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The *Jiuzhang suanshu* ("Arithmetic in nine chapters") is a
mathematical book of the late first century A.D. It gives practical problems
and states algorithms for their solution, with no explanation. A commentary
attributed to Liu Hui , of the third century A.D., gives an explanation of each
algorithm; these explanations satisfy many of the criteria for what we would
call a proof. In this article Liu Hui's explanation of the formula for the
volume of a particular kind of pyramid is translated and discussed.

Das *Jiuzhang suanshu* ("Neun Bücher arithmetischer
Technik" ist ein mathematisches Buch aus dem späten ersten Jahrhundert
unserer Zeitrechnung. Es stellt praktische Probleme der und gibt Algorithmen,
jedoch ohne jede Erklärung, für ihre Lösung. Ein Kommentar, der
Liu Hui aus dem dritten Jahrhundert zugeschrieben wird, erläutert jeden
einzelnen Algorithmus; diese Erläuterungen weisen viele der Merkmale auf,
die bei uns einen mathematischen Beweis charakterisieren. In diesem Artikel
wird Liu Hui's Erläuterung von der Formel vom Rauminhalt einer besonderen
Art Pyramide übersetzt und näher besprochen.

For this Web version I have converted the transcription to Pinyin and inserted the Chinese characters. Otherwise virtually no changes have been made to the original article.

Early Chinese attempts at proofs of mathematical propositions are not well known in the West. Among the mathematicians who attempted to go beyond practical calculation to a more abstract and reasoned mathematics can be named: Liu Hui (third century A.D.), Zhao Shuang (third century A.D.), Zu Chongzhi (429-500), Zu Gengzhi (son of Zu Chongzhi), and Wang Xiaotong (seventh century A.D.).General information on these persons can be found in the usual histories of Chinese mathematics [1].

I have elsewhere translated Zu Gengzhi's derivation of the volume of a sphere
[Wagner 1978a]. Here I translate Liu Hui's treatment of the volume of a solid
called a *yangma* : it is a pyramid with rectangular base and with one
lateral edge perpendicular to the base.

As is well known, it is a consequence of a theorem proved by Max Dehn in 1900 that any proof of the volume of a pyramid must use infinitesimal considerations in one form or another.[2] Liu Hui does in fact use a limit process, though he has considerable conceptual difficulties with it.

Liu Hui's derivation is contained in his commentary on the mathematical
classic *Jiuzhang suanshu* ("Arithmetic in nine chapters").[3] This book appears to contain some very ancient materiel,
but probably reached its present form in the second half of the first century
A.D. [Qian 1964, 32-33]; it is the oldest extant Chinese mathematical book. The
only historical data available on Liu Hui and his edition of the *Jiuzhang
suanshu* is the date 263 A.D., given in a source written four centuries
later [*Jin shu* 1974, 491; *Sui shu* 1973, 409]. This date is
presumably that of a now-lost preface or colophon in an edition in circulation
in the seventh century.

There is some reason to suspect that the commentary attributed to Liu Hui is in fact a conflation of two or more commentaries (a detailed argument has been given in Wagner [1978b]). For example, Section 6.4 in the derivation translated below is clearly a digression and might well be a further comment on Liu Hui's comment by some later writer. It seems fairly sure, however, that the commentary attributed to Liu Hui is no later than the time of the commentator Li Chunfeng (602-670).

Chapter 5 of the *Jiuzhang suanshu* is ostensibly concerned with
earthworks and the amount of labour needed to build them. The only parts of the
chapter which are especially interesting mathematically, however, are those
which give algorithms for calculating the volumes of solids. For each of these
algorithms, Liu Hui gives a derivation. The order of his derivations is
dictated by the order in which the solids are discussed in the *Jiuzhang
suanshu*, but the derivations can be placed in a logical order so that each
depends only-upon those which precede it; Liu Hui is never guilty of circular
reasoning.

In the present article I summarise one of Liu Hui's derivations (Section 4 below) and translate one "Section 6). Below I describe briefly each of the rectilinear solids treated by Liu Hui.[4]

The volume of a box, or rectangular parallelopiped, the product of its three dimensions, is implicitly assumed, and Liu Hui makes no attempt to explain it.

It can be seen from the above that Liu Hui deals only with some very specific geometrical objects; there is no attempt to generalise, even though this would in many cases be rather simple. For example, his normal method could be used to derive the volume of a general pyramid as one-third of the product of the height and the area of the base.[5] This failure to generalise can in part be explained by reference to the nature of Liu Hui's book: it is a commentary on an existing classical work, and therefore its only concern is to explain the statements made in it, not to extend it. Furthermore, Liu Hui's purpose is probably primarily pedagogical. A student who has mastered his methods can easily handle a great variety of geometrical figures as the need arises.

Another feature which will be apparent in the above discussion is that most of the terms for these solids refer to practical objects. This should not lead to the immediate conclusion that ancient Chinese mathematics was only concerned with practical matters: certainly it was more practically oriented than Greek mathematics, but the use of terms like "fodder loft" to denote geometrical figures is not really different from our use of terms like "pyramid."

Before discussing Liu Hui's treatment of the *yangma*, it will be
useful to look at a simpler example of Liu Hui's treatment of the volumes of
rectilinear solids. The example I have chosen here is his derivation of the
volume of a *fangting* : this derivation is mathematically simple and
presents no philological problems.(Liu Hui's proof is translated in full in
Wagner [1975, 15-19].)

Liu Hui's general method in treating a rectilinear solid is to cut it up into parts whose volumes are known. From the formulas for these volumes he then derives the formula for the volume of the solid under discussion. Here he runs into two areas of difficulty: (1) expressing how the solid is to be cut up, and (2) manipulating the formulas for the parts to arrive at the formula for the whole.

Liu Hui solves the first difficulty by describing the division of the solid in
terms of manipulation of a set of standard blocks. These are: a cube, a *qiandu*, a *yangma*, and a *bienao*, each with breadth, length,
and height 1 *chi* (about 21 cm in Liu Hui's time). It is fairly clear
that Liu Hui possessed such a set of blocks, and expected his reader to use
them too. He calls these blocks *qi* : this word normally refers to
chessmen or to pieces in other board games; it is not impossible that these
blocks were part of some game or puzzle with a wider circulation than among
mathematicians.

Liu Hui is less successful in solving the second difficulty. His technical terminology is apparently not well-developed enough for him to refer to the general case; therefore, he derives the formula by reference to a case of specific dimensions, but in such a way that the proof can easily be generalised to the case of arbitrary dimensions. He gives only the broad outlines of the manipulations, leaving the details to the reader.

The *Jiuzhang suanshu* [1963, 164] gives the calculation of the volume of
a *fangting* as

where_{ }

In his derivation of this formula, Liu Hui refers throughout to the specific
case with a = 1 *chi*, b = 3 *chi*, and h = 1 *chi*. He
describes how this solid can be built up with the standard blocks. See Figure
1. The blocks are:

1 cube, ABDCQRML;

4qiandu, CDRQTU, BDRMNS, ABMLIJ, and ACQLKP;

4yangma, CQTGP, DRUHS, BMNFJ, and ALIEK.

He then writes:

The product of [the sides of] the upper and lower squares is [ab=] 3 [square]chi, and multiplying by the height gives [abh=] 3 [cubic]chi. This means there is obtained [the volume of] one central cube and one each of theqianduat the four sides.

[Jiuzhang, 164]

That is to say,

and though the text refers to specific dimensions, this statement is also true in the case of arbitrary dimensions. The statement can be verified geometrically: the five pieces can be fitted together to form a box with dimensionsabh= ABDCQRML + CDRQTU + BDRMNS + ABMLIJ + ACQLKP,

andb^{2}h= ABDCQRML + 2(CDRQTU + BDRMNS + ABMLIJ + ACQLKP) + 3(CQTGP + DRUHS + BMNFJ + ALIEK),

a^{2}h= ABDCQRML.

Again these statements are given in terms of specific dimensions, but both are valid in the case of arbitrary dimensions. Both can be verified geometrically. Liu Hui concludes that

abh+b^{2}h+a^{2}h= 3 ABDCQRML

+ 3 (CDRQTU + BDRMNS + ABMLIJ + ACQLKP)

+ 3 (CQTGP + DRUHS + BMNFJ + ALIEK).

Thus, implicitly using the distributive law,

which was to be proved.

Liu Hui goes on to derive another, equivalent, formula:

Another method: multiply by itself the difference of [the sides of] the squares, multiply by the height, and divide by 3. this gives [the volume of] the fouryangma. then multiply together [the sides of] the upper and lower squares and multiply by the height. This gives [the volume of] the central cube and theqianduat the four sides. Add [these two results] to obtain the volume of thefangting.

[*Jiuzhang*, 164]
This second formula is

_{ }

which may be verified either algebraically from the first formula or by Liu Hui's geometric method.

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.

[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 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.

TheyangmaBDFEC is divided into: 1 box, HILKNDRO; 2qiandu, ILORCP and KLONFQ; and 2yangma, 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* = 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 [xi] are the remaining [dimensions]. The extreme of fineness is called "subtle" [wei]. That which is subtle is without form [

[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 [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 asgong,shang,jue,zheng, oryu[ancient names for musical notes]; and its having no one form [

[Heshang Gong, Chap. 14,juan1, 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?"

- [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 dimensionsa,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.

(4) When the *bienao* have different shapes, then so do
the *yangma*.

See Figure 7a. The sixbienaocan 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.

When the *yangma* have
different shapes, then they cannot be compared [*chunhe*]. When they
cannot be compared, then it is difficult to do it [i.e., derive the formula].[7]

The following passage seems to be an additional comment on the preceding argument.

Why is this? "Dividing the cube slantwise gives
[two] *qiandu*" [abbreviated quotation of (1) above]; here the division is
necessarily in halves. "Dividing [one of the] *qiandu* slantwise gives
[one] *yangma* [and one *bienao*]" [abbreviated quotation of (2)
above]; here again the division is necessarily in halves. One [division] is
vertical, and one is horizontal.

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 twoyangma, FEDCH and CBAFG, and twobienao, FADC and CHGF. Note that the twoyangmaare congruent mirror images, and that the twobienaoare congruent mirror images.

Suppose a *yangma* is on the inside of a division, and
a *bienao* is on the outside.[8] Even if
"the block is long or short, or broad or narrow" [reference to (3) above],
there is still this constant proportion of the divisions. It is only through
this that one knows that the "different shapes'' [reference to (4) above] are
also equal.

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 *yangma* with breadth [*a*], length [*b*], and height
[*h*] each 2 *chi*, use one cubical block, two *qiandu* blocks,
and two *yangma* blocks, all of them black.

Figure 5 shows how the blocks are fitted together.

Joining together the red and black blocks to make a *qiandu*, the breadth, length, and height are each 2 *chi*.

Thisqianduis shown in Figure 8.

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.

Then divide [*xiao* ] [9] the breadth in the middle and divide
[*fen* ] the height in the middle.

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

Fit the red and the black *qiandu* together, in each
case forming a cube with height 1 *chi* and sides [each] 1 *chi*. [10]

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, theseqiandudo 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.

Each division then contains one *bienao* and one *yangma*. [11]

The "divisions" are ABHJMK and KMPNFE in Figure 8. Since theqianduin these divisions have been dealt with, what is left is onebienaoand oneyangmain each.

Each of the remaining items[12] is composed of [blocks with the same form as] the original objects.

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 1/2a, 1/2b, 1/2h.

Thus cubes
[formed of blocks] which are different [from the original *bienao* and *yangma*] occupy a proportion of 3, and cubes [formed of blocks] which
have the same form occupy a proportion of 1.

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 2bienaoandyangmahave 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 .

Even if the cube is elongated,[13] and the blocks change [accordingly], there is clearly a constant situation.

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 [*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?

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. *Daode jing* , Chap. 27: "The
skilled calculator does not use calculating rods."]

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).

Liu Hui's explanations of the statements in the *Jiuzhang suanshu* are not based on an axiomatic system, so it is preferable not to call them
"proofs"; I refer to them as "derivations" in analogy with the somewhat loose
mathematical derivations which one find, for example, in modem engineering
textbooks.

Liu Hui's derivations are based on assumptions, generally unstated, which he clearly considers to be obviously true. Among these assumptions are: that the volume of a box is the product of its dimensions, that the volume of the whole is the sum of the volumes of the parts; and the distributive law of algebra. Other assumptions, which we would not consider obvious, are that "the extreme of fineness . . . is without form," and a special case of Cavalieri's theorem (used in Liu Hui's treatment of curvilinear solids [Wagner 1978a]).

Liu Hui's standard of rigor is high. He is never guilty of circular reasoning; he never invokes numerological or mystical concepts; and in only the one case does he stray from considerations which we would accept as strictly mathematical.

We may imagine the frustration of many attempts to divide up a *yangma* into parts whose volumes are known. Not knowing Dehn's result, he must, in the
beginning, have been convinced that it was possible to derive the volume of the *yangma* in the same way that the volumes of other objects are derived. It
is surprising, and a clear indication of his standard of rigor, that he was not
content to deal with the case of equal dimensions (Section 6.2), but found it
necessary to deal explicitly with the general case. And it is an indication of
his mathematical ability that he succeeded in finding a method of dealing with
the problem.

0f course when I refer to Liu Hui here it is in a broad sense. Not only is there some doubt as to how much of Liu Hui's commentary was actually written by him; more important he was presumably only one in a long oral tradition of masters and disciples who gradually accumulated the mathematical results written down in his book.

In the pages above a number of weaknesses have been noted. Most important, Liu
Hui's terminology and mode of exposition are not adequate to express his
mathematical thought fully: it is clear that he is dealing with general cases,
but he expresses his arguments in terms of specific cases. In his derivation of
the volume of a *xianchu* the mathematical situation is extremely complex,
and he cannot or will not structure his exposition in such a way as to deal
adequately with this complexity. This is a weakness which the early Chinese
mathematical tradition shares with the oral traditions of craftsmen everywhere
(including modem computer programmers). In an oral exposition not everything
need be made explicit; resort can be had instead to enlightening examples and
"hand-waving." The herd work of stating precisely all assumptions and all the
steps in an argument is surely one of the most important driving forces in the
development of mathematics.

Another weakness is the failure to generalise. Liu Hui's conceptual framework
was adequate, for example to deal with a much broader range of geometric solids
than those which he actually considers in his commentary. Had he felt a need to
push his methods to their inherent limits he would surely have contributed a
great deal more to the mathematical tradition. Here we can see the double
influence of the enormous prestige of the *Jiuzhang suanshu*: it provided
a challenge and an inspiration; but it was often a strait jacket which confined
the interests of mathematicians to certain specific problems.

I am grateful to Lis Brock-Berendsen, Albert Dien, Akira Fujieda, Olaf Schmidt, and Nathan Sivin for their guidance and criticism in connection with an earlier version of this paper; to Hui Dong Shin for writing the Chinese characters; and to Erik Skaaning for translating the German summary.

- Qian Baocong , ed. 1963
*Suanjing shi shu*[Ten mathematical classics] Beijing- Qian Baocong 1964
*Zhongguo shuxue shi*[The history of mathematics in China] Beijing.*Jin shu*1974- [Official history of the Jin dynasty, 7th century A.D.], punct. ed. Beijing
*Jiuzhang suanshu*- [Arithmetic in nine chapters, 1st century A.D.] (ed. Qian 1963, 81-258)
- Dehn, M 1900
- Ueber raumgleiche Polyeder
*Nachrichten von der**königl. Gesellschaft der Wissenschaften**zu Göttingen, Mathematisch-physikalischen**Klasse*1900, 345-354 - Dehn, M 1902
- Ueber den Rauminhalt
*Math. Ann.*55, 465-478 - Erkes, Eduard 1958
*Ho-shang- kung's commentary*on*Lao-tse*Ascona- Gauss, Carl Friedrich 1900
*Werke*Leipzig ~- Hilbert, David 1900
- Mathematische Probleme (Vortrag, gehalten auf dem Internationalen
Mathematikerkongress zu Paris, 1900)
*Gesammelten Abhandlungen*3, 290-329 *Heshang Gong*, ed.*Daode jing*[The Way and its Power] (*Sibu congkan*ed.)- Jessen Børge 1939
- Om Polyedres Rumfang (Foredrag holdt i Parentesen den 20/11 1936)
*Mat. Tidsskrift, A*Aarg. 1939, 35-44 - Juschkewitsch, A P 1964
*Geschichte der Mathematik im Mittelalter*translated from Russian by Viktor Ziegler Leipzig- Karlgren, Bernhard 1975
- Notes on Lao-tse
*Bulletin of the Museum**of Far Eastern Antiguities,*No. 47 Stockholm - Lau, D C 1963
*Lao Tzu Tao Te Ching*(Penguin Classics)- Libbrecht, Ulrich 1973
*Chinese Mathematics in the Thirteenth**Century. The Shu-shu Chiu-chang of Ch'in Chiu-shao*Cambridge, Mass. (MIT Press)- Mikami, Yoshio 1913
*The Development of Nathematics in**China and Japan*Leipzig, reprint New York, n.d.- Needham, Joseph 1959
*Science and Civilization in China,*Vol. 3:*Mathematics and the Sciences of the Heavens and the Earth*(Cambridge University Press)*Sui shu*1973- [Official history of the Sui Dynasty, 7th century A.D.], punct. ed. Beijing
- Vogel, Kurt, transl. 1968
*Neun Bücher Arithmetischer Technik*(Braunschweig)- Wagner, Donald B. 1975
*Proof in Ancient Chinese Mathematics:**Liu Hui on the Volumes of Rectilinear**Solids*(unpublished thesis, University of Copenhagen)- Wagner, Donald B. 1978a
- Liu Hui and Tsu Keng-chih on the Volume of a Sphere
*Chinese**Science,*no. 3 - Wagner, Donald B. 1978b
- Doubts Concerning the Attribution of Liu Hui's Commentary on the
*Chiu-chang suan-shu*,*Acta Orientalia*(Copenhagen),**39**, 199-212

[1]Of works in Western languages the following are [in 1979] best: Libbrecht [1973]; Juschkewitsch [1964]; Needham [1959]; Mikami [1913]. By far the best history of Chinese mathematics is Qian [1964].

[2]Gauss asked in 1834 whether a proof of the volume of a pyramid were not possible without the use of infinitesimal considerations. This question was the basis of the third of Hilbert's famous 23 problems for mathematicians of the twentieth century. Dehn solved the problem when he proved that a regular tetrahedron and a prism can in no way be divided into respectively congruent parts. See Gauss [1900, Bd. 8, 244], Hilbert [1900, 301-302], Dehn [1900; 1902], Jessen [1939].

[3]The best available edition of the text [in 1979] is that of Qian Baocong [1963, 81-258]. There is a translation by Kurt Vogel [1968]; this translation is not reliable, but it can give the reader an idea of what the book is like. Vogel translates only the main text, not Liu Hui's commentary.

[4]In an unpublished thesis [Wagner 1975] I have translated Liu Hui's derivations of the volumes of the

[5]The derivation would begin with the fact that
the formula holds for any pyramid with triangular base and with one lateral
edge perpendicular to the base. (Such a pyramid is either the sum or the
difference of two *bienao.) *Next the formula could be derived for the
case of a pyramid whose base is a convex *n*-sided polygon and whose
altitude intersects the base. (Such a pyramid can be divided into n pyramids of
the type considered above.) The general case would be considerably more
complex, but not absolutely beyond Liu Hui's methods.

[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.