Zheng Xuan's Mathematical World: Zheng's mathematical annotations, background and historical contributions
Author: Zhu Yiwen
Source: Philosophy and Culture, Vol. 48, No. 11
Abstract: As a master of the two Han classics, Zheng Xuan annotated the Qunjing, unified the ancient and modern texts, and was proficient in almanac, and was good at the Nine Chapters of Arithmetic. However, in the past, there was no specific study of mathematics in the academic community. Judging from Zheng Xuansan's notes, he quoted Zheng Zhong as saying "nine numbers", implying that the Nine Chapters of Arithmetic originated from the Zhou Li, and repeatedly mentioned the "corn method" and the use of "Pythagorean technique". He often gave the approximate or result of the algorithm without giving the details of the calculation, and used it to eliminate the differences between the classics or versions, and his annotations also used mathematics, thus creating a textual context for the development of mathematics for later scholars. On the occasion of Wang Mang, Liu Xin proposed the idea that mathematics was the basis of musical rhythms, weights and measures, and calendars; in the second year of Guanghe, Dasi Nonghu and Quan Inscription established the authoritative position of this idea and the Nine Chapters of Arithmetic. Compared with Zheng Zhong, Xu Shen, Ma Rong and others, Zheng Xuan was more adept at using mathematics and almanac, so Zheng Shi's citation of the Nine Chapters of Arithmetic had dual political and academic legitimacy. However, on the basis of its study of scripture, later Generations of Confucians developed a tradition of classic arithmetic that was relatively independent of the traditional arithmetic represented by the Nine Chapters of Arithmetic, but deviated from Zheng Xuan's original intention of annotating the scriptures with mathematics.
Keywords: Zheng Xuan, Nine Chapters of Arithmetic, Sutra, Sutra, Liu Xin, Sanli
About author:Zhu Yiwen is an associate professor at the Department of Philosophy and Institute of Logic and Cognition, Sun Yat-sen University in Guangzhou, whose main research interests are the history of mathematics and the philosophy of mathematics, the history of chess and game culture.
I. Introduction
As a master of the two Han classics, Zheng Xuan (127-200) annotated the Qunjing and integrated ancient and modern texts. The Later Han Dynasty Book of Zheng Xuan's Biography of Yun Zheng Shi tong "Three Unified Calendars" and "Nine Chapters of Arithmetic", and Yun Yun "Xuan Shan Arithmetic",[1] from this we can see that Zheng Xuan had certain achievements in astronomy and mathematics[2]. However, academics have done little research on their scientific knowledge and ideas compared to their scriptural studies. [3] However, Zheng Xuan's relationship with ancient Chinese mathematics has been a source of great concern. The modern mathematical historian Qian Baochun (1892-1974) believed that "the Nine Chapters of Arithmetic and Xu Shen's Explanation of Words are similar, part of Confucianism in the early years of the Eastern Han Dynasty, and are closely related to the traditional ideas of Confucianism", and also said that "the compilation of the Nine Chapters of Arithmetic is closely related to the Confucians of the Classical Literature School of the Early Eastern Han Dynasty", and mentioned Zheng Zhong (?). -83), Ma Xu (Brother of Ma Rong), and Ma Rong (79-166) were three scribes. [4] Guo Shuchun argues that Zheng Xuan "actually formed a mathematical center with Liu Hong, Xu Yue, and others", and said that "Liu Hui studied the Zhou Li through Zheng Xuan's annotations, which became one of the direct data that he 'took from what he saw' when he annotated the Nine Chapters." [5] Liu Hong (c. 129-190) wrote the Qianxiang Calendar, Xu Yue (born at the end of the Eastern Han Dynasty) wrote the "Testament of the Number Secretary", and Liu Huijingyuan (263) annotated the Nine Chapters of Arithmetic, all of whom were famous astronomical almanacs at that time. These views of the academic community affirm the relationship between the Eastern Han Classics and the compilation of the Nine Chapters of Arithmetic, affirm the influence of Zheng Xuan on Liu Hui's annotations, and leave room for further study.
In recent years, the author has focused on the mathematical literature in the commentaries on the Confucian classics, and found that the Sui and Tang Confucians of the Southern and Northern Dynasties developed a relatively independent algorithmic tradition (called "jing arithmetic" by the Qing people) in the study of classics, which was different from the Nine Chapters of Arithmetic,[6] and this algorithmic tradition continued until the end of the Qing Dynasty. [7] Specifically, the "art" of the Nine Chapters of Arithmetic relies on arithmetic, has the characteristics of "construction", "mechanization", and "allegory in arithmetic", and its application is extensive; in contrast, the algorithmic tradition in the study of scripture basically does not use calculation, but only relies on writing to calculate and reason, and only occurs in the notes of Zheng Xuan and other predecessors on mathematics. [8] Under this study, we will ask Zheng Xuan whether his own mathematics is also a Confucian tradition. I think the answer is no. But why did Zheng adopt this method of exegesis, and why did this method have such a great impact on the development of mathematics in later generations? In order to answer these questions, we must further analyze Zheng Xuan's relationship with ancient Chinese mathematics or the Nine Chapters of Arithmetic, and advance the history of Chinese mathematics and the history of scripture, which is the purpose of this article.
Qiong and Zheng Xuan quoted the Nine Chapters of Arithmetic to annotate the scriptures
In the process of pervading the Qunjing, Zheng Xuan used many ideological resources. In the past, scholars paid more attention to Zheng Xuan's use of Wei Wei thought in it. [9] In fact, Zheng Xuan also used mathematics extensively. In general, the author believes that Zheng's use of mathematics has three levels: first, he discusses the relationship between mathematics and Zhou Lizhi; second, his mathematical annotations provide a textual context for the development of mathematics in later generations; and finally, his purpose is to use mathematics as a tool to eliminate or bridge the differences between the classics. Let's discuss them in turn.
Zheng Xuan's study of etiquette had a great influence on later generations, so much so that there is a saying that "etiquette is Zheng Xue". In the middle of this, Zheng's note "Zhou Li" "nine numbers" had a great influence on the development of mathematics in the ancient understanding. "Zhou Li Di Guan Bao Shi" Yun: "Raising the sons of the country with the Tao is the six arts of teaching." One is five rites, two are six music, three are five shots, four are five, five are six books, and six are nine numbers. Zheng Xuan quoted Zheng Zhong: "Nine numbers, square field, corn, difference, less broad, shang gong, average loss, equation, insufficient win, side point." Now there are heavy differences, xijie, and sentence stocks. [10] This is to interpret "nine numbers" as nine titles for mathematics. The names of the nine chapters of the present Nine Chapters of Arithmetic are, in order: Fangtian, Corn, Decay, Shaoguang, Shanggong, Average Loss, Profit Deficiency, Equation, and Sentence Stock. The decay point is the difference, and the profit is insufficient, so Zheng Xuan's quotation of Zheng Zhong is highly close to the Nine Chapters of Arithmetic (only the surplus is different from the order of the equations, the side and the sentence stock). Scholars generally believe that this shows the inheritance relationship between the nine numbers of the Zhou Li and the Nine Chapters of Arithmetic. In fact, because of Zheng Xuantong's "Nine Chapters of Arithmetic", he quoted Zheng Zhong's statement, that is, to construct a progressive development relationship from the "nine numbers" of the "Zhou Li" to the "nine numbers" of Zheng Zhong. Through this relationship, Zheng Xuan introduced mathematics to the Zhou Li and implied that the Nine Chapters of Arithmetic was derived from it. Liu Hui's note on the "Nine Chapters of Arithmetic" preface: "According to the Zhou metric system, there are nine numbers, and the flow of nine numbers, then the "Nine Chapters" is a book. [11] Undoubtedly, Zheng Xuan's views were used and affirmed.
Zheng Xuan's commentary repeatedly refers directly to the "corn law" or "corn law". Zheng's note "Zhou Li Kao Gong Ji" (周礼考工記) "Clan for the amount of "cloud" in the present corn method, less than two liters of eighty-one cents of twenty-two. [12] Zheng also noted the Book of Rites and the Book of Funerals: "Twenty-two overflows, in the law of corn, is one of the twenty-four liters of rice." [13] In these two cases, the former is about the conversion between volume and volume, and the latter is about the conversion of weight and volume. Volume II corn in the present nine chapters of Arithmetic is a conversion between various grains, and there is a conversion of volume and volume in the volume five Shang Gong "Wei Su Shu". Therefore, Zheng Xuan's introduction of the "corn method" annotation "Zhou Li" and "Li Ji" also introduced mathematics into the study of scriptures. On the basis of Zheng's notes, the Northern Zhou Zhen Luan, The Early Tang Dynasty Li Chunfeng (602-670) and other commentaries on the Five Classics Arithmetic, the Tang Chu Kong Yingda (574-648) and other compilations of the Book of Rites and Justice, and Jia Gongyan (active in 650-655) and the Zhou Li Notes continue to discuss this. [14]
Zheng Xuan's commentaries often only give an approximation of the algorithm, but no details of the calculation. For example, the Zhou Li Kao Gong Ji Yun: "The length of the bow is respected by one of them." Zheng Note: "Zun, Gao Ye." The bow of six feet, the upper and nearest parts are two feet flat, and the claws are two feet below the lower part. Two feet is a sentence, four feet is a string, ask for its shares. Shares XII. In addition, the surface is three feet and a half also. [15] The Nine Chapters of Arithmetic, Volume IX, said: "Now there are five feet of strings, three feet of sentences, and the question is the geometry of the strands." Key: Four feet. Sentence Strand Technique: ... Also, the sentence is multiplied by itself, and the shares are multiplied by subtraction, and the rest are divided by the opening party, that is, the shares. [16] It can be seen that Zheng Xuan's algorithm is consistent with the Pythagorean of the Nine Chapters of Arithmetic, and also does not give the details of the opening operation. On this basis, Jia Gongyan gave a geometric opening algorithm that was different from the "Nine Chapters of Arithmetic" to calculate the opening formula, and the "Five Classics Arithmetic" annotated by Zhen Luan, Li Chunfeng and others was solved by the calculation of the opening formula. [17] And the Book of Rites and Throwing Pots: "The neck of the pot is repaired seven inches, the abdomen is repaired five inches, the caliber is two and a half inches, and the bucket is five liters..." Zheng Note: "Repair, long also." The belly bucket is five liters, and the three points are one, then it is two buckets, and the image of the circle is obtained, and the accumulation of three hundred and twenty-four inches is also. Repair five inches of belly, and get it. Ask for its round week, circle Tuesday seven inches is strange. It is more than nine inches for the abdomen and neck.... [18] Here Zheng Xuan gives the algorithm for calculating the diameter of the pot, but also without details. Zhen Luan, Li Chunfeng and other commentaries on the "Five Classics Arithmetic", Kong Yingda and other "Ritual Justice" continue to be discussed. [19]
Zheng Xuan sometimes gives the result of the calculation without giving the process. "Ritual and Funeral Dress" Yun: "Tho Tho is widely separated, the left ben is down, go to the five points and think that the belt is ...." Thus, the thickness of the five-service ribbon such as chopping decline, Qi decay, large gong, small gong and hemp is given according to 1/5 of the relationship. Zheng Notes on the Rituals: "Ying Hand Shelving, Shelving, Also." In the middle of the people, the circumference is nine inches. One in five is considered to be a killer, like the number of five suits.... [20] He also noted the Book of Rites: "(Qi Decline) is one of the seven inches and five inches of the Great Club, and (Qi Decline) is one of the five inches and twenty-five quarters of nineteen." (Great gong) the great club of five inches and twenty-five minutes of nineteen, (great merit) with four inches of one hundred and twenty-five quarters of seventy-six. [21] As a result, Zheng Xuan gave four values for the decapitation, qi decay, large gong, and small gong belts, but did not give an algorithm. [22] He also noted in the "Ceremonial And Funeral Dress" ("Drink porridge, overflow rice towards one day, overflow rice at sunset": "Twenty-two overflows, one of the twenty-four liters of rice." [23] Note that the same passage in the Book of Rites and The Book of Funerals states that "the law of corn" is not given, but no details are given in either place.
Zheng Xuan quoted the Nine Chapters of Arithmetic to use mathematics to eliminate differences between the classics or versions, so as to unify the meaning of the scriptures. Zhou Li Kao Gong Ji Yun: "The amount of the fish is thought to be a mane." A deep ruler, an inner square ruler and a circular circle outside. Actually, a fish. Zheng Note: "In the name of its tolerance." Four liters of beans, four beans of the district, four districts of the four-liter, six buckets of four liters also. Ten bells. Square ruler, thousands of inches. In the present corn law, less than two liters of eighty-one cents of twenty-two. The number will be enough. This is a generous statement. He who is outside of him, is the lip of it. [29] Jia Gongyan pointed out that "four liters of beans, four beans of the district, four districts of the four areas, six buckets of four liters." 鬴十則鐘" is quoted from the Spring and Autumn Zuo Clan Biography. Therefore, Zheng Xuan found that there was a difference in the records of the amount of clan in the Zuo Shi Biography and the Zhou Li, and tried to reconcile the two with the calculation of the corn method of the Nine Chapters of Arithmetic. The Zhou Li Examination Worker Chronicle also said: "Ten feet ahead, and half a plan." Zheng Note: "It is said that the length of the dragon before the dragon is also." Strategy, the emperor's strategy also. Ten, or seven. Make seven strings, four feet seven inches as hooks, in order to obtain its strands. The stock is short also, and the seven non-also. [30] Thus, Zheng Xuan discovered that in another version of the Book of Examiners, it was "seven feet before the bow", and explained through the calculation of the Pythagorean that "ten" was correct and "seven" was wrong. The Book of the Later Han Dynasty, The Biography of Zheng Xuan, contains a letter that Zheng Xuan wrote to his son in his later years: "Recite the original meaning of the first saint, think of the unevenness of the whole hundred families, and also exhaust my talents, so I listen to the fate of the disciples." [31] It can be seen from this that unifying the scriptures and restoring the original meaning of the saints is Zheng Xuan's ambition, and mathematics is a favorable tool for realizing his ambition.
From the perspective of the history of scripture, there are many chapters and sentences in the study of the classics at the end of the Han Dynasty, which makes it useless for readers. Zheng Xuan read the Qunjing, studied the sayings of the people, and integrated the ancient and modern texts to complete the unification of the classics. [32] Ma Rong once used mathematics to annotate the scriptures. Zheng Xuan had been under his door in his early years and could not be seen for three years. "It will gather all the students to examine the chart and the weft, and the wenxuan is good at calculation, but it is summoned upstairs." Xuan Yin questioned all the doubts and asked for his resignation. [33] Compared with Ma Rong, Zheng Xuan attached more importance to the role of mathematics, and made extensive use of the Nine Chapters of Arithmetic and its algorithms such as the "Corn Method" and the "Sentence Stock Technique" to complete the unification of scriptures.
See, Zheng Xuan, and zheng xuan use mathematical commentary on the historical context of the scriptures
Qing Ren Pi Xirui (1850-1908) "History of Classics" Yun: "Zheng Jun's erudition and many teachers, the ancient Wen Dao tong into one, seeing that the two families attacked each other at that time, intending to participate in their studies, the words of their own family, although the ancient learning is the sect, but also the modern learning to benefit its meaning." Scholars suffered from the complexity of family law at that time, and saw that Zheng Junbo was broad and all-encompassing, and the public opinion suddenly returned to it, and he did not abandon this and the other. Therefore, Zheng's "Yi Notes" worked, while Shi, Meng, Liang Qiu, and Jing's "Yi" did not work; Zheng's "Book Notes" did not work, while Ouyang and Xiahou's "Book" did not work; Zheng's "Shi Zhi" did not work, while Lu, Qi, and Han's "Poems" did not work; Zheng's "Li Notes" did not work, while The "Li" of Da Dai did not work; Zheng's "Analects of the Analects" did not work; and The Qi and Lu "Analects" did not work. [34] Therefore, we have to ask: Does the belief of other scholars in Zheng's commentary have anything to do with their use of mathematics or the Nine Chapters of Arithmetic as one of the tools for annotating the scriptures?
In fact, this is indeed related to the legal authority of the Nine Chapters of Arithmetic in the late Eastern Han Dynasty. The inscription of the Copper Axe of Guanghe Dasi Nong reads: "The Great Sinong with the Edict of Peng Yin, the Day of the Autumn Equinox, the same measure, the equilibrium stone, the bucket bucket, the right power, and the special copper bucket, the axe, the scale, and the ruler in the states." According to the Huang Zhong Calendar and the Nine Chapters of Arithmetic, with the average length, weight, and size, the Seven Governments of Qi are used to make the hai nei all the same. On the 23rd day of the third month of the leap year of Guanghe II, The Great Si Nong Cao Yu, Yu Chun Yu Gong, Right Cang Cao Tuan Zhu Yin, Shi Han Hongzao. [35] Two other light and two-year copper axes and one of the same year's copper rights have similar inscriptions. [36] "Guanghe" is the era name of the Han Ling Emperor, and the second year is 179, which is the time when Zheng Xuan annotated the scriptures. These inscriptions indicate that the Nine Chapters of Arithmetic had been officially enshrined as a classic at the time, and were an important tool for correcting weights and measures along with the Yellow Bell Calendar. In this context, Zheng Xuan used the Nine Chapters of Arithmetic to examine or integrate data such as length, volume, volume, and weight in different classics to have the legitimacy and authority of official endorsement.
The idea that Dasi Nonghu and Quan Inscriptions placed mathematics in an important position actually came from Liu Xin (50 BC - 23 AD). Ban Gu (32-92) 'Book of Han and The Chronicle of the Laws quotes Liu Xin as playing Shuyun to Wang Mang: "One is a backup number, two is a harmony, three is a degree of judgment, four is a measure, and the fifth is a trade-off." Reference to the five to change, intricate the number, the ancient and modern, the card to the qi, and to the heart and ear, to the scriptures, salty in fact, the door is not coordinated. [37] That is, Liu's "law" includes five parts: "reserve number", "harmony", "evaluation degree", "jia quantity", and "weighing", and "reserve number" is the first, which can be "audited in ancient and modern times" and "examined in the scriptures". The Hejaz "Spare Numbers" begins with the cloud: "The numberers, one, ten, hundred, thousand, and ten thousand." So count things, and follow the principle of life. The Book says, 'First of all, he will tell his fortune.' [38] That is, Liu Xin believes that "number" is a universal existence used by human beings to regulate all things in the universe (including people), and on this basis, the five sounds, weights and measures, the three unifications and three positives, and the calendar number are associated, and the universe thus becomes a system that uses "number" as a connection and specification. [39] "Spare number" continued: "The number of the original yellow bell begins with one and three, three and three products, the number of twelve days, ten has seventy-seven thousand one hundred and forty-seven, and five numbers are prepared." Its algorithm is made of bamboo, one point in diameter, six inches long, and two hundred and seventy-one pieces into six yao, which is a grip. [40] It is pointed out that the "number" originated from the yellow bell, and its calculation method is to rely on the calculation chip, and the 271 calculation chips form a regular hexagon. [41] "Spare number" and cloud: "Fu pushes the calendar law maker, rules and regulations, weight balance, quasi-rope Jia, probe the rope hidden, hook deep and far, do not use it." The length of the degree does not lose a millimeter, the amount of the amount does not lose the gui, and the weight of the weight does not lose the weight. [42] That is, to emphasize the wide range of "numbers", which can be used in music, weights and measures, and calendars. And yun: "Ji is in one, co-is in ten, longer than one hundred, greater than a thousand, derived from ten thousand, and its method is in "Arithmetic". [43] The Hanshu Yiwenzhi, based on Liu Xin's Seven Sketches, records The Xu Shang Arithmetic and Du Zhong Arithmetic. [44] Xu Shang was a famous scientist and the Great Sinong during the reign of Emperor Cheng of Han. [45] It follows from this that there were already books entitled Arithmetic at that time. Thus, "spare numbers" means that the method of counting is published in Arithmetic. [46] "Spare number" and cloud: "Propaganda in the world, primary school is the rule." The job is in Taishi, Xihe and Palm. [47] Liu Shi was xihe at the time of his ascension, which emphasized the power of the official position. In the "JiaLiang" section of the Book of Han and the Chronicle of the Laws, Liu Xin put forward the principle of "judging his appearance with degrees", gave the shape and data of Wang Mang's copper axe, and pointed out that "the position is in Taicang, and the great si nong is in charge." [48] Academics generally believe that he used a value of 3.1547 for his calculations. [49] In short, Liu Xin's discussion constructs the basic role of "numbers" in determining the rhythm, weights and measures, and calendars, and points out that the method of calculating numbers is found in Arithmetic. The Guanghe Dasi Nonghu and Quan inscriptions established the authoritative position of this idea and the Nine Chapters of Arithmetic. Zheng Xuan used mathematics or the Nine Chapters Arithmetic to reconcile the differences between the sutras (such as the volume and volume problems of the Zhou Li's quantity), which is also a continuation of Liu Xin's thoughts and practices.
It is worth noting that the Book of Han and the Chronicle of the Laws proposes to calculate the "six inches long". The Eastern Han Dynasty scholar Xu Shen (c. 58-147) "Commentaries on the Interpretation of Texts" is cloudy: "筭, six inches long." The person who counts the calendars. From the bamboo lane, the words are often not wrong. [50] Apparently inherits the hejaz. Hejaz cloud: "its algorithm uses bamboo" and "its method is in "Arithmetic", indicating that "algorithm" is a specific calculation method, "arithmetic" is a universal "technique", and the relationship between the two is special and general. Zheng Xuan continued this understanding. Xu Shen wrote the "Five Classics of Different Meanings", and Zheng Xuan refuted it, and wrote "Refuting the Objections of the Five Classics", which also used mathematics. For example, Xu Shiyun: "Heterodox: "Ram" says: Yin three thousand princes, Zhou thousand eight hundred princes." The ancient "Spring and Autumn Zuo Clan Biography" says: Yuhui princes yu Tushan, the ruler of the jade warriors of all nations. The land of Tang Yu is ten thousand miles, the land of Rong Baili is ten thousand countries, his hou bo is seventy miles, and the son is more than fifty miles, which is the tianzi matian. Zheng Shi retorted: "How many princes there are, the other world is different." The nations of all nations are also the system of Tang Yu. King Wu had two or eight hundred princes in three parts, and the princes of the end of Yin were two hundred. After the Zhou Metric Ceremony, the quasi-"Royal System" thousand seven hundred and seventy-three kingdoms, as far as the Zhou thousand eight hundred, all of them. [51] This is xu Shen's view that there is a contradiction between the Spring and Autumn Ram Biography and the Spring and Autumn Zuo Clan Biography regarding the number of princes. Zheng Xuan believes that these are just "all of them", that is, taking the whole number, thus reconciling the two theories. Xu Shen believed that there was a difference between the Ram and the Zuo Shi about the leap month, and Zheng Xuan refuted it with the Shang Shu Yao Dian (尚書尧典) that "the leap month is set at four o'clock to become old". And yun: "Today to abolish its greatness and to save its details is to be ridiculed by The Lord." [52] It follows from this that, despite the same recognition of the differences between the sutras, Zheng Xuan was better at reconciling them with mathematics, calendars, and so on than Xu Shen.
In short, compared with scholars such as Zheng Zhong, Xu Shen, and Ma Rong, Zheng Xuan was more adept at using mathematics, calendars, or the Nine Chapters of Arithmetic to annotate the scriptures and unify ancient and modern texts. [53] Liu Xin proposed the idea that mathematics was the basis of musical rhythms, weights and measures, and calendars, and the Inscriptions of Guanghe Ii-Year Grand Si Nong and Tongquan established this idea as the official authority of the Nine Chapters of Arithmetic, and Zheng Xuan quoted the Nine Chapters Arithmetic extensively to coincide with the official position. In the historical context of scholarship and politics, Zheng Xuan gained dual legitimacy with mathematical annotations, which were eventually accepted by scribes.
The influence of Zheng Xuan on future generations with mathematical annotations
From the perspective of the history of scripture, Zheng Xuan's extensive annotation of the Qunjing completed the unification of scripture, which had a great impact on future generations. In fact, from the perspective of the history of mathematics, Zheng Xuan's practice of annotating the scriptures with mathematics also had a great impact on future generations.
Zheng Xuan quoted Zheng Zhong as saying that he had nine numbers, implying that the Nine Chapters of Arithmetic came from the Zhou Li Nine Numbers. It is both a construction of the origin of Chinese mathematics and the concept that mathematics is part of ritual or scripture. On the one hand, Liu Hui's annotation of the Nine Chapters of Arithmetic clearly states that "if there are nine numbers, then the Nine Chapters are", agreeing with Zheng's statement. The Song Dynasty "Arithmetic Sources" talks about the origin of Chinese mathematics, first quoting Li Chunfeng's "Jinshu Vinaya" the Yellow Emperor's envoy to make a calculation, then quoting Liu Xin as a song in the "Book of Han and Vinaya", and then quoting Zheng Xuan's annotation of "Nine Numbers" in the Zhou Li. [54] On the other hand, Zhen Luan wrote the Five Classics Arithmetic to solve mathematical problems in the Confucian classics with traditional arithmetic, which Li Chunfeng and others annotated and established in the Tang Dynasty scholars, and both attempted to continue Zheng Xuan's practice of quoting the Nine Chapters of Arithmetic. Although zhu xi (1130-1200), a great confucian of the Song Dynasty, tended to exclude mathematics from the system of science in the early period, in his later years he incorporated mathematics into his compilation of the "General Interpretation of the Rites and Rites". [55] Scholars' discussions of the origins of Chinese mathematics and the relationship between mathematics and Confucianism during the Ming and Qing dynasties were still influenced by Zheng Xuan.
Zheng Xuan's commentaries often give an approximation or calculation result of an algorithm without the details of calculation, which provides a textual context for later scholars to develop mathematics. In the early Tang Dynasty, he compiled the Five Classics of Justice, and Zheng Xuan's commentary was selected as the standard commentary of mao poems, Zhou Li, Rites of Rites, and Li Ji. Kong Yingda, Jia Gongyan, etc. made annotations on the basis of Zheng Zhu, and added a large number of mathematical practices where they did not give details of calculation, but they were not the same as the "Nine Chapters of Arithmetic" and "Five Classics Arithmetic", thus forming a unique algorithmic tradition in classical research. This tradition was influenced by the Wei and Jin Xuanfeng, and its rise was no later than Huang Kan's (488-545) On Semantic Neglect. [56] Its characteristics of not using arithmetic chips and reasoning with words are related to the confucian way of teaching knowledge under Zheng Xuan and Confucianism that emphasizes classics and light utensils. The result is a deviation from Zheng Xuan's original intention of introducing the Nine Chapters of Arithmetic into the classics--Confucian algorithms are part of the classics, while the traditional arithmetic represented by the Nine Chapters of Arithmetic is a relatively independent field. Zhu Xi had some contempt for the tradition of economic calculation in his early years, but developed it in his later years. During the Ming and Qing dynasties, the algorithm tradition continued to develop.
Zheng Xuan's practice of integrating mathematics and unifying scriptures was also accepted by later scholars. Bao Xian (7-65) differed from Ma Rong's Analects of the Analects, "The Kingdom of the Thousand Multiplications of the Dao", with Bao Shi following the Book of Rites and the "Mencius" and Ma Shi following the Zhou Li. He Yan exists twice. Huang Kan used the Confucian formula to explain the difference between the two. [57] Zhu Xi disagreed with Zheng Xuan's practice of "three points of benefit and one is two buckets" in Zheng Xuan's "Book of Rites and Throwing Pots", and also interpreted it by algorithm. During the Ming and Qing dynasties, huang Zongxi (1610-1695) continued this discussion. [58] In the middle of the Qing Dynasty, Kong Guangsen (1751-1786), Jiao Xun (1763-1820), and Liu Yueyun (1849-1917) in the late Qing Dynasty continued to discuss this.
To sum up, in the context of the numerous chapters and sentences of the classics at the end of the Eastern Han Dynasty, liu Xin's idea that mathematics is the basis of phonology, weights and measures, and the calendar, and the "Nine Chapters of Arithmetic" were established as the authority of revising weights and measures, in order to unify the classics and integrate ancient and modern texts, Zheng Xuan, who was better at the "Nine Chapters of Arithmetic" and the calendar than others, adopted the practice of mathematical annotation, hoping that the Nine Chapters of Arithmetic would become a part of etiquette or scripture. However, Zheng Xuan's commentary often only describes the outline, or gives the result in a straight line without calculating the details. Therefore, later Generations of Confucians (Huang Kan, Kong Yingda, Jia Gongyan, etc.) took advantage of the obscurity of Zheng's language and developed an algorithmic tradition that was relatively independent of traditional mathematics, without the use of calculations and reasoning by words, under the Confucian tradition of Wei and Jin metaphysics and light instruments. Later generations of operators (Zhen Luan, Li Chunfeng, etc.) tried unsuccessfully to use the tradition of traditional arithmetic to annotate the scriptures and unify the two algorithms. In his early years, Zhu Xi, the great Confucian of the Southern Song Dynasty, excluded the two algorithmic traditions from his theory, and in his later years, he incorporated the two together into his etiquette, resulting in a variety of different attitudes and practices of Ming and Qing scholars towards arithmetic and Confucianism. Therefore, Zheng Xuan's practice of mathematical annotation of the scriptures had a great influence on later generations, but the results of this influence deviated from Zheng Xuan's original intention of introducing the Nine Chapters of Arithmetic into the classics.
exegesis:
[1] Southern Song Dynasty. Fan Ye, Book of the Later Han Dynasty (Beijing: Zhonghua Bookstore, 1965), p. 1207. All quotations from the Book of the Later Han Dynasty in this article are based on this, and only the title and page numbers of the following text are listed.
[2] "Mathematics" has the meaning of mathematical in ancient contexts, and the "mathematics" that appears in this article does not take this ancient meaning, but only the present meaning, which is equivalent to the English mathematics.
Chen Meidong, History of Science and Technology in China. Volume of Astronomy (Beijing: Science Press, 2001) and Guo Shuchun, eds., History of Science and Technology in China. None of the volumes of mathematics (Beijing: Science Press, 2010) are devoted to Zheng Xuan's astronomy and mathematics. Wu Cunhao made a general discussion of Zheng Xuan's achievements in natural science, see Wu Cunhao, "A Brief Discussion on Zheng Xuan's Achievements in Natural Sciences", Journal of Changwei Teachers' College, 7.4 (2000): 7-9+17. Academics are interested in Zheng Xuan's Zhou Li. Whether Hooke's law of elasticity was found in the Notes on the Examination of Workers is controversial. See Yi Degang, "Reflections on the Debate of Zheng Xuan's Elasticity Law":"Answer to Mr. Liu Shuyong", Journal of the History of Chinese Science and Technology, 40.1 (2019): 113-116.
[4] Qian Baochun, "The Nine Chapters of Arithmetic and its Relationship with Liu Hui's Notes on Philosophical Thought", Li Yu Qian Baochun's Complete History of Science. Volume IX), Li Yu and Qian Baochun (Shenyang: Liaoning Education Press, 1998), pp. 685-695.
[5] Guo Shuchun, "Liu Hui and the Scholars of the Pre-Qin and Han Dynasties," History of Chinese Philosophy, 2.2 (1993): 3-10.
[6] See my seven articles, in order: "A Preliminary Study of Mathematics and Knowledge in the Confucian Classics: Jia Gongyan's Commentary on Zhou Li. "A Case Study of the History of Natural Sciences" 34.2 (2015): 131-141; "Different Cultures of Computation in Seventh Century China from the Viewpoint of Square Root Extraction," Historia Mathematica 43.1 (2016): 3-25; "Re-examining the Relationship between Ancient Chinese Mathematics and Confucianism: A Case Study of the Different Commentaries on Ritual Numbers by Scholars in the Sixth and Seventh Centuries", Bulletin of Dialectics of Nature, 38.5 (2016): 81-87; Mathematics and Etiquette in the Early Tang Dynasty: A Review of the Book of Rites by Various Schools. Journal of Sun Yat-sen University (Social Sciences Edition), 57.2 (2017): 244-257; "Arithmetic, Confucianism and Institutionalization: The Diversity of Mathematics in the Early Tang Dynasty and Its Relationship with Confucianism", Sinology Research 35.4 (2017): 109-134; "The Diversity of Early Tang Algorithm Cultures from the Unit of Measurement", Journal of the History of Chinese Science and Technology 40.1 (2019): 1-9; "Scholarship and Politics." in Seventh Century China from the Viewpoint of Li Chunfeng’s Writing on Histories,” in Monographs in Tang Official History: Perspectives from the Technical Treatises of the History of Sui (Sui shu ), Daniel Patrick Mongan and Damien Chaussende (eds.) (Switzerland: Springer, 2019), pp.89-116。
[7] See my four articles, in order: "Zhu Xi's Mathematical World: On the Relationship between Mathematics and Confucianism in the Song Dynasty", Philosophy and Culture, 45.11 (2018): 167-182; "The Evolution of the Confucian Kaifang Algorithm: Centered on the Commentaries of The Analects on the Analects of the Tao" "The Land of a Thousand Multiplications", Natural Dialectics Newsletter 41.2 (2019): 49-55; "Mathematics and Yixue in the Song Dynasty: Centered on the Nine Chapters of the Book of Numbers" "The Yarrow and the Micro" of the Zhou Yi Study, Zhou Yi Studies 32.2 (2019): 81-92; "Mathematics, Confucianism, and Western Studies in the Ming and Qing Dynasties: Centered on Huang Zongxi's Mathematical Reality," Journal of Inner Mongolia Normal University (Natural Science Chinese Edition), 48.6 (2019): 538-544. See also Chen Zhihui, "Scholars' Recreation of Two Traditions of Mathematical Commentaries in Late Eighteenth-century China," Historia Mathematica 44.2 (2017): 105-133.
[8] Wu Wenjun (1919-2017) argues that the algorithms of traditional Chinese mathematics are structural and mechanized. See Wu Wenjun, "The Structural and Mechanized Characteristics of Traditional Chinese Mathematics from the Nine Chapters of the Book of Numbers", ed., Qin Jiushao and the Nine Chapters of the Book of Numbers (Beijing: Beijing Normal University Press, 1987), pp. 73-88. Li Jimin (1938-1993) believes that traditional Chinese mathematical theories are characterized by "allegory in calculation" in terms of expression. See Li Jimin, Introduction to and Translation of the Nine Chapters of Arithmetic (Xi'an: Shaanxi Science and Technology Press, 1998), p. 38.
[9] For example, Lü Kai, "Zheng Xuanzhi Weixue" (Taipei: Taiwan Commercial Press, 1982); Shuzo Ikeda and Hong Chunyin, "Introduction to the Study of Zheng Shixue in Weishu", Bibliographic Quarterly 37.4 (2004): 59-78; Jiang Xiren, "On Zheng Xuan's "Qian Chi Du" and "Qiankun Chisel Degree" Annotations on the Shengwang Jing Shi Yi Yun", Zhou Yi Studies 29.5 (2016): 67-74, etc.
[10] Han. ZHENG Xuanzhu,TANG. Jia Gongyanshu, Zhou Li Zhuanshu, Qing Ruan Yuan, Thirteen Commentaries on the Thirteen Classics, vol. 1 (Beijing: Zhonghua Bookstore, 1980), p. 731. All quotations from the Zhou Li Commentary in this article are based on this, and only the title and page numbers are listed later.
[11] Guo Shuchun Hui School, Supplementary Edition of Hui School 'Nine Chapters of Flute Techniques' (Shenyang: Liaoning Education Publishing House/Taipei: Nine Chapters Publishing House, 2004), p. 1.
[12] Zhou Li Notes, p. 971.
[13] Han. ZHENG Xuanzhu,TANG. Kong Yingda et al., "Etiquette and Justice", Qing Nguyen Yuan School Engraving, Thirteen Commentaries on the Thirteen Classics, vol. 1 (Beijing: Zhonghua Bookstore, 1980), p. 1576. All quotations from the Book of Rites and Justices in this article are based on this, and only the title and page numbers of the book are listed later.
[14] For a specific analysis of Zheng Xuan's annotation "Shi as Quantity", see Zhu Yiwen, "A Preliminary Study of Mathematics and Knowledge in Confucian Classics: A Brief Introduction to Jia Gongyan's Zhou Li. The Notes on the Examination of Workers" ("The Name is the Quantity" as an Example" is an example. For Zheng Xuan's note, "Twenty-two overflows, in the method of corn, is one of the twenty-four liters of rice one liter." For a specific analysis, see Zhu Yiwen, "The Diversity of the Early Tang Dynasty Algorithm Culture from the Unit of Measurement and Measures."
[15] Zhou Li Notes, p. 910.
[16] Guo Shuchun, Supplement to the Nine Chapters of the Hui School, pp. 409-410.
[17] For an analysis of Zheng Xuan and Jia Gongyan's algorithms in this example, see Zhu Yiwen, "Different Cultures of Computation in Seventh Century China from the Viewpoint of Square Root Extraction," Historia Mathematica 43.1 (2016).
[18] The Book of Rites and Justice, p. 1666.
[19] For an analysis of the algorithms of Zheng Xuan, Kong Yingda, Zhen Luan, Li Chunfeng, etc., see Zhu Yiwen, "Mathematics and Etiquette in the Early Tang Dynasty: A Review of the Book of Rites by Various Schools. An Example of the Annotation of throwing pots", Journal of Sun Yat-sen University (Social Sciences Edition), 57.2 (2017).
[20] Han. ZHENG Xuanzhu,TANG. Jia Gongyanshu, Commentary on rituals, engraved by Qing Nguyen Yuan, Thirteen Commentaries on the Thirteen Classics, vol. 1 (Beijing: Zhonghua Bookstore, 1980), p. 1097. All quotations in this article from the Commentary on the Rites of Etiquette are based on this, and only the title and page numbers of the book are listed later.
[21] Etiquette and Justice, p. 1499.
[22] For an analysis of this example, see Zhu Yiwen, "Revisiting the Relationship between Ancient Chinese Mathematics and Confucianism: A Case Study of the Different Annotations on Ritual Numbers by Scholars of the Sixth to Seventh Centuries," Studies in the History of Natural Sciences, 34.2 (2015); Zhu Yiwen, "Arithmetic, Confucianism, and Institutionalization: The Diversity of Mathematics in the Early Tang Dynasty and Its Relationship with Confucianism," Sinology Studies, 35.4 (2017).
[23] Liturgical Commentary, p. 1097.
[24] According to calculations, "forty-two" should be "forty-three".
[25] Han. Zheng Xuan's Notes, "Zhou Yi Qian Chisel degree", "Jingyin Wenyuan Ge Siku Quanshu. Vol. 53 (Taipei: The Commercial Press of Taiwan, 1986), p. 877.
[26] Zhang Huiyan of the Qing Dynasty, "Yi Wei Liuyi" (Guangzhou: Guangya Bookstore, 1920), quoted various commentators, arguing that the text here is mostly derived. The author only cites this article here, and will not discuss the issues of Yan Yan and so on.
[27] Ibid., note 25.
[28] Note 25.
[29] Zhou Li Notes, pp. 916-917.
[30] Zhou Li Notes, p. 913.
[31] Book of the Later Han Dynasty, p. 1209.
[32] Ye Chunfang, Outline of the History of Chinese Classics (Beijing: Peking University Press, 2016), pp. 160-161.
[33] Book of the Later Han Dynasty, p. 1207.
[34] Qing. Pi Xirui, History of Sutra Studies (Beijing: Zhonghua Bookstore, 1959), p. 149.
[35] Qiu Long, Qiu Guangming, Gu Maosen, Liu Dongrui, and Wu Hong, eds., Atlas of Ancient Chinese Weights and Measures (Beijing: Cultural Relics Publishing House, 1981), p. 97.
[36] Gao Dalun and Zhang Maorong, "A Study of Han Guang, Hu And Quan," Journal of Northwest University (Social Sciences Edition), 13.4 (1983): 74-83. The article speculates that Cao Yu was the "Tai Servant Cao Ling" mentioned in the Book of Later Han, and that Guang He was a Tai Servant in the first year of the reign, and was reappointed as the Great Si Nong in the second year.
[37] Han. Bangu, Book of Han (Beijing: Zhonghua Bookstore, 1962), p. 956.
[38] Ibid.
[39] Ding Sixin, "The Philosophical Concept of "Numbers" and the Classicization of early Laozi Texts: Sources of Chapters on Lao tzu," Journal of Sun Yat-sen University (Social Sciences Edition), 59.3 (2019): 108-118.
[40] Ibid., note 37.
[41] Li Yu, "Examination of the Calculation System", Shi, "Treatise on the History of Chinese Arithmetic. Episode IV (Beijing: Zhonghua Bookstore, 1955), pp. 1-8.
[42] Ibid., note 37.
[43] Ibid., note 37.
[44] Han. Bangu, Book of Han, p. 1766.
Wu Wenjun, ed., Department of Chinese Mathematical History. Vol. 2 (Beijing: Beijing Normal University Press, 1998), pp. 13-14.
[46] Academics dispute the date of the nine-chapter arithmetic, and it is uncertain whether the nine-chapter arithmetic was written during the Liu Xin period. Therefore, we cannot be sure whether the Arithmetic here refers to the Nine Chapters of Arithmetic. Since this topic is not the main thrust of this paper, I will not discuss it further.
[47] Ibid., note 37.
[48] Han. Bangu, Book of Han, pp. 967-968.
[49] Qiu Guangming, Qiu Long, yang Ping, History of Science and Technology in China. Volume of Weights and Measures (Beijing: Science Press, 2001), pp. 216-230.
[50] Han. Xu Shen,Qing. Duan Yu's Commentary, Notes on the Interpretation of Words and Characters (Shanghai: Shanghai Ancient Books Publishing House, 1981), p. 198.
[51] Qing. Pi Xirui, "Refuting the Objections of the Five Classics", "The Complete Book of the Continuation of the Four Libraries" ( Vol. 171 (Shanghai: Shanghai Ancient Books Publishing House, 2002), p. 206.
[52] Qing. Pi Xirui, "Refuting the Objections of the Five Classics", "The Complete Book of the Continuation of the Four Libraries" ( Vol. 171 (Shanghai: Shanghai Ancient Books Publishing House, 2002), p. 198.
[53] Anonymous censors noted that "in the Qing Dynasty, Ruan Yuan had a book entitled 'Biography of the Domain People', and the Later Han Dynasty II had: Liu Hong, Cai Yong, He Xiu, Zheng Xuan, Xu Yue, Xi Meng, and Zhao Shuang. And I hope that the author can explain the difference between Zheng Xuan and other mathematicians. Ruan Yuan's "Biography of the Domain People" on Zheng Xuanyun "Kang Cheng includes the canon, the network of families, for the ancient Confucian Sect, in astronomical numerology, especially in the ultimate microscopic." For example, the "Mao Poem" is written, according to the rate of corn in the "Nine Chapters"; the "Yi Wei" is annotated, and the number of points is used in the "Qianxiang". Gai Qi has a book, and the Confucians in Tokyo are not caught. "Qing. Ruan Yuan, Luo Shilin, Hua Shifang, Zhu Kebao, Huang Zhongjun, et al., Feng Lisheng, Deng Liang, Zhang Junfeng, And Others, Notes on the Biography of People from The Domain (Zhengzhou: Zhongzhou Ancient Books Publishing House, 2012), p. 61. In fact, Zheng Xuan's introduction of mathematical commentaries and his skillful use of the Nine Chapters of Arithmetic are a major feature that distinguishes him from the Confucians. Liu Hong, Xu Yue, Zhao Shuang and other alpirators have contributed to the mathematics major, but Zheng Xuan is not here. However, if we look at it from today's mathematical perspective, it can be said that other arithmeticists created mathematical knowledge within the body of "questioning, answering, and technique" in the "Nine Chapters of Arithmetic", and Zheng Xuan's quotation from the "Nine Chapters of Arithmetic" actually provides a new mathematical text context (that is, hidden questions), and the new text form will definitely bring new mathematical content, so this can be regarded as Zheng's contribution to mathematics. On this topic, see Zhu Yiwen, "How do We Understand Mathematical Practices in Non-mathematical Fields?" Reflections Inspired by Cases from 12th and 13th Century China,” Historia Mathematica 52(2020): 1-25。
[54] Guo Shuchun, ed., Chinese Science and Technology Classics. Mathematics Volume, Vol. 1 (Zhengzhou: Henan Education Press, 1993), p. 427.
[55] Zhu Yiwen, "Zhu Xi's Mathematical World: On the Relationship between Mathematics and Confucianism in the Song Dynasty," Philosophy and Culture, 45.11 (2018).
[56] Zhu Yiwen, "The Evolution of the Confucian Kaifang Algorithm: Centered on the Commentaries of the Analects on the Analects of the Tao and the Thousand Multiplications"," Bulletin of Dialectics of Nature, 41.2 (2019).
[57] Ibid.
[58] Zhu Yiwen, "Mathematics, Confucianism and Western Studies in the Ming and Qing Dynasties: Centered on Huang Zongxi's Mathematical Reality," Journal of Inner Mongolia Normal University (Natural Science Chinese Edition), 48.6 (2019).
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Zhu Yiwen, "Mathematics and Etiquette in the Early Tang Dynasty: A Review of the Book of Rites by various families. The Case of the Annotation of throwing pots", Journal of Sun Yat-sen University (Social Science Edition), 57.2 (2017): 244-257.
Zhu Yiwen, "Arithmetic, Confucianism, and Institutionalization: The Diversity of Mathematics in the Early Tang Dynasty and Its Relationship with Confucianism," Journal of Sinology, 35.4 (2017): 109-134.
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Zhu Yiwen, "Zhu Xi's Mathematical World: On the Relationship between Mathematics and Confucianism in the Song Dynasty", Philosophy and Culture, 45.11 (2018): 167-182.
Zhu Yiwen, "The Evolution of the Confucian Kaifang Algorithm: Centered on the Commentaries of the Various Schools on the Analects of the Analects of the Tao and the Thousand Multiplications"," Bulletin of Natural Dialectics, 41.2 (2019): 49-55.
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朱一文（Zhu Yiwen），“How do We Understand Mathematical Practices in Non-mathematical Fields? Reflections Inspired by Cases from 12th and 13th Century China,” Historia Mathematica 52(2020): 1-25。
