内容

《测圆海镜》由卷一的圆城图式、说明各个长度名称的总率名号、给出各个长度数值的今问正数、囊括了各个量之间关系的公式总集识别杂记；卷二至卷十二，共一百七十个问题及其解答所组成。书中一共有148问，182种方法是以天元术列出方程以求解，其中列出一次方程31个，二次方程106个，三次方程24个，四次方程20个，六次方程1个

卷一

圆城图式

圆城图式（右图）是全书的总括图解，由一个直角三角形（古时称为勾股形）、它的内切圆以及一些特定的点和直线组成。其中的顶点、圆心和交点都用某个汉字来指代。最大的三角形的三个顶点分别是天、地、乾，天地乾三角形的内切圆圆心称为心。过心的垂直线从上至下分别和三角形、内切圆交于日、南、北三点。过心的水平线从左至右分别和三角形、内切圆交于川、东、西三点。过东的垂直线和过南的水平线都是内切圆的切线，它们分别交天地乾三角形于艮、坤、山、月四点，而相交于巽点。乾坤巽艮构成一个正方形。过月的垂直线交东西水平线于青点，交地乾边于泉点。过山的水平线交南北垂直线于朱点，交天乾边于金点。而这两条线相交于泛点。最后过日的水平线交天乾边于旦点，过川的垂直线交地乾边于夕点。总共22个点。

总率名号

全书所研究的三角形一共有15个，全部是以天地线之间的线段为弦（斜边）的直角三角形。总率名号给出了这些三角形和线段的名称。它们分别是：

其中弦是三角形斜边，股是三角形的长直角边（这里是竖直的），勾是三角形短直角边（这里是水平的）。（a_{1}代表通勾，b_{1}代表通股，c_{1}代表通弦，余类推）。

今问正数

今问正数一节给出了圆城图式中每个线段的长度。其中以内切圆的半径为120步，作为标准。

• 弦

• 勾

• 股

• 勾股和：a + b

• 勾股校：b - a

• 勾弦和：a + c

• 勾弦校：c - a

• 股弦和：b + c

• 股弦校：c - b

• 弦校和：c + (b - a)

• 弦校校：c - (b - a)

• 弦和和：(a + b) + c

• 弦和校：(a + b) - c

例子:「通弦六百八十，勾三百二十，股六百；勾股和九百二十，较（两者的差）二百八十；勾弦和一千，较三百六十；股弦和一千二百八十，较八十；弦较和九百六十，较四百；弦和和一千六百，较二百四十。」

15个勾股形中上高 = 下高；上平 = 下平，因此，15个勾股形中，只有13个勾股形是相异的。

《今问正数》共15个勾股形×13项=195项。

，列表如下。

识别杂记

识别杂记都是关于不同线段之间的几何关系式。一共给出了692个公式。是全书的纲领。

识别杂记包含八项：

• 诸杂名目：是全书的总纲，列出各项定义，例如虚勾虚股相得名为虚率，高股平勾差名为角差，又名远差等等。诸杂名目中还列出三十馀项定理，如凡大差小差相乘为半段径幂，大差勾小差股相乘同上、黄广股黄长勾相乘为经幂等等。

名目

杂用公式

• (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}

•

五和五较

• 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}

此外还有诸弦，大小差，诸差，诸率互见，四位拾遗，拾遗。

一共692关系式，这些关系式完全是几何定理，与具体数值无关。

举例：第三条中「勾股和即弦黄和」一句就是：三角形两直角边之和等于斜边加上内切圆直径（「黄」指内切圆直径）。这个命题可以由直角三角形的勾股定理推出：

:设直角三角形三边分别为a、b、c，其中a^2+b^2=c^2

:内接圆直径d = \frac{2ab}{a+b+c}，因此

:内接圆直径 + 斜边 = d + c = \frac{2ab}{a+b+c} + c = \frac{2ab + c^2 + ac + bc}{a+b+c}

:= \frac{2ab + a^2 + b^2 +ac + bc}{a+b+c} = \frac{(a+b)(a+b+c)}{a+b+c}

:= a+b = 两直角边之和

后面出现的各问题，都根据这些公式中的相等关系而列出方程，然后求解。

李冶的692个公式中，有8个是错误的，只是因为数值吻合而被误认为成立。

新设第一率

新设第二率

新设第三率

新设第四率

第二卷

:正率14问

从第二卷开始，《测圆海镜》中一共出现了一百七十个问题，它们都是围绕著同一个题设背景而展开。

在第二卷开头，李冶作出了以后题目公用的总假设：

这里的圆城就是指天地乾三角形的内切圆，其方向按照圆城图式里面东南西北四个点的位置而定（注意北在下方，东在左边，与现在通用的方位相反），所谓的「乾地」、「坤地」则是指圆城图式里面出现的乾点、坤点等等。以后的每个问题中要求的长度都是圆城的半径或直径。

接下来的问题都是已知某些线段的长度，问圆城的半径或直径。李冶在每一题的题目之后都先写出解法（代数演算），再给出演草（代入数值的计算）。

;洞渊九容

开头十个问题，不需要天元方程。清代数学李善兰认为，第一个问题和《九章算术》的勾股容圆题目一样，第二问至第十问就是《自序》中提到的「洞渊九容」。但李冶原书或《四库全书》李锐较本都没有这九个问题的细草，李善兰在《天算或问》一书中根据相似三角形原理求得各式，并以第二问为例阐明如下：

:{a_{1} \over a_{1}+b_{1}+c_{1

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:

Definitions and formula

Miscellaneous formula

• (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