Ceyuan haijing


Ceyuan haijing[] 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.
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:
This treatise consists of 12 volumes.

Volume 1

Diagram of a Round Town

The monography begins with a master diagram called the Diagram of Round Town. 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

Length of Line Segments

This section 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 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:

Definitions and formula

Miscellaneous formula

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

  11. The Five Sums and The Five Differences

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

Volume 2

This volume begins with a general hypothesis
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 ?
;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 ?
;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

Then subtract tian yuan from eastward paces 200 to obtain:
that is
thus：
Solve the equation and obtain

Volume 3

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

Volume 4

Q1234567891011121314151617
second line segment

Volume 5

18 problems, given。
Q123456789101112131415161718
second line segment

Volume 6

18 problems.
Q123456789101112131415161718-
Given
Second line segment-

Volume 7

18 problems, given two line segments find the diameter of round town
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.
QGiven
1，，
2，，
3，
4，
5，
6，
7，
8，
9，
10，，
11，，
12，
13，，
14，
15，
16，

Problem 14

Algorithm:
Given
：Add these two items, and divide by 2; according to #Definitions and formula, this equals to
HUANGJI difference:
This matches the horizontal of SHANGPING 8th triangle in #Segment numbers.

Volume 9

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

Volume 10

8 problems
ProblemGiven
1，
2，
3，
4，
5，
6，
7，
8，

Volume 11

：Miscellaneous 18 problems：
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
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