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:
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
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 email@example.com 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: firstname.lastname@example.org 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 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
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 place!