## Research note:## Shen Gua and an ignorant editor on the length of an arc12 February 2012
Histories of Chinese mathematics
generally state that Shen Gua 沈括 (1031–1095) in his where (see 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 ‘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. The
text in question is in chapter 18, ‘Arts’ ( 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] 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 measuring 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 Then, without introduction, follows
a method for approximating the length of an arc.
See Figure 1: first Lay out the diameter [ 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 Then the length of the arc is calculated: Multiply the ‘value of the cut’ [ 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 [ 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 The comment gives a concrete
example, with Suppose there is a circular field with
diameter [ 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’ [ 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 This [4 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:
If one cuts again, this method is also
followed. If the diameter is 20 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. A hypothesis 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
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: 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
First, it is interesting to
note that Zhu Shijie 朱世傑 in 1303 improved the formula in 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. References Bréard,
Andrea. 1998. ‘Shen Gua’s cuts’. ———. 1999.
———. 2008.
‘A summation algorithm from 11th century China:
Possible relations between structure and
argument’. In Chemla,
Karine, and Guo Shuchun. 2004. 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). Guo Shuchun 郭书春, et al. (2006). Hoe, John. 1977.
Holzman,
Donald. 1958. ‘Shen Kua and His Meng-ch’i
pi-t’an’. Hu Daojing 胡
道靜, ed. 1962.
Hu Daojing 胡
道靜, et al.
2008. Li Yan 李
儼. 1957. Martzloff,
Jean-Claude. 1997. Qian
Baocong 钱宝琮. 1964. Sivin, Nathan.
1995. ———. 2009.
Wagner,
Donald B. 2008.
[1] On
Shen Gua and his book see especially Sivin
1995; also Holzman 1958. [2] . Chemla
and Guo 2004: 141, 191–193, 773. [3] [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] [7] The
text has [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 [10] [11] |