Abstract
This article starts with an observation. The chapters that compose the earliest extant mathematical book handed down through the written tradition in China, The Nine Chapters on Mathematical Procedures, and that are mentioned in the title of the book, are a key feature distinguishing this book from the manuscripts excavated from tombs sealed in China during, roughly speaking, the Qin and early Western Han dynasties. We argue that as early as the Eastern Han dynasty, the names of these chapters were related to an interpretation of an expression (‘the nine parts of mathematics’) occurring in a text essential for Confucian scholars, Zhou Rites. We thus suggest that these facts allow us to make sense of both the third century commentator Liu Hui’s account of the history of The Nine Chapters, and the difference between The Nine Chapters and the newly discovered manuscripts with respect to the textual parts composing them. We argue that in Liu Hui’s view, the fact that The Nine Chapters was organized into such chapters was a feature manifesting the adherence of this mathematical book to a set of texts to which the first Qin emperor had gone to great lengths to control the access. This hypothesis allows us to interpret Liu Hui’s assertion that the transmission of The Nine Chapters was badly affected by the first Qin emperor’s book burning policy, at the same time as it accounts for why the newly excavated documents are not divided into chapters of this kind. These conclusions suggest that the chapters, and their names, might have played a key role in The Nine Chapters, these textual parts constituting an essential feature distinguishing this Classic from other mathematical writings. In the second part of our argument, we discuss a mathematical interpretation of the meaning of these chapters and the chapterization.
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 ‘Mathematical Sciences in the Ancient World (SAW)’. The results presented in this article were obtained through a discussion between the two authors that started in the spring of 2012. The results expounded here were presented on 10 May 2012 within the framework of the SAW project’s ‘History of Science, History of Text’ seminar. We thank the audience for their questions, which helped us improve our argument, and, in particular, Florence Bretelle-Establet and Micheline Decorps, for their exacting reading. Professors Guo Shirong (郭世荣) and Dai Qin (代钦) provided help for the reference Li Di (2000). We are happy to extend our thanks to them.
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Notes
- 1.
Hereafter, we abbreviate the title as The Nine Chapters . For a critical edition and a French translation of The Nine Chapters, see Chemla and Guo Shuchun (2004).
- 2.
- 3.
The Duke of Zhou (Zhou Gong 周公, fl. mid-eleventh century BCE) played a key part in the establishment of the Zhou dynasty in the eleventh century BCE. Liu Hui refers here to the classic The Offices of the Zhou [Dynasty] (Zhou guan 周官), a book whose authorship has been traditionally credited to the Duke of Zhou, but whose date of completion is unclear. The title of the latter book was changed into Zhou Rites (Zhou li 周禮) while Wang Mang 王莽 (43 BCE–23 CE) acted as regent (his regency lasted between 1 CE and 9 CE; in 9 CE, he overturned the Western Han Dynasty [206 BCE–9 CE] during the time of an interregnum that lasted between 9 CE and 23 CE; the Han dynasty was then reestablished as the Eastern Han dynasty [25–220 CE]). Wang Mang greatly emphasized the importance of Zhou Rites, which then belonged to the set of ritual texts associated with Confucian Canons. Modern scholars published different opinions on the date and author/authors of Zhou Rites. Most scholars now believe it was completed before the establishment of the Qin Empire in 221 BCE (Liu Qiyu 1991; Boltz 1993). Kern (2010a) examines the functions and competences of scribes to which Zhou Rites attests, and shows that parallels with excavated artifacts and other early texts suggest parts of the text do refer to Western Zhou administrative organization. His main emphasis is on literacy. Here, we suggest that a similar argument could be made for numeracy. For greater detail on ritual texts, see Nylan (2001), Chapter 4, ‘The Three Rites Canons’, and Boltz (1993). On Liu Hui’s reference to this text, see Chemla, Footnote 13, in Chemla and Guo Shuchun (2004: 752), and see below. Here, Liu Hui refers more precisely to the mention of ‘nine parts of mathematics’ (jiu shu 九數) in Zhou Rites. To begin with, we translate the expression jiu shu loosely and we return below to our interpretation of its meaning.
- 4.
Liu Hui refers here to China’s first emperor, Qin Shihuang (秦始皇, 259–210 BCE). Qin Shihuang, who reigned between 247 and 210 BCE, was the first emperor to unify the Chinese Empire, in 221 BCE, under the rule of the Qin dynasty (221–206 BCE). In 213 BCE, Qin Shihuang ordered that certain types of books be burnt. We discuss the edict and some of its possible effects below.
- 5.
Zhang Cang (?252–152 BCE) was a civil servant who held quite important positions in the administration of the Han Empire from the beginning of the dynasty onwards. He is said to have lived more than a hundred years, which explains the year 252 and the fact that it is only approximate. See Guo Shuchun, Chap. B, in Chemla and Guo Shuchun (2004: 54–55).
- 6.
Like Zhang Cang but later (in the mid-first century BCE), Geng Shouchang was an official who contributed to the administration of finance, the management of public works and to settling astronomical questions. See Guo Shuchun, Chap. B, in Chemla and Guo Shuchun (2004: 54–55). Related documents are found in Ban Gu (1964, vol. XXIV: 1141; vol. LXX: 3023–3024).
- 7.
Liu Hui uses here another term referring to a section of text. We shall return to it in another publication. In this chapter, we concentrate on the division of the book into ‘nine chapters’.
- 8.
On this episode, and various interpretations to which it lent itself; see Petersen (1995), Neininger (1983), Nylan (2001: 29–30) and Kern (2010b: 111–112). In this chapter, we try to make sense of how Liu Hui ties this episode with the history of The Nine Chapters , and we attempt to assess the reliability of his testimony.
- 9.
On the editorial act as a writing act and Liu Hui’s assessment of its result, see Chemla (2008a).
- 10.
- 11.
The ‘Guardian’ was an official (or the administrative department for which this official was responsible) that dealt with the management of national education and guarding the rulers against wrongdoing while carrying out rites.
- 12.
- 13.
We do not know the exact meaning of pang yao. It probably means surveying by indirect methods. Thus we temporarily translate it as ‘indirectly surveying’. For various explanations about this term, see Zou Dahai (2001a: 157–161).
- 14.
(Sun Yirang 1987: 1010). ‘Base and height’ (gou gu 勾股) refers more precisely to the base and height of a right-angled triangle. In this article, we use this abbreviation.
- 15.
The term guwen literally means ‘ancient script’. For scholars of the Han Dynasty, this expression designated styles of characters used before the unification of the Qin Dynasty and thus quite different and ancient. Guwen was used as an opposite to jinwen (今文), literally ‘modern script’, the current style of characters in Han times. The two expressions were also used to designate two corresponding schools in Han times. Recent scholarship on the history and the historiography of this opposition includes Nylan (1994) and Van Ess (1994).
- 16.
The King of Hejian’s name was Liu De (劉德, ?–129 BCE) and he was the third son of Emperor Jing (景帝). His father conferred this title to him in 155 BCE.
- 17.
Qian Baocong (1964: 31–32) thought the ‘nine parts of mathematics’ mentioned in Zheng Zhong’s gloss represented the branches of mathematics at the end of the Western Han dynasty. In our view, he diverges from Zheng Zhong’s own stand (Zou Dahai 2008). The point is important for our argument, since it supports our view on the date of the formation of the ‘nine parts of mathematics’.
- 18.
‘ci’ (差, rank) and ‘ying’ (贏, excess) were, respectively, changed into ‘cui’ (衰) and ‘ying’ (盈).
- 19.
In addition to the statement about Zhou Rites quoted above, Liu Hui’s preface to his commentary on The Nine Chapters further claims the following about Zhou Rites: ‘Furthermore, mathematics belongs to the Six Arts (liu yi) and in the past, [the government] selected virtuous and talented persons, who were treated as guests, to teach and train the scions of noble families [in these topics].’ (且算在六藝, 古者以賓興賢能, 教習國子) (Chemla and Guo 2004: 126).
- 20.
Li Yan (李儼, 1892–1963) accepted Liu Hui’s statement that before the establishment of the Empire, an old version of The Nine Chapters existed. He further accepted the view that Zhang Cang and Geng Shouchang edited documents to produce The Nine Chapters of the Han dynasty. This was, for him, the book bearing this title that many scholars studied. They included Xu Shang (許商, fl. latter half of the first century BCE), Du Zhong (杜忠, fl. latter half of the first century BCE), Liu Xin, Ma Xu (馬續, fl. latter half of the first century CE to first half of the second century CE), Zheng Xuan, Kan Ze (闞澤, fl. of the third century CE), Liu Hui, etc. Li Yan thought that The Nine Chapters was fixed only after Liu Hui wrote his commentary (Li Yan 1955:11–12, 16–17; Li Yan 1958: 24). Qian Baocong (錢寶琮, 1892–1974) did not accept the validity of Liu Hui’s testimony with respect to the fact that The Nine Chapters , as we see it, was completed by Geng Shouchang in the first century BCE. He was nevertheless convinced that undoubtedly most of Chapters 1–5 dated from before the Qin Dynasty. He further considered as reliable Liu Hui’s statement that The Nine Chapters was primarily based on the ‘nine parts of mathematics’. However, unlike Liu Hui, he dated the ‘nine parts of mathematics’ and The Nine Chapters , respectively, from the end of Western Han Dynasty and the early Eastern Han (Qian 1964: 14, 28–33). Similarly, Guo Shuchun(郭書春) accepted Liu Hui’s interpretation that The Nine Chapters derived from the ‘nine parts of mathematics’. Furthermore, like Liu Hui, he dated the two, respectively, from the Western Han Dynasty and the pre-Qin period (Guo Shuchun 1992: 98–102; 1997). Zou Dahai holds opinions similar to Li Yan’s and Guo Shuchun’s, adding more detailed arguments (Zou Dahai 2001a: 451,497–498; 2001b; 2008).
- 21.
The title means literally ‘Writing on calculations of numbers/quantities’ or ‘Writing on procedures with counting rods’. However, we believe that its actual meaning refers to mathematics, through the fact that mathematics is conceived of as being constituted of mathematical procedures.
- 22.
The title of the book means literally ‘Numbers/Quantities’ or ‘Procedures’; however, we believe that in the title, this term has a wider meaning, which is also attested to.
- 23.
Chen Songchang (2009) doubted whether the set of strips was unearthed from the same tomb. However, the preface of Zhu Hanmin and Chen Songchang (2011) asserts Mathematics was completed no later than the thirty-fifth year of the First Emperor of Qin (212 BCE). This seems to indicate that the collators of Mathematics thought at least these strips were unearthed from the same tomb as the strip recording the thirty-fifth year of the First Emperor. On 6 April 2013, one of two authors of this chapter, Zou Dahai, had an electronic exchange with Xiao Can, the main collator of Mathematics. Xiao and Zou noticed that selling strips from different tombs as one set would earn less money for the antiquarian than selling them separately. It is thus probable that the set of strips was unearthed from the same tomb. This question is still open to further discussion.
- 24.
This book was written not later than 165 BCE (Hu Pingsheng 1998).
- 25.
Indeed, it is highly probable that in ancient China, notions of positive and negative numbers were first introduced in relation to the operations of ‘Measures in square’. ‘Measures in square’ should thus occur before the notions of positive and negative numbers. Writings on Mathematics records a legal standard (cheng 程) for examining effects of doctors’ treatments. In it, concepts of positive and negative numbers similar to those in the chapter ‘Measures in square’ occur. Such a legal standard is probably a quotation from the laws enforced by the Qin State during the Warring States period or perhaps —even though this is less probable—during the Qin Dynasty. Thus Writings on Mathematics provides indirect evidence that ‘Measures in square’ of The Nine Chapters had its origin in pre-Qin times (Zou Dahai 2010b, c).
- 26.
Note that here like below, what we call ‘sections’ derives from the material organization of the manuscript. But the text has no term for it.
- 27.
- 28.
Before Writings on Mathematics was unearthed, many scholars (for example, Qian Baocong 1964: 28–33) expressed distrust with respect to Liu Hui’s statement on the history of The Nine Chapters . After the book was unearthed, scholars like Li Di (1997: 89–95; 2000), Peng Hao (2001b, 2005) and Cullen (2007) continued to hold such views.
- 29.
Erudites (boshi 博士) were officials of the central government, each of whom mastered the teachings of one classic and was expected to answer the Emperor’s questions.
- 30.
The Book of Odes (Shi 詩), sometimes called the Classic of Poetry (Shi jing 詩經), is a collection of poetry from the Western Zhou Dynasty (from the middle of eleventh century to 771 BCE) to the mid-Spring and Autumn Period (770–476 BCE). It is said to have been compiled before Confucius. The Book of Documents (Shu 書), sometimes called the Classic of Documents (Shu jing 書經), is a collection of historical documents ascribed to the Shang (商, ca. sixteenth century BCE to mid-eleventh century BCE) and Zhou dynasties. A few passages deal with the Xia Dynasty (夏, ca twenty-first to seventeenth centuries BCE). Documents claim that Confucius studied these books and revised them (Sima Qian 1963: 3115; Liu Shipei 2006: 3–4; Meng Wentong 1933: 1–3). The ‘sayings of the hundred philosophical schools’ refer to the writings produced in the context of the struggles between many thinkers during the Warring States (475 BCE–221 BCE). Petersen (1995) discusses the meaning of this expression. For lack of space, we do not enter into these details here.
- 31.
- 32.
(Ban Gu 1964: 90). From early on, some scholars have expressed doubts with respect to whether this policy was actually enforced, and how it was applied. We do not want to enter this debate here. First, our aim is to interpret Liu Hui’s preface , in particular in relation to the goal of our chapter, namely, the discussion of textual parts in mathematical writings. Second, in the conclusion to this section, we return to this debate, only to suggest that the evidence of mathematical writings could be used in discussing this issue.
- 33.
Li Di (1997: 89–106) thought many bamboo strips on mathematics were preserved in government offices and were not destroyed during the book prohibition. Such documents (which he called ‘guan jian’ [官簡], official bamboo strips) were gradually assembled and Liu Xin (劉歆) relied on them to eventually compose The Nine Chapters in the early years of the first century CE. Cullen (2007) holds similar views, since he thinks the discovery of Writings on Mathematics cast further doubt on Liu Hui’s viewpoint.
- 34.
For instance, Zhuang Zi (莊子, Master Zhuang), whose title refers to Zhuang Zhou (莊周, ca. 369 – ca. 286 BCE), is a collection of texts by Zhuang Zhou and his followers in the pre-Qin period (Liu Xiaogan 1988: 1–101). In the chapter ‘Movement of the Heaven’ (Tian yun 天運), which is one of the external ones (wai pian) of the Zhuang Zi, the expression 六經 (Six Classics) is used to refer to these six books (Guo Qingfan 1985: 331–332. See an English translation in Watson 1968: 165–166). Similarly, the philosopher Xun Zi (荀子) of the third century BCE also uses the term ‘經’ (classic) in relation to these books (Nylan 2001: 10–11).
- 35.
See, for example, Strips 13–14 of ‘liu de’ (六德 Six virtues); Strips 36–44 of ‘Yu cong yi’ (語叢 – Collection of Aphorisms. Part 1)’ (Jingzhou shi bowuguan 2005: 188, 194–195).
- 36.
Other pieces of evidence document the relationship between Confucian Classics and ‘procedures’. For instance, see ‘The world’ (tianxia 天下) chapter, in Zhuangzi (Guo Qingfan 1985: 1067, and the English translation in Watson 1968: 362–363). Another piece of evidence on the importance of mathematical procedures in relation to the study of Confucian Classics is given by the fact that Zhen Luan (甄鸞, fl. middle of the sixth century CE) compiled a book Mathematical Procedures on Five Classics (Wujing Suanshu 五經算術) , as a commentary on parts of classical texts.
- 37.
Liu Hui’s preface , and excavated documents, evidence several writings that might have belonged to this tradition. To begin with, the original ‘nine parts of mathematics’; then what Liu Hui refers to as the ‘nine parts of mathematics’ in development, which led to the composition of a book in nine chapters before the Qin dynasty; then the various stages that this book underwent (we return to this point below); the damaged parts of this book in nine chapters used by Zhang Cang and later Geng Shouchang ; the ‘Nine Chapters’ they restored and finally, The Nine Chapters handed down to us. A preliminary treatment of this issue was given in Zou Dahai (2004, 2008). Strictly speaking, Liu Hui does not refer to the various stages of the pre-Qin book in nine chapters. However, such a process is evidenced in excavated documents, a typical example being the Lao zi, which was found in two tombs. The copies unearthed from the Han tomb at Mawangdui (馬王堆, Hunan Province) are different from the earlier copies unearthed from the Chu tomb of the Warring States period at Guodian (Hubei Province). Further, the two copies from Tomb 3 at Mawangdui also present differences from each other. In addition, all the unearthed copies are different from the copies handed down to us. Peng Hao (2001a) compared six important editions of Lao zi and discussed their differences.
- 38.
Bao Huanzhi signed his postface to The Nine Chapters in 1200. The printing was carried out in the context of an edition of the Ten Canons of Mathematics, which only partly survived (Qian 1963). Bao Huanzhi seems to have composed a postface for each of the books included. Among the extant postfaces, the latest date recorded is 1213. The remaining portions of this edition were reproduced in Shanghai Tushuguan and Beijing Daxue Tushuguan (1980). In Fig. 5.1, we give a partial translation of the document, using lowercase letters and small capitals to mark the difference between the commentaries and the canon. This convention will be used throughout our chapter.
- 39.
The readers are reminded of the fact that at the time Chinese books were read from right to left and top to bottom. This feature is essential to understand our description of the ancient editions in this chapter. To be more precise, in this edition, the title of the classic has been modified into Mathematical Classic with the Nine Chapters .
- 40.
The only empty columns that occur in the edition are at the end of chapters, since each chapter starts on a new page.
- 41.
On the history of the extant editions of The Nine Chapters , see Guo Shuchun, Chap. C, in Chemla and Guo Shuchun (2004: 71–78).
- 42.
It is reproduced in Guo Shuchun (1993, Vol. I: 95-213). Unless otherwise specified, the other editions belonging to the same family have the same characteristics as those we mention.
- 43.
A nuance must be added to this assertion. As can be seen in Fig. 5.3, in the Wuyingdian juzhen ban edition, the commentaries are not featured in smaller characters and in double columns, between the sentences of the canon. They are, by contrast, laid out in separate columns, starting with a new column and having throughout the commentary an upper margin longer than that for the main text. When a commentary on the name of an operation is inserted between the name and the character ‘procedure,’ which signals the beginning of the text of the procedure, the name occurs alone in a column. By contrast, in that case, in the Song edition, the commentary is placed in the same column as the name of the operation (see Fig. 5.1).
- 44.
Incidentally, the same holds true for the headings of sections in Writings on Mathematical Procedures.
- 45.
We shall see below that Dai Zhen and the nineteenth century editor of another edition of The Nine Chapters , Song Jingchang, apparently both thought in that way.
- 46.
The edition of the various layers of earlier texts Yang Hui included in his Detailed Explanations does not depend on Bao Huanzhi’s printing. Guo Shuchun, Chap. C in Chemla and Guo (2004: 73–74), argues that they share a common prototype. Note that Yang Hui must have known Bao Huanzhi’s reprint, since his subcommentary included the latter’s postface to The Nine Chapters mentioned above. On the chapters extant in this edition, see Guo, Chap. B in Chemla and Guo (2004: 68).
- 47.
The commentaries bear, for the most part, on the procedures.
- 48.
See Guo Shuchun, Footnote 2, in Chemla and Guo 2004: 616). To understand this feature better, it is useful to recall that Dai Zhen had compiled the document that served as the basis for his edition of The Nine Chapters from the edition included in the Grand Classic of the Yongle Period. In this encyclopedia, as far as we can see from the remaining chapters on mathematics, mathematical books had been dissected into parts corresponding to given operations, and the various parts corresponding to the same operation were brought together. So, under a given name of operation, for instance ‘Reducing the width’, a sequence of excerpts about that operation coming from various books had been compiled, each beginning with the title of the book quoted. In that case, under the title of the section ‘Reducing the width’ comes directly the title of the book The Nine Chapters . This title is immediately followed by Liu Hui’s commentary on the title of Chapter 4, ‘Reducing the width’ and then Li Chunfeng’s gloss on the meaning of the expression ‘Reducing the width’, without repeating the expression ‘Reducing the width’ either as the title of Chapter 4 or as the name of the operation (see the reproduction of the remaining mathematical chapters of the Grand Classic in Guo Shuchun (1993, Vol. I: 1414). It is thus highly probable that when recompiling The Nine Chapters from the Grand Classic, Dai Zhen did not understand how the names of operations were used in The Nine Chapters . His 1776 and 1777 editions testify to the fact that in between, he had consulted the 1213 edition. Apparently, this document made him rethink the part played by names of operations in the original book, as is made manifest by the changes he carried out in this respect for the last four chapters as well as for Chapters 2 and 3 in his later editions.
- 49.
- 50.
We do not comment on this problem here. See Guo Shuchun, footnotes to the opening section, in Chemla and Guo (2004: 558–564).
- 51.
In Chinese, ‘Unhusked and husked grains’ are the first two characters in the name of the table.
- 52.
As mentioned above, the case of Chapter 7 is unclear because of the divergence between the various ancient editions in this respect.
- 53.
- 54.
On this issue, see Chemla, introduction to Chapter 6, in Chemla and Guo (2004: 459–474).
- 55.
For a bibliography on these technical terms and first explanations, see Chemla, Glossary, in Chemla and Guo (2004: 910, 933–934).
- 56.
These are measurement units for surfaces.
- 57.
The exegete Li Ji, whose dates are unclear (Tang or Northern Song Dynasty) and who composed, as for the other mathematical classics, Meaning and Pronunciation on The Nine Chapters on Mathematical Procedures (Jiu zhang suanshu yinyi 九章算術音義), links in the same way the ‘nine parts of mathematics’ and the ‘nine pian’, when he writes: ‘The “nine parts of mathematics” … that is, The Nine Chapters . When one explains in terms of procedures of computation [suan], one says “the nine parts of mathematics” and when one explains in terms of in terms of material chapters (pian 篇), one says “the nine chapters .’’ (九 數(…), 即九章是也以算言之,故曰九數;以篇言之,故曰九章)’ (Guo 1990: 457). We return below to this statement.
- 58.
Greater detail and references are given in Chemla, notes to the translation, in Chemla and Guo (2004: 786, Endnotes 13 and 14).
- 59.
Here, we interpret lü as values attached to right-angle triangles mentioned in the problems that follow, and the algorithm attached to them. In this context, the concept of lü designates a set of numbers whose ratios are the same as those between the lengths of the sides of an actual triangle, or between quantities deriving from them. We return below to lü. One can also interpret the statement as follows: “(Base, height, hypotenuse and the procedure linking these terms, that is, a procedural form of the so-called “Pythagorean theorem”) are about to be extended by means of all lü.” In the latter case, the emphasis is placed on the dimensions of the right-angled triangle by means of which the procedure is to be applied, whereas in the former case, it is on those of the triangle to which the procedure is to be applied.
- 60.
On the concept of ‘source/origin ’ and its relation to ‘extension’, and also to the proofs commentators carry out, see Chemla (2008b).
- 61.
The meaning of shu 數 as ‘procedure’ is attested. See the evidence provided in Chemla, Glossary, in Chemla and Guo (2004: 984–986). Volkov (2014: 56, Footnote 2) suggests a similar interpretation of shu as ‘operations with numbers’ or ‘numerical procedures’ and evokes other interpretations (Volkov 2014: 57).
- 62.
- 63.
Liu Hui’s preface also makes clear that he establishes a relationship between finding the source and ‘extending,’ when he explains how he suggests dealing with a problem left unsolved in the version he reads of The Nine Chapters . The third-century commentator on the Han classic The Gnomon of the Zhou , Zhao Shuang , also links placing something at the beginning of a section and stating that it ‘extends’ to what follows. See notes by Chemla, in (Chemla and Guo 2004: 878-879).
- 64.
Here, by derivation, we do not mean ‘historical origin ’, but the possibility of establishing a link of derivation between the ‘fundamental procedure’ and those placed in the same chapter.
- 65.
Tang commentators on Zhou Rites seem to believe that this list was corrupted. See below the list they attribute to Zheng Zhong, as distinct from the list they attribute to Ma Rong (馬融, 79–166) . We already met Ma Rong’s brother, Ma Xu, as someone who had studied The Nine Chapters (see Footnote 20). Ma Rong was a commentator on Confucian Classics, including Zhou Rites. Zheng Xuan counted among his students.
- 66.
- 67.
In this conception, ‘Reiterated differences’ does not play any part. Liu Hui’s preface , and the statement attributed to Zheng Zhong bring the latter ‘fundamental procedure’ into focus.
- 68.
- 69.
We thus see that Jia interprets the items in the list as ‘procedures,’ referring to them as ‘fa’ (method).
- 70.
Again, this remark confirms that the items in the list are mathematical procedures. Moreover, note that, in fact, Jia Gongyan seems to believe that the list of three “modern times” additions should in fact be two lists: one is attributed to Zheng Zhong and another one to Ma Rong , whereas Zheng Xuan seems to have given a compilation of the two. This interpretation further confirms our suggestion that there were competing ideas with respect to the jiu shu. The same idea is indirectly acknowledged by Kong Yingda , in a commentary on another Classic about rituals (Liji); see the quotation in Sun Wenqing (1931: 37–38).
Bibliography
Primary Sources
Ban Gu 班固. 1964. Han shu 漢書 (History of the [former] Han dynasty). Beijing: Zhonghua shuju 中華書局.
Chemla, Karine, and Guo Shuchun. 2004. Les neuf chapitres. Le classique mathématique de la Chine ancienne et ses commentaires. Paris: Dunod.
Fan Ye 范曄. 1973. Hou Han shu 後漢書 (Records on the Later Han dynasty). Beijing: Zhonghua shuju 中華書局.
Guo Qingfan 郭慶藩. ed. 1985. Zhuang Zi Jishi 莊子集釋 (Collected explanations of Master Zhuang). Beijing 北京: Zhonghua Shuju 中華書局.
Guo Shuchun 郭書春. ed. 1990. Hui jiao Jiu zhang suan shu 匯校九章算術. Shenyang 瀋陽: Liaoning jiaoyu chubanshe 遼寧教育出版社.
Guo Shuchun 郭書春. ed. 1993. Zhongguo kexue jishu dianji tonghui. Shuxue juan 中國科學技術典籍通彙.數學卷. Zhengzhou: Henan jiaoyu chubanshe.
Hubei sheng Jingzhou shi Zhouliangyuqiao yizhi bowuguan 湖北省荆州市周梁玉橋遺址博物館. ed. 2001. Guanju Qin Han mu jiandu 關沮秦漢墓簡牘 (Bamboo and wooden strips from Qin and Han tombs at Guanju). Beijing: Zhonghua Shuju 中華書局.
Jia Gongyan 賈公彦. 1987. Xu Zhou li feixing 序周禮廢興 (Preface on the rise and decline of Zhou Rites). In Zhou li zhengyi 周禮正義 (Correct meaning of Zhou Rites). In Shi san jing zhushu 十三經注疏 (Commentaries and explanations on the Thirteen Classics), ed. Ruan Yuan 阮元, 635–636. Beijing: Zhonghua Shuju 中華書局.
Jingzhou shi bowuguan 荆州市博物館. ed. 2005. Guodian chu mu zhujian 郭店楚墓竹簡 (Bamboo strips from a Chu tomb at Guodian). Beijing: Wenwu Chubanshe 文物出版社.
Qian Baocong 錢寶琮. ed. 1963. Suanjing shishu 算經十書 (Qian Baocong jiaodian 錢寶琮校點) (Critical punctuated edition of The Ten Classics of Mathematics). (2 vols.). Beijing 北京: Zhonghua shuju 中華書局.
Ruan Yuan 阮元, ed. 1980. Shi san jing zhushu 十三經注疏 (Commentaries and explanations on The Thirteen Classics). Reprint. Beijing: Zhonghua shuju.
Shanghai tushuguan, and Beijing daxue tushuguan. 1980. Songke suanjing liu zhong 宋刻算經六種 (Six mathematical Canons printed during the Song). Beijing: Wenwu chubanshe.
Sima Qian 司馬遷. 1963. Shi ji 史記 (Historical Records). Beijing 北京: Zhonghua shuju 中華書局.
Sun Yirang 孫詒讓. ed. 1987. Zhou li zhengyi 周禮正義 (Correct meanings of Zhou Rites. Edition and commentary). Beijing: Zhonghua shuju 中華書局.
Yang Jialuo. 楊家駱主編. ed. 1978. Zhongguo tianwen lifa shiliao 中國天文曆法史料 (Historical Material on Chinese Astronomy and Calendar). Taipei: Dingwen shuju 鼎文書局.
Zhangjiashan ersiqi hao hanmu zhujian zhengli xiaozu 張家山二四七號漢墓竹簡整理小組 (Group collating the bamboo strips from tomb no. 247 at Zhangjiashan). eds. 2001. Zhangjiashan hanmu zhujian [ersiqi hao mu] 張家山漢墓竹簡[二四七號墓] (Bamboo strips from the tomb of Han Dynasty excavated from Zhangjiashan (tomb no. 247)). Beijing 北京: Wenwu chubanshe 文物出版社.
Zhu Hanmin 朱漢民 and Chen Songchang 陳松長. eds. 2011. Yuelu shuyuan cang Qinjian (er) 嶽麓書院藏秦簡 (貳) (Bamboo or wooden strips of Qin Dynasty preserved at Yuelu Academy, Vol. II). Shanghai: Shanghai cishu chubanshe 上海辭書出版社.
Secondary Literature
Bielenstein, Hans. 1980. The Bureaucracy of Han times. Cambridge, UK: Cambridge University Press.
Boltz, William. 1993. Chou li 周禮. In Early Chinese texts: a bibliographical guide, ed. Michael Loewe, 24–32. Berkeley, Calif.: Society for the Study of Early China: Institute of East Asian Studies University of California Berkeley.
Chemla, Karine. 1992. Résonances entre démonstration et procédure: Remarques sur le commentaire de Liu Hui (IIIe siècle) aux Neuf chapitres sur les procédures mathématiques (1er siècle). In Regards obliques sur l’argumentation en Chine. Extrême-Orient, Extrême-Occident. Vol. XIV, ed. Karine Chemla, 91–129. Saint-Denis: Presses Universitaires de Vincennes.
Chemla, Karine. 2008a. Antiquity in the shape of a canon. Views on antiquity from the outlook of mathematics. In Perceptions of Antiquity in Chinese Civilization, eds. Dieter Kuhn and Helga Stahl, 191–208. Heidelberg: Edition Forum.
Chemla, Karine. 2008b. Classic and commentary: An outlook based on mathematical sources, Preprint 344/ Max-Planck-Institut für Wissenschaftsgeschicht. Berlin: Max-Planck-Institut für Wissenschaftsgeschichte.
Chemla, Karine, and Guo Shuchun. 2004. Les neuf chapitres. Le Classique mathématique de la Chine ancienne et ses commentaires. Paris: Dunod.
Chen Songchang 陳松長. 2009. Yuelu shuyuan suo cang qinjian zongshu 嶽麓書院所藏秦簡綜述 (Summary on Qin bamboo or wooden strips preserved in Yuelu Academy). Wenwu 文物 (Cultural Relics) 2009(3): 76–88.
Chen Zhenghong 陳正宏 and Tan Beifang 談蓓芳. 2004. Zhongguo jinshu jianshi 中國禁書簡史. Shanghai: Xuelin chubanshe 學林出版社.
Cullen, Christopher. 2004. The Suàn shù shū 筭數書 ‘Writings on reckoning’: A translation of a Chinese mathematical collection of the second century BC, with explanatory commentary. Needham Research Institute Working Papers 1. Cambridge (UK): Needham Research Institute.
Cullen, Christopher. 2007. The Suàn shù shū 筭數書, ‘Writings on reckoning’: Rewriting the history of early Chinese mathematics in the light of an excavated manuscript. Historia Mathematica 34: 10–44.
Fu, Zhengyuan. 1996. China’s legalists. The earliest totalitarians and their art of ruling. Armonk: M. E. Sharpe.
Guo Shuchun 郭書春. 1984. Jiu zhang suanshu he Liu Hui zhu zhong zhi lü gainian ji qi yingyong shixi《九章算術》和劉徽注中之率概念及其應用試析 (Analysis of the concept of lü and its uses in The Nine Chapters on Mathematical Procedures and Liu Hui’s commentary). Kejishi Jikan 科技史集刊 (Journal for the History of Science and Technology) 11: 21–36.
Guo Shuchun 郭書春. 1992. Gudai shijie shuxue taidou Liu Hui 古代世界數學泰斗劉徽 (Liu Hui, a great master of mathematics in the ancient world). Ji’nan濟南: Shandong kexue jishu chubanshe 山東科學技術出版社.
Guo Shuchun 郭書春. 1997. Zhang Cang yu Jiuzhang Suanshu 張蒼與《九章算術》(Zhang Cang and The Nine Chapters on Mathematical Procedures). In Ke shi xinchuan: Qingzhu Du Shiran xiansheng congshi kexueshi yanjiu 40 zhounian xueshu lunwenji 科史薪傳:庆祝杜石然先生從事科學史硏究40周年學術論文集 (Transmission of the flame in the history of science. Collection of articles to celebrate the anniversary of Professor Du Shiran’s Forty years of research in the history of science), eds. Liu Dun 劉鈍, Han Qi 韓琦 et al., 112–121. Shenyang瀋陽: Liaoning jiaoyu chubanshe 遼宁教育出版社.
Hu Pingsheng 胡平生. 1998. Fuyang shuanggudui Han jian shushu shu jianlun 阜陽雙古堆漢簡數術書簡論 (Simple discussion on Han Dynasty books on numbers and procedures written on bamboo strips and unearthed at Fuyang). In Chutu wenxian yanjiu 出土文献研究 (Studies on Unearthed Documents), ed. Zhongguo wenhua yichan yanjiuyuan 中國文化遺產研究院 4, 12–30. Beijing 北京: Zhonghua shuju 中華書局.
Hucker, Charles O. 1985. A dictionary of official titles in Imperial China. Stanford, Calif.: Stanford University Press.
Kern, Martin. 2010a. Early Chinese literature. Beginnings through Western Han. In The Cambridge History of Chinese Literature. Vol. I, eds. Kang-i Sun Chang and Stephen Owen, 1–115. Cambridge (UK): Cambridge University Press.
Kern, Martin. 2010b. Offices of writing and reading in the Rituals of Zhou. In Statecraft and classical learning: The rituals of Zhou in East Asian history, eds. Benjamin Elman and Martin Kern, 65–93. Leiden & Boston: Brill.
Li Di 李迪. 1997. Zhongguo shuxue tongshi. shanggu dao wudai juan 中國數學通史. 上古到五代卷 (General history of mathematics in China. Vol. I: From early ancient times to the Five Dynasties). Nanjing 南京: Jiangsu jiaoyu chubanshe 江蘇教育出版社.
Li Di 李迪. 2000. Guanyu zhujian ‘Suanshu shu’ de ruogan wenti 關於竹簡“算數書”的若干問題 (Several questions on the bamboo strip manuscript Writings on Mathematics on bamboo strips). In 橫地清先生七十大壽紀念志 (Collected papers for the seventieth anniversary of Yokochi Kiyoshi 横地清), ed. Suzuki Masahiko (鈴木正彦), 21–24. Papers of the conference held on December 2, 2000.
Li Jimin 李繼閔. 1982. Jiuzhang suanshu zhong de bilü lilun “九章算術” 中的比率理論 (The theory of ratios in The Nine Chapters on mathematical procedures). In Jiuzhang suanshu yu Liu Hui 九章算術與劉徽 (The Nine Chapters on Mathematical Procedures and Liu Hui), ed. Wu Wenjun, 228–245. Beijing: Beijing Shifan Daxue Chubanshe 北京師範大學出版社.
Li Jimin 李繼閔. 1990. Dongfang shuxue dianji Jiuzhang suanshu ji qi Liu Hui zhu yanjiu 東方數學典籍——《九章算術》及其劉徽注研究 (Research on the Oriental mathematical Classic The Nine Chapters on Mathematical Procedures and on its Commentary by Liu Hui). 西安 Xi’an: 陝西人民教育出版社 Shaanxi renmin jiaoyu chubanshe.
Li Rui 李銳. 2010. Qin fenshu kao 秦焚書考 (Critical research on the “book burning” of the Qin Dynasty). Renwen Zazhi人文雜志 (Journal of Humanities) 5: 140–146.
Li Xueqin 李學勤. 1998. Cong chutu jianbo tandao xieshulü 從出土簡帛談到《挾書律》(Discussion of the statute for keeping books based on unearthed bamboo strips and silks). In Zhou Qin Han Tang yanjiu 周秦漢唐研究 (Studies on Zhou, Qin, Han to Tang Dynasties) 1: 1–7. Xi’an西安: Sanqin chubanshe三秦出版社.
Li Yan 李儼. 1958. Zhongguo shuxue dagang (xiuding ben) 中國數學大綱 (修訂本) (An outline of the history of mathematics in China. Revised edition). Beijing: Kexue chubanshe 科學出版社 (Science Press).
Li Yan 李儼. 1955. Zhongguo suanxue shi 中國算學史 (History of mathematics in China). Shanghai: Shangwu yinshuguan 商務印書館.
Liu Qiyu 劉起釪. 1991. Zhou li zhenwei zhi zheng jiqi shuxie cheng de zhengshi yiju《周禮》真僞之爭及其書寫成的真實依據 (The debate on the authenticity of Zhou Rites and the actual evidence on how it was written). In Gushi xubian 古史續辨 (Continuation of the making of distinctions in ancient history), ed. Liu Qiyu 劉起釪, 619-653. Beijing: Zhonghua shuju 中華書局.
Liu Shipei 劉師培. 2006. Jingxue jiaokeshu 經學教科書 (Textbook of studies on Confucian Classics). Ningwu 寧武: Ningwu nanshi jiaoyin 寧武南氏校印.
Liu Xiaogan 劉笑敢. 1988. Zhuang Zi zhexue jiqi yanbian 莊子哲學及其演變 (The philosophy of Master Zhuang and its development). Beijing: Zhongguo shehui kexue chubanshe 中国社会科学出版社.
Ma Feibai 馬非百. 1982. Qin ji shi 秦集史 (History of the Qin Dynasty based on a collection of documents). Beijing: Zhonghua shuju 中華書局.
Meng Wentong 蒙文通. 1933. Jingxue jueyuan 經學抉原 (Looking for the origin of studies on Confucian Classics). Shanghai: Shangwu yinshuguan 商務印書館.
Neininger, Ulrich. 1983. Burying the scholars alive: on the origin of a Confucian martyrs’ legend. In East Asian civilizations. New attempts at understanding traditions, Vol. II: Nation and Mythology, eds. Krzysztof Gawlikowski, Wolfram Eberhard, Carl-Albrecht Seyschab, 121–136. Munich: Simon & Magiera Publishers.
Nylan, Michael. 1994. The ‘Chin Wen/Ku Wen’ controversy in Han times. T’oung Pao (Second Series) 80 (1–3): 83–145.
Nylan, Michael. 2001. The five ‘Confucian’ Classics. New Haven: Yale University Press.
Peng Hao 彭浩. 2001a. Guodian chujian Lao Zi jiaodu 郭店楚簡《老子》校讀 (Collations and explanations on Master Lao on Chu bamboo strips unearthed from Guodian). Wuhan 武漢: Hubei renmin chubanshe 湖北人民出版社.
Peng Hao 彭浩. 2001b. Zhangjiashan hanjian Suanshu shu zhushi 張家山漢簡算數書注釋 (Commentary on Writings on Mathematics written on the Han bamboo strips excavated at Zhangjiashan). Beijing: Kexue chubanshe 科學出版社.
Peng Hao 彭浩. 2005. Zhangjiashan hanjian Suanshu shu yu Jiuzhang suanshu 張家山漢簡《算數書》與《九章算術》(The Han manuscript on bamboo strips from Zhangjiashan Writings on Mathematics and The Nine Chapters on Mathematical Procedures). In Changsha sanguo wujian ji bainian lai jianbo faxian yu yanjiu guoji xueshu yantaohui lunwenji 長沙三國吳簡暨百年來簡帛發現與研究國際學術研討會論文集 (Proceedings of the International Conference on the Discovery and Study of Wu Strips (Three Kingdoms) Unearthed at Changsha and the Strips and Silks Discovered in the Last Century), ed. Changshashi wenwu kaogu yanjiusuo 長沙市文物考古研究所編, 330–334. Beijing: Zhonghua shuju 中華書局.
Petersen, Jens Østergård. 1995. Which books did the First Emperor of Ch’in burn? On the meaning of Pai chia in early Chinese sources. Monumenta Serica. Journal of Oriental Studies 42: 1–52.
Pian Yuqian 騈宇騫 and Duan Shu’an 段書安. 2006. Ershi shiji chutu jianbo zongshu 二十世紀出土簡帛綜述 (Summary of the unearthed strips and silk manuscripts in the twentieth century). Beijing: Wenwu chubanshe 文物出版社.
Qian Baocong 錢寶琮. 1964. Zhongguo shuxueshi 中國數學史 (History of Mathematics in China). Beijing: Kexue chubanshe 科學出版社.
Sun Wenqing 孫文青.1931. Jiuzhang suanshu yuanliu kao 九章算術源流考 (Examination of the origins and developments of The Nine Chapters on mathematical procedures). Nü shida xueshu jikan 女師大學術季刊 2(1):1-60.
Van Ess, Hans. 1994. The old text/new text controversy. Has the 20th century got it wrong? T’oung Pao 80 (1–3): 146–170.
Vihan, Jan. 2012. Language, likeness, and the Han phenomenon of convergence. PhD. diss. Harvard University.
Volkov, Alexei. 2014. Mathematics education in East- and Southeast Asia. In Handbook on the history of mathematics education, eds. Alexander Karp and Gert Schubring, 55–72, 79–82. New York: Springer.
Watson, Burton. 1968. The complete works of Chuang tzu. New York: Columbia University Press.
Xiao Can 蕭燦. 2010. “Yuelu shuyuan cang qinjian shu yanjiu” 嶽麓書院藏秦簡《數》研究 (A study of the mathematical text Mathematics on the bamboo strips preserved at Yuelu Academy). Ph.D. diss. Changsha 長沙: Hunan University 湖南大學.
Xiao Can 蕭燦 and Zhu Hanmin 朱漢民. 2010. Gougu xinzheng: Yuelu shuyuan cang qinjian shu de xiangguan yanjiu 勾股新證——岳麓書院藏秦簡《數》的相關研究 (New evidence for the Pythagoras theorem: related studies based on the Qin mathematical text Mathematics written on bamboo strips and preserved at Yuelu Academy. Ziran kexueshi yanjiu 自然科學史研究 (Studies in the History of Natural Sciences 29(3): 313–318.
Zhang Qicheng 張啓成. 2000. Dui liu jing wei rujia jingdian shuo de zai sikao 對六經爲儒家經典說的再思考 (Rethinking the viewpoint that the six Classics were Confucian Classics). Guizhou daxue xuebao (shehui kexue ban) 貴州大學學報(社會科學版) (Journal of Guizhou University (Edition of Social Sciences)). 18 (5): 74–78.
Zou Dahai 鄒大海 . 2001a. Zhongguo shuxue de xingqi yu xianqin shuxue 中國數學的興起與先秦數學 (The emergence of mathematics in China and the mathematics in pre-Qin Period). Shijiazhuang 石家莊: Hebei kexue jishu chubanshe 河北科學技術出版社.
Zou Dahai 鄒大海. 2001b. Chutu Suanshu shu chutan 出土《算數書》初探 (A preliminary research on the unearthed Writings on Mathematics). Ziran kexueshi yanjiu 自然科學史研究 (Studies in the History of Natural Sciences) 20(3): 193–205.
Zou Dahai 鄒大海. 2003. Cong Suanshu shu he qinjian kan shanggu liangmi de bilü 從《算數書》和秦簡看上古粮米的比率 (The ratios between different types of grains in high antiquity: A new research based on Writings on Mathematics and Qin strips). Ziran kexueshi yanjiu 自然科學史研究 (Studies in the History of Natural Sciences) 22(4): 318–328.
Zou Dahai 鄒大海. 2004. Cong Suanshu shu yu Jiuzhang suanshu de guanxi kan suanfa shi shuxue wenxian zai shanggu shidai de liuchuan. 從《算數書》與《九章算術》的關係看算法式數學文獻在上古時代的流傳 (The transmission of mathematical documents in the form of computational methods during the early period of ancient times: A research on the basis of the study on the relationship between Writings on Mathematics and The Nine Chapters on Mathematical Procedures). Gannan shifan xueyuan xuebao 贛南師範學院學報 (Journal of Gannan Teachers’ College) 6: 6–10.
Zou Dahai 鄒大海. 2005. Shuihudi bamboo strips of Qin Dynasty and mathematics in pre-Qin period. Kaogu 考古 (Archaeology). 2005 (6): 57–66. An English translation can be found in Frontiers of History in China, Vol.2, no.4, Oct. 2007. Higher Education Press, Springer, pp. 632-654. An abridged English version was published in Chinese archaeology 7, Beijing: China Social Sciences Press, 2007, pp. 132-136.
Zou Dahai 鄒大海. 2007. Cong Suanshu shu ying buzu wenti kan shanggu shidai de ying bu zu fangfa 從《算數書》盈不足問題看上古時代的盈不足方法 (The method of excess and deficit in high antiquity: An investigation on the basis of the problems of excess and deficit in the Suanshu shu). Ziran kexueshi yanjiu 自然科學史研究 (Studies in the History of Natural Sciences) 26 (3): 312–323.
Zou Dahai 鄒大海. 2008. Chutu jiandu yu zhongguo zaoqi shuxueshi 出土簡牘與中國早期數學史 (The unearthed bamboo strips and the history of mathematics in early China). Renwen yu shehui 人文與社會 (Journal of Humanism and Society) 2 (2): 71–98.
Zou Dahai 鄒大海. 2009. Guanyu Suanshu shu, Qinlü he shanggu liangmi jiliang danwei de jige wenti 關於《算數書》、秦律和上古糧米計量單位的幾個問題 (Several questions on Writings on Mathematics, the statutes of Qin and measurement units of grains and husked grains during the early period of ancient time). Neimenggu shifan daxue xuebao (ziran kexue hanwen ban) 內蒙古師範大學學報(自然科學漢文版) (Journal of Neimenggu Normal University (Chinese Edition of Natural Sciences)) 38(5): 508–515.
Zou Dahai 鄒大海. 2010a. Cong chutu zhujian kan zhongguo zaoqi weishu suanti jiqi shehui beijing 從出土竹簡看中國早期委輸算題及其社會背景 (The problems concerning weishu and their social context in early China: A research based on the unearthed bamboo strips). Hunan daxue xuebao (shehui kexue ban) 湖南大學學報(社會科學版) (Journal of Hunan University (Social Sciences)) 24(4): 5–10.
Zou Dahai 鄒大海. 2010b. Cong chutu jiandu wenxian kan zhongguo zaoqi de zhengfushu gainian 從出土簡牘文獻看中國早期的正負數概念 (Concepts of positive and negative numbers in early China: an investigation based on documents written on unearthed bamboo and wooden strips). Kaogu xuebao 考古學報 (Journal of Archaeology (Original English title: Acta Archaeologica Sinica)) 2010(4): 481–504.
Zou Dahai 鄒大海. 2010c. Cong chutu wenxian kan shanggu yishi zhidu yu zhengfushu gainian. 從出土文獻看上古醫事制度與正負數概念 (Medical institutions and concepts of positive and negative numbers in the early Ancient Period: an investigation based on unearthed documents. Zhongguo lishi wenwu 中國歷史文物 (China History and Cultural Relics (Original English title: Journal of National Museum of China)) 2010(5): 69–76.
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Chemla, K., Zou, D. (2018). Parts in Chinese Mathematical Texts. Interpreting the Chapter Form of The Nine Chapters on Mathematical Procedures. In: Bretelle-Establet, F., Schmitt, S. (eds) Pieces and Parts in Scientific Texts. Why the Sciences of the Ancient World Matter, vol 1. Springer, Cham. https://doi.org/10.1007/978-3-319-78467-0_5
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