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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 27, 2005 10:47 AM
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The HM list owner has helpfully pointed out to me that in my last
message, many symbols and formatting features presented problems for the
majordomo software. So I am sending (below) what I hope is a cleaned-up
version of my message.

HM list member Jeff Oaks has kindly posted to his website a copy
of my full paper on this topic. The URL is as follows:

http://facstaff.uindy.edu/~oaks/Biblio/COMHISMA8paper.doc




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 ---> 1 item and 3 more coins
7 coins per person and 4 more coins ---> 1 item

Thus, Liu reasoned,

4(8) coins per person ---> 4 items and 4(3) more coins
3(7) coins per person and 3(4) more coins ---> 3 items

Therefore,

4(8) + 3(7) coins per person ---> 4 + 3 items
[4(8) + 3(7) ]/ [4 + 3] coins per person ---> 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
email rschwart@schoolcraft.edu
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
To: historia-matematica@chasque.apc.org
Subject: [HM] Yin-Yang in the Commentary of Liu Hui

Greetings.

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
1999]

Can anyone help explain what Yin and Yang mean in the mathematics of Liu
Hui?

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.

Cheers

Wann-Sheng from Taiwan
A 228 (February 28) hands in hands peaceful movement is about to take
place!





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