Ancient Chinese mathematics: the(Jiu Zhang Suan Shu) vs Euclid's Elements. Aspects of

proof and the linguistic limits of knowledge

Joseph W. Daubena, b, *aDepartment of History, Herbert H. Lehman College, City University of New York, 250 Bedford Park Blvd. West,

Bronx, NY 10468, USAbPh.D. Program in History, The Graduate Center, City University of New York,

33 West 42nd Street, New York, NY 10036, USA

Abstract

The following is a preliminary and relatively brief, exploratory discussion of the nature of earlyChinese mathematics, particularly geometry, considered largely in terms of one speci®c example: the

(Gou-Gu) Theorem. In addition to drawing some fundamental comparisons with Westerntraditions, particularly with Greek mathematics, some general observations are also made concerningthe character and development of early Chinese mathematical thought. Above all, why did Chinesemathematics develop as it did, as far as it did, but never in the abstract, axiomatic way that it did inGreece? Many scholars have suggested that answers to these kinds of questions are to be found in socialand cultural factors in China. Some favor the sociological approach, emphasizing for example thatChinese mathematicians were by nature primarily concerned with practical problems and their solutions,and, therefore, had no interest in developing a highly theoretical mathematics. Others have stressedphilosophical factors, taking another widely-held view that Confucianism placed no value on theoreticalknowledge, which, in turn, worked against the development of abstract mathematics of the Greek sort.While both of these views contain elements of truth, and certainly play a role in understanding why theChinese did not develop a more abstract, deductive sort of mathematics along Greek lines, a di�erentapproach is o�ered here. To the extent that knowledge is transmitted and recorded in language, oraland written, logical and linguistic factors cannot help but have played a part in accounting for how theChinese were able to conceptualizeÐand think aboutÐmathematics. # 1998 Elsevier Science Ltd. Allrights reserved.

International Journal of Engineering Science 36 (1998) 1339±1359

0020-7225/98/$19.00 # 1998 Elsevier Science Ltd. All rights reserved.PII: S0020-7225(98)00036-6

PERGAMON

* Tel.: 011 212 642 2110; Fax: 001 212 642 1963; E-mail: jdauben@email.gc.cuny.edu.

1. Jiu Zhang Suan Shu

One of the oldest and most in¯uential works in the history of Chinese mathematics is the(Jiu Jang Suan Shu, Nine Chapters on the Art of Mathematics), comprised of nine

chapters and hence its title (Fig. 1).1 Traditionally, this work is believed to include some of theoldest mathematical results of Chinese antiquity. Indeed, the origins of the Jiu Jang Suan Shu

Fig. 1. Title page, from the Southern Song edition of the (Jiu Zhang Suan Shu, Nine Chapters on the Art

of Mathematics), from the copy in the Shanghai Library, facsimile edition: Shanghai: Wen Wu Chu Ban She (WenWu Publishing House), 1981.

1 For Chinese editions of the Nine Chapters, see Refs. [2, 3 and 29], along with the detailed studies in Refs. [1]and [4]; there is also a translation into German in Ref. [5]. A French translation of the Nine Chapters, by KarineChemla and Guo Shu-Chun, is now in preparation and due to be published shortly.

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have been ascribed by some to the earliest period of China's recorded history, where fact shadesinto myth. One tradition says that the Yellow Emperor, Huang Di , who lived in the 27thcentury BC, charged his minister Li Shou with compiling the Jiu Jang Suan Shu.

Unfortunately, the original version of the Nine Chapters no longer exists. One of the ®rstgreat tragedies in Chinese intellectual history occurred in 213 BC, when the Emperor Qin ShiHuang (221±207 BC, famed for his terra-cotta army at Xi'an Yang ) orderedthat all books in the Empire be burned. Although some of the classics may have beensurreptitiously preserved, or memorized and later transcribed, the reconstituted texts producedfor these ``lost'' early documents likely contained inaccuracies or interpolations introduced bytheir rescuers. In the case of mathematical knowledge, later innovations and new techniquesmight well have been incorporated as if they had been part of the original.

The subsequent history of the Nine Chapters is nearly as uncertain as its origins. The earliesttext we have of the Jiu Zhang Suan Shu was compiled by Zhang Cang sometime in the2nd century BC, and revised about 100 years later by Geng Shou Chang . Both ofthese scholars lived in the Western Han Dynasty (206 BC±24 AD), and both were imperialministers who undertook their reconstructions of the Nine Chapters at a time when there weregreat e�orts being made to restore lost classics of any sort [6]. When Liu Hui , amathematician of Wei during the Three Kingdoms Period (220±280 AD), again edited the JiuZhang Suan Shu in 263 AD, this time with an extensive commentary, he began a tradition thatwas repeated after him by Li Chun Feng [ ] in the Tang Dynasty (618±907 AD), whoalso collated and commented on the book.2

The oldest edition of the Nine Chapters to survive is the wood block printing of theSouthern Song Dynasty. Only the ®rst ®ve books (or chapters) are preserved in this edition,which was produced about 1213 AD and is best known from the copy in the Shanghai library.Most other editions are based on the Complete Library of the Four Branches of Literatureedited by Dai Zhen of the Qing Dynasty, who copied it from the Great Encyclopedia ofthe Yong-le Reign Period of the Ming Dynasty (known as the Dai edition). The most famouscommentaries are those by Liu Hui (263 AD), Li Chung-Feng (656 AD), and one by ZuChong-Zhi [ ] (429±500 AD) written during the North and South Dynasties, but nowlost.3

2 One of the Great Masters of ancient Chinese mathematics, Liu Hui is all but an enigma in the history of

Chinese science. Based on nothing more than philological evidence, it is only possible to say that he may have comefrom Zouping in the Shandong Province. Based upon a contemporary report, it can at least be said that he com-posed his commentary on the Nine Chapters in 263 AD. All that is known about Liu Hui, in fact, is his well-known

commentary on the famous Chinese mathematical classic, the Nine Chapters, and a treatise that Liu Hui wrote him-self, the (Hai Dao Suan Jing, The Sea Island Mathematical Manual). This ``manual'', which takes itsname from the ®rst problem devoted to calculating the height and distance of an island at sea, contains only nineproblems. It was originally intended as an additional ``tenth'' chapter to the Nine Chapters, but it later came to be

regarded as a classic, independently, on its own. For further discussion of The Sea Island Mathematical Manual, seeRef. [7], and the recent English translation in Ref. [8]. For studies speci®cally devoted to Liu Hui, see Refs. [1, 4, 9±19].3 The date of Liu Hui's edition and commentary on the Jiu Zhang Suan Shu is based on a sentence in the manu-

script which mentions that it was written in the fourth year of the Jing Yuan reign of King Chen Liu of Wei, whichdates it exactly to 263 AD. See Ref. [7].

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The Jiu Zhang Suan Shu dominated the practice of Chinese administrative clerks for morethan a millennium, and yet in its social origins it was closely bound up with the bureaucraticgovernment system, and was consequently devoted to the problems which ruling o�cials hadto solve. It was also of overwhelming in¯uence on writers in the centuries that followed, and itis no exaggeration to say that virtually all subsequent Chinese mathematics bears its imprint asto both ideas and terminology [20].It is in this sense that the Nine Chapters may be regarded as a Chinese counterpart to

Euclid's Elements, which dominated Western mathematics in the same way the Nine Chapterscame to be regarded as the seminal work of ancient Chinese mathematics for nearly twomillennia.4 In several important respects, however, the two works are more striking for thedi�erences they exhibit rather than their similarities. Euclid's text is renowned for the austerityof its axiomatic method, beginning with abstract, idealized de®nitions and proceeding fromaxioms and postulates to a progressively arranged series of proofs, leading, through thethirteen books that survive, to some remarkable results on the ®ve regular (sometimes calledPlatonic) polyhedra.The Nine Chapters, on the other hand, is a much more down-to-earth (literally) handbook

for the solution of practical problems. However these often led to solutions that were ascomputationally di�cult as they were theoretically subtle. Chinese mathematicians were just ascreative in devising new methods as their contemporaries anywhere in the world, particularly inobtaining solutions of simultaneous equations, in which they had no rivals in antiquity.5

2. Mathematics and the Nine Chapters

Like earlier works of ancient Chinese mathematics upon which the Jiu Zhang Suan Shu wasdoubtless based, it presents a series of problems (246 in number) in a question±answer format.6

Those in the ®rst eight chapters deal with such practical concerns as surveying, commercialproblems, partnerships and taxation rates. Also dealt with are the extraction of square andcube roots, the properties of various solids (including the prism, pyramid, cylinder and cone),and the solution of linear equations in two or more unknowns (in the course of which theChinese were led to introduce for the ®rst time negative quantities).The ®nal chapter of the Jiu Zhang Suan Shu, however, is the most famous. It is in Chapter 9

that the ``Gou-Gu'' theorem, known in the West as the Pythagorean theorem, is introduced.

4 Studies of Chinese mathematics include Needham's classic study, [20], which augments Mikami's much earlier

study, [22]. More recent and reliable studies of Chinese mathematics are given in Refs. [7, 23±25]. For reviews of thelatter, see Refs. [26±28].5 See for example Chapter 8 of the Jiu Zhang Suan Shu, which is devoted to the solution of simultaneous linear

equations by a method known as ``rectangular tabulation'' [ , square tabulation]. The most complicated of

these, problem 18, involves ®ve linear equations in ®ve unknowns; problem 13 involves ®ve equations with sixunknowns, and is indeterminate. See Refs. [29, 30].6 Recent archaeological excavations of tombs at Zhang Jia Shan in Hu Bei have produced a work written on bam-

boo strips, the Suan Shu Shu, dating from the ®rst half of the second century BC or earlier. It is in the question±answer form like the Nine Chapters. More than 60 di�erent types of calculation are included, and some problemsare very close to ones found in the Jiu Zhang Suan Shu. See Ref. [7].

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This theorem states that for any right-angled triangle, the sum of the squares of its sides isequal to the square of the hypotenuseÐa result familiar in algebraic terms as a 2+b 2= c 2.The chapter's 24 problems deal primarily with right triangles and solutions of quadraticequations. One of these is a variation on one of the oldest of China's mathematical problems:

In the middle of a pond that is ten chi in diameter, a reed grows one chi above thesurface of the water. When pulled toward the edge of the pond, the reed just reaches theperimeter. How long is the reed?7

The solution to this problem is a straightforward application of the Gou-Gu theorem (Fig. 2).

3. The Chinese Gou-Gu theorem

Even before the Nine Chapters was written, results dealing with right triangles had beenpresented in an earlier, astronomical±mathematical work, the (Zhou Bi Suan Jing,The Arithmetical Classic of the Zhou Gnomon). In both cases, the written texts clearly state thatfor a right triangle, given the shorter and longer sides enclosing the right angle, the sum oftheir squares is equal to that of the square of the hypotenuse.Although the original explanations of this discovery are lost, it is possible to reconstruct the

general line of reasoning that must have been used from several di�erent texts. In addition tothe Commentary on the Zhou Bi Suan Jing by Zhao Shuang , we also have Liu Hui's ownannotations concerning the results on right triangles in the Nine Chapters. For example, in

Fig. 2. Problem 6, Chapter 9 of the (Jiu Zhang Suan Shu, Nine Chapters on the Art of Mathematics).

7 This is problem 6 in Chapter 9 of the Nine Chapters, in Ref. [29].

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commenting on a passage that reads, ``Combining each square of Gou and Gu, taking thesquare root will be Xian (the hypotenuse),'' Liu Hui explains as follows (See Fig. 5 below):

The Gou-square is the red square [ , Zhu Fang], the Gu -square is the blue square [ ,Qing Fang]. Putting pieces inside and outside according to their type will complement eachother, then the rest (of the pieces) do not move. Composing the Xian-square, taking thesquare root will be Xian (the hypotenuse).8

This entire passage is obscure and problematic. The reference to moving pieces inside andoutside is related to a diagram, no longer extant, and makes use of the so-called ``Out±In''method which was taken as an axiom by ancient Chinese mathematicians. The power of thisaxiom can be seen, however, from the following example. Given a rectangle ABCD dividedhorizontally by aa and vertically by bb, under what circumstances is it possible to prove thatA=B? (Fig. 3). So long as the horizontal line aa and the vertical line bb intersect on thediagonal AC, it will follow that A=B. This can be seen immediately from the followingobservations related to (Fig. 4). Since the diagonal divides the rectangle into two equaltriangles, ABC and ADC, removing the two equal triangles below the horizontal line aa (I andI 0), as well as the two equal triangles above the horizontal line aa (II and II 0), it follows fromthe simple logical principle that equals subtracted from equals are equal, that A=B.9

This now helps to explain Liu Hui's commentary on the Gou-Gu theorem. Applying the``Out±In'' complementary principle to the Xian ®gure, and following Liu Hui's commentary onthe Gou-Gu theorem, the sum of the squares based on each leg of the right triangle ABC,namely the squares ADEB and BFGC, is equal to the square of the hypotenuse (AC), namelythe square AHJC (Fig. 5). In accordance with the ``Out±In'' principle, if we move those partsof the two small squares (ADEB and BFGC) that are on the outside of the large square(AHJC) to its inside, we can see that they ®ll the inside exactly and that the combined areas ofthe two small squares equal that of the larger one. Since the areas are in sum equal to thesquared sides of the triangle, the sum of the squared legs equals the squared hypotenuse.

Fig. 3. Given the rectangle ABCD and the two perpendiculars, the horizontal line aa and the vertical line bb, under

what conditions will the areas A and B be equal?

8 Liu Hui, as quoted in the recent critical edition of the Nine Chapters, [3].9 For a detailed discussion of this ``Xiang bu'' principle, see ``Chu Qu Xiang Bu Yan Li'' (Out±In Principle) in

Ref. [19] as well as Ref. [31].

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The earliest work containing this idea of demonstrating the right-triangle theorem by asuitable rearrangement of areas has already been mentioned: the Zhou Bi Suan Jing, theArithmetical Classic of the Zhou Gnomon, which some scholars have dated to as early as 1100BC (Fig. 6). Astronomical evidence, however, suggests that most of the material must beconsiderably later, dating most likely from the time of Confucius about the 6th century BC.The speci®c case of the right-triangle theorem is given in terms of the 3-4-5 triangle, and thetheorem itself is explained in terms of ``piling up rectangles.''10

Fig. 4. Application of the ``Out±In'' Theorem demonstrating that if aa and bb intersect on any point coinciding with

the diagonal of ABCD, then A = B.

Fig. 5. The [Xian Tu = hypotenuse diagram], based upon the ®gure from the Southern Song edition of the(Zhou Bi Suan Jing, The Arithmetical Classic of the Zhou Gnomon), from the copy in the Shanghai

Library, facsimile edition: Shanghai: Wen Wu Chu Ban She (Wen Wu Publishing House), 1981, p. 3.

10 For analysis of the mathematical contents of the Zhou Bi Suan Jing, see Ref. [32].

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4. Greek mathematics: a dramatic contrast in form and function

Compare these developments of the Gou-Gu theorem in the Chinese mathematical traditionwith what is to be found in Euclid's Elements. The most striking di�erence is certainly theaxiomatic framework of Euclid's work and its abstract, formal character.Book 1 of the Elements begins with careful de®nitions, then introduces axioms and

eventually theorems with proofs that are interconnected and, in general, built upon oneanother in a progressive fashion. Book 9 of the Jiu Zhang Suan Shu, on the other hand, beginsimmediately with a concrete practical problem and wastes no time in providing a modelsolution.However, there is another, less immediately obvious di�erence between the two works as

well. Ask anyone what Euclid's elements is about and the answer will inevitably be

Fig. 6. Title page, from the Southern Song edition of the (Zhou Bi Suan Jing, The Arithmetical Classic ofthe Zhou Gnomon), from the copy in the Shanghai Library, facsimile edition: Shanghai: Wen Wu Chu Ban She

(Wen Wu Publishing House), 1981.

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``geometry.'' And yet the word geometry does not appear at all in Euclid's bookÐa fact that is

as curious perhaps as it is illuminating (for reasons that will be made clear in a moment).

Moreover, whereas the Chinese demonstration of the right-triangle theorem involves a

rearrangement of areas to show their equivalence, Euclid's famous proof of the Pythagorean

TheoremÐProposition I,47Ðdoes not rely on a simple shu�ing of areas, moving A to B and

C to D, but instead depends upon an elegant argument requiring a careful sequence of

theorems about similar triangles and equivalent areas. What Euclid achieves re¯ects an entirely

di�erent approach to mathematics from the more straightforward and concrete Chinese

version.11

De®nitions in Euclid, however, betray earlier origins that bring us much closer to a point of

view re¯ected in the Jiu Zhang Suan Shu. Consider, for example, Euclid's de®nition of a

straight line: ``that which lies evenly with the points on itself,'' (the familiar ``shortest distance

between two points'' de®nition is later, and another story in itself). If one pictures a rope or

cord being stretched between two points, held by surveyors perhaps to measure distances or

plots of land, the points of the cord ``lying evenly'' immediately betrays the kind of concrete

experience from which the Greeks began to think about mathematics, eventually reaching

levels of abstraction that tended to eliminate any trace of the humble, practical origins of what

later was presentedÐand reveredÐas a most sublime achievement of axiomatic, abstract

mathematical proofs.

Of all the lines in Greek geometry, however, among the straight lines the most interesting (if

only for its name) is the hypotenuse. This derives from the Greek word ``teinousa'' meaning

``stretched.'' Plato, in the Meno 48E8±85A2, uses the word teinousa to indicate the line

``stretched across'' between opposite corners of the square, i.e. the diagonal.

The hypotenuse is therefore that which is ``stretched over or across.'' Plato in Parmenides

137E de®nes the straight line as ``that of which the middle covers the ends.'' Heron de®nes the

straight line as ``that stretched to the utmost between both ends.'' Each of these examples

serves to indicate the practical origins of the Greek Pythagorean theorem in earth

measurementÐand the ancient tradition of rope stretching.

This of course drew on experience the Greeks had had with even earlier Egyptian geometry,

and the famous Harpedonaptai, or Egyptian rope stretchers. However, the Egyptians were not

alone in using this technique. The Akkadians, Assyrians, Babylonians, HebrewsÐthey all

carried out basic surveying with the help of ropes, using rope stretchers who served as skilled

land surveyors. Isaiah 35:17, for example, describes the land ``portioned out with the line,'' and

Amos 7:17 speaks of ``land parceled out by the line.''

In keeping with this, the oldest Hebrew geometry, described by one commentator as a sort

of practical handbook for rope stretchers, is the Mishnat-ha-Middot (Theory of Measures).

Here too, the word ``cord'' or ``rope'' is employed to indicate the diagonal of a square, or the

hypotenuse of the right triangle. Similarly, in India the Apastamba Sulva-Sutra of the 5th±4th

11 For studies of Greek mathematics and particularly the signi®cance of Euclid's approach to writing the Elements,see Refs. [33] and [66]; for analysis of Chinese mathematics and the question of proof compared with the methodsand standards re¯ected in Euclid, see Refs. [10] and [16].

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century BC describes right triangles by means of stretched ropes and gives, as examples, suchtriples as 5-12-13, 8-15-17 and even 12-35-37.All of these various, yet similar, experiences with the practical, day-to-day business of

geometry are re¯ected in language directly. Although originally in the hands of those busy withworking or surveying the land, empirical understanding was later passed on to the moresystematic and eventually abstract concerns of mathematicians, who, in turn, extended andgeneralized the scope of geometry. However, more often than not, the words they used werethe words of their practically-minded counterparts.One last example will su�ce to bring all of this into a current frame of reference, for the

word we use today in mathematics for ``that which lies evenly on itself,'' namely the word``line,'' comes from the Latin ``linea.'' Originally, linea meant literally ``linen thread,'' thenominative taken from lineus, meaning ``of ¯ax'' and derived from the Latin word for ¯ax,linum. All of these bear a direct relation to another well-known word in English, linen, whichalso harkens back to the ¯ax from which ®ne linens are made.

5. Chinese geometry and rope stretchers

Chinese geometry, it will now come as no surprise, also seems to have had its origins in arope-stretching, surveying tradition. And again, this legacy is directly re¯ected in mathematicalnomenclature in a natural way. ``Gou'' means leg, ``Gu'' means thigh, and ``Xian'' , thecharacter used for hypotenuse, means lute string.12

Thus Chinese geometry, particularly the Gou-Gu theorem, rests ®rmly in a practical traditionof earth measurement and direct manipulation of physical or visualizable elements used to``demonstrate its results.'' This is very much in keeping with the approach to similar geometricproblems of surveying taken by the earliest mathematical Westerners, the Egyptians andBabylonians.As Lam Lay-Yong has shown in her analysis of the practical rules of arithmetic developed

by the Chinese for land surveying, ``The shape of all things, when broken up into theirappropriate sections, will ultimately yield the shapes of the basic farm ®gures,'' [35]. Chinesemathematics, clearly, was no di�erent from western mathematics in being rooted in practicalconcerns of agriculture and the land. But aside from discovering special mathematical results,particularly very general relations between geometric entities for example (like the right-triangleproperties), what can be said about Chinese interest in ``proofs'' or other forms ofdemonstration? Again, Professor Lam's study of ``The Alpha and Omega of a selection on theApplications of Arithmetical Methods'' is instructive:

12 The Chinese character for ``mathematics'' is itself an interesting one philologically. The character ``Suan'' is

based on a radical meaning cowry for shells or, in a slightly di�erent form, goods. The character above, , is thecharacter for bamboo strips, and is a reference to the bamboo slips used as tallies to sort goods. ``Suan'' originallyreferred to the counting board, with which bamboo counting rods were used to carry out calculations during trans-

actions, and later came to refer to the abacus or any method of calculation, including mathematics generally. Thusthe character ``Suan'' ideogrammatically embodies both the original objects and methods that became the stock intrade of the court and administrative mathematicians. For details, see Ref. [34].

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The working of a problem is selected from various methods, and the method should suit theproblem. In order that a method is to be clearly understood, it should be illustrated by anexample [35].

This was clearly not the Euclidean way. Individual, concrete examples of the Pythagoreantheorem alone would not su�ce to prove, in general, abstractly, universally, a result like theversion as demonstrated in Euclid I,47. However, demonstration by example was the Chineseway. Plausible generalizations were drawn from concrete situations. Consequently, we ®ndmany fascinating problems and examples in Chinese mathematical texts which are meant todemonstrate a wide variety of techniques. From measuring the heights of pagodas, distances ofislands, depths of wells, the similarity of triangles is used as a general method, as is the double-di�erence method, among others, to resolve a host of geometric problems (Fig. 7).13

6. The Chinese versus the Greek mathematical spirit

It is sometimes asked, why did the Chinese not go on to develop a Euclidean axiomaticmathematics? Why not a more abstract proof, for example, of the Gou-Gu theorem? However,this is surely the wrong question. The real question is why should the Greeks have departed

Fig. 7. Determining the height of a pagoda using the double di�erence method and the similarity of triangles.

13 Readers not familiar with these methods, but wishing to know more about the application of similarity of tri-angles and the double di�erence method, should consult the article cited above by Wu Wenjun, particularly the sec-tion on ``Gnomon, Shadow and Double Di�erences,'' in Ref. [31].

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from virtually all other cultures in this respect, namely in their preoccupation with axiomatic,deductive proofs? However, this clearly is a very di�erent problem from determining the originand nature of Chinese mathematics.In a way, however, the Chinese demonstration of the Gou-Gu theorem is general; the

rearrangement of areas holds for any right triangle, not just integer-sided triangles of theEgyptian or Babylonian variety. In working out their many applications of the Gou-Gutheorem, it is clear that the Chinese understood it to hold for any right triangle, and came upwith much more general, non-integer Gou-Gu triples such as 8, 9 1/6, 12 1/6; and 10, 49 1/2, 501/2.It has been argued that geometry never developed further in China than it did with Liu

Hui's commentary because this was su�cient, and comprehensive enough, for Chinese needs.After all, what real, utilitarian purpose is served by abstract mathematical proofs?14 Anotherargument closely related to this one suggests that Chinese mathematics, with its major concernfor practical problems, had little interest in abstract generalizations. Although at ®rst glancethis may indeed seem to have been the case, upon closer examination one soon begins to havedoubts. For example, if one looks at the method Liu Hui follows in his commentary on theNine Chapters, one ®nds that he is very careful to explain each formula given for areas andvolumes, and that these ``explanations'' are very much like simple, basic proofs.Here, much can be learned from D. B. Wagner's study of Liu Hui's commentary on the

volume of a pyramid. At the beginning of his commentary, Liu Hui describes his methodbrie¯y in a short preface:

By properly arranging concepts and propositions through analogous and deductive analysis,they could be put in their proper place. Therefore those branches which grow diversi®ed butshare the main stem of a tree are comprehended to be from one origin. Furthermore toanalyze a theory by proposition, and to illustrate a structure with geometrical ®gures, willthereby make the whole picture of the theory or structure be understood through somesimple principles, and also make that thoroughly apprehensible but not without penetrating.Thus the reader would grasp most of the ideas.15

The character [Zhu] for proposition or judgment has close a�nities here with thephilosophy of Mo-Zi as re¯ected in the Mo-Zi Debates, an ancient work on Chineselogic. Here, ``Zhu'' has a meaning close to ``proposition.'' Since Liu Hui mentions the Debatesin his commentary, he was obviously familiar with Mo-Zi's ideas, and presumably with theterminology and methodological ideas they re¯ected.

14 Actually, there are good grounds for a positive answer to this question, for one of the strongest arguments insupport of pure mathematics has always been the power of its applications. This also constitutes one of the deepestphilosophical puzzles about mathematics, namely the reasons for the important connections between the ideal world

of abstract mathematics and the concrete world to which its applications have proven so signi®cant.15 Liu Hui's commentary on the volume of a pyramid, as translated by Horng Wann-Sheng from Chapter V of theNine Chapters. For details, see Refs. [12±14, 29, 36].

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In this light, particularly of Mo-Zi's ideas, consider Liu Hui's derivation of the volume of apyramid:

The proposition comes out from direct causes; develops with general laws or theory; and ismanipulated by applying to the same class of [things] . . . If these three conditions are allsatis®ed, then it is su�cient to establish the proposition . . .16

In short, Liu Hui does seem to re¯ect an interest in a general systematic approach toestablish his geometric ``demonstrations.'' It is not simply a matter of a few examples givingrise to a generalization, but a realization that some deeper principles serve to establish``causes'' related to ``laws'' that underlie ``propositions.''

7. Chinese values of pp

Moreover, claims of an overriding practical interest of the Chinese only in concrete problemswith useful applications is not borne out in the case of Liu Hui's meticulous approximation ofthe numerical value of p. This example, in fact, illustrates how dramatically the Chinese couldsurpass even the Greeks in accuracy, yet these are matters of no practical value whatsoever.The common practice in China before the Han Dynasty (206 BC±220 AD) was to take the

ratio of p= c/d as 3. The earliest example of a better ®gure than this very crude result (butubiquitous in most ancient cultures with a rudimentary interest in geometry), comes from abronze cylindrical standard measure cast by o�cial order in the early ®rst century which givesa value of 3.1547.By the time of Liu Hui an even better, very sophisticated result had been achieved, and one

that is surprisingly familiar to anyone acquainted with Archimedes, for the Nine Chapters usesinscribed regular polygons to approximate a value of p (as did Archimedes) (Fig. 8).17 This isdescribed in Liu Hui's commentary on the Nine Chapters as follows:

The ®ner we cut the segments, the less will be the loss in our calculation of the area of thecircle. The exact area of the circle is obtained when such segments so cut o� come to bein®nitesimals [37].

Liu Hui ®rst obtained a value for p using a 192-sided polygon, which gave him a value of3.14 64/625 (about 3.141024). He then went on to consider an inscribed polygon of 3072 sides!This extraordinary computation gave him a value for p of 3927/1250 = 3.1416 . . . [37]; but seeas well [67].

16 [29] (translation by Horng Wann-Sheng).17 For further details and analysis of Liu Hui's methods, see Refs. [9, 17, 18, 38±42].

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Certainly, Liu Hui provides a constructive counter-example to the idea that ancient Chinesemathematics stopped at what was only of practical value. It was clearly capable of going wellbeyond what was simply practical, or even physically possible. The theoretical interest ofobtaining greater accuracy, or discovering systematic connections between propositions andtheir demonstrations, does seem to have been an interest of Liu Hui's.Nevertheless, after Liu Hui, Chinese geometry does not seem to have made much further

progress. Although some authors suggest that this was due primarily to the practicalorientation of ancient Chinese mathematics, it may have been its actual success, itscomprehensiveness, that caused the stagnation of any further development. As D. B. Wagnerhas suggested:

Liu Hui's conceptual framework was adequate, for example, to deal with a much broaderrange of geometric solids than those which he actually considers in his commentary. Had hefelt a need to push his methods to their inherent limits, he would surely have contributed agreat deal more to the mathematical tradition. Here we can see the double in¯uence of the

Fig. 8. Slicing p more ®nely: one of Dai Zhen's illustrations for the (Jiu Zhang Suan Shu, The NineChapters on the Art of Mathematics), where Liu Hui's method of approximating the value of p by inscribing regularpolygons is demonstrated.

J.W. Dauben / International Journal of Engineering Science 36 (1998) 1339±13591352

enormous prestige of the Chiu-chang suan-shu: it provided a challenge and an inspiration;but it was often a strait jacket which con®ned the interests of mathematicians to certainspeci®c problems [36].

Like Euclid, Liu Hui summarized his art so successfully that his successors may have feltlittle need, or room, for improvement.

8. Incommensurability and Chinese mathematics

There is, however, another aspect of ancient Chinese mathematics that is also striking, andupon re¯ection, again separates it from Greek, and indeed from some of the most fundamentalprinciples of modern mathematics. Nowhere is there any interest shown in classic Chinesemathematical texts in the ``irrational'' character, for example, of the diagonal of the square.Although certain approximations to its value were explored, neither an analysis byanthyphairesis, or the better-known even±odd analysis to be found in Aristotle's famousdemonstration of the ``incommensurability'' of

���2p

, appear in Chinese mathematical orastronomical texts.In fact, there is an entire class of mathematical arguments missing from Chinese thought,

and its lack is re¯ected in language and logic alike. What is not to be found in any Chinesereasoning about mathematics (or philosophical or logical matters in general) are argumentsbased upon counter-factual reasoning.18

One of the most powerful methods of proof in Western mathematics, however, relies uponcounter-factual arguments of the ``Reductio ad absurdum'' type. Such arguments begin bymaking an assumption, assumed to be true for the sake of argument, and then showing howthe assumption leads to a contradiction. The contradiction in turn establishes the fact that theinitial assumption is indeed false, contradictory to fact. This method does not surprise us, butit would doubtless have surprised and seemed very unnatural to Liu Hui.

9. Recent psycholinguistic research

According to recent research by the linguist Alfred Bloom:

Postulating false premises for the express purpose of drawing implications from them aboutwhat would be the case if they were true is a psycholinguistic act. Hence the development ofa facility for it is likely to be highly contingent on the nature of the incentives that languageprovides [43].

18 Bloom has studied the signi®cance of ``enti®cation'' and counter-factual constructs in Ref. [43]. Other useful lin-

guistic studies include [44±46]. Bloom's work has engendered considerable controversy and discussion; see forexample Refs. [47±51]. For analysis of the mathematical signi®cance of interpreting ancient Chinese approximationsof

���2p

as re¯ecting an understanding of incommensurability, see Ref. [52].

J.W. Dauben / International Journal of Engineering Science 36 (1998) 1339±1359 1353

Counter-factual reasoning, in turn, is closely related to what Bloom terms ``enti®cation,'' thecreation of theoretical concepts by abstraction from speci®c properties or actions. Englishenti®es concepts easily, generalizing from ``soft'' to ``softness,'' ``society'' to ``sociology,''``modern'' to ``modernize,'' but there is no easy way to do this in Chinese, to advance from``white'' to the concept of ``whiteness,'' from ``probably'' to ``probability.''There are, however, several instructive exceptions to this generalization that Chinese does

not make enti®cation easy. A classic example is the 4th century BC philosopher Gong-SunLong [ , Kung Sun Lung], who discussed such concepts as whiteness, horseness, etc.However, his editor/commentator Chan Wing-Tsit notes that he had ``no in¯uencewhatsoever'' after his own time [53]. Here, the exception serves to re¯ect the rule.The importance of enti®cation in mathematics, however, is of considerable importance.

De®ning a ``point'' as ``that which has no dimension,'' or a ``line'' as ``breadth without length,''is to speak of things which are ideal. However, such entities do not in fact exist in the concreteworld, and, therefore, are meaningless from a Chinese point of view.The abstract, in general, falls into this category. It is hardly surprising, therefore, that no

Chinese mathematicians, however facile they may have been with the tools of geometry, thealgorithms of algebra, or with computations in general, ever thought to de®ne points, lines orspace as they are de®ned by Euclid. However, in dealing with actual situations and concreteprocedures, there were no such di�culties. Liu Hui, for example, develops the doubledi�erence method to a high degree of sophistication, in part because it deals ultimately withspeci®c concrete entities.On the other hand, consider a basic counter-factual situation. Suppose a proposition P to be

true when it is not. One of the classic examples of a counterfactual argument from ancientGreek mathematics, beginning with a proposition known to be false, is the proof Aristotlerecounts for the incommensurability of the diagonal of the squareÐwhich establishes (inmodern terms) that

���2p

is irrational.

10. The Pythagorean discovery of incommensurable magnitudes

Aristotle reports the Pythagorean doctrine that all things are numbers and surmises that thisview doubtless originated in several sorts of empirical observation.19 For example, in terms ofPythagorean music theory, the study of harmony had revealed the striking mathematicalconstancies of proportionality. When the ratios of string lengths or ¯ute columns werecompared, the harmonies produced by other, but proportionally similar lengths, were the same.The Pythagoreans also knew that any triangle with sides of length 3,4,5, whatever unit mightbe taken, was a right triangle. This too supported their belief that ratios of whole numbersre¯ected certain invariant and universal properties. In addition, Pythagorean astronomy linkedsuch terrestrial harmonies with the motions of the planets, for which the numerical harmony,or cyclic regularity of the daily, monthly, or yearly revolutions were as striking as the musical

19 For a detailed discussion of the circumstances of the context of Pythagorean mathematics and the unexpecteddiscovery of incommensurable magnitudes, see Ref. [54].

J.W. Dauben / International Journal of Engineering Science 36 (1998) 1339±13591354

harmonies the planets were believed to create as they moved in their eternal cycles. All of theseinvariants gave substance to the Pythagorean doctrine that numbersÐthe whole numbersÐandtheir ratios were responsible for the hidden structure of all nature.The idea that everything could be expressed by such numbers, from musical harmonies to

the size and shape of the heavens, was one of the most notable features of Pythagoreancosmology. It seems to have been virtually an article of faith that literally everything in naturewas thus ``rational,'' expressible as ratios, and could consequently be expressed throughnumbers, either directly in terms of the integers or their fractions. The word the Pythagoreansused to express this rationality was Ðthe ratios that determined all things. However,logos, in an alternative meaning as ``word''Ðwhat is nameable or re¯ecting the essentialcharacter of somethingÐalso meant, as a technical, mathematical termÐthat which is rationaland consequently understandable, at least mathematically.It was, therefore, a shock to the Pythagoreans to discover that despite this basic tenet of

traditional Pythagoreanism, there were nevertheless physical entities that could not beexpressed as numbers or ratios, because there was no ratio of integers a/b that would expresstheir . As a result, such entities were indeed unnameable, indeed ``unspeakable''.Eventually, having been dubbed the ir-rational, these came to be known in mathematics asirrational ``numbers'' associated with the ``incommensurable'' magnitudes discovered by thePythagoreans, i.e.:

���2p 6� a=b:

11. The drowning of Hippasus

Philosophically, discovery that���2p

was irrational would certainly have represented a crisisfor the Pythagoreans. Here was a well-de®ned mathematical object, namely the diagonal of thesquare, that violated the Pythagoreans' most basic assumption that everything could bemeasured by whole numbers or expressed as ratios of numbers as a/b. Having been tempted bythe seductive harmony for generalization, some Pythagoreans had carried their universalprinciple that all things were numbers too far. The complete generalization was inadmissible,and this realization was a major blow to Pythagorean thought, if not to Greek mathematics. Infact, a scholium to Book X of Euclid's Elements re¯ects the gravity of the discovery ofincommensurable magnitudes in the well-known fable of the shipwreck and the drowning ofHippasus:

It is well-known that the man [Hippasus] who ®rst made public the theory of irrationalsperished in a shipwreck in order that the inexpressible ( ) should ever remainveiled . . . , and so the guilty man, who fortuitously touched on and revealed this aspect ofliving things, was taken to the place where he began and there is forever beaten by thewaves.20

20 Scholium to Euclid, Elementa, X,1, in Ref. [55]. For other accounts of the drowning episode, see Refs. [56, 57].Pappus, however, viewed the story of the drowning as a ``parable''; see Ref. [58]. For discussion of ``The discoveryand role of the phenomenon of incommensurability,'' see Ref. [59].

J.W. Dauben / International Journal of Engineering Science 36 (1998) 1339±1359 1355

This was also the gist of a later commentary as well: ``such fear had these men of the theory ofirrationals, for it was literally the discovery of the unthinkable.''21 Later writers like Plutarchand Pappus were equally fascinated. As Plutarch explained the passage, the irrationals were``ine�able because irrational,'' ``unspeakable because secret.'' Burkert, in his translation of 1972(from Lore and Science in Ancient Pythagoreanism), claims:

The fascination of the (arretou) lies in the pretense to indicate the fundamentallimitations of human expression . . . [61].

What deserves attention here, however, particularly with the example of Chinesemathematics and the problem of counter-factual reasoning in mind, are the words``inexpressible'' and ``unimaginable'' in the passage from the scholium to Book X of Euclid'sElements. It is di�cult, if not impossible, for us to appreciate how di�cult it must have been toconceive of something one could not determine or nameÐthe inconceivableÐand this wasexactly the name given to the diagonal: (alogon). This re¯ects the double meaning ofthe word logos as word, and the ``utterable'' or ``nameable,'' and now the irrational, thealogon, as the ``unspeakable,'' the ``unnameable.'' In this context, it is easy to understand thecommentary: ``such fear had these men of the theory of irrationals,'' for it was literally thediscovery of the ``unthinkable,'' the ``unspeakable,'' the ``unnameable.''In dealing with incommensurable magnitudes, ``unfamiliar and troublesome'' concepts as

Morris Kline has described them, the need to formulate axioms and to deduce consequencesone by one so that no mistakes might be made was of special importance [62]. This emphasis,in fact, re¯ects Plato's interest in the dialectic certainty of mathematics and was epitomized inthe great Euclidean synthesis, which sought to bring the full rigor of axiomatic argumentationto geometry. It was in this spirit that Eudoxos undertook to provide the precise logical basisfor the incommensurable ratios, and in so doing, gave great momentum to the logical,axiomatic, a priori ``revolution'' identi®ed by Kant as the great transformation wrought uponmathematics by the Greeks.22

12. Conclusion: the concreteness of Chinese mathematics

Instead of pursuing abstract and logical concerns,23 Chinese mathematics, like Chinesescience, developed a rich tradition of empirical observation. However, theirs was not atheoretical orientation that sought to leave the world of physical experience and practicalapplications behind in order to construct and test purely theoretical models or explanatoryframeworks.

21 For discussion of this passage, see Refs. [60, 61].22 See Refs. [63, 64].23 From the Chinese point of view, one is tempted to write ``abstract and illogical,'' just as the Pythagoreansregarded their discovery of incommensurable magnitudes as illogical, unspeakable and irrational.

J.W. Dauben / International Journal of Engineering Science 36 (1998) 1339±13591356

At a very basic level, one of conceptualization and linguistic construction, the Chinese weredisinclined to pursue abstract notions for their own sake, particularly when such abstractionsmight lead to counter-factual situations, or when their status was only theoretical and bore nocorrespondence to anything that might be con®rmed by empirical evidence.As the 3rd century BC Confucian philosopher Xun-Zi said of the work of Gong Sun

Long [ , Kung Sun Lung]:

There is no reason why problems of ``hardness'' and ``whiteness,'' ``likeness'' and``unlikeness,'' ``thickness'' or ``no thickness,'' should not be investigated, but the superiorman does not discuss them; he stops at the limit of pro®table discourse.24

Such limits, however, as the results of Liu Hui's commentary on the Nine Chapters makeplain, did not preclude the Chinese from discovering and utilizing signi®cant, evenextraordinary results. In particular, in the realm of practical geometry, the Chinese developedtechniques needed to solve quantitative problems in successful, sometimes highly sophisticatedways.

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