The Historical Origin of Ancient Mathematics

The Yellow River Basin and the Yangtze River Basin are the cradles of China culture. Around 2000 BC, the first slave country, the Xia Dynasty, appeared in the middle and lower reaches of the Yellow River. Then the Shang Dynasty and the Yin Dynasty (about 1500B. C- 1027B。 C) and Zhou dynasty (1027B. C-22 1B。 c)。 Historically, it is also called the Spring and Autumn Period and the Warring States Period from the 8th century BC to the establishment of the Qin Dynasty (22 1B). c)。

According to Yi. The copula records: "In ancient times, the knot was used to rule, and later saints used books to do it." There are many numerals in Oracle Bone Inscriptions unearthed in Yin Ruins. From one to ten, as well as hundred, thousand and ten thousand, are special symbolic characters. * * * There are 13 independent symbols, and the notation is written in a combined document, including decimal notation, with a maximum of 30,000.

Calculation is a calculation tool in ancient China, and this calculation method is called calculation. The age of calculation cannot be verified, but it is certain that calculation has been very common in the Spring and Autumn Period.

There are two ways to calculate numbers by counting chips, vertical and horizontal:

123456789

vertical

horizontal

When representing multiple digits, the decimal numerical system is adopted, and the digits of each digit are arranged from left to right, criss-crossing (the rule is: one is vertical and ten is horizontal, one hundred stands upright, one thousand is relative to ten, and ten thousand is equal to one hundred), and a space is used to represent zero. Calculation and financing establish good conditions for addition, subtraction, multiplication and division.

Calculation was not gradually replaced by abacus until the end of Yuan Dynasty in15th century. It was on the basis of calculation that China ancient mathematics made brilliant achievements.

In geometry and historical records. Xia Benji said that Yu Xia has used drawing and measuring tools, such as rules, moments, standards and ropes. And a special case of Pythagorean Theorem (called Pythagorean Theorem in the West) has been discovered. During the Warring States Period, the Work Inspection Book written by Qi people summarized the technical specifications of handicraft industry at that time, including some measurement contents, involving some geometric knowledge, such as the concept of angle.

A hundred schools of thought contended during the Warring States period also promoted the development of mathematics, and some schools also summarized many abstract concepts related to mathematics. As we all know, Mo Jing's definitions and propositions of some geometric terms, such as "a circle, an equal length", "flat, the same height" and so on. Mohist school also gave the definitions of finite and infinite. Zhuangzi records the famous theories of Hui Shi and others, as well as the topics put forward by debaters such as Huan Tuan and Gong Sunlong, emphasizing abstract mathematical ideas, such as "the greatest has no external meaning, the smallest has no internal meaning", "one foot pestle, half a day, inexhaustible" and so on. Many mathematical propositions such as these definitions and limit ideas of geometric concepts are quite valuable mathematical ideas, but this new idea of attaching importance to abstraction and logical rigor has not been well inherited and developed.

In addition, the Book of Changes, which tells the gossip of Yin and Yang and predicts good and bad luck, has sprouted from combinatorial mathematics, reflecting the idea of binary system. This period includes the mathematical development from Qin and Han Dynasties to Sui and Tang Dynasties 1000 years, and the successive dynasties were Qin and Han Dynasties, Wei, Jin, Southern and Northern Dynasties, Sui and Tang Dynasties.

The Qin and Han Dynasties was the formation period of China's ancient mathematical system. In order to systematize and theorize the increasing mathematics knowledge, specialized mathematics books have appeared one after another.

There are two main achievements in mathematics in the astronomical work Zhou Bi suan Jing compiled at the end of the Western Han Dynasty (the first century BC): (1) put forward the special case and universal form of Pythagorean theorem; (2) Chen Zi's method of measuring the height and distance of the sun was a pioneer of gravity difference. In addition, there are more complicated root-finding problems and fractional operations.

Nine Chapters Arithmetic is an ancient mathematical classic that has been compiled and revised by several generations. It was written in the early years of the Eastern Han Dynasty (1st century AD). This book is written in the form of problem sets, and * * * collects 246 questions and their answers, which belong to nine chapters: Tian Fang, Xiaomi, Decline, Shaoguang, Work, Average Loss, Profit and Loss, Equation and Pythagoras. The main contents include four fractional and proportional algorithms, the calculation of various areas and volumes, and the calculation of pythagorean measurement. In algebra, the concept of negative number and the law of addition and subtraction of positive and negative numbers introduced in the chapter of equation are the earliest records in the history of mathematics in the world. The solution of linear equations in the book is basically the same as that taught in middle schools now. As far as the characteristics of Nine Chapters Arithmetic are concerned, it pays attention to the application and integration of theory with practice, and forms a mathematical system centered on calculation, which has had a far-reaching impact on the ancient calculation in China. Some of its achievements, such as decimal numerical system, modern skills and surplus skills, have also spread to India and Arabia, and through these countries to Europe, which has promoted the development of world mathematics.

During the Wei and Jin Dynasties, China's mathematics developed greatly in theory. Among them, the work of Zhao Shuang and Liu Hui is regarded as the beginning of China's ancient mathematical theory system. Zhao Shuang was one of the earliest mathematicians who proved mathematical theorems and formulas in ancient China, and made detailed comments on Zhou Kuai Shu Jing. The Nine Chapters Arithmetic annotated by Liu Hui not only explains and deduces the methods, formulas and theorems of the original book as a whole, but also makes many innovations in the process of discussion, and even writes the island calculation method, which uses gravity difference technology to solve the problems related to measurement. One of Liu Hui's important tasks is to create secant, which lays a theoretical foundation for the study of pi and provides a scientific algorithm.

The society in the Southern and Northern Dynasties was in a state of war and division for a long time, but the development of mathematics was still vigorous. Sun Tzu's Art of War, Xiahou Yang's Art of War and Zhang Qiu's Art of War are all works of this period. Sun Tzu's Mathematical Classics gives the problem of "unknown things" and leads to the solution of a congruence group problem. The "Hundred Chickens Problem" in Zhang Qiujian suan Jing leads to three unknown indefinite equations.

The most representative works in this period are the works of Zu Chongzhi and Zu Rihuan. On the basis of Liu Hui's annotation of Nine Chapters Arithmetic, they greatly promoted traditional mathematics and became a model of attaching importance to mathematical thinking and reasoning. They also made outstanding contributions to astronomy. Their book seal script has been lost. According to historical records, they have made three great achievements in mathematics: (1) Calculate pi to the sixth place after the decimal point and get 3. 14 15926.

Large-scale architecture in Sui Dynasty objectively promoted the development of mathematics. In the early years of the Tang Dynasty, Wang Xiaotong wrote the Sutra of Ji Gu, which mainly discussed the calculation of earthwork in civil engineering, the division of labor and acceptance of engineering, and the calculation of warehouses and cellars.

The Tang Dynasty made great progress in mathematics education. In 656, imperial academy established a Mathematics Museum with doctors and teaching assistants in mathematics. Taishi ordered Li and others to compile and annotate ten calculation books (including Zhou Pi Ai Shu, Jiu Zhang Shu Shu, Dao Shu, Sun Zi Shu, Zhang Qiu Shu, Xiahou Yang Shu, Ji Gu Shu and Sun Zi Shu). It has played an important role in preserving ancient mathematical classics.

In addition, due to the need of calendar in Sui and Tang Dynasties, the quadratic interpolation method was established, which laid the foundation for the higher-order interpolation method in Song and Yuan Dynasties. In the late Tang Dynasty, computing technology was further improved and popularized, and many practical arithmetic books appeared, trying to simplify the multiplication and division algorithm. After the demise of the Tang Dynasty, the Five Dynasties and Ten Kingdoms remained the continuation of the warlord melee. Until the Northern Song Dynasty unified China, agriculture, handicrafts and commerce flourished rapidly, and science and technology advanced by leaps and bounds. 1 1 century to14th century (Song and Yuan Dynasties), computational mathematics reached its peak, which was the heyday of unprecedented prosperity and fruitful achievements in ancient Chinese mathematics. During this period, a number of famous mathematicians and mathematical works appeared, which are listed as follows: Jia Xian's Nine Chapters of the Yellow Emperor (165438+mid-20th century), On the Origin of Ancient Times (65438+mid-2nd century), Several Nine Chapters (1247) and. Yang Huijiu's algorithm (126 1), daily algorithm (1262), Yang Hui's algorithm (1274- 1275) and Zhu Shijie's arithmetic enlightenment (65438).

Mathematics in Song and Yuan Dynasties reached the peak of ancient mathematics in China in many fields, even in the world at that time. The main tasks are:

Numerical solutions of higher order equations;

Celestial arithmetic and quaternion, that is, the legislation and solution of higher-order equations, are the first time in the history of Chinese mathematics to introduce symbols and use symbolic operations to solve the problem of establishing higher-order equations;

The technique of finding the solution of large extension, that is, the solution of a group of congruences, is now called China's remainder theorem;

Raise superposition, that is, high-order interpolation and high-order arithmetic progression summation.

In addition, other achievements include the new development of Pythagorean method, the research of solving spherical right triangle, the research of vertical and horizontal diagram (magic square), the concrete application of decimal, the appearance of abacus and so on.

During this period, folk mathematics education also developed, and the exchange of mathematics knowledge between China and Islamic countries also developed. This period lasted more than 500 years from the establishment of the Ming Dynasty in the middle of14th century to the end of the Qing Dynasty in the 20th century. There are common weaknesses in mathematics except abacus, which involve the limitations of abacus, the deletion of mathematics content in the examination system of13rd century, and the eight-stage examination system of Daxing in Ming Dynasty. Many Chinese and foreign historians of mathematics are still discussing the reasons involved. /kloc-At the end of the 6th century, western elementary mathematics began to be introduced into China, which led to the integration of Chinese and western mathematics research in China. After the Opium War, modern advanced mathematics began to be introduced into China, and China's mathematics turned into a period of mainly studying western mathematics. Until the end of19th century, China's research on modern mathematics really began.

The greatest achievement of the Ming Dynasty was the popularization of abacus, and many abacus readers appeared. It was not until the publication of Cheng Dawei's Command Arithmetic (1592) that the abacus theory was systematized, marking the completion of the transition from preparation to abacus. However, due to the popularity of abacus calculation, calculation almost disappeared, and ancient mathematics based on calculation gradually disappeared, and mathematics stagnated for a long time.

During the Sui Dynasty and the early Tang Dynasty, Indian knowledge of mathematics and astronomy was introduced to China, but it had little influence. By the end of16th century, western missionaries began to enter China and cooperated with China scholars to translate many western mathematical monographs. Among them, the first and most influential one is the first six volumes of Geometrical Elements translated by Italian missionaries Matteo Ricci and Xu Guangqi (1607), whose rigorous logical system and translation methods are highly praised by Xu Guangqi. Xu Guangqi's "Measuring Similarities and Differences" and "The Meaning of Pythagoras" applied the logical reasoning method of "Elements of Geometry" and demonstrated China's Pythagorean observation. In addition, most of the nouns in the textbook "Elements of Geometry" are the first and still in use today. In the imported western mathematics, trigonometry is second only to geometry. Before that, trigonometry had only sporadic knowledge, and then it developed rapidly. The works introducing western trigonometry include DACE (2 volumes, 163 1), Table of Secant Circle and Eight Lines (6 volumes) edited by Deng, and giacomo Rowe's Sense Measurement (10 volumes,163/kloc-). In Xu Guangqi's almanac of Chongzhen (volumes 137, 1629- 1633), the mathematical knowledge about the curve of circular vertebra was introduced.

After entering the Qing Dynasty, Mei Wending, an outstanding representative of Chinese and Western mathematics, firmly believed that China's traditional mathematics must be refined, made in-depth research on ancient classics, and treated western mathematics correctly, which made it take root in China and had a positive impact on the climax of mathematics research in the middle of Qing Dynasty. Contemporary mathematicians include Wang Xizhi and Xirao Nian.

Xu Guangqi et al.

Emperor Kangxi of the Qing Dynasty liked scientific research. His "Essence of Mathematics" (53 volumes, 1723) is a comprehensive elementary mathematics work, which has a certain influence on mathematics research at that time.

During the reign of Ganjia, the school of Ganjia, which was mainly based on textual research, compiled Sikuquanshu, in which the mathematical works included Ten Books of Calculating Classics and the works of Song and Yuan Dynasties, which made important contributions to the preservation of endangered mathematical classics.

In the research of traditional mathematics, many mathematicians have made inventions. For example, Jiao Xun, Wang Lai and Li Rui, who are called "three friends who talk about the sky", have done a lot of important work. Li got the summation formula of triangular self-riding crib in the Stack Ratio Class (about 1859), which is now called "Lie identity". Compared with mathematics in Song and Yuan Dynasties, these works are an improvement. Ruan Yuan, Li Rui and others compiled 46 volumes (1795- 18 10) of Biography of Astronomers and Mathematicians, which initiated the study of the history of mathematics.

1840 After the Opium War, the closed-door policy was forced to stop. The "Arithmetic" Pavilion was added to Wentong Pavilion, and the translation pavilion was added to Shanghai Jiangnan Manufacturing Bureau, which opened the climax of the second translation introduction. The main translators and works are as follows: The last nine volumes of The Elements of Geometry (1857) jointly translated by Li and British missionary William gave China a complete Chinese translation of The Elements of Geometry; Algebra13 (1859); Represents the differential product, volume 18 (1859). Li and the English missionary Ai He translated the Theory of Conic Curves into three volumes, Hua and the English missionary John Flair translated Algebra into 25 volumes (1872), Tracing the Source of Differential Products into eight volumes (1874), Doubt Numbers 10 (/kloc-). In these translations, many mathematical terms and terms were created, which are still used today.

1898, Shi Jing University Hall was established and Wentong Museum was merged. 1905, the imperial examination was abolished and western-style school education was established. The textbooks used were similar to those of other western countries. This period is a period from the beginning of the 20th century to the present, which is often divided into two stages marked by the establishment of 1949 New China.

Modern mathematics in China started from studying abroad in the late Qing Dynasty and the early Republic of China. 1903 Feng Zuxun who studied mathematics earlier, 1908 Zheng who studied in America, 19 10 Hu Mingfu who studied in America,191/kloc-0. 19 13 Chen He studying in Japan 19 15 Xiong Qinglai studying in Belgium, 19 19 Su et al. studying in Japan. Most of them became famous mathematicians and mathematicians after returning to China, and made important contributions to the development of modern mathematics in China. Among them, Hu Mingfu received his doctorate from Harvard University in the United States on 19 17, becoming the first mathematician in China to receive his doctorate. With the return of foreign students, mathematics education in universities all over the world has improved. At first, only Peking University 19 12 set up the Department of Mathematics, Jiang Lifu 1920 set up the Department of Mathematics in Tianjin Nankai University, Xiong Qinglai 1926 set up the Department of Mathematics in Southeast University (now Nanjing University) and Tsinghua University, and soon Wuhan University, cheeloo university University and Zhejiang University were also established. 1930, Xiong Qinglai initiated the establishment of the Mathematics Research Department in Tsinghua University, and began to recruit graduate students. Chen Shengshen and Wu Daren became the earliest mathematics graduate students in China. In 1930s, (1927), (1934), Hua (1936) and (1936) went abroad to study mathematics, and they all became the backbone of modern mathematics development in China. At the same time, foreign mathematicians also come to China to give lectures, such as Russell in Britain (1920), boekhoff in the United States (1934), osgood (1934), Wiener (1935) and Adama in France (/kloc-0). 1935 the inaugural meeting of chinese mathematical society was held in Shanghai, attended by 33 delegates. The publication of 1936 annals of chinese mathematical society and Journal of Mathematics marks the further development of modern mathematics research in China.