Donald B. Wagner

28 October 2012

(stylistic revisions 2023, 2024)

**Click on an image to see it enlarged.**

Figure 1

Figure 2

* ADHE || BCGF ⊥ ABCD || IJKL || EFGH*

Figure 3

*ADHE* || *BCGF* ⊥ *ABCD* || *IJKL* || *EFGH*

Figure 4

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* 河防通議, ‘Comprehensive discussion of Yellow River conservancy’, was edited by Shakeshi 沙克什 (1278–1351, also called Shansi 贍思), a man of Arabic ancestry employed by the Yuan state in posts concerned with river conservancy.
In its present form it consists of six ‘sections’ (*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 *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.

The section *Calculation** *consists of 27 calculation problems under five headings. Some of the problems are quite simple, for example the first:

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

Another problem, the 19th, is only an approximation, though the exact procedure had been known for a millennium:

Suppose an ‘earthen ox’ [*tu** niu* 土牛]
is to be built. The upper length is 2 *zhang* 5 *chi*, the lower length is 3 *zhang* 5 *chi*, the upper breadth is 1 *zhang*,
and the lower breadth is
2 *zhang*.
The height is 1 *zhang* 8 *chi*. What is the volume?

Note that 1 *zhang* = 10 *chi* ≈ 3.1
metres.

Answer: 8100 [cubic] *chi*.

Method: Add together the upper
and lower lengths and halve, obtaining 3 *zhang*. This is the length of
the *ting* 停.

A *ting* in this text is a solid which has
the same volume as a given solid, but whose volume is easier to calculate. In this case the *ting* is a box (rectangular parallelepiped) with dimensions
30 × 15 × 18 *chi*.
But this has only approximately the same volume as the ‘earthen ox’.

Further add together the upper and lower breadths and halve, [obtaining]
1 *zhang* 5 *chi*. This is the breadth of the *ting*. Multiply these together, obtaining 450 *chi*. Multiply by the height in *chi*; the result is the answer.

An ‘earthen ox’, shown in Figure 1, seems
to have been a particular type of earthwork used in river conservancy. The term
for this solid in classical Chinese mathematics is *chutong* 芻童, a frustum of a right wedge. The calculation given above is

But the correct formula, known since the
Han-dynasty *Jiuzhang** suanshu* 九章算術,[3] is

so that the exact calculation would be

The error in the calculation is in the present case 1.8%, which perhaps would have been acceptable to an administrator calculating labour requirements, and the calculation is simpler, involving two multiplications rather than four.

Another, the 18th problem, is simply wrong. See Figure 2:

Suppose
a square terrace is to be built. Above it is 3 *zhang* 2 *chi* square, below it is 5 *zhang* 6 *chi* square, and the height is 48 *chi*.
What is the volume?

Answer: 7936 [cubic] *chi*.

Method: Multiply the upper
square [dimension] by itself, obtaining 1024 [square] *chi*. Multiply the lower square [dimension] by itself, obtaining
3136 [square] *chi*. Multiply the
upper and lower square [dimensions] together, obtaining 1792 [square] *chi*. Add together the three results,
obtaining 5952 [square] *chi*.
Multiply by the height in *chi*,
obtaining 285,696 [cubic] *chi*.
Divide by 36; this is the answer.

This calculation is

Here the divisor should have been 3 rather
than 36; the correct answer is 95,232 *chi*^{3}.
Since the text gives every step of the numerical working, the error is not a
simple scribal error. Someone, either the original author or one of the later
editors, believed that this was a correct calculation.

But my intention in this article is to draw attention to a more interesting case of an ‘ignorant editor’.

Problem 27, translated at the end of this article, concerns the construction of a canal, shown in Figure 3. One group of labourers is to excavate part of it, IJKLHEFG, called the ‘cut’. The dimensions and volume of the whole canal are given, together with the volume of the cut. The dimensions of the cut are required.

This 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

*a*_{1} = 1040 *chi*

*a*_{2} = 890 *chi*

*b*_{1} = 1000 *chi*

*b*_{2} = 850 *chi*

*d* = 1 *zhang* = 10 *chi*

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 *chi*^{3}/labour unit = 23,625,000 *chi*^{3}_{
}*W* = 144,450 labour units × 40 *chi*^{3}/labour unit = 5,778,000 *chi*^{3}

The answers given are:

*x* = 120 *bu*

*a*_{3} = 926 *chi*

*b*_{3} = 886 *chi*

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,

(1)

However, the answers appear to be incorrect, for the volume of the cut is

(1.5)

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:

15

94,500

11,556,000

which is equivalent to the equation

15*x*^{2} +
94500*x* = 11556000

(2)

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.

The widths at the two ends of the *ting* are calculated:

(3)

(4)

The rate of change of the width of the *ting* along its length from west to east
is then

(5)

Let *x* =
the length of the cut in *bu*. Then the
width of the *ting* at the cut is

(6)

Then, including a conversion of *bu* to *chi*, twice the volume of the cut of the *ting* is

*dx* (*Kx* + *c*_{1} + *c*_{2}) × 5 *chi*/*bu* = 2*W*

(7)

15*x*^{2} + 94500*x* = 2*W* = 11556000 *chi*^{3}

(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:

(9)

This quantity could have been calculated more simply,

=

(10)

Calculating further,

(11)

(12)

Where Δ = *a*_{1}–*b*_{1} = *a*_{2}–*b*_{2} = 40 *chi*.

Using (11) and (12) requires that the
difference between widths is the same, Δ, throughout the length of the canal.
If instead *a*_{3} and *b*_{3} had been calculated using
(1), this requirement would have been unnecessary.

*‘Attentive
readers have undoubtedly been able to see that this reasoning is confused.’* (Guo Shuchun 1997: 229).

Guo Shuchun’s treatment of the problem assumes that it was the original author who was confused,
and that, since (7) and (9) include references to *c*_{1}, the cut proceeded from the eastern end of the canal.
Correcting equation (7), he arrives at the equation for this situation,

*dx*(2*c*_{1} – *Kx*) × 5 *chi*/*bu* = 2*W* (13)

(13)

–15*x*^{2} + 102000*x* = 11556000 *chi*^{3}

(14)

and the answers,

*x* ≈ 115.25 *bu*

*a*_{3} ≈ 1005.38 *chi*

*b*_{3} ≈ 965.38 *chi*

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*, *a*_{3} ≈ 949.22 *chi*, and *b*_{3} ≈ 869.22 *chi*.

A third possibility, which I believe is more likely to be correct, is that the original text gave a correct calculation, that a scribal error corrupted it, and that a later editor (perhaps Shakeshi), attempting to make sense of the corrupt text, corrupted it further.

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. Then the volume of the cut was *W*^{*} = 5,328,000 *chi*^{3} (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’ (*c*_{1}) must be taken to be an
error for *xiao kuo* 小闊, ‘smaller breadth’ (*c*_{2}). Then correcting equations (7)–(9) and the given intermediate
results gives the following calculation:

*dx* (*Kx* + 2*c*_{2}) × 5 *chi*/*bu* = 2*W*^{*}

(7′)

(8′)

This has one positive root,

x = 120 bu

And

(9′)

Finally, using either (1) or the method in the text, (11) and (12),

*a*_{3} = 926 *chi*

*b*_{3} = 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 *c*_{3}, *a*_{3},
and *b*_{3}, and discovers that
(9′) and (11)–(12) do not result in the given answers. But he finds that
subtracting *c*_{1} instead of *c*_{2} in (9′) does result in the
given answers. He therefore changes *xiaokuo* to *dakuo* in the statement of (9′), turning it into (9).

The fact that a correct *c*_{3} comes out of a calculation
containing three errors has an interesting explanation. From (5) and (7), and
for simplicity letting *x* be measured
in *chi* rather than *bu*,

Considering similar triangles in the same way as in (1),

So that

Then the calculation (9) gives

So whatever volume *W*^{**} is chosen for *W*,
the solution *x*^{**} of (7),
entered into (9), will give the same value of *c*_{3}.

Scholars have long been aware of *scribal errors* in ancient texts, and
generations of philologists have developed sophisticated ways of dealing with
them. In mathematical texts in particular, the mathematical context is very
often a sure guide in identifying this simple type of error.

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 (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.

In the following see Figure 3.

Suppose
a canal is to be opened. The straight length is [*l* =] 500 *bu*. At
the eastern end the upper breadth is [*a*_{1} =]
1040 *chi* and the lower breadth is [*b*_{1} =] 1000 *chi*. At the western end the upper
breadth is [*a*_{2} =]
890 *chi*, and the lower breadth [*b*_{2} =] is 850 *chi*. The depth is the same
[throughout], [*d* =] 1 *zhang*. The
total volume is [*V* =]
23,625,000 [cubic] *chi*.

Note that 1 *zhang* = 2 *bu* = 10 *chi* ≈ 3.1 metres.

The given total volume is correct:

One
labour unit [*gong* 功], when taking earth at 100 *bu*, is 40 [cubic] *chi*, and it is calculated that 590,625 labour units
[will be used].

It is desired to assign 144,450 labour units. What are the length and breadth of the cut [*jie* 截]?

The
‘cut’ is *IJKLHEFG* in Figure 3.

Here, according to my hypothesis, the original text had 133,200 work units, and a later editor changed this to 144,450.

Answer:
The length of the cut is [*x* =]
120 *bu* and the breadth is cut at [*c*_{3} =]
906 *chi*.

The
dimension *c*_{3} is shown in Figure 4.

(The
upper breadth of the cut is [*a*_{3} =]
926 *chi*, and the lower breadth is
cut at [*b*_{3} =] 886 *chi*.)[4]

Method: Lay out the upper and
lower breadths at the eastern end [*a*_{1}, *b*_{1}], add them together,
and halve, obtaining [*c*_{1} =]
1020 *chi*, which is the larger
breadth of the *ting* 停.

The *ting* is shown in Figure 4.

Further
lay out the upper and lower breadths at the western end [*a*_{2}, *b*_{2}],
add them together, and halve, obtaining [*c*_{2} =]
870 *chi*, which is the smaller
breadth of the *ting*. Subtract this
from the larger breadth of the *ting*;
the remainder, 150 *chi*, is the
difference in breadths. Divide this by the straight length, [*l* =] 500 *bu*, obtaining [*K* =] 3 *cun*, which is the difference
per *bu*.

Let
the *tianyuan* 天元 be [*x* =]
the cut length.

This corresponds to letting the length of the cut be the unknown in a polynomial equation.

Multiply by the difference per *bu* [*K*]; this is the difference in breadths at the place where the
cut stops.

Add
the smaller breadth of the *ting* [*c*_{2}]; this is the breadth [*c*_{3}] of the *ting* at the place of the cut.

Add
the larger breadth [*c*_{1}]
of the *ting*; this is twice the
breadth of
the cut of the *ting*.

According to my hypothesis *dakuo* 大闊, ‘larger breadth’ (*c*_{1}),
is a scribal error for *xiaokuo* 小闊, ‘smaller breadth’ (*c*_{2}).

Multiply
by [*d* =] the depth; this makes
twice the volume per *chi*.

Multiply
by 5 to make the twice the volume per *bu*.

(Move this to the left).[5]

Multiply by the *yuanyi* 元一 [the unknown in the
equation], [*x* =] the length of
the cut. This makes twice the volume of the cut.

Convert
the original labour units [assigned to] the cut, convert to a volume [*W*], and multiply by 2, obtaining
11,556,000 *chi*.

2*W* = 144,450 labour units × 40 *chi*^{3 }/ labour unit
× 2

= 11,556,000 *chi*^{3}

Combine
[? *xiang** xiao* 相消] this with what was moved to the left, obtaining 11,556,000 *chi* as the *shi* 實 [the constant term of
the equation], 94,500 *chi* [as the linear coefficient],
and 15 as the *zongyu* 從隅 [the cubic coefficient].

The equation is

This has one positive root, *x* = 120 *bu*.

Extract
the square root, obtaining [*x* =]
120 *bu*,
which is the length of the cut.

Set
up the labour units of the cut and convert to a volume, obtaining [*W* =] 5,778,000 *chi*.

*W* = 144,450 labour units × 40 *chi*^{3}/labour unit = 5,778,000 *chi*^{3}

Divide
this by the length of the cut [*x*]
converted to *chi*, obtaining 9630 *chi*.

Divide
this by the depth, [*d* =] 1 *zhang*,
obtaining 963 *chi*. Double this and
subtract the larger breadth of the *ting*,
[*c*_{1} =] 1020 *chi*. The remainder, 906 *chi*, is [*c*_{3} =] the cut breadth of the *ting*.

According to my hypothesis the original text had here* xiaokuo* 小闊, ‘smaller breadth’
(*c*_{1}), and an editor changed this to *dakuo* 大闊, ‘larger breadth’ (*c*_{2}).

Double
this, obtaining 1812 *chi*, which is
the sum of the upper and lower breadth of the cut. Subtract the difference
between the upper and lower breadths, 40 *chi*;
halve the remainder, obtaining 886 *chi*,
which is the lower breadth of the cut. Add again 40 *chi*, obtaining 926 *chi*,
which is the upper breadth of the cut. QED.

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 zhuzhi 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.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.