(測圓海鏡 cè yuán hǎi jìng, literally sea mirror of circle measurements) is a treatise on solving geometry problems with the algebra of Tian yuan shu written by the mathematician
. 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.
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
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"(b_{14}-a_{14})+(b_{15}-a_{15})
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 r=120 paces a_{1}=320,b_{1}=640.
The 13 segments of ith triangle (i=1 to 15) are:
• Hypoteneuse c_{i}
• Horizontal a_{i}
• Vertical b_{i}
• :勾股和 :sum of horizontal and vertical a_{i}+b_{i}
• :勾股校: difference of vertical and horizontal b_{i}-a_{i}
• :勾弦和: sum of horizontal and hypotenuse a_{i}+c_{i}
• :勾弦校: difference of hypotenuse and horizontal c_{i}-a_{i}
• :股弦和: sum of hypotenuse and vertical b_{i}+c_{i}
• :股弦校: difference of hypotenuse and vertical c_{i}-b_{i}
• :弦校和: sum of the difference and the hypotenuse c_{i}+(b_{i}-a_{i})
• :弦校校: difference of the hypotenuse and the difference c_{i}-(b_{i}-a_{i})
• :弦和和: sum the hypotenuse and the sum of vertical and horizontal a_{i}+b_{i}+c_{i}
• :弦和校: difference of the sum of horizontal and vertical with the hypotenuse a_{i}+b_{i}
Among the fifteen right angle triangles, there are two sets of identical triangles:
:\triangle TRD=\triangle RMZ,
:\triangle YSG=\triangle BLJ
that is
:a_{6}=a_{7};
:b_{6}=b_{7};
:c_{6}=c_{7};
:a_{8}=a_{9};
:b_{8}=b_{9};
:c_{8}=c_{9};
Segment numbers
There are 15 x 13 =195 terms, their values are shown in Table 1:
• (c_{1}-a_{1})*(c_{1}*b_{1})= 1 \over 2*(d_{1})^2
• a_{10}*b_{11} = 1 \over 2(d_{1})^2
• a_{13}*b_{1} = 1 \over 2(d_{1})^2
• a_{1}*b_{13} = 1 \over 2(d_{1})^2
• b_{2}*b_{15} = (r_{1})^2
• a_{14}*a_{3} = (r_{1})^2
• a_{5}*b_{4} = (d_{1})^2
• a_{8}*b_{6} = a_{9}*b_{7}=(r_{1})^2
• (b_{14}*c_{14})*(a_{15}+c_{15}) = (r_{1})^2
• c_{6}*c_{8} = c_{7}*c_{9})=a_{13}*b_{13}
•
The Five Sums and The Five Differences
• a_{2}+b_{2}+c_{2}=b_{1}+c_{1}
• a_{3}+b_{3}+c_{3}=a_{1}+c_{1}
• a_{4}+b_{4}+c_{4}=2b_{1}
• a_{5}+b_{5}+c_{5}=2a_{1}
• a_{6}+b_{6}+c_{6}=b_{1}
• a_{7}+b_{7}+c_{7}=b_{1}
• a_{8}+b_{8}+c_{8}=a_{1}
• a_{9}+b_{9}+c_{9}=a_{1}
• a_{10}+b_{10}+c_{10}=b_{1}+c_{1}-a_{1}
• a_{11}+b_{11}+c_{11}=c_{1}-b_{1}+a_{1}
• a_{12}+b_{12}+c_{12}=c_{1}
• a_{13}+b_{13}+c_{13}=a_{1}+b_{1}-c_{1}
• a_{14}+b_{14}+c_{14}=c_{1}-a_{1}
• a_{15}+b_{15}+c_{15}=c_{1}-c_{1}
• (b_{7}-a_{7})+(b_{8}-a_{8})+(b_{14}-a_{14})+(b_{15}-a_{15})=2*(b_{12}-a_{12})
• a_{8}+(b_{7}-a_{7})+(b_{8}-a_{8})=b_{7}
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
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 \triangle TLQ
:Algorithm:d={2a_{1} \times b_{1} \over a_{1} + b_{1}+c_{1