Keywords

1 Introduction

The research leading to these results has received funding from the European Research Council under the European Union’s Seventh Framework Programme (FP7/2007–2013)/ERC Grant Agreement No. 269804. We also thank Chinese National Planning Office of Philosophy and Social Science [the National Social Science Fund of China, 21VJXG022] for supporting funding on this research. The results presented in this chapter emerged from the two authors’ cooperation on the textual history of Mathematical Book in Nine Chapters in 2009, when both of us were pursuing our doctoral degrees at the Institute for the History of Natural Sciences, Chinese Academy of Sciences. In 2013, when Zhu Yiwen took up the postdoctoral fellowship in Paris, following discussions with the European project SAW team, we decided to complete a paper in English. These results were initially presented on 17 June 2013, and improved on 3 December 2014 within the framework of the seminars and conferences of the European Project ‘SAW’. We are very grateful to the audiences for their questions and comments, which contributed to improving our study. We continued working on this chapter after the 2014 conference, and would like to thank Karine Chemla and Agathe Keller for their careful reading of this chapter, which helped us greatly. We deeply appreciate their comments and advice. We would also like to thank Sun Yat-sen University and the Chinese Academy of Sciences for their support on this research. We are also grateful to Julia Legh-Smith, Jens Høyrup, Karen Margolis and Christine Mousset for the English editing of this chapter.

Qin Jiushao (秦九韶, 1208–1268) was a scholar-official during the Song dynasty (960–1279) who is regarded as one of China’s most brilliant mathematicians.Footnote 1 His only extant work, titled Mathematical Book in Nine Chapters (Shushu Jiuzhang 數書九章, hereafter Mathematical Book), includes the author’s preface of 1247 and eighty-one problems in nine categories relating to calendar calculations, surveying, tax, construction, military engineering and logistics, commerce, and so forth. Mathematical Book is famous especially for providing the Chinese Remainder Theorem (dayan zongshu shu 大衍總數術, literally the ‘Great inference procedure for all numbers’)—which is the general algorithm for solving simultaneous linear congruences—and a method for solving numerical equations of higher degree (zhengfu kaifang shu 正負開方術, literally the ‘Procedure for extracting roots with positive and negative [numbers]’). This work is also a valuable source for social and economic history. In 1842, the Yijiatang edition of Mathematical Book appeared. This edition was the first printed edition, which was published in Yijiatang Congshu (宜稼堂叢書 Yijiatang Collectanea) in Shanghai with a detailed critical apparatus. The editor and publisher of Mathematical Book were, respectively, Song Jingchang 宋景昌 and Yu Songnian 郁松年.

Modern scholars mainly rely on the 1842 edition to research Qin’s work. Qian Baocong 錢寶琮 (1966) systematically studied Mathematical Book. Then Libbrecht (1973) completed a monograph on Mathematical Book titled Chinese Mathematics in the Thirteenth Century. In the collected papers on Mathematical Book (Wu 1987), Li Di 李迪 (1987b) provided research on the textual history of Mathematical Book, and Shen Kangsheng 沈康身 (1987) corrected some errors in the 1842 edition. Wang Shouyi 王守義 (1992) provided a New Annotated Mathematical Book in Nine Chapters. There are also several studies on Qin Jiushao’s life.Footnote 2

In 2010, we examined a manuscript of Mathematical Book transcribed in 1616 and presented new research on the textual history of Mathematical Book, arguing that the 1616 manuscript is the most complete edition to be found today. This manuscript retains very important features of Qin Jiushao’s mathematics, notably the counting-diagrams with lines for writing procedures of counting rods and numbers with colours representing two opposite numbers (Zheng and Zhu 2010). These features are missing in the 1842 edition. Because of the importance of the 1616 edition, it is thus necessary to revisit the formation of the 1842 edition and evaluate its influence on the study of the history of mathematics in China. This is the aim of the present chapter.

2 Textual History of Mathematical Book in Nine Chapters Before 1800

Some thirteenth-century authors referred to the title of Mathematical Book as Shushu (數術, Mathematical Methods), Shushu Dalüe (數術大略, Outline of Mathematical Methods) or Shuxue Dalüe (數學大略, Outline of Mathematics). From the fifteenth to the seventeenth century, the standard title of Mathematical Book was Shuxue Jiuzhang (數學九章, Mathematics in Nine Chapters), not Shushu Jiuzhang (數書九章, Mathematical Book in Nine Chapters). Although a little different, the meaning is similar. In 1441, the first catalogue of the Imperial Library (Wenyuange Shumu, 文淵閣書目) of the Ming dynasty (1368–1644), which recorded approximatively 5800 titles—notably 24 mathematical works—kept in Beijing, included an entry ‘Shuxue Jiuzhang, one copy, three volumes, complete.’Footnote 3 In 1605, the revised catalogue of the same library only contained 2447 titles. One of the three extant mathematical works was ‘Shuxue Jiuzhang, three volumes, complete, manuscript, written by Qin Jiushao of Lu county, during the Chunyou period [淳祐 1241–1252] of the Song dynasty.’Footnote 4 This manuscript is the master copy for all the editions that still exist today, but it was lost, probably in the turmoil after the fall of Beijing in 1644.Footnote 5

In 1616, a famous bibliophile, Zhao Qimei (趙琦美, 1563–1624), acquired his own copy of Mathematical Book in Beijing and wrote postscripts in this manuscript (1616 [2010]). He called the book Shushu Jiuzhang for the first time and recorded that Wang Yingling had made a manuscript transcribed from the copy kept in the Imperial Library. Zhao was able to borrow Wang’s copy to make this new duplicate.Footnote 6

Another bibliophile, Wang Yingling (王應遴, ?–1644) was especially interested in astronomy (the first two chapters of Mathematical Book are about calendar calculations). In the 1630s, as a minor official, Wang was involved in the calendar reform in Beijing (Zheng and Zhu 2010). His name appears as a reviser on the title page of On Ancient and Modern Eclipses (Gujin Jiaoshi Kao 古今交食考) (Schall von Bell, ca. 1635), a Chinese pamphlet written by a famous Jesuit, Johann Adam Schall von Bell (1592–1666).Footnote 7 But there is no trace of the Jesuits in Beijing ever having had access to any Chinese mathematical work of the thirteenth century. Wang compiled a series of writings which he called Ce Shu (測書, Book of Surveying), Shu Shu (數書, Book of Mathematics), Bei Shu (備書, Book of Defence), etc. (Shi and Song 2006). It is reasonable to assume that Wang changed the title of Mathematical Book from Shuxue Jiuzhang to Shushu Jiuzhang out of personal taste to give all his works similar names (with the form ‘X Shu’).

Wang’s manuscript was lost. Luckily, Zhao Qimei brought his own copy back to his hometown of Changshu 常熟, a city located 80 kilometres northwest of Shanghai. Both Changshu and Shanghai are situated in the Yangtze River Delta, which was the most prosperous region, the centre of literary culture and publishing in the Ming-Qing periods. From the mid-seventeenth to the early twentieth century, Zhao’s manuscript of Mathematical Book was preserved by several local bibliophiles in Changshu. Some duplicates transcribed from Zhao’s manuscript were shared by scholars from the late eighteenth century onward. Since the 1950s, Zhao’s manuscript (or a fairly accurate copy) has been preserved in the National Library of China (Zhao Qimei 1616 [2010]; Zheng and Zhu 2010).Footnote 8

Another lineage of Mathematical Book is more complicated, somewhat like a jigsaw puzzle. At the beginning of the fifteenth century, Zhu Di (朱棣, 1360–1424), the third emperor of the Ming Dynasty, ordered his officials to compile Great Compendium of Yongle Era (Yongle Dadian 永樂大典, 1405–1408 [1986]). This huge encyclopedia, which had 22,937 sections in 11,095 manuscript volumes, drew its material mainly from the collection of the Imperial Library in Nanjing 南京. Mathematical Book was copied into it. In 1421, when Beijing replaced Nanjing as the new capital of the Ming dynasty, the main part of the library’s collection in Nanjing was moved to Beijing and was later housed in the new Imperial Library there. It is thus highly probable that the particular copy of Mathematical Book that Wang Yinglin transcribed in around 1616 is the same as the one transcribed into Great Compendium in around 1408 (Zheng and Zhu 2010).

According to the catalogue of Great Compendium, there were 37 sections (from Sect. 16 329 to Sect. 16 365) listed under ‘Mathematics’ (suan 筭). However, in accordance with the structure of the encyclopedia, the preface and the 81 problems of Mathematical Book were subdivided into various entries and arranged in different sections, which corresponded to different mathematical subjects. The original version of Great Compendium was totally lost in the war in the mid-seventeenth century. Thanks to the Ming government, which made a faithful manuscript copy of Great Compendium around the year 1565, today, there are approximately 408 volumes extant, less than 4% of the total 1565 edition. Three problems of Mathematical Book survive in Sect. 16 343 of the only extant volume under ‘Mathematics’ (Sect. 16 343 and 16 344), which is now held in the library of Cambridge University.Footnote 9

In the turbulent era of the mid-seventeenth century, the 1565 edition of Great Compendium survived almost intact in Beijing and entered the collection of the new Manchu emperors of the Qing dynasty (1644–1911). In 1773, Qianlong Emperor (乾隆帝, 1711–1799) launched another cultural project, compiling Complete Works in the Four Treasuries (Siku Quanshu 四庫全書, hereafter abbreviated to Complete Works). More than ten thousand titles were assembled in Beijing for investigation and censorship.Footnote 10 In the end, the incomparably huge manuscript collection constituted around 3500 titles (approx. 36,000 volumes). All of the books were transcribed according to the same layout. Each of them was prefaced by a summary by the editors.Footnote 11 Around 10% of all the titles in Complete Works were restored from Great Compendium. Many of them were rare or unique editions.

The first manuscript copy of Complete Works (Wenyuange Siku Quanshu 文淵閣四庫全書) was completed in 1781. It comprised 3458 titles, with 386 titles—including Mathematical Book—derived from Great Compendium. There were 25 mathematical works in this edition, including traditional ones like Qin’s Mathematical Book and European ones like the first six books of Euclid’s Elements, which had been translated into Chinese in 1607 by the Italian Jesuit Matteo Ricci (1552–1610) and Xu Guangqi (徐光啟, 1562–1633). By 1787, all seven manuscript copies of Complete Works had been finished. Four of them were housed in different Imperial palaces in the North (Beijing, Rehe, Shenyang), while another three were delivered to the South for the imperial libraries in Hangzhou 杭州, Yangzhou 揚州 and Zhenjiang 鎮江, three prosperous cities in the Yangtze River Delta. Scholars were invited to consult Complete Works in these three libraries. In fact, it was a privilege for high-ranking officials.

All 3500 summaries and a further 6700 summaries of ‘less important but useful books’ were collected together as the Comprehensive Catalogue of Complete Works in the Four Treasuries (Siku Quanshu Zongmu 四庫全書總目) and published in 1795. This Comprehensive Catalogue has been the most influential guide for Chinese classical studies since the nineteenth century. The summary of Mathematical Book criticizes Qin Jiushao’s algorithm for calendar calculation as obsolete but highly praises one of his mathematical methods, which was later named the Chinese Remainder Theorem. The summary describes it as exquisite and ingenious, and even hails it as the origin of the European algebraic technique Jiegenfang (借根方, literally ‘the borrowing of roots and powers [of the unknown]’), the algebraic system the Jesuit missionaries introduced to the Qing court in the late seventeenth century.

Complete Works provided the first critical edition of Mathematical Book. We shall call it the Siku edition. The title page lists the proof readers Gu Zhixiong 古之雄 and Ni Tingmei 倪廷梅 as two subordinate officials from the Imperial Bureau of Astronomy. We know little about them. Meanwhile, according to the extant archive of the Qing dynasty, in 1781 Chen Jixin 陳際新, who was the Imperial astronomer and chief editor for the astronomical and mathematical books in Complete Works, was punished for the clerical errors in the manuscript of Mathematical Book (Zheng and Zhu 2010).Footnote 12 Chen was born into a family that had worked for the Imperial Bureau of Astronomy for several generations. Perhaps Chen was in charge of compiling the new edition of Mathematical Book and wrote the summary.

To retrieve Mathematical Book from Great Compendium is not a simple task. Given that Great Compendium quotes materials without the title or number of the chapters, the editors (who were unaware of the existence of Zhao Qimei’s manuscript) could only rely on Qin Jiushao’s preface, which gave a brief description of each chapter, and reorganized the 81 problems derived from Great Compendium into nine chapters. The sequence of the problems in the Siku edition is therefore different from Zhao Qimei’s manuscript.

The summary of Mathematical Book tells us that in the Siku edition ‘errors are corrected’ and ‘commentaries are added’.Footnote 13 Libbrecht (1973: 43) already remarked on this, saying: ‘The author of the descriptive notes tries to explain some of Ch’in’s [i.e., Qin Jiushao’s] methods, with the result that he proves himself incapable of understanding much of the text; and the greater part of his note is useless talk.’ What is more important is that in the Siku edition the text and the illustrations of the copy-text were always modified (Zhu 2023: 235–241). In fact, the editors reorganized the structure of the first problem of Mathematical Book so as to emphasize the Great inference procedure for all numbers, which was not Qin’s originally meaning. They also mistakenly intrepreted ji 奇 (odd number) and ou 偶 (even number), and thus misguided later scholars’ studies on the reduction of the modules in the Great inference procedure for all numbers. They further believed Qin’s counting-diagrams were usually wrong, hence they changed them and omitted lines within the counting diagrams. Red-colored signs are missing and lines in counting-diagrams are omitted or distorted at random. In a word, it was not an accurate transcription of Mathematical Book from Great Compendium. In fact, casual changes and arbitrary alterations were very common in Complete Works, because of the censorship (which purged any possible anti-Manchu words), as well as carelessness and oversight.Footnote 14

3 Scholarship and Textual Criticism in the Qian-Jia Period

The reigns of Emperors Qianlong and Jiaqing 嘉慶, from 1736–1820, often called the Qian-Jia period, are viewed as the golden age of ‘evidential scholarship’ (kaozhengxue 考證學) during the Qing dynasty.Footnote 15 The project of compiling Complete Works (1773–1781), which brought together rare books and brilliant scholars, was one factor that stimulated the development of scholarship. The late eighteenth and early nineteenth century also witnessed the flourishing of publishing. Classical works were often published in collections of writings. The powerful officials or wealthy gentry were willing to provide financial support to these endeavors and they invited scholars as editors. The principles with which the scholarly editions were carried out in the context of evidential scholarship demanded serious textual criticism.

In this period, the landmark of textual criticism would be the Critical Apparatus for Commentary and Sub-commentary on the Thirteen Classics (Shisanjing Zhushu Jiaokanji, 十三經注疏校勘記, 1808), in 217 sections. The Thirteen Classics are the core literature of Confucianism. They include Book of Changes (Yi Jing 易經), Book of Poetry (Shi Jing 詩經), Analects of Confucius (Lun Yu 論語) and other titles, all compiled before the second century BCE. The commentaries and subcommentaries were written by scholars from the second century CE to the eleventh century. Ruan Yuan (阮元, 1764–1849), a high-ranking official and an excellent scholar, initiated an ambitious project for preparing a new edition of the Commentary and Sub-commentary on the Thirteen Classics (1816) and its Critical Apparatus.Footnote 16 His editorial team included outstanding scholars like Gu Guangqi (顧廣圻, 1770–1839)Footnote 17 and the mathematician Li Rui (李鋭, 1769–1817).Footnote 18 Gu Guangqi was perhaps the most renowned textual critic of the Qing period. He had a famous saying: ‘editing without modification’ (bu jiao jiao zhi 不校校之). The editorial policy followed this dictum. As copy-text, Ruan Yuan tried his best to collect early printed editions which had often been printed in the thirteenth and fourteenth centuries. He republished them without modification of the texts. The Critical Apparatus often quotes more than ten printed and manuscript editions of each commentary and subcommentary, and lists and discusses the variants and readings. However, compared with this strict model, publishing editions of ancient works that were thoroughly revised editions was much more popular. Nevertheless, at that time, it was generally accepted that a reliable new edition of ancient books should be equipped with a critical apparatus.

Partly owing to the project of Complete Works in the Four Treasuries, in the second half of the eighteenth century, a series of ancient mathematical masterpieces were rediscovered and studied by competent scholars. Some critical editions were published subsequently. The leading evidential scholar, Dai Zhen (戴震, 1724–1777), collated Ten Mathematical Classics (Suanjing Shishu, 算經十書).Footnote 19 Dai’s new edition was used as the master copy in Complete Works in the Four Treasuries and it was finally published in Weiboxie Collectanea (微波榭叢書) in 1773. Moreover, the works of a talented mathematician, Li Ye (李冶, 1192–1279), were edited by Li Rui and published in Zhubuzuzhai Collectanea (知不足齋叢書) in 1778. In addition, Gu Guangqi mentioned in a preface (written ca. 1810) for Mathematical Book that a bibliophile, Qin Enfu (秦恩復, 1760–1843), provided his own copy for the republication of Qin Jiushao’s work. But this plan seems to have remained unfulfilled.

4 The First Printed Edition in 1842 of Mathematical Book in Nine Chapters

4.1 Yu Songnian and the Yijiatang Collectanea

At the beginning of the nineteenth century, manuscript duplicates of Qin Jiushao’s Mathematical Book, transcribed from Zhao Qimei’s copy and the Siku edition, circulated among a small community of evidential scholars interested in mathematics. Li Rui, Gu Guangqi, Qian Daxin (錢大昕, 1728–1804), Li Huang (李潢, 1746–1812), Jiao Xun (焦循, 1763–1820) and others acquired their own copies (Li 1987b). They studied the work and shared ideas privately. The publication of Mathematical Book in Yijiatang Collectanea at an auspicious juncture is no surprise.

Yijiatang (宜稼堂, literally ‘favorable farming hall’) was the study room of Yu Songnian (郁松年, 1799–1865), the second son of a very rich family who owned more than seventy cargo ships in Shanghai. Yu was a bibliomaniac rather than a businessman and used his wealth to become one of the most important bibliophiles in the nineteenth century (Matsuura 2012: 201–210).

Between 1840 and 1843, Yu Songnian published the Yijiatang Collectanea, which consisted of critical editions of seven works by six authors, all of whom had lived in the thirteenth and fourteenth centuries (during the Song dynasty and the Yuan dynasty). These included two historical works on the Three Kingdoms period (third century), two anthologies of prose and poetry, and three mathematical works, one by Qin Jiushao and two by Yang Hui (楊輝, fl. 1260). The master copies were rare manuscripts owned by Yu himself or borrowed from his friends. Yu published the revisions with independent volumes of Reading Notes (Zhaji 札記) as critical apparatus. He compiled Reading Notes of the historical works and anthologies by himself and invited Song Jingchang 宋景昌 to edit the three mathematical works. It is noteworthy that editions of the first four titles and of Qin’s Mathematical Book had already been copied into Complete Works. Yu might have felt proud that he could provide a reliable text, which was better than the Emperor’s Complete Works.

This was the period of the first Sino-British war (known as the Opium War, 1839–1842). At the end of 1842, Yu Songnian wrote as follows: ‘When I was preparing Reading Notes for publication, the English barbarians triggered a crisis. I was terribly worried that the manuscript would be destroyed. Then peace returned and I can complete the printing now.’ (Yu Songnian 1842: 1)Footnote 20 The Qing dynasty was defeated. The British army occupied Shanghai county town in June 1842. The year after the Treaty of Nanjing (August 1842), Shanghai became a treaty port officially open to the Western world. It was the beginning of a new era which would have a great impact on the intellectual world. In any case, Yu’s publishing project was suspended after 1843.Footnote 21

4.2 Song Jingchang and His Reading Notes

In March 1842, Yu Songnian wrote the preface for Reading Notes of Mathematical Book in Nine Chapters. In it, he wrote that Mao Yuesheng (毛嶽生, 1791–1841) suggested he should publish Mathematical Book because it contained marvelous mathematical methods unknown for hundreds of years and the extant copies had accumulated a great many errors and omissions due to continuous transcriptions. Mao Yuesheng provided two copies. One was borrowed from a famous scholar Li Zhaoluo (李兆洛, 1769–1841), a duplicate transcribed from Zhao Qimei’s manuscript. Another, owned by Mao himself, was a manuscript transcribed from the Siku edition with the critical notes of the mathematician Li Rui. Mao also recommended Song Jingchang, who specialized in mathematics, as the editor. Song acquired a posthumous manuscript from his teacher Shen Qinpei (沈欽裴, ? –before 1840). It was Shen’s reading notes of Mathematical Book based on Zhao Qimei’s manuscript (probably not the original, but a duplicate).

Song used the duplicate of Zhao Qimei’s manuscript as a copy-text and the duplicate of the Siku edition as a reference text. In Reading Notes, as Libbrecht (1973: 49) summarizes, Song ‘point[ed] out the textual errors’ and where he introduced modifications, he quoted the comments from the Siku edition as well as from his teacher Shen Qinpei, Li Rui and Mao Yuesheng: ‘For some of the problems, he gives a mathematical reconstruction or correction, but as a whole, this work is a textual collation’ (Libbrecht 1973: 49). Reading Notes is a very useful tool for the study of Mathematical Book.

At the beginning of Reading Notes, Song gives a brief description: ‘In the copy-text, every folio has 20 columns and each column has 20 characters.’Footnote 22 The format is the same as that of Zhao Qimei’s manuscript in the National Library of China. Comparing the readings, we discover that some textual errors which Song pointed out in the copy-text are actually correct in Zhao’s manuscript. This shows that Song Jingchang did not actually see Zhao Qimei’s manuscript. More importantly, Reading Notes does not mention the lines in the counting-diagrams, which are a special feature of Zhao’s manuscript. The 1842 edition of Mathematical Book contains only the counting-diagrams without the lines.

We find a small piece of paper with handwritten notes in Chinese ink inserted in Chap. 1 of Zhao Qimei’s manuscript. It reads, ‘It is not necessary to write down the lines [between the numbers]; but it is possible to only write down the signs on both sides.’Footnote 23 It seems to be an instruction for scribes, written in the eighteenth or nineteenth century. This note might have been written by someone who borrowed the manuscript from its owner and wanted to make a new copy for himself. It reminds us that some copies derived from Zhao’s manuscript were produced without lines. We know that Song Jingchang mainly referred to a copy transcribed from Zhao Qimei’s manuscript. It is possible that Song did not find any lines in this copy.

The first printed edition of Mathematical Book and its Reading Notes became the most influential edition. It was republished several times in the twentieth century. It also resulted in Shushu Jiuzhang replacing Shuxue Jiuzhang as the standard title of Mathematical Book. Nearly all the modern studies of Qin Jiushao’s work are based on the 1842 edition. This edition is quite different from Zhao Qimei’s manuscript. We can say that it loses some important information about Qiu Jiushao’s mathematical practice.

5 Divergences: Counting-Diagrams and Text

5.1 The Writing Systems in Mathematical Book in Nine Chapters

A significant feature of Mathematical Book is that Qin Jiushao uses different layers of writing systems to describe the same mathematical procedure. To be precise, in the problems (ti 題), Qin formulated questions (wen 問), which were followed by answers (da 答), procedures (shu 術), detailed solutions (cao 草) and counting-diagrams (suantu 筭圖), in that order (e.g., Fig. 13.1). According to Qin’s preface, the latter three are given in succession, and the counting-diagrams are used to explain the detailed solutions (Zhao Qimei’s manuscript 1616 [2010]: 98).Footnote 24 However, if we check the whole book, the relationships between them differ from what Qin states. Only 45 of all the 81 problems have counting-diagrams. Of these 45 problems, there are 23 in which procedures, detailed solutions and counting-diagrams are separate, while the remaining 22 problems have counting-diagrams presented within the text of the procedures or detailed solutions. This means the relationships are not uniform, and are more complex than Qin says. We have to examine them one by one in order to understand them.

Fig. 13.1
A photo of a typical counting diagram in a manuscript with some texts in a foreign language.

A typical counting-diagram for the rule of three in Zhao Qimei’s manuscript of Mathematical Book. Courtesy of the National Library of China

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Li Di (1991) primarily studied counting-diagrams and the writing system used in Mathematical Book. Karine Chemla (2001) studied the change of diagrams [tu 圖], especially geometrical diagrams, during the Song and Yuan periods. In the present chapter, we aim to show the divergences between different writing layers.

It is important to mention that, as a mathematical book in thirteenth century China, this book was written in Chinese characters except for the counting-diagrams (i.e., the procedures and detailed solutions are in Chinese, but the counting-diagrams include so-called rod-signs; see Table 13.1). Figure 13.1 contains the typical counting-diagrams copied from Zhao Qimei’s manuscript of Mathematical Book. It is clear that the counting-diagram has three writing systems: (1) written numeral systems used to represent numbers (Table 13.1), which derived from counting rods; moreover, colors were sometimes used for positive and negative numbers; (2) lines connecting numbers used to represent operations (Zheng and Zhu 2010; Zhu 2017, 2020); (3) Chinese characters used to explain procedures and give names and measuring units to numbers.

Table 13.1 Writing numeral system in Zhao Qimei’s manuscript of Mathematical Book
Full size table

The three writing systems are all employed in the diagrams. In the 45 problems that include such diagrams, the function of the diagrams depends on which of the three writing systems are used. It also depends on the relationship between diagrams, procedures and detailed solutions. This remark suggests that the functions of diagrams could also vary depending on the cases. Counting-diagrams are a characteristic part of Mathematical Book and tell us something that the procedures and detailed solutions do not reveal.

5.2 Three Representations: The Case of the Chinese Remainder Theorem

The case of the Chinese Remainder Theorem is discussed below to illustrate the use of counting-diagrams and the divergence between counting-diagrams and texts. As mentioned above, the procedure called the ‘dayan zongshu shu’ (大衍總數術, literally the ‘Great inference procedure for all numbers’) occupies the most important position in Mathematical Book and brought fame to Qin Jiushao. In fact, there are ten problems in Mathematical Book relating to the Chinese Remainder Theorem.Footnote 25 A sub-procedure of dayan zongshu shu, called dayan qiuyi shu (大衍求一術, literally the ‘Great inference procedure looking for one’), is at the heart of the Chinese Remainder Theorem. Qin Jiushao used different writing systems to represent this procedure. We will first begin by showing its various representations and analyze them for differences.

Representation A only uses textual description. Qin gives the procedure in Chinese characters in the answer to Problem 1Footnote 26 and a simplified description in the procedure for Problem 1.Footnote 27 These two text descriptions are slightly different, but the key point is that both are used for describing the procedure, which should actually be carried out with counting rods outside the text. In other words, thirteenth-century readers would resort to use of counting rods to understand the text.

Representation B uses counting-diagrams involved in detailed solutions. For instance, in the detailed solution of Problem 1, Qin presents, between the sentences of the text, the counting-diagrams representing the computations with counting rods referred to in the text (Fig. 13.2). By this method, counting-diagrams and text descriptions are written alternately. The whole diagram certainly refers to the same procedure carried out with counting rods, as the one written only in Chinese characters (Case A). However, readers have no need to use counting rods as in Case A, because a part of the procedure is depicted by counting-diagrams.

Fig. 13.2
A photo of a typical counting diagram in a manuscript with some texts in a foreign language.

Counting-diagrams involved in detailed solutions in Zhao Qimei’s manuscript of Mathematical Book (to be read from right to left). Courtesy of the National Library of China

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Representation C uses independent counting-diagrams. This can be illustrated using the other representation of the procedure that Qin gives in the problem called ‘making calendars and computing eras’ in Chap. 2 (Fig. 13.3).Footnote 28 Here, Chinese characters are only used to associate names with numbers. Lines within counting-diagrams are used to indicate an operation applied to the numbers that they connect (e.g., addition, subtraction, multiplication, division). They thus yield a dynamic inscription of the procedure. Therefore, although the counting-diagrams refer to the same use of counting rods as in Cases A and B, readers do not need to be able to operate counting rods to understand the diagrams.

Fig. 13.3
A photo of a Manuscript featuring an autonomous counting diagram with foreign language text, divided into 6 columns across 2 pages.

Independent counting-diagrams in Zhao Qimei’s manuscript of Mathematical Book. Courtesy of the National Library of China

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We can conclude that Qin Jiushao used different writing systems for writing down the so-called Chinese Remainder Theorem. The different writing systems comprise different representations of the same procedure with counting rods. In other words, the same procedure is described using different representations. This means that although there is only one procedure, different representations offer readers different ways to perceive the procedure.Footnote 29 In other words, if only text descriptions were given, readers’ understanding in the thirteenth century would have relied on the operation of counting rods. When counting-diagrams were involved in describing detailed solutions, readers were less dependent on counting rods. When independent counting-diagrams were given, they naturally became a replacement for counting rods for readers.

5.3 Differences Between Zhao Qimei’s Manuscript and the First Printed Edition of 1842

The characteristic of Zhao Qimei’s manuscript is that it preserves lines within counting-diagrams, which offers an opportunity to see a crucial original feature of Mathematical Book. At the same time, lines within counting-diagrams are missing in the first printed edition (1842). Despite the fact that they seem to have been present in the Yongle edition (1565 [1986]), the omission of lines also happened in the Siku edition. Figures 13.4 and 13.5 show the parts in the 1842 edition that correspond to those we have seen in Zhao Qimei’s manuscript in Figs. 13.2 and 13.3. Figure 13.4 is almost the same as Fig. 13.2, because lines are not involved in these diagrams. The absence of lines in Fig. 13.5 makes it look very different from Fig. 13.3.

Fig. 13.4
A photo of a manuscript featuring an autonomous counting diagram with foreign language text divided into multiple columns across 2 pages.

Counting-diagrams involved in detailed solutions in the 1842 edition corresponding to Fig. 13.2. Courtesy of the Institute for the History of Natural Sciences, Chinese Academy of Sciences

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Fig. 13.5
A photo of a manuscript featuring an independent counting diagram with foreign language text, divided into 12 columns across 2 pages.

Independent counting-diagrams in the 1842 edition corresponding to Fig. 13.3. Courtesy of the Institute for the History of Natural Sciences, Chinese Academy of Sciences

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As mentioned previously, Qin used different representations to write the same procedure. These representations actually show different aspects of a procedure. The independent counting-diagrams with lines tell us more about how to use counting rods than text descriptions. However, since lines were omitted in the first printed edition, counting-diagrams became static and lost the function of writing procedures. Thus, the differences between Zhao Qimei’s manuscript and the first printed edition lie not only in the omission of lines in the counting-diagrams, but also in the inclusion of representations by independent counting-diagrams, which helped readers to understand the operations of counting rods in the same procedure.

Li and Feng (1998) suggested that there were two systems used for written calculations in the Qing dynasty. One was based on the Western system introduced in the seventeenth century. The other one was the new rod-signs system derived from the thirteenth-century mathematical writings. A key piece of evidence in favor of the above argument is found in the first printed edition of Mathematical Book. However, the omission of lines in the first printed edition results in the loss of key features of the representation used in Mathematical Book to refer to the operation of counting rods, and it makes the counting diagrams useless and irrelevant asa representation of procedures. In China after the fifteenth century, counting rods were gradually completely replaced by the abacus and other tools for calculation. Therefore, it was difficult for readers of the first printed edition of Mathematical Book to understand the operational procedure of counting rods. Instead, in the nineteenth century, Mathematical Book would be understood in terms of the new mathematical writing system.

6 The Reception of Mathematical Book in the Nineteenth Century

In 1847, a British Protestant missionary, Alexander Wylie (1815–1887), arrived in Shanghai. He supervised the London Missionary Society’s press in the city and engaged in studying Chinese culture. In 1852, Wylie published a series of articles under the title Jottings on the Science of the Chinese. In one article, he translated a passage from Mathematical Book and pointed out that William George Horner’s newly published theorem (1819) ‘for solving Equations of all orders’ had been known to Qin Jiushao (‘Tsin Kew-chaou’).Footnote 30 In another work, Notes on Chinese Literature, a descriptive catalogue of Chinese works, Wylie (1867: 93–94) connected Qin’s ‘Great inference procedure’ to the ‘Hindoo Cuttaca’ as follows:

The first section (of Mathematical Book) contains a new formula for the resolution of indeterminate problems, called 大衍 Ta yen (ZYW & ZC: i.e. ‘Great inference procedure’), being analogous to the better known Hindoo process Cuttaca, which ColebrookeFootnote 31 translates ‘Pulverizer’.

He went on to describe the contents of Mathematical Book briefly, noting that ‘A critical examination and correction of the typographical and other errors in this was published in 1842, by 宋景昌 Sung King-ch’ang, with the title 數書九章札記 Soo shoo kew chang cha ke.’Footnote 32 Wylie’s own copy of the 1842 edition of Mathematical Book and its Reading Notes are now kept in the Weston Library of Oxford University (Helliwell 1985: 51).

Wylie was assisted in his research on Chinese mathematics by his friend and collaborator Li Shanlan (李善蘭, 1810–1882), perhaps the most important Chinese mathematician in the nineteenth century.Footnote 33 Li Shanlan belonged to the generation of scholars who were trained in their youth in the mixture of traditional Chinese mathematics and the old Western mathematics introduced in the seventeenth century. The 1842 edition of Mathematical Book and the special method ‘for solving Equations of all orders’ were also new to him. It is quite likely that Li introduced Mathematical Book to Wylie.

Qin’s reputation in mathematics rests mainly on two phenomenal achievements: the method for solving high-order equations, originally named ‘Procedure for extracting roots with positive and negative [numbers]’ (zhengfu kaifang shu 正負開方術), and the method for solving congruence equations, originally named ‘Great inference procedure for all numbers’ (dayan zongshu shu 大衍總數術). Mathematical Book interested Chinese and Western scholars in the nineteenth century not only because of its great achievements in mathematics but also because they could compare these two methods to analogous procedures in Western and Hindu mathematics. The first method was usually compared to Horner’s or Newton’s method, while the second one was given the name Chinese Remainder Theorem, and compared to the Hindu Cuttaca and later to Gauss’s method.Footnote 34

Qin’s mathematical practice was based on counting rods, and the counting-diagrams in Mathematical Book had guided thirteenth-century readers to follow the operation of counting rods representing the procedures. After the first printed edition of Mathematical Book was published in 1842 and became the main edition, the key representation of procedures using counting-diagrams and the operation of counting rods were lost. In a sense, the first printed edition misled readers and prevented them from understanding the mathematical procedures which Qin carried out with counting rods in Mathematical Book.

However, Chinese and Western scholars in the nineteenth century focused on interpreting ancient mathematics by means of new mathematics, and also compared Chinese mathematics to Western and Hindu mathematics. Although they did not fully understand Qin’s methods of writing mathematics, they successfully applied the new representation for the procedures in Mathematical Book, and evaluated the merits of Mathematical Book by comparison with modern Western mathematics.

7 Conclusion

Mathematical Book in Nine Chapters was written in the context of a practice of mathematics in which practitioners used counting rods, and Qin Jiushao transcribed the counting-diagrams as the new mathematical representation of the operations with counting rods (Zhu 2020). However, from the fifteenth century onward, the abacus gradually replaced counting rods in China. Scholars slowly lost the knowledge of how to use counting rods. In the early years of the seventeenth century, with the compilation of Mathematical Guidance in the Common Language (Tongwen suanzhi 同文算指, 1613) by Matteo Ricci and Li Zhizao (李之藻, 1565–1630), Western written calculations were introduced into China (Chemla 1996; Chemla 1997; Martzloff 2006: 375–376; Zhu 2018). In the nineteenth century, most Chinese scholars understood ancient mathematics in the context of using the abacus and written calculation. Therefore, the transformation from counting rods to the abacus and written calculations changed the context of reading mathematical texts in which counting rods were the main mathematical instrument. These facts had a strong impact not only on the comprehension of the counting-diagrams in Mathematical Book but also on the compilation of its critical edition in 1842.

In the same way as in the nineteenth century, Western and Chinese scholars who studied the history of mathematics in China in the twentieth century focused on applying the new representation with modern signs to traditional mathematics. Generally speaking, the reconstruction of ancient mathematical practices had not received enough emphasis. Although there are a few lines in the fragments of Yongle Dadian (1565 [1986]) and the Siku edition (1781 [1986]), and although Zhao Qimei’s manuscript has been preserved in the National Library of China for half a century, the dominance of the first printed edition of Mathematical Book encouraged the neglect of counting-diagrams. In other words, when scholars created the modern representation for Qin Jiushao’s Mathematical Book, they were shaping a new historical image, which has a strong connection with the first printed edition.