On the Transmission of Mathematical Knowledge in Versified Form in China

Bréard, Andrea

doi:10.1007/978-94-007-5122-4_7

Andrea Bréard⁶

Part of the book series: Boston Studies in the Philosophy and History of Science ((BSPS,volume 301))

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Abstract

Versified mathematical algorithms and problems make their appearance in China around the end of the Song dynasty (eleventh century) and are widely spread among the sources of the Ming (1368–1644) and the Qing (1644–1911) dynasties.

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Notes

1.
A list of Chinese dynasties and their respective dates is given in the appendix.
2.
Rote learning or learning by rote is learning things mechanically by repeating them without thinking about them or trying to understand them.
3.
Elman (2001, 268).
4.
Lee (2000, 623).
5.
Also called Trimetrical Classic, see Giles (1900, iv): “a hornbook for boys”.
6.
See Lee (2000, 439–440, 452–453, 459–461). See also Zhang (1962, 25–27) on the role of these three primers in Ming education.
7.
Citation by Gui Yougang (1507–1571) according to McLaren (2005, 167–168).
8.
Full literacy is sometimes called vocality, meaning the ability to express oneself in writing.
9.
See Elman (2001, 261–270).
10.
In Chinese age reckoning newborns start at 1 year old, and according to the lunar calendar the person’s age is incremented each passing of a New Year.
11.
Cited from Lu Shiyi’s Notes on Reflections and Disputations according to Zhou (1895, 38A), a compilation of citations concerning the reading of books:

.
12.
Elman (2001, 262–263).
13.
The Chinese saying “enough books to make the ox carrying them sweat” (han niu chong dong
) refers to an immense number of books. See Zhou (1895, 38A–40B):

.
14.
McLaren (2005, 167–168).
15.
Fishmongers for example used the Seven-words-a-line song of fish names (Yuming qi yan ge
. See Wang (1994, 55).
16.
The thirteenth-century book on divination Yuanhai ziping
by Xu Sheng
, for example, gives a seven- and a five-syllable-a-line verse for respectively memorizing the sequence of the 12 earthly branches and the 10 heavenly stems, as well as fortune-telling verses. See Wang (1994, 50–51).
17.
In Brard (2010) I do relate this phenomenon to the development of printing and the increasing social mobility under the Ming dynasty. See also Honda (1994).
18.
The now still extant Ten Books of Mathematical Classics (Suanjing shi shu
) were especially compiled to serve as textbooks in a specific sequence. See Siu and Volkov (1999).
19.
See Lee (2000, 517–525).
20.
See Li (1954–1955) for a bibliography of extant and lost Ming dynasty texts.
21.
Cheng (1990, 991) even refers to one book which explicitly indicates versified content in the title, the now lost Sanyuan hua ling ge
.
22.
The texts were flawed by a mixture of simplified and variant writings of characters.
23.
Cheng (1990, juan 1: 1A-1B). The same versified ‘Outline of mathematical methods’ (Suanfa tigang
) can be found earlier in Wu Jing’s book (Wu fl. 1450).
24.
The expression ‘nine numbers’ probably refers to the multiplication table up to nine times nine. See for example the Outline in Zhu (1839, 1127), where this multiplication table is referred to as the ‘Method of the Nine Numbers’ (
).
25.
The Chinese term for heaps refers to an accumulation of discrete objects like grains, in a certain geometric shape, whereas ‘volumes’ refers to a continuous geometric solid.
26.
The terminology here refers to the method of double false position.
27.
This is necessary to solve systems of linear equations.
28.
Liu Hui’s commentary (dated 263) to the algorithm for square root extraction in the Nine Chapters makes use of colors to refer to the geometric entities in a square corresponding to the terms in the algorithm. See Chemla and Guo (2004, 322–329). For a further discussion of the algorithm of square root extraction in its versified form see Sect. 7.3.1.6.
29.
Denoting a set of multiplication jingles for x ⋅y, where x ≤ y and x, y ≤ 9.
30.
Technical term for division by one-digit numbers x ÷ y, where x, y ≤ 9 and x ≥ y.
31.
Mathematically meaning, that in division of 60 by 8, one obtains 7 and a rest of 4, when multiplying 6 by 8 one obtains 48. The smaller characters here reflect the fact that this phrase is typeset as a commentary in the original text.
32.
See Cheng (1990, juan 1: 9B).
33.
Cheng (1990, 800).
34.
Such is the case in the 1716 edition reprinted in Cheng (1990), but not in Cheng (1882).
35.
Translated from the preface to Wang (2008, xxiii).
36.
See the Table of contents (Wang 1524, xxxiii–xxxv). The tunes were Shuixian zi
(Ondine), Xijiang yue
(Moon of the Western River), Luomei feng
(Wind from Falling Plums), Shanpo yang
(Sheep on a Precipice), Chao tianzi
(Son of Heaven), Qing jiang yin
(Clear River, a Prelude), Zui taiping
(Drunk in a Peaceful Time), Hong xiuxie
(The Red Embroidered Shoes), Wu ye’r
(Leaves of Chinese Parasol Tree), Yan er luo
(Falling Swan), Zhegu tian
(A Partridge in the Sky), Pu tian le
(The World being Overjoyed).
37.
Cheng (1990, 888–890).
38.
The name of the tune stems from a Tang dynasty poem by Li Bai
, The Ruin of the Gu-Su Palace
.
39.
The patterns required a certain sequence of level/even tones (ping
) and oblique/non-level/uneven tones (ze
). The tune here follows the pattern:
where the middle tone (zhong
) in brackets signifies an optional choice between even and uneven tone.
40.
A number of variant and simplified character forms rarely used in formal writing before the Ming are included in the printed version I have consulted. I have adopted their nowadays standard form in my transcription.
41.
This is the case for example for the two homophone measure words for bottles: zhi is either printed as
or as
. There are also two different expressions used for a heap (or pile): a variant form of duo
and dui
.
42.
Author of a now lost book from 1424, entitled Entirely Understandable Mathematical Methods from the Nine Chapters (Jiu zhang tongming suanfa
). See Li (1954–1955).
43.
Reference to the Great Compendium of the Yongle Reign-Period (1403–1422) (Yongle dadian
).
44.
At least such is the case in the commented 1836 edition by Luo Shilin, which is the earliest printed edition preserved. There is an earlier manuscript version, dated 1819 in the National Library of China (Ancient Rare Books Section N^∘ 16000), and another manuscript in the National Palace Museum in Taipei, which was copied during the Jiaqing reign (1796–1820), but I have not had a chance to revisit these for this article. See Brard (1999, Chap. 4, n. 19).
45.
Translated from the edition in Zhu and Luo (1937, 429). The puncutation is mine, and results in a pattern, that can be chanted according to the tune Moon of the Western River mentioned in Footnote 40. Zhu and Guo (2006, vol. II: 329), gives a translation, which is only followed partially here.
46.
See Cheng (1990, juan 16: 935).
47.
See Problem 9.6 in Chemla and Guo (2004, 711).
48.
Cf. the situation in medicine as described by Leung (2003, 132):

Yang highly praised the work [the Yijing xiaoxue
(Primary Study on Medicine), an important medical introductory textbook written in 1388], as its rhymes and verses written to facilitate memorization by beginners were based on classics.
49.
Cf. Leung (2003, 141):

Although the verse form could not express the exact composition of the various recipes, it provided the principles of the combination of various drugs in recipes for different purposes.
50.
Here problem 26 of juan 3. I only give the translation of the problem here from Lam and Ang (2004, 219). The Chinese text follows the edition in Qian (1963, vol. 2: 318).
51.
In modern terminology, the problem can be transcribed as a system of simultaneous linear congruence equations:
52.
In Yang (1275, juan
: 1100).
53.
My own translation follows rather literally the edition in Zhou (1969, juan xia
Yin-yang suanshu
: 2a–3a (92–93)). Volkov (2002, 402) based on (Libbrecht 1973, 286) both give more literary translations.
54.
Translated from Cheng (1990, juan 5: 430). Libbrecht (1973, 291–292) translates the passage as follows: Three septuagenarians in the same family is exceptional.
Twenty-one branches of plum-blossom from 5 trees.
seven brides in ideal union precisely the middle of the month.
Subtract 105 and you get it.
Volkov (2002, 401) says the text “may approximately be rendered as follows: Three men walking together, [then] seventy are rare/dispersed (?);
Five trees, 21 branches of plum-blossoms;
Seventh [month’s] gathering is in the middle of the month [=15th day];
Subtract 105 and then [you] can know [the answer].”
55.
The goal of these techniques, referred to as ‘the opening of the square’, is to eliminate successively surfaces of a square, having at the outset a surface equivalent to the number one wants to extract the square root of. The algorithm comes to an end, when one has exhausted the entire surface of the square. For a detailed discussion of the algorithm for square root extraction in the Nine Chapters and Liu Hui’s commentary to it, see Chemla (1994).
56.
Cheng (1990, juan 6: 1A (445)).
57.
Voir (Cheng 1990, juan 6: 1A–1B (445–446)).
58.
I will discuss such kind of jingles with further examples in Sect. 7.3.1.10.
59.
This sentence is printed in smaller characters. This usually marks a commentary or a later addition to the original edition.
60.
See Cheng (1990, juan 17: 6a–b). The corresponding layout of the numbers involved in this kind of written calculation is shown in Fig. 7.4) from Cheng (1990, juan 17: 5a–b).
61.
Translated from Cheng (1990, juan 17: 3b (p. 946)).
62.
Multiplication method of Indian origin. It appears in China as early as in 1450 in Wu Jing’s Great compendium of analogical categories to the Nine Chapters (Jiu zhang suanfa bilei daquan
). See Martzloff (1997, 92)) and Wu (fl. 1450).
63.
See the appendix to Brard (2010), which gives the variant rhymes from four Jianyang imprints.
64.
See Xu (1573).
65.
Illustration from Yu (1599).
66.
Division by estimation of the quotient, originally using three lines on the calculation surface in counting rod calculation.
67.
Translated from Zhu (1839, 1127).
68.
See Cheng (1990, juan 2: 8b (78)).
69.
Ibid. juan 2: 9a (79).
70.
In Yang (1842, juan 1: 2a), translated into French in Friedsam (2003, 60).
71.
A book by the same author, Yang Hui, the Xiangjie jiuzhang suanfa
, written in 1261.
72.
Elman (2001, 268).
73.
See also Lowry (2005, 93):

The bowdlerized written forms suggest that, though perhaps not the most discerning, the reading public for such materials had sufficient familiarity with the verses (as well as recognizing a number of characters) to make sense of variant forms used by printers or copyists.
74.
Whether variations were bound to regional dialects as to maintain the rhyme patterns, still needs to be explored.
75.
Illustration from Cheng (1882), originally published in 1592.

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Laboratoire Paul Painlevé, Université Lille I, Villeneuve d’Ascq Cedex, France
Andrea Bréard

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EHESS-CNRS and UPEC, Centre A. Koyré UMR 8560 and ESPE, Paris, France
Alain Bernard
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Christine Proust

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Bréard, A. (2014). On the Transmission of Mathematical Knowledge in Versified Form in China. In: Bernard, A., Proust, C. (eds) Scientific Sources and Teaching Contexts Throughout History: Problems and Perspectives. Boston Studies in the Philosophy and History of Science, vol 301. Springer, Dordrecht. https://doi.org/10.1007/978-94-007-5122-4_7

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