Abstract
The chapter provides a history of Chinese counting rods (suan 算/筭) and a description of operations performed with them. The author discusses the representation of positive and negative numbers with counting rods of two colors or of two different cross-sections used in the procedure of solution of linear simultaneous equations described in the mathematical treatise Jiu zhang suan shu 九章筭術 (Computational Procedures of Nine Categories) completed no later than the early first century CE and explained in the commentary on it written by Liu Hui 劉徽 in 263. He also describes arithmetical operations performed with the counting rods mentioned in the mathematical treatise of the first millennium CE Sun zi suan jing 孫子筭經 (Master Sun’s treatise on computations) and in the treatise Shen dao da bian li zong suan hui 神道大編曆宗算會 (Grand Compendium of Divine Dao: [Chapters presenting methods] of lineages [of experts] in calendrical computations and of assembly of mathematicians [of the past]) completed by Zhou Shuxue 周述學 in 1558. The author claims that the “algorithmic style” of the traditional Chinese mathematics was intimately related with the use of the counting rods and that the transition from this instrument to the beads abacus (suanpan 算盤) that took place in the first half of the second millennium CE coincided with the general decline of Chinese mathematics.
The author is grateful to two anonymous referees for their useful suggestions. All the remaining problems are his responsibility.
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Notes
- 1.
For Chinese terms, I use the pinyin transliteration of modern Mandarin reading of Chinese characters written in their traditional (not simplified) form.
- 2.
- 3.
Needham (1959, pp. 71–72).
- 4.
- 5.
There exist two Chinese characters used to refer to the counting rods, 算 and 筭; the reading of both of them is “suan” in modern Mandarin dialect. However, their meanings were not fully identical, and the difference is discussed in a section below. Throughout this paper, when citing an original text, I use the version of the character found in the edition cited.
- 6.
Although the historical treatise is authored by Ban Gu 班固 (32–92 CE), the most part of the chapter in question is to be credited to the authorship of Liu Xin 劉歆 (46 BCE–23 CE), as the introductory remarks of Ban Gu suggest; see Vogel (1994, pp. 141, 145, 147).
- 7.
HS, juan 21a, p. 2b. One cun at that time was approximately equal to 2.765 cm, according to Wu (1937, p. 65). The extant specimens of the counting rods currently preserved in Beijing University have round cross-sections; their length varies from 11 to 13 cm; see Han (2012); Dauben et al. (2013, p. 931, n. 4).
- 8.
At first sight, the round cross-section of the rods is suggested by the use of the term jing 徑 found in their description; this term appears in mathematical texts in relation to circles and is conventionally translated as “diameter.” However, the term jing was also used by some ancient authors to refer to the segment cutting a square through its center and parallel to two of its sides; see, for example, the commentary of Zhao Junqing 趙君卿 (fl. ca. 3rd c. CE) on the Zhou bi suan jing 周髀算經 (Computational treatise on the Gnomon of the Zhou [dynasty]) (ZBSJ 2001, p. 33). The term jing thus can be tentatively interpreted as referring to the shortest segment crossing a flat figure and passing through its center of symmetry. The conclusion about the round cross-sections of the rods mentioned in the Han shu is made on the basis of the reconstruction of the structure of the bundle mentioned in this source (see below).
- 9.
Li (1937 [1983], p. 63); see also Needham (1959, p. 71). The pictures provided by Li Yan show a cross-section of the bundle in which the sections of rods are shown as black and white dots, the number of the rods represented with black dots equals to 271, while the number of those represented with white ones equals to 60. The function of the white dots is not explained. Li Di (1997, p. 54) and Lam and Ang (2004 [1992], p. 44, Fig. 2.3) provide versions of Li Yan’s drawing in which the white dots are removed.
- 10.
Wang (1524, p. 348).
- 11.
On this dictionary, see Bottéro and Harbsmeier (2008).
- 12.
SWJZ, juan 5.1, p. 9b.
- 13.
That is, 13.824 cm, since 1 cun at that time was equal to 2.304 cm (Wu 1937, p. 65).
- 14.
Ibid.
- 15.
It is sometimes difficult to decide whether the rods mentioned in earlier sources were used for mathematical computations, for simple “one-by-one” counting, for divination, or for other purposes.
- 16.
- 17.
SJSS (2001, p. 447).
- 18.
SJSS (2001, p. 448).
- 19.
Wu (1937, p. 192).
- 20.
SJSS (1963, p. 542).
- 21.
See Volkov (2000).
- 22.
A translation and discussion of this excerpt is found in Section “The Reason for Alternation of Orientation of the Counting Rods” below.
- 23.
SS (1973, vol. 2, p. 387).
- 24.
Most probably, by the “width” the author means the diameter of the circle in the cross-section if the rods were designed as oblong cylinders, or the side of the square or of the equilateral triangle in the case if their cross-sections were square or triangular, respectively.
- 25.
- 26.
The title of this treatise has various renderings, the suan shu has been understood by some authors as “art of mathematics” (see, for ex., Dauben et al. 2013), while others suggested “computational prescriptions” (Martzloff 1997, p. 124) or “procédures mathématiques” [i.e., “mathematical procedures”] (Chemla and Guo 2004); the latter interpretation is certainly better, given that in the text of the treatise the term shu is systematically used to refer to computational procedures provided for solutions of problems. The term jiu zhang is almost unanimously understood as referring to the number of chapters of the treatise; this interpretation is based upon the modern meaning (“chapter”) of the term zhang. However, as the author of these lines argued in a number of publications, the term jiu zhang historically referred to nine types of mathematical methods, i.e., to the curriculum of “traditional mathematics,” and not to the number of chapters (this interpretation is confirmed by the existence of versions of the treatise bearing the same title but comprising different numbers of chapters).
- 27.
The exact time of compilation of the treatise is unknown . The preface of Liu Hui 劉徽 (263 CE) mentions Zhang Cang 張蒼 (b. before 252–d. 152 BCE) and Geng Shouchang 耿壽昌 (fl. 57–52 BCE) as its compilers, see SJSS (1963, p. 91), Chemla and Guo (2004, pp. 54–55, 127); see also Martzloff (1997, p. 129, nn. 29–30).
- 28.
SJSS (1963, pp. 150, 153).
- 29.
See problem 5 of juan 3 and in problem 4 of juan 6 (SJSS 1963, pp. 134, 124–125; SJSS 2001, pp. 111, 146–147); for translations, comments, and relevant references see Berezkina (1957, pp. 458–459, 481–482, 529, n. 7); Chemla and Guo (2004, pp. 289–291, 499–501, 989); Dauben et al. (2013, pp. 277–279, 671–673). The term certainly could not refer to actual tokens used for operations, since the amounts of suan assigned to administrative units (counties xiang 鄉 and prefectures xian 縣) were rather large: for example, in problem 5 of juan 6, the numbers of suan assigned to six prefectures were equal to 42000, 34272, 19328, 17700, 23040, and 19136, respectively.
- 30.
SJSS (1963, pp. 225, 237–238, 240).
- 31.
According to Cullen (1996).
- 32.
- 33.
The Huainan zi contains one mention of the counting rods suan, in Chapter 2 “Chu zhen xun” 俶真訓; the second occurrence of this term (in Chapter 20 “Tai zu xun” 泰族訓) can be understood as “[result of] computations.” For translations of these excerpts, see Le Blanc and Mathieu (2003, pp. 60 and 949), respectively. Needham (1959, p. 71, note b) reports that there are mentions of the counting rods in Chapter 14 of the same treatise; he apparently means the instrument chou 籌 mentioned twice in this chapter.
- 34.
- 35.
Boltz (1993, p. 237).
- 36.
Couvreur (1951, pp. 142–143).
- 37.
Boltz (1993, p. 236).
- 38.
箭籌八十。長尺有握。握素。(YL, juan 5b, p. 39a), that is, “the [number of] jianchou is 80, the length [of each jianchou] is 1 chi [and it] has a handle (or ‘grip’) [from which the bark] was removed (or ‘without color’).” Couvreur (1951, p. 175) suggests that the length of each jianchou was 20 cm, but the exact length of the object can be determined only if one knows the exact date of compilation of the text, since the length of 1 chi may have varied considerably throughout the first millennium BCE (Wu 1937). Couvreur translates jianchou 箭籌 with the French word “fiches,” while Steele’s translation of this term as “arrows” (Steele 1917, vol. 1, p. 120) looks somewhat misleading since it can make the reader think that he speaks about actual arrows for shooting; the latter hypothesis should be ruled out if one takes into account the length of these objects.
- 39.
Needham referred to the time when the treatise was compiled as “Warring States period (-fourth and -third centuries).” Since the conventional dates for the Warring States period (provided on p. 875 of his volume) are 480–221 BCE, Needham probably meant that he agreed with the opinion shared by a number of sinologists who believed that the compilation of the treatise was completed much later after the conventional dates of life of its legendary author.
- 40.
Needham (1959, p. 70).
- 41.
Needham (1959, p. 71, n. a) claimed that “a misprint in the Thai-Phing Yu Lan [i.e., Tai ping yu lan 太平御覽.-A.V.] encyclopaedia caused de Lacouperie to make nonsense of this reference.” Needham most likely refers to the quote from the Tai ping yu lan found in Albert Terrien de Lacouperie’s (1845–1894) paper (1883) on p. 330. As footnote 20 on p. 302 of the same paper suggests, Terrien de Lacouperie used an edition of 1807 preserved at that time in the Library of Royal Asiatic Society (London); I was unable to get access to this edition. The translation of the phrase under consideration suggested by Terrien de Lacouperie (“good mathematician ought not to use counting stalks”) does not seem particularly wrong, so the misprint mentioned by Needham was most likely related to the source of this quotation. On Terrien de Lacouperie and his work, see de la Grassiere (1896).
- 42.
Two extant versions of the treatise (the so-called Heshang Gong’s 河上公 (fl. ca. 180–157 BCE) and Wang Bi’s 王弼 (226–249 CE) editions, respectively) do not contain the character zhe 者, unlike three other versions (Mawangdui manuscripts A and B, ca. 250 BCE, and Fu Yi’s 傅奕 (ca. 555–639 CE) version).
- 43.
Liu Hui also used the term ce 策 in referring to counting rods elsewhere, see SJSS (1963, p. 237, ln. 4).
- 44.
SJSS (1963, p. 168).
- 45.
Chemla and Guo translate it as “Lorsque l’on cherche à aller jusqu’au bout des quantités (shu), cela signifie que l’on le déduit à l’aide de la situation (géométrique) (qing), cela n’implique pas de calcul avec les baguettes” (2004, p. 433), while Dauben, Guo, and Xu paraphrase the excerpt as “When mathematics is used to deal with something that is inexhaustible, reasoning should be used, not counting rods” (2013, p. 557).
- 46.
- 47.
Cheng (1993).
- 48.
- 49.
See also Chen (1978, pp. 284, 316, notes 110–111).
- 50.
Needham (1959, pp. 14, 70).
- 51.
Djamouri (1994, p. 20).
- 52.
Martzloff (1997, pp. 185–186, Fig. 12.4). Martzloff writes that the rod-symbols similar to those shown in this figure “are also attested well before” the first century CE, but does not provide references to support his claim.
- 53.
Chen (1978, pp. 278–279).
- 54.
Chen (1978, p. 278).
- 55.
- 56.
Hua (1987, p. 58).
- 57.
Needham (1959, pp. 62, 69–72).
- 58.
Libbrecht (1973, p. 488).
- 59.
Libbrecht (1982, p. 207).
- 60.
Libbrecht (1982, p. 207) provides two quotations to support his claim. The first one says “Put (chih [= zhi] 置) on the counting board 36,783” (the source is not specified). Here, Libbrecht makes his conclusion of the use of the hypothetical counting board on the basis of the mere presence of the term zhi 置 in the text, but this term does not specify the instrument used. The second example of Librecht reads “Chih yü shang fang [= zhi yu shang fang] 置於上方, put it on the first board.” Once again, the source of the quotation is not provided, and the phrase quoted undoubtedly means “place [the operand] in the upper row” or “place [the mentioned value] in [the position of] linear term [located] above”; the meaning of the term fang used here depends on the context of the cited instruction, but it certainly refers to a position on the counting surface (“upper side,” i.e., the upper row, or the position of linear term of a polynomial equation), and not to a “board.”
- 61.
- 62.
- 63.
- 64.
Martzloff (1997, p. 209).
- 65.
Hua (1987, p. 58).
- 66.
This treatise contains detailed explanations of operations of multiplication and division performed with counting rods accompanied by numerous diagrams representing the configurations of rods on the counting surface (An 1373a, pp. 1354–1368; An 1373b, pp. 1578–1591). See also Li (1999, pp. 362–365).
- 67.
- 68.
- 69.
Similar mantic figures are shown and discussed in Kalinowski (1994, pp. 43–44, 57, 59).
- 70.
- 71.
According to Smith and Mikami (1914, p. 23), the sangi were wooden prisms 5 cm high with one side of the square base equal to 7 mm.
- 72.
Horiuchi (1994, p. 97).
- 73.
Smith and Mikami (1914, p. 46) claimed that the first edition of the book appeared in 1716. The edition of 1795 used by them remained unavailable to me.
- 74.
- 75.
- 76.
Hua (1987, pp. 58–59).
- 77.
The treatise was compiled most likely between the late third and the early fifth century CE; see Lam and Ang (2004 [1992], p. 28); Martzloff (1997, p. 124). For a detailed presentation of the textual history of the treatise, see Lam and Ang (2004 [1992], pp. 28–32); see also SJSS (1963, pp. 276–277), SJSS (2001, p. 287). The treatise was translated into Russian (Berezkina 1963), French (Schrimpf 1963, pp. 446–456; chapters 2 and 3 only), and English (Lam and Ang 2004 [1992]).
- 78.
Lam and Ang (2004 [1992], p. 193) interpret the universal quantifier fan 凡 as meaning “common [methods of computation ]”; this understanding may suggest existence of some other (“uncommon”) methods. Compare with Berezkina’s rendering “Правило, которое [употребляется] всякий раз при умножении” (Rule that [is used] every time for multiplication) (1963, p. 23); note, however, that the original text does not mention multiplication.
- 79.
The term “position” (wei 位) is ambiguous; it can mean a position of one digit in each operand, but also can refer to a position of an operand on the counting surface (it is used in the latter sense in the methods described in this treatise immediately after the discussed excerpt.) Lam and Ang (ibid.) prefer the latter interpretation and translate wei as “the positions [of the rod numerals]” while Berezkina renders wei as “разряды” (“positions [of digits]”, ibid.) thus preferring the former interpretation. The explanation provided right after the phrase in question seems matching better Berezkina’s interpretation.
- 80.
SJSS (1963, p. 282), SJSS (2001, p. 262). Lam and Ang (2004 [1992], p. 194, note 2) translate this phrase “6 is not accumulation [of rods] and 5 is not a single [rod]” and add that this rule should be placed in the paragraph on the rod numerals, as Qian Baocong suggested (SJSS 1963, p. 558). Berezkina in her translation did not change the position of this phrase (most likely, because it makes sense in the paragraph devoted to multiplication where it is found in received editions of the treatise), see Berezkina (1963, pp. 23; 45, n. 22).
- 81.
There are reasons to believe that this treatise is a later compilation produced by Han Yan 韓延 in the late eighth century; see SJSS (1963, pp. 551–553). There exists an annotated Russian translation of the treatise (Berezkina 1985); see also Schrimpf’s translation of all the problems of the treatise (1963, pp. 468–482) and Berezkina (1980, pp. 52–54).
- 82.
That is, for a digit n > 5 a counting rod representing five should be placed (horizontally or vertically) above the remaining (n − 5) rods.
- 83.
- 84.
P. 3349; Li (1955b, p. 28).
- 85.
Libbrecht (1982, p. 217), compared this line of the Dunhuang treatise only with that found in the Sun zi suan jing.
- 86.
- 87.
- 88.
Brenier (1994, pp. 100 ff).
- 89.
Red color traditionally represented South and thus the (positive) cosmic force Yang 陽, while the black color represented North and the negative force Yin 陰.
- 90.
The numbers of sides (three and four) were numerical symbols of the cosmic forces Yang and Yin, respectively; see below.
- 91.
Needham (1959, p. 71). Apparently, black represented Yin, and white, Yang.
- 92.
SS 1973, juan 16, p. 3b.
- 93.
Qian is the symbol representing the cosmic principle Yang and the Heaven.
- 94.
Kun is the symbol representing the cosmic principle Yin and the Earth.
- 95.
This statement can be understood as referring to the fact that 12 = 3·4, where 3 is the “Number of the Heaven”, and 4, the “number of the Earth”. Another possibility to understand the link between the numbers 3, 4, and 12 is to interpret 12 as the perimeter of the right-angled triangle with the sides (3, 4, 5).
- 96.
The date of compilation of the Xi ci zhuan is uncertain; as W. Peterson suggests: “There is no direct evidence that it existed before the founding of the Han dynasty [i.e., 221 BCE.-A.V.], but it is apparently quoted or paraphrased in writings from the second century B.C. […] [There are pieces of evidence that] strongly suggest that by the middle of the second century B.C. the ‘Commentary on the Attached Verbalizations’ existed and was circulating […].” (Peterson 1982, pp. 75–76)
- 97.
YJ, juan 7, p. 29a.
- 98.
SJSS (1963, vol. 1, pp. 225–226).
- 99.
SJSS (1963, vol. 1, p. 225).
- 100.
- 101.
- 102.
- 103.
Libbrecht (1973, p. 70).
- 104.
There is no description of the algorithm of solution of quadratic equations in the Jiu zhang suan shu; however, the latter treatise contains a technical term that allows to reconstruct the procedure apparently used in Problem 19 of Chapter 9 of the treatise, see SJSS (1963, pp. 255–256); SJSS (2001, p. 191); for translations and analysis, see Berezkina (1957, p. 582, n. 23); Berezkina (1980, pp. 215–216); Chemla and Guo (2004, p. 892); Dauben et al. (2013, pp. 1131–1137). The Zhang Qiujian suan jing 張丘建算經 (Computational treatise of Zhang Qiujian) compiled in the fifth century CE contains two problems to be solved with quadratic equations, see SJSS (2001, pp. 321, 326); for translations see Berezkina (1969, pp. 47, 49), and for an analysis see Berezkina (1969, pp. 75–76; 1980, pp. 216–219). (Full) cubic equations are (numerically) solved in the Qi gu suan jing 緝古筭經 (Computational treatise on the continuation [of traditions] of ancient [mathematical methods]) compiled by Wang Xiaotong 王孝通 in 626 CE, see SJSS (2001, p. 447); for translations, see Berezkina (1975) and Lim and Wagner (2017); see also Berezkina (1980, pp. 227–232).
- 105.
- 106.
- 107.
SJSS (2001, p. 262).
- 108.
The literal meaning of this instruction is not very clear. The words shang xia xiang guan 上下相觀 can mean two different things: (1) this is a requirement stating that the two numbers should “look at” each other; in this case, “look at” is a technical term explained in the subsequent phrases; (2) the word xiang is used to say that the two numbers should “act” or should “be acted upon” similarly, in this case they are “observed together” by the operator. Berezkina renders this instruction with a rather obscure phrase “… [so that the numbers] in upper and lower [lines] would be placed accordingly” ([так, чтобы числа] в верхней и нижней [строках] были соответственно расположены) (1963, p. 23) and provides a word-by-word translation “[in such a way that] their upper and lower [decimal positions] would be mutually seen through” ([чтобы] верхние и нижние [разряды.-A.V.] взаимно просматривались) in note 18 on p. 45 without further explanations. Lam and Ang suggest “the upper and lower position facing each other” (2004 [1992], p. 193).
- 109.
This instruction was apparently misinterpreted by Berezkina (1963, p. 23; 1980, p. 90). Lam and Ang suggested the following interpretation of the verb bu (literally meaning “to walk”, “to move [step by step]”, here “to shift [counting rods] position by position”): “if there are tens in the upper position then the correspondence is with the tens, [i.e., the units of the lower numeral are below the tens of the upper numeral];” they repeated this interpretation for the cases of hundreds and thousands (2004 [1992], pp. 193–194).
- 110.
That is, to multiply them. This is a tentative translation; the term ming had different technical meanings in other texts, in particular, in the Jiu zhang suan shu. Berezkina (1963, p. 23) simply renders it as “multiply”; Lam and Ang suggest “the upper commands the lower” (2004 [1992], p. 194). An example provided in the treatise uses another term (hu) in a similar context; see below.
- 111.
- 112.
Lam and Ang (2004 [1992]) suggest that this instruction should be removed from this algorithm and placed in the section discussing the representation of numbers with counting rods (see above) (p. 194, n. 3).
- 113.
- 114.
Berezkina provides diagrams showing the configuration of rods on the counting surface in the process of solution of the first problem of the treatise (multiplication of 81 by itself) (Berezkina 1980, p. 91); Lam and Ang (2004 [1992]) provide an example of multiplication of 7239 by 23 (p. 58) and solution of the first problem (p. 59); they offer pictures of configurations of counting rods as well as a transcription in modern numerical notation. Martzloff in his monograph (1997) reproduced the multiplication of 81 by 81 with a mistake (or misprint): according to his diagram on p. 217, the partial result obtained at the first step, 80·80, equals to 640 and not to 6400.
- 115.
Berezkina (1963, p. 48, n. 37).
- 116.
This is the opening statement of the procedure described above quoted verbatim.
- 117.
The interested reader can find the complete description of the procedure in Volkov (2018).
- 118.
The complete translation of the description of this method is found in Volkov (2018).
- 119.
This is a commentary printed in smaller characters. Lam and Ang (2004 [1992], p. 195, n. 4) claim that this commentary was added by Li Chunfeng 李淳風 (602–670); however, this is rather unlikely, given that all other commentaries authored by Li Chunfeng were opening with a mention of his name. Instead, it is rather likely that this commentary was added by the author of the text himself.
- 120.
- 121.
Berezkina (1963, p. 23), interprets yi 異 as “opposite” and suggests “Правило, которое [употребляется] всякий раз при делении, прямо противоположно умножению:…” (The rule that [is used] every time for division , is directly opposite to multiplication). Lam and Ang (2004 [1992], p. 194) translate “In the common method of division (fan chu zhi fa凡除之法), this is the reverse of multiplication.” On the use of the term “common” see above; what the word “this” refers to in their rendering remains unclear. As Berezkina before them, they consider yi 異 as referring to an “opposite” operation, while the term zheng 正 vanishes from their translation. The authors of both translations apparently consider yi as the only predicate in this statement, and interpreted zheng as an additional term that qualified the predication (“directly opposite”, as Berezkina suggested) and, in principle, was not very important (this is probably why Lam and Ang omitted it). However, it can be suggested that the terms zheng yi form an expression in which both parts are equally important.
- 122.
For a different interpretation of this phrase see Lam and Ang (2004 [1992], p. 195).
- 123.
The received versions of the treatise read “the divisor retreats (tui 退) for two positions”; however, the previous phrase suggests that the text contains an error, and the word jin 進 (to advance) was erroneously replaced by tui 退 (to retreat). The word “retreat” in this position would have lead to a mathematically wrong procedure.
- 124.
- 125.
- 126.
Needham (1959, p. 70, Fig. 66).
- 127.
Cheng Dawei published two version of his treatise; the one published in 1592 comprised 17 juans (“scrolls”), while the version published in 1593 contained 12 juans.
- 128.
- 129.
Chen (1977, p. 243).
- 130.
For instance, the edition of 1593 produced by the Wensheng tang 文盛堂 publishing house (currently preserved in Waseda University Library, Japan) does not contain the picture of the teacher and disciples, while the edition printed in the Dunhua tang 敦化堂 publishing house in the same year does contain the picture shown in Fig. 34.
Bibliography
Part A: Primary Sources
An Zhizhai 安止齋. 1373a. Xiang ming suan fa 詳明算法 (Detailed Clarifications of Computational Methods). In Guo 1993, vol. 1, pp. 1345–1397.
An Zhizhai 安止齋. 1373b. Xiang ming suan fa 詳明算法 (Detailed Clarifications of Computational Methods). In Jing 1994, vol. 1, pp. 1573–1619.
Cheng Dawei 程大位. 1592. Suan fa tong zong jiao shi 算法統宗校釋 (Unified Origin of Computational Methods, with Emendations and Explanations). Mei Rongzhao 梅荣照 and Li Zhaohua 李兆华 (Eds.). Hefei: Anhui jiaoyu.
Ding Ju 丁巨. 1355. Ding Ju suan fa 丁巨算法 (Computational methods of Ding Ju). In Guo 1993, vol. 1, pp. 1299–1314.
Guo Shuchun 郭書春 (ed.). 1993. Zhongguo kexue zhishu dianji tonghui 中國科學技術典籍通彙 (Comprehensive compendium of classical texts related to science and technology in China). Shuxue juan 數學卷 (Mathematical section). Zhengzhou: Henan jiaoyu chubanshe 河南教育出版社.
HNZ 1989 – Liu Wendian 劉文典 (ed.). Huainan zi honglie jijie 淮南子鴻烈集解 (Assembled Explanations of the Vast and Luminous Book [Compiled by] Masters of Huainan). Beijing: Zhonghua shuju, 1989.
HS – Ban Gu 班固. Han shu 漢書 (History of the [Early] Han [Dynasty]). Qin ding Si ku quan shu hui yao 欽定四庫全書薈要 edition, 1779.
Jing Yushu 靖玉樹 (ed.). 1994. Zhongguo lidai suanxue jicheng 中國歷代算學集成 (Collection of historical [documents on] mathematics in China). Jinan: Shandong renmin chubanshe.
LSQS – Mei Wending 梅文鼎, Li suan quan shu 歷算全書 (Complete Writings on Calendrical Computations). Qin ding Si ku quan shu 欽定四庫全書 edition, 1782.
P. 3349 – Suan jing 笇[=算]經 (Treatise on computations). Bibliotheque Nationale de France, Manuscript Pelliot 3349.
SJSS 1963 – Qian Baocong 錢寶琮 (ed.), Suan jing shi shu 算經十書 (Ten Treatises on Computations). Beijing: Zhonghua shuju.
SJSS 2001 – Guo Shuchun 郭書春 and Liu Dun 劉鈍 (eds.), Suan jing shi shu 算經十書 (Ten Treatises on Computations). Taibei: Jiu zhang.
SS 1973 – Sui shu 隋書 (History of the Sui [dynasty]). Beijing: Zhonghua shuju.
SSS 2001 – Peng Hao 彭浩. Zhangjiashan Han jian ‘Suan shu shu’ zhushi 張家山漢簡算數書注釋 (Commentary and Interpretation of the Han [Dynasty Treatise] Suan shu shu [Written] on Bamboo Strips [and Found at] Zhangjiashan). Beijing: Kexue, 2001.
STS 1698 – Satō Shigeharu 佐藤茂春, Sanpō tengen shinan 算法天元指南 (Compass [for understanding the technique of] ‘Celestial Element’ [used in] computational methods) (preface 1698).
SWJZ – Xu Shen 許慎. Shuowen jiezi 說文解字 (Explanation of graphs and analysis of characters). Qin ding Si ku quan shu hui yao 欽定四庫全書薈要 edition, 1779.
Wang Wensu 王文素. 1524. Suan xue bao jian 算學寳鑑 (Precious Mirror of the Science of Computations). In Guo 1993, vol. 2, pp. 333–971.
Yang Hui 楊輝. 1274. Cheng chu tong bian ben mo 乘除通變本末 (Alpha and Omega of Continuation and Change in [methods] of Multiplication and Division). In Guo 1993, vol. 1, pp. 1047–1072.
YJ – Yi jing tong zhu 易經通注 (The Book of Changes with comprehensive commentaries), Qin ding Si ku quan shu 欽定四庫全書 edition, 1782.
YL – Yi li ji shuo 儀禮集說 (The Rituals and Ceremonies with Anthology of Explanations), Qin ding Si ku quan shu hui yao 欽定四庫全書薈要 edition, 1779.
Zhou Shuxue 周述學. 1558. Shen dao da bian li zong suan hui 神道大編曆宗算會. In: Xuxiu Si ku quan shu 續修四庫全書, vol. 1043, pp. 555–832. Shanghai: Shanghai guji chubanse.
ZBSJ 2001 – Zhou bi suan jing 周髀算經 (Computational treatise on the Gnomon of the Zhou [dynasty]). In SJSS 2001, pp. 29–65.
Part B: Translations of Chinese Texts
Berezkina, Éľvira I. [Березкина, Эльвира И.] (tr.). 1957. Математика в девяти книгах (Mathematics in nine books [= Jiuzhang suanshu], an annotated Russian translation). Историко-математические исследования (Studies in the History of Mathematics), vol. 10, pp. 427–584.
Berezkina, Él’vira I. [Березкина, Эльвира И.] (tr.). 1963. О математическом труде Сунь-цзы. Сунь-цзы: Математический трактат. Примечания к трактату Сунь-цзы (On the mathematical work of Sun zi. Sun zi: Mathematical treatise. Annotations for Sun zi’s treatise). Из истории науки и техники в странах Востока (On the history of science and technology in the countries of the East), Moscow: Nauka, vol. 3, pp. 5–70.
Berezkina, Éľvira I. [Березкина, Эльвира И.]. 1969. О трактате Чжан Цю-Цзяня по математике (On Zhang Qiujian mathematical treatise). Физико-математические науки в странах Востока (Physics and mathematics in the countries of the East), vol. 2, pp. 18–81 (The paper contains the full annotated translation of the treatise).
Berezkina, Éľvira I. [Березкина, Эльвира И.] (tr.). 1975. Ван Сяо-тун. Математический трактат о продолжении древних (методов) (Wang Xiaotong. Mathematical treatise on the continuation of ancient (methods), an annotated Russian translation). Историко-математические исследования (Studies in the History of Mathematics), vol. 20, pp. 329–371.
Berezkina, El’vira I. [Березкина, Эльвира И.] (tr.). 1985. Математический трактат Сяхоу Яна (Mathematical treatise of Xiahou Yang). Историко-математические исследования (Studies in the History of Mathematics), vol. 28, pp. 293–337.
Chemla, Karine, Guo Shuchun. 2004. 九章算術。Les Neuf Chapitres: le classique mathématique de la Chine ancienne et ses commentaires. Paris: Dunod.
Couvreur, Séraphin (tr.). 1951. I-Li: Cérémonial. Paris: Les Belles Lettres (Les Humanités d’Extrême-Orient, Cathasia, série culturelle des Hautes Études de Tien-Tsin). [Reprint; originally published in Tianjin in 1916].
Cullen, Christopher. 1996. Astronomy and Mathematics in Ancient China: the Zhou bi suan jing, (Needham Research Institute Studies, 1). Cambridge, Cambridge University Press.
Cullen, Christopher (tr.). 2004. The Suan shu shu ‘Writing on reckoning’: A Translation of a Chinese Mathematical Collection of the Second Century BC, With Explanatory Commentary. Cambridge, Needham Research Institute (Needham Research Institute working papers, vol. 1).
Dauben, Joseph W. (tr.). 2008. 算數書 Suan shu shu: A Book on Numbers and Computations, Archive for the History of Exact Sciences, vol. 62, pp. 91–178.
Dauben, Joseph W. 道本周, Guo Shuchun 郭書春, and Xu Yibao 徐義保 (tr). 2013. 九章筭术 Nine Chapters on the Art of Mathematics. Shenyang: Liaoning jiaoyu chubanshe.
Lam Lay Yong, and Ang Tian Se (tr.). 2004 [1992]. Fleeting Footsteps. Tracing the Conception of Arithmetic and Algebra in Ancient China, Singapore etc.: World Scientific.
Le Blanc, Charles, et Rémi Mathieu (tr.). 2003. Huainan zi. Paris: Gallimard.
Lim, Tina Su Lyn, and Donald B. Wagner. 2017. The Continuation of Ancient Mathematics. Wang Xiaotong’s Jigu suanjing, Algebra and Geometry in 7th-Century China. Copenhagen: NIAS Press.
Schrimpf, Robert. 1963. La collection mathématique “Souan king che chou”: Contribtion à l’histoire des mathématiques chinoises des origins au VIIe siècle de notre ere. (An unpublished PhD dissertation). Rennes: Universite de Rennes.
Steele, John (tr.). 1917. The I-Li, or Book of Etiquette and Ceremonial. London: Probsthain.
Part C: Secondary Works
Berezkina, Éľvira I. [Березкина, Эльвира И.]. 1980. Matematika drevnego Kitaya [Математика древнего Китая] (Mathematics of Ancient China, in Russian). Moscow: Nauka.
Boltz, William G. 1993. I-Li [Yi li, a bibliographical entry]. In: Michael Loewe (ed.), Early Chinese Texts. Berkeley: The Society for the Study of Early China and The Institute of East Asian Studies, University of California, pp. 234–243.
Bottéro, Françoise, and Christoph Harbsmeier. 2008. The Shuowen Jiezi Dictionary and the Human Sciences in China. Asia Major, vol. 21, pp. 249–271.
Brenier, Joël. 1994. Notation et optimization du calcul des grands nombres en Chine. Le cas de l’échiquier de go dans le Mengqi bitan de Shen Gua (1086). In: Isabelle Ang and Pierre-Étienne Will (eds.), Nombres, Astres, Plantes et Viscères: Sept essays sur l’histoire des sciences et des techqniques en Asie Orientale. Paris: Collège de France, Institut des Hautes Études Chinoises, pp. 89–111.
Chemla, Karine. 1982. Etude du livre “Reflets des mesures du cercle sur la mer” de Li Ye. (An unpublished PhD dissertation). Paris: Univeristy Paris-XIII.
Chemla, Karine. 1996. Positions et changements en mathématiques à partir de textes chinois des dynasties Han à Song-Yuan. Quelques remarques. In: Karine Chemla et Michael Lackner (eds.), Disposer pour dire, placer pour penser, situer pour agir: Pratiques de la position en Chine (Extrême-Orient Extrême-Occident, no. 18), Paris: PUV, pp. 116–147.
Chen Liang-Tso [Chen Liangzuo] 陳良佐. 1978. Xian Qin shuxue de fazhan ji qi yingxiang 先秦數學的發展及其影響 (The development and influence of pre-Qin mathematics). Lishi yuyan yanjiusuo jikan 歷史語言研究所集刊, vol. 49, pt. 2, pp. 263–320.
Chen Liang-Tso [Chen Liangzuo] 陳良佐. 1997. Woguo chousuan zhong de kongwei – ling – ji qi xiangguan de yixie wenti 我國籌算中的空位––零––及其相關的一些問題 (An empty position [used as] zero [when manipulating with] counting rods in our country [in the past] and some related problems). Dalu zazhi 大陸雜誌, vol. 54, no. 5, pp. 238–250.
Cheng, Anne. 1993. Ch’un ch’iu, Kung yang, Ku liang and Tso chuan [Chunqiu, Gongyang, Guliang and Zuozhuan, a bibliographical entry]. In: Michael Loewe (ed.), Early Chinese Texts. Berkeley: The Society for the Study of Early China and The Institute of East Asian Studies, University of California, pp. 67–76.
de la Grasserie, Raoul. 1896. Terrien de Lacouperie. Bulletin de la Société de linguistique de Paris, vol. 9, nos. 39–43, pp. lxxxi–lxxxvi.
des Rotours, Robert. 1932. Le traité des examens, traduit de la Nouvelle histoire des T’ang (chap. 44–45). Paris: Librairie Ernest Leroux.
Djamouri, Redouane. 1994. L’emploi des signes numériques dans les inscriptions Shang. In: A. Volkov (ed.), Sous les nombres, le monde: Matériaux pour l’histoire culturelle du nombre en Chine ancientne (Extrême-Orient, Extrême-Occident, no. 16), pp. 12–42.
Han Wei 韩巍. 2012. Beida Qin jian zhong de shuxue wenxian 北大秦简中的数学文献 (Mathematical texts found among the [documents on] bamboo strips [preserved] in Peking University). Wenwu 文物 (Cultural relics), no. 6, pp. 85–89.
Hoe, John [=Jock]. 1977. Les systèmes d’équations polynomes dans le Siyuan Yujian (1303). Paris: Collège de France.
Horiuchi, Annick. 1994. Les mathématiques japonaises à l’époque d’Edo. Paris: Vrin.
Hua Yinchun 華印椿 (ed.). 1987. Zhongguo zhusuan shigao 中國珠算史稿 (An unofficial history of “computation with beads” in China). Beijing: Zhongguo caizheng jingji chubanshe 中國財政經濟出版社.
Jami, Catherine. 1994. History of Mathematics in Mei Wending’s (1633–1721) Work. Historia Scientiarum, vol. 4, no. 2, pp. 159–174.
Jami, Catherine. 1998. Abacus. In: Robert Bud and Deborah J. Warner (eds.), Instrument of Science: an historical encyclopedia, pp. 3–5. New York/London: Garland.
Kalinowski, Marc. 1994. La divination par les nombres dans les manuscrits de Dunhuang. In: Isabelle Ang and Pierre-Etienne Will (eds.), Nombres, astres, plantes et viscères. Sept essais sur l’histoire des sciences et des techniques en Asie orientale. (Mémoires de l’Institut des Hautes Études chinoises, vol. 35), Paris: Collège de France, pp. 37–88.
Kalinowski, Marc. 1996. Astrologie calendaire et calcul de position dans la Chine ancienne: Les mutations de l’hémérologie sexagésimale entre le IVe et IIe siècle avant notre ère. In: Karine Chemla et Michael Lackner (eds.), Disposer pour dire, placer pour penser, situer pour agir: Pratiques de la position en Chine (Extrême-Orient Extrême-Occident, no. 18), Paris: PUV, pp. 71–113.
Lam Lay Yong, and Ang Tian Se. 2004. Fleeting Footsteps: Tracing the Conception of Arithmetic and Algebra in Ancient China. Singapore: World Scientific.
Li Di 李迪. 1997. Zhongguo shuxue tongshi. Shang gu dao Wudai juan. 中國數學通史。上古到五代卷 (Comprehensive history of Chinese mathematics. Volume [on the period] from ancient times to ‘Five Dynasties’). Nanjing: Jiangsu jiaoyu chubanshe.
Li Di 李迪 (volume editor). (1999). Xi Xia, Jin, Yuan, Ming 西夏金圓明 ([Mathematics of the state of] Xi Xia and of the Jin, Yuan and Ming [dynasties]). Volume 6 of Wu Wenjun 吳文俊 (general editor), Zhongguo shuxue daxi 中國數學史大系 (Encyclopedia of the history of Chinese mathematics). Beijing: Beijing Shifan daxue chubanshe.
Li Yan 李儼. 1931. Zhongguo shuxue dagang 中國數學大綱 (Outline of [the history of] Chinese mathematics).
Li Yan 李儼. 1955a [1935]. Chousuan zhidu kao 籌算制度考 (A study of the system of counting rods). In Li Yan 李儼, Zhong suan shi luncong 中算史論叢 (Collected works on the history of mathematics in China). Beijing: Kexue chubanshe, vol. 4, pp. 1–8.
Li Yan 李儼. 1955b [1954]. Zhongguo gudai shuxue shiliao 中國古代數學史料 (Materials for [study of the history of] ancient mathematics in China). Shanghai: Zhongguo kexue tushu yiqi.
Li Yan 李儼. 1983 [1937]. Zhongguo suanxue shi 中國算學史 (History of Chinese mathematics). Shanghai: Shangwu yinshuguan. [Reprinted facsimile by Taiwan shangwu yinshuguan (Taibei) in 1983].
Li Yan, and Du Shiran. 1987. Chinese mathematics: A concise history. (Translated by John N. Crossley and Anthony W.-C. Lun). Oxford: Clarendon Press.
Libbrecht, Ulrich. 1973. Chinese Mathematics in the Thirteenth Century: The Shu-shu chiu-chang of Ch’in Chiu-shao. Cambridge (Mass.) and London: MIT Press.
Libbrecht, Ulrich. 1982. Mathematical Manuscripts from Dunhuang Caves. In: Li Guohao, Zhang Mengwen, and Cao Tianqin (eds.), Explorations in the History of Science and Technology in China. Shanghai: Shanghai Publishing House, pp. 203–229.
Martzloff, Jean-Claude. 1997. A history of Chinese mathematics. Berlin etc.: Springer.
Mei Rongzhao. 1983. The decimal place-value numeration and the rod and bead arithmetic. In: Ancient China’s Technology and Science, Peking: Foreign Language Press, pp. 57–65.
Mikami, Yoshio. 1911. The influence of abaci on Chinese and Japanese mathematics. Jahresberichte der Deutschen Mathematiker, vol. 20, pp. 380–393.
Mikami, Yoshio 三上義夫. 1926. Shina sūgaku-no tokushoku 支那數學の特色 (Special characteristics of Chinese mathematics). Tōyō gakuhō 東洋學報 (Reports of the Oriental Society of Tokyo), vol. 15, no. 4; vol. 16, no. 1.
Mikami Yoshio. 1934. Zhongguo suanxue zhi tese 支那數學之特色 (Chinese translation of Mikami (1926) by Lin Ketang 林科棠). Shanghai: Shangwu, 1934.
Mikami Yoshio. 1974 [1913]. The development of mathematics in China and Japan. Leipzig; reprinted by Chelsea Publishing Company (New York) in 1974 with an appendix authored by Fujisawa Rikitarō.
Needham, Joseph. 1959. Science and Civilisation in China. (Volume 3: Mathematics and the Sciences of the Heavens and the Earth.) Cambridge: Cambridge University Press.
Peterson, Willard J. 1982. Making connections: ‘Commentary on the Attached Verbalizations’ of the Book of Changes. Harvard Journal of Asiatic Studies, vol. 42, no. 1, pp. 67–116.
Siu Man-Keung, Volkov, Alexei. 1999. Official curriculum in traditional Chinese mathematics: How did candidates pass the examinations? Historia Scientiarum, vol. 9, no. 1, pp. 85–99.
Smith, David E. and Mikami Yoshio. 1914. A History of Japanese Mathematics. Chicago: The Open Court.
Terrien de Lacouperie, Albert Étienne Jean-Baptiste. 1883. The Old Numerals, the Counting-Rods and the Swan-Pan in China. The Numismatic Chronicle and Journal of the Numismatic Society, Third Series, vol. 3, pp. 297–340.
Vissière, A[rnold]. 1892. Recherches sur l’origine de l’abaque Chinois. Bulletin de Geographie Historique et Descriptive, [t. 28,] n. 1, pp. 54–80.
Vogel, Hans Ulrich. 1994. Aspects of metrosophy and metrology during the Han period. In: Alexei Volkov (Ed.), Sous les nombres, le monde: Matériaux pour l’histoire culturelle du nombre en Chine ancientne (Extrême-Orient, Extrême-Occident, no. 16), pp. 135–152.
Volkov, Alexei. 1994. Large numbers and counting rods. In: Alexei Volkov (Ed.), Sous les nombres, le monde: Matériaux pour l’histoire culturelle du nombre en Chine ancienne. (Extrême-Orient Extrême-Occident, no. 16), pp. 71–92.
Volkov, Alexei. 1997. The mathematical work of Zhao Youqin: remote surveying and the computation of pi. Taiwanese Journal for Philosophy and History of Science, no. 8 (1996–1997) (vol. 5, no. 1), pp. 129–189.
Volkov, Alexei. 1998. Counting rods. In: Robert Bud and Deborah J. Warner (eds.), Instruments of Science: an Historical Encyclopedia. London & New York: Garland, pp. 155–156.
Volkov, Alexei. 2000. From Numerology to Arithmetic: the Early Chinese Counting Devices. In: Zhang Jiafeng and Liu Juncan (Eds.), Proceedings of the 1999 Conference on the History of Science (Taipei, Republic of China), Taibei: Academia Sinica, pp. 29–69.
Volkov, Alexeï. Le bacchette [Counting rods]. 2001. In: Sandro Petruccioli (Gen. Ed.), Karine Chemla, Francesca Bray, Daiwie Fu, Yilong Huang and Georges Métailié (Volume Eds.), Storia Della Scienza (Encyclopedia on History of science, in Italian), vol. 2: Cina, India, Americhe (China, India, America), pp. 481–485. Rome: Istituto della Enciclopedia Italiana.
Volkov, Alexei. 2014. History of mathematics education in Oriental Antiquity and Middle Ages. In: Alexander Karp and Gert Schubring (Eds.), Handbook on the History of Mathematics Education, New York etc: Springer, pp. 55–70, 79–82.
Volkov, Alexei. 2018. Visual representations of arithmetical operations performed with counting instruments in Chinese mathematical treatises. In: Fulvia Furinghetti and Alexander Karp (Eds.), Researching the History of Mathematics Education: An International Overview (ICME-13 Monographs). Cham: Springer, pp. 279–304.
Wang Ling. 1956. The Chiu Chang Suan Shu and the History of Chinese Mathematics during the Han Dynasty. (An unpublished PhD dissertation). [Cambridge:] Trinity College.
Wu Chengluo 吳承洛. 1937. Zhongguo duliangheng shi 中國度量衡史 (History of measures, volumes, and weights in China). Shanghai: Shangwu yinshuguan.
Yabuuti Kiyosi. 2000. Une histoire des mathématiques chinoises. Paris: Belin.
Yamazaki Yoemon. 1962. History of instrumental multiplication and division in China - from the reckoning blocks to the abacus. Memoirs of the research department of the Toyo Bunko, no. 21, pp. 125–148.
Yang Qiong-Ru 楊瓊茹. 2003. Ming dai lixuejia Zhou Shuxue ji qi suanxue yanjiu 明代曆算學家周述學及其算學研究 (A study of Zhou Shuxue, an expert in calendrical science of the Ming dynasty, and of his mathematical methods). (An unpublished Master thesis). Taibei: National Taiwan Normal University.
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Appendix: On the picture of a counting instrument from the Suan fa tong zong 算法統宗
Appendix: On the picture of a counting instrument from the Suan fa tong zong 算法統宗
To support his thesis about the existence of the “counting board” in China, J. Needham reproduced a picture from the Suan fa tong zong 算法統宗 (Unified Origins of the Computational Methods, 1592) by Cheng Dawei 程大位 (1533–1606).Footnote 126 The picture has a caption reading “Shi sheng wen nan” 師生問難 (“Master and disciple(s): asking difficult [questions]”); it features one relatively old individual (apparently, the “Master”) and two younger persons gathered around a table on which a rectangular object is placed (Fig. 33).
Picture titled 師生問難 (“Master and disciple(s): asking difficult [questions]”) reproduced from an edition of the Suan fa tong zong of 1883 and identical with Fig. 66 in Needham (1959, p. 70)
The “Master” and the person standing on his right seem to be engaged in a conversation, while the person on the “Master’s” left is standing rather passively; he looks smaller and his picture is not drawn very well, if compared with two others; one may suggest that this person is a servant, while the person on the Master’s right is a disciple. The reason for reproducing this picture in Needham’s volume was related to his statement about the existence in China of a counting instrument he referred to as “counting board”: the object on the table was, according to Needham, a specimen of such a “board.” In this Appendix, I will argue that the object drawn in this picture actually was a beads abacus (suanpan).
Needham’s caption accompanying this picture is somewhat ambiguous; it reads “[…] a view of a counting-board (frontispiece of the Suan Fa Thung Tsung [i.e., Suan fa tong zong], + 1593).” It can be understood in two ways: (1) The picture comes from the original edition of 1593; (2) the picture comes from an unspecified (revised) edition of the version of the treatise originally published in 1593.Footnote 127 It turns out that the correct understanding is the second one. Indeed, the picture reproduced by Needham (and later copied by Libbrecht)Footnote 128 is identical with the one published in the edition of the treatise dated of 1883 and shown in Fig. 33, while the original edition of 1653 contains a somewhat different picture shown in Fig. 34. In the picture of 1883, the object on the table looks like a thick board with painted (or carved) square (or rectangular) cells. The number of columns equals to 10, one of them (on the reader’s left) is not well drawn; the number of horizontal rows clearly equals to five. The horizontal dimension of the object can be roughly estimated as 50–60 cm, and the vertical one, as 30–40 cm. Most interestingly, the object is clearly rather thick; its height can be roughly estimated as 4–5 cm. All cells on the board are empty, and no counting rods are shown either on the table or in the hands of the individuals, even though depicting the rods would not be a difficult task for the block-carver. Chen Liangzuo’s analysis of the picture provided by Needham (that is, the one shown in Fig. 33) led him to the conclusion that it is impossible to decide whether the depicted object is an abacus or a “counting board.”Footnote 129
Picture titled 師生問難 (“Master and disciple(s): asking difficult [questions]”) from the edition of the Suan fa tong zong of 1593 (later also reproduced in the edition of 1784)
The picture published in the edition of 1593 and of 1784 (Fig. 34) is slightly different. It was originally executed much better than the one shown in Fig. 33 (the faces of the “Master” and the disciple on the left look more realistic), but the quality of the picture from 1593 edition and of its copy printed in 1784 available to me did not allow to identify the object on the table. However, given that the pictures shown in Figs. 33 and 34 came from the opening pages of a treatise featuring the operations with the abacus and lacking any explicit mention of the counting rods, and taking into account the fact that by the time of its compilation the counting rods were almost completely forgotten in China, it would be reasonable to conjecture that the object intended to be depicted in the picture was an abacus, and not a conjectural “counting board,” as J. Needham suggested.
The decisive piece of evidence is found in the edition of the treatise of 1758 preserved in Waseda University Library (Japan). This blockprinted picture shown in Fig. 35 is apparently identical with those found in the editions of 1593 and of 1784. The picture leaves no doubts that the instrument placed on the table is a Chinese beads abacus (suanpan) ; the most important detail that allows identify it is a clearly visible horizontal bar separating two sections of the abacus. The vertical bars for sliding beads are also shown rather clearly. The beads, however, are drawn as short horizontal lines between the vertical bars; this may have been the reason why the carvers who produced the printing block for the picture shown in Fig. 33 simplified the original picture and replaced the bars and beads with rectangular grid.
Picture titled 師生問難 (“Master and disciple(s): asking difficult [questions]”) from the edition of the Suan fa tong zong of 1758
Another observation appears relevant here. The extant editions of the treatise in 17 juans contain a picture with the same caption, (“Master and disciple(s): asking difficult [questions]”) shown in Fig. 36, but they do not contain the picture shown in Fig. 35 or its modifications; conversely, the editions in 12 juans do not contain pictures similar to that shown in Fig. 36 and instead often (but not always) provide generic versions of Fig. 34 (such as Figs. 33 and 35).Footnote 130
The composition of this picture is somewhat similar to that of pictures shown in Figs. 33, 34, and 35: the Master is sitting and two disciples are standing; the Master turns to the disciple on his right; the Master is sitting under a large tree; the positions of the two disciples are similar to those shown in the 12-juan version, etc. However, there are some details that make the two pictures different: in particular, the one from the 17-juan edition contains such elements lacking in the shorter one as a deer behind the disciple on the Master’s right and the (immortality?) mushroom growing in front of the Master’s seat; in turn, the desk featured in the 12-juan edition does not appear in the 17-juan one. If the meeting of the Master with the disciples happens, according to the picture from the 12-juan versions, most likely in the inner court of the Master’s house, the picture from the 17-juan version suggests that the meeting is taking place in a secluded place, “among brooks and pine trees,” as the inserted text claims. What is the most important for the present discussion is that the disciple on the Master’s right holds a clearly discernible abacus; interestingly enough, it is executed with some strange irregularities: the numbers of beads on some bars are clearly superior to the conventional two and five. It appears plausible to conjecture that the pictures shown in Figs. 35 (printed in 12-juan versions) and 36 (printed in 17-juan versions) were supposed to illustrate one and the same situation; this conjecture can be supported by the identical captions and similar positions of the portrayed individuals. If this conjecture is correct, it could provide an additional argument for the identification of the instrument depicted in 12-juan versions as an abacus and not as a “counting board.”
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Volkov, A. (2018). Chinese Counting Rods: Their History, Arithmetic Operations, and Didactic Repercussions. In: Volkov, A., Freiman, V. (eds) Computations and Computing Devices in Mathematics Education Before the Advent of Electronic Calculators. Mathematics Education in the Digital Era, vol 11. Springer, Cham. https://doi.org/10.1007/978-3-319-73396-8_7
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