Ceyuan haijing

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The master figure in Sea mirror of circle measurements, that all the problems use. It shows a round town, inscribed in a right triangle and a square. Yuan Cheng Tu Shi .jpg
The master figure in Sea mirror of circle measurements, that all the problems use. It shows a round town, inscribed in a right triangle and a square.

Ceyuan haijing (simplified Chinese :测圆海镜; traditional Chinese :測圓海鏡; pinyin :cè yuán hǎi jìng; lit. : 'sea mirror of circle measurements') is a treatise on solving geometry problems with the algebra of Tian yuan shu written by the mathematician Li Zhi in 1248 in the time of the Mongol Empire. It is a collection of 692 formula and 170 problems, all derived from the same master diagram of a round town inscribed in a right triangle and a square. They often involve two people who walk on straight lines until they can see each other, meet or reach a tree or pagoda in a certain spot. It is an algebraic geometry book, the purpose of book is to study intricated geometrical relations by algebra.

Contents

Majority of the geometry problems are solved by polynomial equations, which are represented using a method called tian yuan shu, "coefficient array method" or literally "method of the celestial unknown". Li Zhi is the earliest extant source of this method, though it was known before him in some form. It is a positional system of rod numerals to represent polynomial equations.

Ceyuan haijing was first introduced to the west by the British Protestant Christian missionary to China, Alexander Wylie in his book Notes on Chinese Literature, 1902. He wrote:

The first page has a diagram of a circle contained in a triangle, which is dissected into 15 figures; the definition and ratios of the several parts are then given, and there are followed by 170 problems, in which the principle of the new science are seen to advantage. There is an exposition and scholia throughout by the author. [1]

This treatise consists of 12 volumes.

Volume 1

Reconstructed Diagram of circular city in alphabets LYYUANCHENTU.png
Reconstructed Diagram of circular city in alphabets

Diagram of a Round Town

The monography begins with a master diagram called the Diagram of Round Town(圆城图式). It shows a circle inscribed in a right angle triangle and four horizontal lines, four vertical lines.

C: Center of circle:

The North, South, East and West direction in Li Zhi's diagram are opposite to our present convention.

Triangles and their sides

There are a total of fifteen right angle triangles formed by the intersection between triangle TLQ, the four horizontal lines, and four vertical lines.

The names of these right angle triangles and their sides are summarized in the following table

NumberNameVerticesHypotenusecVerticalbHorizontala
1通 TONG天地乾 通弦(TL天地)通股(TQ天乾)通勾(LQ地乾)
2边 BIAN天西川 边弦(TB天川)边股(TW天西)边勾(WB西川)
3底 DI日地北 底弦(RL日地)底股(RN日北)底勾(LB地北)
4黄广 HUANGGUANG天山金 黄广弦(TM天山)黄广股(TJ天金)黄广勾(MJ山金)
5黄长 HUANGCHANG月地泉 黄长弦(YL月地)黄长股(YS月泉)黄长勾(LS地泉)
6上高 SHANGGAO天日旦 上高弦(TR天日)上高股(TD天旦)上高勾(RD日旦)
7下高 XIAGAO日山朱 下高弦(RM日山)下高股(RZ日朱)下高勾(MZ山朱)
8上平 SHANGPING月川青 上平弦(YS月川)上平股(YG月青)上平勾(SG川青)
9下平 XIAPING川地夕 下平弦(BL川地)下平股(BJ川夕)下平勾(LJ地夕)
10大差 DACHA天月坤 大差弦(TY天月)大差股(TK天坤)大差勾(YK月坤)
11小差 XIAOCHA山地艮 小差弦(ML山地)小差股(MH山艮)小差勾(LH地艮)
12皇极 HUANGJI日川心 皇极弦(RS日川)皇极股(RC日心)皇极勾(SC川心)
13太虚 TAIXU月山泛 太虚弦(YM月山)太虚股(YF月泛)太虚勾(MF山泛)
14明 MING日月南 明弦(RY日月)明股(RS日南)明勾(YS月南)
15叀 ZHUAN山川东 叀弦(MS山川)叀股(ME山东)叀勾(SE川东)

In problems from Vol 2 to Vol 12, the names of these triangles are used in very terse terms. For instance

"明差","MING difference" refers to the "difference between the vertical side and horizontal side of MING triangle.
"叀差","ZHUANG difference" refers to the "difference between the vertical side and horizontal side of ZHUANG triangle."
"明差叀差并" means "the sum of MING difference and ZHUAN difference"

Length of Line Segments

This section (今问正数) lists the length of line segments, the sum and difference and their combinations in the diagram of round town, given that the radius r of inscribe circle is paces ,.

The 13 segments of ith triangle (i=1 to 15) are:

  1. Hypoteneuse
  2. Horizontal
  3. Vertical
  4. :勾股和 :sum of horizontal and vertical
  5. :勾股校: difference of vertical and horizontal
  6. :勾弦和: sum of horizontal and hypotenuse
  7. :勾弦校: difference of hypotenuse and horizontal
  8. :股弦和: sum of hypotenuse and vertical
  9. :股弦校: difference of hypotenuse and vertical
  10. :弦校和: sum of the difference and the hypotenuse
  11. :弦校校: difference of the hypotenuse and the difference
  12. :弦和和: sum the hypotenuse and the sum of vertical and horizontal
  13. :弦和校: difference of the sum of horizontal and vertical with the hypotenuse

Among the fifteen right angle triangles, there are two sets of identical triangles:

=,
=

that is

;
;
;
;
;
;

Segment numbers

There are 15 x 13 =195 terms, their values are shown in Table 1: [2]

Segment Table 1 Jin Wen Zheng Shu .jpg
Segment Table 1

Definitions and formula

Miscellaneous formula

[3]

  1. = *
  2. =
  3. =
  4. =
  5. =
  6. =
  7. =
  8. =
  9. =
  10. = =

The Five Sums and The Five Differences

  1. [4]

Li Zhi derived a total of 692 formula in Ceyuan haijing. Eight of the formula are incorrect, the rest are all correct [5]

From vol 2 to vol 12, there are 170 problems, each problem utilizing a selected few from these formula to form 2nd order to 6th order polynomial equations. As a matter of fact, there are 21 problems yielding third order polynomial equation, 13 problem yielding 4th order polynomial equation and one problem yielding 6th order polynomial [6]

Volume 2

This volume begins with a general hypothesis [7]

Suppose there is a round town, with unknown diameter. This town has four gates, there are two WE direction roads and two NS direction roads outside the gates forming a square surrounding the round town. The NW corner of the square is point Q, the NE corner is point H, the SE corner is point V, the SW corner is K. All the various survey problems are described in this volume and the following volumes.

All subsequent 170 problems are about given several segments, or their sum or difference, to find the radius or diameter of the round town. All problems follow more or less the same format; it begins with a Question, followed by description of algorithm, occasionally followed by step by step description of the procedure.

Nine types of inscribed circle

The first ten problems were solved without the use of Tian yuan shu. These problems are related to various types of inscribed circle.

Question 1
Two men A and B start from corner Q. A walks eastward 320 paces and stands still. B walks southward 600 paces and see B. What is the diameter of the circular city ?
Answer: the diameter of the round town is 240 paces.
This is inscribed circle problem associated with
Algorithm:
Question 2
Two men A and B start from West gate. B walks eastward 256 paces, A walks south 480 paces and sees B. What is the diameter of the town ?
Answer 240 paces
This is inscribed circle problem associated with
From Table 1, 256 = ; 480 =
Algorithm:
Question 3
inscribed circle problem associated with

Question 4:inscribed circle problem associated with

Question 5:inscribed circle problem associated with

Question 6

Question 7

Question 8

Question 9

Question 10

Tian yuan shu

Ciyuan haijing vol II Problem 14 detail procedure (Cao Yue ) CIYUANHAIJINGXICAO-152-152.jpg
Ciyuan haijing vol II Problem 14 detail procedure (草曰)
From problem 14 onwards, Li Zhi introduced "Tian yuan one" as unknown variable, and set up two expressions according to Section Definition and formula, then equate these two tian yuan shu expressions. He then solved the problem and obtained the answer.
Question 14:"Suppose a man walking out from West gate and heading south for 480 paces and encountered a tree. He then walked out from the North gate heading east for 200 paces and saw the same tree. What is the radius of the round own?"。
Algorithm: Set up the radius as Tian yuan one, place the counting rods representing southward 480 paces on the floor, subtract the tian yuan radius to obtain

Counting rod v-1.png
Counting rod v-4.png Counting rod h8.png Counting rod 0.png

Then subtract tian yuan from eastward paces 200 to obtain:

Counting rod v-1.png
Counting rod v2.png Counting rod 0.png Counting rod 0.png
multiply these two expressions to get:
Counting rod v1.png
Counting rod h6.png Counting rod h-8.png Counting rod 0.png
Counting rod v9.png Counting rod h6.png Counting rod 0.png Counting rod 0.png Counting rod 0.png
Counting rod v2.png
Counting rod 0.png

that is

thus:

Counting rod v-1.png
Counting rod h6.png Counting rod h-8.png Counting rod 0.png
Counting rod v9.png Counting rod h6.png Counting rod 0.png Counting rod 0.png Counting rod 0.png

Solve the equation and obtain

Volume 3

17 problems associated with segment i.e TW in [8]

The pairs with , pairs with and pairs with in problems with same number of volume 4. In other words, for example, change of problem 2 in vol 3 into turns it into problem 2 of Vol 4. [9]

Problem #GIVENxEquation
1direct calculation without tian yuan
2d
3r
4d
5d
6r
7r
8r
9r
10r
11r
12
13
14
15r
16calculate with formula for inscribed circle
17Calculate with formula forinscribed circle

Volume 4

17 problems, given and a second segment, find diameter of circular city. [10]

Q1234567891011121314151617
second line segment

Volume 5

18 problems, given [10]

Q123456789101112131415161718
second line segment

Volume 6

18 problems.

Q1-11,13-19 given,and a second line segment, find diameter d. [10]
Q12:given and another line segment, find diameter d.
Q123456789101112131415161718
Given
Second line segment

Volume 7

18 problems, given two line segments find the diameter of round town [11]

QGiven
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18

Volume 8

17 problems, given three to eight segments or their sum or difference, find diameter of round city. [12]

QGiven
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16

Problem 14

Given the sum of GAO difference and MING difference is 161 paces and the sum of MING difference and ZHUAN difference is 77 paces. What is the diameter of the round city?
Answer: 120 paces.

Algorithm: [13]

Given

:Add these two items, and divide by 2; according to #Definitions and formula, this equals to HUANGJI difference:

Let Tian yuan one as the horizontal of SHANGPING (SG):
=
(#Definition and formula)
Since (Definition and formula)
(diameter of round town),
Now, multiply the length of RZ by
multiply it with the square of RS:
equate the expressions for the two
thus
We obtain:

solve it and we obtain ;

This matches the horizontal of SHANGPING 8th triangle in #Segment numbers. [14]

Volume 9

Part I
Problemsgiven
1
2
3
4
Part II
Problemsgiven
1
2
3
4
5
6
7
8

Volume 10

8 problems [15]

ProblemGiven
1
2
3
4
5
6
7
8

Volume 11

:Miscellaneous 18 problems: [16]

QGIVEN
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17From the book Dongyuan jiurong
18From Dongyuan jiurong

Volume 12

14 problems on fractions [17]

Problemgiven
1=
2=
3
4
5
6
7
8
9
10
11
12
13,
14

Research

In 1913, French mathematician L. van Hoe wrote an article about Ceyuan haijing. In 1982, K. Chemla Ph.D thesis Etude du Livre Reflects des Mesuers du Cercle sur la mer de Li Ye. 1983, University of Singapore Mathematics Professor Lam Lay Yong: Chinese Polynomial Equations in the Thirteenth Century。

Footnotes

  1. Alexander Wylie, Notes on Chinese Literature, Shanghai, p116, reprinted by Kessinger Publishing
  2. Compiled from Kong Guoping p 62-66
  3. Bai Shangshu p24-25.
  4. Wu Wenjun Chapter II p80
  5. Bai Shangshu, p3, Preface
  6. Wu Wenjun, p87
  7. Bai Shangshou, p153-154
  8. Li Yan p75-88
  9. Martzloff, p147
  10. 1 2 3 Li Yan p88-101
  11. Kong Guoping p169-184
  12. Kong Guoping p192-208
  13. Bai Shangshu, p562-566
  14. Footnote:In Vol 8 problem 14, Li Zhi stop short at x=64. However the answer is evident, as from No 8 formular in #Miscellaneous formula: , and from #Length of Line Segments , thus , radius of round town can be readily obtain. As a matter of fact, problem 6 of vol 11 is just such a question of given and, to find the radius of the round town.
  15. Kong Guoping p220-224
  16. Kong Guoping p234-248
  17. P255-263

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References