The tradition of mathematical research bet365 code Song and Yuan Dynasties was interrupted bet365 code Ming Dynasty，But after the Qianjia period bet365 code Qing Dynasty, it became a research hotspot bet365 code academic circle，The results were translated and spread to Europe by Westerners in China。This process also reflects the situation since Western missionaries came to China bet365 code late Ming Dynasty，bet365 code historical context of direct collision between Eastern and Western civilizations，An important aspect of exchanges and mutual learning between Chinese and Western mathematical cultures。

　　Western astronomy and bet365 code introduced in the late Ming and early Qing dynasties

During the Ming and Qing Dynasties，Western Astronomy、bet365 code passed "Chongzhen Almanac" (and its subsequent revised "Western New Almanac" and "New Almanac")、Compilation of large-scale works such as "The Essence of bet365 code" and "The Examination of Lixiang" written by Emperor Kangxi，Acquired the orthodox status of the imperial court。It coincided with the European Renaissance，Relevant research on classical humanities and scientific works has become a common practice。Western missionaries take advantage of convenience，Except for introducing Ptolemy、Tycho and other geometric universe models are used in addition to calculating astronomical calendars，Also translated the first six volumes of "Elements of Geometry" and other ancient Greek mathematical classics into Chinese。Therefore, the Western bet365 code introduced by missionaries was mainly geometry，This branch of bet365 code is characterized by axiomatics，Starting from several "self-evident" axioms，Derivation and proof of a series of propositions，Form a complete knowledge system。

Compared with the long history of geometry，Algebra was learned by Europeans from Arabs，It doesn’t take long，Introduced to China during the Kangxi period，Included in "The Essence of bet365 code"，Known as "borrowing roots"。That is, borrowing the root number、meaning square number，Equivalent to setting unknown numbers x and powers of x。Based on Han Qi’s research，Emperor Kangxi no longer fully trusted missionaries in his later years，Advocate the spirit of "self-reliance"，Strive to get rid of the latter’s control over the calendar。Mei Yingcheng，The grandson of Mei Wending, the “number one in arithmetic in the Qing Dynasty”，First of all, explain the etymology of the Western name "Algebada" (that is, the transliteration of algebra) for "borrowing the root method" as "the method from the East"，It is the method of "establishing Tian Yuanyi" in the Song and Yuan Dynasties that spread to the Western Regions and then spread back again。

　　The Qianjia scholars’ excavation bet365 code Song and Yuan mathematical classics

Now we know，algebra comes from Arabic al-jabr，means restoration，That is, the unknown number in the reduction equation。Therefore, Mei Yingcheng’s etymological explanation is a misunderstanding，He is only based on the "Litian Yuanyi" occasionally mentioned by Mingren，From the overall properties, it is inferred that this is an algebraic method similar to "borrowing the root square"。Has "Jiogenfang" been affected by "Litian Yuanyi"，Requires extensive investigation、Relatively relevant historical materials on bet365 code in Chinese and foreign languages，It has not been proven yet，But whether to prove or falsify，At that time, scholars first had to solve the basic problem of obtaining readable and credible Chinese bet365 code historical materials。Because I experienced the silence of the Ming Dynasty，The science of celestial calculation has become a "unique skill"，What is the specific algorithm of "Li Tian Yuan Yi" method，No one knew it at the time。The turning point in solving the problem lies in the compilation of "Sikuquanshu"，The officials of the Siku compiled Li Ye's "Measuring the Circle Sea Mirror" and "Yi Gu Yan Duan" from the "Yongle Dadian"，Confirmed that "Li Tianyuanyi" is indeed an algorithm for establishing the solution of one-dimensional higher-order equations - Tianyuan Technique。The officials of Siku compiled Qin Jiushao's "Nine Chapters of bet365 code" (also known as "Nine Chapters of bet365 code"，Hereinafter referred to as "Qin Shu")，And "correct those who make mistakes"，Those who are sparse will differentiate，Invert the order，Add your words below”，Made a preliminary review。

In addition，Qianjia scholars far away in Jiangnan，Also able to copy Li from Siku Pavilion、Qin Shu Shu and comparative study，Get deeper conclusions。Qian Daxin, a master of textual criticism, keenly pointed out in "Nine Chapters of Jiushao bet365 code of the Qin Dynasty" that Qin Shu "has one method of establishing Tianyuan"，It is very different from the Tianyuan method derived from Li Ye’s "Measuring the Circle Sea Mirror"。A famous collationist later、Bibliographer Gu Guangqi reviewed the calculation for Yangzhou bibliophile Qin Enfu、Correction of "Nine Chapters of Shushu" in Zhao Qimei's manuscript，Preparing for publication。Gu discovered that “the questions he asked did not correspond to the techniques”，And believes that Qin Jiushao's "Dayan Technique is not the only method of Tianyuan"，There is no way to establish the meaning of Tianyuanyi，Then he should be regarded as the one who is called Tianyuan by Guo Shoujing and Li Ye"，Correctly pointed out that although the Dayan art created by Qin Shu also had "Li Tian Yuan Yi"，But it’s not the Tianyuan Technique in Li Ye’s book。

Although Gu Guangqi made corrections to Qin Shu，But for some reason it was not published。However，The only printed version of "Nine Chapters bet365 code Book of Numbers" was published so far in 1842 (see Figure 1)，The publication process shows the efforts made by Qianjia scholars, masters, apprentices and friends one after another to publish this rare book。Yu Songnian, a wealthy man in Shanghai, once wrote a preface and said it after publishing two kinds of "Yijatang Series"，"Easy to learn the secrets of Song and Yuan people"，I heard that "Nine Chapters bet365 code Book of Numbers" is a book of profound knowledge，Dayan Qiuyi、Positive and negative square root techniques，Especially explains the secret that has not been taught since ancient times"，But publishing still has to face the old problem of correcting errors and omissions。Fortunately, Li Rui had edited the Siku version before，Shen Qinpei collated the beautiful copy of Zhao Qi collected by Zhang Dunren，Thus Song Jingchang, a disciple of Shen Qinpei, was able to stay in Shen、Further collation based on Li’s posthumous manuscript。Now we see that the Yijatang series of books "Nine Chapters of Shushu" is accompanied by "Notes" signed by Song Jingchang，Become an indispensable and important part bet365 code book，Actually gathered Shen Qinpei、Collation results of mathematicians such as Li Rui and Song Jingchang。

Except Qin Shu，The "Yijiatang Series" also includes six works by Yang Hui, another mathematician bet365 code Southern Song Dynasty，Also attached are notes。Yang Huishu Siku has not collected it，Ruan Yuan has been asking for a visit for thirty years without success，I only saw it in "Yongle Dadian" by chance，Transcribed about a hundred items；Later, bibliophile Huang Pilie obtained the loose leaves of Yang Shu，After neatly ranking with Li Rui，Only six volumes of books。Zhu Shijie’s "Four Yuan Jade Mirror" was also acquired by Ruan Yuan when he was in Zhejiang；"Arithmetic Enlightenment" is a Korean reprint obtained by his disciple Luo Shilin from Liulichang Book Store。

Qin Jiushao、Li Ye、Mathematical classics written by Yang Hui, Zhu Shijie and others，Reappeared after hundreds of years of silence，It should be attributed to the excavations of Qianjia scholars、Organization and research。Except for the Qianjia School’s tradition of attaching great importance to textual research on classical documents，Han Qi thinks，The introduction of Western bet365 code, especially "borrowing root squares"，It not only provides knowledge preparation for Qianjia scholars，At the same time, the promotion of the spirit of "independence" stimulated the revival of bet365 code in the Song and Yuan Dynasties。Indeed，The research results of Qianjia scholars on these Song and Yuan mathematical classics，Displays a unique algebraic knowledge system that is ahead of its time in the West。These mathematical (historical) knowledge，It has also become an important material for the next stage of comparison and mutual learning between Chinese and Western bet365 code in the West。

　　Exchanges and mutual learning between Chinese and Western mathematical cultures bet365 code 19th century

In the Kangxi era when "borrowing root prescription" was introduced into China，bet365 code in Western Europe achieved a revolutionary breakthrough。The geometric tradition originating from ancient Greece and the algebraic tradition from Eastern India and Arabs collided and intersected in Europe，Analytical geometry that combines the best of both worlds、Calculus and other branches of bet365 code have developed successively，Become one of the important foundations of the modern scientific revolution。Because of this series of mathematical achievements，Europeans in the 19th century were quite conceited，Even Hindu-Arabic numerals and decimal notation have become signs of "modernity"。John F, the former Governor of Hong Kong. Davis) criticized "Chinese numbers are written very long，Not like the Arabic number system，The powers of a number increase or decrease tenfold according to the number of digits”。This is because he had only read ordinary ancient Chinese books at that time，I haven’t seen the traditional arithmetic book，I don’t know about the private commercial numerals that evolved from arithmetic numerals—the Suzhou code written horizontally like the Indo-Arabic numerals。

With the five-port trade after the Opium War，Alexander Wylie, a British sinologist who came to China in the 1850s, learned from Chinese scholars and collaborated with them to translate foreign mathematical works，Realize that predecessors had deep misunderstandings about Chinese bet365 code。So he wrote a long paper "Jottings on the Science of the Chinese" ("Jottings on the Science of the Chinese") in English. Arithmetic”)，Correction。This article was serialized in 9 issues in "The North-China Herald" in 1852: (1) Introduction、The origin of bet365 code in China、Arithmetic books from 7th century ago；(2) Arithmetic books from the 7th century to the 14th century；(3) Numbers and basic arithmetic operations、"Nine Chapters of Arithmetic" (Part 1)；(4) "Nine Chapters of Arithmetic" (Part 2)；(5) Qin Jiushao Dayan Technique (Part 1)；(6) Dayan Shu (Part 2)；(7) Tianyuan Technique、Prescription；(8) Quaternary Technique、borrow root square、The revival of ancient calculation in the early 19th century；(9) bet365 code research in China at that time、Conclusion。Among them，The four issues (5) to (8) are the bet365 code part of Song and Yuan Dynasties，Accounting for nearly half of the full text。Comparison between China and the West，Wei Lie Yali discovered Qin Jiushao’s prescription technique (see Figure 2)，Equivalent to the European method for solving high-order numerical equations - Horner's method，Thinks that "Qin is the first person we have found who completely developed the solution to complex equations"。This insight，Together with important ancient Chinese bet365 code classics、Mathematicians and their contributions，And the development of bet365 code in China at that time，Also introduced to the Western world。This article has a great response，Soon translated into French、German and other texts are widely spread in European academic circles。

The reason why Wei Lie Yali is interested in the subject of comparative mutual learning between Chinese and Western bet365 code，Except that it was a fashion among Chinese scholars at that time to study bet365 code in the Song and Yuan Dynasties，Also influenced by the British "Cambridge School of Algebra"。This school emerged in the first half of the 19th century，Aims to introduce Leibniz’s notation for calculus to Cambridge，And establish a new algebraic perspective independent of geometry。The representative is the first generation Charles Babbage、George Peacock and the second generation Augustus De Morgan, etc.。Writing the history of bet365 code is one of the unique academic practices of this school，Especially the masters and apprentices of Peacock and De Morgan。French scholar Marie-José Durand-Richard has conducted in-depth research on Peacock，Point out that the latter wrote the article "Arithmetic" in the Encyclopaedia Metropolitana，It describes and compares the arithmetic-algebra history of various nations in the world in detail，And take this as the school’s claim that algebra is a more universal and fundamental science，Not just a clever way to deal with the problem - the main argument。De Morgan is the Penny Cyclopaedia edited by the Society for the Diffusion of Useful Knowledge，Written about 850 astronomy and bet365 code entries，It also involves a lot of history of bet365 code，Wei Lieyali is often quoted in "A Brief Note"。

In addition，Weili Yali believes that ancient Chinese arithmetic and the algebra derived from it are a science，Highlights the wisdom and achievements of traditional Chinese bet365 code, especially arithmetic-algebra，Can guide Chinese people to learn further。Therefore，He later translated De Morgan's Elements of Algebra into "Algebra" together with Li Shanlan。Afterwards，Algebra, which originates from Arabic, is translated as "algebra"。From this，We can see an interesting circulation of mathematical knowledge: the "borrowed root prescriptions" of Western algebra introduced during the Ming and Qing Dynasties stimulated the Qianjia scholars to explore the mathematical classics of the Song and Yuan Dynasties，Then we will compare Chinese and Western mathematical cultures for Western scholars such as Wei Lie Yali、Exchange and mutual learning provide the possibility，Also promotes deeper algebra、The introduction of the latest branches of bet365 code such as calculus。

　　(The author is an associate professor at the Institute of History of bet365 code and Technology, Inner Mongolia Normal University)