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 formulas 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.
The 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(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.
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 Number Name Vertices Hypotenusec Verticalb Horizontala 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:
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
Segment numbers
There are 15 x 13 =195 terms, their values are shown in Table 1:Compiled from Kong Guoping p 62-66
Definitions and formula
Miscellaneous formula
Bai Shangshu p24-25.
The Five Sums and The Five Differences
Wu Wenjun Chapter II p80
Li Zhi derived a total of 692 formula in Ceyuan haijing. Eight of the formula are incorrect, the rest are all correctBai Shangshu, p3, Preface
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 polynomialWu Wenjun, p87
Volume 2
This volume begins with a general hypothesisBai Shangshou, p153-154
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 2Two 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 3inscribed 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
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
：
元 。
Then subtract tian yuan from eastward paces 200 to obtain:
元
multiply these two expressions to get：
元
元
that is
thus：
元
Solve the equation and obtain
Volume 3
17 problems associated with segment i.e TW in Li Yan p75-88 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.Martzloff, p147 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
Volume 4
17 problems, given and a second segment, find diameter of circular city.Li Yan p88-101 。
Q 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 second line segment
Volume 5
18 problems, given。
Q 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 second line segment
Volume 6
18 problems. Q1-11，13-19 given，and a second line segment, find diameter d. 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 18Given Second line segment
Volume 7
18 problems, given two line segments find the diameter of round townKong Guoping p169-184
Q Given 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.Kong Guoping p192-208
Q Given 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:Bai Shangshu, p562-566
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.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.
Volume 9
Part I Problems given 1 ， 2 ， 3 ， 4 ， Part II Problems given 1 ，， 2 ，， 3 ，， 4 ，， 5 ，， 6 ，， 7 ，， 8 ，，
Volume 10
8 problemsKong Guoping p220-224 Problem Given 1 ， 2 ， 3 ， 4 ， 5 ， 6 ， 7 ， 8 ，
Volume 11
：Miscellaneous 18 problems：Kong Guoping p234-248
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
Volume 12
14 problems on fractionsP255-263
Problem given 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
References
Jean-Claude Martzloff, A History of Chinese Mathematics, Springer 1997 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》