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’
Suppose there are
13,500 bundles of straw, each
weighing 13 jin 斤 [ca. 8 kg]. One
wishes to convert to 15 jin
per bundle. How many are the
converted [bundles]? Answer: 11,700
bundles. Method: Set up 13,750
bundles at [position] di
地 [on the counting
board]. Multiply this by 13 jin, obtaining 175,500 jin.
Divide [yue 約] by the weight
of straw to be converted; this
gives the answer. (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.
References
Chemla, 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. |