12 February 2012

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Histories of Chinese mathematics
generally state that Shen Gua 沈括 (1031–1095) in his *Mengxi
bitan*夢溪筆談 (‘Dream Pool essays’)[1] gave this approximation for
the length of an arc of a circle:

where (see Figure 1) *h* is
the sagitta, *b* is the chord, and *d* is
the diameter of the circle. This is historically
plausible, for (1) is
equivalent to an approximation for the area of a
circle segment in the *Jiu zhang
suanshu*九章算術 (‘Arithmetic in nine chapters’,
perhaps 1st century AD),[2] is equivalent to a
proto-trigonometric formula in the 13th-century
calendrical text *Shou shi li* 授時曆 (‘Canon of the season-granting
system’),[3] and was explicitly used by Zhu
Shijie 朱世傑 in a book published in 1303 (Hoe
1977: 297–298).

**Figure
1**

However, this is not precisely the formula given in Shen Gua’s text. There is a phrase in the text which must be removed to obtain (1); but a comment in smaller characters includes this phrase and gives a very odd interpretation.

All modern studies of *Mengxi
bitan* appear to assume that the comments
in smaller characters scattered through the text
are by Shen Gua himself, but I shall argue here
that at least this comment was added by someone
else. I conjecture that the original text,
including the elided phrase, gave a more complex
formula than (1),
that some later edition of the book contained a
corrupted version of this formula, and that
someone published this corrupted version with a
comment which attempted to make sense of it.
This might have been the same person who
committed the original corruption, perhaps
trying to make sense of a text which was too
difficult for him.

‘The
ignorant editor’, who modifies a text which he
does not understand in order to make ‘sense’ of
it, may be a common problem to be considered
when dealing with ancient Chinese technical
texts. 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). Other examples are provided by
the experience of many modern academic and
technical authors with the copy editors employed
by journals and publishers.

The
text in question is in chapter 18, ‘Arts’ (*Jiyi*技
藝) of *Mengxi
bitan*. It is reproduced here from the earliest
extant version, dated 1305 (*Yuan
kan Mengxi bitan* 1975,
18: 4–7; cf. Hu Daojing 1962: 574–587, § 301). None of the versions referred to in
Hu Daojing’s critical edition (which are all
later than this one) has any important textual
variants.

The text starts with an introduction in a form often seen in Shen Gua’s book, with first a statement of what is known or commonly thought on a topic, then the introduction of something new:

In the arts of calculation, the methods for calculating volumes in [cubic]

chi尺 [‘feet’], for example . . . [list of geometric forms], are complete for all object forms. There remains the technique for ‘volumes with interstices’ [xi ji隙 積]. . . . (p. 4, lines 4–6)算術求積尺之法，如芻萌、芻童、方池、 冥谷、塹堵、鱉臑、圓錐、陽馬之類，物形備矣，獨未有隙積一術。…

The text goes on to give methods for calculating the volumes of several geometric forms, then gives a method for ‘volumes with interstices’, i.e. stacked spheres or similar objects. This is equivalent to a method for summation of finite series, but treated as a geometric rather than an algebraic problem.[4]

After this, on page 6b, line 5, comes what may be the start of a new paragraph:

Of methods of measuringmu畝 [‘acres’, i.e., calculating areas], the square, the round, the crooked, and the straight have been perfected. There remains the technique of ‘assembling a circle’ [hui yuan會圓]. Since a circular field can be ‘broken’ [zhe折], it should be possible to assemble [hui會] [the pieces] and restore [fu復] the circle. Among the ancient methods there is only the method of ‘splitting the circle in the middle’ [?zhong po yuan中破圓] to break it, in which the error can be as much as threefold. I have devised a different technique for breaking and assembling [zhe hui zhi shu折會之術]. (p. 6, lines 5–8)

履畝之法，方圓曲直盡矣，未有會圓之 術。凡圓田，既能拆之，須使會之復圓。古法惟以中破圓法拆之，其失有及三倍者。余別為拆會之術。

This
concerns areas, and has no relation to the
preceding text on volumes, so the fact that it
is not a separate paragraph (does not start on a
new line with the initial character raised) may
perhaps be a scribal error.[5] Further,
it has no relation to what follows. We should
expect an explanation of what ‘breaking and
assembling’ means, and how it is done, but
neither breaking nor assembling nor areas are
mentioned again. Clearly something has been
dropped out of the text here, and there appears
to be no way of determining with any certainty
what Shen Gua meant by *hui yuan*.

Then, without introduction, follows
a method for approximating the length of an arc.
See Figure 1: first *b* is
calculated, given *d* and *h*,
using the Pythagorean theorem:

Lay out the diameter [

d] of the circular field and halve it; let this be the hypotenuse [of a right triangle]. Then from the halved diameter subtract ‘the value of the cut’ [suo ge shu所割數, i.e. the sagitta,h], and let the difference be the leg [gu股, the longer leg of the triangle]. Multiply each by itself and subtract [chu除 !][6] the [squared] leg from the [squared] hypotenuse. Extract the square root of the difference to make the base [gou勾, the shorter leg of the triangle]. Double this to make the ‘direct diameter’ [zhi jing直徑, i.e. the chord,b] of the ‘cut field’ [ge tian割田, the circle segment]. (p. 6, line 8 – p. 7, line 1)置圓田徑，半之以為弦，又以半徑減去所割數， 餘者為股；各自乘，以股除弦，餘者開方除為勾，倍之為割田之直 徑。

This calculation is

(2)

Then the length of the arc is
calculated:

Multiply the ‘value of the cut’ [

h] by itself,shift one place[tui yi wei退一位, i.e., divide by 10], and double it. Then divide [chu除] the result by the diameter [d] and add the ‘direct diameter’ [b] to make the arc [s] of the ‘cut field’. (p. 7, lines 1–3)以所割之數自乘，退一位，倍之，又以圓 徑除所得，加入直徑，為割田之弧。

If one chooses to ignore the very odd ‘shift one place’, this calculation is equivalent to (1). Then follows a statement whose meaning is not clear, but may be a reference to some process of successive approximations (see further below):

If it is cut again [

zai ge再割], [the calculation] is the same. Subtracting the previous ‘value of the cut’ [h] gives the ‘value of the second cut’ [zai ge zhi shu再割之數]. (p. 7, lines 3–4)再割亦如之，減去已割之數，則再割之數 也。

Then there is a comment in smaller characters which will be translated and discussed directly below. The text in large characters then concludes:

These two categories are precise techniques which the ancient writers did not reach. My idle ambition lies in this. (p. 7, lines 9–10)

此二類皆造微之術，古書所不到者，漫志 於此。

‘These two categories’ may be ‘volumes with interstices’ and ‘assembling a circle’, or the phrase may refer to something in the missing part of the text.

The comment gives a concrete
example, with *d* = 10 *bu*步 (‘paces’) and *h* = 2 *bu*.
First the chord *b* is calculated:

Suppose there is a circular field with diameter [

d=] 10bu步, and one wishes to cut [ge割] [h=] 2bu. Letting the halved diameter be the hypotenuse, 5bu, and multiplying this by itself gives 25. Subtracting the amount cut, [h=] 2bu, from the halved diameter, letting the difference, 3bu, be the leg, and multiplying this by itself, gives 9. Subtracting this from [the square] outside the hypotenuse [xian wai弦外, i.e., the square of the hypotenuse, 25bu^{2}], one has 16. Extracting the square root gives 4bu, which is the base. Doubling this makes [b= 8bu=] the ‘direct diameter’ of the cut [the chord of the segment]. (p. 7, lines 4–6)假令有圓田，徑十步，欲割二步。以半徑 為弦，五步，自乘得二十五；又以半徑減去所割二步，餘三步為股，自乘得九；用減弦外，有十六，開平方除得四步為勾，倍之為所割直 徑。

This calculation follows (2) above,

So far there have been no difficulties, but from here on the comment is very difficult to explain:

Multiplying the ‘value of the cut’ [

h=] 2bu, by itself gives 4, and doubling this gives 8.Shifting upward one place[tui shang yi wei退上一位[7]] gives 4chi尺. (p. 7, lines 6–7)以所割之數二步自乘為四，倍之得為八， 退上一倍（位）為四尺。

This calculation follows the main text:

after which the commentator
attempts to convert *bu*^{2} to *chi*^{2} by multiplying by 5 *chi* / *bu*,
resulting in 4 (square) *chi.*The
rest is mere nonsense:

This [4chi] is to be divided by the diameter [d], but in this case the diameter, 10 [bu], is an excessive value [ying shu盈數], and it is not possible to divide, so one simply uses 4chi. Adding this to the ‘direct diameter’ [b] gives the arc [s] of the cut [the circle segment]. One obtains in all the diameter of the circle [yuan jing圓徑,sic! i.e. the arc of the segment,s≈], 8bu4chi. (p. 7, lines 7–8)

以圓徑除。今圓徑十，已是盈數，无可 除。只用四尺加入直徑，為所割之孤，凡得圓徑八步四尺也

The commentator seems to believe that, in a division, if the divisor is greater than the dividend, the quotient equals the dividend. This erroneous calculation fortuitously gives the same result as using (1) would give:

The comment concludes:

If one cuts again, this method is also followed. If the diameter is 20

bu, to calculate the value of the arc, one should halve it and then, as stated, ‘divide by the diameter of the circle’. (p. 7, lines 8–9)再割亦依此法。如圓徑二十步求弧數，則 當折半，乃所謂以圓徑除之。

What this might mean is not at all clear to me, and I suspect that it may be further nonsense.

It is unlikely that the astronomer and polymath Shen Gua wrote the strange comment translated here. It is more plausible that a later editor wrote it in order to make a kind of sense of a corrupted version of an original text by Shen Gua.

The most common assumption is that the original text gave the formula (1), and that the corruption consisted of the insertion of the phrase ‘shift one place’. Stranger corruptions have occurred in ancient texts, but this is surely not a very probable scribal error.

The primary purpose of this note has been to point out the problem of ‘the ignorant editor’ and give a clear example. The modifications that such an editor makes will usually be of a very different kind from straightforward scribal errors, and it can be even more difficult than usual to guess the original content of the text. In the present case a hypothesis is possible.

The extant part of the text is explicitly a calculation of the length of an arc, and a possible explanation of the problematic phrase ‘shift one place’ is that it was originally part of a more complex formula. I propose that this formula may have been equivalent to

which is (1) with the addition
of the term 0.2*h*.[8] This is a much better
approximation. See Figure 2:
using (1), the maximum
error is 5.42%; using (3),
the maximum error is 1.86%, and for most of the
range of *h* the error is less than 1%.[9]

**Figure
2**

An ancient Chinese mathematical
writer could have expressed multiplication by
0.2 in a number of ways, but one obvious way
would be to write ‘shift one place and double
it’, and this phrase does in fact occur in the
text: *tui
yi wei bei zhi*退一位 倍之. It is therefore plausible
that Shen Gua’s original formula might have been
equivalent to (3).

There is no historical evidence that a formula like (3) was ever used in ancient China (or anywhere else), and this is a serious argument against my hypothesis. Nevertheless, it was not a difficult formula to discover.

Using modern software it was of
course simple to graph the absolute error of (1) against *h* and
observe that the curve lies close to a straight
line with slope –0.2 (see Figure
3). Would and could Shen Gua have (1) sought and
(2) found the same fact?

**Figure
3**

First, it is interesting to
note that Zhu Shijie 朱世傑 in 1303 improved the formula in *Jiuzhang
suanshu* for the area of a circle segment
by the addition of a corrective term.[10] It is plausible, therefore,
that Shen Gua a bit more than two centuries
before this may similarly have been interested
in improving the related approximation (1).

Chinese astronomers were accustomed to fitting linear, quadratic, and cubic relations to empirical data; in fact Shen Gua appears to mention such an interpolation in one of his jottings.[11] If he had sufficient data on the lengths of arcs in relation to chords and sagittae he would have been able to discover (3) quite easily. Such data could have been acquired empirically, for example by directly measuring arcs of a large circular object: a cartwheel 1 metre in diameter would have allowed sufficient precision. Or Shen Gua could have calculated the lengths of several arcs to any desired precision using Liu Hui’s method of inscribed polygons (Chemla and Guo 2004: 148–149, 193).

The mention of ‘cutting
again’ in Shen Gua’s text suggests that the
original text was in some way concerned with
successive approximations. However, it is
important to note that if Shen Gua used (3) in, for
example, a calculation of π by successive
approximations, he would not have obtained good
results. As can be seen in Figure 2, for very
small arcs the error using (3) is much
larger than that using (1).

Many thanks to Karine Chemla and an anonymous reviewer for criticism of an earlier version of this note. And to the shade of the late Fujieda Akira 藤枝晃, sorely missed, with whom I read this text as an undergraduate some forty years ago.

Bréard,
Andrea. 1998. ‘Shen Gua’s cuts’. *Taiwanese
journal for philosophy and history of science* 10: 141–162. Abstract __http://www.ams.org/mathscinet-getitem?mr=1745099__

———. 1999. *Re-Kreation eines mathematischen Konzeptes im
chinesischen Diskurs: ‘Reihen’ vom 1. bis 19.
Jahrhundert*. (Boethius: Texte und
Abhandlungen zur Geschichte der Mathematik und
der Naturwissenschaften 42). Stuttgart: Franz
Steiner Verlag.

———. 2008.
‘A summation algorithm from 11th century China:
Possible relations between structure and
argument’. In *Logic and theory of algorithms*,
edited by A. Beckmann*, et al.* Berlin /
Heidelberg: Springer, pp. 77–83. __http://www.springerlink.com/content/q129877161q2/#section=237011&page=1&locus=-1__

Chemla,
Karine, and Guo Shuchun. 2004. *Les neuf
chapitres: Le classique mathématique de la
Chine ancienne et ses commentaires*. Paris:
Dunod.

Fu Zong 傅
宗 and Li
Lunzu 李伦祖. 1974. ‘Xijishu he huiyuanshu – Shen Gua
“Mengxi bitan” pingzhu yize’ 隙积术和会圆术—«梦溪笔谈»评注一则 (The methods of
‘volumes of interstices’ and ‘assembling a
circle’ – note on a jotting of Shen Gua). *Xibei
Shifan Daxue xuebao (Ziran kexue ban) *西
北大学学报（自然科学版）(Journal of
Northwestern Normal University [Natural science
edition]), 1974.4: 17–22.

Guo Shuchun 郭书春, et al. (2006). *Jade mirror of the four unknowns*. Shenyang: Liaoning Education Press / Liaoning Jiaoyu Chubanshe.

Hoe, John. 1977. *Les systèmes d’équations polynômes dans le
Siyuan yujian (1303)*. (Mémoires de
l’Institut des Hautes Études Chinoises 6).
Paris: Presses Universitaires de France.

Holzman,
Donald. 1958. ‘Shen Kua and His Meng-ch’i
pi-t’an’. *T’oung Pao* 46.3/5: 260-292. __http://www.jstor.org/stable/20185477__

Hu Daojing 胡
道靜, ed. 1962. *Mengxi bitan jiaozheng *夢溪筆談校證 (Critical
edition of ‘Dream Pool essays’). 2 vols.,
Shanghai. Facs. repr. Shanghai: Shanghai Guji
Chubanshe, 1987. http://book.chaoxing.com/ebook/read_10500697.html

Hu Daojing 胡
道靜, et al.
2008. *Brush talks from Dream Brook*梦
溪笔谈. 2 vols. (Library
of Chinese classics, Chinese–English 大中华
文库 英汉对照).
Chengdu / Shenzhen: Sichuan People’s Publishing
House 四川人民出版社. "Translated
into modern Chinese by Hu Daojing 胡道靜, Jin Liangnian 金良年, and Hu Xiaojing 胡
小静; translated into
English by Wang Hong 王宏 and Zhao Zheng 赵峥."

Li Yan 李
儼. 1957. *Zhong
suanjia de neichafa yanjiu *中算家的內插法
研究 (Studies
of the use of interpolation by Chinese
mathematicians). Beijing: Liaoning Jiaoyu
Chubanshe. __http://book.chaoxing.com/ebook/read_11150779.html__

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.

Qian
Baocong 钱宝琮. 1964. *Zhongguo
shuxue shi*中国数学史 (The history of
Chinese mathematics). Beijing: Kexue Chubanshe.
Facs. repr. 1981.

Sivin, Nathan.
1995. *Science in ancient China: Researches
and reflections*. Aldershot: Variorum.

———. 2009. *Granting the seasons: The Chinese
astronomical reform of 1280, with a study of
its many dimensions and a translation of its
records*. (Sources and studies in the
history of mathematics and physical sciences).
New York: Springer.

Wagner,
Donald B. 2008. *Science and civilisation in
China*, vol. 5,
part 11: *Ferrous metallurgy*.
Cambridge: Cambridge University Press.

*Yuankan Mengxi
bitan*. 1975. 元刊
夢溪筆談 (Facsimile reprint of
a Yuan-period edition of ‘Dream Pool essays’).
Beijing: Wenwu Chubanshe.

[1] On Shen Gua and his book see especially Sivin 1995; also Holzman 1958.

[2] . Chemla and Guo 2004: 141, 191–193, 773.

[3] *h*^{4} + (*d*^{2}–2*sd*)*h*^{2} – *d*^{3}*h* + *s*^{2}*d*^{2} ≈ 0, *h* being approximated by Horner’s method given *s* and *d*.
Sivin 2009: 66–67. Derivations of both
formulas are given by Martzloff (2006:
328–334).

[4] Martzloff (1997: 16 fn. 17) gives a very brief summary of the method. Andrea Bréard (1999: 100–118, 357–360; note also 1998; 2008) gives a full translation of the paragraph and analyzes this first part in detail, but does not deal with the problems discussed here. Translations are also given by Fu Zong and Li Lunzu (1974) and Hu Daojing et al. (2008: 531–537); neither deals with these problems.

[5] But note the ‘two categories’ mentioned further on in the text.

[6] *Chu* 除 is occasionally seen in classical
Chinese mathematical texts, as here, with
the meaning ‘subtract’, but its more usual
mathematical meanings are ‘divide’ and
‘extract a root’. It is a surprise to see
the word used with this meaning here, since
it is used shortly after with the meaning
‘divide’.

[7] The
text has *bei* 倍, ‘double, multiple’, which, following
Hu Daojing (1962: 575), I take to be a scribal
error for *wei* 位. The characters are graphically similar, the
comment refers directly to a parallel sentence
in the main text with *wei*,
and the result of the calculation is in fact a
division by 10.

[8] This is reminiscent of Shen Gua’s use, in the first part, of a known formula plus a corrective term to obtain a new result. Bréard 1998: 116; 1999: 153; 2008: 82.

[9] The
curves in Figure 2 are
independent of *d*.
The curve in Figure 3 is *approximately* independent of *d*:
for all *d* > 1 the curves are so close that they are
covered by one line in the graph.

[10] *S* ≈ ½*h*(*h*+*b*) + (π–3)*b*^{2}/8. Hoe 1977: 295–296; Guo Shuchun et al. 2006: 594–598; Martzloff 1997:
327 (note typographical error). The added term is an exact expression for the error of the* Jiuzhang suanshu* approximation in the case of a semicircle, *b* = *h*/2.

[11] *Yuankan
Mengxi bitan* 1975, 7: 19–22; Hu Daojing 1962: 304–305, §
128; Hu Daojing et al. 2008: 210–215; Li Yan
1957: 77. See also e.g. Qian Baocong 1964:
103–107.