* Ceyuan haijing* (simplified Chinese :测圆海镜; traditional Chinese :測圓海鏡; pinyin :

- Volume 1
- Diagram of a Round Town
- Triangles and their sides
- Length of Line Segments
- Definitions and formula
- Volume 2
- Tian yuan shu
- Volume 3
- Volume 4
- Volume 5
- Volume 6
- Volume 7
- Volume 8
- Problem 14
- Volume 9
- Volume 10
- Volume 11
- Volume 12
- Research
- Footnotes
- References

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.

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.

- TLQ, the large right angle triangle, with horizontal line LQ, vertical line TQ and hypotenuse TL

C: Center of circle:

- NCS: A vertical line through C, intersect the circle and line LQ at N(南north side of city wall), intersects south side of circle at S(南).
- NCSR, Extension of line NCS to intersect hypotenuse TL at R(日)
- WCE: a horizontal line passing center C, intersects circle and line TQ at W(西,west side of city wall) and circle at E (东,east side of city wall).
- WCEB:extension of line WCE to intersect hypotenuse at B(川)
- KSYV: a horizontal tangent at S, intersects line TQ at K(坤), hypotenuse TL at Y(月).
- HEMV: vertical tangent of circle at point E, intersects line LQ at H, hypotenuse at M(山, mountain)
- HSYY,KSYV, HNQ,QSK form a square, with inscribed circle C.
- Line YS, vertical line from Y intersects line LQ at S(泉, spring)
- Line BJ, vertical line from point B, intersects line LQ at J(夕,night)
- RD, a horizontal line from R, intersects line TQ at D(旦, day)

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

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

Number | Name | Vertices | Hypotenuse | cVertical | bHorizontal | a
---|---|---|---|---|---|

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"

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:

- Hypoteneuse
- Horizontal
- Vertical
- :勾股和 :sum of horizontal and vertical
- :勾股校: difference of vertical and horizontal
- :勾弦和: sum of horizontal and hypotenuse
- :勾弦校: difference of hypotenuse and horizontal
- :股弦和: sum of hypotenuse and vertical
- :股弦校: difference of hypotenuse and vertical
- :弦校和: sum of the difference and the hypotenuse
- :弦校校: difference of the hypotenuse and the difference
- :弦和和: sum the hypotenuse and the sum of vertical and horizontal
- :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

- ;
- ;
- ;
- ;
- ;
- ;

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

^{ [3] }

- = *
- =
- =
- =
- =
- =
- =
- =
- =
- = =

^{ [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] }

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

- 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

：

Then subtract tian yuan from eastward paces 200 to obtain:

- multiply these two expressions to get：

that is

thus：

Solve the equation and obtain

- 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 # | GIVEN | x | Equation |
---|---|---|---|

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 |

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

。

Q | 1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 | 10 | 11 | 12 | 13 | 14 | 15 | 16 | 17 |
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|

second line segment |

18 problems, given。^{ [10] }

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

second line segment |

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.

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

Given | |||||||||||||||||||

Second line segment |

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

Q | Given |
---|---|

1 | ， |

2 | ， |

3 | ， |

4 | ， |

5 | ， |

6 | ， |

7 | ， |

8 | ， |

9 | ， |

10 | ， |

11 | ， |

12 | ， |

13 | ， |

14 | ，， |

15 | ， |

16 | ， |

17 | ， |

18 | ， |

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

Q | Given |
---|---|

1 | ，， |

2 | ，， |

3 | ， |

4 | ， |

5 | ， |

6 | ， |

7 | ， |

8 | ， |

9 | ， |

10 | ，， |

11 | ，， |

12 | ， |

13 | ，， |

14 | ， |

15 | ， |

16 | ， |

*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] }

- Part I

Problems | given |
---|---|

1 | ， |

2 | ， |

3 | ， |

4 | ， |

- Part II

Problems | given |
---|---|

1 | ，， |

2 | ，， |

3 | ，， |

4 | ，， |

5 | ，， |

6 | ，， |

7 | ，， |

8 | ，， |

8 problems^{ [15] }

Problem | Given |
---|---|

1 | ， |

2 | ， |

3 | ， |

4 | ， |

5 | ， |

6 | ， |

7 | ， |

8 | ， |

：Miscellaneous 18 problems：^{ [16] }

Q | GIVEN |
---|---|

1 | ， |

2 | ， |

3 | ， |

4 | ， |

5 | ， |

6 | ， |

7 | ， |

8 | ， |

9 | ， |

10 | ， |

11 | ， |

12 | ， |

13 | ，， |

14 | ， |

15 | ， |

16 | ， |

17 | From the book Dongyuan jiurong |

18 | From Dongyuan jiurong |

14 problems on fractions^{ [17] }

Problem | given |
---|---|

1 | ，= |

2 | ，= |

3 | ， |

4 | ， |

5 | ， |

6 | ，， |

7 | ，， |

8 | ，， |

9 | ， |

10 | ， |

11 | ，， |

12 | ，， |

13 | ，，, |

14 | ，，，， |

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。

- ↑ Alexander Wylie,
*Notes on Chinese Literature*, Shanghai, p116, reprinted by Kessinger Publishing - ↑ Compiled from Kong Guoping p 62-66
- ↑ Bai Shangshu p24-25.
- ↑ Wu Wenjun Chapter II p80
- ↑ Bai Shangshu, p3, Preface
- ↑ Wu Wenjun, p87
- ↑ Bai Shangshou, p153-154
- ↑ Li Yan p75-88
- ↑ Martzloff, p147
- 1 2 3 Li Yan p88-101
- ↑ Kong Guoping p169-184
- ↑ Kong Guoping p192-208
- ↑ Bai Shangshu, p562-566
- ↑
**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. - ↑ Kong Guoping p220-224
- ↑ Kong Guoping p234-248
- ↑ P255-263

**Area** is the quantity that expresses the extent of a two-dimensional figure or shape or planar lamina, in the plane. Surface area is its analog on the two-dimensional surface of a three-dimensional object. Area can be understood as the amount of material with a given thickness that would be necessary to fashion a model of the shape, or the amount of paint necessary to cover the surface with a single coat. It is the two-dimensional analog of the length of a curve or the volume of a solid.

A **Pythagorean triple** consists of three positive integers *a*, *b*, and *c*, such that *a*^{2} + *b*^{2} = *c*^{2}. Such a triple is commonly written (*a*, *b*, *c*), and a well-known example is (3, 4, 5). If (*a*, *b*, *c*) is a Pythagorean triple, then so is (*ka*, *kb*, *kc*) for any positive integer *k*. A **primitive Pythagorean triple** is one in which *a*, *b* and *c* are coprime. A triangle whose sides form a Pythagorean triple is called a **Pythagorean triangle**, and is necessarily a right triangle.

In mathematics, the **trigonometric functions** are real functions which relate an angle of a right-angled triangle to ratios of two side lengths. They are widely used in all sciences that are related to geometry, such as navigation, solid mechanics, celestial mechanics, geodesy, and many others. They are among the simplest periodic functions, and as such are also widely used for studying periodic phenomena, through Fourier analysis.

A **triangle** is a polygon with three edges and three vertices. It is one of the basic shapes in geometry. A triangle with vertices *A*, *B*, and *C* is denoted .

A **right triangle** or **right-angled triangle** is a triangle in which one angle is a right angle. The relation between the sides and angles of a right triangle is the basis for trigonometry.

In geometry, an **altitude** of a triangle is a line segment through a vertex and perpendicular to a line containing the base. This line containing the opposite side is called the *extended base* of the altitude. The intersection of the extended base and the altitude is called the *foot* of the altitude. The length of the altitude, often simply called "the altitude", is the distance between the extended base and the vertex. The process of drawing the altitude from the vertex to the foot is known as *dropping the altitude* at that vertex. It is a special case of orthogonal projection.

In mathematics and physics, a **brachistochrone curve**, or curve of fastest descent, is the one lying on the plane between a point *A* and a lower point *B*, where *B* is not directly below *A*, on which a bead slides frictionlessly under the influence of a uniform gravitational field to a given end point in the shortest time. The problem was posed by Johann Bernoulli in 1696.

In geometry, a **hypotenuse** is the longest side of a right-angled triangle, the side opposite the right angle. The length of the hypotenuse of a right triangle can be found using the Pythagorean theorem, which states that the square of the length of the hypotenuse equals the sum of the squares of the lengths of the other two sides. For example, if one of the other sides has a length of 3 and the other has a length of 4, then their squares add up to 25. The length of the hypotenuse is the square root of 25, that is, 5.

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The **Pythagorean trigonometric identity**, also called the **fundamental Pythagorean trigonometric identity** or simply **Pythagorean identity** is an identity expressing the Pythagorean theorem in terms of trigonometric functions. Along with the sum-of-angles formulae, it is one of the basic relations between the sine and cosine functions.

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In geometry, a **pentagon** is any five-sided polygon or 5-gon. The sum of the internal angles in a simple pentagon is 540°.

In mathematics, the **Pythagorean theorem**, also known as **Pythagoras' theorem**, is a fundamental relation in Euclidean geometry among the three sides of a right triangle. It states that the area of the square whose side is the hypotenuse is equal to the sum of the areas of the squares on the other two sides. This theorem can be written as an equation relating the lengths of the sides *a*, *b* and *c*, often called the "Pythagorean equation":

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A **schema for horizontal dials** is a set of instructions used to construct horizontal sundials using compass and straightedge construction techniques, which were widely used in Europe from the late fifteenth century to the late nineteenth century. The common horizontal sundial is a geometric projection of an equatorial sundial onto a horizontal plane.

Wikisource has original text related to this article: |

- Jean-Claude Martzloff,
*A History of Chinese Mathematics*, Springer 1997 ISBN 3-540-33782-2 - Kong Guoping,
*Guide to Ceyuan haijing*, Hubei Education Press 1966 孔国平. 《测圆海镜今导读》 《今问正数》 湖北教育出版社. 1995 - Bai Shangshu: A Modern Chinese Translation of
*Li Yeh Ceyuan haijing*. Shandong Education Press 1985李冶 著 白尚恕 译 钟善基 校. 《测圆海镜今译》 山东教育出版社. 1985 - Wu Wenjun
*The Grand Series of History of Chinese Mathematics*Vol 6 吴文俊主编 《中国数学史大系》 第六卷 - Li Yan,
*A Historic Study of Ceyuan haijing, collected works of Li Yan and Qian Baocong*vol 8《李俨.钱宝琮科学史全集》卷8，李俨《测圆海镜研究历程考》

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