Author | Liu Fusheng
Source | Harmony Math
Editor's note: The recent Guangdong middle school exam questions involved the Helen-Qin Jiushao formula. Here we share an article about Qin Jiushao's mathematical achievements.
The Song Dynasty was an era of highly developed mathematics in China, and Qin Jiushao, who was born in Sichuan, was one of the outstanding representatives. His masterpiece The Nine Chapters of Mathematics (later commonly known as the Nine Chapters of the Book of Numbers) embodies the highest achievements of contemporary mathematics in the world. However, for such a scientist who is at the forefront of the times, the "History of Song" did not establish a biography for him, and the Qing "Four Libraries Summary" said that it was "not detailed why Xu Ren also". Since the last century, Qin Jiushao's mathematical achievements have been revived, which has attracted the attention of the world's scientific history circles. In today's rejuvenation of science and technology, it is of more positive significance to give this mathematical sage the position he deserves.
1. Qin Jiushao and his people
We know very little about Qin Jiushao's family lineage, and there are quite a few contradictions. The Zhou Secret Record at the end of the Southern Song Dynasty: "Qin Jiushao,...... Character Daogu, Qin Fengjian people". And Qin Jiushao dropped the title "Lu County Qin Jiushao" after the "Nine Chapters of Mathematics Self-Introduction". Despite this, if you check the historical data, you can get a rough idea of his life and deeds.
On the famous Fuling "Stone Fish", there is a written inscription on the twelfth day of the first month of the second year of Baoqing (1226), which says: "Gun Shou Li Yu Gong Yu Xin, Tong Chuan Shou Qin Ji Que Hong Father, Gun Gui Cao Tu He Changzong, Ji Wen, Ji Que's son Jiu Shao Daogu, Yu's son Ze Min Zhike, also came to swim stone fish." This is the record of Qin Jiushao following his father to Fuzhou (涪州, in present-day Fuling, Sichuan) and swimming with Taishou in the stone fish. According to the Records of the Pavilion of the Southern Song Dynasty, "Qin Jiushao's father Ji Que Zi Hong's father, "Puzhou Anyue people, Shao Xi four years (1193) Chen Liangbang Tongjin shi birth, rule "Spring and Autumn", seventeen years (1224) September except (secretary shaojian), Baoqing first year (1225) in June except zhixian Mo Pavilion, Zhitong Chuanfu". Tongchuan Prefecture was ruled in present-day Santai, Sichuan. In the seventh year of Chunyou, Lu County was under the rule of the Mongol army, and Qin Jiushao's self-described person should be his ancestral home. When the Song Dynasty and the Qin fenglu southern song dynasty jianyan had already entered the hands of the Jin people, the carefully recorded "Qin Fengjian people" caused some discussions, but they were all speculative words, and there was no accurate historical data to confirm it.
Zhousuming images
Zhou Mi also said that Qin Jiushao "was the head of the volunteer army in the township in the eighteenth year, and he was unruly and unruly, and tried to follow his father to guard the county." According to the Song Shi Ningzong Ji IV, in the twelfth year of Jiading (1219), in March Yihai, "Xingyuan's soldier Quan Xing and others rebelled and committed crimes against Bazhou, and the retainer Qin Jiqiu abandoned the city." "Bazhou is now Bazhong, Sichuan. In Leap March, there were "Xingyuan soldiers Zhang Fu, Mo Jian and other rebels", and in May, Zhang Fu led an army into Puzhou, Tunpu Prefecture's Ming Mountain. In July, it was pacified by the Song Dynasty. During this mutiny, his hometown of Lipe Prefecture (present-day Anyue, Sichuan) was disturbed by him. The "volunteer soldiers" were local armed forces, and when Qin Jiushao was the "head of the righteous soldiers", or not far from the time of the Xingyuan Mutiny, Qin Jiushao was born around the second year of Jiatai (1202). Qin Jiushao once said that "in his early years, he served his relatives in the middle of the capital" and should be about 22 years old when his father was serving as a secretary and a young prisoner.
Qin Jiushao later served as a county lieutenant, and Li Liu (Zi Gongfu, meiting) wrote "Hui Qin County Wei Jiu Shao Xie Cha Correction Qi". Li Liu was a native of Jiangxi, a scholar in the seventh year of Jiading (1214), and an official in Chengdu around the sixth year of Shaoding (1233), but the time of writing is difficult to examine in detail. Li Liu Qizhong said: "Good succession to renzhi, should be the school enemy of Huang Su; Ken from my tour, a small test of Dan lead point survey." The Song Secret Cabinet has yellow and white books, and Qin Jiushao's father was once an official secretary, which is the so-called "Huang Su School Vendetta". It is also carefully said that the Qin clan once learned "Poems of Li Li" from Li Liu, which also means "Willing to travel from me".
In the "Self-Introduction of the Nine Chapters of Mathematics", it is said: "In the intervening times, di di, the years are remotely blocked, and they do not care about themselves all in the yashi, and they taste danger and worry. Titled Chunyou Seven Years (1247) September. In the autumn of the second year of Duanping (1235), the Mongol army invaded Sichuan; in the following autumn, the Mongol army invaded Sichuan again, and in October it invaded Chengdu. In one or two years, most of the prefectures and counties in Sichuan were destroyed. The so-called "Di Yan" refers to the invasion of Sichuan by the Mongol army he encountered in his hometown. Qin Jiushao was about 34 years old.
Nine Chapters of Mathematics
After he left Sichuan, he successively served as a general judge of Pu Prefecture (蕲州, in present-day Puchun, Hubei) and a retainer of Hezhou (和州, in present-day Hexian County, Anhui), as Liu Kezhuang put it, "a pawn in vain and several fierce military coups." Shouhe sells brine, or sells it to the people", but it must not be specific. According to the Jingding Jiankang Zhi, Qin Jiushao was sentenced to Tong Zhilang as a general in August of the fourth year of Chunyou (1244) as a tongzhilang (present-day Nanjing, Jiangsu), and in November left his post for his mother Shouxiao and returned to live in Huzhou (present-day Zhejiang). In the second year of Baoyou (1254), he once went to Jiankang to serve as a senator for the establishment of the yanjiang system.
At the beginning of the first month of the sixth year of Baoyou (1258), Qin Jiushao went to Changsha and held the Huai yanshu to apply for a job at Li Zengbo, who was then the pacifying envoy of Hunan and the envoy of Guangnan. Huai Yan refers to Jia Xiangdao, who was then the ambassador of the two Huai dynasties and the envoy of Huaidong and Huaixi. Soon qin was appointed to guard Qiong Prefecture (琼州, in modern Hainan) and returned to the county for several months. Later, he served as a brief minister of the Jiangdong Council and a farmer. Zhou Mi Jiyun: "When Wu Luzhai was in Yin, he was eager to throw himself into it." Wu Shi will enter the phase and make it a precursor: when thinking about where he is. Qin Fu followed suit. Wu Xuande, Jia Dangguo, Xu Shu, Meizhou. In Mei's administration, he did not quit, and he was killed by Mei. Wu refers to Wu Qian (号履斋), whose resignation took place in July of the first year of Jingding (1260). Qin died in Meizhou (梅州, in modern Meixian County, Guangdong), and his appointment to Meizhou and his death in Meizhou could not be specified, generally in the second year of Jingding, so that he was about 59 years old.
Zhou Mi said that Qin Jiushao was "extremely skillful, and the astrological rhythm, arithmetic, and even the construction were all exquisite, and he learned the poetry of The Piao Li from Li Meiting, and the games, balls, horses, and bows and swords could not be known." Chen Zhensun said: "Qin is erudite and versatile, and all the calendars of recent times have been passed down to Qin. And when titled the "Chongtian Calendar" of the Ji Renzong Dynasty and the "Epoch Calendar" of the Huizong Dynasty, he said: "This second calendar is close to the Shu people Qin Jiu Shao Daogu." It can be seen that he is a knowledgeable scientific and technological talent, especially known for his proficiency in mathematics and calendars, and is the keeper of the "recent calendars", and his contributions have been great!
According to the Records of the Song Dynasty, Qin Jiushao was "sexually extravagant and extravagant, and he was fond of seeking his own life" (Zhou Mi), and even had accusations of "being violent like a tiger and a wolf, and as poisonous as a snake and scorpion" (Liu Kezhuang). The Qing people Jiao Xun discerned the clouds: "Qin Jiushao was scandalized by the thoroughness, as for the unbearable, and his books are also obscure and reproduced, and the talent of the novel is secretly filled in, and the actual learning is not known to him." That is, the so-called friendship with Wu Luzhai, for Jia Xiang to sneak in Meizhou, and the government is not quit, then Qin Zhi is a person, but also a wonderful and useful talent. Lu Xinyuan also agreed with this. Wu Qiansu had good hopes, while the Jia clan was known as a "traitorous minister", and the Qin clan was "unswervingly in power", and Jiao Xunzhi seemed to have its own reasons. However, Jiao and Lu Ershi have no exact materials to prove the character of the Qin clan, and when the Jia clan was in its prime, the Qin clan was actually the same party as them. Mr. Yu Jiaxi, a close friend, has a very detailed analysis of this, believing that Qin Jiushao "does not take care of his own origins, is greedy and unscrupulous, and gives people an attacking end, although those who love him cannot be relieved."
Second, Qin Jiushao's mathematical origins,
The writing and circulation of the Nine Chapters of the Book of Numbers
The accumulation of Qin Jiushao's mathematical knowledge has two origins. One is "Taishi" and the other is "Hidden Gentleman". That is, he said in the "Preface to the Nine Chapters of Mathematics": "In the early years, zhongdu, because he had to visit and learn from Taishi. And taste from the hermit to receive mathematics. ”
Taishi was a court official in charge of the astronomical calendar, in fact, some of the astronomers and mathematicians officially recognized at that time, and Qin Jiushao benefited from them when his father was a secretary and a junior superintendent. Foreword Chen Zhensun's second calendar was also obtained when Qin Jiushao was "serving the middle capital". When Qin Jiushao studied mathematics with the "Hidden Gentleman" in the preface, it was the time of the aforementioned "Ji Shi Di", "tasting dangers and worries, and seeing the changes in current events, his heart was depressed, and he believed in and knew that there were many things". So "in the meantime, the bystander can be explored, and the rough is as good as it gets."
Some people believe that the "hidden gentleman" is a special guide to the Song "Taoist teacher" Chen Yuanliang, which we think cannot be established.
Bronze statue of Chen Yuanliang
Because
First: from the existing historical data, Chen Yuanliang is not a "mathematician". His surviving work, The Chronicle of the Ages, is a work of "festival order", and his friend Zhu Jian (Zhu Xi's grandson) said that it was "a search for the scriptures, and even the wild history of different books, and all those involved in the festival order were collected into a giant and left behind." When looking up to the heavens, he bowed down to examine the personnel"; another friend, Liu Chun, said that it was "the fangrun of the nine streams, the Yinghua of the Hundred Clans, supplemented by the mountains and seas through the sea charts, and the accumulation of exhaustion into a book". Zhu and Liu were both imperial court officials at the time. The Outline of the Four Libraries is classified as a book in the "Seasonal Category", Yun: "In the book, the books of the Moon Order, the Filial Piety, and the Three Tong Calendars are the outlines, and the miscellaneous books are written about the order of the sections." ...... It is more or less designed for Kaiza applications. As for his other two works, the Three Records of Ubmyun and the Chronicle of Shilin Guangji, they are not "mathematical" in terms of the title and some of the contents that remain in the Chronicles of the Years.
Second, Chen Yuanliang's "Chronicle of the Ages of Guang" is titled "Guanghan Immortals", and his ancestor "Mr. Guanghan" was a Fujian Chongren and a disciple of Chen Zhuo. Guanghan Zisun was a jinshi in the fourth year of Shaosheng (1097), while Yuan Liang was a "descendant of Sun", a person during the reign of Emperor Rizong (reigned 1225-1264). "Hidden gentleman" is originally a general term, and any literati who is not a scholar can be crowned with it. Liu Chun said: "At the foot of Guifeng Peak and the bay of Meixi, there is the grandson of Guanghan, a hidden gentleman." "It is a hermit who wanders between the mountains and rivers of Fujian, and the meaning of the text should already be understood." Moreover, apart from his ancestors as a Taoist, he himself seems to have nothing to do with "Taoism" and cannot be seen as a Taoist scholar.
"The Chronicle of the Ages" book shadow
Third: Qin's "servant Zhongdu" has a short time, his father has been serving in the Beijing Division for less than ten months, and he has a "stone fish" swim, and it is certain that he returned to Sichuan with his father. There is no material to show that Qin had or may have had some kind of contact with Chen Yuanliang.
Therefore, we believe that when he receives mathematics from the hermit, it must be the time when he suffers from "Di Affliction" in his hometown, and the so-called "Di Affliction" is called this. In ancient China, "number" science was often associated with Yi Xue, chasing astronomy, looking down on geography, and looking at the middle humanities, and a group of "mathematicians" actually meant that Yi scholars believed that the development of everything in the universe, including human society, was "infinite", that is to say, subject to a mysterious and established "number" agreement. By the Northern Song Dynasty, Shao Yong had gathered its great achievements, and the ancient "mathematics" had a great development. In the place where Qin Jiushao grew up, there was already a saying that "easy to learn in Shu". A group of folk Yi scholars calmly observed the vicissitudes of nature and the human world, and painstakingly searched for the mystery of their changes. This is a group of "hermits" who are unwilling or have no chance to become a scholar, and there are many people with real talents and practical learning, just as the Song Dynasty people said: "Shu many scholars, successful in Taoism." Qin Jiushao had this opportunity, coupled with his own exploration of "wanton", and finally achieved the achievements of surpassing his predecessors. It is Sichuan, the soil with rich cultural atmosphere, that has bred this mathematical wizard.
For mathematics, Qin Jiushao believes: "Zhou teaches six arts, and numbers are realized." ...... To return to it, the number and the Tao are not two books. It is more accurate to say that this "Tao" is the way of "Taoism" than to say that it is the way of the "Six Arts" of Confucianism. "Yi" is shared by the Confucian and Taoist families, so there is no need to force the Qin clan to be Confucian and Suppressed.
Qin Jiushao's preface is titled September of the Seventh Year of Chunyou (1247), and the writing of the Nine Chapters of the Book of Numbers should be traced back ten years here. The Book of Cistern says: "Or to recommend the almanac to the dynasty, it is correct, there is a manuscript and the mathematical outline described." Four years before ChunYou, the imperial court accepted Han Xiang's request and summoned Shan Lin to make a calendar. Recommended its people to the dynasty, when shortly thereafter. The "Outline of Mathematics" should be regarded as the "Nine Chapters of the Book of Numbers".
"Zhizhai Book Record Solution" book shadow
As shown above, this mathematical masterpiece has been recorded in the "Zhizhai Book Record Solution" and belongs to the almanac category. Chen Zhensun, the author of the book, and Qin Jiushao at the same time, have already had a beautiful word about his talents. However, the Song Dynasty did not have a publication, so it was not widely circulated. The Ming Yongle Canon contains this book, entitled "Nine Chapters of Mathematics", a total of 9 volumes. The Nine Chapters of Mathematics in the Qing Dynasty's Siku Quanshu were transcribed and revised from then on, and its Synopsis yun: "Now it is contained in the Yongle Canon, and those who are wrong are correct, and those who are neglectful are discerning." The upside down is secondary. This led to the book being widely circulated later.
Later, there were two manuscripts, one was Li Ruide's four libraries with slight proofreading, and the other was the collector of the Zhao Qi MeiMei Wangguan in Changshu at the end of the Ming Dynasty. In the 45th year of the Wanli Calendar (1616), the Zhao clan was a book called the Nine Chapters of the Book of Numbers, and the cloud was 18 volumes, "The King of Huiji should record it by his father by borrowing the banknotes of the Cabinet and transcribing it." Originally there was no directory, but was added". Ge Ben, also known as the Four Libraries, indicates that the Zhao Family Ben has analyzed the number of volumes of the Four Libraries. This book was owned by Zhang Dunren in the Qing Jiaqing, and at that time the study of Song and Yuan mathematics was extremely popular, and scholars expressed great interest in Qin Jiushao's mathematical works. Gu Guangxi's "Preface to the Nine Chapters of the Book of Numbers" written by Xia Wentao (Zi Fangmi) said that the Hanlin Academy compiled the Jiangdu Qin Enfu (Dunfu Taishi) "school his family Daogu 'Book of Numbers", opened the carving, and belonged to Wen Tao's recalculation", believing that there were many mistakes in the process, "then after the book was written that day, it was not personally overlaid", but its engraving was not later seen and passed down.
Song Jingchang wrote four chapters of the Nine Chapters of the Book of Numbers, and the preface written by the TaiFeng clan of Yu Songnian in Shanghai in the twenty-second year of Daoguang (1842), which described in detail the circulation of the Nine Chapters of the Book of Numbers at that time: "Yu ... Si De Song Yuanren Secret Book, Mao Junshengfu for Yu Yan Qin Daogu "Nine Chapters of the Book of Numbers" thinking of erudition. ...... His books are transposed and recorded, and there are many pseudo-detached. Yuan and Shen Guangwen once obtained the Ming dynasty Zhao Qi's U.S. dollar bills in Yangcheng Zhang Taishou's house, ordered a forgery to make up for it, and had been in office for many years, and had not died of old age and illness. His disciple Jiangyin Song Jun Jingchang was able to pass on his learning, and Yu was mao Junsuo's original. Hui Guangwen was even more ill, and he was subordinate to Wujin Li Taishi. Mao Jun also gave his family Zangyuan and Li Maocai the four library libraries, and belonged to the Song Jun school. Si Guangwen did not, and Song Jun searched his home for several volumes of qin books and periodicals. Therefore, it was mainly Based on Zhao Ben, and it was engraved for it, called the Yijiatang Series. The book collected by the Republic of China's "Series of Books" is this one.
III. The Content of the Nine Chapters of the Book of Numbers and Its Contributions
In the "Nine Chapters of Mathematics Self-Introduction", Qin Jiushao said: "The so-called theosophical being, following life, fixing the skin is not seen." If it is small, it is set as a question and answer for use. Accumulate a lot and abandon it, because take eighty-one questions, divide into nine categories, stand up the art of grass, and send it in the figure. The 9 categories were divided into four libraries, and the four library masters made a "summary" of their contents.
"One day of great derivation, with odd zeros seeking the total number of nine categories." Is a congruence group problem.
"On the second day, with the breath of the lip shadow and the five stars Fushimi." Questions about astronomy, calendars and rain and snowfall.
"Three days of field domain, to push the square circle power product." Questions about field area.
"Four days of surveying to push high, deep and wide." Questions about Pythagorean, weight difference and other measurements.
"Five-day service, with equal taxation." On the issue of land grants and household taxes.
"Six days of money valley, with power and weight in and out." Issues related to the requisition of rice grains and warehouses.
"Seven days to build, to survive the earth." Issues related to building construction.
"Eight Japanese brigades, to determine the line." Issues related to barracks placement and supplies.
"Nine-day market is easy, to rule the trade." Questions about commodity trading and interest calculation.
As can be seen from the above, the scope of the Nine Chapters of the Book of Numbers is quite extensive. Most of the problems are linked to practical needs, and it can be said that the need for practical applications has driven his research. The so-called "theosophical and obedient" are some big and mysterious problems, and the purpose of Qin's book is not this, he explicitly said that it is "intended for use". As Siku Guanchen said: "The ancient law sets its own technique, and Jiushao does not use its ears." Although Qin Jiushao's "Nine Chapters" are different from the Han Dynasty's Nine Chapters of Arithmetic, it is obvious that some of the styles inherited from the latter are inherited. There are 81 questions in 9 categories, and each class uses 9 example questions to illustrate various algorithms. Each question has both a "technique" that explains the method of solving the problem and a "grass" that explains the steps of calculus. All in the form of questions and answers, or illustrated.
Qin Jiushao put forward many interesting and very practical questions in the "Nine Chapters of the Book of Numbers", such as "pushing the earth gong", "Yumi pushing the number", "measuring the rain in the tianchi pond", "counting the land and accommodating the people", "looking at the mountains and high distances", "remote yuancheng", "counting stone dams", "counting cubic camps", etc. Although not necessarily every question has been given the simplest answer, his exploration is undoubtedly very valuable. According to the available data, Qin Jiushao's works may still be the first to use the special number "zero" in China, and "zero" often appears in later mathematical works. For Qin Jiushao's achievements, mathematical historians have made many in-depth discussions, and in general, his contributions are concentrated in the following two issues.
The first is the indefinite analysis discussed in the first part of the Nine Chapters of the Book of Numbers, which Qin Jiushao called the "Great Yan Seeking One" technique. This is a one-time cohom problem, the famous "residual theorem" problem in modern number theory. The earliest proposal of this problem was in the 4th century AD Chinese mathematical work "Sun Tzu Arithmetic Classic", which has the inscription "Things Do Not Know the Number": "Now there are things that do not know the number, three or three numbers are left with two, five or five numbers are left with three, seven or seven numbers are left with two, ask the geometry of things?" "Its solution uses the common solution of three one-time cohomological forms, i.e., the remainder of the three third numbers 2 times 70, the remainder of the five-five number 3 times 21, the remainder of the seven-seven number 2 times 15, the sum of the three numbers, minus the smallest common multiple of 3, 5, 7, multiples of 105, that is, the number is obtained. The columnization equation is:
N=70×2+21×3+15×2-2×105
After this problem entered the folk, there were names such as "Sun Tzu Calculation", "Ghost Valley Calculation", "Han Xin Dianbing" and other names and problem solving songs. This is a problem of indeterminate systems of equations, but its data is relatively simple. The residual theorem reduces the general one-time cohom problem to the selection of a set of numbers that satisfy the condition, which Qin calls "multipliers". He clearly and systematically describes its general calculation steps to satisfy the condition that the remainder 1 eventually appears in the calculations above, which is what he calls the "Great Derivative One Technique". "Dayan" is borrowed from the "I Ching And Genealogy" so-called "fifty of the number of great Yan" that can be deduced and changed in infinity, in short, "seeking one technique".
In the West, it took more than 500 years for the two great European mathematicians, Buler (1707-1783) and Gauss (1774-1855), to study this problem effectively, and regained the same theorem as the "one-finding technique". But it was not until 1852 that British missionaries introduced the ancient Chinese "things do not know the number" problem and Qin Jiushao's solution to Europeans, which caused the surprise and attention of European scholars, no wonder some scholars called the Chinese mathematicians who discovered this method "the luckiest genius". In the history of Western mathematics so far, this solution is called the "Chinese surplus theorem", which shows Qin Jiushao's lofty position in the history of world mathematics.
In ancient China, the study of this problem was clearly driven by astronomical and calendar needs. China's ancient almanacs called the time accumulated from the calendar (the starting time of the calendar, also known as the shangyuan) to the calendar year "shangyuan accumulation year". The extrapolation of the upper elemental product years requires solving a set of one-time congruents. Qin Jiushao was once taught by the "Taishi", which naturally laid a deep foundation for his creative work. There are problems such as "ancient calendar accumulation", which is the practical application of knowledge in this regard.
The second is to find the solution of the higher-order equation. The steps to solve the equations were known in ancient China as "opening the equation", and were studied by two mathematicians who lived in the Northern Song Dynasty. Jia Xian showed the "Origin Diagram of the Opening Method", which for the first time in the world unlocked the knot of opening high powers, which is a new method of opening squares and opening cubes that can be multiplied and added, called the "multiplication method". Later, Liu Yi "quoted the method of positive and negative profit and loss from the opening square", breaking through the limitation that the unknown coefficient can only be a positive number. Although the lives of Jia and Liu are little known today, their creations are centuries earlier than those of the Europeans.
The world's higher than three-dimensional digital equation was first proposed in Qin Jiushao's work, and he successfully handled the following equations:
The first European solution to the quadratic equation was completed by the Italian Ferrari (1522-1565). Qin Jiushao also generalized the "multiplication method" into a numerical solution to any higher-order equation, such as an equation:
the coefficients can be either positive or negative, and there are integers and decimals. In this way, the principle of multiplication and multiplication is implemented to the end. The examples proposed by Qin Jiushao include a total of 25 quadratic equations, cubic equations, and quadratic equations, and 1 ten-dimensional equation. He pushed the numerical solution of higher-order equations in our country to a new stage, and his simple procedure is still common in modern mathematical calculations. This highly creative contribution, known in modern mathematics as the Ruffini-Horner method, was proposed by the Italian Ruffimi (1765-1822) in 1804 and the Englishman Homer (1786-1837) in 1819, more than 500 years after Qin Jiushao's discovery.
American historian of science Sarton
The famous American historian of science Sarton (1884-1956) said in his masterpiece "Introduction to the History of Science": "Qin Jiushao was one of the greatest mathematicians of his time and all periods in the Chinese nation. This evaluation is approved by the current scientific historians, and its significance is naturally far from being limited by the concept of time and space of "Sichuan in the Song Dynasty".
* About author:Liu Fusheng is a professor at the School of History and Culture of Sichuan University. Article/Social Science Research, No. 4, 1996, this article was transferred from the public number: Bashu Quanshu.