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- [6.1. The etymology of the term yangma]
- [6.2. The case of equal dimensions]
- [6.3. The general case]
- [6.4. An obscure digression]
- [6.5 division of the yangma and bienao into smaller pieces]
- [6.6. Carrying the operation to the limit]
- [6. 7. Justification for this concern with impractical matters]

(Translation of *Jiuzhang suanshu *[1963, 167-168]. The Chinese text is
reproduced in Section 8.)

The shape [called] *yangma* is one corner of a *fangzhui*.
[See Section 3 above.] A corner of a hip-gabled roof [*sizhu wu* ] is
called a *yangma*.

Not much is known of architectural terminology in Liu Hui's time, but in the Song the termyangmameant a hip-rafter in a hip-gabled roof.[6]

In the following see Figure 6. Certain phrases here are referred to later in the text. These I have marked (1), (2), etc.

Suppose the
breadth, length, and height are each 1 *chi* . Multiplying these together
gives the volume of a cube [with the same dimensions], 1 [cubic] *chi*.
(1) Dividing the cube [ABCDEFGH] slantwise [along the plane of BCEH] gives two
*qiandu* [BCDAHE and BCFGHE]; (2) dividing [one of the] *qiandu*
[e.g., BCDAHE] slantwise [along the plane of ACFH] gives one *yangma*
[CADEH] and one *bienao* [CABH]. The *yangma* occupies 2 and the
*bienao* occupies 1: this is an unchanging proportion. Fitting together
two *bienao* makes one *yangma*, and fitting together three
*yangma* makes one cube. Hence the division by 3. If this is verified
using blocks, the situation is clear. Cutting all of the *yangma* gives a
total of six *bienao*. Looking at the pieces, it is easy to understand
that the shapes correspond.

It is not completely trivial to "understand that the shapes correspond, for three of thebienaoare mirror-images of the other three.

(3) If the block is long or short, or broad or narrow, so that [the
sides of] the cube are not equal, it can still be cut into six *bienao*.
Their shapes are not the same, but the number [i.e., six] which appears is the
same, and their volumes are in fact equal.

See Figure 7a. When the box ABCDEFGH, with dimensions(4) When thea,b,c, is cut into sixbienaoas described above, the result is the three non-congruentbienaoFADC, FEDC, and FABC (Figures 7b, 7c, and 7d) and their respective mirror images CHGF, CBGF, and CHEF. Each of the six has the same dimensions,a,b, andc, and the volume of each is 1/6abc. But Liu Hui has not yet proved this fact.

See Figure 7a. The sixWhen thebienaocan be put together to form threeyangmaas follows: FADC with FABC to form FABCD (Figure 7e),

FEDC with CHEF to form FEDCH (Figure 7f),

CHGF with CBGF to form FGBCH (Figure 7g). Theseyangmahave the same dimensions, but they are not congruent. There is one other way of fitting thebienaotogether to formyangma(FADC with FEDC, FABC with CBGF, and CHEF with CHGF), and in this case again theyangmaare not congruent.

The following passage seems to be an additional comment on the preceding argument.Why is this? "Dividing the cube slantwise gives [two]

A way to make sense of this passage is as follows. See Figure 7a. If the box ABCDEFGH is cut first on the plane of FDCG and then on the plane of ACHF, then each of the cuts divides the volume of the box in halves; these cuts might reasonably be described as "vertical" and "horizontal," respectively.

The result of these cuts is twoSuppose ayangma, FEDCH and CBAFG, and twobienao, FADC and CHGF. Note that the twoyangmaare congruent mirror images, and that the twobienaoare congruent mirror images.

The intention of this passage might conceivably be to show that mirror imagebienaooryangmaare equal, since for example,

FEDCH + FADC = 1/2 ABCDEFGH,

FEDCH + CHGF = 1/2 ABCDEFGH,but this is unlikely, since Liu Hui does not elsewhere seem concerned about the problem of mirror images.

Here begins the derivation of the formula in the general case. The argument is expressed in terms of the case of equal dimensions, and it is necessary for the reader to extend the argument to the general case.

To make a *bienao* with breadth [*a*], length
[*b*], and height [*h*] each 2 *chi*, use two *qiandu* and
two *bienao* blocks, all of them red.

Figure 4 shows how the blocks are fitted together.To make a

Figure 5 shows how the blocks are fitted together.Joining together the red and black blocks to make a

Thisqianduis shown in Figure 8.

Then divide [The following is probably corrupt; my translation is speculative and involves some emendations to the text (see notes 10-12). Whether or not this translation is completely correct, the interpretation of the mathematical sense of the passage is fairly certain.

The two divisions are on the planes of HJMK and KMPN.Fit the red and the black

The redqianduAGIJML and ILMJCP are fitted together, and the blackqianduFQONKL and CPORIL are fitted together. Two cubes are thus formed, one red and one black.

In the general case, theseEach division then contains oneqiandudo not fit together. However it is clear that in each case the sum of the volumes of the twoqianduis equal to that of a box with dimensions 1/2a, 1/2b, 1/2h.

The "divisions" are ABHJMK and KMPNFE in Figure 8. Since theEach of the remaining items[12] is composed of [blocks with the same form as] the original objects.qianduin these divisions have been dealt with, what is left is onebienaoand oneyangmain each.

The two "items" are the twoqianduBGIHKL and EQOPML, each of which is composed of one redbienaoand one blackyangma.

From this point on there are no major difficulties with the text.These fit together to form a cube.

In the general case these pieces fit together to form a box with dimensions 1/2Thus cubes [formed of blocks] which are different [from the originala, 1/2b, 1/2h.

The situation is now as follows.

Black: 1 cubical block, and 1 cube formed of twoqiandublocks;

Red: 1 cube formed of twoqiandublocks;

Red and black: 1 cube formed of two redbienaoblocks and two blackyangmablocks.

In the general case,

Black: a volume of 1/4abh;

Red: a volume of 1/8abh;

Red and black: 2 redbienaowith dimensions l/2a, 1/2b, 1/2h; 2 blackyangmawith dimensions 1/2a, 1/2b, 1/2h, ; total volume l/8abh.

Thus the original 2 x 2 x 2Even if the cube is elongated,[13] and the blocks change [accordingly], there is clearly a constant situation.bienaoandyangmahave been divided in such a way that three-quarters of the total volume consists of objects whose volumes are known, and one-quarter consists of l x l x lbienaoandyangma. The ratio of the volumes of the known objects is red:black = 1:2, and this ratio holds also in the case of arbitrary dimensions .

The commentator Li Huang (d. 1812 [Qian 1964, 296]) could make no sense of the following and considered it to be corrupt [Qian 1963, 169, n. 1]. Actually the text is fairly clear once one understands what Liu Hui is trying to do.Of the remaining numbers [i.e., the volumes of the pieces resulting from the above manipulations], those which can be definitely determined are separated into one and two parts [one red cube and two black cubes]. Thus, it has been determined that the ratio [of the numbers which can be definitely determined] is 1 to 2. In terms of principle, how could this be arbitrary? To exhaust the calculation, halve the remaining breadth, length, and height; an additional three-quarters can thus be determined. The smaller they are halved, the finer [

On this passage see Section 5 above.Exhausting the calculation is called "calculating with the essence"; one "does not use calculating-rods to calculate it". [Cf.

The object [called] *bienao* has no practical use; the shape
[called] *yangma* can be long or short, or broad or narrow. Nevertheless,
without the *bienao* there is no way to investigate the number [i.e.
volume] of a *yangma*; and without the *yangma* there is no way to
know [the volumes of] such things as *zhui* and *ting* . These are
primary in practical application.

"Zhuiandting" is a condensed form referring to four geometric figures:fangzhui(see Section 3 above),yuanzhui(cone),fangting(see sections 3 and 4 above),yuanting(truncated cone).

[6]I am grateful to Else Glahn for explaining this matter to me. Documentation will be found in her forthcoming book on the Song architectural manual

[7]This passage gives difficulty. It would be
most natural to take *chunhe* to mean "fit together precisely," so that
the translation might be, "When the *yangma* have different shapes, then
they cannot be fitted together precisely. When they cannot be fitted together
precisely, . . . " There are two reasons why this cannot be the correct
interpretation: (1) the *yangma* can in fact be fitted together precisely
to form a box; (2) this is not the problem. Liu Hui's problem is that the three
*yangma* are not congruent, so that it is not proved that their volumes
are equal. The sentence must deal with the volumes of the yang-ma, and
therefore I tentatively translate *chunhe *with the vague word
"compare."

[8]*She yangma wei fen nei, bienao wei fen
wai, qi sui *. . . Qian Baocong's emendation is unnecessary.

[9]The word *xiao* has a variety of
meanings, including "to compare" and "to imitate." I interpret it here as "to
divide," primarily because this is the only interpretation which makes sense in
the context. Note that the word is used parallel with the word *fen* ,
which definitely has this meaning.

[10]Here I emend* gao er chi fang er chi* to* gao yi chi fang yi chi* , i.e. I take the dimensions of the cube to
be 1 x 1 x 1 instead of 2 x 2 x 2.

[11]Here I emend *mei er fen bienao ze yi
yangma* to *mei fen ze yi bienao yi yangma* . Regardless of whether this
emendation is correct, the original sentence is almost certainly corrupt. The
only way to make sense of it is to take *fen* as a measure word, so that
the sentence would be translated, "Each two *bienao *make one
*yangma*." Not only is this sentence pointless here, but since there are
only two *bienao* in the situation at hand, the word "each" is not
necessary.

[12]Accepting the variant *duan* instead
of *qi*.

[13]Reading *tuo* for *sui* . Another
example of this substitution is in *Jiuzhang, *p. 166, line 11.

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