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’ ( ## Hefang tongyi
men 門),
divided into a total of 68 ‘headings’ (mu 目). The first five sections concern
practical engineering, while the last,
‘Calculation’ (Suanfa men
算法門), concerns the mathematical techniques
needed for this work.[1] The complex history of the text has
been studied by Guo Shuchun (1997). Shakeshi had
at hand two versions of a 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 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
## ‘Calculation’
Suppose there are
13,500 bundles of straw, each
weighing 13 Answer: 11,700
bundles. Method: Set up 13,750
bundles at [position] (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
## 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, The given dimensions are:
The
text relates the two volumes to numbers of ‘labour
units’ (
The answers given are:
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 which is equivalent to the equation 15 A root of this equation is found using the ancient Chinese version of Horner’s Method:
The
derivation of (2) uses a
concept seen several times in the chapter
(including Problem 19, above), the
## Figure 4The widths at the two ends of the
Then,
including a conversion of
15 This
equation has one positive root,
Using (11) and (12) requires that the difference between
widths is the same, Δ, throughout the length of
the canal. If instead ## ‘Confusion’
–15 and the answers,
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 ## 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
This has one positive root, x = 120 bu And Finally, using either (1) or the method in the text, (11) and (12),
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 The editor discovers that the
given answers do not satisfy (7): 15 × (120) 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,
He then
calculates ## 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. 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. Chemla, Karine, and Guo
Shuchun. 2004. 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 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 Martzloff,
Jean-Claude. 1997. 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 Qian Baocong 錢寶琮, ed. 1963. Wagner, Donald B. 2008. Wagner, Donald B. 2012.
‘Research note: Shen Gua and an ignorant editor on
the length of an arc’. 12 February 2012. Yabuuchi Kiyoshi 藪內清. 1965. ‘Kabō
tsūgi ni tsuite’ 河防通議について (On the [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. |