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The history of mathematics

Mathematics is an important subject in ancient Chinese science. According to the characteristics of the development of ancient Chinese mathematics, it can be divided into Five periods: germination; formation of the system; development; prosperity and the integration of Chinese and Western mathematics.

The germ of ancient Chinese mathematics

After the emergence of private ownership and goods exchange at the end of the primitive commune, the concepts of number and shape have further developed. The pottery unearthed from the Yangshao Culture period has Engraved with the symbol representing 1234. By the end of the primitive commune, written symbols had begun to be used instead of knotted ropes to record events.

The pottery unearthed in Xi'an Banpo has an equilateral triangle composed of 1 to 8 dots and a square divided into 100 small squares. The foundations of the houses at the Banpo site are all round and square. In order to draw a circle, square it, and determine its straightness, people also created drawing and measuring tools such as rules, squares, standards, and ropes. According to "Historical Records·Xia Benji", Xia Yu had used these tools when controlling floods.

In the middle of the Shang Dynasty, a set of decimal numbers and notation had been produced in oracle bone inscriptions, the largest number of which was thirty thousand; at the same time, the Yin people used ten heavenly stems and twelve earthly branches to form Jiazi. 60 names including Yichou, Bingyin, and Dingmao were used to record the dates of 60 days; in the Zhou Dynasty, the eight trigrams that used to be composed of yin and yang symbols to represent eight kinds of things were developed into sixty-four hexagrams, which represented 64 kinds of things. .

The "Zhou Bi Suan Jing" of the first century BC mentioned the method of using moments to measure height, depth, breadth and distance in the early Western Zhou Dynasty, and cited the Pythagorean three hooks, four legs, five strings and Ring moments can be examples such as circles. The "Book of Rites Nei Ze" mentions that the children of the Western Zhou aristocrats must learn numbers and counting methods from the age of nine. They must be trained in etiquette, music, archery, control, calligraphy, and number. As one of the "six arts", number It has started to become a specialized course.

During the Spring and Autumn Period and the Warring States Period, calculations had been widely used, and the decimal value system had been used in calculation notation. This notation method was of epoch-making significance for the development of world mathematics. During this period, measurement mathematics was widely used in production, and there was a corresponding improvement in mathematics.

The contention of a hundred schools of thought during the Warring States Period also promoted the development of mathematics, especially the debates over the rectification of names and some propositions were directly related to mathematics. Famous experts believe that the noun concepts after abstraction are different from their original entities. They propose that "a square cannot be a square, and a square cannot be a circle." They define "big one" (infinity) as "the greatest without outside", and "small one" (small one) ( Infinitely small) is defined as "the smallest with no interior". He also put forward propositions such as "If you take half of a one-foot stick every day, it will be inexhaustible for eternity."

The Mohists believe that names come from things, and names can reflect things from different aspects and depths. Mohists give some mathematical definitions. For example, circle, square, flat, straight, time (tangent), end (point), etc.

The Mohists did not agree with the proposition of "one foot of stick" and put forward a "not half" proposition to refute: if a line segment is divided into half and half indefinitely, there will be a line that cannot be divided any more. The "not half", this "not half" is the point.

The propositions of famous scholars discuss that finite length can be divided into an infinite sequence, while the propositions of Mohists point out the changes and results of this infinite division. The discussions of mathematical definitions and mathematical propositions by famous scholars and Mohists are of great significance to the development of ancient Chinese mathematical theory.

The formation of the ancient Chinese mathematical system

The Qin and Han Dynasties were the rising period of feudal society, and both the economy and culture developed rapidly. The ancient Chinese mathematics system was formed during this period. Its main symbol is that arithmetic has become a specialized subject and the emergence of mathematical works represented by "Nine Chapters on Arithmetic".

Nine Chapters on Arithmetic is a summary of the development of mathematics during the establishment and consolidation of feudal societies in the Warring States, Qin, and Han Dynasties. In terms of its mathematical achievements, it can be called a world masterpiece of mathematics. For example, the four arithmetic operations of fractions, Jinyoushu (called the three-rate method in the West), square root and cube root (including numerical solutions to quadratic equations), surplus and deficiency method (called the double method in the West), various area and volume formulas, and linear equations The level of solving methods, addition and subtraction rules for positive and negative number operations, Pythagorean solution (especially the Pythagorean theorem and the method of finding the Pythagorean number), etc. is very high. Among them, the method of solving equations and the rules of addition and subtraction of positive and negative numbers are far ahead in the development of mathematics in the world. In terms of its characteristics, it forms an independent system centered on calculation and completely different from ancient Greek mathematics.

"Nine Chapters on Arithmetic" has several notable features: it adopts the form of a collection of mathematical problems divided into chapters; the calculation formulas are all developed from the calculation notation method; it focuses on arithmetic and algebra. The nature of graphics is rarely involved; emphasis is placed on application and lack of theoretical elaboration.

These characteristics are closely related to the social conditions and academic thoughts at that time. During the Qin and Han dynasties, all science and technology must serve the establishment and consolidation of the feudal system and the development of social production, emphasizing the application of mathematics. "Nine Chapters on Arithmetic", which was finally written in the early Eastern Han Dynasty, excluded the discussion of noun definitions and logic by famous scholars and Mohists who appeared in the Hundred Schools of Contest during the Warring States Period, and focused on mathematical problems and their solutions that were closely integrated with production and life at that time. This is completely consistent with the development of society at that time.

Nine Chapters of Arithmetic was spread to Korea and Japan during the Sui and Tang Dynasties, and became the mathematics textbooks in these countries at that time. Some of its achievements, such as the decimal value system, the modern art, and the art of surplus and deficiency, were also spread to India and Arabia, and were spread to Europe through India and Arabia, promoting the development of mathematics in the world.

The Development of Mathematics in Ancient China

The metaphysics that appeared in the Wei and Jin Dynasties was not bound by Han Confucian classics and was more active in thinking; it sought to win through debate and debate, and was able to use logical thinking. Analyzing the principles, these are conducive to improving mathematics theoretically. Zhao Shuang of Wu State annotated "Zhou Bi Suan Jing", Xu Yue wrote annotations of "Nine Chapters of Arithmetic" in the late Han Dynasty and early Wei Dynasty, and Liu Hui in the late Wei and early Jin Dynasty wrote annotations of "Nine Chapters of Arithmetic" and "Nine Chapters of Heavy Differences". during this period. The work of Zhao Shuang and Liu Hui laid the theoretical foundation for the ancient Chinese mathematical system.

Zhao Shuang is one of the earliest mathematicians in ancient China who proved and derived mathematical theorems and formulas. The "Pythagorean Square Diagram and Annotations" and "Sun High Diagram and Annotations" he added to "Zhou Bi Suan Jing" are very important mathematical documents. In "Pythagorean Square Diagram and Notes", he proposed using chord diagrams to prove the Pythagorean theorem and five formulas for solving Pythagorean shapes; in "Sun High Diagram and Notes", he used the area of ??a figure to prove the heavy difference commonly used in the Han Dynasty. Formula, Zhao Shuang's work is groundbreaking and plays an important role in the development of ancient Chinese mathematics.

At the same time as Zhao Shuang, Liu Huiyue inherited and developed the ideas of famous scholars and Mohists during the Warring States Period. He advocated strict definitions of some mathematical terms, especially important mathematical concepts, and believed that mathematical knowledge must be Only through "analysis" can mathematical works be concise and precise, which is beneficial to readers. His notes on "Nine Chapters on Arithmetic" not only provide a general explanation and derivation of the methods, formulas and theorems of "Nine Chapters on Arithmetic", but also have great development in the process of discussion. Liu Hui created the art of cutting circles, used the idea of ??limits to prove the area formula of a circle, and used theoretical methods to calculate the pi ratios of 157/50 and 3927/1250 for the first time.

Liu Hui used the infinite division method to prove that the volume ratio of a right-angled square pyramid and a right-angled tetrahedron is always 2:1, solving the key problem of general three-dimensional volume. When proving the volumes of square cones, cylinders, cones, and truncated cones, Liu Hui proposed the correct way to completely solve the volume of the sphere.

After the Eastern Jin Dynasty, China has been in a state of war and division between the north and the south for a long time. The work of Zu Chongzhi and his son is representative of the development of southern mathematics after the economic and cultural migration to the south. Based on Liu Hui's "Nine Chapters of Arithmetic", they greatly advanced traditional mathematics. Their mathematical work mainly includes: calculating the circumference of pi between 3.1415926 and 3.1415927; proposing the Zu (Riheng) principle; proposing solutions to quadratic and cubic equations, etc.

It is speculated that Zu Chongzhi obtained this result by calculating the areas of regular 6144-sided polygons and regular 12288-sided polygons inscribed in a circle based on Liu Hui's circle-cutting technique. He also used a new method to obtain two fractional values ??of pi, namely the approximate ratio 22/7 and the density 355/113. Zu Chongzhi's work put China about a thousand years ahead of the West in calculating pi;

Zu Chongzhi's son Zu (Riheng) summarized Liu Hui's relevant work and proposed that "power potential is the same as "Then the products are indifferent", that is, if the horizontal cross-sectional areas of two solid bodies of equal height are equal at any height, then the volumes of the two solid bodies are equal. This is the famous Zu (Riheng) axiom. Zu (Riheng) applied this axiom to solve Liu Hui's unresolved spherical volume formula.

Emperor Yang of the Sui Dynasty was very happy with his achievements and carried out large-scale construction projects, which objectively promoted the development of mathematics.

"Ji Gu Suan Jing" written by Wang Xiaotong in the early Tang Dynasty mainly discusses the calculation of earthwork, engineering division of labor, acceptance, and calculation of warehouses and cellars in civil engineering, reflecting the situation of mathematics in this period. Wang Xiaotong established the digital cubic equation without using mathematical symbols, which not only solved the needs of society at that time, but also laid the foundation for the later establishment of Tianyuan Shu. In addition, Wang Xiaotong also used digital cubic equations to solve the traditional Pythagorean solution.

The feudal rulers of the early Tang Dynasty inherited the Sui system and established an arithmetic museum in the Imperial College in 656, with a number of arithmetic doctors and teaching assistants, and 30 students. The "Ten Books of Calculation" compiled and annotated by Taishi Ling Li Chunfeng and others were used as textbooks for students in the School of Calculation. The Ming Dynasty examinations were also based on these books. The "Ten Books on Calculation" compiled by Li Chunfeng and others is of great significance in preserving mathematical classics and providing documentation for mathematical research. Their annotations to "Zhou Bi Suan Jing", "Nine Chapters of Arithmetic" and "Haidao Suan Jing" are helpful to readers. During the Sui and Tang Dynasties, due to the needs of the calendar, astronomicalists created the interpolation method of quadratic functions, which enriched the content of ancient Chinese mathematics.

Calculation is the main calculation tool in ancient China. It has the advantages of simplicity, image, and concreteness. However, it also has the disadvantages that it takes up a large area and is easy to make mistakes when the calculation speed is accelerated. Therefore, Reforms began very early. Among them, Taiyi Suan, Liangyi Suan, Sancai Suan and Abacus are all grooved abacus using beads, which are important technological reforms. Especially "abacus calculation", it inherits the advantages of calculating five-liter decimal and place value system, and overcomes the shortcomings of calculating vertical and horizontal counting and inconvenience of placing chips. Its advantages are very obvious. But at that time, multiplication and division algorithms still could not be performed in a row. The abacus beads have not yet been threaded and are inconvenient to carry, so they are not yet widely used.

After the mid-Tang Dynasty, business prospered and digital calculations increased. There was an urgent need to reform calculation methods. From the bibliography of arithmetic books left in the "New Book of Tang" and other documents, it can be seen that this algorithm reform was mainly to simplify multiplication. , Division algorithm, the algorithm reform in the Tang Dynasty made it possible to perform multiplication and division operations in a row, which is suitable for both calculation and abacus calculation.

The prosperity of ancient Chinese mathematics

In 960, the establishment of the Northern Song Dynasty ended the separation of the Five Dynasties and Ten Kingdoms. In the Northern Song Dynasty, agriculture, handicrafts, and commerce were unprecedentedly prosperous, and science and technology advanced by leaps and bounds. The three major inventions of gunpowder, the compass, and printing were widely used under this economic boom. In 1084, the Ministry of Secretaries printed and published "Ten Books of Suanjing" for the first time, and in 1213, Bao Chuanzhi reprinted it. These have created good conditions for the development of mathematics.

In about 300 years from the 11th to the 14th century, a number of famous mathematicians and mathematical works appeared, such as Jia Xian's "Nine Chapters of the Yellow Emperor's Algorithm and Fine Grass" and Liu Yi's "Discussing the Origin of Ancient Times" , Qin Jiushao's "Nine Chapters of the Book of Numbers", Li Ye's "Measuring the Circle Sea Mirror" and "Yi Gu Yan Duan", Yang Hui's "Detailed Explanation of the Nine Chapter Algorithm", "Daily Algorithm" and "Yang Hui's Algorithm", Zhu Shijie's "Arithmetic" Enlightenment", "Siyuan Jade Mirror", etc., many fields have reached the peak of ancient mathematics, and some of these achievements were also the peak of world mathematics at that time.

From the square root and the cube root to the square root of more than four times, it is a leap in understanding. The person who achieved this leap was Jia Xian. Yang Hui's "Compilation of Nine Chapters on Algorithms" contains Jia Xian's "Method of Adding Square Roots" and "Method of Adding Multiplications to Open Squares"; in "Detailed Explanation of Nine Chapters of Algorithms", he contains Jia Xian's "Origin of Square Root Method" diagram, "The multiplication method to find cheap grass" and the example of using the multiplication method to solve the fourth power. According to these records, it can be determined that Jia Xian discovered the binomial coefficient table and created the multiplication method. These two achievements had a significant impact on the entire Song and Yuan mathematics. Among them, Jia Xian's triangle was proposed more than 600 years earlier than Pascal's triangle in the West.

It was Liu Yi who extended the multiplication method to the solution of numerical higher-order equations (including cases where the coefficients are negative). "Yang Hui's Algorithm" in the "Field Ratio Analog Multiplication and Division Shortcuts" volume introduces 22 quadratic equations and 1 quartic equation in the original book. The latter is the earliest example of using the multiplication method to solve higher-order equations of three or more degrees. .

Qin Jiushao is a master of solving higher-order equations. In "Nine Chapters of the Book of Numbers", he collected 21 problems using the multiplication method to solve higher-order equations (the highest degree is 10). In order to adapt to the calculation procedure of the multiplication method, Zuo Jiushao specified the constant term as a negative number and divided the solutions to higher-order equations into various types.

When the root of the equation is a non-integer, Qin Jiushao continues to find the decimal of the root, or uses subtracting the root to transform the sum of the coefficients of the powers of the equation as the denominator, and the constant as the numerator to represent the non-integer part of the root. This is the "Nine Chapters on Arithmetic" 》 and Liu Huizhu’s development of methods for dealing with irrational numbers. When finding the second digit of the root, Qin Jiushao also proposed a trial division method of dividing the second digit of the root by dividing the coefficient of a linear term by a constant term, which was more than 500 years earlier than the earliest Horner method in the West.

Astronomers Wang Xun, Guo Shoujing and others of the Yuan Dynasty solved the problem of interpolation of cubic functions in the "Time Calendar". Qin Jiushao mentioned the interpolation method (they called it "Zhao Dian Shu") in the question of "Dui Shu Pushing Stars" and Zhu Shijie in the question of "Xiang Moves" in "Four Yuan Jade Mirror". Zhu Shijie obtained an interpolation formula of a quartic function.

Use Tianyuan (equivalent to x) as the unknown number symbol to establish a higher-order equation. In ancient times, it was called Tianyuan Shu. This was the first time in the history of Chinese mathematics that symbols were introduced, and symbolic operations were used to solve the problem of establishing higher-order equations. . The earliest extant Tianyuan Shu work is Li Ye's "Measuring the Circle Sea Mirror".

The extension of Tianyuan Shu to high-order simultaneous equations of two, three and four elements is another outstanding creation of mathematicians in the Song and Yuan Dynasties. The one that has been handed down to this day and systematically discusses this outstanding creation is Zhu Shijie's "Four Yuan Jade Mirror".

Zhu Shijie’s representation method of high-order simultaneous equations of four elements was developed on the basis of Tianyuan Shu. He placed the constants in the center and the powers of the four elements in the upper, lower and left , the four directions to the right, and other items are placed in the four quadrants. Zhu Shijie's greatest contribution was the proposal of the four-element elimination method. The method is to first select one element as the unknown number, and use the polynomials composed of other elements as the coefficients of the unknown number to form a number of one-element higher-order equations, and then gradually eliminate them using the mutual multiplication and cancellation method. This unknown quantity. Repeat this step to eliminate other unknowns, and finally use the multiplication method to solve. This is a major development of the linear method group solution, which is more than 400 years earlier than similar methods in the West.

The Pythagorean solution method had new developments in the Song and Yuan Dynasties. Zhu Shijie proposed the method of solving the Pythagorean form using known chord sums and strand chord sums in the second volume of "Arithmetic Enlightenment", supplementing the "Nine Chapters" "Arithmetic" shortcomings. Li Ye conducted a detailed study on the problem of the Pythagorean circle in "Measuring the Circle of the Sea" and obtained nine formulas of the Pythagorean circle, which greatly enriched the content of ancient Chinese geometry.

Given the angle between the ecliptic and the equator and the constitutive arc of longitude as the sun moves from the winter solstice to the vernal equinox, finding the constitutive arc of right ascension and degrees of declination is a problem of solving a spherical right-angled triangle. The traditional calendar is both Calculations are performed using interpolation. In the Yuan Dynasty, Wang Xun, Guo Shoujing and others used the traditional Pythagorean solution method, and Shen Kuo used Huiyuan Shu and Tianyuan Shu to solve this problem. However, what they obtained was an approximate formula and the result was not precise enough. But their entire derivation step was correct, and mathematically speaking, this method opened the way to spherical trigonometry.

The climax of the reform of computing technology in ancient China also occurred during the Song and Yuan Dynasties. The historical documents of the Song, Yuan and Ming dynasties contain a large number of practical arithmetic books of this period, the number of which is far greater than that of the Tang Dynasty. The main content of the reform is still multiplication and division. At the same time as the algorithm reform, abacus abacus may have appeared in the Northern Song Dynasty. But if modern abacus is regarded as having both abacus and a complete set of algorithms and formulas, then it should be said that it was finally completed in the Yuan Dynasty.

The prosperity of mathematics in the Song and Yuan Dynasties is the inevitable result of the development of social economy and science and technology, and the inevitable result of the development of traditional mathematics. In addition, the scientific and mathematical ideas of mathematicians are also very important. Mathematicians of the Song and Yuan Dynasties were opposed to the Neo-Confucian mysticism of images and numbers to varying degrees. Although Qin Jiushao once advocated that mathematics and Taoism come from the same source, he later realized that mathematics that "connects with the gods" does not exist, only mathematics that "manages world affairs and classifies all things"; Mo Ruo said in the preface to "Siyuan Jade Mirror" The proposed method of "using illusions to appear true and using imaginary to inquire into reality" represents a highly abstract thinking method; Yang Hui's research on the structure of vertical and horizontal diagrams reveals the essence of Luo Shu and effectively criticizes the mysticism of xiangshuo. All these are undoubtedly important factors in promoting the development of mathematics.

Integration of Chinese and Western Mathematics

China entered the late stage of feudal society from the Ming Dynasty. The feudal rulers implemented totalitarian rule, promoted idealist philosophy, and implemented the Eight-Part Examination System. Under these circumstances, with the exception of abacus, mathematical development gradually declined.

After the end of the 16th century, Western elementary mathematics was gradually introduced into China, which led to the integration of Chinese and Western mathematics in Chinese mathematics research. After the Opium War, modern mathematics began to be introduced into China, and Chinese mathematics was transformed into a study-based system. A period dominated by Western mathematics; it was not until the end of the 19th century and the beginning of the 20th century that modern mathematics research really began.

From the early Ming Dynasty to the middle Ming Dynasty, the commodity economy developed. In line with this commercial development, abacus became popular. The emergence of "Kuiben Xiangxiang Siyan Zazi" and "Lu Ban Mu Jing" in the early Ming Dynasty shows that abacus has become very popular. The former is a textbook for children to learn how to read pictures, while the latter includes abacus as a necessary household item in general wooden furniture manuals.

With the popularity of abacus, the abacus algorithm and formula are gradually becoming more and more perfect. For example, Wang Wensu and Cheng Dawei added and improved formulas for collision and return; Xu Xinlu and Cheng Dawei added formulas for addition and subtraction and widely used reduction and division in division, thereby realizing the formulaization of all four abacus arithmetic operations; Zhu Zaiyi and Cheng Dawei applied the method of calculating the square root and cube root to abacus. Cheng Dawei used abacus to solve numerical quadratic and cubic equations, etc. Cheng Dawei's works have been widely circulated at home and abroad and have great influence.

In 1582, the Italian missionary Matteo Ricci came to China. After 1607, he and Xu Guangqi translated the first six volumes of "Elements of Geometry" and one volume of "The Meaning of Measurement", and compiled "Ruan" with Li Zhizao. "Rong Jie Yi" and "Tong Wen Shuan Zhi". In 1629, Xu Guangqi was appointed by the Ministry of Rites to supervise the compilation of the calendar. Under his leadership, 137 volumes of the "Chongzhen Almanac" were compiled. "Chongzhen Almanac" mainly introduces the geocentric theory of the European astronomer Tycho. As the mathematical basis of this theory, Greek geometry, several European trigonometry, and calculation tools such as Napier's arithmetic and Galileo's rule of proportion were also introduced.

Among the incoming mathematics, the most influential was "Elements of Geometry". "Elements of Geometry" is China's first translated work of mathematics. Most of the mathematical terms are originated, and many of them are still in use today. Xu Guangqi believes that there is "no need to doubt" and "no need to change" about it, and "there is no one in the world who cannot learn from it." "Elements of Geometry" was a must-read mathematics book for mathematicians of the Ming and Qing dynasties, and had a great influence on their research work.

The second most widely used is trigonometry. The books introducing Western trigonometry include "The Great Survey", "The Table of Eight Lines Cutting a Circle" and "The Complete Meaning of Measurement". "Daqi" mainly explains the properties of the eight lines of triangles (sine, cosine, tangent, cotangent, secant, cosecant, sine vector, and covector), as well as the methods of making and using tables. In addition to adding some plane triangles that are missing in "The Complete Measurement", the more important ones are the product and difference formula and spherical trigonometry. All of these were translated and used in the calendar work at that time.

In 1646, the Polish missionary Muni Ge came to China. Among those who followed him to study Western science were Xue Fengzha, Fang Zhongtong and others. After Munige's death, Xue Fengzha compiled "Li Xue Tong" based on what he learned, hoping to integrate Chinese and French methods. The mathematical contents in "Calendar Society" mainly include Proportional Logarithm Table, "New Proportional Four-Line Table" and "Trigonometric Algorithm". The first two books introduce the invention of modified logarithms by British mathematicians Napier and Briggs. In addition to the spherical triangle introduced in "Chongzhen Almanac", the latter book also contains half-angle formulas, half-arc formulas, Derich's proportions, Nessler's proportions, etc. "Shuduyan" written by Fang Zhongtong explains the theory of logarithms. The input of logarithms is very important and is used immediately in calendar calculations.

There were many scholars in the Qing Dynasty who had experience in studying Chinese and Western mathematics and wrote books that have been handed down to the world. The ones with greater influence include Wang Xichan's "Illustrations" and Mei Wending's "Mei Congshu Yao" (including 13 mathematical works). ***Volume 40), Nian Xiyao's "Visual Science", etc. Mei Wending is the master of Western mathematics. He organized and researched the solutions to linear equations, the Pythagorean solution and the method of finding positive roots of higher-order powers in traditional mathematics, which brought vitality to the dying Ming Dynasty mathematics. Nian Xiyao's "Perspective" is the first work in China to introduce Western perspective.

Emperor Kangxi of the Qing Dynasty attached great importance to Western science. In addition to personally studying astronomy and mathematics, he also trained some talents and translated some works. In 1712, Emperor Kangxi appointed Mei Yucheng as the compiler of Mengyangzhai, and together with Chen Houyao, He Guozong, Ming Antu, Yang Daosheng and others compiled astronomical algorithm books. In 1721, 100 volumes of "Lü Li Yuan" were completed and published in 1723 under the name of Kangxi "Yu Ding".

Among them, "The Essence of Mathematics" is mainly responsible for Mei Yuncheng, and is divided into two parts. The upper part includes "Elements of Geometry" and "Elements of Algorithm", both of which are translated from French works; the lower part includes arithmetic, algebra, plane geometry, plane triangles, and solids. Elementary mathematics such as geometry, with tables of prime numbers, logarithms and trigonometric functions. Because it is a relatively comprehensive encyclopedia of elementary mathematics and has the title of "Yu Ding" under Kangxi, it had a certain influence on mathematics research at that time.

From the above, it can be seen that mathematicians in the Qing Dynasty did a lot of work on Western mathematics and achieved many original results. Compared with traditional mathematics, these achievements are progressive, but compared with contemporary Western mathematics, they are obviously lagging behind.

After Yongzheng came to the throne, he closed himself off to the outside world, which led to the stop of the import of Western science into China and the implementation of a high-pressure policy internally. As a result, ordinary scholars could neither have access to Western mathematics nor dare to interfere with the knowledge of managing the world and applying it, so they immersed themselves in Study ancient books. During the Qianjia period, a Qianjia school of thought that focused on textual criticism gradually formed.

With the collection and annotation of "Ten Books of Suan Jing" and mathematical works of Song and Yuan Dynasties, there was a climax of studying traditional mathematics. Among them, those who can break through the old framework and create inventions include Jiao Xun, Wang Lai, Li Rui, Li Shanlan, etc. Compared with the algebra of the Song and Yuan dynasties, their work was better than the previous one; compared with Western algebra, it was a little later, but these results were obtained independently without the influence of modern Western mathematics.

At the same time as the traditional mathematics research reached its climax, Ruan Yuan, Li Rui and others compiled a biography of astronomical mathematicians - "Chou Ren Biography", which collected the deceased astronomers from the Huangdi period to the fourth year of Jiaqing. and more than 270 mathematicians (including less than 50 whose mathematical works have been handed down to the world), and 41 missionaries who introduced Western astronomy and mathematics since the late Ming Dynasty. This work is entirely composed of "picking up historical books, gathering various books, and carefully recording them." It collects completely first-hand original data and is quite influential in the academic world.

After the Opium War in 1840, modern Western mathematics began to be introduced into China. First, the British established the Mohai Library in Shanghai to introduce Western mathematics. After the Second Opium War, Zeng Guofan, Li Hongzhang and other bureaucratic groups launched the "Westernization Movement". They also advocated the introduction and study of Western mathematics and organized the translation of a number of modern mathematical works.

Among the more important ones are "Algebra" and "Ten Levels of Calculus" translated by Li Shanlan and Wei Lie Yali; "Algebra" and "Tracing the Source of Calculus" co-translated by Hua Hengfang and the Englishman Frya "Mathematics for Doubts"; "Preparing Purposes for Forms", "Preparing Purposes for Algebra", and "Mathematics for Writing Arithmetic" compiled by Zou Liwen and Di Kaowen; "Combining References with Forms" and "Eight Lines Preparing Purposes" co-translated by Xie Honglai and Pan Shenwen, etc.

"Ten Levels of Algebra" is China's first translation of calculus; "Algebra" is a translation of symbolic algebra written by the British mathematician De Morgan; "Caesical Mathematics" is the first Translation of Part Theory of Probability. In these translations, many mathematical terms and terms were created, which are still in use 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 some of the above-mentioned works became the main textbooks.

While translating Western mathematical works, Chinese scholars also conducted some research and wrote some works. The more important ones are Li Shanlan's "Explanation of the Apical Cone Transformation Method" and "Testing the Root Method"; Xia Wanxiang "Illustrations of Dong Fang Shu", "Zhiqu Shu", "Zhiqu Illustrations", etc. are all research results that integrate Chinese and Western academic thoughts.

Because the imported modern mathematics required a process of digestion and absorption, and the rulers of the late Qing Dynasty were very corrupt. Under the impact of the Taiping Rebellion and the plunder of the imperialist powers, they were too overwhelmed to take care of mathematical research. It was not until after the May 4th Movement in 1919 that the research on modern mathematics in China really began.