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5. Summary of the derivation of the volume of a yangma

Problem 15 of chapter 5 of the Jiuzhang suanshu is as follows:

A yangma has breadth 5 chi, length 7 chi and height 8 chi. What is the volume?
Answer: 93 1/3 [cubic] chi.
Method: multiply the breadth and the length together; multiply by the height; divide by 3.
[Jiuzhang, 166-167]
See Figure 2; the formula given here is V = abh/3, which is correct. Substituting a = 5 chi, b = 7 chi, and h = 8 chi, the result is as stated.

Liu Hui's comment on this problem is translated in Section 6 below. Since this translation is somewhat difficult to follow, I give here a summary in modern terms.

If a yangma has dimensions a, b, and h as shown in Figure 2, then it can be fitted together with a bienao to form a qiandu as shown in Figure 3. Here the yangma is BDFEC and the bienao is BACE. Let

C = the volume of the qiandu ABDCEF,
Y = the volume of the yangma BDFEC,
P = the volume of the bienao BACE.

Liu Hui has already proved that C = abh/2. Therefore to prove that Y = abh/3 it is only necessary to show that Y = 2P. Divide up the bienao and yangma as in Figures 4 and 5, respectively, with: IJML perpendicular to ACE, bisecting AC; and HILK perpendicular to BDF, bisecting BD. Then we have the following situation.

The bienao BACE is divided into: 2 qiandu, AGIJML and ILMJCP; and 2 bienao, BGIL and EPML.
The yangma BDFEC is divided into: 1 box, HILKNDRO; 2 qiandu, ILORCP and KLONFQ; and 2 yangma, BHILK and LOPEQ.
Now the sum of the volumes of the two qiandu pieces of the bienao is clearly one-half of the sum of the volumes of the one box and two qiandu pieces of the yangma. Thus it is left to prove that the two bienao pieces of the bienao together have half the volume of the two yangma pieces of the yangma.

These smaller bienao and yangma can again be divided up as in Figures 4 and 5. This division again yields some parts whose volumes have the desired ratio, plus four smaller bienao and four smaller yangma. These can again be divided up in the same way. Continuing the process to the limit, we have Y = 2P, which was to be proved.

To complete the proof in modern terms, we need only note that the process converges, since the total volume of the remaining pieces is reduced by a factor of 4 after each cut.

As might be expected, Liu Hui has difficulty expressing the idea of carrying the process to the limit. He states:

The smaller they are halved, the finer [xi ] are the remaining [dimensions]. The extreme of fineness is called "subtle" [wei ]. That which is subtle is without form [xing ]. When it is explained in this way, why concern oneself with the remainder?
[Jiuzhang, 168]
The terms used in this statement, xi ("fine"), wei ("minute," with overtones of "subtle, mysterious"), and xing ("form"), demand further study. That these are important concepts in ancient Chinese metaphysics is certain, but the systematic investigation of this metaphysics has scarcely begun. In order to give a glimpse of the wider background of Liu Hui's statement, we may consider a passage from the Daode jing and an explication of it by an early commentator, where the same relationship between wei and xing appears. The Daode jing has:
We look for it [the Way], but we do not see it: we name it the Equable. We listen for it, but we do not hear it: we name it the Rarefied. We feel for it, but we do not get hold of it: we name it the Subtle [wei]. These three we cannot examine. Thus they are One, indistinguishable.
Its upper part is not bright, its lower is not dark. It is endless and unnameable. It returns to where there are no things. That is called the shape [zhuang ] without a shape, the appearance [xiang ] without a thing [wu ]. This is called the confused; you meet it but you do not see its head, you follow it but you do not see its rear.
[Daode jing, Chap. 14]
(The translation is mine, following [Karlgren 1975; Lau 1963].)

The commentator Heshang Gong 's dates are very uncertain, but he certainly lived within a century of Liu Hui, most probably in the second century A.D. [Erkes 1958, 5-12]. His comment explicates this passage in terms of the necessity to perception of distinguishing features:

That which is without colour is called "equable"; the text says that the One [the Way] is without colour, so that we cannot see it.
That which is without tone [sheng ] is called "rarefied"; the text says that [the sound of] the One is without tone, so that we cannot hear it.
That which is without form [xing ] is called "subtle" [wei ]; the text says that the One is without form, so that we cannot grasp it...
"These three" are the Equable, the Rarefied, and the Subtle. That "we cannot examine" them refers to their being without colour, without tone, and without form. We cannot speak of them, we cannot write of them; they must be received through quiescence and sought through the spirit. We cannot attain to them through the senses . . .
"Endless" refers to its inexhaustibility in extent. "Unnameable" refers to its having no one colour, so that it cannot be distinguished as blue or yellow, white or black; its having no one tone, so that it cannot be listened for as gong, shang, jue, zheng, or yu [ancient names for musical notes]; and its having no one form [xing], so that it cannot be measured as long or short, large or small.
[Heshang Gong, Chap. 14, juan 1, 7a]
In the light of this almost contemporary text, it appears that the passage in which Liu Hui carries the process of division to a limit can be interpreted as follows. The limiting case is not (as we might first assume) a collection of yangma and bienao with zero dimensions, but a collection of objects with no form; they are no longer yangma and bienao, and it is meaningless (or beyond human reason) to speak of their dimensions. These objects cannot be "examined," so "why concern oneself with the remainder?"

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