=0\\4y^{2}-8xy+3x^{2}-8yz+6xz+3z^{2}=0\\y^{2}+x^{2}-z^{2}=0\\2y+4x+2z-w=0\end{cases}}}$ 

Sources