Mathematics is a subject that studies concepts such as quantity, structure, change and spatial model. By using abstract and logical reasoning, the shape and motion of objects are counted, calculated, measured and observed. Mathematicians have extended these concepts in order to express new conjectures with formulas and establish strictly deduced truths from properly selected axioms and definitions.

Mathematical attribute is the measurable attribute of anything, that is, mathematical attribute is the most basic attribute of things. The existence of measurable attributes has nothing to do with parameters, and the result depends on the selection of parameters. For example, time, whether measured in years, months, days or hours, minutes and seconds; Space, whether measured in meters, microns, inches or light years, always has their measurable properties, but the accuracy of the results is related to these reference coefficients.

Mathematics is a science that studies quantitative relations and spatial forms in the real world. Simply put, it is the science of studying numbers and shapes. Due to the needs of life and labor, even the most primitive people know simple counting, and it has developed from counting with fingers or objects to counting with numbers.

The knowledge and application of basic mathematics will always be an indispensable part of individual and group life. The refinement of its basic concepts can be found in ancient mathematical documents of ancient Egypt, Mesopotamia and ancient India. Since then, its development has made small progress until the Renaissance in16th century, and the mathematical innovation generated by the interaction with new scientific discoveries led to the acceleration of knowledge, until today.

Today, mathematics is used in different fields of the world, including science, engineering, medicine and economics. The application of mathematics in these fields is usually called applied mathematics, and sometimes it will lead to new mathematical discoveries and the development of new disciplines. Mathematicians also study pure mathematics with no practical value, even though its application is often discovered later.

French Bourbaki School, founded in 1930s, thinks that mathematics, at least pure mathematics, is a theory to study abstract structures. Structure is a deductive system based on initial concepts and axioms. Boone School believes that there are three basic abstract structures: algebraic structure (group, ring, domain …), ordered structure (partial order, total order …) and topological structure (neighborhood, limit, connectivity, dimension …).

Edit the etymology of this paragraph.

Mathematics (mathematics; Greek: μ α θ η μ α κ? In the west, this word comes from the ancient Greek word μ? θξμα(máthēma) has learning, learning and science, and it has another narrow and technical meaning-"mathematical research", even in its etymology. Its adjective μ α θ η μ α κ? (mathēmatikós), which means related to study or hard work, can also be used to refer to mathematics. Its surface plural form in English and its surface plural form in French, les mathématiques, can be traced back to the neutral plural mathematica in Latin, which is Cicero's name from the Greek plural τ α α θ η μ α ι κ? (ta mathēmatiká), a Greek word used by Aristotle, refers to the concept of "everything counts".

(Latin: Mathemetica) means counting and counting technology.

Ancient mathematics in China was called arithmetic, also called arithmetic, and later it was changed to mathematics.

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Chip, a counting tool used by the Inca Empire. Mathematics, which originated from the early production activities of human beings, is one of the six great arts in ancient China, and is also regarded as the starting point of philosophy by ancient Greek scholars. Mathematical Greek μ α θ η μ α κ? (mathematikós) means "the basis of learning", which comes from μ? θξμα(máthema) ("science, knowledge and learning").

The evolution of mathematics can be regarded as the continuous development of abstraction and the extension of subject matter. The first abstract concept is probably number, and its cognition that two apples and two oranges have something in common is a great breakthrough in human thought. In addition to knowing how to calculate the number of actual substances, prehistoric humans also learned how to calculate the number of abstract substances, such as time-date, season and year. Arithmetic (addition, subtraction, multiplication and division) will naturally occur. Ancient stone tablets also confirmed the knowledge of geometry at that time.

In addition, writing or other systems that can record numbers are needed, such as Mu Fu or chips used by the Inca Empire to store data. There are many different counting systems in history.

Since the historical era, the main principles in mathematics have been formed, which are used for tax and trade calculation, for understanding the relationship between numbers, for measuring land and predicting astronomical events. These needs can be simply summarized as the study of quantity, structure, space and time in mathematics.

By16th century, elementary mathematics, such as arithmetic, elementary algebra and trigonometry, had been basically completed. The appearance of the concept of variables in the17th century made people begin to study the relationship between variables and the mutual transformation between graphs. In the process of studying classical mechanics, the method of calculus was invented. With the further development of natural science and technology, set theory and mathematical logic, which are produced for studying the basis of mathematics, have also begun to develop slowly.

Mathematics has been continuously extended since ancient times, and has rich interaction with science, and both of them have benefited a lot. There are many discoveries in mathematics in history, and they are still being discovered today. According to Mikhail B. Sevryuk's record in the Bulletin of the American Mathematical Society of June 5438+ 10, 2006: "Since 1940 (the first year of mathematical review), the number of papers and books in the database of mathematical reviews has exceeded1900,000, with an annual increase of more than 750,000. Most of this learning sea is a new mathematical theorem and its proof. "

Edit this history of Chinese mathematics

Mathematics, called arithmetic in ancient times, is an important subject in ancient Chinese science. According to the characteristics of the development of ancient mathematics in China, it can be divided into five periods: the germination period; The formation of the system; Development; Prosperity and the integration of Chinese and western mathematics.

The Germination of Ancient Mathematics in China

At the end of primitive commune, after the emergence of private ownership and commodity exchange, the concepts of number and shape developed further. The pottery unearthed during Yangshao culture period has been engraved with the symbol representing 1234. By the end of the primitive commune, written symbols had begun to replace knotted notes.

Pottery unearthed in Xi 'an Banpo has an equilateral triangle composed of 1 ~ 8 dots, and a pattern of 100 small squares divided into squares. The houses in Banpo site are all round and square. In order to draw circles and determine straightness, people have also created drawing and measuring tools such as rulers, moments, rulers and ropes. According to Records of Historical Records Xia Benji, Yu Xia used these tools in water conservancy.

In the middle of Shang Dynasty, a set of decimal numbers and notation had been produced in Oracle Bone Inscriptions, the largest of which was 30 thousand; At the same time, the Yin people recorded the date of 60 days with 60 names, including Jiazi, Yechou, Bingyin and Dingmao, which were composed of ten heavenly stems and twelve earthly branches. In the Zhou Dynasty, eight kinds of things were previously represented by eight diagrams composed of yin and yang symbols, which developed into sixty-four hexagrams, representing sixty-four kinds of things.

The book Parallel Computation in 1 century BC mentioned the methods of using moments of high, deep, wide and distance in the early Western Zhou Dynasty, and listed some examples, such as hook three, strand four, chord five and ring moments can be circles. It is mentioned in the Book of Rites that the aristocratic children of the Western Zhou Dynasty have to learn numbers and counting methods since they were nine years old, and they have to be trained in rites and music, shooting, controlling, writing and counting. As one of the "six arts", number has begun to become a special course.

During the Spring and Autumn Period and the Warring States Period, calculation has been widely used and decimal notation has been used, which is of epoch-making significance to the development of mathematics in the world. During this period, econometrics was widely used in production, and mathematics was improved accordingly.

The contention of a hundred schools of thought in the Warring States period also promoted the development of mathematics, especially the dispute of rectifying the name and some propositions were directly related to mathematics. Famous experts believe that the abstract concepts of nouns are different from their original entities. They put forward that "if the moment is not square, the rules cannot be round", and defined "freshman" (infinity) as "nothing beyond the maximum" and "junior" (infinitesimal) as "nothing within the minimum". He also put forward the idea that "one foot is worth half a day, which is inexhaustible".

Mohism believes that names come from things, and names can reflect things from different sides and depths. Mohist school gave some mathematical definitions. Such as circle, square, flat, straight, sub (tangent), end (point) and so on.

Mohism disagreed with the proposition of "one foot" and put forward the proposition of "non-half" to refute: if a line segment is divided into two halves indefinitely, there will be a non-half, which is a point.

The famous scholar's proposition discusses that a finite length can be divided into an infinite sequence, while the Mohist proposition points out the changes and results of this infinite division. The discussion on the definition and proposition of mathematics by famous scholars and Mohists is of great significance to the development of China's ancient mathematical theory.

The Formation of Ancient Mathematics System in China

Qin and Han dynasties were the rising period of feudal society, with rapid economic and cultural development. The ancient mathematical system of China was formed in this period, and its main symbol was that arithmetic became a specialized subject, and the emergence of mathematical works represented by Nine Chapters of Arithmetic.

Nine Chapters Arithmetic is a summary of the development of mathematics during the establishment and consolidation of feudal society in the Warring States, Qin and Han Dynasties. As far as its mathematical achievements are concerned, it is a world-famous mathematical work. For example, the operation of quartering, the present skills (called the three-rate method in the west), square roots and square roots (including the numerical solution of quadratic equations), the skills of surplus and deficiency (called the double solution in the west), various formulas of area and volume, the solution of linear equations, the principle of addition and subtraction of positive and negative numbers, the Pythagorean solution (especially the Pythagorean theorem and the method of finding Pythagorean numbers) and so on are all very high levels. Among them, the solution of equations and the addition and subtraction of positive and negative numbers are far ahead in the development of mathematics in the world. As far as its characteristics are concerned, it forms an independent system centered on calculation, which is completely different from ancient Greek mathematics.

"Nine Chapters Arithmetic" has several remarkable characteristics: it adopts the form of mathematical problem sets divided into chapters according to categories; Formulas are all developed from counting method; Mainly arithmetic and algebra, rarely involving graphic properties; Attach importance to application and lack of theoretical explanation.

These characteristics are closely related to the social conditions and academic thoughts at that time. In Qin and Han dynasties, all science and technology should serve the establishment and consolidation of feudal system and the development of social production at that time, emphasizing the application of mathematics. Nine Chapters of Arithmetic, which was finally written in the early years of the Eastern Han Dynasty, ruled out the discussion of famous scholars and Mohists in the Warring States period on the definition and logic of nouns, but focused on mathematical problems and their solutions closely combined with production and life at that time, which was completely consistent with the development of society at that time.

Nine Chapters Arithmetic spread to Korea and Japan in Sui and Tang Dynasties, and became the mathematics textbook of these countries at that time. Some of its achievements, such as decimal numerical system, modern skills and surplus skills, have also spread to India and Arabia, and through India and Arabia to Europe, which has promoted the development of mathematics in the world.

The Development of Ancient Mathematics in China

Metaphysics, which appeared in Wei and Jin dynasties, was not bound by Confucian classics in Han dynasty and was active in thought. It can argue and win, use logical thinking and analyze truth, all of which are conducive to improving mathematics theoretically. During this period, the Nine Chapters Heavy Difference Diagram appeared in Wu and Zhao's annotation of Zhou Huishu, Xu Yue's annotation of Nine Chapters Arithmetic in the late Han Dynasty and early Wei Dynasty, and Liu Hui's annotation of Nine Chapters Arithmetic in the Wei and Jin Dynasties. The work of Zhao Shuang and Liu Hui laid a theoretical foundation for the ancient mathematical system of China.

Zhao Shuang was one of the earliest mathematicians who proved and deduced mathematical theorems and formulas in ancient China. The Pythagorean Grid Diagram and Annotations and Sunrise Diagram and Annotations added by him in Zhou Kuai Shu Jing are very important mathematical documents. In Pythagorean Square Diagram and Notes, he put forward five formulas to prove Pythagorean theorem and Pythagorean shape with chord diagram; In Sunrise Picture, he proved the weight difference formula widely used in Han Dynasty with the graphic area. Zhao Shuang's work was groundbreaking and played an important role in the development of ancient mathematics in China.

Liu Jicheng, who was contemporary with Zhao Shuang, developed the thoughts of famous scholars and Mohists in the Warring States Period and advocated strict definition of some mathematical terms, especially important mathematical concepts. He believes that mathematical knowledge must be "analyzed" in order to make mathematical works concise, compact and beneficial to readers. His Notes on Nine Chapters of Arithmetic not only explains and deduces the methods, formulas and theorems of nine chapters of arithmetic as a whole, but also gets great development during the discussion. Liu Hui created secant, proved the formula of circle area by using the idea of limit, and calculated the pi as 157/50 and 3927/ 1250 by theoretical method for the first time.

Liu Hui proved by infinite division that the volume ratio of right-angled square cone to right-angled tetrahedron is always 2: 1, which solved the key problem of general solid volume. When proving the volume of square cone, cylinder, cone and frustum, Liu Hui put forward the correct method to solve the volume of sphere completely.

After the Eastern Jin Dynasty, China was in a state of war and north-south division for a long time. The work of Zu Chongzhi and his son is the representative work of the development of mathematics in South China after the economic and cultural shift to the south. On the basis of Liu Hui's Notes on Nine Chapters of Arithmetic, they greatly promoted traditional mathematics. Their mathematical work mainly includes: calculating pi between 3.1415926 ~ 3.1415927; Put forward the principle of ancestor (constant sky); The solutions of quadratic and cubic equations are put forward.

Presumably, Zu Chongzhi calculated the area of the circle inscribed by the regular polygon 6 144 and the regular polygon 12288 on the basis of Liu Hui secant method, and thus obtained this result. He also obtained two fractional values of pi by a new method, namely the approximate ratio of 22/7 and the density ratio of 355/ 1 13. Zu Chongzhi's work made China lead the west in the calculation of pi for about one thousand years.

Zu Chongzhi's son Zu (Riheng) summed up Liu Hui's related work and put forward that "if the potentials are the same, the products cannot be different", that is, two solids with the same height, if the horizontal cross-sectional areas of any height are equal, the volumes of the two solids are equal, which is the famous Zu (Riheng) axiom. Zu (Riheng) applied this axiom to solve Liu Hui's unsolved spherical volume formula.

Emperor Yang Di was overjoyed and made great achievements, which objectively promoted the development of mathematics. In the early Tang Dynasty, Wang Xiaoyu's "Jigu Shujing" mainly discussed earthwork calculation, division of labor, acceptance and calculation of warehouses and cellars in civil engineering, which reflected the mathematical situation in this period. Wang Xiaotong established the cubic equation of number without using mathematical symbols, which not only solved the needs of the society at that time, but also laid the foundation for the establishment of the art of heaven. In addition, for the traditional Pythagorean solution, Wang Xiaotong also used the digital cubic equation to solve it.

In the early Tang Dynasty, the feudal rulers inherited the Sui system, and in 656, they set up the Arithmetic Museum in imperial academy, with 30 students, including arithmetic doctors and teaching assistants. Ten arithmetic classics edited and annotated by Taishiling Li are used as teaching materials for students in the Arithmetic Museum and as the basis for verifying arithmetic. Ten Books of Calculating Classics compiled by Li and others is of great significance in preserving classical works of mathematics and providing literature for mathematical research. Their notes on Zhoupian suan Jing, Nine Chapters Arithmetic and Island Suan Jing are helpful to readers. During the Sui and Tang Dynasties, due to the need of calendar, celestial mathematicians created quadratic function interpolation method, which enriched the content of ancient mathematics in China.

Calculation and compilation were the main calculation tools in ancient China. It has the advantages of simplicity, image and concreteness, but it also has some shortcomings, such as large compilation area, improper handling when the operation speed is accelerated and so on. So the reform was carried out very early. Among them, Taiyi, Ermi, Sancai and Abacus are all abacus with beads, which is an important technical reform. In particular, "abacus calculation" inherits the advantages of calculating five liters and decimal places, overcomes the shortcomings of inconvenient calculation and preparation of vertical and horizontal numbers, and its advantages are very obvious. But at that time, the multiplication and division algorithm could not be performed continuously. The abacus beads have not been worn and are not easy to carry, so they are still not widely used.

After the middle Tang Dynasty, the prosperity of commerce and the increase of digital calculation urgently required the reform of calculation methods. It can be seen from the list of books left by New Tang Shu and other documents that this algorithm reform is mainly to simplify the multiplication and division algorithm. The algorithm reform in the Tang Dynasty enabled multiplication and division to be operated in parallel, which was suitable for both calculation and abacus calculation.

The Prosperity of Ancient Mathematics in China

In 960, the establishment of the Northern Song Dynasty ended the separatist regime of the Five Dynasties and Ten Countries. Agriculture, handicrafts and commerce in the Northern Song Dynasty flourished unprecedentedly, and science and technology advanced by leaps and bounds. Gunpowder, compass and printing are widely used in this situation of rapid economic growth. 1084, the secretariat printed and published the Ten Books of Calculating Classics for the first time. 12 13, Bao Ganzhi reprinted it. All these have created good conditions for the development of mathematics.

During the 300 years from 1 1 to14th century, a number of famous mathematicians and mathematical works appeared, such as Jia Xian's Nine Chapters on Arithmetic and Fine Grass of the Yellow Emperor, The Theory of Ancient Roots, The Book of Nine Chapters, The Survey of the Sea Mirror Circle and Yi Gu Yan Duan.

From square root, square root to square root is more than four leaps in understanding, which was realized by Jia Xian. Yang Hui included Jia Xian's Kaiping Method of Multiplication and Kaiping Method of Multiplication in the Collection of Algorithms in Nine Chapters. Detailed Explanation of Algorithms in Chapter Nine contains Jia Xian's "Root Flow", "Seeking Base Grass by Multiplication" and examples of using multiplication to open the fourth power. According to these records, we can determine the binomial coefficient table discovered by Jia Xian, and create the methods of increasing, multiplying and opening. These two achievements had a great influence on the whole mathematics of Song and Yuan Dynasties, among which Jia Xian Triangle was put forward more than 600 years earlier than Pascal Triangle in the west.

It was Liu Yi who extended multiplication and multiplication to the solution of digital higher-order equations (including the case of negative coefficients). In the volume "Fast Method of Multiplication, Division and Ratio of Field and Mu" of Yang Hui Algorithm, 22 quadratic equations and 1 quartic equations in the original book are introduced, and the latter is the earliest example of solving higher-order equations by adding, multiplying and opening methods.

Qin is an expert in solving higher-order equations. He collected 2 1 questions in Shu Shu Jiu Zhang (the highest number is 10). In order to adapt to the calculation program of multiplication, multiplication and division, Jiu Shao defined the constant term as a negative number and divided the solutions of higher-order equations into various types. When the root of the equation is non-integer, Qin continued to find the decimal of the root, or used the sum of the coefficients of the reduced root transformation equation as the denominator and the constant as the numerator to represent the non-integer part of the root, which is the development of the method of dealing with irrational numbers in Nine Chapters Arithmetic and Liu Hui's notes. When seeking the second root, Qin also proposed a second trial and error method based on dividing the coefficient of the first term by the constant term, which was more than 500 years earlier than Horner's earliest method in the West.

Astronomers Wang Xun and Guo Shoujing in Yuan Dynasty solved the problem of cubic function interpolation in the calendar method. Qin mentioned the interpolation method in the title of "Composition Pushing Stars", Zhu Shijie mentioned the title of "Elephant Trick" in "Four Lessons" (they called it "Call Difference"), and Zhu Shijie got an interpolation formula of quartic function.

Using Tianyuan (equivalent to x) as the symbol of unknown number, the equation of higher order is established, which was called Tianyuan technique in ancient times. This is 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 earliest extant celestial art work is Ye Li's Rounding Sea Mirror.

It is another outstanding creation of mathematicians in Song and Yuan Dynasties to extend celestial sphere to higher-order simultaneous equations of binary, ternary and quaternary. What has been handed down to this day is Zhu Shijie's Meet with Siyuan, which systematically discusses this outstanding creation.

Zhu Shijie's representation of high-order four-element simultaneous equations is developed on the basis of celestial body theory. He put the constant in the center, the powers of four elements in four directions: up, down, left and right, and the others in four quadrants. Zhu Shijie's greatest contribution is the four-element elimination method. The method is to select an element as an unknown quantity, use polynomials composed of other elements as the coefficients of this unknown quantity, then list several high-order equations of an element, and then gradually eliminate the unknown quantity by the method of mutual multiplication and elimination. By repeating this step, we can eliminate other unknowns, and finally we can get the solution by multiplying and opening. This is an important development of linear method group solution, which is more than 400 years earlier than similar methods in the west.

The Pythagorean solution had a new development in the Song and Yuan Dynasties. Zhu Shijie put forward the known pythagorean sum, chord sum and pythagorean formula under the volume of "Arithmetic Enlightenment", which supplemented the shortcomings of "Nine Chapters Arithmetic". Ye Li made a detailed study of Pythagorean inclusion in "Measuring the Circle of the Sea Mirror", and obtained nine inclusion formulas, which greatly enriched the contents of ancient geometry in China.

Knowing the angle between the ecliptic and the equator and the back arc of the ecliptic when the sun runs from the winter solstice to the vernal equinox, it is a problem to solve the spherical right triangle. Traditional calendars are calculated by interpolation. In the Yuan Dynasty, Wang Xun, Guo Shoujing and others used the traditional Pythagorean method to solve this problem, while Shen Kuo used the skills of meeting the circle and Tianyuan. But what they got was an approximate formula and the result was not accurate enough. But their whole calculation steps are correct. In the mathematical sense, this method opens up a way for spherics.

The climax of China's ancient computing technology reform also appeared in the Song and Yuan Dynasties. The historical documents of the Song, Yuan and Ming Dynasties contain a large number of practical arithmetic bibliographies of this period, far more than those of the Tang Dynasty. The main content of the reform is still multiplication and division. At the same time as the algorithm reform, the abacus of piercing beads may have appeared in the Northern Song Dynasty. However, if the modern abacus calculation is regarded as both a threading abacus calculation and a set of perfect algorithms and formulas, it should be said that it was finally completed in the Yuan Dynasty.

The prosperity of mathematics in Song and Yuan Dynasties is the inevitable result of the development of social economy, science and technology and traditional mathematics. In addition, mathematicians' scientific thinking and mathematical thinking are also very important. Mathematicians in Song and Yuan Dynasties all opposed the mysticism of image number in Neo-Confucianism to varying degrees. Although Qin once advocated the homology of several ways, he later realized that there is no mathematics that "connects the gods", only mathematics that "manages everything"; In the preface to the encounter with Siyuan, Mo Ruo put forward the idea of "taking the virtual image as the truth and asking the truth with the virtual image", which represents a highly abstract thinking method; Yang Hui studied the structure of vertical and horizontal diagrams, revealed the essence of Luo Shu, and strongly criticized the mysticism of image numbers. These are undoubtedly important factors to promote the development of mathematics.

Integration of Chinese and western mathematics

China entered the late feudal society from the Ming Dynasty. The feudal rulers practiced totalitarian rule, propagated idealistic philosophy and practiced stereotyped examination system. In this case, in addition to abacus, the development of mathematics gradually declined.

After 16, western elementary mathematics was introduced into China, which led to the integration of Chinese and western mathematics research in China. After the Opium War, modern mathematics began to be introduced into China, and China's mathematics turned into a period of mainly learning western mathematics; It was not until the end of 19 and the beginning of the 20th century that the study of modern mathematics really began.

From the early Ming Dynasty to the middle Ming Dynasty, the development of commodity economy and the popularization of abacus were adapted to this commercial development. In the early Ming Dynasty, the appearance of Kuibn Du Xiang's four-character miscellaneous son and Luban Mu Jing showed that abacus had become very popular. The former is a textbook for children to read pictures, and the latter is to include abacus as a family necessity in the manual of general wooden furniture.

With the popularization of abacus calculation, the abacus calculation algorithm and formula are gradually improving. For example, Wang Wensu and Cheng Dawei increased and improved collisions and made formulas; Xu Xinlu and Cheng Dawei add and subtract formulas and are widely used in division, thus realizing all the formulas of four abacus calculations; Zhu Zaiwen and Cheng Dawei applied the method of calculating square root and square root to abacus calculation, and Cheng Dawei used abacus calculation to solve quadratic and cubic equations. Cheng Dawei's works are widely circulated at home and abroad and have great influence.

1582, Italian missionary Matteo Ricci went to China. 1607, he translated the first six volumes of Geometry Elements and one volume of Measuring the Meaning of Law with Xu Guangqi, and compiled Yi Rong's Pen Meaning with Li Zhizao. 1629, Xu Guangqi was appointed by the Ministry of rites to supervise the revision of the calendar. Under his auspices, he compiled the almanac of Chongzhen (137). The almanac of Chongzhen mainly introduces the geocentric theory of European astronomer Tycho. As the mathematical basis of this theory, it also introduces Greek geometry, some trigonometry of Yushan in Europe, Napier's calculation, Galileo's scale specification and other calculation tools.

Among the introduced mathematics, geometric elements have the greatest influence. The Elements of Geometry is China's first mathematical translation. Most mathematical terms are the first, and many are still in use today. Xu Guangqi thinks there is no need to doubt it and change it, and thinks that "there is no one in the world who can't learn well". The Elements of Geometry is a must-read for mathematicians in Ming and Qing Dynasties, which has a great influence on their research work.

Secondly, trigonometry is the most widely used, and the works introducing western trigonometry include Great Survey, Table of Secant Circle and Eight Lines, and Significance of Measurement. The survey mainly explains the properties, tabulation methods and table usage methods of the eight lines of triangle (sine, cosine, tangent, cotangent, secant, cotangent, orthovector and cotangent). In addition to adding some plane triangles that are missing in the big survey, the more important ones are the product sum and difference formula and spherical triangle. All these were used together with translation in the calendar work at that time.

1646, Polish missionary Monig came to China, and his followers were Xue Fengzuo and Fang Zhongtong. After Mounigo's death, Zuo compiled the General Theory of Calendar Societies based on what he had learned, so as to integrate China, France and western France. The mathematical contents in Sydney Huitong mainly include proportional logarithm table, new proportional four-line table and trigonometric algorithm. The first two books introduced the logarithm invented and modified by British mathematicians Napier and Briggs. In addition to the spherical triangle introduced by Chongzhen almanac, the latter book also includes half-angle formula, half-arc formula, German proportional formula, Nestor proportional formula and so on. Fang Zhongtong's "Several Words" explains the logarithmic theory. The introduction of logarithm is very important and is immediately applied in calendar calculation.

There are many books handed down from generation to generation by beginners in the Qing Dynasty by studying Chinese and Western mathematics. Among them, Wang Xichan's Illustration, Mei's Collected Works (including 13 kinds of mathematical works ***40 volumes) and Visual Research have great influence. Mei Wending is a master of western mathematics. He sorted out and studied the solution of linear equations, Pythagorean solution and the method of finding positive roots of higher powers in traditional mathematics, which brought vitality to the dying mathematics of Ming Dynasty. Xirao Nian's Vision is the first book in China to introduce the western perspective.

Emperor Kangxi of Qing Dynasty attached great importance to western science. Besides studying astronomy and mathematics by himself, he also trained some talents and translated some works. 17 12, Emperor Kangxi appointed Mei Li as the assembler of Ren Meng Yangzhai, and compiled astronomical algorithm books together with Chen Houyao, He Guozong, Ming Jiatu and Yang Daosheng. 172 1 year, fayuanli was completed in volume 100, and published in the name of Kangxi "Yu Ding" on 1723. Among them, Mei Gaocheng is mainly responsible for the essence of mathematics, which is divided into two parts. The first part includes geometrical features and algorithm elements, which are all translated from French works. The second part includes elementary mathematics such as arithmetic, algebra, plane geometry, plane triangle and solid geometry, including prime number table, logarithm table and trigonometric function table. Because it is a comprehensive encyclopedia of elementary mathematics, and it is known as "King James" by Kangxi, it has certain influence on the mathematical research at that time.

To sum up, we can see that mathematicians in the Qing Dynasty did a lot of work on western mathematics and achieved many original results. These achievements, if compared with traditional mathematics, are progressive, but compared with contemporary western countries, they are obviously backward.

After Yongzheng acceded to the throne, he closed his door to the outside world, which led to the cessation of importing western science into China and the implementation of a high-handed policy at home. As a result, ordinary scholars can't get in touch with western mathematics and dare not ask themselves what they have learned, so they bury themselves in studying ancient books. During the reign of Ganjia, the Ganjia school, which mainly focused on textual research, gradually formed.

With the collection and annotation of Ten Books of Calculating Classics and mathematics works in Song and Yuan Dynasties, there appeared a climax of learning traditional mathematics. Among them, Wang Lai, Li Rui and Li. Can break the old rules and have inventions. Compared with Song and Yuan Algebra, their work is better than Chen Wenzhao's. Compared with western algebra, it is a little late, but these achievements were obtained independently without being influenced by modern western mathematics.

At the same time that the research of traditional mathematics reached its climax, Ruan Yuan and Li Rui compiled The Biography of Astronomical Mathematicians-Biography of People in the Domain, which collected more than 270 astronomers and mathematicians who died from the Yellow Emperor to Jiaqing four years ago (among them, less than 50 were handed down from generation to generation), and 4 1 person of missionaries who introduced western astronomical mathematics since the late Ming Dynasty. This book is composed of "collecting history books, collecting group records and recording them", which is completely first-hand original information and has great influence in academic circles.

1840 after the opium war, modern western mathematics began to be introduced into China. First of all, the British set up the Mohai Library in Shanghai and introduced western mathematics. After the Second Opium War, Zeng Guofan, Li Hongzhang and other bureaucratic groups launched the "Westernization Movement", also advocated the introduction and study of western mathematics, and organized the translation of a number of modern mathematics works.

The most important one is Algebra translated by Li and Li. Algebra, Trace of Differential Product and Suspicious Mathematics translated by John Flair, a Briton; Zou He edited Metaphysics, Algebra and Mathematical Writing; Xie Hongtai and Pan He translated Dai Shen and Eight Acts.

A Generation of Differential Calculus is China's first translation of calculus. Algebra is a translation of Symbolic Algebra written by British mathematician Augustus de Morgan. Doubtful mathematics is the first translation of probability theory. In these translations, many mathematical terms and terms have been created, which are still used today, but the mathematical symbols used have generally been eliminated. After the Reform Movement of 1898, new law schools were established in various places, and these works became the main textbooks.

While translating western mathematics works, China scholars have also done some research and written some works, the most important of which are Li's Solution to the Reform of the Sharp Cone and Solving Several Roots. Xia Wanxiang's Picture Book of Cave, Music Induction Techniques, Music Induction Picture Book, etc. , are the research results that will connect Chinese and western academic thoughts.

Because the imported modern mathematics needs a process of digestion and absorption, and the rulers in the late Qing Dynasty are very corrupt, overwhelmed by the impact of the Taiping Heavenly Kingdom Movement and plundered by imperialist powers, they have no time to take care of mathematical research. It was not until the May 4th Movement of 19 19 that the study of modern mathematics in China really began.