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Published online by Cambridge University Press: 13 September 2022
Before the mathematical manuscript titled Writings on Mathematical Procedures (Suanshu shu 筭數書) was found at Zhangjiashan, historians of mathematics could trace mathematics in early imperial China only on the basis of the received canonical literature, notably The Nine Chapters on Mathematical Procedures (Jiuzhang suanshu 九章算術). After the Zhangjiashan and other mathematical manuscripts were found, they were mainly compared with The Nine Chapters, in the belief that these were all early imperial mathematical works and therefore adequate objects of comparison. As such, The Nine Chapters was transmitted with layers of commentaries and subcommentaries. This article argues that Writings on Mathematical Procedures presents important parallels with the commentarial literature on The Nine Chapters. This sheds light on how such exegeses were composed. The article further demonstrates that examination of these commentaries and subcommentaries allows us to perceive parallels between Writings on Mathematical Procedures and The Nine Chapters that to date have not been considered.
在張家山漢簡《筭數書》被發掘出來之前, 數學史家只能在傳世文獻, 尤其是《九章算術》的基礎上研究秦丶漢數學。《筭數書》這部漢簡出土後, 主要用來與《九章算術》進行比較。作為一部經典,《九章算術》是與劉徽注和唐朝李淳風等注釋一起流傳下來的。筆者認為,《筭數書》與《九章算術》的注和注釋之間有重要的相似之處, 這為注和注釋如何形成提供了一些有趣的線索。筆者還認為, 注和注釋使我們能夠察覺到《筭數書》與《九章算術》之間某些至今尚未為人考慮的相似之處。
I am grateful to Michael Nylan, Vanessa Davies, Sarah Allan, and Wen-Yi Huang for their help in the preparation of this article. I extend my thanks to the two referees, whose comments helped me improve my argument.
1. All dates are c.e. unless otherwise noted. Recent critical editions of this anthology include Suanjing shi shu 算經十書, ed. Qian Baocong 錢寶琮, 2 vols. (Beijing: Zhonghua, 1963); Suanjing shi shu 算經十書, ed. Guo Shuchun 郭書春 and Liu Dun 劉鈍, 2 vols. (Shenyang: Liaoning jiaoyu, 1998).
2. For an outline of the context and the project, see McMullen, David, State and Scholars in T’ang China (Cambridge: Cambridge University Press, 1988), 67–112Google Scholar. A precise description of the canonical texts in mathematics, the selected commentaries, and the state of the surviving editions is given in Chemla, Karine and 朱一文, Zhu Yiwen, “Contrasting Commentaries and Contrasting Subcommentaries on Mathematical and Confucian Canons: Intentions and Mathematical Practices,” in Mathematical Commentaries in the Ancient World: A Global Perspective, ed. Chemla, Karine and Most, Glenn W. (Cambridge: Cambridge University Press, 2022)CrossRefGoogle Scholar.
3. To make this assertion, I set aside The Gnomon of the Zhou (Zhoubi 周髀), which was probably completed in the first century, before The Nine Chapters, as this work provides the mathematical knowledge needed for cosmography and the calendar. Since it does not present any significant parallel with excavated texts, it does not allow me to consider the issue addressed in this article. For The Gnomon of the Zhou’s date of completion, and an English translation, see Cullen, Christopher, Astronomy and Mathematics in Ancient China: The Zhou Bi Suan Jing (Cambridge: Cambridge University Press, 1996)CrossRefGoogle Scholar. The date of completion is still debated, however, as summarized in Feng Ligui 馮禮貴, “Zhoubi suanjing chengshu niandai kao” 《周髀算經》成書年代考, Guji zhengli yanjiu xuekan 古籍整理研究學刊 1986, 37–41.
4. About this translation, see Chemla and Zhu, “Contrasting Commentaries.” A critical edition and a French translation of The Nine Chapters, Liu Hui’s commentary, and the exegetical material prepared under Li Chunfeng’s editorial supervision, appear in Chemla, Karine and 郭書春, Guo Shuchun, Les Neuf Chapitres: Le Classique Mathématique de la Chine Ancienne et ses Commentaires (Paris: Dunod, 2004)Google Scholar. In what follows, unless otherwise specified, I rely on the critical edition published in this book. I have argued that The Nine Chapters was completed in the first century (Chemla, “Présentation du chapitre 6,” in Chemla and Guo, Neuf Chapitres, 475–78).
5. This interpretation of the title derives from the analysis of the document given in Daniel Morgan and Karine Chemla, “Writing in Turns: An Analysis of Scribal Hands in the Bamboo Manuscript Suan shu shu 筭數書 (Writings on Mathematical Procedures) from Zhangjiashan Tomb No. 247,” Silk and Bamboo 1.1 (2018), 152–90. The first annotated critical edition of this manuscript was given in Peng Hao 彭浩, Zhangjiashan Han jian “Suanshu shu” zhushi 張家山漢簡《算數書》注釋 (Beijing: Kexue, 2001). Unless otherwise noted, I rely on this edition. The work of critical edition continues, and, where necessary, I will mention important contributions. See Guo Shuchun 郭書春, “Suanshu shu jiaokan” 筭數書校勘, Zhongguo keji shiliao 中國科技史料22.3 (2001), 202–19; Guo Shirong 郭世榮, “Suanshu shu kanwu” 算術書勘誤, Neimenggu shida xuebao ziran kexue (Han wen) ban 內蒙古師大學報–自然科學 (漢文)版30.3 (2001), 276–85. A new critical edition, with translations into modern Chinese and Japanese, appears in Chōka zan Kankan Sansū sho kenkyūkai, 張家山漢簡『算數書』研究会, Kankan Sansû sho 漢簡算數書 (Kyoto: Hōyū, 2006). The same year, an annotated translation into modern Chinese, with another critical edition, was published: Horng Wann-sheng 洪萬生, Lin Cangyi 林倉億, Su Huiyu 蘇惠玉, and Junhong, Su 蘇俊鴻, Shu zhi qiyuan 數之起源 (Taipei: Taiwan shangwu, 2006)Google Scholar. An English translation, based on Peng Hao’s edition (modified), appeared in Christopher Cullen, The Suan Shu Shu 筭數書 ‘Writings on Reckoning’: A Translation of a Chinese Mathematical Collection of the Second Century BC, with Explanatory Commentary (Cambridge: Needham Research Institute, 2004). Another English translation is Joseph Dauben, “算數書 Suan Shu Shu (A Book on Numbers and Computations). English Translation with Commentary,” Archive for History of Exact Sciences 62 (2008), 91–178. More recently, Rémi Anicotte, Le Livre sur les Calculs Effectués avec des Bâtonnets. Un Manuscrit du —IIe Siècle Excavé à Zhangjiashan (Paris: Presses de l’Inalco, 2019) has provided a French translation.
6. Chemla, Karine, “Numerical Tables in Chinese Writings Devoted to Mathematics: From Early Imperial Manuscripts to Printed Song-Yuan Books,” EASTM (East Asian Science, Technology and Medicine) 44 (2016), 69–123CrossRefGoogle Scholar, esp. 83–89.
7. The expression is used in Cullen, Christopher, “The Suan Shu Shu 筭數書 ‘Writings on Reckoning’: Rewriting the History of Early Chinese Mathematics in the Light of an Excavated Manuscript,” Historia Mathematica 34.1 (2007), 10–44, 28CrossRefGoogle Scholar.
8. The prominence of this question is acknowledged in Cullen, “The Suan Shu Shu: Rewriting,” 40.
9. Guo Shuchun 郭書春, “Shilun Suanshu shu de shuxue biaoda fangshi” 試論《筭數書》的數學表達方式, Zhongguo lishi wenwu 中國歷史文物 2003.3, 28–38.
10. Guo Shuchun 郭書春, “Suanshu shu yu Suanjing shishu bijiao yanjiu” 《算數書》與《算經十書》比較研究, Ziran kexueshi yanjiu 自然科學史研究 23.2 (2004), 106–20.
11. Comparative remarks are pervasive in the footnotes and annotations given in the critical editions and translations. Sections are devoted to them in, e.g., Horng et al., Shu zhi qiyuan, 104–26, as well as in Dauben, “Suan Shu Shu,” 94–100.
12. Material analyses of the document led Morgan and me to suggest rather that Writings assembled notes produced by several hands, in a context of mathematical education. Importantly, we discerned two separate hands (Hand B and Hand A), with the former guiding the learning process of the latter. For part of the argument, see Mo Zihan 墨子涵 (Daniel Morgan), and Lin Lina 林力娜 (Karine Chemla), “Ye you lunzhe xie de: Zhangjiashan Han jian Suanshu shu xieshou yu pianxu chutan” 也有輪著寫的：張家山漢簡《筭數書》寫手與篇序初探, Jianbo 簡帛 12 (2016), 235–52. See also Morgan and Chemla, “Writing in Turns,” 181–82.
13. Anicotte, Le Livre sur les Calculs, 18, mentions this issue. Peng, Zhangjiashan Han jian, 25–32, actually describes how The Nine Chapters was produced using Writings. In Guo Shuchun 郭書春, “Shilun Suanshu shu de lilun gongxian yu bianzuan” 試論算數書的理論貢獻與編纂, Faguo hanxue 法國漢學 (French Sinology) 6 (2002), 505–37, Guo Shuchun opposes views of this kind. For him, The Nine Chapters and Writings are intimately connected but in no way is the latter a basis on which the former was composed (see n. 14). For Zou, the converse holds true: The Nine Chapters existed as a pre-Qin canon, and Writings derived from it; Zou Dahai 鄒大海, “Cong Suanshu shu yu Jiuzhang suanshu de guanxi kan suanfashi 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, Gannan shifan xueyuan xuebao 贛南師範學院學報 2004.6, 6–10.
14. Peng, Zhangjiashan Han jian, 26, and Dauben, “Suan Shu Shu,” 94–95, both see in Writings a reference work for administrators. I return to this point below. Guo considers that, with the diversity of its procedures and the luxuriance of its terminology, Writings derives from mathematical sources that reflect the disunity of pre-Qin mathematics, whereas The Nine Chapters is a product of early imperial China order and of the standardization that was applied to every subject (Guo Shuchun, “Shilun Suanshu shu de lilun gongxian yu bianzuan,” 530). For Zou, Writings belonged to a set of works that were not canonical (he sees two types of books of this type: some specialized for professionals, and others practical), whereas, in pre-Qin times, The Nine Chapters belonged to the set of canonical works, and it represents pre-Qin mathematics more faithfully than Writings; see Zou Dahai 鄒大海, “Cong Suanshu shu yu Jiuzhang suanshu de guanxi” and “Zai lun Suanshu shu yu Jiuzhang suanshu de guanxi” 再論《算數書》與《九章算術》的關係. Xin fajia 新法家, January 26, 2007, www.xinfajia.net/2830.html, accessed on September 3, 2021.
15. This is a major point addressed in Cullen, “The Suan Shu Shu: Rewriting,” 27, 35–38. Cullen’s thesis is that The Nine Chapters was put together during Wang Mang’s reign, on the basis of “textlets” such as those forming the sections of Writings, to fit the jiu shu 九數 (Nine Fundamental Procedures) in the Zhou li 周禮 (Zhou Rites). On other views, see n. 14. Chemla, Karine, “Documenting a Process of Abstraction in the Mathematics of Ancient China,” in Studies in Chinese Language and Culture—Festschrift in Honor of Christoph Harbsmeier on the Occasion of His 60th Birthday, ed. Anderl, C. and Eifring, H. (Oslo: Hermes Academic Publishing and Bookshop A/S, 2006), 169–94Google Scholar, addresses the question of the composition of The Nine Chapters from a different angle.
16. See Dauben’s “The Evolution of Mathematics in Ancient China: From the Newly Discovered Shu and Suan Shu Shu Bamboo Texts to the Nine Chapters on the Art of Mathematics,” Notices of the ICCM (International Congress of Chinese Mathematicians) 2.2 (2014), 29. This is also the focus in Cullen, “The Suan Shu Shu: Rewriting.”
17. Hubei sheng wenwu kaogu yanjiusuo 湖北省文物考古研究所 and Yunmeng xian bowuguan 雲夢縣博物館, “Hubei Yunmeng Shuihudi M77 fajue jianbao” 湖北雲夢睡虎地M77發掘簡報, Jiang Han kaogu 江漢考古 109 (2008), 31–37 and Plates 11–16. NB: photos of only ten slips have been published to date. For an edition and an annotated translation of these slips, see Chemla, Karine and Biao, Ma 馬彪, “Interpreting a Newly Discovered Mathematical Document Written at the Beginning of Han Dynasty in China (before 157 B.C.E.) and Excavated from Tomb M77 at Shuihudi 睡虎地,” Sciamvs 12 (2011), 159–91Google Scholar.
18. This document, acquired in December 2007 on the Hong Kong antiquities market, is now housed in the Yuelu Academy 嶽麓書院 (Hunan University). In its editors’ view, Mathematics was probably composed no later than 212 b.c.e. On the date, see Zhu Hanmin 朱漢民 and Chen Songchang 陳松長 eds., Yuelu shuyuan cang Qin jian (er) 嶽麓書院藏秦簡（貳） (Shanghai: Shanghai cishu, 2011), 3, and Xiao Can 蕭燦, “Yuelu shuyuan cang Qin jian Shu yanjiu” 嶽麓書院藏秦簡《數》研究 (PhD diss., Hunan University, 2010), 16. They all argue in favor of this terminus ante quem. The title is on the verso of slip 0956 (editors’ number 1 verso), and its photograph is reproduced in Yuelu shuyuan cang Qin jian, 3.
19. See Zhangjiashan Han jian, 24, 29, 47, for only a few examples.
20. “The Suan shu shu was almost certainly designed as a work of ready reference for government bureaucrats of the Qin and early Han dynasties”; Dauben, “Suan Shu Shu,” 92. See also Peng, Zhangjiashan Han jian, 26.
21. Peng, Zhangjiashan Han jian, 6–12. On this level of the administration, see chapter 3, “Provincial and Local Government,” in Michael Loewe, The Government of the Qin and Han Empires (Indianapolis: Hackett Publishing Company, 2006), 37–55. About this kind of official, and their need of mathematical knowledge, see chapter 6, “The Officials,” in ibid., 71–85.
22. This was first noticed by Peng Hao, Zhangjiashan Han jian, 80–81 (slips 88–90). See the parallel regulation on the slips 41–43 of Shuihudi Qin mu zhujian 睡虎地秦墓竹簡, ed. Shuihudi Qin mu zhujian zhengli xiaozu 睡虎地秦墓竹簡整理小組 (Beijing: Wenwu, 1990), 29–30. Translation and analysis are given in Hulsewé, A., Remnants of Ch’in Law (Leiden: Brill, 1985), 42CrossRefGoogle Scholar.
23. Peng Hao, “Official Salaries and State Taxes as Seen in Qin-Han Manuscripts, with a Focus on Mathematical Texts,” in Mathematics, Administrative and Economic Activities in Ancient Worlds, ed. Cécile Michel and Karine Chemla (Cham: Springer Nature, 2020), 125–55. See also Horng et al., Shu zhi qiyuan, 141–53.
24. Loewe, Michael, “The Measurement of Grain During the Han Period,” T’oung Pao, 2nd ser. 49.1–2 (1961/62), 64–95Google Scholar.
25. Cullen, “The Suan Shu Shu: Rewriting,” 38.
26. Cullen, “The Suan Shu Shu: Rewriting,” 38.
27. Cullen, “The Suan Shu Shu: Rewriting,” 38.
28. Cullen, “The Suan Shu Shu: Rewriting,” 38.
29. Guo Shuchun, “Shilun Suanshu shu de lilun gongxian yu bianzuan,” 530, 510–12, respectively.
30. See Chemla, “Documenting a Process of Abstraction.”
31. Chemla, Karine, “Generality above Abstraction: The General Expressed in Terms of the Paradigmatic in Mathematics in Ancient China,” Science in Context 16.3 (2003), 413–58Google Scholar.
32. Neuf Chapitres, 526–27. I use small caps to distinguish the text of the canon from that of the commentaries.
33. The group of superiors has two persons and that of inferiors, five: the “difference” 差 is now of three persons (five minus two), and no longer “one,” as above.
34. Neuf Chapitres, 528–29. I have inserted paragraphs in the translation to make the argument clearer.
35. On the commentators’ understanding of the similarity between problems, see Chemla, “Qu’est-ce qu’un problème dans la tradition mathématique de la Chine ancienne?” Extrême-Orient, Extrême-Occident 19 (1997), 91–126.
36. Chemla, “Qu’est-ce qu’un problème.”
37. Neuf Chapitres, 282–83, and related endnotes.
38. On these terms, see Karine Chemla, “Glossaire des expressions techniques,” in Chemla and Guo, Neuf Chapitres, 936–37, 940.
39. Neuf Chapitres, 428–33, plus the endnotes.
40. Chemla, Karine, “Geometrical Figures and Generality in Ancient China and Beyond: Liu Hui and Zhao Shuang, Plato and Thabit Ibn Qurra,” Science in Context 18.1 (2005), 123–66CrossRefGoogle Scholar. The article contains a full bibliography on the topic.
41. See problem 5.17, Neuf Chapitres, 432–37.
42. Two pieces of information are needed to read. First, as length measurement units, 1 zhang 丈 = 10 chi 尺. Second, the unit chi is also used to state volumes, with the following convention: A volume of two chi is the volume of a right cuboid (or rectangular parallelepiped) with two chi height and a square face of one chi side. The square base does not change, while the height will expressthe designated amount of volume.
43. Zhangjiashan Han jian, 101–3. For other editorial remarks and suggestions of restoring of the text, see Guo Shirong, “Suanshu shu kan wu,” 283; Guo Shuchun, “Suanshu shu jiaokan,” 214; Cullen, The Suan Shu Shu: A Translation, 138; Kankan Sansû sho, 34–35 as well as Plate 6, editors’ number 14; Horng et al., Shu zhi qiyuan, 70–72, 257. Note, however, that the research group Chōkazan Kankan Sansū sho Kenkyūkai worked on photos of the manuscript taken after Peng Hao had completed his transcription and that for them, after the first two characters, the greatest part of the procedure at the bottom of slip 141 was illegible (see Plate 6).
44. Zhangjiashan Han jian, 101n1, considers that 美 is a copy mistake for 羡. Kankan Sansū sho, 34, note 1, suggests it is simply an abridged form. Peng (Zhangjiashan Han jian, 101) discusses the evidence enabling us to interpret the restored expression as a path towards a tomb.
45. Zhangjiashan Han jian, 101n2, suggests reading 定 as 頂. Cullen, The Suan Shu Shu: A Translation, 138n83, follows, without, however, translating in agreement with this interpretation (p. 89). The term has been interpreted in two ways: either as designating the most advanced face of the solid, which contains its deepest part (Peng, Zhangjiashan Han jian, 101, for example, interprets in this way), or as referring to a solid that is contrasted with the chu, the two making the yanchu. To my knowledge, this suggestion was first put forward by Su Yiwen 蘇意雯, Su Junhong 蘇俊鴻, Su Huiyu 蘇惠玉, Chen Fengzhu 陳鳳珠, Lin Cangyi 林倉億, Huang Qingyang 黃清陽, and Ye Jihai 葉吉海, “Suanshu shu jiaokan” 《算數書》校勘, HPM Tongxun 通訊 3.11 (Nov. 2000), 1–20 (16n152). In my view, it better fits the way the text describes the solid. The authors suggest the solid described was a combination of a half-parallelepiped (the chu, in this case) and a parallelepiped (ding). However, their publication depended on Peng’s incorrect reading of a numerical value in Zhangjiashan Han jian, 101 (see below), and the authors thought their interpretation did not correspond to the value of the volume given in Writings. Kankan Sansū sho, 34n3, gives the correct reading of the value, thereby highlighting that the hypothesis put forward by Su et al. in fact fits with the volume as stated in the manuscript. Kankan Sansū sho, 34, adopts the latter interpretation of the solid. Anicotte, Le Livre sur les Calculs, 250, also follows this interpretation and translates ding “fond,” signaling the meaning of “extrémité.”
46. Here, Zhangjiashan Han jian, 101, reads “九.” Kankan Sansū sho, 34n3, corrects the reading to “六.”
47. For those who followed the first interpretation of ding (see n. 45), the solid described was only the left part of what Figure 1 shows. They thus considered that either the volume or one of the data given was wrong. Zhangjiashan Han jian, 101n4, suggests correcting the result of the volume. Dauben, “Suan Shu Shu,” 153, also adopts this solution. However, in note d (p. 154), he points out the alternative interpretations. Cullen, The Suan Shu Shu: A Translation, 138n84, suggests correcting the value of the length as “五丈六尺.” Guo Shirong, “Suanshu shu kan wu,” 283, points out that one might indifferently correct one or the other. No correction is needed with an interpretation of the solid as shown on Figure 1.
48. That a scribe dropped a portion of the procedure, which I restore here, has been suggested in Horng et al., Shu zhi qiyuan, 70–72, 257; they presumed that the passage “以高及袤乘之，六而一, 即” was omitted. I suggest a slightly different emendation, whose rationale can be summarized as a saut du même au même. See, below, in the paragraph after the translation, for further evidence supporting my emendation.
49. Zhangjiashan Han jian, 101n5, considers that, in contrast with the previous one, this 定 ding should not be understood as referring to 頂 ding. As a result, the two occurrences of the same character on the same slip would have two different meanings. This is probably incorrect. Anicotte, Le Livre sur les Calculs, 252, suggests that the two occurrences should be interpreted alike. I adopt this interpretation. However, I do not adopt the way he understands the whole sentence. Clearly, from the emendation of the text in Horng et al., Shu zhi qiyuan, 70–72, 257, Horng et al., consider that the two ding have the same meaning. In-text slip numbers are enclosed in slashes.
50. On the fact that the terms designating the fundamental dimensions of a solid state its shape, see Chemla, Chapter D, in Chemla and Guo, Neuf Chapitres, 101–4. This is yet another practice that Writings shares with The Nine Chapters.
51. The alternative emendation suggested here relies on a proposition put forward by Guo Shirong, “Suanshu shu kan wu,” 283, who corrects the text of the procedure as follows (numbers 1–3 in brackets indicate numbered comments below): “朮（術）曰：廣積卅（三十）尺，(1) 以其/141 [高] (2)、袤乘之，[六而一] (3)，即定。/142/.” (1) Here, Guo Shirong considers 除高 as an interpolation, whereas I suggest alternatively that only 高 was interpolated. (2) Here, Guo Shirong considers that 廣 was erroneously written instead of 高. (3) Finally, Guo Shirong suggests that the division by 6 (六而一) was omitted. Guo Shuchun, “Suanshu shu jiaokan,” 214, considers the mistake occurred following another scenario. Here is the text as he corrects it, into which again, I have inserted note references to describe the scenario: “朮（術）曰：廣積卌（四十）尺，以除高 (1) /141/袤乘之，[六而一] (2)，即定。/142/.” (1) Here, Guo Shuchun suggests that the three characters 以其/141/廣 were interpolated. (2) Guo also considers that the division by 6 was omitted. Dauben, “Suan Shu Shu,” 153, translates a text modified along these lines, opting for a formulation closer to that of the procedure in The Nine Chapters. However, a typo led him to repeat the multiplication by the width erroneously.
52. Note that in the context of this alternative suggestion, 定 is not read as 頂 here. However, there is no other such occurrence of the character 定 with the intended meaning in Han mathematical texts. This remark undermines the likelihood of this emendation.
53. Neuf Chapitres, 432–33. In modern terms, this solid is a pentahedron with a trapezoidal base and a face perpendicular to the base. Note that in Writings, the solid has a “height” and not a “depth,” as in The Nine Chapters. Why exactly this is the case needs to be clarified.
54. Guo Shuchun, “Shilun Suanshu shu de lilun gongxian yu bianzuan,” is devoted to describing the lack of “standardization” of Writings and to interpreting this feature from the standpoint of the history of mathematics in early imperial China (see n. 14).
55. Slip 0977, editors’ number 193, Xiao Can, “Yuelu shuyuan cang Qin jian,” 96; Yuelu shuyuan cang Qin jian, 27. Also see Anicotte, Le Livre sur les Calculs, 253.
56. On this specificity, see Karine Chemla, The Practices of Generality in Various Epistemological Cultures (Uppsala: Salvia Småskrifter, 2019), www.idehist.uu.se/digitalAssets/775/c_775182-l_1-k_2019motley-practices-of-generality--final-versionrausinglecture2017originalcorrected.pdf.
57. Slips 183–84, Zhangjiashan Han jian, 123–24. Kankan Sansū sho, 9–10 and pl. 3.
58. As above (see n. 50), the shape of the rectangle is indicated by the names of the fundamental dimensions of the figure.
59. Kankan Sansū sho, 9, considers that one of the damaged characters in the statement of the problem might be tian 田.
60. Neuf Chapitres, 170–73.
61. Neuf Chapitres, 172–73, emphasis added. The similarity between the two procedures was noted notably by Zhangjiashan Han jian, 124, Cullen, The Suan Shu Shu: A Translation, 109–10, Dauben, “Suan Shu Shu,” 95.
62. This property of procedures is analyzed in greater detail in Chemla, Karine, “Describing Texts for Algorithms: How They Prescribe Operations and Integrate Cases. Reflections Based on Ancient Chinese Mathematical Sources,” in Texts, Textual Acts and the History of Science, ed. Chemla, Karine and Virbel, Jacques (Dordrecht: Springer, 2015), 317–84CrossRefGoogle Scholar, esp. 341–43, 359–60.
63. For them, this was precisely what its name expresses, and so I follow their interpretation and translate da guang 大廣 as “the greatest generality.” I thus interpret here guang 廣 as “general,” and not as “width,” which in my view does not make sense here. In particular, da guang has been considered as opposite to shao guang 少廣. However, this does not hold either—da 大 is opposed to xiao 小, and shao 少 is opposed to duo 多—and hence the names of these two operations need not be translated as a pair. I therefore take da as enhancing the generality to which guang 廣 refers. Both options lead me to translate differently from Cullen, The Suan Shu Shu: A Translation, 109–10 (“The greater breadth”) and from Dauben, “Suan Shu Shu,” 165–66 (“General widths”). One can find a compilation of all the translations of this expression that have been offered in Guo Shuchun 郭書春, Joseph W. Dauben 道本周, and Xu Yibao 徐義保, Nine Chapters on the Art of Mathematics. With the Annotations by Liu Hui [State of Wei] and Notes and Annotations of Li Chunfeng and Associates [Tang Dynasty] 九章算術，魏劉徽注，唐李淳風注釋. A Critical Edition and English Translation Based upon a New Collation of the Ancient Text and Modern Chinese Translation by Guo Shuchun. English Critical Edition and Translation, with Notes by J. Dauben and Xu Yibao 郭書春校勘并譯注。道本周，徐義保英譯并注, 3 vols. (Shenyang: Liaoning jiaoyu, 2013), vol. 1, 79.
64. Exactly the same situation recurs with the name of the operation jing fen 徑分 (directly sharing). This is the title in Writings (Zhangjiashan Han jian, 48) and the one glossed by Li Chunfeng et al.; see Neuf Chapitres, 166–67. However, in all the ancient editions of The Nine Chapters, the name of the operation is recorded as jing fen 經分. Rarely have authors addressed how Writings could help us assess the reliability of the information about The Nine Chapters given in the editions of the work that have come down to us. The comparison between the manuscript and the commentaries might help us restore states of parts of the canon closer to what the ancient commentators saw.
65. In this, I differ from the assumption tacitly adopted by most historians (e.g., Cullen, The Suan Shu Shu: A Translation, 24–25; Anicotte, Le Livre sur les Calculs, 69–74). My argumentation is sketched in Chemla, Karine, “Observing Mathematical Practices as a Key to Mining Our Sources and Conducting Conceptual History. Division in Ancient China as a Case Study,” in Science after the Practice Turn in Philosophy, History, and the Social Studies of Science, ed. Soler, Léna et al. (New York: Routledge, 2014), 238–68Google Scholar, esp. 242–48, 257–62.
66. See Chemla and Ma, “Interpreting a Newly Discovered Mathematical Document,” and Chemla, “Numerical Tables.”
67. Cullen, “The Suan Shu Shu: Rewriting,” 28.
68. Zhangjiashan Han jian, 50–52. Kankan Sansū sho, 147–48 and plate 22.
69. Zhangjiashan Han jian, 51n2, considers that after 共， 買 was omitted.
70. Zhangjiashan Han jian, 51n3, suggests that after 三， 錢 was omitted.
71. That is: giving back the money in proportion of what each paid. On the name of this operation, see below.
72. Cullen, “The Suan Shu Shu: Rewriting,” 22, notes this fact.
73. I argue the text embodies an understanding of abstraction shared by Han actors in Chemla, Karine, “Writing Abstractly in Mathematical Texts from Early Imperial China,” in Technical Arts in the Han Histories: Tables and Treatises in the “Shiji” and “Hanshu,” ed. Csikszentmihalyi, Mark and Nylan, Michael (Albany: State University of New York Press, 2021), 307–38Google Scholar.
74. The operation “suppose” is the form of “rule of three” contained in The Nine Chapters (Neuf Chapitres, 222–23). The expression occurs in both Liu Hui’s commentary and Li Chunfeng’s subcommentary, after problem 6.24 (Neuf Chapitres, 538–39).
75. Neuf Chapitres, 158–59.
76. Emphasis added. Zhangjiashan Han jian, 45–47.
77. My emphasis. Here and below, I rely on Neuf Chapitres, 158–59, references being indicated in the related footnotes.
78. On this, see Chemla, “Glossaire des expressions techniques,” 948–49. As a rule, the commentators use the concept of “category” in two main ways: they either discuss how mathematical objects sharing the same category can interact with each other or how they can be all known in the same respect in virtue of sharing the same category. The uses of the concept in Writings and in the commentary on “Gathering parts” both fall under the former type of use.
79. Neuf Chapitres, 364–66.
80. Neuf Chapitres, 456–57, as well as 36–39.
81. Chemla, “Glossaire des expressions techniques,” 924–25.