Jade Mirror of the Four Unknowns

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.

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 pacesSet 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： ( a + b − c ) zhi ji： a b therefore ( a + b − c ) a b = 24 Further, the sum of vertical and hypothenus isSet up the unknown unitary tian as the vertical

x = a

We obtain the following equation

Solve it and obtain x=3

equation： − 2 y 2 − x y 2 + 2 x y + 2 x 2 y + x 3 = 0 ;

from the given

equation： 2 y 2 − x y 2 + 2 x y + x 3 = 0 ;

we get：

and

by method of elimination, we obtain a quadratic equation

solution: x = 4 。

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。

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

x 4 − 6 x 3 + 4 x 2 + 6 x − 5 = 0

Solve to this 3rd order equation to obtain x = 5 ；

Change back the variables

We obtain the hypothenus =5 paces

Simultaneous of the Four Elements

This section deals with simultaneous equations of four unknowns。

1： 0 0 − 2 y + x + z = 0 ;2： 0 0 − y 2 x + 4 y + 2 x − x 2 + 4 z + x z = 0 ;3： 0 0 x 2 + y 2 − z 2 = 0 ;4： 0 0 2 y − w + 2 x = 0 ;

Successive elimination of unknowns to get

Solve this and obtain 14 paces

Problems of Right Angle Triangles and Rectangles

There are 18 problems in this section.

Problem 18

Obtain a tenth order polynomial equation:

16 x 10 − 64 x 9 + 160 x 8 − 384 x 7 + 512 x 6 − 544 x 5 + 456 x 4 + 126 x 3 + 3 x 2 − 4 x − 177162 = 0

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

Surveying with Right Angle Triangles

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

x 4 + 480 ∗ x 3 − 270000 ∗ x 2 + 15552000 ∗ x + 1866240000 = 0 。

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：

x 4 + 360 ∗ x 3 − 270000 ∗ x 2 + 20736000 ∗ x + 1866240000 = 0 。

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.

Summon Men According to Need

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

men summoned : n ∗ a + 1 2 ∗ 1 ∗ n ∗ ( n − 1 ) ∗ b + 1 3 ∗ 2 ∗ 1 ∗ n ∗ ( n − 1 ) ∗ ( n − 2 ) ∗ c

+ 1 4 ∗ 3 ∗ 2 ∗ 1 n ∗ ( n − 1 ) ∗ ( n − 2 ) ∗ ( n − 3 ) ∗ d 。

In whicn

a=1st order difference

b=2nd order difference

c=3rd order difference

d=4th order difference

Fruit pile

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

Problem 1

Find the sum of triangular pile ： 1 + 3 + 6 + 10 + . . . + 1 2 n ( n + 1 )

and value of the fruit pile is:

v = 2 + 9 + 24 + 50 + 90 + 147 + 224 + ………… 1 2 n ( n + 1 ) 2

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

and obtained the formular： v = 1 2 ∗ 3 ∗ 4 ( 3 x + 5 ) ∗ x ∗ ( x + 1 ) ∗ ( x + 2 )

From given condition v = 1320 , hence

3 ∗ x 4 + 14 x 3 + 21 x 2 + 10 x − 31680 = 0

Solve it to obtain x = n = 9 。

Therefore,

v = 2 + 9 + 24 + 50 + 90 + 147 + 224 + 324 + 450 = 1320 。

Equation of Four Unknowns

Six problems of four unknowns。

Question 2

Yield a set of equations in four unknowns：.

− 3 ∗ y 2 + 8 ∗ y − 8 ∗ x + 8 ∗ z = 0 4 ∗ y 2 − 8 ∗ x ∗ y + 3 ∗ x 2 − 8 ∗ y ∗ z + 6 ∗ x ∗ z + 3 ∗ z 2 = 0 y 2 + x 2 − z 2 = 0 2 ∗ y + 4 ∗ x + 2 ∗ z − w = 0