Research note: Another ignorant editor? Note on a mathematical problem in a
14th-century Chinese treatise on river
conservancy 28 October 2012
Correct results from an incorrect calculation In an
earlier article, ‘Shen Gua and an ignorant editor
on the length of an arc’ (Wagner
2012), I analyzed a text by Shen Gua 沈括 (1031–1095) and concluded that it had
been tampered with by an editor who did not
understand it. I was roundly criticized by friends
and others for claiming that a certain part of the
text was ‘nonsense’. It seems to be a general
assumption among historians of Chinese mathematics
that an ancient text always makes correct
mathematical sense or, if it does not, the problem
is the result of scribal errors rather than an
initial error by the author or intentional
tampering by a later editor. In the present article I consider
another mathematical text which clearly is
‘confused’ (hunluan 混乱) according
to the eminent historian of mathematics Guo
Shuchun 郭书春. Hefang tongyi
The complex history of the text has been studied by Guo Shuchun (1997). Shakeshi had at hand two versions of a Hefang tongyi by Shen Li 沈立, completed shortly after 1048.[2] These he called the ‘Directorate version’ (jian ben 監本) and the ‘Kaifeng version’ (Bian ben 汴本). (The Directorate version had previously been in the possession of the Directorate of Waterways, Dushui jian 都水監, of the Jin 金 dynasty.) He writes in his preface that both versions were badly organized and difficult to consult; therefore, ‘I have removed redundancies, corrected errors, reduced the number of sections, and organized it in categories.’ Shakeshi’s book was completed in 1321. There seems to be no way of knowing whether other editions were printed. It was copied into the great Ming-dynasty encyclopedia Yongle dadian 永樂大典 (completed 1408), and this copy was copied into the Qing collectaneum Siku quanshu 四庫全書 (completed 1782). All extant versions are ultimately based on the Siku quanshu version; no earlier version is now extant. However this version contains many banal scribal errors which are corrected in the Shoushan’ge congshu 守山閣叢書 and Congshu jicheng 叢書集成 editions. All three editions are available at http://www.scribd.com/collections/3809180/. Comments in smaller characters are
scattered throughout the text. They occasionally
include clues to their origin: Some clearly
originate in the Directorate version, and some –
those which explicitly compare the Directorate
version with the Kaifeng version – are clearly by
Shakeshi. Some refer to events after 1321, and
thus are by some later editor, perhaps the Yongle dadian or Siku
quanshu editors. The few comments in the
‘Calculation’ section do not happen to provide
such clues, and may originate from any of these
three sources. ‘Calculation’
(Bundles of straw were of great importance in dealing with flooding emergencies.) This calculation is
Figure 1
Another, the 18th problem, is simply
wrong:
Figure 2This calculation is
But my
intention in this article is to draw attention to
a more interesting case of an ‘ignorant editor’. Construction of a canal
ADHE || BCGF ⊥ ABCD || IJKL || EFGH Figure 3This is a simplified version of a practical problem of construction administrators: the available labour determines the volume to be excavated, and the labourers must be told how far they are to dig, x in Figure 3. The given dimensions are: l = 500 bu = 2500 chi ≈ 780
metres The text relates the two volumes to numbers of ‘labour units’ (gong 功), which seem to correspond to man-days. By the particular administrative norm invoked in the text, one labour unit corresponds to 40 cubic chi of the canal, and the volumes of the canal and the cut are: V = 590,625 labour
units × 40 chi3/labour
unit = 23,625,000 chi3 The answers given are: x = 120 bu The text does not state explicitly whether the work starts at the western or the eastern end of the canal, but these answers indicate that the cut is at the western end, for they satisfy the equations derived by consideration of similar triangles,
The text arrives at the given answers using the classical Chinese algebraic method known as Tianyuan yi 天元一. Briefly, a column of numbers on the counting board represents what we would call the coefficients of a polynomial equation (see e.g. Mei Rongzhao 1966; Chemla 1982; Martzloff 1997: 143–149; Yabuuchi 1965: 303–304). The manipulations described in the text result in a column of numbers on the counting board: which is equivalent to the equation A root of this equation is found using the ancient Chinese version of Horner’s Method: x = 120 bu The derivation of (2) uses a concept seen several times in the chapter (including Problem 19, above), the ting 停, a solid which has the same volume as a given solid, but whose volume is easier to calculate. In this case the ting is shown in Figure 4. ADHE || BCGF ⊥ ABCD || IJKL || EFGH Figure 4The widths at the two ends of the ting are calculated:
Then, including a conversion of bu to chi, twice the volume of the cut of the ting is dx (Kx + c1 + c2) × 5 chi/bu = 2W (7) 15x2 + 94500x = 2W = 11556000 chi3 (8) This equation has one positive root, x = 120 bu. The breadth of the ting at the cut is then calculated in a curiously roundabout way:
Using (11) and (12) requires that the difference between
widths is the same, Δ, throughout the length of
the canal. If instead a3 and b3 had been calculated using (1),
this requirement would have been unnecessary. ‘Confusion’
dx(2c1 – Kx) × 5 chi/bu = 2W (13) –15x2 + 102000x = 11556000 chi3 (14) and the answers, x ≈ 115.25 bu If we
follow Guo Shuchun’s reasoning, but assume that
the cut proceeded from west to east rather than
east to west, the calculation requires correction
of two equations, (7) and (9). Then the answers are x ≈ 129.9 bu, a3 ≈ 949.22 chi, and b3 ≈ 869.22 chi. An alternative hypothesis
Under this assumption it is a reasonable inference that the answers have not been corrupted, for they satisfy equation (1), and this also indicates that the work proceeded from west to east, as shown in Figures 3 and 4 above. Then the volume of the cut was W* = 5,328,000 chi3 (equation (1.5)), and therefore the number of work units assigned was given in the original text as 5,328,000 / 40 = 133,200 rather than the 144,450 of the present text. Therefore, in two places in the text (noted in the translation), the phrase da kuo 大闊, ‘larger breadth’ (c1) must be taken to be an error for xiao kuo 小闊, ‘smaller breadth’ (c2). Then correcting equations (7)–(9) and the given intermediate results gives the following calculation: dx (Kx + 2c2) × 5 chi/bu = 2W* (7′) This has one positive root, x = 120 bu And Finally, using either (1) or the method in the text, (11) and (12), a3 = 926 chi b3 = 886 chi These are the answers given in the text. How may the text have reached its present state? My hypothesis is that the original text gave the number of work units as 133,200 and gave a calculation equivalent to (7′)–(9′). At some point in its history a scribal error crept in: a substitution of dakuo 大闊, ‘larger breadth’, for xiaokuo 小闊, ‘smaller breadth’, in the statement of (7′). This amounts to changing (7′) to (7). The editor discovers that the given answers do not satisfy (7): 15 × (120)2 + 94500 × 120 = 11556000 ≠ 2 × 40 × 133200 He therefore changes the number of work units to 144,450 = 11,556,000 / (2×40). Now the root of the equation is the given answer, x = 120 bu. He then
calculates c3, a3,
and b3,
and discovers that (9′) and (11)–(12) do not
result in the given answers. But he finds that
subtracting c1 instead of c2 in (9′) does result in the given answers. He
therefore changes xiaokuo to dakuo in the statement of (9′), turning it into (9). Correct results from an incorrect calculation
Considering similar triangles in the same way as in (1), So that
Concluding remarks
But it seems that historians of Chinese science and technology must begin to take seriously the possibility of more complex corruptions of their texts, in which an ancient editor, encountering a text which he does not understand, ‘corrects’ the text to make ‘sense’ to him. Many ancient Chinese technical texts were difficult to read even in their own time. These became increasingly difficult as the centuries passed between then and now; scribes and editors preparing new editions must often have had difficulties in dealing with them, and most often these later editions are all that we have today. I have pointed out several possible examples of this problem in my study of ancient Chinese ferrous metallurgy (Wagner 2008: 51 n. g, 52 n. n, 217, 274 n. 126, 346) and in the above-mentioned note on a text by Shen Gua. Other examples are provided by the experience of many modern academic and technical authors with the copy editors employed by journals and publishers. In
dealing with editor-introduced textual errors the
proper procedure would seem to be: (1) propose a
hypothesis as to the intention of the original
text, and argue for its historical plausibility;
(2) propose a hypothetical course of events which
produced, from this, the text as it now appears,
suggest how the editor may have interpreted it,
and argue for the historical plausibility of the
hypothesis. Both requirements are difficult, and
will often be impossible. In the present article I
believe I have been moderately successful in
satisfying these requirements. On the other hand,
my hypothetical reconstruction of Shen Gua’s
calculation of the length of an arc (Wagner 2012), while at
the moment it seems (in my judgement) to be the
best so far proposed, is less sure.
ReferencesChemla, Karine. 1982. Étude du livre ‘Reflets des mesures du cercle sur la mer’. Dissertation, University of Paris XIII. Chemla, Karine, and Guo Shuchun. 2004. Les neuf chapitres: Le classique mathématique de la Chine ancienne et ses commentaires. Paris: Dunod. Guo Shuchun 郭书春. 1997. ‘“Hefang tongyi • Suanfa men” chutan’ «河防通议•算法门»初探 (‘An elementary study on the Suan fa men of He fang tong yi’). Ziran kexue shi yanjiu 自然科学史研究 (‘Studies in the history of natural sciences’) 16.3: 223–232. http://www.scribd.com/doc/105105547 Guo Tao 郭涛. 1994. ‘Shuxue zai gudai shuili gongcheng zhong de yingyong – «Hefang tongyi • Suanfa» de zhushi yu fenxi’ 数学在古代水利工程中的应用—«河防通议•算法»的注释与分析 (The application of mathematics in ancient hydraulic engineering – commentary and analysis of the ‘Calculation’ section of Hefang tongyi). Nongye kaogu 农业考古 (‘Agricultural archaeology’) 1994.1: 271–778, 285. Martzloff, Jean-Claude. 1997. A history of Chinese mathematics. Translated by S. S. Wilson. Berlin / Heidelberg: Springer-Verlag. ‘Corrected second printing’, 2006. Orig. Histoire des mathémathiques chinoises, Paris: Masson, 1987. Mei Rongzhao 梅荣照. 1966. ‘Li Ye ji qi shuxue zhuzuo’ 李冶及其数学著作 (Li Ye and his mathematical works). In Song Yuan shuxue shi lunwenji 宋元数学史论文集 (Essays on the history of mathematics in the Song and Yuan periods, AD 960–1368), edited by Qian Baocong 錢寶琮. Beijing: Kexue Chubanshe, pp. 104–148. Qian Baocong 錢寶琮, ed. 1963. Suanjing shi shu 算經十書. Beijing: Zhonghua Shuju. Wagner, Donald B. 2008. Science and civilisation in China, vol. 5, part 11: Ferrous metallurgy. Cambridge: Cambridge University Press. Wagner, Donald B. 2012. ‘Research note: Shen Gua and an ignorant editor on the length of an arc’. 12 February 2012. http://donwagner.dk/Shen-Gua-arc.htm Yabuuchi Kiyoshi 藪內清. 1965. ‘Kabō tsūgi ni tsuite’ 河防通議について (On the Hefang tongyi). Seikatsu bunka kenkyū 生活文化研究 13: 297–304. http://www.scribd.com/doc/105105664 [1] Besides Guo Shuchun the ‘Calculation’ section has been discussed by Yabuuchi Kiyoshi (1965), Guo Tao (1994), and three others, cited by Guo Shuchun (1997), whose publications have not been available to me. None deals with the ‘confusion’ noted by Guo Shuchun. [2] Other interpretations of Shakeshi’s preface are possible, but here I follow Guo Shuchun. [3] Ch. 5, problems 18–19. Qian Baocong 1963: 169–170; Chemla & Guo 2004: 390, 437–439. [4] Comment in smaller characters in the text. [5] Comment in smaller characters in the text. |