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.
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.
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.
C: Center of circle:
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
In problems from Vol 2 to Vol 12, the names of these triangles are used in very terse terms. For instance
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:
Among the fifteen right angle triangles, there are two sets of identical triangles:
There are 15 x 13 =195 terms, their values are shown in Table 1:
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
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.
The first ten problems were solved without the use of Tian yuan shu. These problems are related to various types of inscribed circle.
Then subtract tian yuan from eastward paces 200 to obtain:
Solve the equation and obtain
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.
|1||，||direct calculation without tian yuan|
|16||，||calculate with formula for inscribed circle|
|17||，||Calculate with formula forinscribed circle|
|second line segment|
18 problems, given。
|second line segment|
|Second line segment|
18 problems, given two line segments find the diameter of round town
17 problems, given three to eight segments or their sum or difference, find diameter of round city.
：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.
：Miscellaneous 18 problems：
|17||From the book Dongyuan jiurong|
|18||From Dongyuan jiurong|
14 problems on fractions
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。
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