Is there any information about the history of the development of mathematics in the past 60 years

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

First, the budding of ancient Chinese mathematics, the end of the primitive commune, private ownership and the exchange of goods, after the concept of number and shape has been further developed, the Yangshao culture period of excavated pottery, which has been engraved with symbols indicating 1234. At the end of the primitive commune, the use of writing symbols had begun to replace the knotted rope to keep track of things.

The pottery unearthed in Xi'an Half-slope has a pattern of equilateral triangles composed of 1 to 8 dots and divided squares as 100 small squares, and the house bases in the Half-slope ruins are all circles and squares. In order to draw a circle as a square, to determine the level, people also created a gauge, rectangle, quasi, rope and other drawing and measuring tools. According to the Records of the Grand Historian, these tools were used by Xia Yu when he ruled over the water.

In the middle of the Shang Dynasty, a set of decimal numbers and notation were produced in the oracle bones, the largest number of which was 30,000; at the same time, the Yin people used ten celestial stems and twelve geodesic branches to form the A Zi, B Chou, C Yin, Ding Mao, and other 60 names to remember the date of the 60 days; in the Zhou Dynasty, the eight trigrams, previously composed of symbols of yin and yang, represented the development of eight things into the sixty-four hexagrams, which represented 64 things.

The Zhou Thigh Calculation Scripture of the first century BC mentions the use of moments to measure height, depth, breadth, and distance in the early Western Zhou Dynasty, and cites the examples of the hook three, four, and five strings of the hook and strand shape, as well as the fact that the ring moment can be a circle. It is mentioned in the chapter of "Ritual Records - Internal Rules" that the children of the nobles in the Western Zhou Dynasty had to learn the number and the method of counting from the age of nine, and they had to be trained in rituals, music, archery, harnessing, writing and counting, and counting, which was one of the six arts, had already started to become a specialized course.

During the Spring and Autumn period and the Warring States period, the counting has been widely used, and the counting system has used the decimal value system, which is epoch-making for the development of the world's mathematics. In this period, measurement mathematics was widely used in production, and there was a corresponding improvement in mathematics.

The Warring States period also promoted the development of mathematics through the Hundred Schools of Thought, especially the debate on the proper names and some propositions directly related to mathematics. Famous scholars believe that after the abstraction of the concept of the noun and their original entity is different, they put forward "rectangles are not square, the gauge can not be round", the "big one" (infinity) is defined as "to the big no outside The "big one" (infinity) is defined as "the biggest is not outside", and the "small one" (infinity) is defined as "the smallest is not inside". Also put forward "a foot of whip, the day to take its half, the world will not be exhausted" and other propositions.

On the other hand, Mohists believe that the name comes from the object, and the name can reflect the object from different aspects and different depths. Mohist gives some mathematical definitions. For example, circle, square, flat, straight, second (tangent), end (point) and so on.

The Mohists disagreed with the proposition of "one-foot whip", and put forward a "non-half" proposition to refute it: a line segment will be divided indefinitely by half and half, and there will be a "non-half" that can not be divided again. This "non-half" is the point.

The propositions of the Naimists state that a finite length can be divided into an infinite sequence, and the propositions of the Mohists point out the variations and consequences of this infinite division. The discussion of the mathematical definitions and mathematical propositions of the Famous School and the Mohist School is of great significance to the development of ancient Chinese mathematical theory.

Second, the formation of the ancient Chinese mathematical system

The Qin and Han dynasties were the rising period of feudal society, with rapid economic and cultural development. Ancient Chinese mathematical system was formed in this period, its main symbol is that arithmetic has become a specialized discipline, as well as the emergence of mathematical works represented by the "nine chapters of arithmetic".

The Nine Chapters of Arithmetic is a summary of the mathematical development of the period of the Warring States, Qin and Han feudal society and the consolidation of the creation of the mathematical achievements, can be regarded as the world's mathematical masterpieces. For example, the four operations of fractions, the art of presenting (the Western method of three rates), open square and open cube (including the numerical solution of quadratic equations), the art of surplus and deficit (the Western method of double managed), a variety of area and volume formulas, the solution of systems of linear equations, the rules of adding and subtracting positive and negative numbers, and the solution of collinear shapes (in particular the collinear theorem and the method of solving for the collinear number), etc., the level of which are all very high. Among them, the method of solving systems of equations and the rules for adding and subtracting positive and negative numbers are far ahead in the development of mathematics in the world. As far as its characteristics are concerned, it formed an independent system centered on arithmetic and completely different from ancient Greek mathematics.

The Nine Chapters of Arithmetic has several notable features: the use of the form of mathematical problem sets by category and chapter; arithmetic are developed from the chip notation; arithmetic, algebra-based, rarely involved in the nature of the graphics; emphasis on application, the lack of theoretical elaboration and so on.

The Qin and Han Dynasties emphasized the applicability of mathematics. Written in the early years of the Eastern Han Dynasty, the Nine Chapters of Mathematics, excluding the Warring States period in the Hundred Schools of thought in the emergence of the famous and the Mohist emphasis on the definition of the term and the discussion of logic, favoring the close combination of production and life at the time of the mathematical problems and their solutions.

The Nine Chapters of Arithmetic was transmitted to Korea and Japan during the Sui and Tang dynasties, and became the math textbook of those countries at that time. Some of its achievements, such as the decimal value system, the art of present-day availability, and the art of surplus and deficit, were also transmitted to India and Arabia, and through India and Arabia to Europe, promoting the development of world mathematics.

Third, the development of mathematics in ancient China

Wei, Jin period of the emergence of metaphysics, not for the Han Confucian Confucianism bound, the idea is more active; it is cross-examination of the defense of victory, and can use logical thinking, analysis of the righteousness of the reasoning, which is conducive to the improvement of mathematics from the theoretical point of view. Wu Zhao Shuang note "Zhou Thigh Calculation Classic", the end of Han and early Wei Xu Yue compiled "nine chapters of arithmetic" note, the end of Wei and early Jin Liu Hui compiled "nine chapters of arithmetic" note, "nine chapters of the weight of the difference map" all appeared in this period. The work of Zhao Shuang and Liu Hui laid the theoretical foundation for the ancient Chinese mathematical system.

Zhao Shuang was one of the earliest mathematicians in ancient China to prove and derive mathematical theorems and formulas. His additions to the book Zhou Thigh Calculating Scripture, the "Chart and Note on the Circle and Square" and the "Chart and Note on the Height of the Sun", are very important mathematical documents. In the "Chart and Note on Choke-Shaft Circle and Square", he proposed to prove the Choke-Shaft Theorem and five formulas of unchoke-shaft shape by the string diagram; in the "Chart and Note on Sun Height", he used the graphic area to prove the formula of redifference, which was commonly used in the Han Dynasty, and Zhao Shuang's work was groundbreaking and occupied an important position in the development of mathematics in ancient China. In the development of mathematics in ancient China occupies an important position.

Liu Hui about the same time as Zhao Shuang, he inherited and developed the ideas of the Warring States period of the celebrities and Mohists, advocating a number of mathematical terms, especially important mathematical concepts to give a strict definition, that mathematical knowledge must be "analyzed", in order to make mathematical writings concise and rigorous, conducive to the reader. His "Nine Chapters on the Art of Arithmetic" is not only a general explanation and derivation of the methods, formulas and theorems of the "Nine Chapters on the Art of Arithmetic", but also a great development in the process of exposition. Liu Hui created the art of cutting circles, using the idea of limit to prove the formula for the area of a circle, and for the first time used the theoretical method to calculate the circumference of the circle as 157/50 and 3927/1250.

Liu Hui proved that the ratio of the volume of a right-angled square cone to that of a right-angled tetrahedron is constant at 2:1 by the method of infinite division, which solved the key problem of the volume of the three-dimensional volume in general. In proving the volumes of square cones, cylinders, cones, and circular tables, Liu Hui proposed the right way to completely solve the volume of the ball.

After the Eastern Jin Dynasty, China was in a state of prolonged war and division between the north and the south. The work of Zu Chongzhi's father and son is representative of the development of mathematics in the south after the economic and cultural shift southward, and they built on Liu Hui's notes on the Nine Chapters of the Mathematical Art to take traditional mathematics a step forward considerably. Their mathematical work is mainly: calculated pi between 3.1415926 ~ 3.1415927; put forward the Zu Yi principle; put forward the solution of quadratic and cubic equations and so on.

It is hypothesized that Zu Chongzhi got this result by calculating the area of the square 6144 sides and the square 12288 sides inside a circle on the basis of Liu Hui's circle cutting technique. He also used a new method to obtain the two fractional values of pi, i.e., the approximate rate of 22/7 and the dense rate of 355/113, which put China about a thousand years ahead of the West in the calculation of pi;

Zu Yi, son of Zu Chongzhi, summarized Liu Hui's work and put forward the idea that "the power of two cubes of equal heights is the same, but the product is not the same", i.e., two cubes of equal heights can not be different.

The son of Zu Chongzhi summarized the work of Liu Hui and proposed that "if the power potential is the same, the volume can not be different", that is, the two three-dimensional equal height, if any of the height of the horizontal cross-sectional area of the same, the two three-dimensional volume is equal, which is the famous Zu Yi axiom. This is the famous axiom of Zu Yi. Zu Yi applied this axiom to solve the ball volume formula that Liu Hui had not yet solved.

Emperor Yang of Sui Dynasty built a large construction, objectively promote the development of mathematics. In the early Tang Dynasty, Wang Xiaotong's "Ancient Calculation Scriptures", which mainly discusses the calculation of earthworks in civil engineering, the division of works, acceptance, and the calculation of warehouses and cellars, reflects the situation of mathematics in this period. Wang Xiaotong set up numerical cubic equations without the use of mathematical symbols, which not only solved the needs of the society at that time, but also laid the foundation for the establishment of the Tianyuan technique later. In addition, for the traditional solution of the hook-and-square form, Wang Xiaotong also solved it with numerical cubic equations.

The rulers of the early Tang Dynasty inherited the Sui system, and established the Arithmetic Pavilion in the State Academy in 656, with a doctor of arithmetic and an assistant professor, and 30 students. Li Chunfeng and other compilers by the Imperial Historian annotated "ten books of calculation", as the textbook for students of arithmetic hall, Ming arithmetic examination is also based on these books of arithmetic. Li Chunfeng and other compilers of the Ten Books of Mathematical Scriptures, the preservation of the classical works of mathematics, mathematical research for the provision of documentary information is very meaningful. The annotations they gave to the Zhou Thigh Calculating Scripture, the Nine Chapters of the Mathematical Art, and the Sea Island Calculating Scripture are helpful to the readers. During the Sui and Tang dynasties, due to the need for calendars, the celestial arithmeticists created the method of interpolation of quadratic functions, which enriched the content of ancient Chinese mathematics.

Calculator is the main calculation tool in ancient China, it has the advantages of simplicity, image, specificity, etc., but there are also cloth chip occupies an area of large, easy to fiddle with the speed of the speed of the chip is not correct and cause errors and other shortcomings, so it is very early to start the reform. Among them, Taiyi calculation, two meters calculation, three talents calculation and bead calculation are all groove abacus with beads, technically is an important reform. Especially the "bead counting", it inherited the advantages of the chip counting five liters ten into the system with the value of the place, and overcome the shortcomings of the chip counting vertical and horizontal counting and the inconvenience of placing chips, the superiority is very obvious. However, the multiplication and division algorithm still could not be carried out in a horizontal column. The beads have not been worn file, carry inconvenient, so there is still no universal application.

After the middle of the Tang Dynasty, commercial prosperity, increased number of calculations, the urgent need to reform the calculation method, from the "New Book of the Tang Dynasty" and other documents left behind in the bibliography of the calculator, it can be seen that this algorithmic reform is mainly to simplify the multiplication and division algorithms, algorithmic reform of the Tang Dynasty so that multiplication and division of the law can be carried out in a cross-column arithmetic, which applies to the chip calculation, but also applies to bead calculator.

Fourth, the prosperity of ancient Chinese mathematics

In 960, the establishment of the Northern Song Dynasty ended the situation of the Five Dynasties and Ten States. The Northern Song Dynasty's agriculture, handicrafts, business unprecedented prosperity, science and technology soared, gunpowder, compass, printing three major inventions is in this economic upsurge is widely used. 1084 Secretary of the Province of the first printing and publication of the "ten books of the arithmetic scriptures", 1213 Bao rolling of the re-engraving. All these created good conditions for the development of mathematics.

From the 11th to 14th century about 300 years period, there are a number of famous mathematicians and mathematical works, such as Jia Xian's "Huang Di nine chapters of the algorithm of the fine grass", Liu Yi's "discussion of the ancient roots", Qin Jiushao's "book of nine chapters of the number", Li Ye's "measurement of the circle of the Sea Mirror" and "Yigu evolution of the paragraph", Yang Hui's "Detailed explanation of the algorithm of the nine chapters of the algorithm of the Daily Algorithm and Yang Hui algorithms, Zhu Shijie's "arithmetic Enlightenment", "The Four Yuan Yujian" and so on. Siyuan Yujian", etc. Many fields reached the peak of ancient mathematics, and some of these achievements were also the peak of world mathematics at that time.

It is a leap in understanding to go from square and cubic to more than four times square, and it was Jia Xian who realized this leap. Yang Hui in the "nine chapters of the algorithm codification class" contains Jia Xian "multiplication of the open flat method", "multiplication of the open method"; in the "detailed explanation of the nine chapters of the algorithm" contains Jia Xian's "open the square method of the original source" diagram, "multiplication of the method to find the square", "multiplication of the method to find the square", "multiplication of the method to find the square". Based on these records, it can be determined that Jia Xian had discovered the binomial coefficient table and created the method of multiplication. These two achievements had a significant impact on the entire Song and Yuan mathematics, in which Jia Xian triangle was proposed more than 600 years before the Western Pascal's triangle.

It was Liu Yi who generalized the multiplication-opening method to the solution of numerical higher order equations (including cases with negative coefficients). In the volume of Yang Hui Algorithm, "Tian Mu Bi Classification Multiplication and Division Jie Fa", 22 quadratic equations and 1 quadratic equation were introduced, and the latter is the earliest example of solving high order equations of more than three times by the method of multiplying and multiplying.

Qin Jiushao, a master of higher-order equation solving, collected 21 problems solving higher-order equations (up to a maximum of 10) by multiplication-expansion methods in "Nine Chapters of the Book of Numbers". In order to accommodate the computational procedures of the increasing-multiplying-opening method, Zao Jiushao specified the constant term as a negative number, and classified the solutions of higher-order equations into various types. When the root of the equation is a non-integer, Qin Jiu-shao takes to continue to find the decimal of the root, or use the sum of the coefficients of the powers of the reduced-root transformation equation as the denominator, and the constant term as the numerator to represent the non-integer portion of the root, which is a development of the Nine Chapters of the Art of Arithmetic and Liu Hui's note on the method of dealing with irrational numbers. In finding the second digit of the root, Qin Jiushao also proposed the trial division method with the coefficient of the primary term divided by the constant term as the second digit of the root, which is more than 500 years earlier than the earliest Horner's method in the West.

Yuan dynasty astronomers Wang Xun and Guo Shoujing solved the problem of interpolation of the cubic function in the "Calendar of Time". Qin Jiushao in the "Suffixing Art to push the star" question, Zhu Shijie in the "Four Yuan Yu Jian" "such as the image of the trick" question mentioned the interpolation method (they are called the trick difference), Zhu Shijie obtained a quadratic function of the interpolation formula.

The use of Tianyuan (equivalent to x) as the symbol of the unknown, to set up the higher equations, known as Tianyuan art in ancient times, is the first time in the history of Chinese mathematics to introduce symbols, and the use of symbolic operations to solve the problem of setting up higher equations. The earliest surviving work on Tianyuan technology is Li Ye's Measuring Round Sea Mirror.

The generalization of the technique from Tianyuan to higher-order systems of associative equations in binary, ternary, and quaternary is another outstanding creation of Song and Yuan mathematicians. The one that has survived to this day, and which provides a systematic discussion of this outstanding creation, is Zhu Shijie's Four Elements of Yujian.

Zhu Shijie's representation of the quadratic system of higher quadratic associative equations was developed on the basis of the Tianyuan technique, in which he placed the constants in the center, the powers of the quadratic in the four directions of up, down, left, and right, and the other terms in the four quadrants. Zhu Shijie's greatest contribution is to put forward the quadratic elimination method, which is to choose one element as the unknown first, and the polynomials composed of other elements as the coefficients of this unknown, which are listed into a number of one-element higher equations, and then apply the multiplication elimination method to eliminate this unknown step by step. Repeating this step will eliminate the other unknowns, and finally use the multiplication method to solve. This is a major development of the linear method of group solving method, more than 400 years earlier than the Western similar methods.

Pythagoras have a new development in the Song and Yuan dynasties, Zhu Shijie in the "Arithmetic Enlightenment" volume of the known hook chord and strings, strings and solving the Pythagoras method, supplementing the "nine chapters of the arithmetic" of the shortcomings. Li Ye in the "measuring circle sea mirror" on the hook stock tolerance circle problem carried out a detailed study, get nine tolerance circle formula, greatly enriched the content of ancient Chinese geometry.

Knowing the angle between the ecliptic and the equator and the remainder arc of the yellow meridian of the sun's orbit from the winter solstice to the equinoxes, finding the remainder arc of the equinoxes and the number of degrees of equinoxes is a problem of solving a spherical right triangle, and traditional calendars use the interpolation method to calculate. In the Yuan Dynasty, Wang Xun and Guo Shoujing solved the problem by using the traditional hook-and-square solution, and Shen Kuo solved the problem by using the technique of meeting the circle and the technique of celestial elements. However, what they got was an approximate formula and the result was not precise enough. But their entire projection step is correct, in the mathematical sense, this method opens the way to the spherical trigonometry.

The culmination of the reform of computing technology in ancient China also occurred in the Song-Yuan period. The historical documents of the Song, Yuan, and Ming dynasties contain a large number of bibliographies of practical arithmetic from this period, far more than in the Tang dynasty, and the main content of the reform is still multiplication and division. At the same time as the algorithmic reforms, the bead-piercing abacus may have appeared in the Northern Song Dynasty. But if the modern bead counting is seen as both wearing a bead abacus, but also a set of perfect algorithms and mnemonics, then it should be said that it was finally completed in the Yuan Dynasty.

Song and Yuan mathematicians were opposed to the mysticism of the rationalists to varying degrees. Although Qin Jushao had advocated mathematics and Taoism from the same source, but he later realized that "through the gods" of mathematics does not exist, only "by the world class of all things" of mathematics; Muo Ruo in the "four yuan yu jian" preface to the "with the Mo Ruo in the preface to the Four Elements of the Jade Guide proposed "use the false to look like the true, ask the real with the false" represents the ideological method of highly abstract thinking; Yang Hui's research on the structure of the vertical and horizontal diagrams revealed the nature of the Luoshu, which powerfully criticized the mysticism of the Elephantine numbers. All these are undoubtedly important factors in promoting the development of mathematics.

The integration of Chinese and Western mathematics

China entered the late feudal society from the Ming Dynasty, and after the end of the 16th century, Western elementary mathematics was introduced to China one after another, which led to the emergence of a fusion of Chinese and Western mathematical research; after the Opium War, modern mathematics began to be introduced to China, and Chinese mathematics was transformed into a period dominated by the study of Western mathematics; by the end of the 19th century, and at the beginning of the 20th century, modern mathematical research had been carried out, and it was only in the late 19th and early 20th centuries that modern mathematical research was carried out. In the late 19th and early 20th centuries, the study of modern mathematics really began.

From the beginning of the Ming Dynasty to the middle of the Ming Dynasty, the commodity economy has developed, and this commercial development is compatible with the popularization of the bead calculator. In early Ming Dynasty, "Kui Ben to phase four words and words" and "Luban wood scripture" appeared, indicating that bead calculations have been very popular. The former is a textbook for children to read and write, and the latter included the abacus as a household necessity in the general wood furniture manual.

With the popularization of bead counting, the algorithms and mnemonics of bead counting also gradually tended to be perfected. For example, Wang Wensu and Cheng Dawei increased and improved the phrase of "bumping into" and "starting with"; Xu Xinlu and Cheng Dawei added the phrase of "adding" and "subtracting" and widely used in division, thus realizing the entire phrase of the four operations of the bead calculator; Zhu Zaimu and Cheng Dawei applied the method of calculating the square and cube to the bead calculator, and Cheng Dawei solved the numerical two- and three-dimensional equations with the bead calculator, etc. The works of the bead calculator have been circulated both at home and abroad. Cheng Dawit's writings were widely circulated at home and abroad and had a great influence.

In 1582, the Italian missionary Matteo Ricci came to China, after 1607, he has with Xu Guangqi translated the first six volumes of the original Geometry, Measurement of the Law, a volume, and Li Zhizao compilation of Won Yung more than the meaning of the "and" the same text calculation refers to the "1629, Xu Guangqi was appointed by the Ministry of Rites to oversee the revision of the calendar, under his auspices, the compilation of "Chongzhen Calendar" 137 volumes. Chongzhen Calendar" is mainly an introduction to the European astronomer Dagu's geocentric doctrine. As the mathematical basis of this doctrine, the Greek geometry, a number of European Yushan trigonometry, as well as Napier chips, Galileo proportionality and other computational tools are also introduced at the same time.

Among the imported mathematics, the most influential was Geometry Originally. Xu Guangqi believed that it was "not necessary to doubt", "not necessary to change", "there is no one in the world is not learning". After the Manchu invasion of the Central Plains, science was once again put into the "cold palace". Not only was the second half of the book delayed in translation, even the first half of the book translated by Xu Guangqi was no longer released. Western missionaries to bring scientific and technological works, become Kangxi, Yongzheng or Qianlong emperor exclusive hobby.

The second most widely used is trigonometry, the introduction of Western trigonometry works are "big measurement", "cut circle eight line table" and "measurement of the whole meaning". The "Great Survey" mainly explains the nature of the eight lines of trigonometry (sine, cosine, tangent, cotangent, tangent, cotangent, cotangent, cotangent, cotangent, cotangent, cotangent), the method of making the table and the method of using the table. In addition to adding some plane triangles which are missing in "The Great Measurement", the more important ones in "The Measurement of the Whole Measurement" are the formula of accumulation and difference and spherical triangles. All of these, in the calendar work at that time were translated with the use.

In 1646, the Polish missionary Munigaku came to China, and he was followed by Xue Fengzhu, Fang Zhongtong, and others who studied Western science. After the death of Munigaku, Xue Fengzhu, according to his learning, compiled the "Calendar Society", trying to integrate Chinese, French and Western law. The main mathematical contents of the book were Proportional Logarithmic Table, New Table of Proportional Quadratic Line, and Trigonometric Algorithm. The first two books are the introduction of British mathematicians Napier and Briggs invented the logarithm. The latter book in addition to the "Chongzhen Calendar" introduced the spherical triangle, but also half-angle formula, half-arc formula, German proportional formula, Na proportional formula. The book by Fang Zhongtong, "Diffraction of Numbers", explains the logarithmic theory. The introduction of logarithms was very important, and it was immediately applied in calendar calculations.

Early Qing scholars studied Chinese and Western mathematics and wrote a lot of books, the impact of Wang Xie expound "diagrams", Mei Wending "Mei's series of books" (including mathematical works of 13 kinds of *** 40 volumes), Nian Xiyao, "visual science" and so on. Mei Wending was a master of centralized Western mathematics. He organized and researched the traditional mathematical method of solving systems of linear equations, the method of solving collinear forms, and the method of finding the positive root of a higher power, etc. Nian Xiyao's Visual Science was the first Chinese work to introduce Western perspective.

Kangxi attached importance to Western science, but only as a hobby. 1712 Kangxi ordered Mei Liucheng to be the compilation officer of Meng Yangzhai, together with Chen Houyao, He Guozong, Ming Antu, Yang Daosheng, etc. compiled astronomical algorithms, and in 1721, completed the 100 volumes of the "Origin of the Calendar," which was published in 1723 in the name of the Kangxi "Royal Decree". It was published in 1723 under the name of Kangxi's "Royal Decree". One of the "mathematical essence" is mainly by Mei enough to be responsible for, divided into upper and lower two series, the upper series includes "geometry originally", "algorithm originally", are translated from the French work; the lower series includes arithmetic, algebra, plane geometry plane trigonometry, three-dimensional geometry and other elementary mathematics, with a table of prime numbers, logarithmic tables and trigonometric function tables. Because it is a more comprehensive encyclopedia of elementary mathematics, and has the name of Kangxi "Royal Decree", so the mathematical research had a certain impact.

Qing mathematicians did a lot of work on Western mathematics, and achieved many original results. These results, such as and traditional mathematics comparison, there is progress, but and contemporaneous comparison of the West is obviously backward.

After the reign of Emperor Yongzheng, the external closed, resulting in the stop of Western science imported into China, the internal implementation of high-pressure policy, resulting in the general scholars can not contact the Western mathematical, and do not dare to ask the study of the world, and therefore buried in the study of ancient texts. During the Qianjia years, a Qianjia school of thought was gradually formed, which was mainly based on the study of evidence.

With the collection and annotation of the Ten Books of Arithmetic and Song and Yuan mathematical works, there was a climax in the study of traditional mathematics. Among them, those who could break through the old boxes and inventions are Jiao Zhuan, Wang Lai, Li Rui, Li Shanlan, etc. Their work is similar to that of the Song and Yuan dynasties. Their work, compared with the algebra of the Song and Yuan dynasties, was progressive; compared with Western algebra, it was a little later in time, but these results were obtained independently without the influence of Western modern mathematics.

At the same time as the culmination of the study of traditional mathematics, Ruan Yuan and Li Rui wrote a biography of astronomers and mathematicians - "Cultivator Biography", which collected more than 270 astronomers and mathematicians from the period of the Huangdi Emperor to the fourth year of the Jiaqing period (of which fewer than 50 mathematical works have been handed down), and 41 missionaries since the end of the Ming Dynasty to introduce the Western astronomical and mathematical missionaries. This work is composed of "pick up the history of the book, Tsuen Chui group of books, screening and recording" and become, the collection is completely first-hand primary source, in the academic community is quite influential.

In 1840, after the Opium War, Western modern mathematics began to be imported into China. First of all, the British set up the Mo Hai Bookstore in Shanghai to introduce Western mathematics. After the Second Opium War, Zeng Guofan, Li Hongzhang and other bureaucratic groups to carry out the "foreign affairs movement", but also advocate the introduction and study of Western mathematics, the organization translated a number of modern mathematical works.

One of the more important Li Shanlan and Weili Yali translation of "Algebra", "generation of micro-products to pick up the level"; Hua Hengfang and the British Fu Lanya translation of "Algebra", "micro-products of the traceability" "to determine the mathematical"; Zou Lifeng and Di Kauwen compiled "form of the purpose of science," "algebra," "mathematical calculations," Xie Honglai and Pan Shenwen translation of the "generation of form of the reference," "the eight lines of the purpose of preparation," and so on. In these translations, many mathematical terms and terminology were coined and are still used today, but the mathematical symbols used have generally been eliminated. After the Hundred Days' Reform, schools of the New Law were organized everywhere, and some of the above works became the main textbooks.

In the translation of Western mathematical works at the same time, Chinese scholars have also conducted some research, wrote some works, the more important Li Shanlan's "pointed cone variation solution", "examining the number of root method"; Xia Bentxiang's "hole Fangjiao illustration," "to the curve of the art," "to the curve of the illustration," and so on, are all through the Chinese and Western academic thinking research results.

Due to the input of modern mathematics requires a process of digestion and absorption, coupled with the late Qing rulers were very corrupt, under the impact of the Taiping Heavenly Kingdom movement, under the plunder of the Great Powers, burnt out, and have no time to take care of mathematical research. Until after the May Fourth Movement in 1919, the study of modern mathematics in China really began.