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

In 1969, 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.

內容

張邱建算經三卷，現存92題，內容多取材自《九章算術》，加以擴充而成。每道問題大致按九章算術格式，多以「今有……」開首，以「問……若干」結尾。隨即是答案「答曰：……」，接著是甄鸞加注的解釋計算程序的「術曰：……」，有些術後帶有小字「臣淳風等謹按」，是李淳風所加的註解。隨後是比「術曰」詳細的劉孝孫細草。

全書內容可分為幾大類：

• 分數的四則運算，

• 開平方與開立方，

• 正比例，反比例，

• 等比級數，等差級數

• 線性方程

• 不定方程：百雞問題

分數的四則運算

卷一第二問：「以二十一七分之三乘三十七九分之五，問：得幾何？」。答曰：八百四二十一分之十六。

「草曰：置二十一以分母七乘之內子三得一百五十又置三十七以分母九乘之內子五得三百三十八二位相乘得五萬七百為實，以二分母七九相乘得六十三而一得八百四，餘六十三分之四十八，各以三約之，得二十一分之十六，合前問。」

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

開平方與開立方

卷一19問：「今有圓材，徑頭二尺一寸，欲以為方，問：各幾何？」。「答曰一尺五寸」。「術曰：置徑尺寸數，以五乘之為實，以七位法，實如法而一」。「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，合前問。」

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

等差級數和等比級數

卷上第23問：「今又女子，不善織，日減功，初日織五尺，末日織一尺，今三十日織訖，問：織幾何？」。「術曰：並初末日織數，半之，除以織訖日即得。」

「答曰二疋一丈。」

織布數=（初日織數+末日織數）/2*織訖日數。

「草曰：置初日五尺訖日一尺並之得六，半之得三，以三十日稱之得九十尺，合前問。」

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

百雞問題

《張邱建算經》第三十八問：是中國數學史上最早的不定方程問題：「今有雞翁一，值錢五，雞母一、值錢三、雞雛三，值錢一；凡百錢，買雞百隻；問，雞翁、雞母、雞雛各幾何？

答曰，雞翁四、雞母十八、雞雛七十八、

又答曰：雞翁八，雞母十一，雞雛八十一

又答曰 ：雞翁十二，雞母四，雞雛八十四

清代數學家駱騰風將《張邱建算經》中的百雞問題化為不定方程組

x+y+z=100

5x+3y+z/3=100

版本

• 南宋嘉定六年（1213年）鮑澣之刻本 存上海圖書館

• 戴震-孔繼涵微波榭本校勘本

• 錢寶琮校點本 《張邱建算經》 《算經十書》《李儼.錢寶琮科學史全集》卷9

• 郭書春，劉純校點，《張邱建算經》《算經十書》第二冊 遼寧教育出版社 1998