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Topic: Re: [HM] Yin-Yang in the Commentary of Liu Hui
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Randy K. Schwartz

Posts: 49
Registered: 12/3/04
Re: [HM] Yin-Yang in the Commentary of Liu Hui
Posted: Oct 26, 2005 2:53 PM
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Dear colleagues,
Although more than a year has passed since the query from
Christopher Baltus and the reply from Wann-Sheng Horng were
posted to this list (and I have placed copies of both of these
below for your convenience), more recently I've done some
thinking related to this topic, and thought I would share that.
In what follows, I rely on the same source as did Christopher
Baltus, namely:
Shen Kangshen, John N. Crossley and Anthony W.-C. Lun, The Nine
Chapters on the Mathematical Art: Companion and Commentary
(Oxford: Oxford University Press, 1999).
For the purpose of this message, I want to call attention to
Chapter 7 of the Nine Chapters (Jiuzhang Suanshu). This is
the chapter devoted mainly to solving problems of the type
involving “too much” (ying) or “not enough” (bu tsu). These
problems include many of the same kind that were later solved
in the medieval Arab world by hisab al-khata’ayn, i.e., the
method of “double false position”.
When I examined Chapter 7 in the course of a recent project,
I reached the conclusion that the philosophical outlook that
seems to most strongly underlie Liu Hui’s commentary is that
of balancing excess against deficit to achieve harmony,
which is a key interpretation of the Confucian doctrine of
yin and yang.
To illustrate that point, I want to make a detailed analysis
of Problem 1 from Chapter 7, which exemplifies the “joint
purchase” situation:
Now an item is purchased jointly; everyone contributes 8
[coins], the excess is 3; everyone contributes 7, the deficit
is 4. Tell: The number of people, the item price, what is each?
Answer: 7 people, item price 53. (Shen et al. 1999: 358)
The original text provided just the final answers and a formulaic
procedure for solving such problems on a standard Chinese
counting board, with rods to represent numbers. There was no
attempt to justify the steps of the procedure. Liu Hui’s later
explanatory comments (c. 260 CE) are thus invaluable to us
because they reflect how the Chinese viewed these problems. I
have taken Liu’s annotations for Problem 1 (from Shen et al.
1999: 359-360) and distilled them to the following chain of
deductions. We are given:
8 coins per person à 1 item and 3 more coins
7 coins per person and 4 more coins à 1 item
Thus, Liu reasoned,
4(8) coins per person à 4 items and 4(3) more coins
3(7) coins per person and 3(4) more coins à 3 items
4(8) + 3(7) coins per person à 4 + 3 items
[4(8) + 3(7) ]/ [4 + 3] coins per person à 1 item
So, each person must contribute 53/7 the value of one coin.
The rest of the solution follows from that point. But we can
take note already of the character of Liu’s reasoning. We
would have to say that his approach in the above is based on
the notion of balancing excess (4 times 3 coins) against
deficit (3 times 4 coins) and thereby eliminating or canceling
these quantities from consideration, as if on a balance sheet.
Indeed, the Sinologist and historian of science Joseph Needham
reached the conclusion that Chapter 7’s emphasis on balancing
excess and deficit reflects the Confucian doctrine of balancing
yin and yang to achieve harmony: see Needham, Science and
Civilization in China, Vol. 3, “Mathematics and the Sciences
of the Heavens and the Earth” (Cambridge: Cambridge University
Press, 1959), p. 119.
While I haven’t examined all of the Nine Chapters treatise in
detail, I did make a comparative analysis of Chapter 7 as part
of a larger recent investigation into the possible origins of
the Arab hisab al-khata’ayn. In this context, it’s significant
that the Chinese discussion relies heavily on the doctrine of
balance rather than on Greek-type arguments of proportionality.
The latter formed the unifying theme in the early Arabic
explanations of hisab al-khata’ayn, whether these explanations
were stated in an arithmetical or geometric form.
The mathematical operations that Liu invokes in the above
problem (multiplying both sides of a balance-expression by a
constant; adding one balance-expression to another; canceling
terms common to both sides) remind us of those that are used
in the very next chapter of the Jiuzhang Suanshu, which instructs
how to manipulate a fangcheng (rectangular array of numbers) in
order to solve a system of linear equations. For that matter,
they are also reminiscent of some operations used in medieval
Arab algebra (al-jabr wa’l-muqabala). Yet the Chapter 7
operations are fundamentally arithmetical, not algebraic,
inasmuch as they are carried out on numbers only, not on
expressions involving unknown quantities (such as, in Arabic,
shai’, mal, or jidhr). Liu’s term for the cross-multiplication
4(8) + 3(7) is qi, or “homogenization” of the suppositions 8 and
7, and his term for the equalization 4(3) = 3(4) is tong, or
“uniformization” of the excess and deficit. (I rely on Shen
et al. for these translations.) But these are arithmetical
operations par excellence. In fact, Liu used this exact same
terminology earlier in his commentary to explain the procedure
for solving arithmetic problems like 8/3 + 7/4.
My basic assessment of Liu Hui’s discussion of ying bu tsu is
therefore­ philosophically, it represents the Confucian doctrine
of balance rather than the Greek doctrine of proportion;
mathematically, it represents the arithmetic of antiquity,
rather than the algebra of the Middle Ages.
Prof. Randy K. Schwartz
Department of Mathematics
Liberal Arts Building
Schoolcraft College
18600 Haggerty Road
Livonia, MI 48152-2696 USA
voice 734/462-4400 extn. 5290
fax 734/462-4558
-----Original Message-----
From: Christopher Baltus [mailto:baltus@Oswego.EDU]
Sent: Friday, February 20, 2004 11:03 AM
Subject: [HM] Yin-Yang in the Commentary of Liu Hui
In his Preface to his commentary (263 CE) to The Nine Chapters on the
Mathematical Art, Liu Hui said
I read the Nine Chapters as a boy, and studied it in full detail
when I was older. [I] observed the division between the dual natures
of Yin and Yang [the positive and negative aspects] which sum up the
fundamentals of mathematics. . . .
[from the English version by Shen Kangshen et al.,Oxford University Press
Can anyone help explain what Yin and Yang mean in the mathematics of Liu
Thank you for your help.
Christopher Baltus
Oswego, NY USA
-----Original Message-----
Date: Feb 25, 2004 9:42 PM
Author: Wann-Sheng Horng
Subject: [HM] Yin-Yang metaphor in Liu Hui's commentary
Dear All,
The contrast of Yin and Yang, two very popular terms of natural
Chinese philosophy even today in the Chinese community, was used by
Liu Hui as a metaphor to indicate that diversified (natural) phenomena
change in a dualistic "Yin-Yang" way. One is tempted to interpret
that Liu Hui meant in this passage to emphasize mathematical law
underlying the changing phenomena. Yet, well, perhaps this is only a
rhetoric of ancient Chinese scholars like Liu Hui to justify the
status of mathematical study, an expertise which Confucians basically
did not pay due respect. Similar rhetoric can also be seen in the Sunzi
Suanjing (Mathematical Canon of Master Sun, 4th or 5th century AD),
where the so called Chinese Remainder Theorem originated.
Wann-Sheng from Taiwan
A 228 (February 28) hands in hands peaceful movement is about to take

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