Ancient Chinese Mathematics: the (Jiu Zhang Suan Shu) Vs Euclid's Elements

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

Ancient Chinese mathematics: the (Jiu Zhang Suan Shu) vs Euclid's Elements. Aspects of proof and the linguistic limits of knowledge

Joseph W. Dauben a, b, *

aDepartment of History, Herbert H. Lehman College, City University of New York, 250 Bedford Park Blvd. West, Bronx, NY 10468, USA bPh.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 early Chinese mathematics, particularly geometry, considered largely in terms of one speci®c example: the (Gou-Gu) Theorem. In addition to drawing some fundamental comparisons with Western traditions, particularly with Greek mathematics, some general observations are also made concerning the character and development of early Chinese mathematical thought. Above all, why did Chinese mathematics develop as it did, as far as it did, but never in the abstract, axiomatic way that it did in Greece? Many scholars have suggested that answers to these kinds of questions are to be found in social and cultural factors in China. Some favor the sociological approach, emphasizing for example that Chinese 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 stressed philosophical factors, taking another widely-held view that Confucianism placed no value on theoretical knowledge, 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 the Chinese did not develop a more abstract, deductive sort of mathematics along Greek lines, a dierent approach is oered here. To the extent that knowledge is transmitted and recorded in language, oral and written, logical and linguistic factors cannot help but have played a part in accounting for how the Chinese were able to conceptualizeÐand think aboutÐmathematics. # 1998 Elsevier Science Ltd. All rights reserved.

* Tel.: 011 212 642 2110; Fax: 001 212 642 1963; E-mail: [email protected].

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 the oldest 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 (Wen Wu 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 Karine Chemla and Guo Shu-Chun, is now in preparation and due to be published shortly. J.W. Dauben / International Journal of Engineering Science 36 (1998) 1339±1359 1341 have been ascribed by some to the earliest period of China's recorded history, where fact shades into myth. One tradition says that the Yellow Emperor, Huang Di , who lived in the 27th century 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 ®rst great tragedies in Chinese intellectual history occurred in 213 BC, when the Emperor Qin Shi Huang (221±207 BC, famed for his terra-cotta army at Xi'an Yang ) ordered that all books in the Empire be burned. Although some of the classics may have been surreptitiously preserved, or memorized and later transcribed, the reconstituted texts produced for these ``lost'' early documents likely contained inaccuracies or interpolations introduced by their rescuers. In the case of mathematical knowledge, later innovations and new techniques might 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 earliest text we have of the Jiu Zhang Suan Shu was compiled by Zhang Cang sometime in the 2nd century BC, and revised about 100 years later by Geng Shou Chang . Both of these scholars lived in the Western Han Dynasty (206 BC±24 AD), and both were imperial ministers who undertook their reconstructions of the Nine Chapters at a time when there were great eorts being made to restore lost classics of any sort [6]. When Liu Hui ,a mathematician of Wei during the Three Kingdoms Period (220±280 AD), again edited the Jiu Zhang Suan Shu in 263 AD, this time with an extensive commentary, he began a tradition that was repeated after him by Li Chun Feng [ ] in the Tang Dynasty (618±907 AD), who also collated and commented on the book.2 The oldest edition of the Nine Chapters to survive is the wood block printing of the Southern 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 Literature edited by Dai Zhen of the Qing Dynasty, who copied it from the Great Encyclopedia of the Yong-le Reign Period of the Ming Dynasty (known as the Dai edition). The most famous commentaries are those by Liu Hui (263 AD), Li Chung-Feng (656 AD), and one by Zu Chong-Zhi [ ] (429±500 AD) written during the North and South Dynasties, but now lost.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 come from 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 its name from the ®rst problem devoted to calculating the height and distance of an island at sea, contains only nine problems. 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, see Ref. [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, which dates it exactly to 263 AD. See Ref. [7]. 1342 J.W. Dauben / International Journal of Engineering Science 36 (1998) 1339±1359

The Jiu Zhang Suan Shu dominated the practice of Chinese administrative clerks for more than a millennium, and yet in its social origins it was closely bound up with the bureaucratic government system, and was consequently devoted to the problems which ruling ocials had to solve. It was also of overwhelming in¯uence on writers in the centuries that followed, and it is no exaggeration to say that virtually all subsequent Chinese mathematics bears its imprint as to 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 Chapters came to be regarded as the seminal work of ancient Chinese mathematics for nearly two millennia.4 In several important respects, however, the two works are more striking for the dierences they exhibit rather than their similarities. Euclid's text is renowned for the austerity of its axiomatic method, beginning with abstract, idealized de®nitions and proceeding from axioms and postulates to a progressively arranged series of proofs, leading, through the thirteen books that survive, to some remarkable results on the ®ve regular (sometimes called Platonic) 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 as computationally dicult as they were theoretically subtle. Chinese mathematicians were just as creative in devising new methods as their contemporaries anywhere in the world, particularly in obtaining 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 was doubtless 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, commercial problems, partnerships and taxation rates. Also dealt with are the extraction of square and cube 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 the Chinese 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 the latter, 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 six unknowns, 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 bamboo 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 dierent types of calculation are included, and some problems are very close to ones found in the Jiu Zhang Suan Shu. See Ref. [7]. J.W. Dauben / International Journal of Engineering Science 36 (1998) 1339±1359 1343

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

This theorem states that for any right-angled triangle, the sum of the squares of its sides is equal 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 quadratic equations. 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 the surface of the water. When pulled toward the edge of the pond, the reed just reaches the perimeter. 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 been presented 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 that for a right triangle, given the shorter and longer sides enclosing the right angle, the sum of their 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 dierent texts. In addition to the Commentary on the Zhou Bi Suan Jing by Zhao Shuang , we also have Liu Hui's own annotations concerning the results on right triangles in the Nine Chapters. For example, in

7 This is problem 6 in Chapter 9 of the Nine Chapters, in Ref. [29]. 1344 J.W. Dauben / International Journal of Engineering Science 36 (1998) 1339±1359

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? commenting on a passage that reads, ``Combining each square of Gou and Gu, taking the square 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 each other, then the rest (of the pieces) do not move. Composing the Xian-square, taking the square root will be Xian (the hypotenuse).8

This entire passage is obscure and problematic. The reference to moving pieces inside and outside is related to a diagram, no longer extant, and makes use of the so-called `Òut±In'' method which was taken as an axiom by ancient Chinese mathematicians. The power of this axiom can be seen, however, from the following example. Given a rectangle ABCD divided horizontally by aa and vertically by bb, under what circumstances is it possible to prove that A = B? (Fig. 3). So long as the horizontal line aa and the vertical line bb intersect on the diagonal AC, it will follow that A = B. This can be seen immediately from the following observations related to (Fig. 4). Since the diagonal divides the rectangle into two equal triangles, ABC and ADC, removing the two equal triangles below the horizontal line aa (I and I0), as well as the two equal triangles above the horizontal line aa (II and II0), it follows from the 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 `Òut±In'' complementary principle to the Xian ®gure, and following Liu Hui's commentary on the 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), namely the square AHJC (Fig. 5). In accordance with the `Òut±In'' principle, if we move those parts of 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 of the two small squares equal that of the larger one. Since the areas are in sum equal to the squared sides of the triangle, the sum of the squared legs equals the squared hypotenuse.

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]. J.W. Dauben / International Journal of Engineering Science 36 (1998) 1339±1359 1345

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.

The earliest work containing this idea of demonstrating the right-triangle theorem by a suitable rearrangement of areas has already been mentioned: the Zhou Bi Suan Jing, the Arithmetical Classic of the Zhou Gnomon, which some scholars have dated to as early as 1100 BC (Fig. 6). Astronomical evidence, however, suggests that most of the material must be considerably 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 the theorem itself is explained in terms of ``piling up rectangles.''10

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]. 1346 J.W. Dauben / International Journal of Engineering Science 36 (1998) 1339±1359

Fig. 6. Title page, 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.

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

Compare these developments of the Gou-Gu theorem in the Chinese mathematical tradition with what is to be found in Euclid's Elements. The most striking dierence is certainly the axiomatic 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 one another in a progressive fashion. Book 9 of the Jiu Zhang Suan Shu, on the other hand, begins immediately with a concrete practical problem and wastes no time in providing a model solution. However, there is another, less immediately obvious dierence between the two works as well. Ask anyone what Euclid's elements is about and the answer will inevitably be J.W. Dauben / International Journal of Engineering Science 36 (1998) 1339±1359 1347

``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 shuing 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 dierent 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 methods and standards re¯ected in Euclid, see Refs. [10] and [16]. 1348 J.W. Dauben / International Journal of Engineering Science 36 (1998) 1339±1359 century BC describes right triangles by means of stretched ropes and gives, as examples, such triples 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 with working or surveying the land, empirical understanding was later passed on to the more systematic and eventually abstract concerns of mathematicians, who, in turn, extended and generalized the scope of geometry. However, more often than not, the words they used were the words of their practically-minded counterparts. One last example will suce 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,'' the nominative 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, which also 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 a rope-stretching, surveying tradition. And again, this legacy is directly re¯ected in mathematical nomenclature in a natural way. ``Gou'' means leg, ``Gu'' means thigh, and ``Xian'' , the character used for hypotenuse, means lute string.12 Thus Chinese geometry, particularly the Gou-Gu theorem, rests ®rmly in a practical tradition of 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 geometric problems of surveying taken by the earliest mathematical Westerners, the Egyptians and Babylonians. 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 their appropriate sections, will ultimately yield the shapes of the basic farm ®gures,'' [35]. Chinese mathematics, clearly, was no dierent from western mathematics in being rooted in practical concerns of agriculture and the land. But aside from discovering special mathematical results, particularly very general relations between geometric entities for example (like the right-triangle properties), what can be said about Chinese interest in ``proofs'' or other forms of demonstration? Again, Professor Lam's study of ``The Alpha and Omega of a selection on the Applications 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 dierent form, goods. The character above, , is the character for bamboo strips, and is a reference to the bamboo slips used as tallies to sort goods. ``Suan'' originally referred 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. Thus the character ``Suan'' ideogrammatically embodies both the original objects and methods that became the stock in trade of the court and administrative mathematicians. For details, see Ref. [34]. J.W. Dauben / International Journal of Engineering Science 36 (1998) 1339±1359 1349

Fig. 7. Determining the height of a pagoda using the double dierence method and the similarity of triangles.

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

This was clearly not the Euclidean way. Individual, concrete examples of the Pythagorean theorem alone would not suce to prove, in general, abstractly, universally, a result like the version as demonstrated in Euclid I,47. However, demonstration by example was the Chinese way. Plausible generalizations were drawn from concrete situations. Consequently, we ®nd many fascinating problems and examples in Chinese mathematical texts which are meant to demonstrate a wide variety of techniques. From measuring the heights of pagodas, distances of islands, depths of wells, the similarity of triangles is used as a general method, as is the double- dierence 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 axiomatic mathematics? 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

13 Readers not familiar with these methods, but wishing to know more about the application of similarity of triangles and the double dierence method, should consult the article cited above by Wu Wenjun, particularly the sec- tion on ``Gnomon, Shadow and Double Dierences,'' in Ref. [31]. 1350 J.W. Dauben / International Journal of Engineering Science 36 (1998) 1339±1359 from virtually all other cultures in this respect, namely in their preoccupation with axiomatic, deductive proofs? However, this clearly is a very dierent problem from determining the origin and 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 the Egyptian or Babylonian variety. In working out their many applications of the Gou-Gu theorem, it is clear that the Chinese understood it to hold for any right triangle, and came up with much more general, non-integer Gou-Gu triples such as 8, 9 1/6, 12 1/6; and 10, 49 1/2, 50 1/2. It has been argued that geometry never developed further in China than it did with Liu Hui's commentary because this was sucient, and comprehensive enough, for Chinese needs. After all, what real, utilitarian purpose is served by abstract mathematical proofs?14 Another argument closely related to this one suggests that Chinese mathematics, with its major concern for practical problems, had little interest in abstract generalizations. Although at ®rst glance this may indeed seem to have been the case, upon closer examination one soon begins to have doubts. For example, if one looks at the method Liu Hui follows in his commentary on the Nine Chapters, one ®nds that he is very careful to explain each formula given for areas and volumes, 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 method brie¯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 but share the main stem of a tree are comprehended to be from one origin. Furthermore to analyze a theory by proposition, and to illustrate a structure with geometrical ®gures, will thereby make the whole picture of the theory or structure be understood through some simple 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 anities here with the philosophy of Mo-Zi as re¯ected in the Mo-Zi Debates, an ancient work on Chinese logic. Here, ``Zhu'' has a meaning close to ``proposition.'' Since Liu Hui mentions the Debates in his commentary, he was obviously familiar with Mo-Zi's ideas, and presumably with the terminology 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 in support of pure mathematics has always been the power of its applications. This also constitutes one of the deepest philosophical 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 the Nine Chapters. For details, see Refs. [12±14, 29, 36]. J.W. Dauben / International Journal of Engineering Science 36 (1998) 1339±1359 1351

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

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

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

7. Chinese values of p

Moreover, claims of an overriding practical interest of the Chinese only in concrete problems with useful applications is not borne out in the case of Liu Hui's meticulous approximation of the numerical value of p. This example, in fact, illustrates how dramatically the Chinese could surpass 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 (but ubiquitous in most ancient cultures with a rudimentary interest in geometry), comes from a bronze cylindrical standard measure cast by ocial order in the early ®rst century which gives a 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 uses inscribed regular polygons to approximate a value of p (as did Archimedes) (Fig. 8).17 This is described 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 the circle. The exact area of the circle is obtained when such segments so cut o come to be in®nitesimals [37].

Liu Hui ®rst obtained a value for p using a 192-sided polygon, which gave him a value of 3.14 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 see as 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]. 1352 J.W. Dauben / International Journal of Engineering Science 36 (1998) 1339±1359

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

Certainly, Liu Hui provides a constructive counter-example to the idea that ancient Chinese mathematics stopped at what was only of practical value. It was clearly capable of going well beyond what was simply practical, or even physically possible. The theoretical interest of obtaining greater accuracy, or discovering systematic connections between propositions and their 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 practical orientation of ancient Chinese mathematics, it may have been its actual success, its comprehensiveness, that caused the stagnation of any further development. As D. B. Wagner has suggested:

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

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 certain speci®c problems [36].

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

8. Incommensurability and Chinese mathematics

There is, however, another aspect of ancient Chinese mathematics that is also striking, and upon re¯ection, again separates it from Greek, and indeed from some of the most fundamental principles of modern mathematics. Nowhere is there any interest shown in classic Chinese mathematical texts in the ``irrational'' character, for example, of the diagonal of the square. Although certain approximations to its value were explored, neither an analysis by anthyphairesis, or the better-known even±odd analysisp to be found in Aristotle's famous demonstration of the ``incommensurability'' of 2, appear in Chinese mathematical or astronomical 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 Chinese reasoning about mathematics (or philosophical or logical matters in general) are arguments based upon counter-factual reasoning.18 One of the most powerful methods of proof in Western mathematics, however, relies upon counter-factual arguments of the ``Reductio ad absurdum'' type. Such arguments begin by making an assumption, assumed to be true for the sake of argument, and then showing how the assumption leads to a contradiction. The contradiction in turn establishes the fact that the initial assumption is indeed false, contradictory to fact. This method does not surprise us, but it 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 about what would be the case if they were true is a psycholinguistic act. Hence the development of a facility for it is likely to be highly contingent on the nature of the incentives that language provides [43].

18 Bloom has studied the signi®cance of ``enti®cation'' and counter-factual constructs in Ref. [43]. Other useful linguistic studies include [44±46]. Bloom's work has engendered considerable controversy and discussion; see for examplep Refs. [47±51]. For analysis of the mathematical signi®cance of interpreting ancient Chinese approximations of 2 as re¯ecting an understanding of incommensurability, see Ref. [52]. 1354 J.W. Dauben / International Journal of Engineering Science 36 (1998) 1339±1359

Counter-factual reasoning, in turn, is closely related to what Bloom terms ``enti®cation,'' the creation of theoretical concepts by abstraction from speci®c properties or actions. English enti®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-Sun Long [ , Kung Sun Lung], who discussed such concepts as whiteness, horseness, etc. However, his editor/commentator Chan Wing-Tsit notes that he had ``no in¯uence whatsoever'' 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 concrete world, 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, the algorithms of algebra, or with computations in general, ever thought to de®ne points, lines or space as they are de®ned by Euclid. However, in dealing with actual situations and concrete procedures, there were no such diculties. Liu Hui, for example, develops the double dierence method to a high degree of sophistication, in part because it deals ultimately with speci®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 ancient Greek mathematics, beginning with a proposition known to be false, is the proof Aristotle recounts for the incommensurabilityp of the diagonal of the squareÐwhich establishes (in modern terms) that 2 is irrational.

10. The Pythagorean discovery of incommensurable magnitudes

Aristotle reports the Pythagorean doctrine that all things are numbers and surmises that this view doubtless originated in several sorts of empirical observation.19 For example, in terms of Pythagorean music theory, the study of harmony had revealed the striking mathematical constancies of proportionality. When the ratios of string lengths or ¯ute columns were compared, 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 might be taken, was a right triangle. This too supported their belief that ratios of whole numbers re¯ected certain invariant and universal properties. In addition, Pythagorean astronomy linked such 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 unexpected discovery of incommensurable magnitudes, see Ref. [54]. J.W. Dauben / International Journal of Engineering Science 36 (1998) 1339±1359 1355 harmonies the planets were believed to create as they moved in their eternal cycles. All of these invariants gave substance to the Pythagorean doctrine that numbersÐthe whole numbersÐand their 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 Pythagorean cosmology. It seems to have been virtually an article of faith that literally everything in nature was thus ``rational,'' expressible as ratios, and could consequently be expressed through numbers, either directly in terms of the integers or their fractions. The word the Pythagoreans used 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 essential character of somethingÐalso meant, as a technical, mathematical termÐthat which is rational and 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 be expressed as numbers or ratios, because there was no ratio of integers a/b that would express their . As a result, such entities were indeed unnameable, indeed ``unspeakable''. Eventually, having been dubbed the ir-rational, these came to be known in mathematics as irrational ``numbers''p associated with the ``incommensurable'' magnitudes discovered by the Pythagoreans, i.e.: 2 6 a=b:

11. The drowning of Hippasus p Philosophically, discovery that 2 was irrational would certainly have represented a crisis for the Pythagoreans. Here was a well-de®ned mathematical object, namely the diagonal of the square, that violated the Pythagoreans' most basic assumption that everything could be measured by whole numbers or expressed as ratios of numbers as a/b. Having been tempted by the seductive harmony for generalization, some Pythagoreans had carried their universal principle 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. In fact, a scholium to Book X of Euclid's Elements re¯ects the gravity of the discovery of incommensurable magnitudes in the well-known fable of the shipwreck and the drowning of Hippasus:

It is well-known that the man [Hippasus] who ®rst made public the theory of irrationals perished in a shipwreck in order that the inexpressible ( ) should ever remain veiled..., and so the guilty man, who fortuitously touched on and revealed this aspect of living things, was taken to the place where he began and there is forever beaten by the waves.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 discovery and role of the phenomenon of incommensurability,'' see Ref. [59]. 1356 J.W. Dauben / International Journal of Engineering Science 36 (1998) 1339±1359

This was also the gist of a later commentary as well: ``such fear had these men of the theory of irrationals, for it was literally the discovery of the unthinkable.''21 Later writers like Plutarch and Pappus were equally fascinated. As Plutarch explained the passage, the irrationals were ``ineable 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 fundamental limitations of human expression... [61].

What deserves attention here, however, particularly with the example of Chinese mathematics and the problem of counter-factual reasoning in mind, are the words `ìnexpressible'' and `ùnimaginable'' in the passage from the scholium to Book X of Euclid's Elements. It is dicult, if not impossible, for us to appreciate how dicult it must have been to conceive of something one could not determine or nameÐthe inconceivableÐand this was exactly the name given to the diagonal: (alogon). This re¯ects the double meaning of the word logos as word, and the `ùtterable'' or ``nameable,'' and now the irrational, the alogon, as the `ùnspeakable,'' the `ùnnameable.'' In this context, it is easy to understand the commentary: ``such fear had these men of the theory of irrationals,'' for it was literally the discovery of the `ùnthinkable,'' the `ùnspeakable,'' the `ùnnameable.'' In dealing with incommensurable magnitudes, `ùnfamiliar and troublesome'' concepts as Morris Kline has described them, the need to formulate axioms and to deduce consequences one 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 in the great Euclidean synthesis, which sought to bring the full rigor of axiomatic argumentation to geometry. It was in this spirit that Eudoxos undertook to provide the precise logical basis for 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 upon mathematics by the Greeks.22

12. Conclusion: the concreteness of Chinese mathematics

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

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 Pythagoreans regarded their discovery of incommensurable magnitudes as illogical, unspeakable and irrational. J.W. Dauben / International Journal of Engineering Science 36 (1998) 1339±1359 1357

At a very basic level, one of conceptualization and linguistic construction, the Chinese were disinclined to pursue abstract notions for their own sake, particularly when such abstractions might lead to counter-factual situations, or when their status was only theoretical and bore no correspondence 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 superior man 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 make plain, did not preclude the Chinese from discovering and utilizing signi®cant, even extraordinary results. In particular, in the realm of practical geometry, the Chinese developed techniques needed to solve quantitative problems in successful, sometimes highly sophisticated ways.

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