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

Ayangmahas breadth 5chi, length 7chiand height 8chi. 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.See Figure 2; the formula given here is

[Jiuzhang, 166-167]

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 theqianduABDCEF,Y= the volume of theyangmaBDFEC,P= the volume of thebienaoBACE.

Liu Hui has already proved that *C* = *abh*/2.
Therefore to prove that *Y* = *abh*/3 it is only necessary to show
that *Y* = 2*P*. 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.

ThebienaoBACE is divided into: 2qiandu, AGIJML and ILMJCP; and 2bienao, BGIL and EPML.

TheNow the sum of the volumes of the twoyangmaBDFEC is divided into: 1 box, HILKNDRO; 2qiandu, ILORCP and KLONFQ; and 2yangma, BHILK and LOPEQ.

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* = 2*P*, 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 [The terms used in this statement,xi] are the remaining [dimensions]. The extreme of fineness is called "subtle" [wei]. That which is subtle is without form [

[Jiuzhang, 168]

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 [(The translation is mine, following [Karlgren 1975; Lau 1963].)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 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 [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 asIn 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 ofgong,shang,jue,zheng, oryu[ancient names for musical notes]; and its having no one form [

[Heshang Gong, Chap. 14,juan1, 7a]

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