內容

《測圓海鏡》由卷一的圓城圖式、說明各個長度名稱的總率名號、給出各個長度數值的今問正數、囊括了各個量之間關係的公式總集識別雜記；卷二至卷十二，共一百七十個問題及其解答所組成。書中一共有148問，182種方法是以天元術列出方程以求解，其中列出一次方程31個，二次方程106個，三次方程24個，四次方程20個，六次方程1個

卷一

圓城圖式

圓城圖式（右圖）是全書的總括圖解，由一個直角三角形（古時稱為勾股形）、它的內切圓以及一些特定的點和直線組成。其中的頂點、圓心和交點都用某個漢字來指代。最大的三角形的三個頂點分別是天、地、乾，天地乾三角形的內切圓圓心稱為心。過心的垂直線從上至下分別和三角形、內切圓交于日、南、北三點。過心的水平線從左至右分別和三角形、內切圓交于川、東、西三點。過東的垂直線和過南的水平線都是內切圓的切線，它們分別交天地乾三角形于艮、坤、山、月四點，而相交于巽點。乾坤巽艮構成一個正方形。過月的垂直線交東西水平線于青點，交地乾邊于泉點。過山的水平線交南北垂直線于朱點，交天乾邊于金點。而這兩條線相交于泛點。最後過日的水平線交天乾邊于旦點，過川的垂直線交地乾邊于夕點。總共22個點。

總率名號

全書所研究的三角形一共有15個，全部是以天地線之間的線段為弦（斜邊）的直角三角形。總率名號給出了這些三角形和線段的名稱。它們分別是：

其中弦是三角形斜邊，股是三角形的長直角邊（這裡是豎直的），勾是三角形短直角邊（這裡是水平的）。（a_{1}代表通勾，b_{1}代表通股，c_{1}代表通弦，余類推）。

今問正數

今問正數一節給出了圓城圖式中每個線段的長度。其中以內切圓的半徑為120步，作為標準。

• 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，較（兩者的差）二百八十；勾弦和一千，較三百六十；股弦和一千二百八十，較八十；弦較和九百六十，較四百；弦和和一千六百，較二百四十。」

15個勾股形中上高 = 下高；上平 = 下平，因此，15個勾股形中，只有13個勾股形是相異的。

《今問正數》共15個勾股形×13項=195項。

，列表如下。

識別雜記

識別雜記都是關于不同線段之間的幾何關係式。一共給出了692個公式。是全書的綱領。

識別雜記包含八項：

• 諸雜名目：是全書的總綱，列出各項定義，例如虛勾虛股相得名為虛率，高股平勾差名為角差，又名遠差等等。諸雜名目中還列出三十餘項定理，如凡大差小差相乘為半段徑冪，大差勾小差股相乘同上、黃廣股黃長勾相乘為經冪等等。

名目

雜用公式

• (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}

•

五和五較

• 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}

此外還有諸弦，大小差，諸差，諸率互見，四位拾遺，拾遺。

一共692關係式，這些關係式完全是幾何定理，與具體數值無關。

舉例：第三條中「勾股和即弦黃和」一句就是：三角形兩直角邊之和等于斜邊加上內切圓直徑（「黃」指內切圓直徑）。這個命題可以由直角三角形的勾股定理推出：

:設直角三角形三邊分別為a、b、c，其中a^2+b^2=c^2

:內接圓直徑d = \frac{2ab}{a+b+c}，因此

:內接圓直徑 + 斜邊 = 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 = 兩直角邊之和

後面出現的各問題，都根據這些公式中的相等關係而列出方程，然後求解。

李冶的692個公式中，有8個是錯誤的，只是因為數值吻合而被誤認為成立。

新設第一率

新設第二率

新設第三率

新設第四率

第二卷

:正率14問

從第二卷開始，《測圓海鏡》中一共出現了一百七十個問題，它們都是圍繞著同一個題設背景而展開。

在第二卷開頭，李冶作出了以後題目公用的總假設：

這裡的圓城就是指天地乾三角形的內切圓，其方向按照圓城圖式裡面東南西北四個點的位置而定（注意北在下方，東在左邊，與現在通用的方位相反），所謂的「乾地」、「坤地」則是指圓城圖式裡面出現的乾點、坤點等等。以後的每個問題中要求的長度都是圓城的半徑或直徑。

接下來的問題都是已知某些線段的長度，問圓城的半徑或直徑。李冶在每一題的題目之後都先寫出解法（代數演算），再給出演草（代入數值的計算）。

;洞淵九容

開頭十個問題，不需要天元方程。清代數學李善蘭認為，第一個問題和《九章算術》的勾股容圓題目一樣，第二問至第十問就是《自序》中提到的「洞淵九容」。但李冶原書或《四庫全書》李銳較本都沒有這九個問題的細草，李善蘭在《天算或問》一書中根據相似三角形原理求得各式，並以第二問為例闡明如下：

:{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