ADHE || BCGF ⊥ ABCD || IJKL || EFGH
ADHE || BCGF ⊥ ABCD || IJKL || EFGH
Donald B. Wagner
28 October 2012
(stylistic revisions 13 October 2023)
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.
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. 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, 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:
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 chi3. 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
a1 = 1040 chi
a2 = 890 chi
b1 = 1000 chi
b2 = 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 chi3/labour unit = 23,625,000 chi3
W = 144,450 labour units × 40 chi3/labour unit = 5,778,000 chi3
The answers given are:
x = 120 bu
a3 = 926 chi
b3 = 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,
However, the answers appear to be incorrect, for the volume of the cut is
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
15x2 + 94500x = 11556000
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:
The rate of change of the width of the ting along its length from west to east is then
Let x = the length of the cut in bu. Then the width of the ting at the cut is
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
15x2 + 94500x = 2W = 11556000 chi3
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:
This quantity could have been calculated more simply,
Where Δ = a1–b1 = a2–b2 = 40 chi.
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.
‘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 c1, the cut proceeded from the eastern end of the canal. Correcting equation (7), he arrives at the equation for this situation,
dx(2c1 – Kx) × 5 chi/bu = 2W (13)
–15x2 + 102000x = 11556000 chi3
and the answers,
x ≈ 115.25 bu
a3 ≈ 1005.38 chi
b3 ≈ 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, a3 ≈ 949.22 chi, and b3 ≈ 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 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*
This has one positive root,
x = 120 bu
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).
The fact that a correct c3 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),
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 c3.
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 above.
Suppose a canal is to be opened. The straight length is [l =] 500 bu. At the eastern end the upper breadth is [a1 =] 1040 chi and the lower breadth is [b1 =] 1000 chi. At the western end the upper breadth is [a2 =] 890 chi, and the lower breadth [b2 =] 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 [c3 =] 906 chi.
The dimension c3 is shown in Figure 4.
(The upper breadth of the cut is [a3 =] 926 chi, and the lower breadth is cut at [b3 =] 886 chi.)
Method: Lay out the upper and lower breadths at the eastern end [a1, b1], add them together, and halve, obtaining [c1 =] 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 [a2, b2], add them together, and halve, obtaining [c2 =] 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 [c2]; this is the breadth [c3] of the ting at the place of the cut.
Add the larger breadth [c1] of the ting; this is twice the breadth of the cut of the ting.
According to my hypothesis dakuo 大闊, ‘larger breadth’ (c1), is a scribal error for xiaokuo 小闊, ‘smaller breadth’ (c2).
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).
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.
2W = 144,450 labour units × 40 chi3 / labour unit
= 11,556,000 chi3
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 chi3/labour unit = 5,778,000 chi3
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, [c1 =] 1020 chi. The remainder, 906 chi, is [c3 =] the cut breadth of the ting.
According to my hypothesis the original text had here xiaokuo 小闊, ‘smaller breadth’ (c1), and an editor changed this to dakuo 大闊, ‘larger breadth’ (c2).
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. http://staff.hum.ku.dk/dbwagner/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
 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.
 Other interpretations of Shakeshi’s preface are possible, but here I follow Guo Shuchun.
 Ch. 5, problems 18–19. Qian Baocong 1963: 169–170; Chemla & Guo 2004: 390, 437–439.
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