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## ‘An early Chinese derivation of the volume of a pyramid: Liu Hui, third century A.D’. Historia mathematica, 1979, 6: 164-188.

Donald B. Wagner

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### Summaries

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.

### 1. Introduction

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.

### 2. The text

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).

### 3. Rectilinear solids treated by Liu Hui

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.

 A qiandu (lit., "moat-wall") is a right prism with right-triangular base [Jiuzhang, 166]. A yangma (lit., "male horse," but originally an architectural term; see Section 6.1 below) is a pyramid with rectangular base and with one lateral edge perpendicular to the base [Jiuzhang, 166-168]. Liu Hui's derivation of its volume is translated in Section 6 below. A bienao (lit., "turtle's shoulder-joint") is a pyramid with right-triangular base and with one lateral edge perpendicular to the base. The perpendicular edge is not to be at the right-angled vertex of the base [Jiuzhang, 168]. Liu Hui's derivation of its volume is an integral part of his derivation of the volume of a yangma; see Section 6. In one of his derivations (that of the volume of a xianchu , see below), Liu Hui uses a pyramid with right-triangular base and with the lateral edge at the right-angled vertex of the base perpendicular to the base. This he unfortunately also calls a bienao; but he is aware that his derivation of the volume of a bienao does not apply to this object, and he gives a separate derivation of its volume [Jiuzhang, 168-169]. A fangzhui (lit., "square awl") is a right pyramid with square base [Jiuzhang, 165}. A chumeng (lit., "fodder loft") is a right wedge with rectangular base [Jiuzhang, 169-170]. A chutong (lit., "fodder boy", probably originally some sort of agricultural implement or receptacle) is a truncated right pyramid with rectangular base [Jiuzhang, 170]. This object is also referred to by the terms panchi (lit. "winding moat"), quchi ("crooked moat"), and minggu ("dark valley"). [Jiuzhang, 170-173]. The etymologies of these terms are obscure. A fangting (lit., "square pavilion") is a truncated right pyramid with square base [Jiuzhang, 164]. Obviously the fangting is a special case of the chutong; Liu Hui does not use this fact, but treats the two independently. Liu Hui's derivation of the volume of a fangting is summarised in Section 4 below. A xianchu (lit., "drain"?) is a wedge with trapezoidal base and with one lateral face perpendicular to the base [Jiuzhang, 168-169]. In treating the volume of a xianchu, Liu Hui runs into a mathematical situation which is too complex for his methods and conceptual framework; the result is a very obscure text. A very tentative translation of this derivation has been given in [Wagner 1975].

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."

## 4. The fangting: an example of Liu Hui's method

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 a, b, and h are as shown in Figure 1. This formula is correct.

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;
4 qiandu, CDRQTU, BDRMNS, ABMLIJ, and ACQLKP;
4 yangma, 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 the qiandu at the four sides.
[Jiuzhang, 164]

That is to say,

abh = ABDCQRML + CDRQTU + BDRMNS + ABMLIJ + ACQLKP,
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 dimensions a, b, h. In the same way Liu Hui states:
b2h = ABDCQRML + 2(CDRQTU + BDRMNS + ABMLIJ + ACQLKP) + 3(CQTGP + DRUHS + BMNFJ + ALIEK),
and
a2h = 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 + b2h + a2h = 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 four yangma. 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 the qiandu at the four sides. Add [these two results] to obtain the volume of the fangting.

[Jiuzhang, 164] This second formula is

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

## 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?
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?"

## 6. Translation of the derivation

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

### [6.1. The etymology of the term yangma]

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 term yangma meant a hip-rafter in a hip-gabled roof.[6]

### [6.2. The case of equal dimensions]

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 the bienao are mirror-images of the other three.

### [6.3. The general case]

(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 a, b, c, is cut into six bienao as described above, the result is the three non-congruent bienao FADC, 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, and c, and the volume of each is 1/6 abc. 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 six bienao can be put together to form three yangma as 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). These yangma have the same dimensions, but they are not congruent. There is one other way of fitting the bienao together to form yangma (FADC with FEDC, FABC with CBGF, and CHEF with CHGF), and in this case again the yangma are 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]

### [6.4. An obscure digression]

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 two yangma, FEDCH and CBAFG, and two bienao, FADC and CHGF. Note that the two yangma are congruent mirror images, and that the two bienao are 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 image bienao or yangma are 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.

### [6.5 division of the yangma and bienao into smaller pieces]

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.

This qiandu is 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 red qiandu AGIJML and ILMJCP are fitted together, and the black qiandu FQONKL and CPORIL are fitted together. Two cubes are thus formed, one red and one black.
In the general case, these qiandu do not fit together. However it is clear that in each case the sum of the volumes of the two qiandu is equal to that of a box with dimensions 1/2 a, 1/2 b, 1/2 h.

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

The "divisions" are ABHJMK and KMPNFE in Figure 8. Since the qiandu in these divisions have been dealt with, what is left is one bienao and one yangma in each.

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

The two "items" are the two qiandu BGIHKL and EQOPML, each of which is composed of one red bienao and one black yangma.
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 1/2 a, 1/2 b, 1/2 h.

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 two qiandu blocks;
Red: 1 cube formed of two qiandu blocks;
Red and black: 1 cube formed of two red bienao blocks and two black yangma blocks.
In the general case,
Black: a volume of 1/4 abh;
Red: a volume of 1/8 abh;
Red and black: 2 red bienao with dimensions l/2 a, 1/2 b, 1/2 h; 2 black yangma with dimensions 1/2 a, 1/2 b, 1/2 h, ; total volume l/8 abh.
Thus the original 2 x 2 x 2 bienao and yangma have 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 l bienao and yangma. 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.

### [6.6. Carrying the operation to the limit]

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."]

### [6. 7. Justification for this concern with impractical matters]

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.

"Zhui and ting" 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).

## 7. Conclusions

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.

## Acknowledgments

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.

## References

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
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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 fangting, yangma, and xianchu.

[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 Yingzao fashi .

[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.