Abstract
The earliest mathematical books written in Chinese that have been handed down through the written tradition share a common feature: they were all granted the status of “canons (jing)” some time after their completion Moreover, in correlation with this status, in the seventh century as well as, in some cases, some centuries earlier, commentaries were composed on them. This chapter focuses on one of these canons, The Nine Chapters on Mathematical Procedures, as well as its two earliest extant commentaries. The Nine Chapters is one of the oldest canons that has come down to us. It has played a central role both in the history and the historiography of mathematics in China. The main questions addressed in this chapter were inspired by some remarks regarding the way in which The Nine Chapters and its commentaries have been handed down.
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Notes
- 1.
Evidence supporting this assertion is supplied in Appendix 1 in (Chemla 2008b, 41–44). The assertion requires qualification, which is also provided there. These nuances are not important here.
- 2.
I opt for this dating, see my arguments in (Chemla and Guo Shuchun 2004, 475 sq.). However the date of composition of The Nine Chapters —this is how I shall abbreviate the title in this chapter— is still a matter of debate. For a presentation of the various stands, see chapter B, by Guo Shuchun, in (Chemla and Guo Shuchun 2004). The latter book provides a critical edition and translation on which I shall rely here. I have pleasure in thanking all the other authors in this volume for the remarks they made on my chapter. They considerably helped me clarify my argument. I am also grateful to Maurice Shukla and Richard Kennedy for their help in polishing the English in the paper. I am nevertheless fully responsible for the remaining shortcomings.
- 3.
Most of what we know concerning Liu Hui’s biography derives from the chapters entitled “Monograph on the musical scale and the calendar,” in the History of the Jin and the History of the Sui. Both chapters were composed by Li Chunfeng (602–670). These sources give further important information: see note 133, in my annotations to the French translation, in (Chemla and Guo Shuchun 2004, 774–775). Li Chunfeng’s name is associated to the second extant commentary (see below). On the process of the compilation of official histories of previous dynasties in the seventh century, see Chapter 5 in (McMullen 1988).
- 4.
Arguments for this claim are given in Appendix 1 in (Chemla 2008b).
- 5.
Appendix 1 gives the list of books brought together in the anthology. On the general issue of editing canons, selecting earlier commentaries and composing new commentaries in seventh century China, see Chapter 2 in (McMullen 1988). The author analyzes the involvement of state institutions in this process. The anthology was to be used, shortly after the commentary had been completed, as a set of textbooks for mathematical education and examinations in the Tang dynasty state teaching institutions. To complete the set, two additional treatises were employed as textbooks. Let us mention the only one that has survived, since it will be useful below as an example of text that, at least at the beginning, has been handed down in relation to the institution of the University and not its inclusion in the set of canons: the Memoir on the Methods of Numbering (Shushu jiyi 數術記遺) composed by Xu Yue (fl. ca. 220). The dating and authorship of the book is discussed in (Volkov 1994). Except for that writing, all the mathematical books that have been handed down in China were canons selected to be included in this anthology and used in the government education system at the time. The use of these textbooks for teaching and examining students is discussed in (Siu and Volkov 1999). A table giving information about these texts and a description of their main features can be found in (Martzloff 1997, 123–141). In recent decades, two critical editions of the anthology have appeared: (Qian Baocong 錢寶琮 1963; Guo Shuchun 郭書春 and Liu Dun劉鈍 1998). In what follows, I shall abbreviate the title of the anthology into Ten Canons.
- 6.
For the sake of clarity, in what follows, I shall restrict the use of the term “collection” to refer to this set of texts, whereas the term “anthology” will systematically designate the Ten Canons with the commentaries included in the seventh century.
- 7.
From the above description, it is clear that the same questions could be raised with respect to the Ten Canons. Indeed, this anthology also represents a set of texts compiled, not by historians but by actors and, as we see below, handed down in the form of an anthology. Note, however, that, although the same research program could be followed, we can expect the answers to be different, since the cohesion of the anthology does not translate in the same way in the set of extant sources as it does for The Nine Chapters and its commentaries. From early on, as we shall see, separate editions of our collection of texts appeared. These editions show that the collection was more strongly felt to be a coherent and independent whole. Addressing the question first in relation to The Nine Chapters is an indispensible prerequisite before studying the Ten Canons from a similar perspective. I plan to address the latter issue in another publication. In addition, it is only among the Ten Canons, for The Gnomon of the Zhou [Dynasty], and perhaps also The Mathematical Procedures of the Five Canons, which appears to have been much less important historically, that we could carry out a research program similar to the one sketched out in this chapter for The Nine Chapters. The reason is that these two canons are the only other ones for which layers of commentaries have accumulated and have been handed down as such. I also plan to return to these canons in subsequent publications.
- 8.
To help the reader follow my analysis, in Appendix 2, I have provided a sketch of the earliest stages in the transmission of the collection. A more detailed account is given in chapter B, by Guo Shuchun, in (Chemla and Guo Shuchun 2004). I limit myself to what is essential for me here.
- 9.
On the Imperial Library during the Northern Song, see (Winkelman 1974, 5–10). More generally, the reader is referred to this publication for a description of the imperial holdings and cataloguing of books during the Song. Source material on the various printings of the mathematical canons during the Song dynasty is collected in (Li Yan 1954, 89–92). Li Yan emphasizes in particular the connection that existed, in his view, between the project of resuming examination in mathematics during the Northern Song dynasty and the 1084 printing of the canons. However, finding conflicting evidence regarding whether the auxiliary book Memoir on the Methods of Numbering (Shushu jiyi) was also printed in 1084 or not, he refrains from concluding on this point. (Qian Baocong 錢寶琮 1963, vol.1, 1–9) describes the history of the editorial projects that regularly republished the Ten Canons. In particular, Qian explains how in the eleventh century, the text of two canons, the Zhuishu and the Xiahou Yang suanjing, had already been irretrievably lost and how a mathematical book probably composed in the eighth century was mistakenly substituted for the latter, see (Qian Baocong 錢寶琮 1963, vol. 2, 551–552). This last detail shows the impact of the intention of retrieving the canons on the printing of historical mathematical texts.
- 10.
The “Monograph on the musical scale and the calendar” of the History of the Song (Songshi) ((Yang Jialuo (ed.) 楊家駱主編 1978, 4: 2892–2894 ) documents how at the end of the twelfth century and the beginning of the thirteenth century, the official Bao Huanzhi worked within the framework of state institutions on a reform of the calendar and designed the Kaixi calendar, enacted in 1207. This calendar was an improvement to the Tongtian calendar, which had been designed by Yang Zhongfu 杨忠辅 and enacted in 1199. It was a few years later that Bao Huanzhi dealt with the reedition of the anthology. More information on Bao Huanzhi’s career as a civil servant is provided in the introduction to the reprint of the extant sections of his edition of the Ten Canons, in (Shanghai tushuguan and Beijing daxue tushuguan 1980), placed at the beginning of the reprint of the Shushu jiyi.
- 11.
The postface to The Gnomon of the Zhou has survived in the part of the Southern Song woodblock print that has been handed down. The same holds true for the postface to the Memoir on the Methods of Numbering, a book used as a textbook complementing the Ten Canons in Tang dynasty teaching institutions (see Footnote 5) and reprinted by Bao Huanzhi as appended to the anthology of Canons. This postface is published with a text entitled Source and development of mathematics (Suanxue yuanliu 算學源流), which sketches a history of mathematical examinations and teaching institutions in China. The postface to The Nine Chapters was handed down in the 1261 subcommentary by Yang Hui 楊輝 as well as through Dai Zhen’s copy from the Grand Classic of the Yongle period (see below). For a critical edition, see (Guo Shuchun 郭書春 1990, 491–492). Like the postface to The Nine Chapters, the postface to the Mathematical Canon of the Sea Island was copied, probably, as Yan Dunjie showed, from the Grand Classic, and included in the nineteenth century in the manuscript entitled Zhujia suanfa ji xuji 諸家算法及序記 « Mathematical methods and records of prefaces from all schools »; see its reproduction and the introduction in (Guo Shuchun 郭書春 1993, 1: 1451–1452). In addition to the chapter of the History of the Song (Songshi), already mentioned, the four postfaces provide information on different official positions held by Bao Huanzhi.
- 12.
(Siu and Volkov 1999). Since the Memoir on the Methods of Numbering (Shushu jiyi) was one of the two additional books (see footnote 5), its inclusion in Bao’s editorial project manifests this intention, which the postface makes explicit. Bao’s postface to that book explains that he could not find a copy of the book in the imperial libraries and tells the story of its recovery from the library of a Daoist temple. Moreover, the postface shows Bao’s awareness that one canon, the Zhuishu, and one auxiliary book, the Sandengshu, had been lost. See the reprint of the Song Shushu jiyi, (Shanghai tushuguan and Beijing daxue tushuguan 1980, 11a).
- 13.
Incidentally, this information provides evidence regarding the 1084 edition of The Nine Chapters and shows that it consisted of the whole collection. In the surviving copies of Bao Huanzhi’s edition of The Gnomon of the Zhou [dynasty], the Mathematical Canon of the five Bureaus and the Mathematical Canon of Master Sun, the final pages record evidence of the 1084 editorial work done by the imperial administration for publishing these books. The same holds true for the canons for which only copies of that edition made by Mao Yi in the 1680s were handed down: the Mathematical Canon continuing the Ancients and the Mathematical Canon by Xiahou Yang. These pieces of evidence indicate that, for his enterprise, Bao Huanzhi reproduced a larger part of the 1084 official printing than only The Nine Chapters. The final pages of the edition of The Nine Chapters and probably of the Mathematical Canon by Zhang Qiujian are no longer extant. However, the postface to the Haidao suanjing, the southern Song printing of which did not survive, testifies to the fact that Bao Huanzhi did not have access to the Northern Song print of it. The postface explains how Bao Huanzhi assessed the documents he found and how he relied on historical documents to go against the evidence of the manuscripts handed down and ascribe to a book a different author.
- 14.
We shall come back to the interpretation of this “meaning (yi)” in the subsection III.1 of the chapter. It is interesting that Bao looks for an edition with a commentary and a subcommentary, and not an interpretation authored by a single person.
- 15.
Bao Huanzhi relied on the bibliographical treatises of the dynastic histories of the Jin and the Tang, as well as on Liu Hui’s preface to his own commentary, to establish, in the final section of his postface, why, in his view, the text of the commentaries to which the 1084 edition gave him access had deteriorated through the written transmission.
- 16.
What in Yang Hui’s text is to be ascribed to Jia Xian is unclear, as shown by (Guo Shuchun 郭書春 1988). We know almost nothing about Jia Xian’s or Rong Qi’s biography. As for Yang Hui 楊輝, we have little evidence about him, except the testimony provided by the writings he composed between 1261 and 1275.
- 17.
Guo Shuchun, in chapter C of (Chemla and Guo Shuchun 2004, 73–74), argues that they share a same prototype. Note that Yang Hui must have known Bao Huanzhi’s reprint, since he included in his subcommentary the postface to The Nine Chapters (see Footnote 11).
- 18.
Rong Qi’s preface states: “When the government instituted the examinations in mathematics to select officials, they chose The Nine Chapters to be the foremost (literally, the head) of the mathematical canons, since, indeed, it is like the six canons of the Confucians, the (Canon of) difficulties and the (Grand) Simplicity of the medical schools, the Book of Master Sun of military schools! 以國家嘗設算科取士選 九章,以為算經之首, 蓋猶儒者之六經,醫家之‘難’,‘素’,兵家之‘孫子’歟 ! ” (The preface was reproduced in Yang Hui’s Detailed explanations, who also quoted this passage in one of his own prefaces in the book. Critical editions of these texts can be found in (Guo Shuchun 郭書春 1990, 489, 495).) Likewise, in his 1200 postface to the collection, Bao Huanzhi claims that “Among the books of mathematical procedures, there are altogether ten schools. One can only consider The Nine Chapters as being the foremost of the canons. With the methods of its nine parts of mathematics (jiu shu), there is nothing which is not complete. Although the procedures established by the various schools present variation, when one looks for the original meaning (yi), they all come from them” (My emphasis, (Guo Shuchun 郭書春 1990, 491).) Note the use here of the term “meaning (yi).” This term is the same one as the one we saw Bao Huanzhi use to express the reason of his interest in the commentaries on The Nine Chapters: they highlighted in his view the yi of the canon. Both statements imply that for Bao Huanzhi, the reader looks for the “meaning (yi)” of the canons and that this search is made easier, in the case of The Nine Chapters, if one can rely on the commentaries. Moreover, inquiring into the yi of the other canons shows why The Nine Chapters has in fact a higher status. These facts, which are essential to capture the kind of reading actors made of the collection as such, incite us to expect a relation between the content of the commentaries and the special status enjoyed by The Nine Chapters within the anthology.
- 19.
See, for example, the prefaces to the Mathematical writings in nine chapters (1247), by Qin Jiushao, Deploying the pieces for the [Collection] augmenting the ancient (methods) (1259), by Li Ye (Guo Shuchun 郭書春 1993, 1: 439, 875), and the prefaces to Jade mirror of the four origins (1303) by Zhu Shijie (Guo Shuchun 郭書春 1993, 1205–1206).
- 20.
The Zhang Qiujian suanjing and the Jigu suanjing were not contained in it, nor was the Shushu jiyi, see tables 1 and 3 in (Chu Pingyi (Zhu Pingyi) 2010).
- 21.
See Guo Shuchun, Chapter C, in (Chemla and Guo Shuchun 2004, 72–73).
- 22.
One should take also into consideration the excerpts contained in Yang Hui’s final chapter, “Compiling the Categories of The Nine Chapters (Jiuzhang zuanlei). However, I shall leave these details out here.
- 23.
In what follows, regarding the Ten Canons, I rely on (Chu Pingyi (Zhu Pingyi) 2010), Chapter C by Guo Shuchun in (Chemla and Guo Shuchun 2004, 74–79) as well as the introduction to (Guo Shuchun 郭書春 and Liu Dun 劉鈍 1998). Moreover, for the various editions of the collection published by Dai Zhen, see Chapter C, by Guo Shuchun, in (Chemla and Guo Shuchun 2004).
- 24.
In his postface to the printing of one of his editions of The Nine Chapters by Qu Zengfa 屈曾發 in 1776 (see (Guo Shuchun 郭書春 1990, 503)), Dai Zhen recalls how he had been looking for a copy of The Nine Chapters for twenty years when in 1767 he managed to have access to the text, in the edition of the Grand Classic. The passage is recorded with differences in (Duan Yucai 1980, 228–229). Moreover, it is quoted in (Guy 1987, 62). However, in my view, the translation is not completely accurate. The reader will find in this book a general introduction to the project of the Complete libraries.
- 25.
If this fact manifests Dai Zhen’s perception of the collection as a cohesive whole, by contrast the Complete Libraries did not place emphasis on the cohesiveness of the anthology. Even though the notice on the collection in the “General catalogue” of the Complete Libraries mentioned its belonging to the “Ten Canons of Mathematics,”—it was restated to be “the foremost of the Ten Canons of Mathematics 为《算经十书》之首”—, the corresponding mention was by no means systematically made for the other components of the Tang anthology; see (Ji Yun, Lu Xixiong and Sun Shiyi 1997, 1: 1386–1387, 1401–1405) for the relevant notices.
- 26.
- 27.
The extant copy contained the first five chapters of The Nine Chapters with its commentaries as well as some of the other canons. We know, for instance, that Mei Wending had access to it, thanks to Huang Yuji 黃虞稷. I thank Catherine Jami for having provided me with evidence from Mei’s book Wu’an lisuan shuji 勿庵曆算書記 (Siku quanshu, p. 50a) that Mei mainly saw the first chapter of the Southern Song reprint.
- 28.
In addition to The Nine Chapters, Dai Zhen included in the Complete Libraries six other of the Ten Canons on the basis of the edition of the Grand Classic. The remaining two canons (Mathematical Canon by Zhang Qiujian and Mathematical Canon Continuing the Ancients) as well as the Memoir on the Methods of Numbering, which had not been copied in the Grand Classic, were included in the Complete Libraries thanks to copies found otherwise through the book search done in the Empire in relation to the compilation of the new imperial project. Reproductions of the southern Song reprint played a key part in this. However, in his editorial work, Dai Zhen sometimes used more than one edition, see the introduction of (Guo Shuchun 郭書春 and Liu Dun 劉鈍 1998). The mathematical writings selected for inclusion in the prestigious Impression of the Assembled Gems from the Palace Wuying followed another principle of selection: they were only those of the Ten Canons that could be copied from the Grand Classic. See table 2, in (Chu Pingyi (Zhu Pingyi) 2010).
- 29.
On the dating of the publication, see note 3 in (Guo Shuchun 郭書春 1990, 506). Mikami Yoshio gives a short biography for Kong Jihan and describes the impact of the publication in (Mikami Yoshio 1913, 122–123). In his book, Mikami clearly relies on Dai Zhen’s historical work carried out in relation to the latter’s involvement in the Complete Libraries. This is one of the manifold manifestations of how Dai Zhen’s work in relation to the compilation of the Complete Libraries shaped the subsequent historiography.
- 30.
The first translation of The Nine Chapters was carried out into Russian in the 1950s: (Biérëzkina 1957). It served as a basis for the German translation: (Vogel 1968).
- 31.
For a sketch of his biography and an analysis of his translation, see (Martija-Ochoa 2002).
- 32.
(Van Hée 1933, 269–282). The doubts Van Hée casts on the authenticity of the text are formulated in a disparaging tone that permeates through the whole paper. Interestingly, Van Hée complements his translation of each of Liu Hui’s problems and procedures, with the details of the operations taken from Li Huang’s 1820 commentary. The same canon was translated into English: (Swetz 1992). The author translates the statement of the problems and the procedures provided afterwards and solving them. However, Swetz explained (p. 19), “Li Chunfeng’s commentary is omitted because it does not aid in understanding the rationale of Liu Hui’s solution methods.” The question remains open to understand why the text has been handed down in that way.
- 33.
(Hideki Kawahara 1980).
- 34.
The same attitude can be met in the most recent critical edition of the Ten Canons: (Guo Shuchun 郭書春 and Liu Dun 劉鈍 1998). Zhen Luan’s level being judged as low and his errors multiple (see introduction), his commentary on The Gnomon of the Zhou, as well as Li Chunfeng’s subcommentary on it, were deleted from the edition (p. 34).
- 35.
(Wagner 1978). My introduction to Chapter 6 in (Chemla and Guo Shuchun 2004, 472–473) summarizes my conclusions on this topic and provides a more detailed and recent bibliography. I cannot, within the context of this chapter, tackle this issue any further. Suffice it to notice that part of what has been done on a commentary that was thought to be Liu Hui’s may have in the future to be reconsidered in this perspective. If, as we shall see, in recent decades, Liu Hui has attracted the greatest attention, this may be a consequence of editorial problems that will need to be addressed anew in the future. Let us for now keep in mind that this name may refer to a complex historical reality and concentrate in what follows on “commentaries,” without stressing too much who their author may be. This has, however, a consequence for our project: we can raise questions concerning the relationship between the canon and its commentaries, taken together. But we will not be in a position to discuss accurately the difference in kind between Liu Hui’s commentary and Li Chunfeng’s subcommentary.
- 36.
(Shen Kangshen, Crossley and Lun 1999).
- 37.
See the reference and analysis in (Youschkévitch 1982).
- 38.
Although Schrimpf’s focus was the anthology, he emphasized those canons and commentaries —or the parts in them— that consisted of problems and algorithms. His thesis, rooted in sinology, aimed at restoring conceptual specificities of the mathematics expounded in the anthology, translating or describing its content in three parts (arithmetic, geometry, algebra). I shall come back to it in a publication on the historiography of the anthology.
- 39.
(Wylie [1852] 1882).
- 40.
It should be stressed that the scholars just mentioned come from backgrounds that are dramatically different and from quite distinct places of the planet.
- 41.
Typically, he would write: “We, therefore, believe that our analysis may represent the original thought-process of the author. This conjecture is confirmed by the commentary of Liu Hui (third century A.D.) who remarked that it is actually a case of the rule of proportion, Chin Yu Shu” (that is, jinyou shu, the procedure of “Suppose,” which is the rule of three), see (Wang Ling 1956, 182). It is interesting to note that, except marginally, such an approach does not seem to have had a strong impact on (Needham and Wang Ling 1959).
- 42.
In addition to the themes usually found in previous publications and summarized above, the way in which Liu Hui established the correctness of the algorithms for the volumes of solids is also emphasized in Chapter 3 of (Yabuuti Kiyosi 1974). For analyzing how this book approached the commentaries, I was able to rely on the translation into French published in (Yabuuti Kiyosi 2000).
- 43.
(Li Yan and Du Shiran 1963, 77). It is worth stressing that we find in the original text the first occurrence of the claim that the commentator “established the correctness of algorithms,” by contrast to the claim that one would have proofs comparable to those recorded in Euclid’s Elements. As far as I can tell, it required the influence of Wu Wenjun’s approach to the ancient Chinese mathematical sources from an algorithmic point of view, developed in the 1980s, for this approach to the commentaries to resume developing much later on. In relation to this claim, the authors gave a richer set of illustrations than Li Yan could give a few years earlier. This book was translated into English. However, note that the English translation of this precise statement, which is quite important for what it reveals of the understanding that was developing in China at the time, is misleading: “We know that the Nine Chapters listed general methods of calculation but the explanations and discussions are very brief. The commentary by Liu Hui manages to make up for those deficiencies. We can go one step further and say that the commentary and explanations for The Nine Chapters gave brief proofs for the various types of calculation and verified the accuracy of the calculation.” (Li Yan and Du Shiran 1987).
- 44.
I am grateful to my colleague, Professor Tian Miao, for having copied Li Yan’s first edition for me. To be precise, in the two editions, Li Yan comes back to The Nine Chapters amongst the other canons, when dealing with the procedure for root extraction, the solution of systems of simultaneous linear equations, areas and volumes. There Liu Hui is mentioned for the new procedures he introduced.
- 45.
- 46.
(Wagner 1975 (1 March 1976)).
- 47.
As is suggested by the development on this term below, this may also be understood in the plural: “its meanings/intentions.” Alternatively, if the anaphora refers to the mathematical procedures, this can also be interpreted as “their meaning/intentions.” We come back to this point. The end of Liu Hui’s preface uses this term yi again in relation to the content of his commentary and that which he attempts at recovering. On the interpretation of this term, see the Glossary I compiled and published in (Chemla and Guo Shuchun 2004, 1018–1022).
- 48.
See also the statement quoted in footnote 18.
- 49.
In the 1980s, Wu Wenjun played a key part to promote the idea that these procedures had to be read as algorithms and not as ordinary statements. Such a shift in the research work on The Nine Chapters was a prerequisite for developments of the kind that follow.
- 50.
In this case, the sequence amounts to three operations: multiplying each numerator by the denominator of the other fraction, subtracting the greatest of the resulting numbers from the smallest; lastly, taking this as dividend and the product of the denominators as divisor yields the difference.
- 51.
We come back below to this second kind of “meaning.” I transcribe it as “yi’” to distinguish it from the first one.
- 52.
My use of quotation marks in the translation is the result of a convention: quotation marks indicate that the Chinese text is quoting the canon. The translation of both occurrences can differ for reasons of differences of syntax between English and Classical Chinese.
- 53.
The whole passage on which we rely here can be found in (Chemla and Guo Shuchun 2004, 378–385). The argument outlined here is developed in greater detail in (Chemla 2008a).
- 54.
- 55.
I dealt with this question from different viewpoints in (Chemla 1997, 2000, 2003a, 2009). I only make use of the conclusions here without repeating the argument. The second paper gives reasons to believe that, in the commentators’ eyes, the problems provided in The Nine Chapters were meant to clarify the yi of the procedure. It also suggests that there may have existed a culture of mathematical problems in ancient China, elaborated for that use.
- 56.
The best example is that of “Multiplying parts” (1.21), see the papers mentioned in the previous footnote.
- 57.
On the question of the generality of figures in ancient China, see (Chemla 2005).
- 58.
- 59.
This quest of Liu Hui’s is strongly emphasized by (Guo Shuchun 郭書春 1992, 301–320).
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Appendices
Appendix 1: The Ten Canons of Mathematics
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1.
The Gnomon of the Zhou [dynasty] (Zhoubi). Composed around the beginning of the Common Era, it expounds the mathematics at the basis of calendar-making, within the framework of the cosmographical theory “Heaven like a chariot-cover (Gaitian).” (Cullen 1996) provides a translation of the canon. In conformity with the front pages of the edition printed during the Song dynasty, the received version contains a third century commentary by Zhao Shuang 趙爽, a sixth century subcommentary by Zhen Luan 甄鸞 (fl. ca 566) and a subcommentary on all previous layers composed under Li Chunfeng’s supervision (Qian Baocong 錢寶琮 1963, 3–6).
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2.
The Nine Chapters on Mathematical Procedures (Jiuzhang suanshu), a title abbreviated to The Nine Chapters. The received version contains a commentary by Liu Hui and a subcommentary written under Li Chunfeng’s supervision. The canon is mainly composed of problems and algorithms solving them. By contrast, the commentaries deal with the correctness of the algorithms and more generally offer reflections on mathematics.
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3.
The Mathematical Canon of the Sea Island (Haidao suanjing), written by the third century mathematician, Liu Hui. Liu Hui’s preface to his commentary on The Nine Chapters diagnosed that the book, as restored by Han scholars, failed to cover all the categories of mathematical problems (Chemla 2008a). The Mathematical Canon of the Sea Island, devoted to measuring lengths at a distance, filled the gap indicated by the preface and was first composed as a complement to Chapter 9 of The Nine Chapters concerning the right-angled triangle. It was turned into an independent book in the Tang dynasty. The commentary that Liu Hui composed on it appears to have been lost at the time when the commentary written under Li Chunfeng’s supervision was compiled (Qian Baocong 錢寶琮 1963, 261–263). (Swetz 1992) contains a translation.
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4.
The Mathematical Canon by Master Sun (Sunzi suanjing). (Qian Baocong 錢寶琮 1963, 275–276) argues that the book was composed around 400 CE, but stresses that the received version displays hints of later, Tang, changes. Moreover, in the Song dynasty printing, the front page of each chapter indicates that a commentary on it was composed under Li Chunfeng’s supervision, but the printed version does not appear to contain it. (Lam and Ang 2004) provides a translation of the canon. Like The Nine Chapters, its second and third chapters present mathematical knowledge within the framework of problems and algorithms. They mainly deal with themes treated in The Nine Chapters. The first chapter, however, adopts another presentation, introducing sequences of measure units or of procedures, the basic algorithms for common arithmetical operations as well as rhyme tables. Due to the evidence provided by archaeology, we know that such modes of writing down mathematics go back to, at least, the Han dynasty (Peng Hao 彭浩 2001).
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5.
The Mathematical Canon by Zhang Qiujian (Zhang Qiujian suanjing). (Qian Baocong 錢寶琮 1963, 325–327) argues it was composed in the second half of the fifth century, whereas in the sixth century, Liu Xiaosun added detailed explanations about the execution of its procedures. Its shape and topics basically follow those of The Nine Chapters. Among the two commentaries that the edition printed in the Song dynasty mentions on the front pages of each chapter, only the commentary composed under Li Chunfeng’s supervision is still extant.
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6.
The Mathematical Canon of the five Bureaus (Wucao suanjing). (Qian Baocong 體錢寶琮 1963, 409) attributes its composition to Zhen Luan (sixth century). The book contains elementary procedures given in the framework of problems and useful for local government officials. It is divided according to the nature of the duties of various bureaus (fields, soldiers, assemblies, granaries, and finance). Despite the mentions on the front pages of the Song edition, no commentary composed under Li Chunfeng’s supervision appears in the edition.
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7.
The Mathematical Procedures of the five Canons (Wujing suanshu). In this book, Zhen Luan (sixth century) compiles commentaries on passages of the Confucian canons requiring mathematical knowledge and adds technical explanations to them. The received version contains a commentary composed under Li Chunfeng’s supervision (Qian Baocong 錢寶琮 1963, 437–438).
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8.
The Mathematical Canon continuing the Ancients (Qigu suanjing) was written by Wang Xiaotong in the first decades of the seventh century. Except for a first problem concerning an astronomical conjunction, the book gathers problems dealing with volumes or right-angled triangles, and to be solved by cubic, or biquadratic equations. The extant copy of the Song printed edition mentions that Wang Xiaotong wrote the book and commented himself on it. Ancient bibliographies mention the existence of a commentary written by Li Chunfeng. If it existed, it does not seem to have survived.
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9.
Zhui shu 綴術 by Zu Chongzhi (fifth century). The canon did not survive. It has been suggested that this was due perhaps to its high level. (Yan Dunjie 嚴敦傑 2000, 125–126, 139) gathers evidence showing that Li Chunfeng also produced a commentary on this book.
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10.
The Mathematical Canon by Xiahou Yang (Xiahou Yang suanjing), the composition of which (Qian Baocong 錢寶琮 1963, 551–553) dates to the eighth century. Qian argues that the book handed down was not the one that was originally included in the anthology, the substitution having occurred in the Song dynasty. The book describes algorithms to perform the basic arithmetical operations in a more convenient way, and introduces decimal fractions. Its topics follow on those dealt with in The Nine Chapters. Its presentation is a mixture of algorithms described within and without the context of problems. In sum, it is not surprising that Li Chunfeng’s name is not associated with the book.
It is important to keep in mind that if we set aside the case of the Memoir on the Methods of Numbering (see footnote 5), the only mathematical books handed down through the written tradition were these canons.
Appendix 2
Historical analysis of the early shaping and transmission of the collection
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An arrow that is dotted indicates that it is unclear by means of which documents the new edition relates to the older one.
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Chemla, K. (2010). A Chinese Canon in Mathematics and Its Two Layers of Commentaries: Reading a Collection of Texts as Shaped by Actors. In: Bretelle-Establet, F. (eds) Looking at it from Asia: the Processes that Shaped the Sources of History of Science. Boston Studies in the Philosophy of Science, vol 265. Springer, Dordrecht. https://doi.org/10.1007/978-90-481-3676-6_6
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