Contents

In its surviving form, the book has a preface and three chapters. There are two missing bits, one at the end of Chapter 1 and one at the beginning of Chapter 3. Chapter 1 consists of 32 problems, Chapter 2 of 22 problems and Chapter 3 of 38 problems. In the preface, the author has set forth his objectives in writing the book clearly. There are three objectives: The first is to explain how to handle arithmetical operations involving fractions; the second objective is to put forth new improved methods for solving old problems; and, the third objective is to present computational methods in a precise and comprehensible form.

Here is a typical problem of Chapter 1: "Divide 6587 2/3 and 3/4 by 58 ı/2. How much is it?" The answer is given as 112 437/702 with a detailed description of the process by which the answer is obtained. This description makes use of the Chinese rod numerals. The chapter considers several real world problems where computations with fractions appear naturally.

In Chapter 2, among others, there are a few problem requiring application of the rule of three. Here is a typical problem: "Now there was a person who stole a horse and rode off with it. After he has traveled 73 li, the owner realized theft and gave chase for 145 li when thief was 23 li ahead before turning back. If he had not turned back but continued to chase, find the distance in li before he reached thief." Answer is given as 238 3/14 li.

In Chapter 3, there are several problems connected with volumes of solids which are granaries. Here is an example: "Now there is a pit the shape of the frustum of a pyramid with a rectangular base. The width of the upper rectangle is 4 chi and the width of the lower rectangle is 7 chi. The length of the upper rectangle is 5 chi and the length of the lower rectangle is 8 chi. The depth is 1 zhang. Find the amount of millet that it can hold." However, the answer is given in a different set of units. The 37th problem is the "Washing Bowls Problem": "Now there was a woman washing cups by the river. An officer asked, "Why are there so many cups?" The woman replied, "There were guests in the house, but I do not know how many there were. However, every 2 persons had cup of thick sauce, every 3 persons had cup of soup and every 4 persons had cup of rice; 65 cups were used altogether." Find the number of persons." The answer is given as 60 persons.

The last problem in the book is the famous Hundred Fowls Problem which is often considered as one of the earliest examples involving equations with indeterminate solutions. "Now one cock is worth 5 qian, one hen 3 qian and 3 chicks 1 qian. It is required to buy 100 fowls with 100 qian. In each case, find the number of cocks, hens and chicks bought."

English translation

Ang Tian Se, a student of University of Malaya, prepared an English translation of Zhang Qiujian Suanjing as part of the MA Dissertation. But the translation has not been published.

内容

张邱建算经三卷，现存92题，内容多取材自《九章算术》，加以扩充而成。每道问题大致按九章算术格式，多以「今有……」开首，以「问……若干」结尾。随即是答案「答曰：……」，接著是甄鸾加注的解释计算程序的「术曰：……」，有些术后带有小字「臣淳风等谨按」，是李淳风所加的注解。随后是比「术曰」详细的刘孝孙细草。

全书内容可分为几大类：

• 分数的四则运算，

• 开平方与开立方，

• 正比例，反比例，

• 等比级数，等差级数

• 线性方程

• 不定方程：百鸡问题

分数的四则运算

卷一第二问：「以二十一七分之三乘三十七九分之五，问：得几何？」。答曰：八百四二十一分之十六。

「草曰：置二十一以分母七乘之内子三得一百五十又置三十七以分母九乘之内子五得三百三十八二位相乘得五万七百为实，以二分母七九相乘得六十三而一得八百四，馀六十三分之四十八，各以三约之，得二十一分之十六，合前问。」

21\frac{3}{7} * 37\frac{5}{9}=\frac{150}{7} * \frac{338}{9}=804\frac{48}{63}=804\frac{16}{21}

开平方与开立方

卷一19问：「今有圆材，径头二尺一寸，欲以为方，问：各几何？」。「答曰一尺五寸」。「术曰：置径尺寸数，以五乘之为实，以七位法，实如法而一」。「草曰：置二尺一寸以五乘之得一百五寸，以七除之得一尺五寸，合前问。」

\frac{21*5}{7}=\frac{105}{7}=15

等差级数和等比级数

卷上第23问：「今又女子，不善织，日减功，初日织五尺，末日织一尺，今三十日织讫，问：织几何？」。「术曰：并初末日织数，半之，除以织讫日即得。」

「答曰二疋一丈。」

织布数=（初日织数+末日织数）/2*织讫日数。

「草曰：置初日五尺讫日一尺并之得六，半之得三，以三十日称之得九十尺，合前问。」

\frac{5+1}{2}*30=3*30=90

百鸡问题

《张邱建算经》第三十八问：是中国数学史上最早的不定方程问题：「今有鸡翁一，值钱五，鸡母一、值钱三、鸡雏三，值钱一；凡百钱，买鸡百只；问，鸡翁、鸡母、鸡雏各几何？

答曰，鸡翁四、鸡母十八、鸡雏七十八、

又答曰：鸡翁八，鸡母十一，鸡雏八十一

又答曰 ：鸡翁十二，鸡母四，鸡雏八十四

清代数学家骆腾风将《张邱建算经》中的百鸡问题化为不定方程组

x+y+z=100

5x+3y+z/3=100

版本

• 南宋嘉定六年（1213年）鲍浣之刻本 存上海图书馆

• 戴震-孔继涵微波榭本校勘本

• 钱宝琮校点本 《张邱建算经》 《算经十书》《李俨.钱宝琮科学史全集》卷9

• 郭书春，刘纯校点，《张邱建算经》《算经十书》第二册 辽宁教育出版社 1998