Chapter 4 Section 4: The Climax of Chinese Mathematics from the Mid-Tang Dynasty to the Song and Yuan Dynasties

After the great development of the prosperous Tang Dynasty, after the middle of the Tang Dynasty, new substantive changes gradually occurred in production relations and all aspects of society. In the second half of the 10th century, Zhao Kuangyin established the Song Dynasty and unified China. The Chinese feudal society entered a new stage. The land ownership system Mainly state-owned changed to privately owned, and tenant farmers replaced the Wei and Tang dynasties who had the status of serfs and apprentices.Agriculture, handicrafts, commerce and science and technology have been greatly developed.Three of the four great inventions in ancient China—the wide application of printing and movable type printing, gunpowder for war, and compass for navigation—were completed in the mid-Tang Dynasty to the Northern Song Dynasty.In the seventh year of Yuanfeng (AD 1084), Secretary of the Song Dynasty published for the first time ten calculation scriptures including "Nine Chapters of Suanshu" (at that time, "Xiahou Yang Suanjing" and "Zhushu" had been lost, because one in the second half of the 8th century At the beginning of Han Yan's "Shushu" there is "Xia Houyang's saying" and so on, and he mistakenly thought it was the former and engraved it, and the latter had to be left out), which is the first printed mathematics work in the world.Later, Bao Huanzhi, a mathematician in the Southern Song Dynasty, reprinted and engraved these engraved editions, including "Nine Chapters on Suanshu" (half), "Zhoubi Suanjing", "Sun Tzu Suanjing", "Five Cao Suanjing", "Zhang Qiujian Suanjing" Wuzhong and "Shu Shu Ji Yi" and other isolated copies have been handed down to the present, and are the earliest printed mathematics works in the world.Most of the works of Song and Yuan mathematicians Jia Xian, Li Ye, Yang Hui, and Zhu Shijie were published shortly after they were written.With the help of printing, mathematical works can be widely circulated unprecedentedly, which is of far-reaching significance to the dissemination and popularization of mathematical knowledge.

The climax of mathematics in the Song and Yuan Dynasties had already been seen in the middle of the Tang Dynasty.With the vigorous development of commerce and trade, people improved the method of multiplication and division. The new and old "Book of Tang" recorded a large number of such books, but most of them were lost. Only Han Yan (life unknown) "Math" (8th century) and " The name of Xia Houyang Suan Jing has been handed down. This book proposes several shortcut algorithms for converting multiplication and division into addition and subtraction, and uses decimals in calculations, which is extremely valuable.

In the first half of the 11th century, Jia Xian (whose life is unknown) wrote "Nine Chapters of the Yellow Emperor's Computational Classics and Fine Grass", which is the most important mathematical work in the Northern Song Dynasty.Jia Xian served as Zuoban Dianzhi (low-level military officer) and was a student of Chu Yan, a famous astronomer and mathematician at that time.He also wrote the second volume of "Algorithm (Xuepo) Ancient Collection", which has been lost.He abstracted most of the technical texts in "Nine Chapters of Arithmetic" that did not leave the topic and set specific objects or even numerical values ​​into general arithmetic texts, which improved the theoretical level of "Nine Chapters of Arithmetic"; he summarized certain types of mathematical problems, such as proposing The origin of the method of prescribing is the Jiaxian Triangle. As the calculation table of the method of establishing and releasing the lock (that is, prescribing) proposed by him, this is the outline of the problem of prescribing; he has proposed several new important methods, the most prominent of which is to create a The method of multiplication is proposed, and the program of the fourth power is proposed.Jia Xian's thoughts and methods had a great influence on Song and Yuan mathematics, and he was one of the main promoters of Song and Yuan mathematics. Most of "The Yellow Emperor's Nine Chapters of Suanjing and Fine Grass" was copied by Yang Hui's "Detailed Explanation of the Nine Chapters Algorithm" (Que Juan 1, 2 and the upper part of Juan 3, and a part of Juan 5).

The great scientist Shen Kuo (AD 1031-1095) made unique contributions to mathematics.In "Mengxi Bi Tan", he first created the gap accumulation technique, pioneered the summation problem of high-order arithmetic series, and proposed the rounding technique, and first proposed the approximate formula for finding the arc length of a bow. In the 12th century, Liu Yi (biography unknown) in the Northern Song Dynasty wrote "Yigu Yuanyuan", which was also lost.Yang Hui's "Field Mu Comparison, Multiplication and Division Method" quoted several of its topics and methods. After "Zishu" was lost, the coefficients of the method of extraction were still all positive numbers. Liu Yi broke through this limitation, first introduced the equation of negative coefficients, and created the method of formulating formulas of benefit and subtraction to find their positive roots. Yang Huiyu's It is "the real crown before the ancient".

In 1127, the Jin Dynasty entered the Central Plains, and Zhao and Song moved south, which was known as the Southern Song Dynasty in history. In 1234, the Mongolian nobles destroyed the gold and later established the Yuan Dynasty. In 1279, the Yuan Dynasty destroyed the Southern Song Dynasty and unified the whole country. From the middle of the 13th century to the beginning of the 14th century, it was a concentrated expression of the upsurge of mathematics in the Song and Yuan Dynasties, and it was also the half century that left the most important mathematical works in Chinese history. Side two math centers.

The South Center, represented by Qin Jiushao and Yang Hui, focuses on the research of numerical solutions of higher-degree equations, congruence solutions, and improved multiplication and division algorithms.The northern center is represented by Li Ye, and mainly focuses on the Tianyuan technique of high-order equations and its solutions.Zhu Shijie, after the unification of China in the Yuan Dynasty, gathered the great achievements of the two mathematics centers in the north and the south, reaching the highest level of Chinese calculation. In 1247, Qin Jiushao compiled 18 volumes of "Nine Chapters of Shu Shu".Qin Jiushao, courtesy name Daogu, claimed to be a native of Lujun (now Shandong Province), and was born in Anyue County, Puzhou (now Sichuan Province) around 1202.He lived in the last years of the Southern Song Dynasty when the Song and Yuan struggled fiercely, and was involved in the battle between the ruling group and the two factions in the Southern Song Dynasty. He supported Wu Qian, an anti-war faction, and was repeatedly impeached by Liu Kezhuang and others.After Jia Sidao took power, he was demoted to Meizhou (now Guangdong Province), and soon (about 1261 AD) died in office, and was slandered by Zhou Mi who followed Jia Sidao after his death.He is talented and studious, and is very proficient in mathematics, astronomy, civil engineering, poetry, rhythm, bow and horse, etc.He has repeatedly called on the rulers to implement benevolent governance, and regards mathematical knowledge as a powerful tool to open up sources of income, implement benevolent governance, and benefit the country and the people. "Nine Chapters of Shushu" is divided into 81 questions in nine categories: Dayan, Tianshi, Tianyu, Surveying, Taxation, Money and Valley, Construction, Military Brigade, and Market Change. There are 88 conditions in the question, and some answers have as many as 180. The number of military questions is also unprecedented, reflecting the concern of the Qin family against the Yuan War.Dayan's total arithmetic system solves the solution method of a congruence group; positive and negative square extraction develops the method of finding positive roots of higher-order equations dominated by the multiplication and opening method to a very complete level, and some equations are as high as ten times; linear The solution of equations completely replaces the direct division method with the method of mutual multiplication and cancellation; a triclinic quadrature formula equivalent to Heron's formula is proposed; a complete decimal notation is used, etc., are all outstanding achievements.

Yang Hui wrote a total of five works on mathematics, four of which have been handed down, ranking first among mathematicians before the Yuan Dynasty.Yang Hui, whose name is Qianguang, was born in Qiantang (now Hangzhou City).Compared with others, his works focus on education and popularization. In 1261, Yang Hui solved problems and made analogies on the basis of Liu Hui's annotations, Li Chunfeng's annotations, and Jia Xianxicao's "Nine Chapters of Arithmetic", and added three volumes of diagrams, multiplication and division, and compilation. Algorithm", the present map, multiplication and division, Fangtian, corn, the first half of the decline, and a part of Shang Gong have been lost.The stacking technique in Shang Gongzhang's analogy developed Shen Kuo's gap producting technique; "Compilation" broke the classification pattern of "Nine Chapters of Arithmetic", and divided it into multiplication and division, interchange, combination ratio, division ratio, and decline according to the method. There are nine categories of points, accumulation, surplus and deficiency, equation, and Pythagorean shares. In 1262, he wrote "Daily Algorithm", which focused on improving the multiplication and division algorithm, and only a small number of topics survived. In 1274, he wrote three volumes of "Multiplication and Division Tongbian Fundamentals and Moments".The "Compendium of Learning Calculation" on the volume is a mathematics teaching plan from the enlightenment to the main methods of "Nine Chapters".This book also summarizes the nine-return equal multiplication and division algorithm and its formulas.In the next year, he compiled the second volume of "The Method of Multiplication and Elimination of Field Mu Comparison", quoted Liu Yi's method and title, and criticized the mistake of seeking the method of four different fields in "Five Cao Suanjing".In the same year, he compiled the second volume of "The Algorithm of Resuming the Ancients and Picking Strange Things", which made a great contribution to the study of the vertical and horizontal graph, that is, the magic square.The latter three books are often collectively referred to as "Yang Hui's Algorithm".

In the twelfth and thirteenth centuries, many works on Tianyuanshu appeared in the north, but most of them were lost. The earliest works handed down to this day with Tianyuanshu as the main method are the 12 volumes of "Cheyuan Haijing" (AD 1248), " Yigu Yanduan" three volumes (AD 1259).Li Ye (1192-1279 A.D.), courtesy name Renqing and nickname Jingzhai, was born in Luancheng (now Hebei Province), Zhending, and was born in Daxing (now Beijing).His father was a clean and honest official. Li Ye received a good education since he was a child, and was fond of mathematics. He became a well-known scholar in the Central Plains when he was young, and he was a Jinshi in Jinci Fu.After entering the Yuan Dynasty, he lived in seclusion in the area of ​​Xin and Nao [Guo Guo] (now northern Shanxi Province), studying mathematics and various knowledge under extremely difficult conditions. Since 1251, he has presided over Fenglong Academy (now Hebei Province). In 1257 and 1260, he was summoned twice by Kublai Khan, the ruler of the Yuan Dynasty, and published political views on legislation, rectification of discipline, promotion of gentlemen, retreat of villains, reduction of punishment, termination of war, and opposition to racial prejudice.He was hired as a Hanlin bachelor.However, he was ashamed of being an imperial literati who only obeyed the orders of the emperor and the prime minister, so he soon returned to Fenglongshan with the excuse of old illness.He wrote a lot of literature and history in his life, only "Jingzhai Ancient and Modern" survives. "Measuring Yuanhai Jing" considers 10 basic relationships between Pythagorean shapes and circles on the basis of the nine volumes of Dongyuan. Long question, the answer is of course all the same.Most of these problems need to use Tianyuan technique to formulate equations.Volume 1 is the theoretical basis of the whole book, including the round city schema, recognition miscellaneous notes and other parts.The round city schema uses Chinese characters such as heaven, earth, qian, and kun to represent points, which is a pioneering work.Recognition Miscellaneous Notes puts forward 692 formulas, all of which are correct except eight, which is a collection of the achievements of the research on the relationship between Pythagorean and circle in the past dynasties. "Yigu Yanduan" has 64 questions. This is a work that uses Tianyuan technique to explain the equations of Jiang Zhou (probably from the Northern Song Dynasty) "Yigu Ji".Among them, several titles and old techniques (methods) of "Yigu Collection" are preserved.

Zhu Shijie has two important works "Enlightenment of Mathematics" (AD 1299) and "Siyuan Yujian" (AD 1303) handed down to the world.Zhu Shijie, courtesy name Hanqing, nicknamed Songting, was born in Yanshan (now Beijing), his life is unknown.At the end of the 13th century, he traveled around the country as a famous mathematician for more than 20 years, and many people learned mathematics from him. "Enlightenment of Mathematics" has 20 subjects and 259 questions, including multiplication and division and its shortcut algorithm, multiplication and opening method, Tianyuan technique and other aspects of mathematics at that time, forming a relatively complete system. "Siyuan Yujian" has 24 gates and 288 questions. The front of the volume gives four kinds of five diagrams including the seven-fold square diagram of the ancient method (improved Jiaxian triangle), as well as Tianyuanshu, Binaryshu, Sanyuanshu, Quaternary An example of a solution.The creation of the four-element elimination method, which solved the problem of multivariate high-order equations, the summation of high-order arithmetic series, and the problem of high-order difference method, is the greatest contribution of this book.This book is the highest level of mathematics in ancient China.

Yang Hui, Zhu Shijie and others improved and summarized the algorithm of multiplication and division, which led to the emergence of abacus and abacus arithmetic (about the middle of the Yuan Dynasty), and completed the reform of computing tools and computing technology in China.In the middle and late Yuan Dynasty, there appeared "Ding Ju Algorithm", Jia Heng's "Algorithm Almighty Collection", He Pingzi's "Detailed Algorithm" and other works that improved the multiplication and division algorithm.