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General History of Chinese Mathematics

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The book was planned in the winter of 1994, and the writing plan was implemented in 1995. The book is titled "General History of Chinese Mathematics". Published in 4 volumes, one volume was submitted every year from 1995 to 1998. The first volume "Ancient to Five Dynasties Volume" was finalized in December 1995 and submitted to the publishing house in January 1996. When I started to write the second volume "Song Yuan Juan", there were too many other things (especially hosting international conferences and going abroad to attend conferences), which took up a lot of time. Can only write and stop, until now it is finally completed.

The time span of the second volume is 408 years from 960 to 1368. Compared with the first volume, the time span is much shorter. But the complexity is quite large, and it is quite thoughtful to divide the chapters. The most important thing is to take care of the chronological order, while also concentrating some similar problems. After repeated consideration, the mathematics of the Southern Song Dynasty will be discussed in chronological order from the second section of the second chapter to the end of the third chapter. Chapters 4 and 5 discuss the mathematics of the Yuan Dynasty before and after the fall of the Jin Dynasty and the unification of the whole country by the Yuan Dynasty. The last two chapters are Yuan. Algebra. Why consider this? The main point is the cohesion of dynasties and the emergence of the north-south features of mathematics itself. After the first chapter, we will talk about the Southern Song Dynasty first, because it is mathematically inherited from the Northern Song Dynasty, especially in terms of publishing and algorithms. Later, I talked about the north because the country was unified by the north, and the Mongolia and Yuan dynasties in the north became the Yuan dynasty of the whole country. Then, the mathematics of the Yuan dynasty was logical. I think this is a reasonable method of division. Zhu Shijie is a great mathematician who integrates the north and the south, and he focuses on the northern elements. However, Qin Jiushao's "Mathematics of Numbers" does not seem to have been exposed to him, so such things as "Dayan Summoning Technique" and "Dayan Qiuyishu" etc. No one will discuss this kind of issue in Qin's future.

A work on the history of Chinese mathematics that runs through more than four hundred years. If you only write a few first-rate mathematics and the work of some second- and third-rate mathematicians, it seems a bit "lonely" and a bit sudden, then It is best to link the work of these characters together so that there is a certain connection between them, or present a transitional "intermediary." The difficulty in this area far exceeds the difficulty of chapter division. In order to solve this problem, I mainly adopted the following four methods:

First, like picking up sesame seeds, pick up the small historical materials that people can't see and write them in the appropriate places in the book. Some small historical materials can also reflect some larger problems. For example, the Yuan Dynasty stipulated that officials should "know arithmetic" as an example, and so on.

Secondly, pay special attention to the application of mathematics in other works besides mathematics works. Calendar works are the focus of more general attention, and this book is no exception. But at the same time, attention was paid to "Hefang General Discussion" and "Building Method", which showed that mathematics was highly developed in the Song and Yuan dynasties on the one hand, and on the other hand was widely used. Mathematics is not something that several mathematicians do in isolation, but is inextricably linked to the entire society.

Thirdly, pay attention to the unknown little people and dig out their deeds bit by bit. Some of them can be connected into a piece. For example, there were many people who mastered mathematics in the Yuan Dynasty and said how advanced they are in mathematics. The level is not up to the point, but the time and area are relatively wide, which is a social phenomenon. Everyone in mathematics is by no means an isolated person, but the top of many people of the same kind.

Fourth, each chapter introduces the social situation at the time, especially the regime change, the ruler’s attitude towards mathematics, and the relevant trends of intellectuals. This is to a certain extent The mathematics work and its characteristics are brought out, and at the same time, the isolation of the work of mathematics is weakened.

Whether this can achieve the intended purpose is up to the majority of readers to judge. In fact, this has already been done in the previous volume, but it is not stated in the preface.

In addition, in the "General History of Chinese Mathematics" with more than one million words, if readers are not provided with more original materials, it will cause readers to have some misunderstandings, such as solving higher-order equations and solving Linear equations, Tian Yuan Shu, and Four Yuan Shu, etc., are given as they are as far as possible. Readers can see from this: how many and how different the expression methods of the ancients and modern mathematics are. In other words, readers cannot just listen to what the author said, but at the same time they must take out some of the originals of the ancients, and let the readers analyze and compare them to get their own opinions.


Book Catalog


Chapter One Mathematics in the Northern Song Dynasty

Section One The Calendar and Mathematical Studies in the Northern Song Dynasty

Section 2 Liu Yi and "The Origin of the Ancient Chinese"

Section 3 Jia Xian’s Contributions

Section 4 Shen Kuo’s Dream The Mathematics Contents in "Xi Bi Tan"

Section 5 Mathematics Education and Publication of Mathematics Works in the Northern Song Dynasty

Section 6 Mathematics Works in the Northern Song Dynasty and Mathematics in the Liao Dynasty

Chapter Two The Mathematics of the Early and Mid-South Song Dynasty in Xixia, Jin and the Contributions of Qin Jiushao

Section 1 The North-South Bifurcation of Mathematical Development and Xixia Jin Mathematics

The Second Section of Mathematics in the Early and Mid-South Song Dynasty


Section 3 Qin Jiushao and His Mathematics Research

Section 4 Dayan Summarization and Dayan Qiuyishu

Section 5 Qin Jiushao’s Research on Equation Theory< /p>

Other mathematics achievements and socio-economic data contained in the sixth section "Mathematics of Mathematics"

Chapter 3 Southern Mathematics in the Late Southern Song Dynasty

First Section Yang Hui and His Mathematics Works

Section Two "Compendium of Exercises" and Intuitive Figures

Section Three Simplified Algorithms and Stack Product Problems

Section Four The vertical and horizontal graphs in "Continuing the Ancient and Difficult Algorithms"

The fifth section of Ding Yidong’s vertical and horizontal diagrams

Chapter Four Tianyuanshu and Li Ye

Section 1 The origin and development of Tianyuanshu

Section 2 Li Ye and his works

Section 3 "Measurement of Circle and Sea Mirror" with geometric properties

Fourth Section "Measuring the Round Sea Mirror" and Tian Yuan Shu

Section 5 "Yigu Performance" Achievements

Chapter 5 Official Calendar Mathematical Studies of Mongolia and the Early Yuan Dynasty

The first section of Sino-foreign exchanges in mathematics

Section two is an intellectual group headed by Liu Bingzhong

Section III Wang Xun Guo Shoujing’s calendar reform

Four Sections of Wang Xun’s Mathematical Achievements

Chapter VI Zhu Shijie and the Confluence of North and South Mathematics

Section One Zhu Shijie and His Mathematics Enlightenment

Section Two by Tian Yuan Shu to quaternion surgery

Section 3 Zhu Shijie’s higher-order equations

Section 4 Zhu Shijie’s arithmetic progression and tricks

……< /p>


Book Digest

Section 5 Mathematics Education in the Northern Song Dynasty and Publishing Mathematics Works

Chinese mathematics education reached its extreme in the Tang Dynasty In its heyday, the melee period of the Five Dynasties did not completely stop. After the founding of the Northern Song Dynasty, the political situation changed a lot. The country's mathematics education should be restored soon, but in fact it did not do so. Only the doctors of mathematics were established among the officials in the team system after Jianlong (960-963). , But no teaching assistant①. There is no further relevant information, and it is estimated that no mathematics education facilities have been established. It was not until Shenzong a hundred years later that mathematics was established in the Imperial College in the seventh year of Yuanfeng (1084). According to Song Dynasty person Sun Fengji's record: "Guo Zi Jian, in charge of the five studies of Guo Zi, Taixue, Wuxue, Law, and Arithmetic. It was carried out by order in the sixth year of Yuanfeng." ②The sixth year of Yuanfeng may be the era of approval. In the second year, there was an edict: "Four election officials are all scholars, and Xu Yu's department will take the test. Those who pass the test are the first-class excluding doctors, and the second is the academic edict." (3) It will be carried out in February, with the purpose of making up for the lack of Ph.D. in mathematics ④. Because of the need for mathematics talents, he "established mathematics" on December 7th of Yuanfeng's seventh year.

But this did not go well. After the approval of the establishment of mathematics, there should have been teaching places and school buildings, but soon after the death of Shenzong Zhao Xu, construction was not started until the first year of Zhezong Yuanyou (1086), and "The selection of academic officials is also qualified." Some people thought it was a waste and requested that the construction should be stopped ⑤. On June 28 of the same year, "Strikes on Mathematical Studies" ⑥.

In the two or three years from the seventh year of Yuanfeng to the first year of Yuanyou, although the establishment of mathematics was not smooth, one thing was indeed done. This is the first time that a batch of mathematics works of predecessors have been published. This work is under the responsibility of the Secretary Province. The secretary and supervisor Zhao Yanruo is responsible for this task. He and some others have checked the printed calculations in advance. The purpose is obviously to prepare textbooks for mathematics students.

The publication work was carried out in September of the seventh year of Yuanfeng. The historical records were not clear. After Li Yan summarized and sorted out, the following eight types were identified: "Zhou Tu Suan Jing", "Sun Zi Suan Jing" ", "Five Cao Suan Jing", "Ji Gu Suan Jing", "Island Suan Jing", "Nine Chapter Suan Jing", "Xia Hou Yang Suan Jing" and "Zhang Qiu Jian Suan Jing". They have a unified way of signing, that is,

Secretary Province

So-and-so counts as a ××total × volume

September Day School, Yuanfeng Seventh Year Xuandelang secretary to the province

School book Langchen Ye Zuqia advances

School booker Langchen secretary Provincial school book Langchen Wang Zhongxiu

School book Lang Xing Secretary, Provincial School Book Lang Chen Qian Changqing

Feng Yi Lang Shou Secretary Cheng Chen Han Zonggu

Dong Xing Secretary Lang Shi Secretary and Junior Supervisor Sun Jue

Relegated to Zhao Yanruo, secretary and supervisor of the Chaosan Lang Examination ②.

However, the signature of Wang Guowei Kao's "Xiahouyang Sutra" is different from the above: "September 28, Yuanfeng, 7th year, entered into the Imperial Treasure and approved Yiyi to set the template", Sima Guang and other 9 people, excluding the above 6 people, have relatively high status, and the lowest is "Chobong Lang's secretary, Cheng Qiu, bestowed on the fisherman Han Zhi" ③. In fact, this is not a general signature, but a reporting official who is one level higher than Zhao Yanruo. That is, Zhao Yanruo waited to do the specific work, and the task should be issued by Sima Guang, etc., and after completion, they would present the emperor.

In the 7th year of Yuanfeng, the arithmetic books were printed by the State Secretary and Provincial Government. This is a major event in the history of Chinese mathematics and the publishing history of arithmetic books. In the Tang Dynasty, Li Chunfeng and others commented on the "Ten Sutras" which was the first stipulation of mathematics textbooks. According to research, there was already printing in the Tang Dynasty①, most of which consisted of printing Buddhist scriptures, poems, etc., in the form of single sheets of paper or rolls of paper. No one has made it clear when the thread-bound books that were bound into a volume appeared. As for the printing of thread-bound abacus books in the early days, it was undoubtedly the earliest batch of thread-bound abacus books printed in the seventh year of Yuanfeng. This has played an inestimable role in spreading mathematical knowledge and preserving ancient arithmetic, and is an important achievement and contribution of Yuanfeng Li arithmetic. It should be highly appraised.

Almost 20 years have passed since the first year of Yuanyou’s dismissal of mathematics, and the establishment of mathematics in the fourth year of Huizong ascended the throne and the third year of Chongning (1104). There are more detailed records:


Mathematics: The school was founded in Chongning three years ago, and the number of students was 210, and Xu Mingguan and the generals did it. His industry uses "Nine Chapters", "Zhou Ti" and hypothetical doubts as calculation questions, and still combines "Island", "Sun Tzu", "Five Cao", "Zhang Qiujian", "Xiahou ②" algorithm and calendar calculation, Three styles and astronomy books are undergraduates. Outside the undergraduate course, people occupies a small scripture, and those who wish to occupy a large scripture listen to it. Public and private examinations, three homes and strategies like Taixue. The third class Tweet in the upper house, with Tongshi, Dengshi, and Jiangshilang as the second ③. The textbook used is the calculation book published in the seventh year of Yuanfeng. It also stipulates examinations, learning content and post-graduation employment, etc. The number of recruits is 210, which is not a small number.

Only more than a year later, in the first month of the Chongning 5th year (1106), he stopped studying, and was later restored by Xue Ang's performance. Although Yuanfeng School of Mathematics had not yet been established, there were already regulations on mathematics at that time. In the third year of Chongning, the Yuanfeng Mathematical Regulations were revised into an imperial decree. In November of the sixth year (1107), the "Capital Province Zhazi..." . In the name of the imperial order format of Chongning Guozi Supervising School, begging for implementation, follow it". In addition, "The format of the imperial order of Chongning Guozi Supervising School of Huizong, and a detailed review of the revision." This shows that the "Chongning Guozi Supervising Mathematical Order Format" that was formed was revised on the basis of the Yuanfeng Mathematical Regulations. It is a national document, which has been preserved, and the "Review on Comprehension" has long since disappeared. . The existing ones are divided into three parts: "Arithmetic Order", "Arithmetic Pattern" and "Arithmetic Versus Chinese Book Province Pattern". As follows:

Chongning Guozi Supervision and Calculation Order:

All the students studied the meanings of "Nine Chapters" and "Zhou Ti" and calculated and asked the predicates and hypothetical numbers, and also learned the "Island" and " "Sun Tzu", "Five Cao", "Zhang Qiujian", "Xiahouyang" algorithm and calendar, three formulas, and astronomical books.

All trials should be counted as one pass. Counting is based on the merits of the right, with a pass and three points or more as qualified. The calendar calculation is the five-star faint sleepiness of the previous quarter, or the eclipse of the sun and the moon, and the time and the eclipse score are still calculated. The three forms are shot and cover and Yu occupies yin for three days.

Yang, wind and rain. Astronomy means that one month or one season is decided. And to prepare the grass according to the classics, and the room is connected.

Chongning Guozi Supervisor's Qualifications:


The four members of the Ph.D. and the two members are divided into "Nine Chapters" and "Zhou Ti", and the two members are divided into two members. Practice ③ calendar calculation, three formulas and astronomy.

Xuezheng is a member of academic regulations.

Ministry entry

The only one who has learned the record and corrected the insufficiency.

Learning encyclical teaches all living beings by the deeds learned.

The director is in charge of eating and drinking for one person.

Two of them are straight learning and writing books and the students are willing to go in and out.

Si Shuzhang alone.

Zhai Chang corrects those who are not as good as the rules in the fast, Zhai Yuzhang Zuo Zhaichang Dao to all the students, one person each.


There are thirty people in the house.

Eighty people in the house.

One hundred and fifty people in the outer house.

The supplementary test is the same as the official test of "Nine Chapters".

As a two-way question.

Private test of Meng Yue

Make up the first session of the inner house

The "Nine Chapters" and "Zhou Pi" three links.

As a two-way question.

Private trial in the middle of the month

Make up for the second session of the inner house

Calculate one calendar.

Private Trial Season Month

Make up for the third session of the inner house

Triple or astronomical one.

Chongning Guozi Supervising School of Mathematics and Completing the Middle School Book Province:

Autumn audition to mathematics promotion to the upper house and the next item in Tune

The upper house Wait for Tong Shi Lang.

The middle-class Deng Shi Lang on the upper house.

Upper and lower general Shi Lang.


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