The ˚rst Chinese translation of the last nine books of...

Historia Mathematica 32 (2005) 4–32www.elsevier.com/locate/hm

The �rst Chinese translation of the last nine booksof Euclid’s Elements and its source

Yibao Xu

The Graduate School and University Center, City University of New York, 365 Fifth Avenue, New York, NY 10016, USA

Available online 16 March 2004

Abstract

Books VII to XV of the Elements (Books VII to XIII by Euclid and Books XIV and XV by Hypsiclesof Alexandria) were �rst translated into Chinese by the British missionary Alexander Wylie and the Chinesemathematician Li Shanlan between the years 1852 and 1856. The translation was subsequently published in 1857.Neither of the translators in their prefaces to the translation or in their other writings mentioned the speci�c originalsource. Accordingly, historians have pondered this question ever since. Some took a bold guess that its source wasIsaac Barrow’s English translation. This article provides solid evidence to show that the guess is wrong, and arguesthat the �rst English translation of Euclid’s Elements of 1570 by Henry Billingsley was the actual source. 2003 Elsevier Inc. All rights reserved.

2003 Elsevier Inc. All rights reserved.

MSC: 01A25; 01A55

Keywords: Euclid; Elements; Henry Billingsley; Isaac Barrow; Alexander Wylie; Li Shanlan; China

E-mail address: [email protected].

0315-0860/$ – see front matter 2003 Elsevier Inc. All rights reserved.doi:10.1016/j.hm.2003.12.002

http://www.elsevier.com/locate/hm

Y. Xu / Historia Mathematica 32 (2005) 4–32 5

1. Introduction: Euclid’s Elements in China

It is widely known among historians that Euclid’s Elements may �rst have been known in China asearly as the Yuan dynasty, sometime between 1250 and 1270. This has at least been the case ever sincethe historian of Chinese mathematics Yan Dunjie pointed out in 1943 that a book mentioned inthe “Catalogue of the Muslim Books (Huihui shuji )” included in chapter seven of the History ofthe Imperial Archive and Library (Mishujian zhi ), which was compiled in the mid-14th century,might be interpreted as a version in 15 books of Euclid’s Elements [Yan, 1943, 35].1 In addition, inthe same article Yan argued that one sentence in the Jami` al-tawar�kh (Compendium of Chronicles)written by Rash�d al-D�n (ca. 1247–1318), a high functionary at the Mongolian court of Iran,2 praises theintelligence of Möngke Khan (Mengge Han , reigned 1251–1259) for “solving some propositionsof Euclid,” which clearly suggests that Euclid’s Elements was known in the Mongolian court by theearly 1250s [Yan, 1943, 36].3 One year before Yan’s publication, the Japanese scholar Tasaka Kodo, inhis study of the introduction of Islamic culture into China during the Yuan dynasty, also discussed thepossible meanings of the recorded book title in the Catalogue of the Muslim Books. But Tasaka held thatit might refer to a work of Abu Ja`far Muhammad ibn Musa al-Khwarizmi [Tasaka, 1942, 444–445].Fifteen years later, in his English article, “An Aspect of Islam Culture Introduced into China,” Tasakachanged his view, and also concluded that the title refers to an Arabic version of Euclid’s Elements, byusing a sounder linguistic analysis than Yan had used [Tasaka, 1957, 103–104].

Although there is little doubt about the book in question being Euclid’s Elements, it is unclear whetherany Chinese scholar of that time was familiar with any of the contents of the book. It is also unknownwhether there may have been a Chinese translation of all or only part of the copy of Euclid’s Elementswhich was in the library of the Northern, or Islamic, Observatory (Bei sitiantai ) built in 1271by the Mongolians in the upper capital (Shangdu ), situated near Dolon Nor (Zhenglanqi ) inInner Mongolia. Moreover, Rash�d’s brief and ambiguous description is really not suf�cient to prove thatMöngke Khan had studied Euclid’s Elements as Yan suggested. Nevertheless, the Elements, in either anArabic or a Persian version, certainly existed in China no later than 1273, the year in which the IslamicObservatory reported on its collection of books to the Mongol authorities.4

1 See Wang and Shang [1992, 129]. The Chinese term, , may also be pronounced or romanized in Pinyin as Bishujian.I am grateful to one of the referees for pointing out this alternative, which is attested, for instance, by Hucker [1985, 376b–377b]; Luo [1991, 72]. But it seems that Mishujian is more popular than Bishujian, at least in the Chinese-speaking academicworld. The Mishujian is rendered here as “imperial archive and library,” but it functioned much more broadly than an archiveor a library does, and undertook astronomical observations, oversaw calendar-making, and even organized grand book projects,among others. For a relatively detailed discussion of the history of this institution and its main functions in the 12th and 13thcenturies, see Hucker [1985, 376b–377b], as well as the introduction to the latest Chinese edition of the Mishujian zhi by GaoRongsheng in Wang and Shang [1992, 1–2].

2 For a detailed biography of Rash�d al-D�n, see Morgan [1995, 443a–444b].3 The original words in Persian may be found in Rash�d al-D�n [1836, 324]. The French translation of the sentence by

Etienne Marc Quatremère is: “Mangou-kaân se distinguait entre tous les monarques Mongols par ses lumières, son esprit, saprudence, son tact, sa sagacité, sa �nesse ; il était si instruit, qu’il avait expliqué plusieurs des �gures d’Euclide” [Rash�d al-D�n, 1836, 325]. An alternative English translation of the sentence by Wheeler McIntosh Thackston reads: “Mänggü Qa’an wasdistinguished among Mongol rulers for his intelligence, perspicacity, sharp wit and mind, and he even solved some problemsleft by Euclid” [Rash�d al-D�n, 1998–1999, 502a].

6 Y. Xu / Historia Mathematica 32 (2005) 4–32

Euclid’s Elements was again brought to China some 300 years later. This time it was in the form ofa Latin version by Christopher Clavius, professor of mathematical sciences at the Collegio Romano (theJesuit seminary founded in Rome by Ignatius Loyola in 1551). It was the Jesuit Matteo Ricci, a studentof Clavius, or one of his Jesuit brothers who brought a copy of Clavius’s Euclid with them to China.Unlike the Muslims before him, Ricci was determined to translate the book, along with other Westernworks, in hopes of gaining the support that he and his fellow Jesuits sought from the Chinese literati. Indoing so the Jesuits expected to secure a footing in pagan China, and eventually to convert the Chineseto Christianity. Not long after his arrival in China, Ricci sought help from one of his early converts andtogether they began to translate the Elements. But at the time he was only able to �nish the �rst book, themanuscript of which was later lost [Engelfriet, 1998, 59–66].

Ricci never lost interest in translating the Elements. In 1604, when he eventually settled in Beijingafter experiencing many years of hardship in the south of China, he approached Xu Guangqi ,who not only showed a religious inclination toward Christianity, but was a well-educated member ofthe literati who was on the verge of getting the highest degree the Ming dynasty could offer.5 Ricciproposed they collaborate in translating the Elements. Fascinated by its style and the entirely differentapproach to geometry it represented in contrast to traditional Chinese geometry, Xu Guangqi devotedhimself wholeheartedly to assisting Ricci in translating Euclid’s Elements.6 After three years of hardwork, they had �nished the �rst six books. Although Xu Guangqi wanted to continue to translate the restof the book, Ricci thought it would be better to publish their partial translation �rst. Depending uponits reception, they could then decide whether to continue the project or not. Accordingly, they publishedtheir Chinese translation of the �rst six books in 1607 [Engelfriet, 1998, 460].

Reactions to the publication of the Chinese Euclid were somewhat mixed. On the one hand, its exoticstyle and formality (not its axiomatic–deductive system) was warmly accepted, and the newly inventedterm Jihe for geometry was widely adopted in titles of Chinese mathematical books published soonthereafter. Moreover, a number of books were written to elucidate parts of the Ricci/Xu translation. Onthe other hand, some mathematicians, including Mei Wending , who was one of the most prominentmathematicians in the early Qing period, held that traditional Chinese mathematics, represented by theNine Chapters on Mathematical Procedures (Jiuzhang suanshu ), included all of the methods tobe found in the translation, with but one possible, if notable, exception—namely, Euclid’s introductionof extreme and mean ratios.7 Moreover, Chinese mathematicians regretted that the translation was onlypartial, and some, notably Mei, even suspected that Ricci was trying to hide something important from theChinese in the later books of the Elements [Liu, 1989, 57]. As we shall see below, Mei’s false accusationwas to become a major factor in the decision, made early in the 1850s by a later missionary, AlexanderWylie, to complete the work left undone by Ricci.

4 For recent studies of scienti�c transmission between Chinese and Arabs under the Mongols, see, for instance, Li [1999,135–151]; van Dalen [2002].

5 For a biography of Xu Guangqi, see, for example, Hashimoto [1988].6 There are three editions of Clavius’s Elements, namely, those of 1574, 1591, and 1603. Which of these was actually used

by Matteo Ricci and Xu Guangqi in preparing their translation is not clear. Only the last two editions have ever been found inChina. See Verhaeren [1969, 376].

7 For reactions to the Chinese Euclid in the 17th century, see Martzloff [1981, 1993b]; Mei et al. [1986/1990]; and Engelfriet[1998, 289–448].


During Mei’s time, the Emperor Kangxi (reigned 1662–1722) was especially interested inWestern science in general and mathematics in particular. In order to learn Euclidean geometry, sometimearound 1690 he requested two French Jesuits, Joachim Bouvet and Jean-François Gerbillon, who hadrecently arrived in China, to lecture at his court. Bouvet and Gerbillon, who both held the title “King’sMathematician” granted by the Sun King, Louis XIV, compiled their lessons for the Emperor in Manchufrom a short textbook by their countryman Ignace-Gaston Pardies: Elémens de Géométrie (�rst publishedin Paris in 1671).8 The resulting lecture notes for Emperor Kangxi were subsequently translated intoChinese and later incorporated into a mathematical encyclopedia, the Essence of Mathematics (Shulijingyun ), which was compiled under the auspices of Kangxi, but not published until 1723, theyear immediately after his death.9

The Chinese translation (with some additions and deletions) of Pardies’ Elémens, when it appeared inthe Essence of Mathematics, bore the same title as the one used by Ricci and Xu, but its contents andstyle were very different. The Pardies translation was divided into 12 chapters (juan ), the �rst �ve ofwhich covered, respectively, lines and angles, triangles, quadrilateral �gures and polygons, circles, andsolids. Pardies’ Book VI, on proportion, constituted the contents of chapters six to nine in the translation.However, Books VII and VIII of Pardies’ Elémens, devoted to incommensurables, progressions, andlogarithms, were left out, presumably because of their dif�culty. The last book of the original text,Book IX, which included 48 practical geometrical problems, provided the contents for chapters 11 and12 of the translation. The contents of chapter 10 were devised by Bouvet and Gerbillon themselves orother missionaries [Han, 1994, 3–4].

The Essence of Mathematics version of the Elements was not, in reality, even a close translation ofEuclid’s Elements, nor was it complete. But the drastic negative change in Qing dynasty policy toward themissionaries certainly did nothing to encourage either the Chinese or the Jesuits—or indeed any othermissionaries—to complete Ricci’s un�nished project for another 130 years. The Emperor Yongzheng

, Kangxi’s immediate successor, adopted harsh measures against the missionaries in China as a directresult of the rites controversy that had started during the Kangxi reign. The controversy was mainly aboutwhether ancestor worship should be an allowable practice for Chinese converts to Catholicism or whetherthe Jesuit missionaries should be allowed to follow the practice of ancestral rites and thus “accommodate”themselves to Chinese rites.10 The ruling of Pope Benedetto XIV against this in 1742 eventually led theQing court to close China’s doors to all foreigners, but especially to missionaries, and this situation wasonly reversed by British gunboats in the 1840s as a result of the so-called Opium Wars.

With China’s doors reopened to foreigners in 1842, missionaries from the West—especially fromGreat Britain—�ooded in. Among them was Alexander Wylie from Scotland, who was sent to Shanghai

8 See Martzloff [1997, 26–27]; Liu [1991, 90]; and Jami [1994, 233]. For the King’s mathematicians, see Landry-Deron[2001].

9 A Manchu manuscript of Euclid’s Elements, three volumes (san ce ) in bulk, which undoubtedly came from thelecture notes by Joachim Bouvet and Jean-François Gerbillon, has survived and is preserved in the Palace Museum in Beijing.The manuscript had been subsequently translated into Chinese. There are three different copies of the Chinese manuscripts, onepreserved in the National Library in Taipei and the other two in the Palace Museum in Beijing. Each of these three Chinesemanuscripts represents a different stage of a working version which was eventually published in the Essence of Mathematics.For the relations among Pardies’ Elémens de Géométrie, the Manchu and Chinese manuscripts, and the published Chineseversion in the Essence of Mathematics, see Chen [1931]; Li [1984]; Liu [1991]; and Martzloff [1993a].

10 For details and historical meanings of the Chinese rites controversy, see, for example, Mungello [1994].


Fig. 1. Alexander Wylie (1815–1887). From the Chinese Researches by Alexander Wylie (Shanghai, 1897).

by the London Missionary Society in 1847 (Fig. 1).11 Wylie soon found that the work left un�nished byRicci and Xu Guangqi was worth pursuing. He described his thoughts about this as follows:

After arriving in China, I read the translation of the �rst six books of the Elements by the European [Matteo] Ricci in the Mingdynasty. The translation was highly regarded by most [Chinese] mathematicians. They knew it was incomplete, and therefore,expressed their dissatisfaction. Mei [Wending] of Xuancheng once said that [Ricci] had hidden something from the translation.[But that incompleteness] was due to the sophistication of the theories and to the dif�culty of translating. The nature of knowledgeis to provide means to all people. Why hide it and not reveal it?. . . My translation has just been completed. It has not only ful�lledRicci’s wish, but also dispelled Mei’s doubts. [Wylie and Li, 1857/1865, Wylie’s preface, 3a, and 4a; translation by the author]

Wylie went on to explain that whereas Ricci believed the teachings of Jesus Christ were fundamental,he also emphasized that calendrical science, mathematics, and the other sciences, though far lessimportant, were nevertheless useful [Wylie and Li, 1857/1865, Wylie’s preface, 4a].

Like Ricci, Wylie carried out his translation in collaboration with a Chinese scholar, Li Shanlan ,who later proved to be the ablest and most prominent Chinese mathematician in the second half of the

11 For Alexander Wylie’s life and his contributions to Chinese studies, see Edkins [1897]; Thomas [1897]; Cordier [1897];Han [1998]; and Wang [1998].


Fig. 2. Li Shanlan (1811–1882). From The Chinese Scienti�c and Industrial Magazine (its Chinese title is Gezhihuibian), founded by John Fryer in Shanghai in 1876, the issue for July, 1877.

19th century (Fig. 2).12 Wylie and Li started their translation of Books VII through XV of the Elementsin 1852, and took about four years to complete their work. The translation was published under thepatronage of Han Yingbi in Shanghai in 1857, and later was reprinted along with the �rst six books byRicci and Xu Guangqi in 1865.

Over the next half a century, Chinese interest in the last nine books of the Elements was exclusivelytechnical. As time passed by, historical interest in the translation gradually arose. Historians quitenaturally wanted to know about the original source, or sources, that Wylie and Li had used to preparetheir translation in order to evaluate the Chinese translation itself, assess the transmission of Westernmathematics into China in the second half of the 19th century, and study the in�uence of Euclid’sElements upon Chinese mathematicians, especially the translator Li Shanlan, among other historicalinterests. However, neither Wylie nor Li had said anything, speci�cally, in their prefaces to the translationabout its original source. The only information Wylie offered was the following:

This book commonly used in Europe is of six or eight books, none of which is complete. I had searched for a copy of the completedbook in Shanghai since my arrival here, and yet found none. Then I turned to my own country and �nally got a copy. The book is

12 For Li Shanlan’s life and his contributions to mathematics, introduction of Western mathematics and mathematicaleducation in China, see, for instance, Fang [1943]; and Horng [1991].


a translation from Greek into my native tongue. It has not been reprinted recently. The copy I have is an old version, but not wellproofread and has mistakes. Such mistakes, even minor, would cause great dif�culties in understanding for readers. [Wylie and Li,1857/1865, Wylie’s preface, 3a]

Though this provides an important, but not entirely correct, clue (namely, the original text was anold English translation of a complete Greek version of the Elements), the just-quoted passage does notmention the actual source of the translation done by Wylie and Li. Accordingly, early historians of Chi-nese mathematics, including Li Yan , Zhang Yong , and Yan Dunjie, all searched for the originalsource in vain [Li, 1940, 801; Yan, 1943, 35]. However, another historian, Qian Baocong , venturedthe following hypothesis: “The Englishman Isaac Barrow (wrongly spelled as “Issac” in the original text)�rst translated a Latin version from the Greek in 1655, and then rendered it into English in 1660. Barrow’sEnglish translation seems to be the original source of the Chinese translation” [Qian, 1981, 324]. Qian’shypothesis has been widely cited ever since, but the present paper will show that Qian’s hypothesis iswrong. After describing in the next section the essential features of Barrow’s English version of the El-ements, and evidence that contradicts Qian’s claim, this paper will argue, based on more solid evidence,that the true source of the translation done by Wylie and Li was the �rst English translation of Euclidby Henry Billingsley of 1570. A conjecture about the actual copy of Billingsley’s book that AlexanderWylie and Li Shanlan used, as well as its current whereabouts, is offered at the end of this paper.

2. Isaac Barrow and his English translation of Euclid’s Elements

Isaac Barrow was educated at Trinity College, Cambridge, where he was made a fellow in 1649.Inspired by the Elementa geometriae planae et solidae (The Elements of Plane and Solid Geometry),a popular Latin edition of the Elements (comprising only Books I–VI and XI–XII) by the Frenchgeometer, André Tacquet, Barrow decided to produce his own Latin version of the Elements. This resultedin a translation in 15 books, which was printed in London in 1655. Meanwhile, having been forced toleave the University because of his family’s Royalist connections, with the Restoration of the monarchyBarrow was appointed to the professorship of Greek at Trinity College four years later. It was then thathe prepared the second edition of his Latin version of Euclid, Euclidis Elementorum libri xv (1659), andthen translated it, with corrections, into English. This �rst appeared in a small octavo edition in 1660.

In the English version of Barrow’s Elements, like the previous two Latin editions, in order to achievehis goal of “[conjoining] the greatest compendiousness of demonstration with as much perspicuity as thequality of the subject would admit,” Barrow used many symbols borrowed from the Clavis Mathematicae(Key of Mathematics) by William Oughtred, who devised new symbols, in particular, for his account ofBook X of the Elements [Barrow, 1660/1983, preface; Murdoch, 1971, 451–452]. For instance, Barrowused × for multiplication; + for addition; − for subtraction; √ for a root of any degree; −: for thedifference of the two quantities or magnitudes on the two sides of the symbol; Q&q for a square; C&cfor a cube; and Q.Q. for the ratio of two square numbers [Barrow, 1660/1983, preface].13

While employing symbols for the sake of brevity, Barrow preserved the total number and order of thepropositions and also tried to provide the same demonstrations of the propositions as Euclid himself did.Barrow wrote:

13 For a list of the major symbols used by Isaac Barrow in his English Elements, see “The Explication of the Signs orCharacters,” which immediately precedes Book I.


I have altered nothing in the number and order of the propositions, nor taken the liberty to leave out any one of Euclid’s as lessnecessary, or to reduce certain of the easiest into the Classis of Axiomes. . .

I had a different purpose from the beginning; not to compose Elements of Geometry any wise at my discretion, but to demonstrateEuclide himself, and all of him, and that with all possible brevity. . .

I purposed to use generally no other than Euclide’s own demonstrations, contracted into a more succinct form, saving perchance inthe second and thirteenth, & sparingly in the seventh, eighth, and ninth Books, where it seem’d convenient to vary something fromhim. [Barrow, 1660/1983, preface]

The basic character of Barrow’s treatment of the Elements was well summarized by John Keill,Savillian professor of astronomy at Oxford and a supporter of Isaac Newton in the calculus prioritycontroversy. In the preface to his own treatment of the Elements (containing Books I–VI, XI, and XIIonly), which �rst appeared in 1702, Keill wrote:

Barrow’s demonstrations are so very short, and are involved in so many notes and symbols, that they are rendered obscure anddif�cult to one [who is] not versed in geometry. There, many propositions which appear conspicuous in reading Euclid himself, aremade knotty, and scarcely intelligible to learners, by his algebraical way of demonstration; as is, for example, prop. 13 Book I. Andthe demonstrations which he lays down in Book II, are still more dif�cult: Euclid himself has done much better, in shewing theirevidence by the contemplations of �gures, as in geometry should always be done. The Elements of all science ought to be handledafter the most simple method, and not to be involved in symbols, notes, or obscure principles, taken elsewhere. [Keill, 1708/1782,A4]

Despite Keill’s criticisms, Barrow’s English version of the Elements was well received in Britain. Overthe course of the century since it �rst appeared in 1660, the book was reproduced (reprinted or reset) atleast six times, �rst in 1686, and then in 1696, 1705, 1714, 1722, and 1732. The last edition appeared in1751. The editions of 1705 (and after) also included as an appendix Archimedes’ Theorems on the Sphereand Cylinder. The 1714 edition added Euclid’s Data, as well as François Foix-Candale’s Brief Treatise ofRegular Solids.14 The next edition of 1722 also included additional supplements, namely some corollariesBarrow had deduced himself from the propositions of Euclid’s Elements. The 1732 edition was issuedafter Barrow’s death, and thus added his portrait. The last edition was virtually the same as that of 1732,except for a new supplement on logarithms [Simpkins, 1966, 242–244]. Despite slight differences in thelayouts of these editions, the content of the Euclidean geometry in all editions remained virtually thesame. Consequently, in what follows, Barrow’s �rst English edition is used for comparison with Wylieand Li’s Chinese translation.

3. Barrow’s version of the Elements contrasted with the translation by Wylie and Li

Before comparing Isaac Barrow’s English Euclid and Wylie/Li’s Chinese translation of Euclid (here-after referred to as IBE and WLC, respectively), to determine whether the former was really the sourceof the latter, it is necessary to describe the format and major characteristics of the latter. As alreadymentioned above, Alexander Wylie considered it his mission to ful�ll the un�nished project left behind

14 François Foix-Candale’s full French name is François de Foix, Comte de Candale, known in Latin as Franciscus Flussatusand in English as Flussas. I am grateful to one of the referees for this information. For a brief biography of François Foix-Candale, see Morembert [1979].


by Matteo Ricci. Thus it was natural for him to adopt the terminology and format created by Ricci andXu Guangqi for their translation of the �rst six books.

Two basic formats, as shown in a full English translation of two typical examples taken from Ricciand Xu’s translation by Peter M. Engelfriet [1998, 142–145], were followed in the arrangement ofpropositions in WLC. One of these used frequently begins with the description of a proposition, then“explanations of the proposition” (jie yue ), which is immediately followed by a “proof” (lun yue

). Since some propositions contain more than one argument, this format was sometimes extended toinclude “another explanation” (you jie yue ) and “another proof” (you lun yue ). The otherbasic format starts with the description of a proposition followed immediately by “the [construction]method says” (fa yue ). Corollaries, lemmas, and porisms were all translated under the headingxi (literally, “connected” or “subordinate”), which usually came at the end of a proposition. Somepropositions also include a “note” (an ), an “example” (li ), or a “comment” (zhu yue ), andsometimes all of these. In WLC, everything was articulated in words without use of mathematical symbolsor formulas, and most of the propositions were illustrated with diagrams. The Roman and Greek lettersused in the diagrams were replaced by Chinese characters borrowed from the 10 heavenly stems and 12earthly branches.15

With the above brief description of the basic format and characteristics of WLC in mind, it is nowpossible to compare it with IBE. The most obvious contrasts between the two books are that (1) IBEmakes liberal use of symbols, whereas WLC is totally free of symbols, and (2) IBE provides no comments,examples, or notes after the proof of a proposition, whereas WLC supplies a considerable amount ofsuch additional explanatory commentary. For example, consider Book X, Proposition 42, which plainlyillustrates these differences (see Figs. 3 and 4). In order to facilitate this comparison, and make whatfollows below clearer, the following is a literal translation, back into English, of Proposition 42 as it wasrendered in WLC:

Book X, Proposition XLII:

Two straight lines which are incommensurable in square, on which the sum of the squares is a medial. A rectangle made of the linesis also medial. The rectangle is incommensurable with the sum of the two squares. Thus, the sum of the two lines is irrational, namingit the side of the two medials.

Explanation: The two straight lines AB, BC are incommensurable in power. As the Problem stated, adding them gives AC. TheProblem says AC is irrational.

Proof: Let DE be a commensurable line, on which construct a rectangle DF, making it equal to the sum of the squares AB and BC(Book I, 45). Construct another rectangle GH, making it equal to twice the rectangle AB, BC. Then the rectangle DH is equal to thesquare AC (Book II, 4). The sum of the squares AB and BC is a medial, and is equal to the rectangle DF, therefore DF must be amedial. But it is applied to the rational line DE, so DG is rational and incommensurable with DE (this Book, 23). GI is also rationaland is incommensurable with FG, that is, DE. Same reason as above. Since the sum of the squares AB, BC, is incommensurable withtwice the rectangle AB, BC, the two rectangles DF, GH are incommensurable (Book VI, 1; this Book, 10). The two lines [DF andGH] are rational. Therefore, DF and GI are rational lines commensurable only in square. So, DI is irrational and called binomial (thisBook, 37). While DE is rational, so the rectangle DH is irrational (this Book, a note in Proposition 39), therefore the side of a squarewhich has the same area is irrational. The rectangle DH is equal to the square AC, therefore, AC is irrational, and is called the line ofthe two medials.

15 For a list of the correspondence between English and Greek letters and their associated Chinese characters, see Martzloff[1997, 121].


Note: A line, whose square is equal to the sums of the squares on AB, BC, and of twice the rectangle AB, BC, is equal to the twomedials. Therefore, it is called a line of the two medials.

Another Note: An irrational line can be divided into two lines at one point only. The following problems will illustrate this matter.One example is given here �rst.

Example: Suppose the point D divides AB evenly, and AC is greater than EB. Subtract the common part CE, the remainder AE isgreater than CB. Since AD is equal to DB, therefore ED is less than DC. The distances between the midpoint D and the points C andE are not equal. The sum of the rectangle AC, CB, and the square CD is equal to the square DB (Book II, 5). The sum of the rectangleAC, EB and the square ED is also equal to the square DB. Therefore, the sum of the rectangle AC, CB and the square CD is equalto that of the rectangle AE, EB, and the square ED. Since the square ED is less than the square CD, the remaining rectangle AC, CBis less than AE, EB. Therefore, twice the rectangle AC, CB is less than twice AE, EB. Their difference is greater than the differencebetween the sum of the squares on AC, CB, and that of the squares AE, EB. [Wylie and Li, 1857/1865, Book X, part I, 47b–49a]

Comparing this literal translation with the corresponding proposition in IBE, note the followingdiscrepancies:

(1) IBE’s description is concise and replete with symbols such as =, +, and � , whereas WLC’sdescription is lengthy and only verbal;

(2) The lettering used in the two diagrams does not correspond to each other;(3) References to previous propositions upon which X, 42 depends are provided in the margin of IBE,

but are incorporated into the text of WLC; and(4) The most dramatic difference is that WLC has an explanation (jie yue ) of the proposition, along

with two short notes (an and you an ), as well as a lengthy example (li ) after the proof, noneof which can be found in IBE.

If IBE were the original source of WLC, Wylie and Li would have had not only to translate the symbolsof IBE into verbal language, but then to rearrange the text, providing their own notes and examples,which in Book X are also given for Propositions 10, 17, 19, 22, 31, 32, 38–41, 54, 60, 73, 88, and 117;in Book XII for Propositions 2 and 4; and in Book XIII for Propositions 1, 2, 13, and 18, and indeed, formany other propositions throughout WLC. Doing all of this would have required tremendous effort andconsiderable ingenuity.

Further comparisons of the two books reveal other major discrepancies. For instance,

(1) Book VII of IBE includes 23 de�nitions, the last of which states: “One number is said to measureanother, by that number, which, when it multiplies, or is multiplied by it, it produces” [Barrow,1660/1983, 143]. This de�nition cannot be found in WLC, and the three postulates and twelve axiomsimmediately following the de�nition in IBE are also missing from WLC [Barrow, 1660/1983, 143–144; Wylie and Li, 1857/1865, Book VII, 2b–3a].

(2) Again, in Book VII of IBE, Proposition 3 has only one corollary [Barrow, 1660/1983, 146], but thesame proposition in WLC has two [Wylie and Li, 1857/1865, Book VII, 4a]; moreover, Propositions18, 19, and 36 all have a scholium or corollary, but there are none for the corresponding propositionsin WLC [Barrow, 1660/1983, 154, 159, and 161; Wylie and Li, 1857/1865, Book VII, 13b–15a, and25a–26a].

(3) The de�nitions for a parallelepiped and inscribed and circumscribed solid �gures (De�nitions XXX,XXXI, and XXXII) in Book XI of IBE are not to be found in WLC [Barrow, 1660/1983, 270; Wylieand Li, 1857/1865, Book XI, 4].


Fig. 3. Alexander Wylie and Li Shanlan’s Chinese translation of Euclid’s Elements, Book X, 42. From the 1865 edition of Jiheyuanben , Book X, part I, pp. 47b–49a.


Fig. 4. Propositions 41 and 42 from Isaac Barrow’s English Elements, Book X (1660 edition). From Early English Books,1641–1700, 1383: 9, p. 224 (Ann Arbor, Michigan: University Micro�lm International, 1981–1982).

(4) At the beginning of Book XIV of WLC, the translators state clearly that neither Book XIV nor XVwas written by Euclid, but both were added to the Elements by Hypsicles of Alexandria. Wylie andLi also rendered the preface by Hypsicles into Chinese [Wylie and Li, 1857/1865, Book XIV, 1] butIBE does not include the Hypsicles preface [Barrow, 1660/1983, 345].


Many more discrepancies, major and minor, between IBE and WLC could easily be identi�ed. Butthe comparisons made above suf�ce to cast considerable doubt on Qian Baocong’s hypothesis thatIsaac Barrow’s English version of Euclid’s Elements served as the basis for the Wylie/Li translationof Books VII–XV into Chinese. The question that now arises concerns the book Alexander Wylieand Li Shanlan actually used in preparing their Chinese translation of the last nine books of Euclid’sElements, and whether the exact edition of Euclid’s Elements they used can be identi�ed. The nextsection establishes beyond doubt that the source of the Wylie/Li translation must have been the �rstEnglish edition of the Elements, published in 1570 by Billingsley.

4. Billingsley’s version of the Elements compared with the translation by Wylie and Li

By 1850 many English editions of Euclid’s Elements were available when Alexander Wylie decided totranslate the last nine books into Chinese. Nevertheless, most included only the �rst six books, althoughsome added XI and XII as well [Heath, 1956, 109–112]. Among English versions with 15 or even 16books of the Elements are the �rst English translation in 15 books by Henry Billingsley of 1570 (hereafterreferred to as HBE) and a translation by John Leeke and George Serle of 1661 (hereafter referred to asLSE). Of these, the New York Public Library’s Rare Book Division has an original copy of HBE, andColumbia University’s Butler Library has a micro�lm copy of LSE. Comparing both of these with theChinese translation leads to some interesting conclusions, but �rst it is necessary to introduce brie�ythese two English editions.

The �rst English translation of Euclid’s Elements, by Billingsley, was made from the 1558 Latinedition: Euclidis Megarensis mathematici clarissimi Elementorum geometricorum libri XV (Fig. 5). Thisin turn was the �rst Latin translation based upon a Greek text, and was made by Bartolomeo Zambertiwho published it in Venice in 1505 [Archibald, 1950, 445–448]. A brief description of HBE can be foundin Charles Thomas-Stanford’s Early Editions of Euclid’s Elements, which provides a facsimile of the titlepage [Thomas-Stanford, 1977, 13–16, 43–44, Plate X]. More detailed information may also be found inWalter F. Shenton’s “The First English Euclid” [Shenton, 1928]. In addition to including a facsimile ofthe same title-page as shown in Thomas-Stanford’s book, the article provides seven more illustrations,including a portrait of John Daye (also Day), the printer. Three-dimensional geometric models made outof folded pieces of paper serve to illustrate the text. Shenton also reprints the preface to the book in itsentirety [Shenton, 1928, 505–512].16

Following “The Translator to the Reader,” HBE has a lengthy introduction, 24 folios in all, by theRenaissance polymath John Dee, who was misattributed as the “true” translator of the book by AugustDe Morgan [De Morgan, 1837, 38].17 The translation of the 13 books by Euclid cover 414 folios andare followed by the 14th and 15th books by Hypsicles (29½ folios). The book ends with an appendixas Book XVI (13 folios) and a “Brief Treatise of Mixed and Composed Regular Solids” (5 folios) byFrançois Foix-Candale.18 At the beginning of each book there is an outline, or as Billingsley puts it, “the

16 The copy of Euclid that Walter F. Shenton consulted, now in the collection of the American University in Washington, DC,is a folio bound in two volumes, the �rst of which contains the �rst nine books of the Elements. The copy in the New YorkPublic Library is of the same size, but bound in a single volume.

17 A facsimile of the preface by John Dee, together with an introduction by Allen G. Debus, was published by Science HistoryPublications in New York. See Dee [1975].


Fig. 5. Title page of the Henry Billingsley Elements. Stuart Collection, Rare Books Division, The New York Public Library,Astor, Lenox, and Tilden Foundations.


argument” of the main contents of the book in question. For Books VII, X, and XI, explanations arealso provided after every de�nition. A double numbering system, for both theorems and propositions, isused in the book. The translation is not a mere word-for-word translation, but contains “all the earlierand current commentary of importance” [Archibald, 1950, 449]. Many corollaries, assumptions, andnotes by Apollonius, Archimedes, Campanus, John Dee, Eudoxus, François Foix-Candale, Hypsicles,Montaureus, Pappus, Plato, Proclus, Pythagoras, Regiomontanus, and Theon of Alexandria, amongothers, are also incorporated [Archibald, 1950, 449].

The LSE resembles the HBE in many ways. For instance, each book begins with “The Argument.” Or-namental initial letters at the beginning of “The Argument” decorate the book throughout. Explanations,though shorter, are provided for each de�nition. A dual number system is adopted and descriptions ofthe propositions and their proofs are entirely verbal. Many corollaries and comments are also provided.Because of these and many other similarities, the LSE was mistaken for the second edition of HBE byRobert Potts, a 19th-century expert on Euclid [Heath, 1956, 110: 22, 233]. Potts’ view was widely ac-cepted by many in�uential scholars, including August De Morgan and Thomas L. Heath [De Morgan,1837, 38; Heath, 1956, 110; Archibald, 1950, 451]. His misattribution was only refuted later, in 1950, byRaymond C. Archibald. To quote Archibald again:

“It [the 1661 English edition] is simply an independent, and inferior, edition of the Elements in the preparation of which L[eeke] &S[erle]—the obscure ‘students in mathematicks’—perhaps consulted B[illingsley’s edition] constantly.” [Archibald, 1950, 451]

Archibald provides some compelling evidence to make clear the differences between HBE and LSE[Archibald, 1950, 450–451], but these do not help much in deciding whether HBE or LSE was the modelfor WLC. Nevertheless, there is direct evidence that shows that LSE could not have been the basis forWLC, and at the same time establishes the link between HBE and WLC. Book X, Proposition 42, whichwas used above to compare IBE and WLC, again offers a constructive basis for comparison.

Proposition 42 in Book X is numbered as “PROP. 42. TEOR. 30.” in LSE, and as “The 29. Theoreme.The 41. Proposition.” in HBE (the discrepancy of these numberings is explained below). Comparing theproposition in LSE (Fig. 6) with that in WLC as translated above (but see Fig. 3 as well), the followingdifferences can be observed: (1) The letterings G, F , and H in the diagrams do not correspond to oneanother; (2) LSE does not use words similar to those found in the section jieyue in WLC; and(3) like IBE, LSE does not include anything corresponding to the sections under the headings anand li [Leeke and Serle, 1661/1982, 288–289; Wylie and Li, 1857/1865, Book X, part I, 47b–49a].But the jieyue appears in WLC throughout. Not one proposition in LSE provides such explanationsaccompanying the statement of each proposition. Moreover, the contents of the an and li in otherplaces of WLC, some of which have been pointed out above, would have been added by Alexander Wylieand Li Shanlan themselves if LSE were the book they used for preparing WLC. But when we turn tocompare Book X, Proposition 42 in WLC with its counterpart in HBE, we shall see there is no suchpossibility.

As the illustration (Fig. 7) shows, the contents of “The 29. Theoreme. The 41. Proposition.” correspondvery well to the literal translation of WLC provided in the previous section. In particular, note the

18 The original pagination is not always correct. Certain numbers were skipped or repeated. For instance, there is a jumpfrom folio 203 to folio 205, and in some cases the pages are wrongly numbered, for example, the three folios after 319 werenumbered as 340, 341, and 317, respectively. But following these errors, the correct pagination is restored.

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Fig. 6. Book X, 41 of the Elements translated by John Leeke and George Serle and published in 1661. From Early English Books, 1641–1700, 1283: 17,pp. 288–289 (Ann Arbor, Michigan: University Micro�lm International, 1981–1982).

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Fig. 7. Book X, 41 of the Henry Billingsley Elements. From Henry Billingsley’s English translation (printed by John Daye in 1570), folios 260 verso, 261 recto.Stuart Collection, Rare Books Division, The New York Public Library, Astor, Lenox, and Tilden Foundations.


sentences “Let these two right lines AB and BC. . . Then I say that the whole line AC is irrational,”beginning at the ornately designed initial letter L, the words in the jieyue of WLC convey the samemeaning. Just as “jieyue ” is part of the standard format of WLC, “Suppose that. . . Then I say,” orless frequently “Let there be. . . Then I say,” can be found throughout HBE. Furthermore, compare the�rst note an with the following words:

A line whose power is two medials, is an irrationall line which is composed of two right lines incommensurable in power, the squaresof which added together, make a medial super�cies, and that which is contained under them is also mediall, and moreover it isincommensurable to that which is assumptes of the two squares added together. [Billingsley, 1570, folio 261 recto]

Similarly, compare the second note you an with this paragraph from HBE:

And that the said irrationall lines are divided one way onely, that is, in one point onely, into the right lines of which they are composed,and which make every one of the kinds of those irrationall lines, shall straight way be demonstrated, but �rst will we demonstrate twoassumptes here following. [Billingsley, 1570, folio 261 recto]

These comparisons clearly show that the two notes in WLC were doubtlessly translated from thesetwo passages. Moreover, the last sentence of the previous quotation indicates that two “assumptions” areprovided immediately after. Having compared the �rst “assumption” with the li, we must conclude thatthe contents accord directly with each other.

The above example is not an isolated case. Compare the last proposition of Book X, which is numberedas the 117th problem in WLC, and “The 92. Theoreme. The 116. Proposition.” in HBE [Billingsley, 1570,folio 310 recto]. (This proposition cannot be found in IBE—another solid piece of evidence that Barrow’sElements was not the basis for WLC.) At the end of the problem, WLC includes three notes (one an andtwo you an), which may be translated, once again literally, as follows (Fig. 8):

Note: For any two incommensurable lines, A, B, the squares on which must be incommensurable. If C is the mean proportional of Aand B (Book VI, 13), the ratio of A to B is equal to that of any similar �gures: squares, rectangles, or circles, on A and C ( Book VI,20). The ratio of two circles is equal to that of the two squares on their diameters (Book XII, 2). Therefore, from two incommensurablelines, many incommensurable areas can be deduced.

Another Note: Any two areas, whether they are commensurable or incommensurable, can be judged according to the previous note.For any two solids, whether they are commensurable or incommensurable can be deduced from the similar solids, cylinders or cones,which have the same heights, constructed on lines A and B. The ratio of the two solids is equal to that of the two bases (Book XI, 32;Book XII, 56). If their bases are commensurable, so are the solids (this Book, 10). If their bases are incommensurable, so are the twosolids.

Another Note: On two circles, A and B, construct two cones with the same height. The ratio of the cones is equal to that of thecircles. If the two circles are commensurable, so are the two solids. If incommensurable, so are their solids (this Book, 10). Therefore,whether solids are commensurable or not, is just like the cases of areas and lines. [Wylie and Li, 1857/1865, Book X, part III, 52]

In HBE, there are the following related passages (Fig. 9):

Here Follows an instruction by some studious and skilful Grecian (perchance Theon) which teaches us of farther use and fruit of theseirrational lines

Seing that there are founde out right lines incommensurable in length the one to the other, as the lines A and B, there may also befounde out many other magnitudes having length and breath (such as are playne super�cieces) which shall be incommensurable theone to the other. For if (by the 13. of the sixth) between the lines A and B there be taken the mean proportionall line, namely C, then


Fig. 8. Alexander Wylie and Li Shanlan’s Chinese translation of Euclid’s Elements, Book X, 117. From the 1865 edition of Jiheyuanben , Book X, part III, p. 52.

(by the second corollary of the 20. of the sixth) as the line A is to the line B, so is the �gure described upon the line A to the �guredescribed upon the line C, being both like and in like sort described, that is, whether they be squares (which are alwayes like the oneto the other), or whether they be any other like rectiline �gures, or whether they be circles aboute the diameters A and C. For circleshave that proportion the one to the other, that the squares of their diameters have (by the 2. of the twelfth). Wherfore (by the secondpart of the 10. of the tenth) the �gures described upon the lines A and C being like and in like sort described are incommensurable theone to the other, wherfore by this means there are founde out super�cieces incommensurable the one to the other. In like sort theremay be founde out �gures commensurable the one to the one, if ye put the lines A and B to be commensurable in length the one tothe other. And seing that it is so, now let us also prove that even in solids also or bodyes there are some commensurable the one tothe other, and other some incommensurable the one to the other. For if from each of the squares of the lines A and B, or from anyother rectiling �gures equal to these square be erected solids of equall altitude, whether those solids be composed of equidistancesuper�cieces, or whether they be pyramids, or prismes, those solids so erected shal be in that proportion the one to the other thattheyr bases are (by the 32. of the eleventh and 5, and 6, of the twelfth) Howbeith there is no such proposition concerning prismes.And so if the bases of the solids be commensurable the one to the other, the solids be commensurable the one to the other, the solidsalso shall be commensurable the one to the other, and if the bases be incommensurable the one to the other, the solids also shall beincommensurable the one to the other (by the 10. of the tenth) and if there be two circles A and B and upon ech of the circles beerected Cones or Cylinders of equal altitude, those Cones & Cilinders shall be in that proportion the one to the other that the circlesare, which are their bases (by the 11. of the twelfth): and so if the circles be commensurable the one to the other, the Cones andCilinders also shall be commensurable the one to the other. But if the circles be in commensurable the one to another, the Cones alsoand Cilinders shall be incommensurable the one to the other (by the 10. of the tenth). Wherfore it is manifest that not only in linesand super�cieces, but also in solids or bodies is found commensurable or incommensurability. [Billingsley, 1570, folio 310 verso andfolio 311 recto]

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Fig. 9. Book X, 116 of the Henry Billingsley Elements. From Henry Billingsley’s English translation (printed by John Daye in 1570), folios 310 verso, 311recto. Stuart Collection, Rare Books Division, The New York Public Library, Astor, Lenox, and Tilden Foundations.


The three notes in WLC are all apparently translations from the “instruction” added by the prominentGreek mathematician Theon of Alexandria.19 The above lengthy quotation not only provides furtherevidence to establish that HBE is indeed the source of WLC, but it also makes clear that an and you anare translations of the notes, comments, assumptions, and other supplements that Billingsley inserted inhis book. They are not comments originally written by Alexander Wylie or Li Shanlan themselves.

Some historians have misattributed the three notes, as well as many others, to Li Shanlan, and wronglypraised him for having extended the theory of irrationals from straight lines to planes and solids [Wang,1990, 405–406; 1994, 1149]. Moreover, the contents under the commentary zhu yue are also translations.The commentary zhu yue in WLC Book X, Proposition 9, for instance, is a translation of “An Assumpt.”of the same proposition in HBE [Billingsley, 1570, folio 240 verso].

In comparing WLC and IBE above, it was pointed out that there is a preface by Hypsicles, the authorof Books XIV and XV, before Book XIV in WLC. Although missing from IBE, it is to be found inHBE [Billingsley, 1570, folio 416 recto–verso]. Nevertheless, WLC is not a cover-to-cover translationof HBE. In order to follow in general the format created by Ricci and Xu Guangqi, Wylie and LiShanlan did not translate the outlines of each book provided in HBE, nor the lengthy explanation forevery de�nition. Even the two de�nitions of Book XV, and many propositions, theorems, corollaries,assumptions, comments, and notes by scholars other than Euclid were omitted. Twenty propositionsafter Hypsicles’ work, the entire Book XVI, and the appendix by François Foix-Candale in HBE arealso missing from WLC [Billingsley, 1570, folios 437 recto–463 recto]. In addition, the total numberof propositions in Book X, and of the de�nitions in Book XI, is not identical. Book XI of WLC has29 de�nitions, but HBE only 25. This difference was caused by breaking down certain original de�nitionsinto separate parts. For instance, the �rst de�nition in HBE was translated as the �rst two de�nitions inWLC [Billingsley, 1570, folio 312 verso; Wylie and Li, 1857/1865, Book XI, part I, 1a]. As noted above,Book X, Proposition 42 in WLC actually corresponds to Proposition 41 in HBE. This difference wascaused by “An Assumption” added after Proposition 12 in HBE: “If there be two magnitudes comparedto one and the self same magnitude, and if the one of them be commensurable unto it, and the otherincommensurable; those magnitudes are incommensurable the one to the other,” which was translated,but numbered as Proposition 13 in WLC [Billingsley, 1570, folio 241 verso, 242 recto; Wylie and Li,1857/1865, Book X, part I, 14a]. All of this evidence strongly suggests that HBE was the original sourceof WLC.

5. Where is the book Alexander Wylie used?

We have just given very strong evidence that Billingsley’s English Elements was the original source forthe �rst Chinese translation of the last nine books of Euclid’s Elements. We may ask ourselves one �nalquestion related to the Chinese translation, namely, where is the book Wylie and Li used? We suspectthat the copy preserved at the Rare Book Division of the New York Public Library may very well be thecopy in question, strange and unlikely as this may seem at �rst. But this surmise is not the product ofsheer imagination, but the result of several factors based upon certain speci�c concrete evidence.

Wylie in his preface brie�y mentions how he conceived the idea for his translation in the �rst place.His words suggest that the entire project was a personal matter, and in no way connected with the

19 For a biography of Theon of Alexandria, see Toomer [1976].


London Missionary Society for which he worked between 1847 and 1860. When Wylie purchased the“old version” English copy of Euclid in England, it was at his own expense. Accordingly, Wylie actuallyowned the copy of the book that he and Li Shanlan were translating. The Catalogue of the London MissionLibrary, Shanghai, which Wylie edited himself and printed in 1857, the year in which his translation waspublished, con�rms that the Society did not itself have a copy of the Billingsley Euclid.20 In the sectionof the Catalogue devoted to “Books Belonging to the London Missionary Society,” Wylie recordedtwo copies of “Euclid’s Elements, Cambridge edition.” These were copies of Robert Pott’s edition, butthere was also one copy of Dionysius Lardner’s Elements [London Mission Library, 1857, 13, 14, 17].Both Pott’s and Lardner’s Elements are not full versions of Euclid, as Wylie noted. Since the LondonMissionary Society in Shanghai did not at the time have its own library, the books were distributed amongits missionaries, who took responsibility for their safe keeping. The missionaries Alexander Williamsonand Grif�th John each had a copy of Pott’s Elements, while the Lardner edition was in the care of WilliamLockhart. Under Wylie’s own name in this section, only four books are listed, namely, Robert Morrison’sA Dictionary of the Chinese Language, A View of China for Philological Purposes, Joshua Marshman’sClavis Sinica, and Matthew Henry’s Commentary for the NIV: Genesis to Revelation [London MissionLibrary, 1857, 15]. No edition of Euclid’s Elements is listed under Wylie’s name, con�rming that thecopy of Billingsley’s Euclid was his own possession.

Soon after his translation was published, Shanghai was engulfed in the political and social turmoilcaused by the Taiping Rebellion. In 1860, the city was in great danger of being captured by the rebels,and given their hostility toward foreigners, Wylie decided in November to return to England. Before heleft, Wylie sold his collections of Chinese books to the Asiatic Society, although it is reasonable to suspectthat Wylie may have taken the precious Billingsley Euclid back with him to England. Billingsley’s book,so far as this author knows, has never been reported in any catalogues of libraries, either public or private,in China. For the next two and a half years, Wylie stayed in Britain, and during this time he may havesold the Billingsley Euclid for �nancial reasons.

There is some evidence to link Wylie’s copy with a copy of the Billingsley Euclid currently in thepossession of the New York Public Library. This was a copy donated to the Lenox Library in 1892 bythe widow of the millionaire Robert L. Stuart (1806–1882). Stuart was an important philanthropist inNew York City and president of the American Museum of Natural History from 1872 to 1881. Stuarthad made his fortune in business, but had a passion for collecting natural history specimens, �ne art,manuscripts, and rare books. Beginning in the 1850s, he made numerous trips to Europe to build uphis collections, and in 1867 he purchased Billingsley’s book in London, only a few years after Wyliemay have sold his copy [Stuart, 1860, 88]. According to a note under the entry of the book by a personwho identi�ed himself as “Thomas” in the published Catalogue of the Library of Robert L. Stuart, thebook “belonged to Schumacher” [Stuart, 1884, 176]. Further information is revealed in a clipping ofthe original advertisement of the book which is now pasted inside the front cover of the book itself,which reads: “EUCLID’S ELEMENTS, by the celebrated DR. JOHN DEE, the FIRST ENGLISHEDITION. . . with Latin MSS. Notes in a contemporary hand, in the ORIGINAL BINDING, . . . £L126s. SCHUMACHER’S COPY” [Billingsley, 1570].

20 The copy of the Catalogue of the London Mission Library, Shanghai in the New York Public Library Oriental Division,is catalogued as “Legge 1348,” which indicates that the catalogue was in the possession of James Legge, the renowned 19th-century sinologist who actually helped Wylie start his long and successful career as a missionary in China. See London MissionLibrary [1857].


Fig. 10. Detail of the recto of the front �yleaf of the Henry Billingsley English Elements (printed by John Daye in 1570). StuartCollection, Rare Books Division, The New York Public Library, Astor, Lenox, and Tilden Foundations.

Fig. 11. The cover of the �rst volume of the Chinese translation of Elias Loomis’s The Elements of Analytical Geometry andof the Differential and Integral Calculus (London Mission Society Press, 1859), preserved at the Prentis Library of ColumbiaUniversity. Courtesy of the C.V. Starr East Asian Library, Columbia University.

Who was this Schumacher? Was he a rare book dealer to whom Wylie may have sold his copy inLondon, or was he a previous owner of the book from whom Wylie purchased it, or someone else? Whatis known is that on the recto of the front �yleaf of the book there is a mark which reads “A No. 15”(Fig. 10). This handwritten letter can be compared with Wylie’s autograph on the cover page of a bookentitled Daiweiji shiji (The Elements of Analytical Geometry and of the Differential andIntegral Calculus), which is in the Prentis Library of Columbia University. This was the �rst Chinesetranslation, also by Wylie and Li Shanlan, of a calculus book by the American mathematician EliasLoomis. The mark “A No. 15” looks very similar, and may well have been written by Wylie’s hand


(Fig. 11).21 Although there is no further evidence to prove that the copy of Billingsley’s Elements in theNew York Public Library was in fact the copy Wylie actually used, this nevertheless remains a distinctpossibility.

6. Conclusion

It took about six centuries from the time the title of a Persian or Arabic version of Euclid’s Elements�rst appeared in a Chinese record until a complete translation of all 15 books of the Elements wasavailable in Chinese. Many people—pious religious men and pagans; Muslims, Westerners, Mongolians,Manchus, and Chinese; pure scholars, missionaries, and emperors—were all directly involved in thevarious attempts to provide versions in Chinese of Euclid’s Elements. One interesting fact is that eachtranslator chose for his original source a version of Euclid as close to his own culture as possible. MatteoRicci chose a Latin version by his teacher Christopher Clavius; the French Joachim Bouvet and Jean-François Gerbillon selected a French version by Ignace-Gaston Pardies; and Alexander Wylie based histranslation upon an English version of the Elements.

The speci�c English version of Euclid’s Elements that Wylie used to prepare the �rst Chinesetranslation of Books VII to XV of the Elements was not the one by Isaac Barrow as some historianshave speculated, but the one published in 1570 by Henry Billingsley, as this paper has argued. Thisconclusion also coincides with Wylie’s own brief description of the source he used. As he noted inhis preface, “The book is a translation from Greek into my native tongue. It has not been reprintedrecently. The copy is an old version, but it was not well proofread and has mistakes” [Wylie andLi, 1857/1865, Wylie’s preface, 3a]. Billingsley’s translation of the Elements was published in 1570,and has never been reprinted since. Thanks to Raymond C. Archibald, we now know that Billingsleytranslated his edition from a 1558 Latin edition by Bartolomeo Zamberti. But Wylie’s contemporary,the prominent mathematician August De Morgan mistakenly wrote, “This translation [Billingsley’s] ofEuclid was either made from the Greek, or corrected by the Greek” [De Morgan, 1837, 39], and thiswidely held view may have led Wylie to believe that the source of the English Euclid he was usingwas based directly upon a Greek version. Wylie did not mention the name of the translator of the bookhe used, which may be explained by the fact that most writers of the 19th century did not bother toidentify the sources they used, and also by the fact that De Morgan doubted that Billingsley was thetrue translator, and instead took John Dee to have been the translator in question [De Morgan, 1837,39]. Probably in order to avoid a possible mistake, Wylie may have thought it best to say nothingabout the authorship of the English Euclid he was using. Billingsley’s translation was beautifullyprinted by John Daye, but like any book, it was not free of printing errors. By saying that “it was notwell proofread and has mistakes,” Wylie may have intended to exaggerate the dif�culties he and LiShanlan had faced in translating Books VII through XV of Billingsley’s Euclid for the �rst time intoChinese.

21 The full name of I.J. Roberts is Issachar Jacob Roberts (1802–1871). He was sent to China in 1837 by the AmericanBaptist Missionary Union and in 1847 taught Hong Xiuquan , who was to become the leader of the Taiping Rebellion(1851–1864), the essence of Christianity.


7. Epilogue

In closing, having argued that it was the Billingsley version of Euclid and not the Barrow translationas others have suggested that served as the basis for the Chinese translation completed by Wylie and Liin the 1850s, several further questions remain to be addressed. Did it make a difference that Wylie andLi used the Billingsley translation, rather than the Barrow or possibly the Leeke–Serle translation? Andwhy should they have focused on translating Euclid rather than more cutting-edge geometry that from aWestern perspective would surely have been of greater mathematical interest, for example, the works ofCarl F. Gauss, Jakob Steiner, Nikolai I. Lobachevsky, Julius Plücker, and others that were transformingEuropean geometry in the 19th century?

As for the signi�cance of the Billingsley translation, as already noted, the value of this particularversion of Euclid lay in its notes and explanatory comments, which Wylie and Li duly included intheir Chinese version. Not only was the notation of the Barrow edition at odds with the translationalready available in Chinese of the �rst six books, which would have made it an inappropriate choicefor Wylie and Li, but the Leeke–Serle version, lacking the more extensive commentary of the Billingsleytranslation, would not have been as useful for Chinese readers, nor perhaps as easy to penetrate. This, ofcourse, is speculation, and it may simply have been that Wylie used the Billingsley translation becausethis is the one that most easily came to hand; but even if it was a choice due to expedience, it was afortuitous one.

As for why Wylie and Li chose to translate Euclid �rst rather than other, more modern geometersof the 19th century, there may be several factors worth considering. First and foremost, the translationof Euclid into Chinese was un�nished business. Chinese familiar with the Ricci/Xu translation knewonly the �rst six books, and completing the translation indeed made available for the �rst time inChinese the most fundamental work in the history of geometry, a prerequisite for moving on tothe more sophisticated geometries of the 19th century, including non-Euclidean geometry. As thevocabulary and methods of Euclidean geometry became increasingly familiar to Chinese studyingWestern mathematics at the end of the 19th century, they not only studied modern mathematics in Europe,but began to translate and communicate more advanced results in the decades following the appearanceof the Wylie/Li completion of the Euclidean Elements. But as Jean-Claude Martzloff has observed, incommenting on European mathematical works that had been translated into Chinese by the end of the19th century:

. . .this literature does not contain echoes of the philosophical controversies about in�nitesimal calculus, imaginary numbers andparallelism. It does not contain any information whatsoever about the theories of Cauchy, Gauss, Riemann or Dirichlet and algebraicsymbolism similar to that of Descartes did not appear in China until the second half of the 19th century. Similarly, differentialequations, analytic geometry, and mechanics are all absent from the Chinese mathematics before this period. [Martzloff, 1997,112]

Horng Wannsheng accounts for this, at least in part, by noting that until well into the 20th century,Chinese interest in Western mathematics remained focused on application, not on abstract or overlytheoretical mathematics:

Some texts like [William] Wallace’s Algebra were more easily appreciated than other texts like [Augustus] De Morgan’s Algebra,but this had nothing to do with their contents, [but] with their different styles of exposition. In general, mathematical texts with the“logarithmic” features were more easily linked to pragmatic uses. Under such circumstances, it is easy to understand why Wallace’sAlgebra was more acceptable. [Horng, 1991, 383]


Horng then goes on to cite Chang Hao, who also explains that “To the extent that they [Chinesescholar–of�cials] responded to the West at all, their responses re�ected basically the same pragmaticapproach and technically rational mode of thinking they had long applied to tackling domestic problemsof statecraft” [Chang, 1976, quoted from Horng, 1991, 382].

Whether the subject is algebra, calculus, geometry, or any other branch of mathematics, theincreasingly powerful and abstract concerns of European theoretical mathematicians would not becomeprominent interests in China until well into the 20th century, when the �rst generation of professional,research mathematicians began to appreciate, teach, and publish their own work along lines they hadstudied �rst in Europe or America, and then brought back to China. But this is a more complex story aboutthe transmission of mathematics from West to East, and of the emergence of an indigenous mathematicalcommunity in China, than can be recounted here.22 Nevertheless, it is certainly an important feature ofthe assimilation of Western mathematics that one of its essential works, the Euclidean Elements, was atlast available to Chinese mathematicians in its entirety, thanks to Alexander Wylie and Li Shanlan, whenthey �nally made their translation of Euclid, Books VII–XV, available in 1857.

Acknowledgments

The author is grateful to Joseph W. Dauben and Andrea Bréard for their valuable comments andsuggestions made on earlier drafts of this article. He is equally grateful to Adrian C. Rice for his help inlocating Auguste De Morgan’s article on Euclid. He thanks the New York Public Library for allowinghim to consult the �rst English edition of Euclid’s Elements (1570), translated by Henry Billingsley,and for permission to reproduce certain pages from it. He also thanks the staff, especially John F. Ratheand Elie Weitsman of the Rare Rook Division of the New York Public Library, for their assistance. Anearlier version of this paper was presented in the Special Sessions for History of Mathematics at the jointannual meeting of the American Mathematical Society and the Mathematical Association of America(AMS/MAA), Baltimore, Maryland, January 17, 2003. He is grateful to the organizers of this specialsession, Joseph W. Dauben and David Zitarelli, for inviting him to participate, and to the audience for itscomments and questions. Last but not least, the author thanks the anonymous referees and the editors ofthis journal for their comments and criticism, which have also helped to improve the published versionof this paper.

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Historia Mathematica 34 (2007) 10–44www.elsevier.com/locate/yhmat

The Suàn shù shu , “Writings on reckoning”:Rewriting the history of early Chinese mathematics

in the light of an excavated manuscript ✩

Christopher Cullen

Needham Research Institute, 8 Sylvester Road, Cambridge CB3 9AF, United Kingdom

Available online 3 April 2006

Abstract

The Suàn shù shu is an ancient Chinese collection of writings on mathematics approximately 7000 characters in length,written on 190 bamboo strips, recovered from a tomb that appears to have been closed in 186 B.C. This anonymous collection isnot a single coherent book, but is made up of approximately 69 independent sections of text, which appear to have been assembledfrom a variety of sources. Problems treated range from elementary calculations with fractions to applications of the Rule of FalsePosition and �nding the volumes of various solid shapes. The Suàn shù shu is now the earliest datable extensive Chinese materialon mathematics. This paper discusses its relation to ancient works known through scribal transmission, such as the so-called “NineChapters,” Ji�u zhang suàn shù , which is �rst mentioned in connection with events around A.D. 100, but may have beencompiled about a century earlier. It is proposed that the evolution of Chinese mathematical literature in the centuries that separatethese two texts may be understood through comparison with what is known to have taken place during that time in another area ofChinese technical literature, that of medicine.

© 2005 Elsevier Inc. All rights reserved.

MSC: 01A25

Keywords: China; Hàn dynasty; Ji�u zhang suàn shù; Suàn shù shu

✩ I explain my use of tone-marked romanization in the appendix, in which I also discuss the status and signi�cance of the title used here.E-mail address: [email protected].

0315-0860/$ – see front matter © 2005 Elsevier Inc. All rights reserved.doi:10.1016/j.hm.2005.11.006

http://www.elsevier.com/locate/yhmat

mailto:[email protected]

http://dx.doi.org/10.1016/j.hm.2005.11.006

C. Cullen / Historia Mathematica 34 (2007) 10–44 11

Contents

0. Chronology . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 111. Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 122. Background and context . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 133. The physical and formal structure of the Suàn shù shu . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13

3.1. Physical form, reconstitution, and transcription . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 133.2. The formal structure of the Suàn shù shu . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16

3.2.1. The individual characters . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 163.2.2. The clause and the sentence . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 163.2.3. The section . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 163.2.4. Sequence and grouping of sections . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19

4. Mathematical content of the Suàn shù shu . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 194.1. Mathematical language . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 194.2. Topics and techniques . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20

5. The origins of the Suàn shù shu . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 265.1. The received tradition of ancient Chinese mathematics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 275.2. Mathematicians of the early Chinese empire and their texts . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 305.3. Liú Hu� and the history of the Nine Chapters . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 325.4. The commentators and the “Nine Reckonings” . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 335.5. The Nine Chapters and the Suàn shù shu . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35

5.5.1. Problems in constructing a narrative . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 355.5.2. A parallel: Medicine in the Western Hàn and the format of the Suàn shù shu . . . . . . . . . . . . . . . . . . . 355.5.3. The role of Wáng M�ang . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 375.5.4. The Suàn shù shu and the purpose of mathematical study in the Western Hàn . . . . . . . . . . . . . . . . . . 38

6. Conclusion and summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 40Acknowledgments . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41Appendix. Suàn shù shu—title or label? . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42

Pre-modern works . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42Modern works . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 43

0. Chronology

In reading this article, the following summary listing may be helpful to readers not familiar with early Chinesehistory:

1046 B.C. Beginning of Western Zhou dynasty.

771 B.C. Fall of Western Zhou; the kings of Zhou continued in titular power under the protection of powerfulvassals until 256 B.C. The period from 771 to 256 B.C. is often referred to as “Eastern Zhou.” TheEastern Zhou includes the “Spring and Autumn” period (722–481 B.C.), and most of the “WarringStates” period (481–222 B.C.).

221 B.C. State of Qín destroys all rivals and founds a uni�ed centralized empire.

206 B.C.–A.D. 9 Western Hàn dynasty.

A.D. 9–23 Reign of Wáng M�ang .

A.D. 23–221 Eastern Hàn dynasty.

I refer to the period from the beginning of the Qín empire to the end of the Hàn as “early imperial China.” Given thetendency of some scholars to use the word “ancient” to refer to China up to near the end of the �rst millennium A.D.,I should also note that my more limited usage of this term does not extend much beyond the end of the Hàn.

12 C. Cullen / Historia Mathematica 34 (2007) 10–44

1. Introduction

The Suàn shù shu , “Writings on reckoning,” is an ancient Chinese collection of writings on mathematicsapproximately 7000 characters in length, written on 190 bamboo strips, which were originally bound side by sidewith string and rolled up into a scroll-like bundle. It was discovered in 1983 when archaeologists opened a tomb atZhangjiashan in Húb�ei province [Zhangjiashan, 2001]; since the binding strings had long since perished,the strips were scattered in disorder. From documentary evidence this tomb is thought to have been closed in 186 B.C.,early in the Western Hàn dynasty. It is in any case certainly an early second century B.C. burial. The unknown occu-pant of this tomb appears to have been a minor local government of�cial who had begun his career in the service ofthe Qín dynasty but started work for its successor, the Hàn, in 202 B.C. [Péng, 2001, 11–12]. The work discussed herewas not the only one deposited in this tomb; in addition to material containing administrative regulations there werealso writings on medicine and therapeutic gymnastics, all of which have been published and widely discussed else-where [Harper, 1998, 30–33; Tsien, 2004, 227]. A full translation of the Suàn shù shu into English, with explanatorycommentary and a critical edition of the Chinese text, has recently been published [Cullen, 2004].

Until the discovery of the Suàn shù shu the most ancient extant Chinese mathematical work was the Ji�u zhang suànshù . The title of this book has been variously rendered. While I would prefer something like “Mathematicalmethods [or procedures]1 under a ninefold classi�cation,” it is more common to call it “Nine chapters on mathematicalprocedures,” “Nine chapters on the mathematical arts” or “The Nine Chapters” for short. For simplicity I shall followthe latter practice.

For centuries the Nine Chapters played in China a role similar to that of the Elements of Euclid (�. c. 300 B.C.)in Europe, in that it was both a paradigm for the learned practice of mathematics and the earliest surviving majormathematical text. There is, however, no direct evidence of the existence of the Nine Chapters before the �rst cen-tury A.D. (see below). For historians of Chinese mathematics the impact of the Suàn shù shu is therefore similar towhat would follow from the discovery of an ancient Greek manuscript with substantial material by such pre-Euclideanmathematicians as Hippocrates (c. 485 B.C.) and Archytas (c. 385 B.C.). The importance of the Suàn shù shu for thehistory of world mathematics is indisputable.

This article outlines the nature and signi�cance of this text and indicates some of the ways in which its discoverychanges our views of the beginning of the ancient Chinese mathematical tradition, especially in relation to the placeof the Nine Chapters in that tradition. I have divided my account into sections as follows:

(a) A very brief sketch (Section 2 of relevant features of the historical and cultural context in which the Suàn shùshu belonged. Early imperial China was a world very different from the ancient cultures more familiar to mostEnglish speaking historians of mathematics, and some preliminary orientation is indispensable to understandingwhat follows.

(b) The physical and formal structure of the text (Section 3). Here I discuss the implications of the fact that the Suànshù shu took the uniquely Chinese form of a bundle of bamboo strips, and I outline the way that the writings itcontains are structured and divided. Such issues will be of major importance in analyzing the historical place ofthis material in the evolution of later Chinese mathematical writing.

(c) The mathematical content of the text (Section 4). Here I give a summary of the mathematical methods used inthe Suàn shù shu and give samples to illustrate the way they are applied. Readers who require more details canof course refer to my full translation and commentary published elsewhere [Cullen, 2004].

(d) The historical implications of the discovery of the Suàn shù shu (Section 5). The striking new evidence nowavailable demands a radical reassessment of our view of early Chinese mathematics and mathematical literature.In the last sections of this article I sketch what I believe to be the essential features of such a reassessment.Up to now, all accounts of the �rst few centuries of the history of Chinese mathematics have been dominatedby the presence of the Nine Chapters. It was taken for granted that the Nine Chapters represented a naturalor at least unproblematic form for mathematical writing in ancient China. No other early examples of extensivemathematical writing were known, and it was claimed that the ninefold classi�cation of mathematics on which thebook is structured dated back to well before the imperial age. As a result, the early history of Chinese mathematics

1 In this paper I use “method” rather than “procedure” for shù , not because I think the latter rendering is wrong, but simply from a preferencefor using less technical terms unless the original text renders it impossible.


was in effect simply the history of the Nine Chapters. The Suàn shù shu liberates us from these assumptions,and opens the way for us to sketch a vision of the early imperial age as an epoch of mathematical change andcreativity, in which we do not simply see texts like the Nine Chapters as intellectual monuments, but can analyzethe processes which created them. In the light of the evidence provided by the Suàn shù shu I suggest that theNine Chapters is a more recent text than has commonly been supposed, and I identify a plausible set of historicaland intellectual circumstances that might have led to its creation.

2. Background and context

The Suàn shù shu was found in a tomb closed early in the second century B.C., and while some of the material inthis collection was no doubt composed considerably earlier, the consensus among all scholars who have studied theSuàn shù shu so far is that in broad terms it belongs in the cultural milieu of around 200 B.C. Unlike the Rhind/Ahmosepapyrus, it is probably not a copy of a much more ancient document. If we want to understand the nature and purposeof the Suàn shù shu, it will be helpful to say something about what kind of place China was at the time this collectionwas placed in the tomb.

China as a uni�ed and centrally governed imperial state is not very old compared to (say) the cultures of ancientMesopotamia. The �rst emperor who uni�ed the Chinese world under one centralized government took the throneof his short-lived Qín dynasty in 221 B.C., only 35 years before the Zhangjiashan tomb was closed. For the �rsttime much of what is now modern China was governed by a civil service responsible to one man at the center.For the preceding �ve centuries there had been no entity with a realistic claim to hegemony over the multitude ofaristocratically governed states into which China was divided. When we do go back far enough to �nd such a claimbeing maintained, the central power of the rulers of the Western Zhou kingdom (1046–771 B.C.) was nothing likeas strong and �ne-meshed as the imperial system which the Qín inaugurated. The Suàn shù shu is a product of thenew bureaucratic empire founded by Qín, a point that will be underlined when we consider its contents in detail.Nevertheless what was imagined to have been the Zhou system of government was still seen as a worthy model bysome writers in the early imperial age. As we shall see, some scholars of that period who discussed the origins ofmathematics believed that the mathematics of their own age could be traced back to Zhou models. This view willrequire careful evaluation in relation to the new evidence provided by the Suàn shù shu.

After the fall of Qín, the Hàn dynasty (206 B.C. to A.D. 221) ruled for more than four centuries (apart fromthe break under Wáng M�ang from A.D. 9–23), and its government was largely founded on a continuation of Qíninstitutions. Under the Hàn emperors a new and self-conscious social group took deep root and �ourished: this was thescholar-of�cial gentry who staffed the new empire. The rich grave-goods in their tombs are witness to the prosperitybrought to them by service to the emperor. For our present purposes, the most signi�cant point about Hàn funerarypractices is that for at least the �rst two centuries of the dynasty of�cials were sometimes buried with collections ofbooks to take into the next world [Tsien, 2004, 207–232]. Now before such collections began to be excavated in recentdecades, there were only two ways of getting a picture of the range of topics and formats of ancient Chinese literature.One was to rely on the surviving texts from a given epoch, transmitted to us initially through the process of scribalcopying, and then after about 1000 A.D. preserved through printing. The Nine Chapters is the oldest mathematicalliterature to reach us by that route, but our earliest surviving (and incomplete) copy of this book represents an early13th century reprint of an edition of 1084 [Chemla and Guo, 2004, 71–72]. The other method, less dependent on theaccidents of survival, was to look at the lists of titles in ancient bibliographies extant from the �rst century onward,but in many cases the books listed there are now utterly lost and we can only guess at their contents. Now, however,the texts excavated from Hàn tombs give us direct access to some of the actual writings that imperial scholar-of�cialsread 2000 years ago. The Suàn shù shu is one of those.

3. The physical and formal structure of the Suàn shù shu

3.1. Physical form, reconstitution, and transcription

The Suàn shù shu originally took a form common to many excavated Chinese writings from the late �rst millen-nium B.C., that of a roll of bamboo strips [Tsien, 2004, 96–125]. In the case of the Suàn shù shu each strip of bamboo


was about 30 cm long and 6 or 7 mm wide and bore a single column of characters written in ink so as to be read fromtop to bottom with the strip held vertically: see Fig. 1 for some examples. Each strip was cut from the original stalkso that there is a “node” or thickening marking the division between sections of growing bamboo about 15 mm belowthe top of the strip, and about 20 mm above the lower end. The main body of writing on any given strip lies betweenthe two nodes. To make the roll, the strips of bamboo were laid side by side, and strings were then knotted roundthe strips near the top and bottom to hold them in place so that they could be rolled up like a mat. When the bundlewas unrolled, the columns of characters on successive strips were read from top to bottom and right to left, as wasalso the case with all Chinese texts on silk or paper until the horizontal and left to right arrangement for writing wasadopted under Western in�uence in recent times. Since there are 190 strips in the reconstituted collection, allowing afew millimeters for the knotted cord between strips, it seems that the text would have been somewhat less than 2 mlong when fully unrolled.

When the Suàn shù shu was found, its strings had long since perished and its strips had spread out in disorderthrough the action of soil and water movement on the original bundle during two millennia. A drawing showing thewidely dispersed layout of the strips as found was prepared by the team who excavated them [Péng, 2001, fold-outfollowing p. 134]. The �rst scholars who studied the text could only attempt to restore the correct order of the stripsusing such clues as continuity of text and proximity when excavated [Péng, 2001, 1–4]. The dif�culty of this task maybe judged from the comparative table of excavation numbers of the strips (which roughly re�ect proximity as found)and the numbers used for the strips in the order of the published transcription, which re�ect the editors’ judgments ofthe original order of the strips in the bundle [Péng, 2001, 129–130]. While the dispersion of the original bundle of stripshad clearly not produced a completely random order on the ground, it was rare for the excavators to �nd a sequence ofstrips lying together in just the right order. Added to that is the fact that although most of the strips were quite legible anumber were damaged or partly obliterated. Transcribing and ordering the strips was therefore no easy task, althoughthat in itself seems insuf�cient to explain why it took 17 years before the �rst transcription of the text appeared inpublished form. It seems that several different arrangements of the strips were made before the version now beforeus was decided on [Péng, 2001, 133]. Slight differences between the successive transcriptions published show thatrevisions were still being made after the �rst version appeared. During the long period when the work of transcriptionwas in hand, only brief references to the contents of the text appeared in books and periodicals. The delay that hasalready occurred makes it all the more urgent to make this text accessible to historians of mathematics in general aswell as to specialists in Chinese mathematics.

The �rst full publication of the transcribed text of the Suàn shù shu was given in the archaeological peri-odical Wénwù 9 (2000) 78–84 under the title “Jianglíng Zhangjiashan Hànji�an ‘Suàn shù shu’ shìwén”

(English title given in original publication as “Transcription of bamboo Suàn shùshu or a book of arithmetic from Jianglíng”). This work [Jianglíng, 2000] was ascribed to the Study Group of theJianglíng Documents. The transcription from early Hàn script in this publication was made using modern simpli�edcharacters rather than the full form modern characters that would have more closely paralleled the ancient orthog-raphy, and there was no indication of where one strip ended and another began. Its value for study was thereforelimited.

A further publication appeared the following year [Péng, 2001]. This gave a transcription using full form moderncharacters, indicated the start and end of strips, and included editorial notes on the transcription and a commentary onthe mathematical content. Although Péng’s book included photographs of some of the strips, many were not included,which made it impossible to come to an independent judgment on the transcriptions of doubtful characters by theeditors. Later in the same year a full set of full-sized photographs of all the Zhangjiashan strips was �nally published[Zhangjiashan, 2001]. This included a transcription of the Suàn shù shu text with editorial notes, substantially identicalto the work of Péng Hào.

The edition used in the present study is based on a study of the published photographs of the strips [Cullen, 2004,113–145]. Comparison with the versions published in China will show that I have been rather less willing to suggestemendations to the text than others to date. In part this is perhaps linked to the fact that I have not felt justi�ed inassuming that where the text is irregular or elliptic, this is because a scribe has failed to reproduce fully a hypotheticalsource text that did not have such features. The Suàn shù shu is what it is: it does not deserve to be treated as animperfect version of some other shadowy document now lost.


Fig. 1. Some of the bamboo strips on which the Suàn shù shu was written. Counting from the right, the �rst strip shows the label Suàn shù shu, “Writings on Reckoning,” that described the contents of the original bundle. The second, third, fourth, and eighth strips show section

titles above the upper node of the bamboo, and the second and �fth strips have the names Wáng and Yáng below their lower nodes. Theninth strip has the words Yáng y�� chóu , “Checked by Yáng,” below the lower node. In the numbering system used for the translation in[Cullen, 2004], the strips shown here are numbered as 6 (reverse side shown here), 119, 148, 113, 102, 101, 134, 133, and 56. Reproduced withpermission from [Péng, 2001].


3.2. The formal structure of the Suàn shù shu

In analyzing the structure of the Suàn shù shu, I suggest that we may begin by looking at the most basic elementsof the text, and moving up from that level step by step to the overall structure of the entire collection.

3.2.1. The individual charactersThese are written fairly clearly throughout in a scribal style that is similar to that of many other excavated Hàn

documents. While the appearance of the characters used is super�cially quite different from modern standard script,it does not take very long for a reader familiar with classical Chinese to begin reading this material with relative ease.I cannot myself distinguish obvious changes of handwriting style from one part of the text to another. This may meanthat a single scribe copied all the text we see, or it may mean that a number of scribes working on the text had beentrained in ways that produced a fairly standardized style. It may also mean that I am not suf�ciently familiar withearly Hàn documents to detect differences that may be obvious to the more expert. It is signi�cant, however, thatthe group of Chinese scholars who examined the text after its excavation also concluded that the uniformity of thewriting suggested that it was all the work of a single copyist [Péng, 2001, 3]. A small number of characters found inthis text are unknown to dictionaries based on the printed literary tradition as it existed in late imperial times, suchas in strip 80 for what appears to be a type of fat (here as in [Cullen, 2004] I follow the strip numbering in thefull photographic reproduction [Zhangjiashan, 2001]). In a few other cases characters of standard form are used inways that later ceased to be common; for instance, is used frequently for shù “procedure, method” where latermathematical texts have . The text is not always consistent in the use of one form of character for one meaning.These variations are not necessarily a sign that different scribes have been at work, since in the example of strip 13 wehave zeng in the title, followed two characters later by zeng in the same sense of “increase.” An indication of theattitude of the scribe or scribes to the effort required to write down characters is shown by the frequent use of the dittomark = indicating that the preceding character is to be repeated. No consideration of elegance or reader-friendlinessseems to stand in the way of saving ink and reducing the labor of copying wherever possible.

3.2.2. The clause and the sentencePunctuation was not a major feature of pre-modern written Chinese, although it was not altogether absent [Guan,

2002]. On the whole punctuation was a task left to the reader, who marked red circles with the writing brush in orderto indicate pauses between sentences or clauses. Punctuating the text was thus part of the task of comprehending it.Accordingly, large parts of the Suàn shù shu have to be read using only the guidance to clause structure provided byobvious units of sense, by introductory formulae such as shù yue , “the method says,” and by familiar particlesin classical Chinese such as zh�e (which mainly functions to nominalize a preceding verbal expression) and y�e(which roughly speaking marks the end of an assertion). However, in parts of the Suàn shù shu one frequently �ndsthat the scribe has already provided the punctuation, in the form of the “hook-mark” �, sometimes written clearly asshown here, but sometimes little more than a slightly bent streak. The usage of this mark does not seem to follow anyobvious pattern. Sometimes it seems quite super�uous, as in several cases where it occurs only after y�e has alreadymade the clause division clear. Elsewhere its role in dividing groups of �gures from one another does sometimesmake the task of grasping where separate numbers begin and end a little easier. A less frequent punctuation mark isthe round blob •, which may perhaps mark a more signi�cant pause than the hook-mark. But for large portions of thetext punctuation does not occur at all.

3.2.3. The sectionIn most cases, any text forming part of the Suàn shù shu can be identi�ed as part of what I have called a “section,”

a self-suf�cient piece of writing possibly extending over several strips, beginning with a title written by the scribeabove the upper node of a strip. The end of a section may coincide with the end of a strip, and the scribe may squeezecharacters together to avoid running on to the next strip. Often, however, there is an obvious gap in the form of alength of blank bamboo between the end of the text and the lower node of the strip on which the section ends. Wemust not forget that such a section as we see it today in a published transcription has been reconstituted from scatteredstrips by the original Chinese editors. However, the continuity of text recording calculations or specifying methodsis often so strong from one strip to the next that one can feel considerable con�dence that the editors have made theright decision in ordering strips to reconstitute a section. In some cases the continuous text of a section can be seen


to consist of a number of subsections, which may be variants on the main theme apparently from the same source, ormay even say the same thing in language different enough to suggest different origins.

Some of the sections and subsections of the Suàn shù shu simply make statements, such as “1/4 times 1/4 is1/16,” or tell us what amount of a given commodity has the same exchange value as a given amount of some othercommodity. The majority, however, follow some variant of the following form, in which a problem is stated andsolved:

(a) a description of a situation, sometimes including the expressions j�n y �ou , “Now we have/there is/there are”(12 instances), or j�n yù , “Now it is desired to” (5 instances);

(b) a question about the situation, often in the form wèn X j�� hé X , “It is asked: how much is X?” (26 in-stances), or simply X j�� hé X , “How much is X?,” where X is the name of some quantity (14 instances);

(c) a statement of the result, sometimes simply introduced by yue , “It says” (27 occasions), but also by dé yue, “The result says” (13 instances), and once by qí dé yue , “Its result says”;

(d) a statement of the method used, introduced once simply by yue , “It says” (Section 53), but mostly as shù yue, “The method says” (47 instances), and sometimes qí shù yue , “Its method says” (6 occasions).

In the examples above I have translated yue literally as “says” for the sake of clarity. Yue is, however, a ratherweak word that often functions in classical Chinese as no more than a divider between one block of direct speechand the next. In translating the above expressions as part of continuous prose it seems to me that it is usually betterrendered as a simple colon “:” so that (e.g.) shù yue is rendered as “Method:”.

The Suàn shù shu as a whole shows no consistent usage of the expressions listed here, and indeed it is hard tosee any consistent pattern even within individual sections, which were presumably likely to have been copied from asingle source. This looseness is a striking contrast to the Nine Chapters, in which the format of problems, answers,and stated methods is, though �exible, generally quite uniform.

In a few cases the section occupies a single strip only. In the transcription below the �rst three characters are a title,written above the upper node on the original strip. To represent this separation they are placed here on a separate line.The rest of the text begins below the node, and the presence of a long gap between the end of the text and the end ofthe strip suggests strongly that what we have here is a complete unit of writing.

Strip 13

Increasing or decreasing parts [i.e., fractions]When increasing a part one increases its numerator [lit. “child”]; when decreasing a part one increases its denominator

[lit. “mother”].

More frequently the section continues over more than one strip:

Strip 34

Strip 35

(6 character gap toend of strip)

(12) The fox goes through a customs-postA fox, a wildcat, and a dog go through a customs-post; they are taxed 111 cash. The dog says to the wildcat, and the

wildcat says to the fox, “Your skin is worth twice mine; you should pay twice as much tax!” Question: how much is paidout in each case? Result: the dog pays out 15 cash and 6/7 cash; the wildcat pays out 31 cash and 5 parts; the fox pays


out 63 cash and 3 parts. Method: let them be double one another, and combine them [into] 7 to make the divisor; multiplyeach by the tax to make the dividends; obtain one for [each time] the dividend accommodates the divisor.

Where a section continues from one strip to the next, the transition is evidently of no importance to the scribe, whowill happily split a clause between strips. In one case (strips 1 and 2), the scribe actually places a “punctuation hook”at the top of one strip, although it follows a clause that ends at the bottom of the preceding strip. As already noted,there may also be signi�cant divisions within a section, as in the following example. The letters (a), (b), and (c) havebeen inserted in the Chinese text to mark the divisions I have made in my translation.

Strip 17

Strip 18

(long gap to end of strip)

(7) Simplifying parts(a) The method for simplifying parts: Take the numerator from the denominator; in turn take the denominator from the

numerator. When the numbers [on the sides of] the numerator and denominator are equal to one another, then you can goon to simplify.

(b) Further, the method for simplifying parts: What can be halved, halve it. Where one [can be counted for eachmultiple] of some amount, [count] one for [each multiple of] that amount.

(c) One method: Take the numerator of the part from the denominator. [If that is] the lesser take the denominator fromthe numerator. When [the numbers on the sides of] the numerator and denominator are equal, take that [number] as thedivisor. For numerator and denominator complete one for [each time] they accommodate the divisor.

The title “Simplifying parts [i.e., fractions]” appears above the node of strip 17, and is clearly an overall titlefor this section. The long gap at the end of strip 18 suggests that at this point the scribe felt that he had �nisheda unit of writing, although the original editors place after this two further strips with related fragments of material.Subsection (a) gives a method for �nding the highest common factor of two numbers (in this case the numerator anddenominator of a fraction to be simpli�ed) similar to that found in Euclid, Book 7, Proposition 2. Subsection (b) thengives another method for simpli�cation, based on simply dividing numerator and denominator by any factors one can�nd. Subsection (c) �nally gives a differently phrased version of the rule already given in (a), evidently taken froma different source. This eclectic assembling of variant statements of related methods is a frequent feature of the text.The fact that this was thought to be worthwhile is signi�cant (though somewhat cryptic) evidence as to what the aimsof early Chinese mathematical writing may have been. I believe that the hypothesis I introduce later in this article goessome way toward explaining such features of the text.

Two different persons, with the common surnames Wáng and Yáng , appear to be associated with the text,since one or other of their names are written below the lower nodes of 14 strips. Wáng’s name appears on strips 42(Section 15 using the numbering of sections in my translation [Cullen, 2004]), 88 (Section 36a), and 119 (Section 47a);in the case of strip 42 the name occurs at the end of a well-de�ned section in the form of the announcement Wángy�� chóu , “Checked by Wáng.” In the other two instances the name alone appears. As for Yáng, his name isfound at the end of strips 1 and 3 (Section 1), 56 (Section 21), 98 (Section 40a), 101 (Section 40b), 105 (Section 41a),107 (Section 41b), 109 (Section 42a), 111 (Section 42b), 121 (Section 47b), and 123 (Section 47c). Strip 56 endsSection 21, which is on a theme (rates of weaving in geometrical progression) similar to Section 15. We read hereYáng y�� chóu , “Checked by Yáng,” a clear parallel to the earlier Wáng y�� chóu . In this case the exampleis incorrectly worked. Apart from Sections 15 and 21, the rule seems to be that the name is given on the �rst stripof a section of material associated with the person in question. It is only in Section 47, which divides naturally intosubsections, that both names occur. The names Wáng and Yáng are never found on the same strip, nor do they everoccur on consecutive strips that are linked by continuous text. Of course such named strips are a minority: 14 bearnames, but that leaves 176 out of 190 unaccounted for. Sections bearing the names of Wáng and Yáng seem to differ


from one another in their names for grains, and also in their terminology for fractions (see below). Who this Wángand Yáng were or what role they played is not clear. If the text as we have it today really is the work of a single hand,they would seem to be more than mere copyists labeling strips on which they had worked: in that case why would thelast copyist bother to preserve their names, and why only label 14 strips out of 190? And if they were no more thanscribes, it is hard to see why their names should be associated in a distinctive way (“checked by . . . ”) with the twoweaving problems. On the whole the most probable solution seems to be that Wáng and Yáng were in some sensemathematical teachers and (to a limited extent at least) authors of mathematical problems, and that material by themhas been copied into the present collection.

3.2.4. Sequence and grouping of sectionsEven if we do not always feel certain that every section is correctly and fully reconstituted from the scattered strips,

the existence of scribal section titles makes it plain that the division of the Suàn shù shu into sections does actuallyrepresent an original feature of the collection as deposited in the tomb. The same cannot, however, be said of the orderof sections within the transcription. Looking again at the table of excavation numbers and transcription numbers forslips, together with the diagram showing the strips as found, we can see straight away that the order in which thesections are placed in the transcription does not relate in any obvious way to the positions and sequence of the stripsas excavated. Thus, for instance, the fact that the sections dealing with elementary operations of arithmetic are placed�rst in the transcription is largely a matter of an editorial decision that the simplest topics come �rst, and that similartopics belong together—although there is the additional factor that what is clearly the title of the whole collection(or perhaps we should say its label: see Appendix) is found on the back of strip 6. This strip must therefore havecome early enough in the series for it to be visible when the bundle was wrapped up with the text facing inward andthe beginning of the roll on the outside of the bundle. We can therefore be sure that the topic covered on strip 6—the multiplication of fractions—was placed quite early in the collection, thus con�rming to some extent the notionthat elementary topics preceded the rest.

Under these circumstances, however, we cannot deduce much about the order of the sections in the original textfrom the order in which the editors have chosen to place them here. When therefore in my translation I propose the 14rough groupings of sections under broad topic headings listed below, I merely claim that these groupings give a usefulidea of the topics that are covered within the collection as a whole. I certainly cannot prove that the relevant sectionswere so grouped in the undamaged roll of strips, or that even if they were so grouped the groups stood in the orderin which they are currently found. On the other hand it does seem unlikely that a collection of mathematical materialwould have been compiled in a completely random order, and as we have already seen, material on similar topics wasdeliberately grouped together within sections. It seems likely, therefore, that sections bearing related material wouldalso have been grouped with one another in the collection as a whole.

However, even on the most favorable rearrangement of the sections, it is clear that the Suàn shù shu can never havebeen a systematic and ordered treatise. It is, even at the level of the individual section, a patchwork compilation ofmaterial from different sources rather than something cut from whole cloth by an individual with a didactic plan inmind.

4. Mathematical content of the Suàn shù shu

4.1. Mathematical language

The ways in which the Suàn shù shu expresses numbers and speci�es mathematical processes are not markedlydifferent from what we know of early Chinese mathematics through the received tradition. One striking feature of thecollection is, however, its lack of standardization of mathematical vocabulary and syntax compared with later texts. Totake one obvious example, the concluding phrase of a problem frequently takes the form of an instruction to performan operation in which we �nd the number of times one number (the f �a , a word I render in this context as “divisor”)is contained in (literally “like” or “accommodated by,” rú ) another. The following variants may easily be noted inthe opening sections alone. I translate as literally as possible for the sake of clarity; the small roman letters after eachsection number designate subsections.


Section 7 (c) (strip 18): , “Complete one as it accommodates the divisor.”Section 8 (b) (strip 22): , “Accommodating the divisor, complete one.”Section 8 (e) (strip 25): , “One as it accommodates the divisor,” in a different statement of the rule already

stated in Section 8 (b).Section 11 (strip 33): , “Accommodating the divisor, obtain one cash.”Section 12 (strip 35): , “Accommodating the divisor, obtain one.”

More illuminating, because more systematic, is the difference in the way that fractions are expressed in the variousparts of the text. Sections that contain fractions, use one of two basic formats. Taking 1/4 as an example, we have

(i) , “four parts, one” or , “four parts [of an] X, one,” where X is a unit of measurement;(ii) , “one of four parts” or , “one of four parts [of an] X,” where X is a unit of measurement.

There are 34 sections in which fractions consistently follow the �rst of these formats, but only nine (Section 7(strip 20), Section 9 (strip 26), Section 21 (strip 55), Section 40 (strips 98–104), Section 41 (strips 105–108), Sec-tion 49 (strip 130), Section 57 (strip 146), Section 66 (strip 162), and Section 67 (strips 168–182)) follow the secondpattern. In general, a section that uses one of these patterns uses it consistently. In the two instances where a singlesection contains examples of both, this happens in different subsections. While Section 33 (a) and (b) use the �rstform, the second appears only in (c), which seems to be a later note added to the end of a completed text. In Sec-tion 47, (a) (strip 119) uses the �rst form and (b) (strip 121) uses the second, while (c) has no fractions. However,there seems to a be good case for treating (b) as the start of a new section [Cullen, 2004, 76].

One of the most interesting aspects of this difference of mathematical vocabulary takes us back to the names ofWáng and Yáng. It is striking that of the sections using the second form, none of them bears Wáng’s name and threeof them (Sections 21, 40, 41) bear Yáng’s. If we turn to Section 47, which has both usages, we �nd that (b) with thesecond form bears Yáng’s name, while (a) with the �rst form bears Wáng’s. We cannot simply say that Yáng neveruses the more common form, since his name appears twice in Section 1 (strips 1, 3), where the �rst form is used.But we can at least say that Yáng’s sources included examples of both forms, whereas Wáng’s did not. And we maynote that Yáng prefers the second fractional form for his (incorrect) weaving problem in Section 21, while Wáng usesthe �rst form for his (correct) weaving problem in Section 15. All this certainly strengthens the impression that thematerial in the collection comes from several different hands, even if a single scribe wrote out the material as we haveit today.

4.2. Topics and techniques

In the listing below the mathematical techniques used in the Suàn shù shu are summarized in relation to the fourteengroupings of sections that I have proposed. It will be evident that in a few cases I have changed the published order ofstrips slightly.

Group 1: Elementary operation (Sections 1–8, strips 1–25)Multiplying whole numbers and fractions; simplifying fractions; adding fractions.

Group 2: Sharing in proportion; progressions (Sections 9–17, strips 26–47)Division of a mixed number by a whole number; subtraction of a fraction from a mixed number, and of one fraction

from another; sharing out a common pool of pro�t or liability in proportion to the amount contributed by a numberof persons; the case of contributions in geometrical progression; �nding the original amount that has been repeatedlydiminished in a given proportion to produce a given result; value of a given amount of some commodity, given a unitprice.

Group 3: Wastage (Sections 18–19, strips 48–51)Finding the amount to be allowed to obtain a given amount of product in a process involving a �xed proportion

that is wasted; amount wasted out of a given initial amount.

Group 4: Sharing, contributions, and pricing (Sections 20–26, strips 52–67)


Sharing; reaching a total amount through contributions at different rates; �nding the cost of some quantity of acommodity from the price of a given amount; �nding interest payable on a loan for a given time on the basis of agiven monthly rate.

Group 5: Changes in rates (Sections 27–29, strips 68–73)Correcting tax payable when there has been an error in the tax rate; change in the amount of product when the

amount of raw material changes; change in amount of tax when there is a change in the taxable amount.

Group 6: Rating by unit (Sections 30–33, strips 74–82)Calculation of price of a unit amount from price of a given amount; �nding amounts of ingredients in a mixture of

a given total amount.

Group 7: Wastage and equivalents (Sections 34–37, strips 83–92)Allowance for the amount wasted in drying a commodity; exchange of one commodity for another.

Group 8: Allowing for mistakes (Sections 38–39, strips 93–97)Dealing with use of an incorrect tax rate by changing the nominal area of the �eld to be taxed.

Group 9: Converting grains (Sections 40–47, strips 98–125)The use of standard ratios to calculate the amount of one type of grain equivalent to another type, or the amount of

product when grain is processed; problems of sharing and mixing involving grains.

Group 10: Rationalizing and checking tasks (Sections 48–51, strips 126–132)Calculation of a rate of unit production from the rate at which parts of the production task are completed; �nding

expected amount of processed silk produced from raw silk; checking the time taken for a journey given initial and�nal sexagenary day numbers.

Group 11: Rule of false position (Sections 52–54, strips 133–140 and 185–186)Use of the Rule of False Position to solve problems of sharing and mixtures, and the extraction of an approximate

square root.

Group 12: Shapes and volumes (Sections 55–61, strips 141–152)Calculation of the volume of various three-dimensional shapes.

Group 13: Circle and square (Sections 62–64, strips 153–158)Relative dimensions of a square and its inscribed circle.

Group 14: Sides and areas with mixed numbers (Sections 65–69, strips 159–184 and 187–190)Calculation of the unknown side of a rectangle, given its area and one side; divisions involving the sum of several

different unit fractions; multiplication of mixed numbers; interconversion of area units.

Such a listing can of course give only a rough idea of the contents of the Suàn shù shu. Let us now look at a fewtranslated samples in order to illustrate the way in which the mathematics found in this collection actually works.Fuller discussions of all these examples will be found in my published translation [Cullen, 2004].

First, here is an example from Group 1 that illustrates the most elementary type of material to be found in thetext—a simple listing of mathematical facts:

Strip 1 (has name “Yáng” at bottom), Strip 2 (has long blank gap between end of text and end of strip)

(1) Multiplying together(a) a cùn multiplying a cùn is a cùn; multiplying a chí , [it] is one tenth of a chí; multiplying ten chí, [it] is one

chí; multiplying a hundred chí, [it] is ten chí; multiplying a thousand chí, [it] is a hundred chí;(b) a half cùn multiplying a chí is one twentieth of a chí; one third of a cùn multiplying one chí is one thirtieth of a

chí; one eighth of a cùn multiplying one chí is one eightieth of a chí.


The labeling convention that I use here encodes the fact that the material translated runs continuously from stripnumber 1 to strip number 2. The name Yáng appears below the lower node of the �rst strip, and there is a long gapafter the end of the material on strip 2, suggesting that what we have here is a complete unit of text. The title translatedhere as “Multiplying together” is written above the upper node of strip 1. Like the section numbers, the lower caseletters identifying subsections are my own, and are not paralleled in the original text. The cùn and chí are units oflength corresponding roughly to the inch and foot of British measures, respectively, although there are 10 cùn to 1 chí.However, from the material above it is clear that the same names are also given to the area measures that would bedescribed in English as a “square cùn” or a “square chí.”

From Group 2, we may choose the following problem:

Strips 40, 41, 42 (�ve-character gap after end of text; “checked by Wáng” at bottom)

(15) The woman weavingIn a neighboring village there is a woman good at weaving, who doubles her [production each] day. In weaving, [she]

says: “By doubling up on myself for �ve days I [had] woven �ve chí.” Question: on the day she began weaving and thesubsequent ones, how much [was produced] in each case? Reply: At the start she wove 1 cùn and 38/62 cùn; next 3 cùnand 14/62 cùn; next 6 cùn and 28/62 cùn; next 1 chí 2 cùn and 56/62 cùn; next 2 chí 5 cùn and 50/62 cùn. Method: setout 2; set out 4; set out 8; set out 16; set out 32; combine them to make the divisor; multiply them by 5 chí on one side,each to make its own dividend; obtain a chí for [each time] the dividend accommodates the divisor; what does not �ll achí, 10-fold it; [count] 1 cùn for [each time the result] accommodates the divisor; for what does not �ll a cùn, designatethe part by the divisor.

This is one of two “weaving rates” problems in this collection. This one is linked with the name of Wáng by thecharacters Wáng y�� chòu , “checked by Wáng,” below the lower node of the third strip, while Section 21 has asimilar problem which mentions the name Yáng in the same phrasing used here. The solution starts from the notionthat the production of successive days can be said (in modern terms) to be in the ratio 2 : 4 : 8 : 16 : 32. The presenttext contains no name for such a sequence of proportions, but the term used in the third section of the Nine Chapters(titled Cu� fen , “Differential allocation”) is liè cu� , “separate differentials” [Chemla and Guo, 2004, 288–289; Shen, 1999, 162–163]. The total of 5 chí is thus divided into (2 + 4 + 8 + 16 + 32) = 62 parts, and these areallocated to days in accordance with the ratios listed. It is not clear why the ratios start from 2 rather than 1, nor whythe fractions in the result are not reduced to their lowest terms.

The similar problem associated with the name Yáng is as follows:

Strips 54, 55, 56 (�ve character gap after end of text; “checked by Yáng” at bottom)

(21) Women weavingThere are 3 women; the eldest one weaves 50 chí in 1 day; the middle one weaves 50 chí in 2 days; the youngest

one weaves 50 chí in 3 days. Now their weaving produces 50 chí. Question: how many chí does each deliver? The result:The eldest delivers 25 chí; the middle one delivers 16 chí and 12/18 chí; the youngest delivers 8 chí and 6/18 chí. Method:set out 1; set out 2; set out 3; then let each [contribute] as many to make the divisor. Then 10- and 5-fold them to make thedividends; one chí results from [each time the dividends] accommodate the divisor; for what does not �ll a chí, designatethe parts according to the divisor. 3 is the dividend for the eldest one; 2 is [the dividend] for the middle one; 1 is [thedividend] for the youngest one.

It seems possible that Yáng may have set out to construct a distinctive weaving problem of his own involving aseries of numbers based on rates of work, leading to a division of a total length of cloth, 50 chí in both cases. Thereis, however, a �aw in the reasoning which has led to the wrong answer. The Nine Chapters [ Chemla and Guo, 2004,542–543; Shen, 1999, 343–345] has a problem involving the same basic principle of several sources contributing to aknown total at different rates, in which a pool is �lled by �ve separate streams, and in each case we are told how manydays each would take to �ll the pool on its own. The correct step is then to take the reciprocals of these rates to �ndhow many times each day the pool would be �lled by each stream alone. These are then amalgamated and divided intoone day to �nd how long all the streams together take to �ll the pool. Clearly Yáng’s calculation would have workedin the same way if he had taken the number of times each woman wove 50 chí in one day as his basic rates, rather


than the days each took to produce 50 chí. Thus, for all the women weaving together to produce 50 chí would take

1 day/(1 + 1/2 + 1/3) = 6/11 day.

From this it follows easily that the actual productions of each of the three women during this time will be

50 × 6/11 chí = 27 3/11 chí,

50/2 × 6/11 chí = 13 7/11 chí, and

50/3 × 6/11 chí = 9 1/11 chí.

It seems that the author’s check of his answer must have been limited to seeing whether the total was 50 chí—whichit certainly is. He cannot have asked himself whether the ratios (or perhaps we should say liè cu�, “separate differen-tials”) of the contributions made sense, since otherwise he would have been warned by the fact that the production ofthe middle woman is not half of the production of the eldest, as it should have been.

Most of the problems in Groups 2 to 8 involve nothing more complicated than the elementary arithmetic used here.The following example from Group 5 is interesting not so much because of the mathematics used but because of thecontext in which the calculation is set:

Strip 70 (four character gap after end of text)

(26) Norms for bamboo(a) The norm: A bamboo of 8 cùn [circumference] makes 183 strips of 3 chí. Now strips are made from a 9 cùn bamboo.

How many strips should there be? Reply: it makes 205 strips and 7/8 of a strip. Method: Take 8 cùn as the divisor.

The strips referred to here are of course the type of writing material on which the text we are studying was written,although the length of 3 chí does not correspond to one of the standard lengths for of�cial documents [ Tsien, 2004,115–118]. The heading of this section contains the �rst occurrence so far in this text of the term chéng , familiarfrom Qín administrative documents with the sense of “regulation,” “standard,” or (as suggested here) “norm” as astandard for productivity [Hulsewé, 1985, 61]. Thus we �nd three items in the Shuìh �udì regulations labeled gongrén chéng , “chéng for workmen,” detailing standards for production by workers [Shuìh�udì, 2001, 45–46]. Aneditors’ note to Shuìh�udì strip 108 interprets chéng in this context as the amount of work done per day. This word isfound 20 times in the present text, usually in circumstances suggesting that the writer is quoting some of�cial standard.It is therefore �tting that the �rst instance of its use should refer to the production of the standard of�ce stationery ofthe early imperial bureaucrat.

In mathematical terms, this problem simply looks at the consequences of changing the circumference of the bambooto be used. The width of strips and the length of bamboo required do not enter into the calculation. Unusually, the“method” section gives only a brief gesture toward the calculation that is to be made, which is of course

9 × 183/8 = 1647/8 = 205 7/8.

Group 9 is mainly concerned with the subject of the exchange of quantities of different types of grain. Naturallywe �nd some straightforward listings of equivalent values, as in

Strip 98 (“Yáng” at end), Strips 99, 100 (long gap after end of text)

(40) Bài and hu��(a) Hulled grain, a diminished half sheng , makes bài [hulled grain], 3/10 sheng; 9-fold it and take 1 for 10. Hulled

grain, a diminished half sheng, makes hu�� hulled grain 4/15 sheng. 8-fold it and take 1 for 10. Hulled grain, a diminishedhalf sheng, makes wheat, half a sheng. 3-fold it and take 1 for 2.

(b) Wheat, a diminished half sheng, makes unhulled grain, 10/27 sheng. 9-fold the denominator and 10-fold thenumerator, [that is,] 10-fold it and take 1 for 9.

Wheat, a diminished half sheng, makes hulled grain, 2/9 sheng. 3-fold the denominator and double the numerator,[that is,] 2-fold it and take 1 for 3.


Wheat, a diminished half sheng, makes bài [hulled grain], 1/5 sheng. 15-fold the denominator and 9-fold the numera-tor, [that is,] 9-fold it and take 1 for 15.

Wheat, a diminished half sheng, makes hu�� [hulled grain], 8/45 sheng. 15-fold the denominator and 8-fold the numer-ator.

The “hulled [= decorticated] grain” m�� referred to here is almost certainly millet, which was a common foodgrain in ancient northern China. The volume measure sheng in ancient China was about 1/5 litre, which is aboutthe amount of grain one needs to cook a meal for two persons. The expression “diminished half,” sh�ao bàn ,simply means one-third; elsewhere in the text we also �nd “augmented half,” dà bàn , meaning two thirds. Sothe �rst sentence of the passage given here is simply saying that 1 /3 sheng of hulled millet is the same value as 3/10sheng of bài grain, which is a slightly more highly milled form of millet. And clearly 1/3 × 9/10 = 3/10, as statedhere. The form described as hu�� takes the milling a step further. Once again, the material presented here does notseem homogeneous.

Much of Group 9 is taken up with lists of conversion ratios of the type sampled here. One problem in this groupruns as follows:

Strips 117, 118 (nine character gap after end of text)

(46) Hulled and unhulled grain combinedThere is 1 shí of hulled grain and 1 shí of unhulled grain. They are combined. Question: [the owners of] hulled and

unhulled should each take how much? Reply: the owner of the hulled grain takes 1 shí 2 d�ou 8/16 d�ou; the owner of theunhulled grain takes 7 d�ou 8/16 d�ou. Method; Set out hulled grain 10 d�ou and 6 d�ou, and combine to make the divisor;separately multiply what has been set out by 2 shí so that each makes a dividend by itself. The 6 d�ou is the number of theunhulled grain in hulled [terms].

Measures of grain volume are related as follows:

1 shí = 10 d�ou , and 1 d�ou = 10 sheng (about 2 litre).

Elsewhere we are told that unhulled grain produces 3/5 the volume of hulled grain, so that 1 shí or 10 d�ou of unhulledgrain produces 6 d�ou of hulled grain (as indeed the last sentence notes). Clearly then, the whole 2 shí of mixed grainis worth 16 shí of hulled grain, to which the contributors are entitled to shares in the proportions of 10 to 6. The 2 shíis thus divided by 16 and multiplied separately by 10 and 6 to get the shares required.

Group 10 is mainly concerned with problems involving the management of work, of which the following is atypical example:

Strip 131 (10 character gap after end of text)

(50) Feathering arrowsThe norm: 1 man in 1 day makes 30 arrows; he feathers 20 arrows. Now it is desired to instruct the same man to make

arrows and feather them. In one day, how many does he make? Reply: he makes 12. Method: Combine the arrows andfeathering to make the divisor; take the arrows and the feathering multiplied together to make the dividend.

The calculation speci�ed is (30 × 20)/(30 + 20) = 12. It is interesting to think how this result could have beenarrived at without using modern algebra. I suggest that the multiplication 30×20 indicates that we begin by imaginingthe man being told to make 30 × 20 feathered arrows. How long will he take to do this? Clearly, since he can make 30arrows a day he will take 20 days to make the arrows, and since he can feather 20 arrows a day he will take 30 daysto feather these arrows. Hence the daily rate of production of the �nished product will indeed be (30 × 20)/(30 + 20)

as speci�ed.In Group 11 we �nd four problems all using the method of Yíng bù zú , “excess and de�cit,” which

is equivalent to the method known much later in the West as the “Rule of False Position.” The last of these isparticularly interesting, since the method is there applied to a problem which involves nonlinear functions of theunknown:


Strips 185, 186 (long gap after end of text)

(54) Squaring a �eldThere is a �eld of one m�u: how many bù is it square? Reply: it is square 15 bù and 15/31 bù. Method: If it is square

15 bù it is in de�cit by 15 bù; if it is square 16 bù there is a remainder of 16 bù. Reply: combine the excess and the de�citto make the divisor. Let the numerator of the de�cit multiply the denominator of the excess, and the numerator of theexcess multiply the denominator of the de�cit; combine to make the dividend. Reverse this like the method for revealingthe width.

The m�u (approximately 460 m2 or 0.11 English acres) is a measure of land area, while the “double pace” bù(approximately 1.4 m) is commonly used for the linear measurements of �elds, etc. It is clear that in some contextsthe “bù” referred to is a square bù as was the case with the cùn and the chí; there are 240 square bù in a m�u. It isintriguing to �nd the “false position” method used here to �nd a good approximation to a square root, thereby perhapssuggesting the possibility that at the time of writing the algorithm for �nding square roots had not yet been discovered.The algorithm for square roots is given explicitly in the fourth section of the Nine Chapters [Chemla and Guo, 2004,362–367; Shen, 1999, 204]. The main calculation is

length of side (“numerator”) 15 16

excess/de�cit (“denominator”) 15 (de�cit) 16 (excess)

(15 × 16 + 16 × 15)/(15 + 16) = 15 15/31

Of course the “false position” method only works exactly in linear problems, but in fact

(15 + 15/31)2 = 239 + 761/931,

which is quite close to the 240 we are aiming for.Group 12 consists of six problems on the volumes of a number of three-dimensional forms. These include an

excavation in the form of a wedge, a wall in the form of a triangular prism, a rectangular hopper in the form of aninverted frustum of a hipped roof, a pile of grain in conical form, an earthwork in the form of a frustum of a cone,and a piece of timber or a pit in the form of a cylinder. All the methods stated are accurate, apart from the use ofthe usual approximation π = 3. For all but one of the sections in this group, the language used is consistent enoughto suggest that the problems come from a common source. In the case of the cone, however, we have two differentsections dealing with the same material, the second of which differs in language from the remainder of the group:

Strips 146, 147 (long gap)

(58) A whirl of grainA whirl of hulled grain has a height of 5 chí; its lower circumference is 3 zhang; the volume is 125 chí. [There is]

1 shí for 2 chí 7 cùn. [So] it makes 46 shí 8/27 shí of hulled grain. The method for this: Let the lower circumference bemultiplied by itself; multiply it by the height; make 1 [from] 36. The greater volume is 4500 chí.

Strip 148

(59) A granary coverA granary cover has a lower circumference of 6 zhang; its height is 2 zhang; this makes a volume in chí of 2000

chí. Method for multiplying: set out [a number] identical to the circumference; let them be multiplied together. Furthermultiply it by the height; let 36 complete 1.

The length unit zhang met here for the �rst time is equivalent to 10 chí. In section 58 we are also told how tocalculate the amount of grain in a given volume. But the real interest of that section is that the thinking behind thealgorithm used in the Suàn shù shu is revealed in the reference to the “greater volume.” This quantity is the volumeof a cuboid of the same height as the cone, but with a side equal in length to the circumference. As Liú Hu�, thethird century A.D. commentator on the Nine Chapters, points out [Chemla and Guo, 2004, 429; Shen, 1999, 267], thefactor of 1/36 follows from the fact that the volume of a pyramid on that square base would be 1/3 of the volume


of the cuboid, and the area of the circular base of the cone is taken as 1/12 of the area of the base of the cuboid.Clearly all this was known at least �ve centuries earlier. It is noteworthy too that in its method for �nding the volumeof the rectangular hopper (Section 57), the Suàn shù shu problem clearly bases itself on the system of dissectingcomplex forms into elementary units such as cubes, wedges and pyramids which also underpins the methodologyof the Nine Chapters in such cases [Chemla and Guo, 2004, 391–405; Shen, 1999, 251–306]. Interestingly the waythe algorithm is presented suggests strongly that the writer of the Suàn shù shu “hopper” problem was thinking of adifferent dissection from that used in the Nine Chapters [Cullen, 2004, 93–99].

After a pair of problems dealing with the cutting of a square timber from a round log and vice versa, the lastsubstantial part of the text as edited today is Group 14, which is mainly concerned with problems involving the �eldswhere the area is given as 1 m�u and one side is given whose length has a fractional part. The problem is then to �ndthe other side. The core of this group is a sequence of 15 strips in which we begin with a side of length 1 + 1/2 bù,then consider in turn sides of 1 + 1/2 + 1/3 bù, 1 + 1/2 + 1/3 + 1/4 bù, and so on up to a series ending in 1/10.It would be an anachronism to treat this material as showing an interest in summing harmonic series. From the waythe problems are posed and solved is it clear that the point is to �nd a fraction by which one can multiply a group offractions with different denominators to produce a whole number for convenience of calculation. We may illustratethis by the last problem in the series:

Strips 179, 180, 181 (fragmentary)

If in the lowest [place] there is 1/10; take 1 as 2520; a half as 1260; 1/3 as 840; 1/4 as 630; 1/5 as 504; 1/6 as 420;1/7 as 360; 1/8 as 315; 1/9 as 280; 1/10 as 252; joining them: 7381 to be the divisor; one obtains a length of 81 bù and6939/7381 bù; multiply it to form a �eld of 1 m�u.

Now there are 240 square bù in a m�u. So if the known side is of length 1 + 1/2 + 1/3 + 1/4 + 1/5 + 1/6 + 1/7 +1/8+1/9+1/10 (which is evidently what is meant by saying there is 1/10 “in the lowest place”) we face the division

240/(1 + 1/2 + 1/3 + 1/4 + 1/5 + 1/6 + 1/7 + 1/8 + 1/9 + 1/10).

As the text indicates, multiplying the divisor by 2520 and summing produces a total of 7381. Clearly we will alsoproceed to “take 1 as 2520” in the case of the dividend, which becomes 240 × 2520 = 604,800. Finally we calculate604,800/7381 = 81 and 6939/7381 as stated.

Before concluding this outline of the mathematical content of the Suàn shù shu, it is worth making a �nal point.While the text tells us some very elementary mathematical facts (such as, for example, that 1 times 10 is 10, 10 times amyriad is 10 myriad, and so on [Cullen, 2004, 37]), it never actually tells us how to carry out the four basic operationsof arithmetic—addition, subtraction, multiplication, and division. Nor is there any direct reference to a physical meansfor carrying out calculations. We can be sure that the reader was not supposed to be able to do sums in his head,since in Hàn times the ability to carry out mental calculations, x�n jì , was considered a striking enough feat toqualify one for special government employment—this is what happened in the case of the 13-year-old Sang Hóngyáng

(c. 140–80 B.C.), who was later to play a decisive and controversial role in the shaping of the imperialeconomy [Hàn shu , 24b, 1164; Twitchett and Loewe, 1986, 163 and 602–607]. All historians do, however,agree that in the early imperial age and for many centuries afterward computations would have been carried outusing small wooden, metal, or ivory counting-rods (variously named chóu , cè , suàn , etc.) arranged on ahorizontal surface to represent decimal digits, and manipulated according to rules to represent the new numbersgenerated as calculation proceeds. Descriptions of the way this was done are available from the �rst few centuries ofthe Christian era [Chemla and Guo, 2004, 15–20; Lam and Ang, 1992, 20–73], but the details need not concern ushere.

5. The origins of the Suàn shù shu

The essential features of the form and content of the Suàn shù shu have now been outlined. In addition, as wehave seen, the known date of closure of the tomb in the early second century B.C. gives us a �rm chronologicalcontext for this document. But the task of the historian of mathematics is not ended when a text has been translated,and a date assigned to it. As Eleanor Robson has pointed out in the case of the famous Babylonian tablet Plimpton


322 it is not the case that “[like] the scenarios of detective �ction, pieces of mathematics are self-contained worlds,whose mysteries can be solved by close analysis of nothing but themselves.” Rather “[mathematics] is, and always hasbeen, written by real people within particular mathematical cultures which are themselves the products of the societyin which those writers of mathematics live” [Robson, 2001, 168]. In a discussion of ancient Egyptian mathematics,Annette Imhausen has concluded “Traditional approaches to Egyptian mathematics have provided only a super�cialaccount of mathematical practices, and almost no information about the role of mathematics within Egyptian culture”[Imhausen, 2003, 367]. To understand the Suàn shù shu at more than a super�cial level, we need to situate it withinthe mathematical culture of early imperial China. As I hope to show, the attempt to �t the Suàn shù shu into what wealready know about that culture poses problems that cannot be solved unless we widen our perspective to include thewhole question of how technical knowledge was transmitted at the time this collection was entombed.

I shall begin by reviewing the parts of the “received tradition” of Chinese mathematical writing relevant to the agewhich concerns us here. Then I shall go on to say a little about the few people known to us who were renowned formathematical skills in the early imperial age. That will lead us back to the received textual tradition, and at that pointwe shall be in a position to evaluate the earliest attempt to write a history of that tradition, which dates from the thirdcentury A.D. I shall then turn back to the Suàn shù shu and ask how its existence changes our view of that tradition,and how the Suàn shù shu can be understood in a wider cultural and historical context. It would be useful at this pointto be able to refer the reader to a wider synthetic discussion of the role and content of mathematical activity in earlyimperial China, but unfortunately no one has yet attempted successfully to write generally on this topic in English. Itwould be an invidious task to list books that might have been expected to succeed in this aim, but do not. Commonlyread texts such as [Martzloff, 1997; Needham and Wang, 1959] fall into this category. There is, however, a succinctand helpful introduction to ancient Chinese mathematics in French, translated from the writing of one of the 20thcentury’s greatest historians of the exact sciences in East Asia [Yabuuti, 2000, 1–41].

5.1. The received tradition of ancient Chinese mathematics

According to traditional thinking on such matters, the oldest Chinese mathematical works known to us througha continuous literary transmission are the “The gnomon of Zhou,” Zhou bì (in which “Zhou” is probably areference to the capital of the Western Zhou dynasty) and the “Nine Chapters,” Ji�u zhang suàn shù . There isa full translation of the Zhou bì into English [Cullen, 1996], and translations of the Nine Chapters have been producedin English [Shen, 1999] and French [Chemla and Guo, 2004]. Both these books are anonymous, and we have nodirect accounts of the circumstances of their composition. Neither of them is mentioned in any source from beforethe Christian era, and even after that we have only hints of their history before the third century A.D., when their�rst known commentators worked, a little after the end of the Eastern Hàn dynasty (A.D. 25–220). The texts we havetoday seem to be largely identical to those seen by their �rst commentators.

The Zhou bì and the Nine Chapters are, however, very different in form and content. The Zhou bì is, I have argued[Cullen, 1996, 138–156], a collection of texts by different hands, written with a variety of purposes in mind, and atdates which may extend from as early as the second century B.C. to the early �rst century A.D. Although the Zhoubì has been listed as the earliest of the “Ten Mathematical Classics” since the seventh century A.D., the primaryaim of those whose writing is included in the book does not appear to have been to expound mathematical methodssystematically, but rather to apply available methods to calendrical calculation and to working out the dimensions ofthe cosmos. The mathematical content of the Zhou bì may be described quite brie�y. In general it is simply assumedthat the reader can add, subtract, multiply, divide, square, and extract square roots using fairly large numbers withmixed units and fractional parts. Two geometrical principles are used as the basis for calculating cosmic dimensions.The �rst of these is equivalent to the modern notion of similar right triangles, based on sightings on distant objectsusing a vertical gnomon. The second is equivalent to the Pythagorean relation between the lengths of the side of aright triangle. There is also a certain amount of what may be called “metamathematical discourse,” most of it in adialogue between a (possibly �ctional) master and pupil who discuss the dif�culties of mathematical study, and theways in which methods of problem solution may be studied and developed [Cullen, 1996, 175–178; see also below].

There is little point in attempting a detailed comparison between texts as different as the Zhou bì and the Suàn shùshu. In the case of the Nine Chapters, however, there is a clear family resemblance that demands further analysis. Likethe Suàn shù shu, the Nine Chapters consists largely of problems, for each of which a solution is provided togetherwith an explanation of the method to be used. However, unlike the looseness and variety we found in the Suàn shù


shu, the format of the Nine Chapters is highly regular, though not rigid. Every chapter consists mainly of a series ofproblems, and each problem is introduced by “Now we have . . . ,” j�n y �ou (or yòu y�ou , “Further we have” ifmore than one problem is enunciated). The result is always given as da yue . “The answer says,” and the method(sometimes given at the end of a group of similar problems) is always introduced by shù yue , “The method says”(sometimes with a preceding descriptive phrase specifying what the method is for, especially when several problemsare followed by one overall method statement).

As its name implies, the Nine Chapters is divided into nine sections, each of which deals with a broadly relatedfamily of problems. The problems treated range from the handling of elementary fractions up to more advancedprocedures involving quantities such as the surface area and volumes of various geometrical �gures, the methodnamed in English “The Rule of False Position,” the solution of problems involving (in modern terms) simultaneouslinear equations in several unknowns, and �nally applications of the Pythagorean relation to geometrical problems.A summary listing of the contents of the Nine Chapters will be useful here:

Chapter 1: Fang tián , “Rectangular �elds”Areas of �elds of various shapes; manipulation of fractions.

Chapter 2: Sù m�� , “Millet and rice”Exchange of commodities at different rates; pricing.

Chapter 3: Cu� fen , “Differential distribution”Distribution of commodities and money at proportional rates.

Chapter 4: Sh�ao gu�ang , “The lesser breadth”Division by mixed numbers; extraction of square and cube roots; dimensions, area and volume of circle and sphere.

Chapter 5: Shang gong , “Consultations on works”Volumes of solids of various shapes.

Chapter 6: Jun shu , “Equitable transport”More advanced problems on proportion.

Chapter 7: Yíng bù zú , “Excess and de�cit”Linear problems solved using the principle known later in the West as the “Rule of False Position.”

Chapter 8: Fang cheng , “The rectangular array”Linear problems with several unknowns, solved by a principle similar to Gaussian elimination.

Chapter 9: Gou g�u , “Base and altitude”Problems involving the principle known in the West as the “Pythagoras theorem.”

It is striking how often one can �nd quite close parallels between problems in the Suàn shù shu and in the NineChapters. Here for instance is a problem from the Nine Chapters that is clearly very close to the �rst of the two“weaving” problems quoted above; the example cited comes from the “Differential allocation,” Cu� fen section[Chemla and Guo, 2004, 288–289; Shen, 1999, 162–163].


Now there is a girl good at weaving who doubles her [production] every day; in 5 days she weaves �ve chí. Question: in[each] day how much does she weave?Answer:

On the �rst day she weaves 1 cùn 19/31 cùn; on the next day she weaves 3 cùn 7/31 cùn; On the next day she weaves6 cùn 14/31 cùn; on the next day she weaves 1 chí 2 cùn 28/31 cùn; On the next day she weaves 2 chí 5 cùn 25/31 cùn.Method:

Set out 1, 2, 4, 8, 16 as separate differentials; adjointly combine them to make the divisor; multiply the uncombined[differentials] by 5 chí, each to make its own dividend; obtain 1 chí for [each time] the dividend accommodates the divisor.

Note that unlike the example cited earlier, the Nine Chapters begins its doubling with 1 rather than 2; also thefractions are reduced to their simplest terms. Other similar examples of parallels could easily be cited [Cullen, 2004,passim]. However, although the content of the Suàn shù shu is often quite closely related to that of the Nine Chapters,the two texts are by no means identical in coverage. Taking the Nine Chapters in order, we may identify materialpresent there but not represented in the Suàn shù shu as follows:

1. Fang tián , “Rectangular �elds”Mean value of a group of fractions; areas of plane �gures other than rectangles and circles (such as triangles);areas of spherical caps.

2. Sù m�� , “Millet and rice”Problems of �nding combinations of prices of goods purchased in one lot.

3. Cu� fen , “Differential distribution”No major omissions.

4. Sh�ao gu�ang , “The lesser breadth”Algorithm for extraction of square roots; algorithm for extraction of cube roots; volume of sphere.

5. Shang gong , “Consultations on works”Volumes of yángm�a and bienáo (special forms required for volume dissection techniques); volume ofsquare pyramid (although the more complex chútong , a frustum of a rectangular pyramid, is treated, as isthe circular cone).

6. Jun shu , “Equitable transport”The conspicuous omission here is the basic concept of Jun shu “equitable transport” itself, that is, the adminis-trative technique of apportioning tax liability by taking account of population and the distance over which thedelivery of tax has to be made. Large-scale arrangements of this kind probably date back no further than about115 B.C. under Hàn W�udì [Chemla and Guo, 2004, 475–478; Twitchett and Loewe, 1986, 602–607], althoughanother document from the Zhangjiashan tomb may possibly bear the words jun shu on one rather obscure strip[Zhangjiashan, 2001, 256]. Apart from the extensions of the jun shu principle to the fair sharing of labor tasks,other omissions include problems of pursuit and mutual approach, for instance by travelers at different speeds.

7. Yíng bù zú , “Excess and de�cit”No major omissions.

8. Fang cheng , “The rectangular array”Absent.

9. Gou g�u , “Base and altitude”Absent.

The fact that only two books apart from the Suàn shù shu have been mentioned here does not mean that they arethe only other signs of mathematical activity in China in the early centuries of the empire. For instance such obvioussources as the �rst two standard Chinese histories, the Sh�� jì , “Records of the Historian” by S�m�a Qian(completed c. 90 B.C.) and the Hàn shu , “History of the [Western] Hàn Dynasty” by Ban Gù (largelycomplete by his death in A.D. 92) both contain substantial discussions of such topics as astro-calendrical calculationand the mathematics of the standard sets of musical pitch-pipes. Sources of this kind are, however, concerned withapplications of fairly elementary arithmetic rather than with discussing methods of calculation and problem-solving in


their own right. The same may be said of the excavated fragments of Western Hàn administrative records on woodenstrips showing the results of calculation. A selection of these is transcribed and discussed by L�� Dí [1997, 98–103],who makes use of earlier work by Guo Shìróng . They show us examples in which the scribe must haveperformed addition, subtraction, multiplication, and division using whole numbers and some mixed numbers, withone example of the calculation of the area of a rectangular �eld. However, English-speaking readers may also referto the material in Michael Loewe’s study of excavated strips recovered from the sites of Western Hàn frontier posts,which give many details of the kind of accounting required by the daily work of scribes in the military administration[Loewe, 1967]. The two examples given here are in both studies. In the �rst we have a record of grain issued tosoldiers’ families (I modify Loewe’s translations to be consistent with the practices adopted in this article, and followthe Chinese text given by L�� Dí):

Fù Fèng, private, “Grasping the Barbarians” beacon post: wife Juny��, adult, aged 28, grain consumed 2 shí 1 d�ou 6 and 2/3sheng; daughter Sh��, child age 7, grain consumed 1 shí 6 d�ou 6 and 2/3 sheng; daughter Jì, infant age 3 grain consumed1 shí 1 d�ou 6 and 2/3 sheng. Total of grain consumed 5 shí. [Loewe, 1967, 86–87]

Here the scribe had to do nothing more dif�cult than simple addition. In the next example division is required toproduce the result 510 bundles/17 men = 30 bundles/man. The signi�cance of the �nal �gure of 5520 is unclear.

Eleventh month, sexagenary day 54, 24 men. Of these, 1 man acted as supervisor, 3 prepared meals, 1 was sick, 2 menpiled reeds: 7 men released as above. 17 men were detailed for working tasks, cutting 510 [bundles of] reeds. The rate lü

was 30 for a man. This gave 5520. [Loewe, 1967, 134–135]

Such records are practical notes of of�cial data and their processing, with no discussion of method, but theycertainly indicate how much accounting work went on at the lowest levels of the Hàn state machine. It is not surprisingto �nd that in three examples of what appear to be pro�ciency certi�cates issued to military of�cers, it is said of allthree néng shu kuài jì , “he is able to write and keep accounts” [Loewe, 1967, 178–179]. So far I know ofonly one example of a bamboo strip from a government site of Hàn date that shows signs of going beyond the mererecording of accounts to indicate how particular problems are to be tackled. The fragment in question is one of manyunearthed from Juyán on the northwest frontier, and may date from some time between 100 B.C. and A.D. 100;it is broken at top and bottom:

. . . 5 d�ou 2 sheng 26/27 sheng. Method: combine the upper and lower. . . . [Xiè, 1987, 206, strip 126.5; Loewe, 1961, 70]

This material clearly raises the tantalizing possibility that at least one Hàn frontier of�cial had by him a collectionof materials that might have had much in common with the Suàn shù shu.

5.2. Mathematicians of the early Chinese empire and their texts

Clearly numeracy as well as literacy was a basic requirement for ancient Chinese of�cials. The problems in theSuàn shù shu would have been comprehensible to many of them, even if they did not often meet them in daily workor could not have tackled them unaided. But so far as named individuals famous for their outstanding mathematicalskills are concerned, the historical record of the early empire is quite sparse. One major �gure, Zh ang Cang(c. 250–152 B.C.), began his career under the Qín. Of him we are told:

He was perspicuous in his studies of the empire’s charts, writings, accounts and records. Cang also excelled in the useof calculating rods, and in mathematical harmonics and the calendar. [Sh�� jì, 96, 2676]

With the change of dynasties, Zhang Cang went on to become a high of�cial under the Hàn, with overall respon-sibility for aspects of government activities involving calculation. One of his successors was the economic reformerSang Hóngyáng, already mentioned above for his skills in mental calculation. For another high Western Hàn of�cial,


G�eng Shòuchang (�. c. 57–52 B.C.), we have good evidence of his skill in mathematical questions relating tocivil engineering and administration:

He excelled in calculation, and was able to evaluate the bene�ts of works . . . [It was said of him] “He is practiced inthe evaluation of works, and in matters of the allocation of money.” [Hàn shu, 24a, 1141]

The phrase shang gong , “evaluation of works,” appears later as one of the headings of the Nine Chapters. In histime G�eng seemed to have enjoyed a role as a mathematical expert only a few degrees below the eminence accordedto Zhang Cang in the previous century. Shortly after the words quoted here, G�eng is credited with the innovation ofthe institution of cháng píng cang , “Ever normal granaries,” to even out variations in the price of grain by statepurchase and resale. G�eng Shòuchang was also an observational astronomer: his work on the motions of the moon isdiscussed by Ji�a Kuí in A.D. 92 [Hòu Hàn shu, zhi 2, 3029]. The Hàn shu bibliography, based on a listing ofthe holdings of the imperial library as they were c. 10 B.C., records the presence in the library of 32 juàn, “rolls,” ofcharts and 2 juàn, of numerical data by him on this topic [Hàn shu, 30, 1766].

Another of�cial said to have been skilled in reckoning generally, as well as in “evaluation of works” was X �u Shang(�. c. 30 B.C.) [ Hàn shu, 29, 1688–1689]. The Hàn shu bibliography records a book (long lost) under his name:

X�u Shang’s mathematical methods, 36 juàn. [Hàn shu, 30, 1766]

Another similarly titled book from the same source, also lost, is

Dù Zhong’s mathematical methods, 16 juàn. [Hàn shu, 30, 1766]

Of Dù Zhong nothing whatever is known apart from this entry.We cannot be sure that these books contained anything similar to the mathematical methods found in the Nine

Chapters and the Suàn shù shu. Both of the books just cited, like the works of G�eng Shòuchang mentioned earlier, arefound in the section of the bibliography labeled lì p�u , “[astronomical] systems and listings.” The titles of otherbooks in that section suggest that they are concerned with mathematical astronomy, an impression which is in linewith the editorial note that follows the section. Indeed it is striking that nowhere in the whole Hàn shu bibliography isthere any group of books described by the editors as being speci�cally related to practical arithmetical calculations ofthe kind that currently concern us. The genre of which the Nine Chapters was the �rst example in the received traditionappears not to have been recognized as a distinct form of writing at the time the original imperial bibliography wascompiled.

The earliest �rm evidence of a scholar having studied any mathematical book known today is found in a bio-graphical note on M�a Xù , the son of M�a Yán (A.D. 17–98) and elder brother of the famous scholar andcommentator on the classics M�a Róng (A.D. 79–166). Hence he presumably �ourished c. A.D. 100. Accordingto the Hòu Hàn shu,

He was widely acquainted with the mass of documentary sources, and excelled in the Nine Chapters on MathematicalMethods. [Hòu Hàn shu, 24, 862]

M�a Róng’s most distinguished student was the great classical scholar Zhèng Xuán (A.D. 127–200); in anaccount of his youthful studies it is said elsewhere

He began by going through . . . the Nine Chapters on Mathematical Methods. [Hòu Hàn shu, 35, 1207]


It seems very likely, therefore, that the Nine Chapters was a well-known book by the beginning of the secondcentury A.D. Later we hear brief mentions of other students of this book, such as Xú Yuè (�. c. 210?) and K �anZé (?–243) [Chemla and Guo, 2004, 59]. Zhào Shu�ang , writing in his commentary on the Zhou bì in thethird century, says of a particular mathematical principle shù zài ji�u zhang , “The method is in the NineChapters” [Guo and Liú, 2001, 36]. But none of these records takes us back before around A.D. 100, when M�a Xù“excelled” in its study.

Even if we allow a whole century after its presumed compilation for the Nine Chapters to become widely enoughknown for such a claim to be meaningful, we would still be no further back than the beginning of the �rst century A.D.,nearly two centuries later than the entombment of the Suàn shù shu. We may recall there is no mention of the NineChapters in the Hàn shu bibliography, which re�ects the state of the imperial collection around 10 B.C. There are thusno positive grounds for believing that the Nine Chapters was in existence before the beginning of the Eastern Hàn.This conservative view is strengthened by the fact that the Suàn shù shu proves that some material of the kind laterincluded in the Nine Chapters was known about three centuries before the Nine Chapters appears in the historicalrecord. It is therefore clear that mathematical writing of a very different kind from the Nine Chapters did exist in theWestern Hàn, and was quite capable of conveying all the mathematical skills needed by early imperial administrators.

We shall now turn, however, to the work of an ancient writer whose views run quite counter to the cautious positionon the origin of the Nine Chapters set out here.

5.3. Liú Hu� and the history of the Nine Chapters

At some time around A.D. 263, Liú Hu� wrote a lengthy and detailed commentary on the Nine Chapters thataccompanies all extant editions of the main text [Chemla and Guo, 2004, 57]. Liú’s commentary is a major contributionto the development of Chinese mathematics, and he is undoubtedly the earliest identi�able Chinese mathematician ofdistinction from whom we have a substantial extant body of writing. In his preface to the Nine Chapters, he gives hisviews on the history of mathematics before his own day, and one obviously turns to his account with considerableinterest, in the hope that it will tell us something about the milieu in which both the Suàn shù shu and the NineChapters originated.

Liú Hu� begins his preface by claiming that the arts of reckoning go back to the time of the legendary sageBao X� (more commonly known as Fú X� ), who �rst drew the eight trigrams of the “Book of Change,”Yì j�ng , and made the ji �u ji�u , “Nine nines”—the multiplication table. But after a few more such gestures atremote antiquity, he tells us disarmingly enough—and quite convincingly—qí xiáng wèi zh� wén y�e ,“As for the details of this, I have never heard them.” Then we move into more historical territory:

Now when the Duke of Zhou laid down the Rituals the Nine Reckonings came into existence. It is the Nine Chaptersthat are the descendant of the tradition of the Nine Reckonings.

[But] when long ago the violent Qín [dynasty] burned the books, canonical methods were scattered and ruined. Afterthat time, Zhang Cang the B�eipíng Marquis, and the Grand Supervisor of Agriculture G�eng Shòuchang, [men of the] Hàn,both won fame in their age through excellence in reckoning. [Zhang] Cang and others took the remaining fragments ofthe old texts, and each was equal to [the task of ] making deletions and additions. So if one compares the headings, thereare certain differences from the ancient [version], and the discussions contain many modern expressions. (My translation;cf. [Chemla and Guo, 2004, 126–129; Shen, 1999, 52–53].)

The “burning of the books” referred to here was ordered in 213 B.C. [Twitchett and Loewe, 1986, 69–72] as partof a campaign to eliminate ideological dissent from the newly established Qín empire. Liú Hu�’s view is, therefore,that:

(a) In historical times the basis of mathematical learning may be traced back to what he calls the “Nine Reckonings”(see below on these) which he believes originated in the time of the Duke of Zhou (�. c. 1040 B.C.).


(b) The Nine Chapters came into existence on the basis of the “Nine Reckonings” tradition (how close this was to thetime of the Duke of Zhou he does not indicate), but the transmission of the text was interrupted by the destructionof books under the Qín, and Western Hàn scholars (including Zhang Cang and G�eng Shòuchang) had thereforebeen obliged to piece the book together again from surviving fragments.

Despite the fact that it runs counter to the impression given by all the other evidence that the origins of the NineChapters are relatively late, Liú Hu�’s account is broadly accepted by several modern historians of Chinese mathemat-ics [Chemla and Guo, 2004, 54–56]. However, other opinions on this subject are also widely held [L��, 1982], and Isuggest that as well as being dubious in itself, Liú’s view is cast into further doubt by the discovery of the Suàn shùshu. My reasons for this may be stated brie�y here, with detailed argument being reserved for a separate publication:

(i) The gap of time: Liú Hu� is writing around A.D. 263, three and four centuries respectively after the times ofG�eng Shòuchang and Zhang Cang. That is comparable to a reader in the year 2000 contemplating the activitiesof persons in 1700 and 1600. Between Liú Hu� and the events he is describing lie the major losses of documentsin the destruction with which both the Western and Eastern Hàn ended. If he had named as the reconstitutersof the Nine Chapters a person or persons unknown to the historical sources we have today, we might have beenforced to conclude provisionally that Liú Hu� had access to sources now lost to us. But the fact that he names theonly two Western Hàn �gures said in extant records to have enjoyed real eminence as mathematicians suggestshis information may have been no better than ours, and he is simply looking for likely candidates to �t into a storybased on conjecture. Other extant sources on the mathematical activity of Zhang Cang and G�eng Shòuchang saynothing about the Nine Reckonings or the Nine Chapters.

(ii) The allegedly massive Qín destruction of ancient literature was something that later Chinese scholars were in thehabit of lamenting, and indeed exaggerating when possible. But it is exceedingly unlikely that the Nine Chapterswould have been among the books destroyed by the Qín, had it existed at the time. Such destruction as there wasonly targeted on ideologically dangerous works that “used the past to criticize the present,” and books on usefulsubjects were explicitly exempted; moreover, books held by high-ranking state advisers like Zhang Cang werealso exempted [Sh�� jì, 6, 255]. If the Nine Chapters had existed in Zhang Cang’s day he would undoubtedly havehad a copy. If Liú Hu� was anything of a historian he could not have missed the very well-known source in whichall this is made clear. The fact that despite this he believes the Qín could have burned the Nine Chapters does notsay much for his judgment as a historian.

(iii) Quite apart from the exemption of “useful” books, it is impossible to imagine the Qín dynasty—obsessed as itwas by calculation and accountancy—destroying a text such as the Nine Chapters, or for that matter the Suànshù shu, some of whose material bears a close resemblance to what we know to have been the content of Qínadministrative regulations [Cullen, 2004, 67].

5.4. The commentators and the “Nine Reckonings”

But what grounds did Liú Hu� have for claiming the Ji�u shù , “Nine Reckonings,” had any connection withthe Nine Chapters? And what if anything did the Nine Reckonings have to do with the Suàn shù shu?

The work from which Liú Hu� takes the expression “Nine Reckonings” is the Zhou l�� , “Ritual of the Zhou[dynasty],” which supposedly describes the state organization of the Western Zhou dynasty c. 1000 B.C. In the time ofLiú Hu� in the late third century A.D., the Zhou l�� was a famous and widely studied text, which was assumed to give atrue account of the most ancient institutions of Chinese government. The modern view is, however, that it is probablya work of the later Warring States period, perhaps of the fourth century B.C. The Zhou l�� is not mentioned before themiddle of the second century B.C., when it is said to have been presented to a younger brother of the emperor W�udì bya scholar who had rediscovered it. It was not then highly regarded. However, in A.D. 9, after having been the effectiveruler for several years, the powerful minister Wáng M�ang received the abdication of a boy emperor, and beganto rule as the �rst emperor of what he called the X �n , “New” dynasty, which lasted until his overthrow in A.D. 23.During his reign, a reform of state institutions was carried out on the ostensible basis of Zhou models, and the Zhoul�� was given an honored place as the basis of these innovations—which were of course presented as the restorationof antique purity. From that time onward this book was given much attention by scholars [Loewe, 1993, 24–32]. Inthe Zhou l��, the “Nine Reckonings” are prescribed as part of the “Six Arts,” liù yì , which form the educational


curriculum of young aristocrats. The other �ve subjects are the Five Rituals, the Six Musics, the Five Archeries, theFive Horsemanships, and the Six Writings [Zhou l��, 212–213]. The Zhou l�� itself does not give us any idea of what theNine Reckonings (or the Five Rituals, etc.) actually were. No ancient text of any kind currently known mentions theNine Reckonings before the Eastern Hàn. If we look for other ancient examples of expressions involving the numbernine and something to do with calculation, possibly the earliest comes from the Gu�an Z�� book, an eclectic textcontaining material dating from the late Warring States to the early Western Hàn [Loewe, 1993, 246–249]. Here themultiplication table is referred to as ji �u ji�u zh� shù , “the nine nines reckoning” [Rickett, 1998, 499]. If thatwas all we had to go on, it would seem quite plausible that the expression “Nine Reckonings” of the Zhou l�� wassimply an abbreviated version of this phrase; the multiplication table is about the limit of mathematical knowledgeone can imagine being in�icted upon young lordlings otherwise preoccupied with their practice of archery, ritual, etc.

So where then did Liú Hu� get the idea that there was a link between the Nine Reckonings and the Nine Chapters?The answer lies in the activity of a group of scholars in the early Eastern Hàn, who tried to deal with the problemposed by the fact that many ideologically important ancient texts such as the Zhou l�� contained terms and expressionsthat were no longer comprehensible. In response they wrote voluminous commentaries which aimed to provide thestudent with solutions—sometimes based on evidence, sometimes clearly conjectural—to (almost) all such dif�culties.Perhaps the most renowned of all such commentators was Zhèng Xuán—who as we have seen was the second personin Chinese history known to have studied the Nine Chapters. Zhèng Xuán wrote a commentary on the Zhou l��, and LiúHu�’s quotations from his work show that he was familiar with it [Zhou l��, 619; Chemla and Guo, 2004, 453; Shen,1999, 297–298]. For the Nine Reckonings, Zhèng Xuán gives the following explanation:

The Nine Reckonings are Fang tián, Sù m��, Cha fen, Sh�ao gu�ang, Shang gong, Jun shu, Fang chéng, Yíng bù zú, Pángyào; nowadays we have the Chóng cha, Xì jié and Gou g�u. [Zhou l��, 1815, 212–213]

The list of Nine Reckonings given here is almost identical to the chapter titles of the present text of the NineChapters. Apart from trivial differences of wording and the fact that the seventh and eight topics are interchanged, theonly noteworthy difference is that the present text has Gou g�u instead of Páng yào (for brevity, I leave to one sidewhat the topics Páng yào, Chóng cha, and Xì jié might refer to). It is clear, therefore, that Liú Hu�’s claim that theNine Chapters originates from the Nine Reckonings is based on Zhèng Xuán’s statement. It appears from adjacenttext that Zhèng Xuán is himself quoting from Zhèng Zhòng (?–A.D. 83), who is roughly of the same generationas M�a Xù, the �rst known student of the Nine Chapters at a time when it was apparently already well known. Thecritical point in evaluating Liú Hu�’s view on the origins of the Nine Chapters is therefore what Zhèng Xuán’s (orrather Zhèng Zhòng’s) basis might be for their identi�cation of the Nine Reckonings with the headings of the NineChapters. It is clearly highly signi�cant that the �rst reference to the Nine Chapters as a well-known book appearssimultaneously with the �rst attempt to assimilate the Nine Reckonings to the Nine Chapters, and that both events takeplace in the �rst century of the Eastern Hàn. The coincidence of the appearance of the book at the same time as theexplanation is striking enough. I shall shortly suggest a possible reason for this coincidence. For the moment, we maysimply note that the existence in their own day of a well-known book on mathematics in nine sections would seem tohave been suf�cient reason for the two Zhèngs to explain the Nine Reckonings in the way they did, without any needto make implausible assumptions about their access to otherwise lost traditions of the meaning of an ancient text.

To sum up the position on Liú Hu�’s attempt to write history of mathematics, we may say that where he makes aclaim not supported elsewhere, as in his theory that the Nine Chapters was a pre-Qín text, lost and later reconstituted,his views seem to be based on a mixture of uncritical acceptance of historical cliché and straightforward conjecture.Where he bases himself on the opinions of others (the two Zhèngs) his acceptance of their views is irrelevant to thereliability of the explanations he quotes, which will turn on our analysis of the period in which the original writersworked. From the point of view of the Suàn shù shu Liú Hu�’s writing does nothing to convince us that the NineChapters were (in whatever form) in existence at the time the Suàn shù shu was placed in the tomb, nor does it makeit seem likely that the Nine Reckonings were an accepted way of organizing mathematics at that period. Under thesecircumstances we can treat the Suàn shù shu as a collection in its own right, rather than as an imperfect attempt to dothe same job as the Nine Chapters.


5.5. The Nine Chapters and the Suàn shù shu

In the �nal sections of this paper I shall attempt to resolve some of the historical problems that are evident fromthe account of the Suàn shù shu and its context given above. The task is essentially to construct an evidence-basednarrative that can deal with the fact that the two earliest texts of Chinese mathematics can be securely dated no closerthan two or three centuries from each other, and while clearly belonging to the same tradition differ very signi�cantlyin format as well as in detailed content. What follows is directed toward those who are not satis�ed by the simpleassumption that mathematical progress is inevitable, but who are interested in seeing how far mathematical activityand mathematical innovation can be seen as part of a wider context of social and intellectual change.

5.5.1. Problems in constructing a narrativeThe problems faced by any narrative of the kind required include the following:

(1) If we take the Suàn shù shu and the Nine Chapters as representative of the actual state of mathematical literature inthe early Western Hàn and early Eastern Hàn, we have to explain how the practice of mathematics and the modesof transmission of mathematical knowledge can have changed suf�ciently to move us from a world in which theSuàn shù shu was seen as normal to one in which the Nine Chapters were seen as normal. We have to explain howmathematical writing changed from being a matter of generating short problem-sized passages to being a matterof writing a lengthy and well-organized book—and a book whose organization depended crucially on the numbernine.

(2) However, as we have seen, we have rather little direct information on the activities of those who used mathematicsin the intervening centuries, or on the social framework within which they operated. The only ancient attempt atwriting something like the history of mathematics—Liú Hu�’s preface to the Nine Chapters—is, as we have seen,of little independent value.

Clearly the present article is not the �rst attempt that has been made to solve such problems. This is not the time toattempt a systematic review of the opinions of Chinese scholars on the origins of the Suàn shù shu and its connectionswith the Nine Chapters. Brie�y, however, we may say that two main currents of thought are clear at the time of writing.One of them, represented by the leading historian of mathematics Guo Shuchun , takes Liú Hu�’s account ofthe origins of the Nine Chapters fairly literally [Chemla and Guo, 2004, 54–56; Guo, 2003]. It is thus assumed that theNine Chapters actually did exist in some form as a book before the Qín, and that it was damaged or scattered and laterreconstituted as Liú Hu� tells us. The question therefore arises whether the Suàn shù shu is in some way in the trueline of descent that leads to the Nine Chapters as we have it today, and the answer is negative. The polymath historianof science L�� Dí , on the other hand, discounts Liú Hu� as a reliable historian so far as the story of the “�resof Qín” is concerned, and believes that mathematical knowledge in the Western Hàn circulated in the form of whathe calls guan ji�an , “of�cial bamboo strips.” It was from such material that the Nine Chapters was assembledand edited [L��, 1997, 88–138]. L��’s views have developed from conjectures expressed before the text of Suàn shù shuwas fully published and widely discussed, but he has recently stated that they remain basically unchanged [privatecommunication, 2004]. It will be seen that my views are closer to those of L�� than to those of Guo; I must, however,bear responsibility for the arguments I put forward in the next two sections, which (so far as I know) are mine alone.

5.5.2. A parallel: Medicine in the Western Hàn and the format of the Suàn shù shuThese problems just outlined are not trivial ones, and might well cause one to hesitate before trying to construct a

history of Hàn mathematics capable of bridging the gap that confronts us. It is therefore very fortunate that we alreadyhave the main outlines of the history of another technical �eld that showed major changes in the writing down andtransmitting of knowledge between Western and Eastern Hàn—changes similar to those that would take us from theSuàn shù shu to the Nine Chapters. The �eld in question is that of medicine. In medicine, as in mathematics, we �ndourselves contemplating the impact of major �nds of manuscript material from early Western Hàn tomb deposits,in the context of a canonical literature that cannot be reliably traced back further than the start of the Christian era.Understanding the development of one text-based technical �eld is clearly likely to be helpful in making sense of whathappened in another such �eld.


For a description and evaluation of the major portions of the Western Hàn medical material, the reader may turn �rstto the work of Harper [1998] to which may be added the research results of Vivienne Lo, embodied in her unpublishedPh.D. thesis [1998]. A pioneering and original discussion was contributed by Yamada [1979] and there is also animportant study by Sivin [1995]. But for our present purpose it is the work of Keegan [1988] that is most relevant.What Keegan’s groundbreaking study achieved was to give us a new picture of what technical medical literaturewas like under the Western Hàn, and to elucidate some of the ways the Western Hàn heritage was transformed insucceeding centuries to produce the canonical literature of the received tradition.

In summary, Keegan’s study of a group of Western Hàn medical manuscripts, mainly the so-called “vessel texts”from a tomb at M�awángduì as well as from Zhangjiashan, led him to the conclusion that the elementary unit ofthat literature was not to be seen as the “text” in its usual meaning of an extensive piece of writing equivalent to whatwe would call a book, but rather what we may call a “textlet,” a shorter piece of writing capable of being transmittedon its own. Different extended texts might contain overlapping but not identical collections of textlets, and one textmight separate textlets contiguous in another text, while bringing together textlets separated in other collections. Instudying medicine, one increased one’s knowledge base in part by receiving more material from one’s teachers, whomight sometimes pass on textlets only after an impressive ritual requiring a commitment not to transmit them tothe unworthy. Such rituals have been reconstructed and evidenced from historical texts by Sivin. Clearly differentdoctors within a given tradition of medicine would tend to have overlapping but often differently ordered collectionsof material at their disposal.

By the Eastern Hàn, however, Keegan claims that this process had led to the formation of more than one largeossi�ed collection of material that was no longer subject to “textlet” transmission in the old way. This is his explanationof the origin of the different recensions of the so-called Huángdì , “Yellow Emperor,” corpus of which signsappear for the �rst time in the Hàn shu bibliography. An examination of three of these early medical compilationsshows that they do indeed embody much material common to one another, and that some of their contents closelyresemble “textlets” from the M�awángduì and Zhangjiashan material, but in an order and arrangement different enoughto witness to the ability of the “textlet” to be transmitted independently.

The extant representatives of the Huángdì corpus are by no means chaotic works, although any attempt to readthem as deliberately composed and systematic treatises will lead rapidly to a sense of confusion on the part of theconscientious reader. A recent detailed study of one part of the corpus discusses related problems [Unschuld, 2003].However, more systematized works do exist: one of them, the Huángdì ji�a y�� j�ng , “Huángdì’s ABCcanon,” was composed by Huángf�u Mì around A.D. 256–282, so that he might have been a contemporary ofLiú Hu�. This uses material from the Huángdì corpus to give a systematic account of acupuncture and moxibustion.We may also note the existence of a major text which is held by some to stand outside the Huángdì corpus, the Nánj�ng , “Canon of Dif�culties”: this is a highly formalized and ordered book in which 81 sections each raise aquestion in the form of a “dif�culty” which is then answered. With the Nine Chapters in mind, we may note that9 × 9 = 81. It is usually held that the Nán j�ng dates from at the latest the second or �rst centuries A.D., since it isquoted shortly thereafter.

Turning back indeed to the Nine Chapters and its relation to the Suàn shù shu, it does seem that a pattern similarto the one sketched above for the case of medicine can be detected. Even without the work of the scholars mentionedabove, an inspection of the Suàn shù shu suggests that in one technical �eld, that of mathematics, the independentlycirculating unit of knowledge in the early Western Hàn—the “molecule of written information,” so to speak—was atextlet rather than an extended and orderly treatise. In many cases the textlets of the Suàn shù shu take the form ofa complete section beginning with a title. In other cases, a section with a title contains what is clearly a deliberatelycollected group of related textlets, which make it plain that an individual scribe has felt that gathering textlets in thisway was just what his reader would expect him to have done. The frequent duplications and repetitions that resultfrom this practice make it plain how much the accumulation of transmitted textlets was valued for its own sake, evenif each addition to the collection did little or nothing to increase the sum of mathematical knowledge already gathered.Here of course we are speaking of what we can deduce from the activity of the �nal and apparently anonymous handto work on this material, ignoring any intervening process of simple copying.

Unlike the case of the medical texts, however, the presence of the names of Wáng and Yáng gives us the chance tolook back a little further than the manuscript itself, though our ability to draw any reliable conclusions is reduced byour inability to say just what sort of people they were, and what roles they played. But we can note that the way theirnames appear does not suggest any association with long passages of text: the longest continuous passages marked


with their names are the two “weaving problems” (discussed above), at the end of which we are told that they were“checked,” chóu , by Wáng and Yáng respectively. This pattern is consistent with the notion that at this period amathematically active individual was more likely to generate mathematics in the form of textlet-sized packages thanto write a discursive treatise. In other words, Western Hàn scholars who wrote mathematics managed their knowledgebases in ways similar in part to the practices of contemporary literate healers. We may recall too, that when Liú Hu�tried to characterize the nature of mathematical writing before his imagined “reconstitution” of the Nine Chapters hesaid there were only “remaining fragments of the old texts,” jiù wén zh� yí cán , available to scholars. Thiscertainly seems like a re�ection, perhaps in distorted form, of the fact that Western Hàn mathematical writing reallydid take a form that Liú Hu� would have regarded as fragmentary—the short sections of the Suàn shù shu on bamboo.But if the Suàn shù shu �ts easily into the pattern of Western Hàn technical writing, what can have happened duringthe transition to the Eastern Hàn to cause the change from the Suàn shù shu format to that of the Nine Chapters?

5.5.3. The role of Wáng M�angThe �rst appearance of the Nine Chapters in the historical record in the early Eastern Hàn may certainly be seen as

generally parallel to the emergence of such works as the Huáng dì corpus, the Nán j�ng, and the Huángdì ji�a y�� j�ng atthe same period or a little later. To get from the Suàn shù shu to the Nine Chapters we need to apply the same kindsof transformations that take us from the somewhat disorderly Western Hàn medical material to the more systematicmedical texts of the Eastern Hàn—one of which, as we have seen, also used a division of its subject based on thenumber 9. Reversing the argument, we may say that given the existence of the Nine Chapters in the Eastern Hàn, acollection such as the Suàn shù shu is (given the evidence from medicine) just the kind of mathematical text we wouldexpect to �nd from the early Western Hàn. We may now return to the fact that the �rst evidence of the existence ofthe Nine Chapters appears at the same time as an explanation of the Zhou l��’s Nine Reckonings that ties them �rmlyto the headings of the Nine Chapters. What historical process could have created this link?

The answer, I suggest, lies in the activities of the person who raised the Zhou l�� from being an obscure text to arevered guide to government. As mentioned above, this was Wáng M�ang, who during his reign as emperor from A.D. 9to 23 consciously presented himself as a new Duke of Zhou, and modeled his governmental structure on the Zhou l��. Itis well known that Wáng M�ang wanted his reign to be signalized by new learning in the numerical disciplines, rangingfrom calendrical astronomy to divination of all kinds. We know that in A.D. 5, when he was still regent, Wáng M�angheld a great gathering of those interested in such topics and learned in ancient texts [Hàn shu, 1962, 21A, 955], whichwas attended by several thousand scholars [Hàn shu, 1962, 12, 359]. There is persuasive evidence that Wáng M�ang’spatronage may have led to the compilation of the Zhou bì, the text dealing with astronomical calculation mentionedabove, which begins with a dialogue involving the Duke of Zhou [Cullen, 1996, 153–156]. That dialogue also stressesthe importance of the number nine as the basis of numerical data about the cosmos:

. . . The patterns for these numbers come from the circle and the square. The circle comes from the square, the squarecomes from the trysquare, and the trysquare comes from [the fact that] nine nines are eighty-one. [Cullen, 1996, 174]

If Wáng M�ang had sponsored the compilation of a treatise on methods of calculation, it would not have beensurprising to �nd that it based its structure on the only phrase in the Zhou l�� which could be interpreted as a referenceto the divisions of mathematics, which is of course the cryptic reference to the Nine Reckonings. It would thus havebeen likely to have used a ninefold division of its subject. Another famous work by one of Wáng M�ang’s supportersis divided into 9 × 9 = 81 sections: this is the Tàixuán j�ng of Yáng Xióng (53 B.C.–A.D. 18) [Loewe,1993, 460–466]. And the medical work known as the Nán j�ng, “Canon of Dif�culties,” mentioned above, likewisedivided into 81 sections, may also originate from the same period [Unschuld, 1986, 424]. Another feature of the NineChapters that points to the time of Wáng M�ang is the widespread use of the hú measure for grain in that text, aunit entirely absent from the Suàn shù shu. According to Michael Loewe’s careful examination of datable excavatedtexts bearing government accounting records, these point to the conclusion that “the term hu [as a unit of capacity]was introduced of�cially from the Wang Mang period” [ Chemla and Guo, 2004, 202; Loewe, 1961, 73].

If the Nine Chapters originated in the time of Wáng M�ang, that would �t perfectly with its �rst appearance in thehistorical record as a well-known book studied by M�a Xù about eighty years after the fall of Wáng M�ang’s regime. Theopprobrium later heaped on Wáng and on those scholars associated with him would have ensured that even if EasternHàn scholars found such a book as the Nine Chapters too useful to ignore, they might not have wished to advertise its


origins. The fact that Zhèng Zhòng gave an explanation of the Nine Reckonings close to the chapter headings of theNine Chapters would then follow from the fact that, like Liú Hu�, he simply assumed that the Nine Chapters were anauthentic descendant of the Nine Reckonings tradition—whereas in fact they were merely an attempt to reconstructit by dividing the content of mathematics into nine arti�cial categories based on the fact that the Zhou l�� speci�ednine “reckonings.” Indeed it is noticeable that after Zhèng has listed the Nine Reckonings, he records the existenceof three other mathematical topics known in his own day that were not then counted in the Nine Reckonings—whichwould suggest that the ninefold division had not had very long to take hold, and that it was by no means natural formathematicians at that time to think of mathematical methods as subject to a ninefold classi�cation. We may recallthat the only books with titles referring to “mathematical methods” in the of�cial listing of c. 10 B.C. were in 36 and16 sections, respectively.

5.5.4. The Suàn shù shu and the purpose of mathematical study in the Western HànOn the basis of the discussions above, it seems that we must interpret the aims and methods of mathematics in the

Western Hàn without referring to the special features of the Nine Chapters, or to the organizing principle of the “NineReckonings.” But what then can we say about the aims of those who wrote the material copied and assembled in theSuàn shù shu, which is now our only substantial and reliable witness to early Western Hàn mathematical practice?What did they see themselves as trying to do, and what criteria did they use to decide whether they were successful?

In deciding how to respond to this challenge, some would say that we need to confront an obvious problem: isthis material simply a practical handbook of reckoning, or is it written in part by or for people whose interest inmathematics goes beyond its use for administrative purposes? That question is somewhat crudely put, but rather thantrying to add in all the possible quali�cations, let us try to confront it directly.

Some sections of the Suàn shù shu do certainly give no more than elementary information on how to do calculations,and useful facts about such matters as how much of one type of grain is equivalent to another sort—but then the samecould be said of the Nine Chapters, which no one would nowadays claim is simply a practical guide for administrators.Without having been a Qín or early Hàn of�cial, it is in any case dif�cult to be sure whether the more complexproblems about amounts of grain, and elaborate arrangements about sharing and mixtures requiring treatment by theRule of False Position, really do re�ect any practical needs likely to be met in the course of one’s work. However,there is no doubt in my mind that in the pair of “weaving” problems found in Sections 15 and 21, discussed above, weare faced by texts where the main interest is in problem-solving structures rather than any conceivable administrativereality. And since it is notable that these are the two cases where named persons are linked with problems in theclearest and most formal manner, it does seem probable that at least two persons near the beginnings of imperialChina were for at least part of the time interested in displaying their ability to create (or at least to pass on) problemsand methods of solution whose interest lay in their structure and ingenuity rather than in their practical value. Thefact that one of the methods is faulty shows that we are not just reading a recital of standard methods, but are inthe presence of real attempts at mathematical creativity—which can sometimes go wrong. It therefore seems safe toconclude that some early Western Hàn scholars were to some extent interested in clever methods of calculation forreasons other than their straightforward usefulness in administration.

The intriguing question that then presents itself is who the audience for such mathematical virtuosity might havebeen, what criteria that audience operated in deciding what counted as good mathematics, and what rewards followedfrom a reputation for mathematical skill beyond the call of of�cial duty. For the �rst and third parts of that questionwe are currently far from being able to give a well-based answer. But for the second part of the question, we maybe able to hear some Western Hàn mathematicians telling us what they thought good mathematics was about, andthat evidence will prove directly relevant to the Suàn shù shu itself, and to its relation to the Nine Chapters. This ispossible because we have a piece of metamathematical writing that probably dates from the Western Hàn dynasty, andthat sums up the essentials of the work in which Chinese mathematicians were then engaged on the basis of collectionssuch as the Suàn shù shu.

The material in question comes from the Zhou bì. In part of this book, a part which I have argued is probably fromthe �rst century B.C., a teacher, Chén Z �� , is represented as telling his student Róng Fang how to learnmathematics. Neither of these characters is known to history. It is notable that as in the case of the dialogue betweenSocrates and the slave-boy in Plato’s Meno, the �rst piece of Chinese writing about mathematics is in the form of aconversation between master and student.


Chén Z�� said, “Yes, these are all things to which calculation methods [suàn shù ] can attain. In regard to calcu-lation, you have the ability to understand these matters, if only you give sincere and repeated thought to them [ . . . ] Inrelation to numbers, you are not as yet able to generalize categories [it is categories of problems that are meant here]. Thisshows there are things your knowledge does not extend to, and there are things that are beyond the capacity of your spirit.Now in the methods of the Way [that I teach], illuminating knowledge of categories [is shown] when words are simple buttheir application is wide-ranging. When you ask about one category and are thus able to comprehend a myriad matters,I call that understanding the Way. Now what you are studying is the methods of reckoning [suàn shù zh� shù ,in which the �rst two characters are those in the title of the collection we are discussing, the third is a possessive particle,and the fourth is “methods”], and this is what you are using your understanding for. But still you have dif�culties, whichshows that your understanding of the categories is too simple. The dif�cult part about understanding the Way, is that whenone has studied it, one has to worry about broad application of it. Once it has been broadly applied, one has to worry aboutnot [being able to] put it into practice. Once one has put it into practice, one worries about not being able to understandit. So similar methods are studied comparatively, and similar problems are comparatively considered. This is what sortsthe stupid scholar from the clever one, and the worthy from the worthless. So being able to categorize in order to unitecategories—this is the substance of how the worthy will devote themselves to re�ning practice and understanding. (See[Cullen, 1996, 175–178], from which this translation is slightly modi�ed.)

We may note that in telling his student how to study mathematics, Chén Z�� says nothing about the Nine Reckoningsas a means of organizing mathematical knowledge, nor about the Nine Chapters or indeed about any kind of standardmathematical book to be studied. That is consistent with the view suggested above, according to which such thingswere not yet part of the intellectual repertoire of Chinese mathematics before the �rst century A.D. The key to thispassage lies in Chén Z��’s insistence on the need to “unite categories,” hé lèi . In his preface to the Nine Chapters,Liú Hu� in the third century A.D. was even more ambitious than Chén Z�� in setting out a program for the uni�cationof mathematical methods:

The categories under which the matters [treated herein fall] extend each other [when compared], so that each bene�ts[from the comparison]. So even though the branches are separate they come from the same root, and one may know thatthey each show a separate tip [of the same tree]. (My translation; cf. [Chemla and Guo, 2004, 126–127; Shen, 1999, 53].).

The implication here is that, properly understood, all the problems in the Nine Chapters are in essence solvedby one and the same method. In his actual commentary, Liú Hu� gets nowhere near being able to realize this idealprogram. But the overall ambition to connect and to unify mathematical methods is certainly an aspiration that LiúHu� and Chén Z�� have in common. I have quoted the passage from the Zhou bì at length because I believe it sums upthe spirit of early Hàn dynasty mathematical learning that eventually �owed into a systematic and powerful book asthe Nine Chapters, in which “words are simple but their application is wide-ranging [and] when you ask about onecategory [you] are thus able to comprehend a myriad matters.” That is what I believed when I �rst translated thosewords of Chén Z��, at a time when the Suàn shù shu was just being unearthed. But now that I have seen the Suànshù shu, I am all the more sure that in Chén Z��’s advice to his student we can hear the authentic voice of the earlyHàn mathematicians who assembled collections such as the Suàn shù shu. It is obvious that the aim of those whogathered the material in each section of that collection was precisely to bring together related problems and methodsfrom various sources so that as Chén Z�� says “similar methods are studied comparatively, and similar problems arecomparatively considered.”

To underline the distinctive nature of the program of algorithmic generalization that underlies the mathematicaltexts we are discussing, an East–West comparison will be helpful. It was the fate of the Nine Chapters to be spoken ofin later centuries as a paradigmatic work that summed up the essential spirit and content of mathematics in China. Thesame thing, more or less, happened to Euclid’s Elements in the West, and with about as little justi�cation, as can easilybe seen by considering the contrast between Euclid and (for instance) Diophantos or Heron. But even stereotypes maytell us something interesting, and there is an illuminating contrast here.


The Elements, as we know, treat mathematics with a well-de�ned program in mind, which may in part be describedas follows. We start from the smallest possible number of statements which the author has to ask us to accept as true.From these we attempt to derive logically the largest possible number of true propositions. So impressively is thisprogram executed that it is not surprising that some of Euclid’s later readers were tempted to think that this waswhat all “real” or “true” mathematics should be like, and that anything else was in some sense a falling-short. Nowjudged in that way, much of early Chinese mathematics including the Nine Chapters is a lamentable failure to do realmathematics at all.

However, on the view set out above, the Nine Chapters had an aim that was in some sense orthogonal to that ofEuclid: whereas the Elements sought to move from a few assumptions to a potentially unlimited number of true propo-sitions by logical deduction, the Nine Chapters sought to move from the in�nite variety of mathematical problems tothe smallest number of general algorithms that could solve them all, grouped under the nine main headings of itschapters. And it seems clear that some part of that program was already implicit in the way the writers of the Suàn shùshu organized their material, and in the way their approximate contemporaries thought of the way one should studymathematics.

6. Conclusion and summary

We may now sum up the results of the investigation outlined in this article.

(a) With the discovery of the Suàn shù shu we can move on from the previous situation when the Nine Chaptersrepresented the only substantial example of a mathematical text from the early imperial age. The history of earlymathematics in China largely consisted of attempts to �nd some way of dating the Nine Chapters, a processconducted by searching for the earliest dates at which the techniques set out in that book could have existed. Thehistory of early Chinese mathematics was, essentially, the history of the Nine Chapters, and writing that historywas not so much a matter of analyzing how Chinese mathematics was created, but rather concerned itself withtrying to �nd where a unique mathematical monument should be set in place on a timeline. After the Suàn shùshu things are different: we are now able to sketch a vision of the early imperial age as an epoch of change andcreativity, in which we do not simply see mathematical texts as intellectual monuments, but can begin to analyzethe processes and in�uences through which they were created.

(b) In purely mathematical terms, the Suàn shù shu enables us to locate the origins of certain techniques to at leastas far back as c. 200 B.C., given that the tomb in which it was found was closed not long after that date. Thesetechniques include the Rule of False Position and methods for calculating the volumes of a number of solid bodies.Clearly many of these techniques might have been known since a considerably earlier date. We may, however, notethe absence of certain other techniques, such as the extraction of square roots, the right angled triangle theoremand the solution of linear equations in several unknowns, all of which are found in the Nine Chapters. Nor isthere any evidence, from the Suàn shù shu or elsewhere, that Western Hàn (or earlier) scholars regarded the “NineReckonings” of the Zhou l�� as a normal way to structure mathematical knowledge.

(c) The styles and formats of the Suàn shù shu and the Nine Chapters differ greatly from one another, but an examina-tion of their contents shows clearly that they belong to the same mathematical tradition. These differences of styleand format are not mere matters of the idiosyncrasies of different writers, but can be shown to relate closely to thechanging nature of the creation and transmission of technical knowledge in ancient China, and to the in�uenceof political changes at the highest levels of the Chinese imperial state. Despite these differences, both texts showclear signs of a commitment to a particular style of mathematical learning that is advocated in more than oneancient text, and appears to be distinctively Chinese.

(d) The appearance of the Suàn shù shu enables us to see the Nine Chapters in a new light. Once, given the factthat the Nine Chapters was the earliest known text on mathematics, it may have been natural to assume that thebook must have been in existence in some form or other from the earliest period when the techniques it containscould reasonably have been thought to be in use, certainly as far back as the Warring States. The existence ofthe Suàn shù shu early in the Western Hàn with its comparatively rich content in a format quite different fromthe Nine Chapters liberates us from that assumption, and allows us to confront the historical evidence withoutpresuppositions. Given that before the early Eastern Hàn there is no sign whatsoever of the Nine Chapters orindeed of a tradition of dividing mathematical methods into nine sections, our working hypothesis has to be that


the Nine Chapters originated early in the �rst century A.D. And given that we can demonstrate the existence ofspecial historical circumstances that could have generated such a book at that time, the burden of proof is now�rmly on those who would claim an earlier date. It is hard to see on what evidence such a contention couldcurrently be based.

The views I outline here are intended to make the most economic and cautious use of the historical and textualevidence actually before us. Nevertheless, no conclusions in this �eld can ever be �nal. Exciting new �nds fromChinese archaeological excavations are now reported at monthly intervals, and we may hope that it is only a matterof time before more mathematical material from the early imperial age comes to light. When that happens, it may bepossible to test the hypotheses I have put forward in this article.

Acknowledgments

Like many western scholars I had long been aware, from hints and short references in Chinese writings on thehistory of mathematics, that the Suàn shù shu existed, and that it was an early Hàn text on mathematics, but that wasall I knew or could know, since for 17 years after its discovery only fragments of a transcription were published. I was�nally able to begin the study of the Suàn shù shu in early 2001, using the transcription published in the mainlandChinese journal of archeology Wénwù [Jianglíng, 2000]. When I began to read it, I found that the only wayto understand the harder sections was to write down a rough version in English. The process continued until by thespring of 2001 I had completed a draft translation of the whole text, with a methodological commentary and citationsof parallels from other mathematical works, principally the Nine Chapters. I also drafted an essay on the origins andsigni�cance of the Suàn shù shu. Professor Sir Geoffrey Lloyd and Professor Nathan Sivin very kindly read this draftmaterial in its earliest form and gave me detailed comments. It was not, however, possible to make further progresswithout sight of a better transcription. In late summer 2001 I obtained a copy of the book by Péng Hào [Péng, 2001],and later in the same year I saw the full publication in [Zhangjiashan, 2001], which enabled me to resolve a numberof outstanding points.

The substance of my work was therefore largely complete in draft by early 2002. In the summer of that year I alsohad the chance to discuss my views with Karine Chemla during a visit to Paris, and I recall �nding her suggestionson the dissection of solid forms particularly helpful. My work was however in no �t state for publication, becauseit represented no more than the results of my own obsession with the text, still largely based on single combat withthe Wénwù transcription, translated and annotated in the light of what I felt I already knew about early Chinesemathematics. One pressing but unaccomplished task before me was to take account of the work of Chinese scholars,and indeed of the detailed notes and commentary accompanying the transcription in [Péng, 2001]. But 2002 was morefull of major distractions from scholarship than 2001, and I was able to do little more than �ddle with minor detailsof what I had already written. It was not until the autumn of 2003 that I began the �nal efforts that led the project toits present state. The results of my translation work so far have now been published [Cullen, 2004]. I should like toacknowledge the great help I have derived from the generous practical assistance and critical input given by CatherineJami in the preparation of that work, as well as in the present article. I am also grateful to the editors of HistoriaMathematica for advice and encouragement in preparing a complex text for publication. Thanks are due to Péng Hào

for permitting the reproduction of the photograph of bamboo strips in this article.The present article is intended for a readership using modern Western languages. I have not therefore referred to

the large amount of excellent relevant scholarship in East Asian languages beyond the necessary minimum.

Appendix. Suàn shù shu—title or label?

In this discussion, I have referred to the ensemble of material before us as “the Suàn shù shu,” while in my articletitle I have added the gloss “Writings on Reckoning.” In this appendix I should like to discuss the signi�cance ofthese decisions. Firstly we may turn to the relatively trivial point of the pronunciation of the name itself. “Suàn shùshu” is a representation in Roman script of the sounds of the characters in modern standard Chinese, which isan idealized version of the northern Chinese language sometimes known as “Mandarin.” The accents mark the toneswhich are an integral part of the language. The character is simply an archaic form of the more common , withidentical meaning and pronunciation. However, since the material we are studying is over 2000 years old, one wouldnot expect the original owner to have pronounced the characters in the modern fashion. Various attempts have been


made to reconstruct the ways in which Chinese pronunciation has evolved in past millennia: one of the best-knownand most accessible is still that of Karlgren [1964]. Karlgren reconstructs two pronunciations, an “archaic” schemerepresenting the �rst millennium B.C., and an “ancient” scheme centering around A.D. 600. In the “archaic” schemeand using Karlgren’s diacritic markings, would be read . For interest, we may note that in the samescheme the title of the Nine Chapters, Ji�u zhang suàn shù , would have been read . Theuseful point of such reconstructions is when, as here, we note that two words with the same modern pronunciationsuch as and , both shù today, were formerly quite distinct as and respectively. But no argument basedon such considerations is central to my present discussion.

There are, however, some signi�cant issues involved with decisions of how we translate the heading of the doc-ument we are studying. First, what is meant by the expression Suàn shù ? Now this certainly is an expressionrather than just a pair of separate characters that happen to follow one another: an electronic search of the entire textsof the 25 standard histories gives 51 separate examples of this pairing. It also, as we have seen above, occurs in theZhou bì. Following the entry for in the now standard Hàny�u dà cídi�an [Luò, 1997, 5231] we mayagree that a broad characterization of this usage over Chinese literature as a whole would be that it means about thesame as the now identically pronounced but differently written expression suàn shù , “methods/procedures ofcalculation/mathematics” (leaving to one side the few instances in which should be read as suàn sh�u, with themeaning “count”). It seems likely that the present text shows the earliest datable example of the expression; in choos-ing a single English word to render it, I have allowed myself to be in�uenced by the elementary nature of much of thematerial before us, and have gone for the slightly archaic and (at one time) demotic term “reckoning.”

Rather more hangs on the question of how to render shu . All translations of the words Suàn shù shu intowestern languages so far appear to have used “book” or in the case of the one French example known to me theequivalent “livre.” While not denying that shu can be used in Chinese in ways equivalent to English “book,” I feelvery uncomfortable with this rendering. As I have argued above there is no way that the material written on the originalscroll of 190 bamboo strips from the Zhangjiashan tomb can be thought of as once having constituted a continuouspiece of writing with some kind of large-scale order to it—which is surely a minimum requirement for something tobe called a “book” in English. A mere collocation of short pieces of textual material, such as a collection of papersin a binder, or an electronic �le holding pages downloaded from the Internet, would not, I suggest, usually qualifyas a “book” in the view of most native English speakers. Or if I was to copy notes on (say) French grammar fromvarious sources onto a collection of sheets permanently bound together in hard covers, no one would normally say Ihad composed a book on that subject. However, as everybody familiar with classical Chinese will acknowledge, shumeans much more than just “book.” As a verb it means “to write,” and its basic sense as a noun is indeed the broaderone of “things written, writings, the act of writing.” I have therefore adopted “writings” as my preferred sense of thisword in the present context, since it seems to me to avoid the misleading implications of “book.”

By this point, the reader may see why I join L�� Dí in being reluctant to say that the characters Suàn shù shushould be described as a book title [L��, 1997, 91]. Like him, I feel that these words are best viewed as a descriptivelabel telling the potential reader what the collected material in the relevant bundle of strips is about. The test here is,of course whether the same label could have been put on a different collection of material on the same topic, and I feelthat it could well have been. I suggest that the same applies to the characters written on the back of strips from fourother rolls from the same tomb, such as Èr nián lü lìng , “Statutes and ordinances of the second year,” Màishu , “Writings on the vessels,” Gài Lú , “[Questions by] Gài Lú,” and Y��n shu , “Writings on stretching[as medical therapy].” The two other rolls reconstituted by the editors have no visible labels.

References

Pre-modern works 2

Hàn shu , “History of [Western] Hàn dynasty” by Ban Gù , largely complete on his death in A.D. 92, edn.1962, Beijing, Zhonghua press.

2 The listing of editions used is given here for convenience of reference. In the case of all Hàn and pre-Hàn texts, readers will �nd substantialdiscussions of content, text history, and associated questions in [Loewe, 1993].


Hòu Hàn shu , “History of Later [i.e. Eastern] Hàn dynasty” by Fàn Yè , c. A.D. 450, edn. 1963, Beijing,Zhonghua press.

Ji�u zhang suàn shù , “Nine Chapters on mathematical methods,” �rst attested c. A.D. 100, edn. [ Guo, 1990];see also edition in [Guo and Liú , 2001]; translated into English in [Shen, 1999] and into French in[Chemla and Guo, 2004].

Sh�� jì , “Records of the Historian” by S�m�a Qian , c. 90 B.C., edn. 1962, Beijing, Zhonghua press.Yì j�ng , “Book of change” [originally a manual of divination (pre-Qín), with later accretions dating as late as

Western Hàn], edn. 1815, Shí san j�ng zhù shù collection, repr. 1973, Taipei, Yiwen press.Zhou bì , “The gnomon of Zhou” [compiled near beginning of Christian era]; edition in [Guo and Liú ,

2001]; translated [Cullen, 1996].Zhou l�� , “Rituals of the Zhou dynasty,” Shí san j�ng zhù shù edn. 1815, Shí san j�ng zhù shù

collection, repr. 1973, Taipei, Yiwen press.

Modern works 3

Chemla, K., Guo, Shuchun, 2004. Les neuf chapitres: Le classique mathematique de la Chine ancienne et ses com-mentaires. Dunod, Paris.

Cullen, C., 1996. Astronomy and Mathematics in Ancient China: The Zhou bi suan jing. Cambridge Univ. Press,Cambridge, UK.

Cullen, C., 2004. The Suàn shù shu , “Writings on Reckoning”: A Translation of a Chinese MathematicalCollection of the Second Century B.C., with Explanatory Commentary. Needham Research Institute WorkingPapers 1, Needham Research Institute, Cambridge. This publication has been made available in both hard copy andelectronic form: See the details available on the Website of the Needham Research Institute, http://www.nri.org.uk.

Guan, X�hua , 2002. Zhongguó g�udài biaodi�an fúhào fazh�an (A History of the Devel-opment of Punctuation Marks in Pre-modern China). Bashu, Chengdu.

Guo, Shuchun , 1990. Ji�u zhang suàn shù ([Critical Edition of the] Nine Chapters). Liaoning EducationPress, Shenyang.

Guo, Shuchun , 2003. ‘Suàn shù shu’ chutan (A preliminary investigation of the Suàn shù shu).Guóxué yánjiu 11, 307–349.

Guo, Shuchun , Liú, Dùn , 2001. Suànj�ng shíshu ([Critical Edition of the] Ten MathematicalClassics). Jiuzhang Publishers, Taipei.

Harper, D.J., 1998. Early Chinese Medical Literature: The Mawangdui Medical Manuscripts. Sir Henry WellcomeAsian Series. Royal Asiatic Society, London.

Hulsewé, A.F.P., 1985. Remnants of Ch’in Law: An Annotated Translation of the Ch’in Legal and AdministrativeRules of the 3rd Century B.C. Discovered in Yün-meng Prefecture, Hu-pei Province, in 1975. Brill, Leiden.

Imhausen, A., 2003. Egyptian mathematical texts and their contexts. Science in Context 16 (3), 367–389.Jianglíng [anonymous collective work by team in charge of Jianglíng documents], 2000. Jianglíng Zhangjiashan

Hànji�an ‘Suàn shù shu’ shìwén (English title given in original publication as ‘Tran-scription of Bamboo Suan shu shu or A Book of Arithmetic from Jiangling’). Wénwù 9, 78–84.

Karlgren, B., 1964. Grammata Serica Recensa. Museum of Far Eastern Antiquities, Stockholm.Keegan, D., 1988. ‘The Huang-ti nei-ching’: The Structure of the Compilation; The Signi�cance of the Structure.

Ph.D. thesis in history. University of California at Berkeley.Lam, L.Y., Ang, T.S., 1992. Fleeting Footsteps: Tracing the Conception of Arithmetic and Algebra in Ancient China.

World Scienti�c, Singapore. Reprinted and revised in 2004.L��, Dí , 1982. ‘Ji�u zhang suàn shù’ zhengmíng wèntí de gàishù (An outline account

of controversies about the Nine Chapters). In [Wú, 1982, 28–59].L��, Dí , 1997. Zhongguó shùxué tongsh��: Shàngg �u dào W�udài juán (General History

of Chinese Mathematics: Volume on Early Times to the Five Dynasties). Jiangsu Educational Press, Nanjing.

3 The use of full or simpli�ed Chinese characters in this listing follows the style of the original publication. Names of Chinese authors in westernlanguage publications are given as in those publications.

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Lo, V., 1998. The in�uence of Yangsheng culture on early Chinese medical theory. Ph.D. thesis in history. Universityof London.

Loewe, M., 1961. The measurement of grain during the Han period. T’oung Pao 49, 64–95.Loewe, M., 1967. Records of Han Administration, Vol. 2: Documents. Cambridge Univ. Press, Cambridge, UK.Loewe, M. (Ed.), 1993. Early Chinese Texts: A Bibliographical Guide. Cambridge Univ. Press, Cambridge, UK.Luò, Zhufeng (Ed.), 1997. Hàny�u dà cídi�an (The Great Chinese Dictionary). Great Chinese Dictio-

nary Press, Shanghai.Martzloff, J.-C., 1997. A History of Chinese Mathematics. Springer-Verlag, Berlin.Needham, J., Wang, Ling, 1959. Science and Civilization in China, Vol. 3: Mathematics and the Sciences of the

Heavens and the Earth. Cambridge Univ. Press, Cambridge, UK.Péng, Hào , 2001. Zhangjiashan Hànji�an ‘Suàn shù shu’ zhùshì (The Hàn Dynasty

Book on Wooden Strips Suàn shù shu Found at Zhangjiashan with a Commentary and Explanation). Science Press,Beijing.

Rickett, W.A., 1998. Guanzi: Political, Economic and Philosophical Essays from Early China, Vol. 2. Princeton Univ.Press, Princeton, NJ.

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Shen, Kangshen, 1999. The Nine Chapters on the Mathematical Art: Companion and Commentary (Crossley, J.N.,Lun, A.W.-C, Trans.). Oxford Univ. Press, Oxford.

Shuìh�udì [anonymous collective work by team in charge of Shuìh�udì documents], 2001. Shuìh�udì Qínmùzhuji�an (Bamboo Strips from the Qín Tomb at Shuìh�udì). Wénwù Publishing, Beijing.

Sivin, N., 1995. Text and Experience in Classical Chinese Medicine. In: Bates, D.G. (Ed.), Knowledge and the Schol-arly Medical Traditions. Cambridge Univ. Press, Cambridge, UK, pp. 177–204.

Tsien, Tsuen-Hsuin, 2004. Written on Bamboo and Silk: The Beginnings of Chinese Books and Inscriptions, secondedition. Chicago Univ. Press, Chicago.

Twitchett, D., Loewe, M. (Eds.), 1986. The Cambridge History of China, Vol. 1: The Ch’in and Han Empire. Cam-bridge Univ. Press, Cambridge, UK.

Unschuld, P.U., 1986. Nan-Ching: The Classic of Dif�cult Issues. Univ. of California Press, Berkeley, CA.Unschuld, P.U., 2003. Huang Di nei jing su wen: Nature, Knowledge, Imagery in an Ancient Chinese Medical Text;

with an appendix, The Doctrine of the Five Periods and Six Qi in the Huang Di nei jing su wen. Univ. of California,Berkeley, CA.

Wú, Wénjùn (Ed.), 1982. ‘Ji�u zhang suàn shù’ yú Liú Hu� (Liú Hu� and the Nine Chapters).Beijing Shifan Daxue Press, Beijing.

Xiè, Guìhua et al. (Eds.), 1987. Juyán Hànji�an shìwén héjiào (Complete Transcriptions ofHàn Strips from Juyán). Wénwù Press, Beijing.

Yabuuti, K., 2000. Une histoire des mathématiques chinoises (Baba, K., Jami, C., Trans.). Berlin, Paris.Yamada, K., 1979. The formation of the Huang-ti Nei-ching. Acta Asiatica 36, 67–89.Zhangjiashan [anonymous collective work by team in charge of Zhangjiashan documents], 2001. Zhangjiashan

Hànmù zhuji�an: Èr sì q� hào mù (Bamboo Strips from the Hàn Tombs atZhangjiashan: Tomb 247). Wénwù Publishing, Beijing.


On mathematical problems as historically determined artifacts:Re�ections inspired by sources from ancient China

Karine Chemla ∗

CNRS, UMR 7596, REHSEIS, Paris, FranceUniversité Paris 7, 3 square Bolivar, 75019 Paris, France


For Menso FolkertsOn the occasion of his 65th birthday

As an expression of friendship and appreciation

Abstract

Is a mathematical problem a cultural invariant, which would invariably give rise to the same practices, independent of the socialgroups considered? This paper discusses evidence found in the oldest Chinese mathematical text handed down by the writtentradition, the canonical work The Nine Chapters on Mathematical Procedures and its commentaries, to answer this question inthe negative. The Canon and its commentaries bear witness to the fact that, in the tradition for which they provide evidence,mathematical problems not only were questions to be solved, but also played a key part in conducting proofs of the correctness ofalgorithms.© 2008 Elsevier Inc. All rights reserved.

Résumé

Un problème mathématique est-il un invariant culturel, qui déclencherait les mêmes pratiques quel que soit le groupe socialconsidéré ? Cet article analyse des témoignages prélevés dans les sources chinoises les plus anciennes à avoir été transmises par latradition écrite, l’ouvrage canonique Les neuf chapitres sur les procédures mathématiques et ses commentaires, pour répondre par lanégative à cette question. Ces documents attestent que, dans la tradition dont ils témoignent, les problèmes n’étaient pas seulementdes énoncés à résoudre, mais qu’ils jouaient un rôle clef dans la conduite des démonstrations de correction d’algorithmes.© 2008 Elsevier Inc. All rights reserved.

MSC: 01A25; 01A35; 01A85

Keywords: Mathematical problem; Ancient China; The Nine Chapters on Mathematical Procedures; Liu Hui; Proof of the correctness ofalgorithms

* Fax: +33 1 44 27 86 47.E-mail address: [email protected].

0315-0860/$ – see front matter © 2008 Elsevier Inc. All rights reserved.doi:10.1016/j.hm.2008.11.005




214 K. Chemla / Historia Mathematica 36 (2009) 213–246

Introduction

The main goal of this paper is methodological.1 It derives from the general observation that historians have oftenworked under the assumption that the main components of scienti�c texts—problems, algorithms and so on—areessentially ahistorical objects, which can be approached as some present-day counterparts. To be clear, by this diag-nosis, I do not mean that when problems re�ect tax systems or civil works, historians have failed to recognize thattheir statements adhere to a given historical context. Rather, I claim that they have mainly viewed problems in the waywe commonly use them today—or as we commonly think we do—that is, as formulations of questions that requiresomething to be determined and call for the execution of this task. In contrast, by focusing on mathematical problemsfound in ancient Chinese sources, this paper aims at establishing that we cannot take for granted that we know apriori what, in the contexts in which our sources were produced or used, a problem was. Moreover, besides providingevidence to support this claim, it suggests a method that can be used to describe the nature of mathematical problemsin a given historical tradition. My conclusion is that developing such descriptions should be a prerequisite to settingout to read sources of the past.

To substantiate these statements and illustrate this method, I shall concentrate on the earliest extant mathematicaldocuments composed in China. These sources are of various types, a point that will prove essential for my argument.The book that factually is the oldest mathematical writing that has survived from ancient China, the Book of Mathe-matical Procedures (Suanshushu ), was recently discovered in a tomb sealed ca. 186 B.C.E.2 In contrast to thisdocument, which was not handed down, the earliest book that has come down to us through the written tradition, TheNine Chapters on Mathematical Procedures (Jiuzhang suanshu ), to be abbreviated as The Nine Chapters,was probably completed in the �rst century C.E. and considered a “Canon ( jing )” soon thereafter.3 Both booksare for the most part composed of particular problems and algorithms solving them. What kind of texts were they?How should we read them? These are the main questions I have in view here. The outlines of the problems often echothe concrete problems that bureaucrats or merchants of that time might have confronted in their daily practice. It hashence often been assumed that mathematics in ancient China was merely practical, oriented, as it seems, toward solv-ing concrete problems. In fact, such a hasty conclusion conceals two implicit assumptions regarding problems. The�rst assumption, speci�cally attached to sources such as the Chinese ones, is that the situation used to set a problemshould be taken at face value: learning how to handle the concrete task presented by the problem would precisely bewhat motivated its inclusion in a text. The second assumption, much more generally held, is that the sole aim of aproblem is to present a mathematical task to be executed and to provide means to do so. In this paper, I shall castdoubt on the second assumption, showing that it distorts our view of the way in which problems were actually used inancient China. As a side result, the �rst assumption will also be undermined. This conclusion implies that we shouldbe careful in the ways in which we use the evidence provided by the particular situations described in the problems

1 A �rst version of this paper was presented at the conference organized by Roger Hart and Bob Richards, The Disunity of Chinese Science,which was held in Chicago on May 10–12, 2002. It is my pleasure to take the opportunity to thank the organizers of this meeting and the audiencefor their comments. A revised version was prepared at the invitation of Della Fenster, for the conference “Exploring the History of Mathematics:How Do We Know What Questions To Ask?” that was held at the University of Richmond, Richmond, VA (USA), on May 12–15, 2004. May shebe thanked for her generosity and encouragement. Markus Giaquinto, the referees, and the editors of Historia Mathematica spent much time tryingto help me formulate the argument of this paper more clearly. I owe them a huge debt of gratitude.

2 The �rst critical edition was published in Peng Hao [2001]. Since then, several papers have suggested philological improvements, suchas Guo Shirong [2001], Guo Shuchun [2001]. Two translations into English are forthcoming: [Cullen, 2004], the �rst version ofwhich was published on the internet, and [Dauben, 2008]. A new critical edition and translation into Japanese and Chinese was published recently:[ Chôka san kankan Sansûsho kenkyûkai. Research group on the Han bamboo slips from Zhangjiashan Bookof Mathematical Procedures, 2006]. Recently, Cullen [2007] presented his rewriting of the mathematics of ancient China based on the discovery ofthe manuscript.

3 I argue in favor of this dating in Chemla and Guo Shuchun [2004, 475–481]. In chapter B [Chemla and Guo Shuchun, 2004, 43–56], GuoShuchun presents the various views on the date of completion of the book held by scholars in the past and argues that the composition of The NineChapters was completed in the �rst century B.C.E. In what follows, I shall regularly rely on the glossary of mathematical terms I composed toback up the translation into French of The Nine Chapters and the commentaries (see below) [Chemla and Guo Shuchun, 2004, 897–1035]. In it,the entry jing , “Classic, Canon,” provides evidence countering the commonly held view that only in the seventh century were such books asThe Nine Chapters considered to be “Canons.” In Chemla [2001 (forthcoming), 2003b], on the basis of the extant evidence regarding The NineChapters, I discuss the kind of scripture a Canon constituted in ancient China. In addition to separate contributions, [Chemla and Guo Shuchun,2004] contains joint work, such as the critical edition of The Nine Chapters and the commentaries on which this paper relies.

K. Chemla / Historia Mathematica 36 (2009) 213–246 215

inserted in mathematical documents. We cannot conclude merely from their outer appearance that they were writtenonly for practical purposes.

How can we devise a method for determining what a mathematical problem was in a given historical context? Thesuggestion I develop in this paper is to look for readers of the past who left evidence regarding how they read, orreacted to, our sources. This is why the Chinese case is most helpful: in the form of commentaries, it yields sourcematerial documenting early readings of some mathematical sources. This is also where the difference in type betweenthe two Chinese documents mentioned above is essential. In relation to its status as a Canon, The Nine Chapterswas handed down and several commentaries were composed on it. Among these commentaries, two were selectedby the written tradition to be handed down together with the text of the Canon: the one completed by Liu Huiin 263 and the one presented to the throne by Li Chunfeng in 656. As we have already stressed above, TheNine Chapters is for the most part composed of problems and algorithms solving them. The commentaries bear onthe algorithms and, less frequently, on the problems. They are placed at the end of the piece of text on which theycomment or between its statements. In this paper, I shall gather evidence from Liu Hui’s commentary with respect tohow he used problems and also how he read those included in The Nine Chapters. This will provide us with sourcematerial to reconsider in a critical way the nature of a mathematical problem in ancient China and to establish thatproblems were submitted to a mathematical practice differing, on many points, from the one we spontaneously attachto them.

What does such evidence tell us? To start with, it shows us why the questions we raise on problems are essentialand cannot be dismissed if we are to read our sources in a rational way. We shall select two examples of the dif�cultieswhich face the historian and which are illustrated by the commentary. First, The Nine Chapters describes a procedurefor multiplying fractions after the following problem:

(1.19) Suppose one has a �eld which is 4/7 bu wide and 3/5 bu long. One asks how much the �eld makes., , 4 [Chemla and Guo Shuchun, 2004, 170–171]

Why is it that, in the middle of his commentary on the procedure for multiplying fractions, the third centurycommentator Liu Hui introduced another problem that can be formulated as follows:

1 horse is worth 3/5 jin of gold. If a person sells 4/7 horse, how much does the person get?5

Seen from our point of view, the two problems are identical: their solution requires multiplying the two fractionsby one another. Obviously, from Liu Hui’s perspective, they differ. Otherwise, he would not need to change one forthe other. Inquiring into their difference as perceived by Liu Hui should hence disclose one respect in which thepractice of problems in ancient China differs from the one we would be spontaneously tempted to assume. This willbe our �rst puzzle.

In the example above, the numerical values remain the same. The �rst dif�culty hence regards the interpretation ofthe situation chosen to set a problem. In addition, the commentator also regularly changes the numerical values withwhich a problem is stated, without changing the situation, before he comments on the procedure attached to it. Forinstance, Problem 5.15 in The Nine Chapters, which requires determining the volume of a pyramid, reads as follows:

4 The shape of the �eld is designated by the names of its dimensions: a �eld with only a length (north–south direction) and a width (east–westdirection) is rectangular. I discuss the names of geometrical �gures and their dimensions in Chemla and Guo Shuchun [2004, chapter D, 100–104].“1.19” indicates the 19th problem of Chapter 1 of The Nine Chapters. The same convention is used for designating other problems in this paper.Note that at the time when The Nine Chapters was composed, the bu was between 1.38 and 1.44 m. The values of the length and width do not seemconcrete.

5 Section 3 of this paper describes the text in greater detail. I refer the reader to this section to see precisely how the commentator formulatesthis problem. Note that procedures for multiplying fractions similar to the one in The Nine Chapters occur in several contexts in the Book ofMathematical Procedures. In some cases, the procedure is given apparently without the context of any problem (bamboo slips 6, 7 [Peng Hao

, 2001, 38, 40]). In another case, its statement follows a problem of the type Liu Hui introduces in his commentary (selling a fraction of anarrow and getting cash in return, bamboo slips 57–58 [Peng Hao , 2001, 65]). In yet other cases, the procedure is formulated in relation tocomputing the area of a rectangular �eld and showing how it inverts procedures for dividing (bamboo slips 160–163 [ Peng Hao , 2001, 114]).


(5.15) Suppose one has a yangma, which is 5 chi wide, 7 chi long and 8 chi high. One asks how much the volume is., , , 6 [Chemla and Guo Shuchun, 2004, 428–429]

At the beginning of his commentary, which, as always, follows the procedure, Liu Hui states:

Suppose7 the width and the length are each 1 chi, and the height is 1 chi , , . . .

Again, the same question forces itself upon us: for which purpose does the commentator need to change thenumerical values here, as he also regularly does when he discusses other surfaces and solids? This will be our secondpuzzle.

My claim is that it is only when we are in a position to argue for an interpretation of the differences between, on theone hand, the situations and, on the other hand, the numerical values that we may think we have devised a non-naivereading of mathematical problems as used in ancient China.

This paper develops an argument for solving the two puzzles. Here are the main steps of the argument:First, I shall show that we must discard the obvious explanation that one could be tempted to put forward, that is,

that a problem in ancient China only stood for itself. We can put forward evidence showing that the commentatorsread a particular problem as standing for a class (lei) of problems. Moreover, they determined the extension of theclass on the basis of an analysis of the procedure for solving the particular problem rather than simply by a variation ofits numerical values. An even stronger statement can be substantiated: one can prove that these readers expected thatan algorithm given in The Nine Chapters after a particular problem should allow solving as large a class of problemsas possible [Chemla, 2003a]. In making these points, we shall thereby establish the �rst elements of a description ofthe use of problems in mathematics in ancient China.

Section 1 thus proves that our puzzles cannot be easily solved and require explanations of another kind. In Section 2,I introduce some basic information concerning the practice of proving the correctness of algorithms as carried out bycommentators like Liu Hui, since this will turn out to be necessary for our argumentation.

On this basis, Section 3 concentrates on the situations used to set problems. It argues that, in fact, far from beingreduced to questions to be solved, the statements of problems, and more speci�cally the situations with which theyare formulated, were an essential component in the practice of proof as exempli�ed in the commentaries. Viewing thesituations from this angle allows accounting for why the problem with cash was substituted for that about the area ofa �eld.

Section 4 focuses on the numerical values given in the problems and suggests that, if we set aside cases in which achange of values aims at exposing the lack of generality of a procedure, it is only within the framework of geometrythat Liu Hui changed the values in the statements of problems. In Section 4, I argue that the reason for this is thatthe commentators introduce material visual tools to support their proofs and that the numerical values given in theproblems refer to these tools. This leads me to suggest a parallel between the role played by visual tools in thecommentaries and the part devoted to the situations described by problems.

Sections 1 to 4 concentrate on how the evidence provided by the commentary allows solving the two puzzles putforward. To be sure, these puzzles could also be grasped only thanks to the commentaries. As a result, my argumentestablishes how the practice with problems attested to by the commentators differs from our own and how this de-scription accounts for the dif�culties presented. Even if this is to be considered as the only outcome of the paper,we would have ful�lled our aim of providing an example of a practice with problems that does not conform to ourexpectations. Now, the question is: can we go one step further and transfer the results established with respect to theuse of problems by the commentators to The Nine Chapters itself, even though the earliest extant commentary on The

6 The yangma designates a speci�c pyramid, with a rectangular base. Its shape is de�ned by the fact that its apex is above a vertex of therectangular base. See Section 4. I opt for not translating the Chinese term, to avoid a long expression that would express the yangma as a kind ofpyramid, whereas the term in Chinese does not link the shape to that of other geometrical solids. We do not know exactly by means of which objectthe Chinese term designates the pyramid. Nor do we know whether, at the time when The Nine Chapters was composed, the term had acquired atechnical meaning or was only designating a speci�c object with the geometrical shape required. Note also that, at that time, 1 chi was between0.23 and 0.24 m.

7 The expression for “suppose” is not the same as in The Nine Chapters itself; i.e., jinyou has been replaced by “suppose (jialing),” which in thecommentaries as well as in some later treatises is more frequently used to introduce a problem.


Nine Chapters was composed probably some two or three centuries after the completion of the Canon? This questionis addressed somewhat brie�y in Section 5.

The reason that we must proceed in this indirect way relates to the fact that commentators wrote in a style radicallydifferent from that of the Canons. More precisely, commentaries express expectations, motivations, and second-orderremarks, all these elements being absent from The Nine Chapters, which is mainly composed of problems and algo-rithms. Commentaries hence allow us to grasp features of mathematical practice that are dif�cult to approach on thebasis of the Canon itself, at least when one demands that a reading of ancient sources be based on arguments. Onepoint must be emphasized: Unless we �nd new sources, there is no way to reach full certainty about whether what wasestablished on the basis of third-century sources holds true with respect to writings composed some centuries earlier.However, this having been said, two remarks can be made.

First, relying on commentaries composed more than two centuries after the Canon to interpret the latter appears tobe a less inadequate method than relying on one’s personal experience of a mathematical problem. It seems to me tobe more plausible that the practice of problems contemporary with the compilation of The Nine Chapters is related toLiu Hui’s practice than that it is related to ours. However, here we are in the realm of hypothesis rather than certainty.

Second, once we restore the practice of problems to which the commentaries bear witness, we can �nd manyhints indicating that some conclusions probably hold true for The Nine Chapters itself and even for the Book ofMathematical Procedures. Section 5 is devoted to discussing such hints. With these warnings in mind, let us turn toexamining our evidence.

1. How does a problem stand for a class of problems?

1.1. A �rst description of the statement of problems

Problem 1.19 quoted above illustrates what, in general, a problem in The Nine Chapters looks like. It is particularin two respects. The statement of the problem refers to a particular and most often apparently concrete situation, suchas, in this case, computing the area of a �eld. Moreover, it mentions a particular numerical value for each of the datainvolved—in this case, 3/5 bu and 4/7 bu for the data “length” and “width,” respectively.8 However, some problemsare only particular in this latter respect. An example of this is Problem 1.7, one of three that precede the procedure forthe addition of fractions:

(1.7) Suppose one has 1/3 (one of three parts), 2/5 (two of �ve parts). One asks how much one obtains if one gathersthem. , ,

Although numerical values are given, the fractions to be added, or in other terms, the units out of which parts aretaken, are abstract.9 All problems in The Nine Chapters are of one of the two types exempli�ed by 1.19 and 1.7.

The procedures associated with the problems in The Nine Chapters also show some variation. Problem 1.19 isfollowed by a procedure for the multiplication of parts, which amounts to a

b· c

d= ac

bd. It is expressed in general,

abstract terms, since it makes no reference to the situation in the problem, the area of a �eld:

Multiplying partsProcedure: the denominators being multiplied by one another make the divisor. The numerators being multiplied by one

8 The same remark holds true for the problems in the context of which the procedure amounting to the “Pythagorean theorem” is discussed.Further, in fact, there is evidence showing that at the latest in the third-century the term “�eld” came to designate a geometrical shape in general. Se eChemla and Guo Shuchun [2004, Glossary, 992–993]. It is hence dif�cult to determine whether the statement of the problem in The Nine Chaptersstill uses the term with a concrete sense or already with a technical meaning. The same problem was raised above regarding the interpretation ofyangma. More generally, the quali�cation of the statement of a problem as “concrete” should be manipulated with care. Vogel [1968, 124–127]describes the general format of a problem in The Nine Chapters. He suggests that the problems can be divided into two groups: those dealingwith problems of daily life and those that can be considered as recreational problems. In my view, this opposition is anachronistic and does not �tthe evidence we have from ancient China (see the following discussion). Moreover, Vogel fails to point out that there are problems formulated inabstract terms.

9 For a general discussion on the form of these problems, see Chemla [1997a]. Compare [Cullen, 2007, 17; Guo Shuchun , 2002, 514–517] for a description of the form of problems in the Book on mathematical procedures.


Fig. 1. The problem of the log stuck in the wall.

another make the dividend. One divides the dividend by the divisor.

, , [Chemla and Guo Shuchun, 2004, 170–171].

However, in other cases, the procedure given in The Nine Chapters is expressed with respect to the concretesituation and values described by the problem. An example of this is the following problem from Chapter 9, “Basisand height (gougu),” which is devoted to the right-angled triangle (cf. Fig. 1):

(9.9) Suppose one has a log with a circular section, stuck into a wall, with dimensions unknown. If one saws it with a sawat a depth of 1 cun (CD), the path of the saw (AB) is 1 chi long. One asks how much the diameter is.Answer: the diameter of the log is 2 chi 6 cun.Procedure: half the path of the saw being multiplied by itself, one divides by the depth of 1 cun, and increases this (theresult of the previous operation) by the depth of 1 cun, which gives the diameter of the log.

, , ,

, , , 10

In modern terms, the procedure amounts to the formula11 AC2

CD + CD = 2AO. Since both the problem and theprocedure are formulated in the same concrete terms, we might, for such cases at least, be tempted to assume that theyare to be read as standing only for themselves. Interestingly, as we will see below, we can �nd evidence allowing us todetermine how Liu Hui read this problem. It clearly shows that even in such cases this assumption must be discarded.Moreover, it also reveals how Liu Hui used this problem as a general statement, and not as a particular one.

1.2. The commentator Liu Hui’s reading and use of problems

The piece of evidence on which we can rely to approach the commentator’s reading of the latter problem comesfrom Liu Hui’s commentary on Problems 1.35/1.36 and the procedure included in The Nine Chapters for the deter-mination of the area of a circular segment. The key point is that, in this piece of commentary, Liu Hui refers to theproblem of the log stuck in a wall from the Canon. The Problems 1.35 and 1.36 are similar. The second one reads asfollows (cf. Fig. 2):12

10 [Chemla and Guo Shuchun, 2004, 714–715]. In the translation, for the sake of my argument, I have inserted references to a geometrical �guredrawn by myself (Fig. 1). Needless to say, neither the �gure nor the references are to be found in The Nine Chapters. More generally, the Canondoes not refer to any visual tool.11 In the right-angled triangle OAC, the difference of the hypotenuse and the side OC is equal to CD. Dividing AC2 by CD yields the sum of thehypotenuse and the side OC; hence the result.12 Again, I have drawn the �gure for the sake of commenting. No such �gure is to be found in the original sources. Moreover, I have added to thetranslation of the original text references to the �gure between brackets.


Fig. 2. The area of the circular segment.

(1.36) Suppose again13 that one has a �eld in the form of a circular segment, whose chord (AB) is 78 bu 1/2 bu,and whose arrow (CD) is 13 bu 7/9 bu. One asks how much the �eld makes. , ,

[Chemla and Guo Shuchun, 2004, 190–191]

The Nine Chapters then provides an algorithm to compute the area of this �eld, which amounts to the formula(AB.CD + CD2)/2. In his analysis of this procedure, Liu Hui �rst shows that when the circular segment is half of thecircle, in fact the algorithm computes the area of the half-dodecagon inscribed in the circle. Moreover, he stresses thatthe imprecision increases when the circular segment is smaller than the half-circle. This imprecision motivates him toestablish a new procedure, which derives from tiling the circular segment with triangles and computing its area as thesum of their areas.14 It is within this context that Liu Hui �rst needs to compute the diameter of the circle containingthe circular segment from the two data of Problem 1.36, the chord (AB) and the arrow (CD). For this, he refers toProblem 9.9 on the log stuck in a wall as follows:

(. . .) It is appropriate then to rely on the procedure of the (problem) where one saws a log with a circular section in the(Chapter) “Basis (gou) and height (gu)” and to look for the diameter of the corresponding (circle) by taking the chord ofthe circular segment as the length of the path of the saw, and the arrow as the depth of the piece sawn. Once one knowsthe diameter of the circle, then one can cut the circular segments in pieces. (. . .) , ,

, , (My emphasis). [Chemla and Guo Shuchun, 2004, 192–193]

This piece of evidence shows that Liu Hui does not read the problem of the log stuck in a wall and the procedureattached to it as merely standing for themselves, but as expressing something more general. It thus reveals that, evenin a case like Problem 9.9, where the procedure is expressed with reference to the concrete situation of a log stuck ina wall and particular values, the third-century commentator reads its meaning as exceeding this particular case. Thisholds true in the entire commentary, in which one can �nd other pieces of evidence con�rming this conclusion, as wewill see below.

1.3. The procedure de�nes the extension of the class meant by the problem

Another interesting example of the generality the commentator attaches to a problem and the corresponding pro-cedure is the following one:

(6.18) Suppose that 5 persons share 5 coins (units of cash) in such a way that what the two superior persons obtain is equalto what the three inferior persons obtain.15 One asks how much each obtains. , ,

[Chemla and Guo Shuchun, 2004, 526–529]

13 This is the usual beginning for problems after the �rst one, when a mathematical question is dealt with through a sequence of problems.14 This reasoning implies covering an area with an in�nite number of tiles and it has been the topic of much discussion in the literature. It fallsoutside the topic of this paper. For a concise exposition, compare [Li Yan and Du Shiran, 1987, 68–69].15 As the commentator makes clear, it is assumed that the �ve persons have unequal ranks de�ned by the integers 5 to 1 and that the share theyobtain depends on their rank. We may feel that the statement of the problem is incomplete, but this would mean that we project our own expectationsof how a problem should be formulated. Actual description of how problems recorded in historical documents were formulated should replace thisanachronistic approach. However, dealing with this topic would exceed the scope of this paper.


The procedure following the answers to the problem is also expressed with reference to the particular situationand values mentioned in the statement. However, in this case, through the analysis of the procedure that he thendevelops, Liu Hui brings to light that the procedure is not general. To be more precise, the procedure adequatelysolves the problem, but cannot solve all similar problems, because it uses features speci�c to the situation describedin Problem 6.18. Note that this is the only case when this happens for a procedure given in The Nine Chapters andthat the commentator immediately exposes the lack of generality. This reaction betrays his expectation that a problemstands not only for itself, but also for a class (lei). Liu Hui reacts to this situation in several steps.

First, his analysis determines criteria that enable him to know to which problems the procedure of The Nine Chap-ters can be successfully applied.16 In a sense, Liu Hui inquires into the class of problems for which the particularproblem stands, and he does so through an examination of the procedure.

Second, he formulates another problem similar to that of the Canon, as follows:

Suppose17 7 persons share 7 coins and they want to do this in such a way that (what) the two superior persons (obtain) isequal to (what) the �ve inferior persons (obtain). , [Chemla and Guo Shuchun,2004, 528–529]

The criteria previously put forward immediately show why the procedure of The Nine Chapters does not apply tothis problem.18 Despite the appearances, on the basis of the procedure stated, the two problems do not belong to thesame class. In other terms, the category of problems for which a problem stands is not determined by a variation of itsnumerical values, but rather by the procedure provided to solve it.

Third, Liu Hui suggests modifying the procedure in such a way that it solves all similar problems. Seen fromanother angle, Liu Hui aims at stating a procedure for which all similar problems belong to the same class. We seethat the commentary on a procedure analyzes it in such a way as to inquire into the extension of its validity andmodi�es it to extend the class of problems that can be solved by it (and for which a particular problem stands). Asa result, on the basis of the sections of Liu Hui’s commentary examined so far, and in fact of others, we can thusstate that, in his view, a problem stands for a class (lei) of problems that is determined on the basis of the proceduredescribed after it. It is not so much the similarity of structure between the situations described by different problemsthat allow considering them as sharing the same category, but, most importantly, the fact that they are solved by thesame procedure.19 We hence reach the conclusion that far from being only the sequence of operations allowing a givenproblem to be solved, the procedure is read beyond the speci�c context within which it is formulated, and, further, iteven determines the scope of generality of a given problem. We shall come back to this issue in Section 3.

In the fourth step, the one we are most interested in here, Liu Hui suggests an entirely different, more generalprocedure for dealing with Problem 6.18. In fact, the commentator does this simply by suggesting “to imitate theprocedure” given in the Canon for the next problem, 6.19. This problem reads as follows:

16 These criteria are as follows: the number of inferior persons must exceed the number of superior ones by only 1; moreover, the sum of thecoef�cients attached to the superiors (5 and 4) must be greater than that attached to the inferiors (3,2,1). I do not enter into any detail here,referring the interested reader to Chemla [2003a].17 Here too, the expression jinyou for “suppose” in The Nine Chapters has been replaced by jialing. In this paper, I do not discuss the numericalvalues chosen to set a problem. However, clearly they call for comment. The �gures used in Problem 6.18 in The Nine Chapters are the simplestpossible with which the mathematical question can be formulated. In his commentary, Liu Hui introduces values that are the simplest possible tomake his point. Also, the �gures occurring in Problem 1.19 (3, 4, 5, 7; see above) are probably chosen on purpose. Compare my introduction toChapter 9 in Chemla and Guo Shuchun [2004, 663–665, 684–689]. Further research is needed in this respect.18 There are three more persons among the inferiors than among the superiors. Moreover, the sum of the coef�cients attached to the superiors(7 + 6) is smaller than the sum of those attached to the inferiors (5 + 4 + 3 + 2 + 1).19 On this point, I refer the reader to Chemla [1997a], where I analyze how Liu Hui uses the term lei “class, category” with respect to problems. Asrightly stressed by C. Cullen, in the earliest known theoretical discussion on the modes and methods of inquiry in mathematics and cosmography,i.e., in the opening sections of The Gnomon of the Zhou (Zhou bi), which he dates to the beginning of the common era, the concept and practiceof “categories” in mathematics are central [Cullen, 1996, 74–75, 177]. In my glossary of mathematical terms, I discuss more generally the varioususes of the term lei in the commentaries on The Nine Chapters and in philosophical texts of antiquity [Chemla and Guo Shuchun, 2004, 48–949]. Itis interesting that the Book of Mathematical Procedures also attests to the use of the term lei in the description of mathematical procedures as earlyas the second century B.C.E. (see bamboo slip 21 [Peng Hao , 2001, 45]).


(6.19) Suppose that a bamboo has 9 internodes20 and that the 3 inferior internodes have a capacity of 4 sheng whereasthe 4 superior internodes have a capacity of 3 sheng. One asks, if one wants that between two (neighboring) inner in-ternodes capacities be uniformly distributed,21 how much they each contain. , ,

,

Again, the procedure described after the statement of Problem 6.19 refers to the particular situation and valuesdisplayed in the statement. However, despite the differences on both counts between Problems 6.18 and 6.19, Liu Huidirectly imitates the procedure solving 6.19 (that is, transfers it step by step) to solve Problem 6.18, and, beyond, theproblems that now belong to the same class. As a result, the new procedure shapes Problem 6.18 as standing for amuch larger class.

In this case, as in the case of the commentary following Problem 1.36 quoted above, the same phenomenon recurs:the procedure circulates from one context to the other, disregarding the change in situation and in numerical values.22

What is particularly noteworthy, however, is how Liu Hui does so in both contexts. Let us explain on the example ofthe commentary on the circular segment (Problem 1.36) what we mean by the “circulation” of a problem. Liu Hui doesnot feel the need to express a more abstract statement or procedure that would capture the “essence” of Problem 9.9and could be applied to similar cases such as Problem 1.36. On the contrary, he directly makes use of the proceduregiven after 9.9, with its own terms, in the context of 1.36, by establishing a term-to-term correspondence, “taking thechord of the circular segment as the length of the path of the saw, and the arrow as the depth of the piece sawn.” Thisseems to indicate that the situation described in Problem 9.9 can be directly put into play in other concretely differentsituations. The particular appears to be used to state the general in the most straightforward way possible.

Further, to describe a more general procedure for problems of the same class as 6.18, Liu Hui imitates the procedurefor 6.19 within the context of the most singular of all problems (6.18) and not that of the more “generic” one, whichhe introduced as a counterexample. More importantly, for all problems such as 6.18, he could simplify the proceduregiven for 6.19 to make it �t certain speci�c features that these problems all share—for problems like 6.18, in contrast tothe bamboo problem and its middle internodes, there are no persons who do not belong to either the group of inferiorsor that of superiors. Instead, Liu Hui prefers to keep the procedure with the higher generality that characterizes it. Theconclusion of the previous paragraph can be stated in a stronger way: the most particular of all paradigms is used toformulate the most general of all algorithms and consequently it now stands for a much wider class.

A remark concerning the use of problems in the context of commentaries is here in order and will prove usefulbelow. Liu Hui uses the procedure solving Problem 9.9 in a commentary in which he describes a new procedure forcomputing the area of the circular segment. At the same time as he shapes the procedure, he shows why it is correct.Such a concern for the correctness of algorithms drives the greatest part of the commentaries, which systematicallyestablish that the procedures in The Nine Chapters are correct. Problems play a key role in achieving this goal. In thecontext examined, the use of Problem 9.9 is signaled by the verb “to look for qiu .”23 More generally, this termsignals the use of a problem in a proof. The commentary on the area of a circular segment illustrates the following useof a problem in a proof: The task of establishing the new procedure is divided into subtasks, which are identi�ed withproblems known—to start with, Problem 9.9. On the one hand, using the procedure that The Nine Chapters gives tosolve Problem 9.9 yields the �rst segment of the procedure now sought for. On the other hand, since the correctnessof the procedure solving 9.9 has already been established, the commentator can rely on the fact that it yields themagnitude needed at this point of the reasoning, that is, the diameter of the circle. In the terms Liu Hui uses to speakof the proof of the correctness of a procedure, the “meaning yi ” of the result of the �rst segment of the procedurehas been ascertained. More generally, let us stress the fact that, within the proof of the correctness of an algorithm heis shaping, Liu Hui uses problems and procedures solving them to determine step by step the “meaning” of the wholesequence of operations, that is, to determine that the procedure he establishes yields the area of a circular segment.

20 The trunks of bamboos have nodes. The Chinese term refers here to the space between two nodes and considers the “capacity” of the volumethus formed. The problem hence deals with nine terms of an arithmetic progression—the capacities of the successive spaces between the nodes.The sums of the �rst three terms and of the last four ones, respectively, are given and it is asked to determine all the terms.21 As in the previous problem, the capacities of the cavities form an arithmetic progression.22 Compare the analysis of transfer between situations in Volkov [1992, 1994].23 Note that the verb occurs in the statement of problems and tasks in the Book of Mathematical Procedures; see bamboo slips 160–163 [PengHao , 2001, 114].


1.4. A similar way of reading and using problems in later sources

In ancient China, this way of using procedures stated in one context directly in another, illustrated above forthe use of the procedure solving Problem 9.9 for dealing with the area of a circular segment, was not speci�c toLiu Hui. In fact, one can �nd a similar piece of evidence four centuries later, in a seventh-century commentaryon the Mathematical Canon Continuing the Ancients (Qigu suanjing), written by Wang Xiaotong in the �rst halfof the seventh century.24 The commentary relates the �rst problem of the book, which is devoted to astronomicalmatters, to a problem dealing with a dog pursuing a rabbit, indicating that the latter problem is included in The NineChapters.25 In this case, too, the latter problem is not reformulated in astronomical terms, nor is a third and abstractdescription of it introduced as a middle term, to allow the result concerning the dog and the rabbit to be applied inastronomy. In exactly the same way as described above, although The Nine Chapters presents the problem within aparticular concrete context, the �rst reader that we can observe, namely the seventh-century commentator, reads it asexemplifying a set of problems sharing a similar structure and solved by the same algorithm. Furthermore, as in theprevious example, the commentator feels free to make the problem and procedure, which apparently do not relate toastronomy, “circulate” as such into a different, astronomical context. This seems to indicate that there was an ongoingtradition in ancient China that did not mind discussing general mathematical procedures in the particular terms ofthe problems in which they had been formulated, although the questions discussed exceeded the case illustrated bythe particular situation. The above evidence from the third and the seventh centuries leads to the same conclusions.This indicates that it would not be farfetched to assume that this was also the way in which the authors of The NineChapters conceived of the problems that they included in the Canon. This seems all the more reasonable because, aswe have indicated above, except for one case (6.18), all procedures following the statement of problems are general.26

We shall come back to this issue in greater detail in Section 5.Even though it is perhaps less striking in comparison with our own uses of problems, let us stress that, in fact, the

same pieces of evidence show that these conclusions hold true with respect to the numerical values. Although Liu Huiregularly comments on a given problem and procedure on the basis of particular numerical values, he understands themeaning of his discussion as extending beyond this particular set and as, in fact, general. Again, in this respect, thecommentator thus proves to discuss the general in terms of a particular [Chemla, 1997a].

The problems of logs stuck in walls and dogs pursuing rabbits that can be found in The Nine Chapters maybe perceived as recreational by some readers of today, because of the terms in which they are cast. The evidenceexamined proves that things are not so simple. The historian is thus warned against the assumption that the categoryof “mathematical problem” remained invariant in time. Such a historical reconstruction guards us against mistakinga problem as merely particular or practical, when Chinese scholars read it as general and meaningful beyond its owncontext, or mistaking it as merely recreational when it was put to use in concrete situations.

Now that we have seen that, in Liu Hui’s practice, a problem did practically stand for a category of problems, andhow it did so, we have discarded the simple solution that could have accounted for our puzzles. We are hence left withthe question: Why is it that, within the context of his commentary on the procedure for “multiplying parts,” or on thatfor the volume of the yangma, Liu Hui feels it necessary to substitute one situation for another, or one set of values foranother, although both the original and the substituted problem seem to us to share the same category? One may even

24 Volume 2 of Qian Baocong [1963] contains a critical edition of Wang Xiaotong’s Qigu suanjing. The problem and commentary mentioned areto be found in Qian Baocong [1963, 2, 495–496]. Eberhard (Bréard) [1997] discusses this example. Part of her discussion and her translation arepublished in Bréard [1999, 41–43, 333–336]. For a further study concerning transmissions of problems of this type, see Bréard [2002].25 Although The Nine Chapters contains similar problems, the extant editions do not contain precisely the one quoted. Since we are only interestedhere in how a problem dealing with a given situation is used in another context, this textual problem can be left aside.26 This fact is only one feature among the many hints indicating that generality was a key epistemological value that inspired the composition ofThe Nine Chapters. Another hint is provided by how the “chapters zhang ” are composed. They each embody a part of mathematics that derivesfrom a unique procedure, and in correlation with this fact, their text is organized around the generality of the procedure placed at the beginning,which by derivation commands the whole chapter. Alexei Volkov stresses this fact in his translation of the title of the Canon as ComputationalProcedures for Nine Categories [of Mathematical Problems] [Volkov, 1986, 2001]. Compare the translations of the title as Nine Chapters on theMathematical Art [Needham and Wang Ling, 1959, 19], Nine Books on Arithmetical Techniques [Vogel, 1968], Nine Chapters of Calculation [WangLing, 1956, 15], Mathematical Methods in a Nine-Fold Categorization [Cullen, 2002, 784], and Mathematical Procedures under Nine Headings[Cullen, 2004, 1]. Among the meanings that may be intended by the title, the term zhang designates a division in a writing or a stage in a processof development, as well as, more generally, a distinction. Whatever the interpretation of the title may be, the division of the book into nine chaptersmanifests the in�uence of the value of generality.


say that after examining the evidence presented so far, our puzzles look even more intriguing. Elaborating a solutionfor these puzzles will compel us to enter more deeply into the practice of mathematical problems as exempli�ed inLiu Hui’s commentary.

2. Proving the correctness of procedures in order to find out general formal strategies in mathematics

The interpretation of Liu Hui’s commentary on the procedure for “multiplying parts” or computing the volume ofthe yangma requires that we recall some basic information regarding the mathematical practices linked to the exegesisof such a Canon as The Nine Chapters.

As has been recalled above, after virtually every procedure given by The Nine Chapters for solving a problem or aset of problems, the commentators systematically establish its correctness. However, the way in which they deal withthe issue of correctness manifests a speci�c practice of proof, which can be linked to the context of exegesis withinwhich it develops.27 I shall sketch its main characteristics, since it will prove useful for solving our puzzles.

To this end, I shall �rst evoke Liu Hui’s commentary on the algorithm given by The Nine Chapters for adding upfractions, which appears to be a pivotal section in his text.28

2.1. Proving the correctness of the procedure for the addition of fractions

In Chapter 1, where the arithmetic of fractions is dealt with, the Canon presents three problems similar to Prob-lem 1.7, after which it offers the following general and abstract algorithm for adding up fractions, equivalent toab

+ cd

= ad+cbbd

:

Gathering partsProcedure: The denominators multiply the numerators that do not correspond to them; one adds up and takes this as thedividend (shi). The denominators being multiplied by one another make the divisor (fa). One divides the dividend by thedivisor. (. . .)

, , [Chemla and Guo Shuchun, 2004, pp. 156–161].

Let us outline how, in his commentary on this section of the Canon, Liu Hui establishes the correctness of thisprocedure.

The Canon and the commentaries approach the object “fraction,” or in Chinese terms: “parts,” in two ways. Theexpression for m/n used in The Nine Chapters, “m of n parts” (n fen zhi m n m), gives the fraction as beingcomposed of “parts.” This dimension is the one emphasized in what I call the “material” approach to fractions. Theexpression also displays a numerator and a denominator (“m of n parts”), which are the basis for what I designateas the “numerical” approach to fractions. These two approaches to fractions appear to have been combined by themathematicians of ancient China. On the one hand, the problem asking us to add up fractions requires gathering vari-ous disparate parts together to form a single quantity, which must hence be evaluated ( a

b+ c

d=?). On the other hand,

the algorithm prescribes computations on numerators and denominators—the numerical dimension of the fractionsinvolved—to yield a value as the result of a division ( ad+cb

bd—where ad + cb is the dividend and bd the divisor).

Establishing the correctness of the algorithm requires proof that the value obtained ( ad+cbbd

) measures the quantityformed by bringing together the various parts ( a

b+ c

d).

In his commentary on the simpli�cation of fractions, a topic dealt with immediately before the addition of fractions,Liu Hui had approached the fractions as entities to be manipulated by the procedure concerned, i.e., as a pair consistingof a numerator and a denominator, and he had stressed the potential variability of their expression: one can multiply, or

27 The question of the relationship between the exegesis of a Canon, on the one hand, and Liu Hui’s or Li Chunfeng’s speci�c practice of proof, onthe other hand, is addressed in Chemla [2001 (forthcoming)]. In Chapter A of Chemla and Guo Shuchun [2004, 27–39], I discuss the fundamentaloperations that the commentaries put into play when proving the correctness of procedures.28 I discuss in detail this annotation of Liu Hui’s commentary in Chemla [1997b]. For further argumentation regarding what is stated in thissection, the reader is referred to this paper.


divide, the numerator and the denominator by the same number, he had stated, without changing the quantity meant.In this context, to divide (e.g., 2/4 becoming 1/2) is to “simplify yue ” the fraction. The opposed operation (e.g.,2/4 becoming 4/8), which Liu Hui had then introduced and called by opposition “to complicate fan ,” is neededonly for the sake of proving the correctness of algorithms dealing with fractions.

At the beginning of his commentary on the procedure for “gathering parts,” Liu Hui, then, considers the counter-parts of these operations with respect to the fractions regarded as parts: “simpli�ed” fractions correspond to “coarserparts” (halves instead of fourths), “complicated” ones to “�ner” parts (eighths instead of fourths). At this level, LiuHui again stresses the invariability of the quantity, beyond possible variations in the way of composing it (using halves,fourths, or eighths).

Now to prove that

a

b+ c

d= ad + bc

bd,

the commentator shows that the algorithm carries out the following program: by multiplication of both the numeratorand the denominator, it re�nes the disparate parts to make them have the same size ( a/b becoming ad/bd and c/d

becoming cb/bd , so that all parts are bd ths). Quoting the Canon, Liu Hui expounds the actual “meaning” of eachstep, in terms of both parts and numerators/denominators, making clear how they combine to ful�ll the programoutlined.29 When “the denominators are multiplied by one another”—an operation that, in the course of the proof, LiuHui names “to equalize”—this computes the denominator common to all fractions (bd) and de�nes the size that thedifferent parts can have in common: when parts have a common size, the fractions can be added. Moreover, when “thedenominators multiply the numerators that do not correspond to them,” to yield ad and bc, the numerators, he says,are made homogeneous with the denominators to which they correspond. The overall operation is a “complication.”It has been previously shown to be valid and it ensures that the “original quantities are not lost.” Here, too, in passing,Liu Hui confers a name on this set of operations: “to homogenize.”

“Equalizing” the denominators and “homogenizing” the numerators, the algorithm thus actually yields a correctmeasure of the quantity formed by gathering the various fractions.

2.2. The correctness of algorithms and the search for fundamental operations

We, contemporary readers, may read Liu Hui’s commentary on the procedure for “gathering parts” as establish-ing the correctness of an algorithm. But what were the aims pursued by the commentator when writing it? They arehighlighted by the following part of this commentary, which continues with highly abstract and philosophical consid-erations, concluded by a key declaration: “Multiply to disaggregate them, simplify to assemble them, homogenize andequalize to make them communicate, how could those not be the key points of computations/mathematics (suan)?”[Chemla and Guo Shuchun, 2004, 158–159].30

Opaque as it may seem, this declaration is essential. It clearly shows that something else is at stake in the previousproof besides establishing the correctness of a procedure. The proof had exhibited operations at play in the algorithm:multiplying numerators and denominators to disaggregate the parts they represent, dividing them to assemble the partsinto coarser parts, equalizing denominators, homogenizing numerators. These operations constitute the topic of thesubsequent considerations: exhibiting them appears to be one motivation for carrying out the proof. Moreover, allthese operations were introduced in relation to fractions, for which they referred to precise operations on numeratorsand denominators. However—and this is the point dif�cult to understand—Liu Hui’s concluding declaration indicates

29 As above, in the discussion of the commentary on Problem 9.9, the term “meaning” refers to what the operations carry out, with respect to thesituation in which they are applied. One term in Liu Hui’s terminology can be interpreted to correspond to this concept: “meaning, intention yi .”See my glossary in Chemla and Guo Shuchun [2004, 1018–1022].30 In the English translation of The Nine Chapters by Shen Kangshen et al., various critical parts of the work are not translated accurately and theirimportance is hence overlooked. For the passage considered, compare the translation by Shen Kangshen et al.: “Multiplying [the denominators]means �ne division and reducing means rough division; the rules of homogenizing and uniformizing are used to get a common denominator. Arethey not the key rules of arithmetic?” [Shen Kangshen et al., 1999, 72]. In addition to inaccuracies (such as, for instance, translating “assemble”by “rough division”), the theoretical import and generality of the statement, which is one of the most important of the commentary, is completelymissed. The consequences should appear clearly with what is explained below, in this section and in the following one.


that their relevance far exceeds this limited context, since they are now listed among the “key points of mathematics.”How can we interpret this claim?

In fact, when we read Liu Hui’s commentary as a whole, we observe that these operations recur in several otherproofs that the commentator formulates to establish the correctness of other procedures described in the Canon[Chemla, 1997b]. Let us allude to an example.31 Chapter 8 of the Canon is devoted to solving systems of simul-taneous linear equations. If we represent a system by the equations

bx + ay = e,

dx + cy = f,

the fundamental algorithm given in The Nine Chapters amounts to transforming them into

bdx + ady = ed,

bdx + bcy = bf.

Then, by subtraction of the two equations, one eliminates x and determines the value of y, after which the valueof x is easily obtained. When Liu Hui accounts for the correctness of the algorithm, he brings to light that one canmultiply and divide the coef�cients of an equation by the same number without altering the relationship it expressesbetween the unknowns. Moreover, he points out that the algorithm “equalizes” the values of the coef�cients of x,whereas it “homogenizes” the values of the other coef�cients. This is how all the operations included in the keydeclaration quoted above occur again in the proof of the correctness of another algorithm. The same holds true forother cases, as we shall shortly see.

This fact explains why Liu Hui’s declaration can be so general and why he makes a statement, the validity of whichgoes beyond the context of fractions. However, if we compare the two situations alluded to, in which Liu Hui identi�es“equalization,” clearly in the context of equations we do not have an “equalization of denominators,” since what is“equalized” is the coef�cients of x. In other words, what is common between the two contexts is not the concretemeaning that “equalization” takes, although in each context the speci�c concrete meaning of “equalization” is what isrequired for the proof to work.

The declaration invites us to �nd something else that is common to the various contexts in which the operationsidenti�ed occur and which would justify its validity. This leads us to note that in each proof in which they occur, theterms designating the operations have in fact two meanings. Let us explain this point for the term “equalization.”

In the context of adding up fractions, equalization was interpreted as the operation equalizing denominators. Inthat of equations, it was interpreted as the operation that made the coef�cients of x equal. Similarly, in all the othercontexts in which “equalizing” occurs, it can be interpreted in terms of its effect with respect to the particular situationto which it is being applied. This material effect constitutes the �rst meaning of “equalization,” one that changesaccording to the context. However, the fact that the operation recurs in different contexts reveals that the term takesa second meaning, a formal one that is common to all contexts: the term “equalizing” points to how the algorithmswork. All algorithms for which the proof of correctness highlights that “equalizing” and “homogenizing” are at playproceed by “equalizing” some quantities while “homogenizing” others. The second meaning of the two terms capturesand expresses the strategy followed by the procedure. And the parallel between the proofs discloses that, in fact, thealgorithms follow the same formal strategy of equalizing and homogenizing. Even though the concrete meanings ofequalizing and homogenizing vary according to the contexts in which they are at play, formally, they operate in thesame way. These remarks reveal a key feature shared by the proofs: They bring to light that the same fundamentalalgorithm underlies various procedures. It is on the basis of the actual reasons accounting for the correctness ofthe algorithms that, through the proofs, a concealed formal connection between them is unveiled. This conclusion

31 For the purpose of clarity the example has been simpli�ed in its detail: cf. [ Chemla, 2000]. In Section 3, I shall describe with greater detail how“equalizing” and “homogenizing” occur in another context.


Fig. 3. The transformation of the trapezoid prism.

reveals that while proving the correctness of an algorithm, the commentator concentrates on formal dimensions in theprocedure.32

This concern relates to one of the reasons for Liu Hui to carry out proofs, i.e., bringing to light such fundamentalformal strategies common to the various procedures provided by The Nine Chapters.33 Such key algorithms, such as“equalizing/homogenizing,” allow a reduction of the variety of procedures of the Canon, uncovering a small numberof strategies systematically used in designing all its procedures. Multiplying or dividing all numbers in an adequateset, as well as equalizing and homogenizing, appear to underlie many of the algorithms of the Canon in domainsthat for us belong to arithmetic or algebra. This is why, when they �rst occur, in the context of the procedure for“gathering parts,” Liu Hui immediately stresses their importance. Thus his declaration appears to gather together themost fundamental algorithms underlying the procedures of The Nine Chapters, those procedures being brought tolight by the proofs contained in his commentary.

2.3. A fundamental operation in geometry

The same motivation of disclosing fundamental operations common to various algorithms appears to permeate theproofs that Liu Hui develops in the context of geometry. This can be deduced from the fact that the proofs establishingthe correctness of the most important algorithms related to geometrical shapes all bring to light that these algorithmsuse the same formal strategy, which Liu Hui captures with the expression “one uses the excess to �ll up the void.” 34

Let us illustrate this point with the example of the trapezoid prism (see Fig. 3—again no �gure is to be found for thisproblem either in the commentary or in The Nine Chapters). This solid is the �rst one considered in Chapter 5, inwhich most problems of that type are gathered.

The procedure given in The Nine Chapters to compute the volume of solids of this shape reads as follows:

Procedure: one adds the upper and lower widths and halves this (the result). One multiplies this (the result) by the heightor the depth. Again, one multiplies this (the result) by the length, hence the chi of the volume

, , [Chemla and Guo Shuchun, 2004, 410–413]35

32 We could capture this point in the way in which the commentary unfolds. Several hints indicate that the authors of The Nine Chapters alsoconsidered procedures from a similar perspective. However, dealing with this issue would exceed the scope of this article.33 Such a motivation appears to be driving later commentators as well [Chemla, 2001 (forthcoming), 2003b].34 This point was �rst discussed in great detail in Wu Wenjun [1982]. See further developments in Volkov [1994].35 For reasons that will be presented in Section 4, I have drawn the solid in a position different from the one the procedure refers to. Whatcorrespond to the “lower” and “upper width” are shown on the �gure as the front and rear width. The following argument is not affected by thisrotation.


To establish its correctness Liu Hui writes:

In this procedure, the reason that “one adds the upper and lower widths and halves this” is that if one uses the excessto �ll up the void , this yields the average width. “Multiplying this (the result) by the height or the depth” yields theerected surface of a front. The reason that “again, one multiplies this, (the result) by the length” is that it yields thevolume corresponding to the solid; this is why this makes “the chi of the volume” , ,

“ ”, “ ” , , (My emphasis) [Chemla andGuo Shuchun, 2004, 412–413]

Given the position in which I have drawn the trapezoid prism, what Liu Hui calls a front is represented in theupper and lower planes. The �rst steps of the procedure are interpreted as computing the area of a face, by meansof its transformation into a rectangle. With the expression “one uses the excess to �ll up the void,” Liu Hui indicatesthe concrete transformation of the solid that is at the basis of the proof (it is illustrated in Fig. 3) and that allowshim to interpret the “meaning” of the successive steps of the procedure. In addition, most importantly for us here,the commentator refers to this transformation by the same sentence that he uses to designate other different concretetransformations that make the proof work in other geometrical contexts. In each context, the actual transformationsdiffer. However, the recurrence of the same formula to refer to them reveals that viewed from a certain angle, they areformally the same.

This conclusion con�rms what we have seen with “multiplying,” “dividing,” “equalizing,” and “homogenizing”:the proof again appears as a means for bringing to light formal patterns that are common to various algorithms despitethe apparent difference between them that derives from the fact that they prescribe different computations.36

How does the proof ful�ll the function of revealing such formal patterns? The example of the procedure for “multi-plying parts,” which we will analyze in the following section, highlights that the problems play a key part in enablingthe proof to ful�ll this function and, in this case, disclose the hidden action of “equalizing” and “homogenizing” in itsprocess. It is from this perspective that we can now go back to our �rst puzzle and offer a solution for it.

3. The situation in the statement of a problem as a condition for exhibiting formal strategies

As has already been mentioned, Problem 1.19, after which The Nine Chapters states the procedure for “multiplyingparts,” requires computing the area of a rectangular �eld, 3/5 bu long and 4/7 bu wide. However, the procedure itself isformulated without reference to any concrete situation. Liu Hui’s commentary on the procedure provides key evidencefor understanding the part played by problems for proofs to ful�ll the function brought to light in the previous section.Let us analyze it in greater detail.

3.1. The �rst proof of the correctness of the procedure for multiplying fractions

In the �rst part of his commentary on this procedure, Liu Hui develops abstract reasoning to account for its correct-ness. This argument shows one way in which multiplication and division are at play in the design of the procedure. Itsopening section can be translated as follows:

In each of the cases when a dividend does not �ll up a divisor, 37 they hence have the name of numerator and denom-inator.38 If there are parts (i.e., a fraction), and if, when expanding the corresponding dividend by multiplication, then,

36 For the sake of clarity, we opposed the �rst set of general operations, presented in Liu Hui’s key declaration, to the transformation “one usesthe excess to �ll up the void,” which occurs only in relation to geometrical shapes. Most probably, this type of transformation was conceived of asone of the general patterns with which the operations identi�ed in the key declaration could take shape.37 This technical expression refers to the case when the dividend is smaller than the divisor. Note that “dividend” designates the content of aposition on the calculating surface, and not a determined number—such a way of employing terms corresponds to the assignment of variables,whose use for the description of algorithms is characteristic in ancient China, in contrast to other ancient traditions. As a result, in what follows,the word “dividend” will designate different values, depending on the operations that have been applied to the value put in the position at each stepof the computation. I have respected this technical use of terms in the translation.38 In such cases, the result of the division is the fraction whose numerator and denominator are respectively the dividend and the divisor. “Nu-merator” and “denominator” refer to the numbers as constituting a fraction; “dividend” and “divisor” refer to them as the terms of the operationyielding the fraction. The commentary alternately uses the two sets of terms, with the greatest precision.


correlatively, it (i.e., the dividend produced by the multiplication) �lls up the divisor, the (division) hence only yieldsan integer.39 If, furthermore, one multiplies something by the numerator, the denominator must consequently divide (theproduct) in return (baochu). “Dividing in return,” this is dividing the dividend by the divisor. ,

, , , , , [Chemla andGuo Shuchun, 2004, 170–171, 768, footnote 176]

Before translating the end of the argument, let us explain the meaning of some of the technical expressions. Theexpression of “dividing in return” (baochu) is particularly important to note. The commentator introduces it here forthe �rst time in the whole text (Canon and commentaries). Seen as an operation, as Liu Hui makes it clear, it consistsin a division. However, the quali�cation “in return” adds something to the prescription of a division: it makes explicitthe reasonfor dividing. Using “dividing in return” means that it was needed, for some reason, earlier in the procedure,to carry out a multiplication, which was super�uous with regard to the sought-for result: this division compensatesfor the earlier multiplication, deleting its effect. This general idea clearly makes sense in the passage of Liu Hui’scommentary translated above. The use of the technical term indicates that multiplying by the numerator is to beinterpreted as follows: instead of multiplying “something” by a fraction a/b, one multiplies the “something” by itsnumerator a. Multiplying the fraction a/b by b, one obtains the numerator a. Having multiplied the “something” bythe numerator a, instead of a/b, one has multiplied by a value that was b times what was desired. Consequently, onehas to “divide in return” by that with which one had super�uously multiplied, that is, by b. Multiplying something bya/b is hence shown to be the same as multiplying by a alone, and dividing the result by b.

Most importantly for our purpose, “dividing in return” is one of the quali�cations of division that one can �nd inThe Nine Chapters itself too. This fact calls for two remarks. First, the expression clearly adheres to the sphere ofjusti�cation, since the very prescription of the division indicates a reason for carrying it out. In other words, suchhints prove that The Nine Chapters does refer to arguments supporting the correctness of algorithms in some speci�cways. Second, we can �nd the expression both in The Nine Chapters and in the commentaries. This fact points tocontinuities between the two, which will be useful in our argumentation below.

The continuation of Liu Hui’s �rst proof of the correctness of the procedure for multiplying fractions interprets themeaning of the procedure as follows:

Now, “the numerators are multiplied by one another,”40 the denominators must hence each divide in return.[Chemla and Guo Shuchun, 2004, 170–171]

For each of the numerators of the fractions to be multiplied, the argument developed above applies. If one multipliesthe numerators instead of multiplying the fractions, one must divide the product by each of the denominators. The lastsentence of Liu Hui’s �rst proof transforms the sequence of operations just established to carry out the multiplicationof fractions (multiplication of the numerators, division by a denominator, division by the other denominator) into theprocedure given in the Canon, as follows:

Consequently, one makes the “denominators multiply one another” and one divides altogether (lianchu) (by their product).[Chemla and Guo Shuchun, 2004, 170–171]41

39 If the division of a by b yields a/b, then, if a is multiplied by kb, the new dividend akb divided by b yields the integer ak.40 This is a quotation from the procedure of the Canon.41 Shen Kangshen et al. [1999, 82] translate the whole passage as follows: “When the dividend is smaller than the divisor, then [one] gets a [proper]fraction. When the numerator is multiplied by the dividend, the product may be larger than the denominator (divisor), thus yielding an integer. If thenumerator is multiplied, the product should be divided by the denominator. Since the product of [all the] numerators is taken as dividend, it shouldbe divided by the product of [all the] denominators, i.e., take the continued product of the denominators as divisor.” I do not see what, in the secondsentence, “multiplying the numerator by the dividend” may mean. Moreover, there is no word for “numerator” in the text at this point. Later, “ifthe numerator is multiplied” does not conform to the syntax of the Chinese and the translation expresses a meaning unclear to me, leaving aside(as below) the key word “divide in return.” In correlation with this, the following sentence is not translated. The transformation of two divisionsinto a division by the product is hidden by the addition, in many places, of the term “product,” which is not in the original text. The mathematicalexplanation given in footnote 1 (p. 83) does not seem to me to �t with the meaning of the text. It takes the commentary as distinguishing variouscases in the multiplication and fails to read the argument that Liu Hui makes in it.


Transforming a sequence of two divisions into a unique division by the product of the divisors is a valid transfor-mation because the results of division are exact. Liu Hui proves to be aware of the link between the two facts, thatis, between the validity of the transformation and the exactness of the results of division [Chemla, 1997/1998]. Asabove, the formulation “to divide altogether” prescribes a division in a way that indicates the reason why the divisionprescribed is to be carried out in this particular way. This expression recurs regularly in Liu Hui’s commentary. Thequali�cation adheres to the sphere of justi�cation. In contrast to the “division in return,” however, the expressionlianchu never occurs in The Nine Chapters itself.

This sentence concludes Liu Hui’s �rst proof of the correctness of the procedure. One can see how the proofdiscloses that multiplication and division, two of the fundamental operations listed in Liu Hui’s declaration, are putinto play, as opposed operations, for designing the procedure as it stands. For our purpose, it is interesting to note thatthis proof of the correctness of the algorithm develops independently of the framework of the problems in the contextof which the “procedure for multiplying parts” was formulated, that is, the problems about rectangular �elds. Thearguments only make use of general properties of dividend and divisor, numerator and denominator. Moreover, theybring into play the properties of multiplication and division with respect to each other.

However, Liu Hui does not end here his commentary on this procedure. He goes on to develop a second proof,which highlights how, seen from another angle, this procedure also puts into play “equalizing” and “homogenizing.”This is where we go back to the �rst puzzle that we have presented.

3.2. The second proof of the correctness of the procedure for multiplying fractions

The sentence linking the two parts of the commentary is important for our purpose. Liu Hui states:

If here one makes use of the formulation of a �eld with length and width, it is dif�cult to have (the procedure) beunderstood with a greater generalitynan yi guang yu , [Chemla and Guo Shuchun, 2004, 170–171]

In fact, this sentence justi�es why the commentator discards Problem 1.19 in the context of which the procedurefor “multiplying parts” is described in The Nine Chapters and why he introduces, immediately thereafter, anotherproblem equivalent to the original one: 1 horse is worth 3/5 jin of gold. If a person sells 4/7 horse, how much doesthe person get? Let us analyze more closely the assertion introducing the alternative proof that the commentatordevelops in the sequel. Since it is a key assertion for my argument, I shall insist on it and distinguish all the facts thatit reveals.

First, Liu Hui’s above statement establishes a link between the context of a problem—here the problem providedby The Nine Chapters, i.e., that of a rectangular �eld—and the aim of understanding the procedure. It appears herethat a given problem can prevent one from gaining a wider understanding of the procedure. Second, this seems toindicate that Liu Hui also perceives the preceding passage as contributing to an understanding of the procedure. Thislink between establishing the correctness of a procedure and understanding it is not fortuitous. It will recur again inthe passage of the commentary following the above statement.

Now, the change of problem is justi�ed by the attempt to gain a more general understanding of the procedure.Strikingly enough, if we bring all these facts together, the statement indicates a possible link between the contextof a problem and the proof of the correctness of a procedure. Again, this link will be con�rmed in what follows.In fact, Liu Hui next introduces a sequence of three new problems, all formulated with respect to a single kindof situation, which make it possible to develop the second proof of the correctness of the procedure conformingto the practice of demonstration sketched in Section 2. Let us stress right at the outset the essential consequencethat can be derived from both the above statement and the analysis developed below: problems appear here notto be reduced to questions that require a solution—as we would readily assume—but they also play a part inproofs.

Observing Liu Hui’s next development should hence allow us to progress on two fronts. It should provide evidenceshowing how problems intervene in proving the correctness of a procedure. And this is where the difference betweenthe problem with the �eld and those with the persons, the horses, and the gold should become apparent. Moreover, itshould help us grasp how Liu Hui conceives of “understanding a procedure.”


Let us �rst indicate, in modern terms, the main idea of the second proof of the correctness of the procedure. It canbe represented by the following sequence of expressions:

a

b· c

d= ac

bc· bc

bd= ac

bd.

The second proof brings to light that the computations of ac and bd , with which the procedure for “multiplyingparts” begins, have in fact the meaning of “homogenizations.” This meaning can be seized if the “equalizing” of bc,which is essential for disclosing the pattern, is revealed. These are the key points of the second argument. We shallexamine below how Liu Hui puts them into play in providing his second proof.

In this, an essential part will be played by an operation the importance of which was already stressed with respect toproofs: interpreting the “meaning” (yi) of the computations prescribed by an algorithm, that is, expressing the “mean-ing” of their results—their “semantics”—in terms of the situation described by the problem. This remark enables usto infer why the �rst problem cannot be used here. The situation of the �eld with length and width is unsuitable from asemantic point of view: it does not offer possibilities of interpretation rich enough for the “meaning” of the computa-tion of bc to be expressed in a natural way. As a consequence, the problem of computing the area of a rectangular �elddoes not allow bringing to light the “equalization” that underlies the procedure for “multiplying parts.” The schemeof “equalizing” and “homogenizing” cannot be unfolded in this context.

In contrast, the situation in which the sequence of three new problems is formulated, with the three componentsconstituted by the quantities of gold, persons, and horses, offers richer possibilities for interpreting the effects of oper-ations. It is semantically rich enough for disclosing that the scheme of “homogenizing” and “equalizing” underlies theprocedure for “multiplying parts.” Consequently, the new situation allows revealing in another way how the procedurefor “multiplying parts” relates to the fundamental operations identi�ed by Liu Hui and discussed in Section 2. This iswhat is at stake in the second proof.

The key point here is that the interest in bringing to light how equalization is at play in “multiplying parts” belongsonly to the sphere of proving. The equalization plays no role in the actual computation of the result. This is why theproblem in the context of which the algorithm is described differs from the problems in relation to which the proofis carried out. A link is thereby established between bringing to light a formal strategy accounting for the correctnessof the procedure and interpreting the “intentions,” the “meanings” (yi) of the operations in the �eld of interpretationoffered by the situation of a problem. The example of the multiplication of fractions, in which the description of thealgorithm and the development of its proof require a sequence of different problems, reveals the essential part playedby problems in establishing the correctness of algorithms. Note, however, that the numerical values are common to allproblems: Liu Hui introduced the new problem mentioned above in such a way that the task to be carried out is still tomultiply 3/5 by 4/7. As a consequence, the computations leading to a solution of the problem included in The NineChapters (3 times 4, 5 times 7, dividing the former result by the latter) are an actual subset of the operations involvedin proving the correctness of the procedure for “multiplying parts” (which furthermore includes computing 4 times 5).

With these observations in mind, let us examine in detail how Liu Hui uses problems to conduct his second proof.

3.3. The part played by mathematical problems in a proof

Liu Hui’s second proof of the correctness of the procedure for “multiplying parts” consists in articulating, in anadequate way, a sequence of equivalent problems that are transformed one into the other. What would be done inmodern mathematics by formal computations is here carried out by interpreting the results of operations with respectto a succession of problems.

In the alternative proof, Liu Hui’s �rst step consists in formulating a �rst problem, the solution of which requiresuse of the last operation of the procedure for “multiplying parts,” i.e., dividing 12 (ac) by 35 (bd). It reads as follows:

Suppose that one asks: 20 (bc) horses are worth 12 (ac) jin of gold. If one sells 20 (bc) horses and if 35 (bd) persons sharethis [the gain], how much does a person get? Answer: 12/35 jin. To solve it, one must follow the procedure for “directlysharing (jingfen )”42 and take 12 jin of gold as dividend and 35 persons as divisor. , ,

42 This procedure, which is described in The Nine Chapters just before the procedure for “multiplying parts,” covers all possible cases of divisioninvolving integers and fractions [Chemla, 1992a]. Note that division is dealt with before multiplication and that, in correlation with this fact, the


, , , , ,[Chemla and Guo Shuchun, 2004, 170–171]

The fact that the procedure given to solve this problem is correct was established by Liu Hui in his commentaryon the preceding section of The Nine Chapters, devoted to “sharing parts.” This problem is immediately followed byanother problem, presented as a transformation of the former one:

Suppose that, modifying (the problem), one says: 5 (b) horses are worth 3 (a) jin of gold. If one sells 4 (c) horses and if7 (d) persons share this (the gain), how much does a person get? Answer: 12/35 jin. , ,

, [Chemla and Guo Shuchun, 2004, 170–171]

The key point here is that the procedure for solving this second problem is �rst described in terms of “homogeniz-ing”:

To do it, one has to homogenizethese quantities of gold (ac) and persons (bd). They then all conform to the �rst problemand this is solved by the (procedure for) “directly sharing.” , , , [Chemlaand Guo Shuchun, 2004, 170–171]43

The use of the term “homogenize” implies that, in parallel, an equalization is carried out. Only the operationof equalization can confer the meaning of “homogenization” on the other operations. Liu Hui will make this pointexplicitly in one of the subsequent sentences of his proof. “Homogenizing” quantities of gold and persons goes alongwith “equalizing” quantities of horses. This is why, as Liu Hui states, the homogenizations prescribed lead to the “�rstproblem”: indeed, they yield the two following statements, in which we recognize the �rst problem:

20 (bc) horses are worth 12 (ac) jin of gold.If one sells 20 (bc) horses and if 35 (bd) persons share [the gain], how much does a person get?

This transformation of the second problem into the �rst one involves computing ac and bd , which amount to the�rst part of the procedure for “multiplying parts.” The transformation is clearly correct: it does not alter the meaning ofthe relationship between the values considered. The procedure solving the �rst problem can then conclude the solutionof the second one. Now, if one considers the whole procedure that correctly solves the second problem, throughtransforming it into the �rst one, one realizes that the procedure for “multiplying parts” is embedded in it. Whatmakes the difference between the two is that the procedure for “multiplying parts” does not prescribe the computationof bc. However, if we pay closer attention to the way in which Liu Hui formulates the transformation of the secondproblem into the �rst one, we observe that there, too, he evokes the computation of bc only in an allusive way, byreferring to the computation of ac and bd as “homogenizations,” and by stating that the second problem is reducedto the �rst one. In fact, the sequence of operations that is concretely given to solve the second problem correspondsexactly to the procedure given by The Nine Chapters for “multiplying parts.” To recapitulate, Liu Hui describes theprocedure that correctly solves the second problem as being the list of operations that constitute the procedure for“multiplying parts.” While stressing in his following statements the identity between the two procedures step by step,most importantly, Liu Hui transfers the interpretation in terms of homogenization and equalization into the procedurefor “multiplying parts.” He writes:

If this is so, “multiplying the numerators by one another to make the dividend (ac)” is like homogenizingthe correspond-ing gold. “Multiplying the denominators by one another to make the divisor (bd)” is like homogenizingthe correspondingpersons. Equalizing the corresponding denominators makes 20 (bc). But the fact that the horses be equalized plays

exegete makes use of the procedure for division in his commentary on multiplication. To make my argument clearer, I have inserted in the translationmodern symbolic expressions between parentheses.43 Shen Kangshen et al. translate this sentence as follows: “By the Homogenization and Uniformization Rule one can get the same answer as bythe rule of division.” [Shen Kangshen et al., 1999, 83], which does not �t with the Chinese. Again, the argument developed by Liu Hui cannot beunderstood from the translation.


no role. One only wants to find the homogenized (quantities) and this is all. , ;, , , [Chemla and Guo Shuchun, 2004, 170–171]

At this point, it is established that, on the one hand, the procedure for “multiplying parts” solves the secondproblem correctly and, on the other hand, the procedure involves only homogenizations. Liu Hui devotes the followingstatement to considering the ambiguous status of “equalization.” The computation of bc as the equal quantity in thestatement of the �rst problem is essential to ascertain that ac and bd correctly correspond to each other in a patternsimilar to that of the �rst problem. In other words, it is essential for the proof. But this computation is useless inobtaining the result: once one knows the reason why ac and bd correspond to each other, it suf�ces to divide oneby the other to yield the solution. This explains why Liu Hui emphasizes that the “equalization” plays no part in theprocedure itself, except for allowing an interpretation of its �rst steps as “homogenizations.” In turn, the interpretationof these steps as “homogenizations” is what lies at the basis of the correctness of the procedure.

The key point to note here is that exhibiting the homogenizations and equalizations can only be done within theframework of the new situation with persons, gold, and horses, in which the equalization can be interpreted andthereby brought to light, and not within the framework of computing the area of a rectangular �eld.

Liu Hui’s last step is to show that, as far as the question raised is concerned, the second problem is equivalent to athird problem, itself strictly identical to Problem 1.19. The idea is the following: we now know that the procedure for“multiplying parts” correctly solves the second problem. We want to show that this procedure correctly solves 1.19.The step to be taken is to show that the two problems amount to the same. Liu Hui establishes the equivalencebetween the two by showing the equivalence between the second problem and one strictly identical to Problem 1.19,but formulated in terms of horses, gold and persons. He states:

Moreover, that 5 horses are worth 3 jin of gold, these are lü in integers. If one expresses them in parts, then this makes1 horse worth 3/5 jin of gold. That 7 persons sell 4 horses is that 1 person sells 4/7 horse. , , , ;

, , [Chemla and Guo Shuchun, 2004, 170–171]

Qualifying as lü the data in each of the two statements of the quote designates the ability of the pairs to be possiblymultiplied or divided by a common number, without altering the meaning of the relationship [Guo Shuchun ,1984b]. If we use this property, as Liu Hui suggests, to transform the outline of the second problem into an equivalentproblem, we obtain the third problem, the one with which we have formulated our �rst puzzle:

1 horse is worth 3/5 (a/b) jin of gold. If one sells 4/7 (c/d) horse, how much does the person get?

This quotation brings to a close the passage of Liu Hui’s commentary on “multiplying parts” that we needed toanalyze to solve our �rst puzzle and to account for why Liu Hui substituted the third problem for Problem 1.19. Wenow return to the two questions we have raised in our analysis above.

First, how can we qualify the understanding of the procedure for “multiplying parts” yielded by Liu Hui’s com-mentary here? In fact, as was the case for the procedure for “gathering parts,” Liu Hui proves the correctness of theprocedure by bringing to light how the “procedure of homogenizing and equalizing” underlies it.

As we have emphasized above, the “meanings” of “equalizing” and “homogenizing” differ in the two contexts.In the procedure for “gathering parts,” “equalizing” meant equalizing the denominators. Here, “equalizing” refers tothe fact that a denominator and a numerator are made equal so that the procedure can ful�ll its task. Even thoughthe concrete meanings differ, the procedures use the same strategy formally: in both contexts, it is by making somequantities equal and then making other quantities homogeneous that one can obtain the solution. This is what theproofs disclose. This remark indicates yet again the sense in which, in Lu Hui’s eyes, the fundamental operations helisted can be deemed fundamental for mathematics.

Each proof sheds a different light on the procedure. Through highlighting how the �rst operations prescribed by theprocedure for “multiplying parts” amount to homogenizing some quantities, the second proof discloses a new meaningfor the algorithm, that is, another way of conceiving its formal strategy.44 The new problems introduced appear to be

44 The commentators use another term to designate this second type of “meaning.” To avoid confusion with the �rst one, I transcribe it as yi’ .See the corresponding entry in my glossary in Chemla and Guo Shuchun [2004, 1022–1023].


an essential condition for formulating the new understanding, based on the pattern of homogenizing and equalizing.This is because they allow the disclosure of the part played by “equalization” by providing the means to interpret themeaning of the operation. It is in this way that we can interpret Liu Hui’s introductory statement, where he claims thathe needs to discard the problem of the area of the rectangular �eld in order to “have (the procedure) be understoodwith a greater generality.”

The second proof requires introducing a procedure—the procedure of homogenization and equalization—in whichthe procedure for “multiplying parts” is embedded. This brings us back to the remark I made in Section 1, on thecommentary Liu Hui devoted to Problem 1.36 and the procedure for the area of the circular segment: proofs of thecorrectness of procedures regularly include establishing, or formulating, new procedures. For multiplying fractions,the interpretation of the operations of a procedure developed for the sake of the proof required the introduction of anew problem. In the case of the commentary on the topic of Problem 1.36, in order to formulate his new procedure, LiuHui introduces new elements in the �gure of the circular segment: the circle from which it derives, and then trianglestiling its surface. These elements are in fact the topics of other problems in The Nine Chapters and are thus associatedwith procedures that the commentator uses to establish his new procedure for the area of the circular segment. Inparticular, Liu Hui employs these new elements to express the “meaning” of the steps of the new procedure. Wehence see here a parallel emerging between the uses of, on the one hand, geometrical �gures and, on the other hand,problems. In what follows, we shall analyze further this parallel between problems and geometrical elements in a�gure.

The distinction between the procedure required by the proof and the procedure for solving a problem is importantfor explaining the substitution of one problem for the other in the case of the multiplication of fractions. The differencebetween the new problem and the one described in The Nine Chapters now appears precisely to lie in the fact that theprocedure developed within the context of the proof can be “interpreted” only with respect to the new problem. Thekey point is that the operation of “equalization,” which is part of the procedure needed by the proof, is not neededby the procedure that computes the result. In correlation with this, the new problem, and not the one in The NineChapters, allows an interpretation of the “meaning yi” of the procedure. These remarks highlight how problems playa key part in Liu Hui’s practice of proof. They account for the fact that problems that appear to be the same to us arein fact different for him.

This is how I suggest that our �rst puzzle can be solved. Its solution reveals that, in the ancient Chinese mathe-matical practice to which these documents bear witness, problems did not boil down to being statements requiring asolution, but were also used as providing a situation in which the semantics of the operations used by an algorithmcould be formulated in order to establish its correctness. This second function becomes apparent when the problemful�lling the �rst function cannot ful�ll the second. This aspect appears to be an essential component of the practiceof proof as carried out by the commentators. What is most interesting, furthermore, as the case discussed here shows,is that the interpretation of the operations with respect to the situation described to formulate a problem is used tocarry out the proof and thereby disclose the latter’s most formal dimensions.

Second, the previous development allows us to go back to the question of determining the class of problems forwhich a particular problem stands. In Section 1, we have shown that a problem stands for a class of problems sharingthe same category. But this conclusion seemed to be contradicted by the need to replace the problem about the areaof a rectangular �eld by a problem with horses, gold, and persons, both problems sharing exactly the same particularvalues: the two problems looked identical to each other. In fact, there is no contradiction if we recall that the category(lei) associated to a problem is de�ned by examining the procedure attached to it. With respect to the procedurefor “multiplying parts,” these problems belong to the same category, but, with respect to the procedure developedfor the sake of the proof, they can no longer be substituted one for the other. However, what is essential is that allproblems similar to the one with horses, gold, and persons, in which the procedure of the proof, i.e., equalization andhomogenization, can be unfolded share the same category. Consequently, even though the proof is discussed in termsof horse, gold, and person, with particular values, it is meant to be read as general.45 This conclusion holds true for a

45 In his unpublished dissertation, Wang Ling [1956, 179–181] discusses the commentary on the multiplication of fractions in relation to theproblem with horses, people, and cash. He mentioned neither the change of problem, nor the pattern of “homogenizing”/“equalizing.” Moreover,his entire analysis develops in terms of “proportion,” which does not �t with the concepts in the text. However, his conclusion of the use of thecontext of a problem is worth quoting (p. 181): “Evidently, he [Liu Hui] chose these speci�c numbers, 3, 4, 5, 7 and problems about horses, money,and persons in a representative sense. Thus we have here an example demonstrating all the logical steps to prove a rule.”


Fig. 4. The pyramid called yangma.

procedure as well as for a proof: here as above, when the problem of the piece sawn was used within the context of the�eld with the shape of a circular segment, the general is discussed in terms of the particular. This conclusion cannotbe underestimated. Proofs developed by the commentators were regularly written down in ways that con�ict with ourexpectations, since they seem to be formulated only with respect to particular cases. As a result, they were interpretedas lacking generality. Such an interpretation is in my view anachronistic, because it reads our sources in ways thatproject our modes of interpretation onto documents that require other readings. I think that the previous discussionestablishes why they must be read as expressing general arguments.

The same conclusion can be drawn not only for the situation in which a problem is formulated, but also for theparticular values it involves. Proofs relying on problems with particular values are also meant to be general in thissense: the example of the pyramid with a square basis (yangma) provides a clear illustration. We shall now use thisexample to establish this point.

4. The values in the statement of a problem: a parallel between problems and visual tools

To analyze the relevance of the numerical values used in the statement of a problem, we shall use the same methodas we did in the previous paragraph, in which we focused on changes in the situations used to formulate a problem.Here, we shall examine cases in which the commentator modi�es the numerical values of a problem presented in TheNine Chapters, before he begins commenting on a given topic.

In our discussion of Problem 6.18 in Section 1, we have seen that Liu Hui changed the values of the problem toexpose the lack of generality of the procedure given by The Nine Chapters. This use employs problems as counterex-amples, in a fashion quite common today. The change of numerical values in Problem 5.15, which deals with thevolume of the pyramid yangma, already mentioned in the Introduction, is more dif�cult to account for. We will nowexamine the context in which it is carried out and the way it is used.

4.1. The proofs for the singular and the general cases

Chapter 5 of The Nine Chapters deals with the determination of the volumes of various kinds of solids. The typeof pyramid called yangma , in Problem 5.15, is represented in Fig. 4. Also in this case, there is no diagram in thesources. However, Liu Hui’s commentary does refer to a visual tool of a different kind, namely “blocks qi .” This isthe only type of visual tool the commentators use for space geometry, whereas for plane geometry they use “�gures tu

.” In the diagrams below, I have tried to picture only what the text says about blocks without adding more modernways of dealing with visual aids. In particular, I have not added letters by which one could refer to the vertices of ablock. Moreover, I have drawn some of the diagrams in such a way as to give a sense that the actual visual tools usedwere composed of solid pieces assembled in space.

The procedure stated by The Nine Chapters for solving Problem 5.15 prescribes multiplying all three dimensions(“width guang ,” “length zong ,” and “height gao ”) by each other and then dividing the result by 3 to yield the


Fig. 5. Three yangma with equal dimensions form a cube.

volume. In his commentary, Liu Hui sets out to establish the correctness of this very algorithm.46 The beginning ofhis proof is the most important part for us here.

To present his proof, as we have seen in the Introduction, Liu Hui �rst suggests modifying the values in theproblem by considering a yangma with dimensions (width, length, height) equal to 1 chi. He then remarks that threesuch yangma form a cube, as is shown in Fig. 5.

It follows immediately that the volume of the yangma is one-third of the volume of the cube with the same dimen-sions, which is obtained by multiplying its length, width, and height by one another.

However, Liu Hui notes—and this is the key point for our purpose—that this reasoning does not extend to thegeneral case, when the dimensions are not all equal. The reason for this is that when the dimensions are different, thethree pyramids into which the parallelepiped with the same dimensions can be decomposed are not identical, as wasthe case for the cube. Thus from this decomposition, one cannot conclude that the volumes of the three pyramids areequal before the procedure under study has been proved correct. As a result, Liu Hui discards the �rst reasoning andstarts a second one, general and much more complex.

So far, one may think that the change of the three numerical values of Problem 5.15 to 1 chi was intended to focusthe reader’s attention on a speci�c preliminary case. Surprisingly, in the general reasoning, instead of making use ofthe values of Problem 5.15, or of no values at all, Liu Hui again introduces three other values for the length, the width,and the height, this time making each equal to 2 chi. Why does he change the values of the problem at all? Why doeshe change the values he has just introduced to new values that are also equal to each other? We shall sketch the �rstand main part of Liu Hui’s general proof—the only one important for the points we want to make—to �nd answers tothese questions.

For the second reasoning, Liu Hui considers simultaneously a yangma and a speci�c tetrahedron named bienao, which together form a half-parallelepiped called qiandu (see Fig. 6). Both solids, the qiandu and the

bienao, are the topics of problems of The Nine Chapters itself (5.14 and 5.16, respectively). On the basis of theqiandu thus formed, Liu Hui sets himself a new goal: establishing that the yangma occupies twice as much volumein the qiandu as the bienao. From this property, the correctness of the procedure for computing the volume of thepyramid can be derived immediately.

However, we do not need to examine this part of the reasoning here, since to make our points, it suf�ces to evokethe beginning of Liu Hui’s reasoning, whereby he establishes the proposition that is his new target.

46 This commentary was the topic of quite a few publications, among which are Li Yan [1958, 53–54] (I could not consult the �rst edition of thisbook, but the second sketches the essential points of the proof), Wagner [1979], Guo Shuchun [1984a, 51–53], and Li Jimin [1990,295–303]. See also Chemla [1992b], Chemla and Guo Shuchun [2004, 396–398, 428–433, 820–824], in which a French translation is provided.Note that Wagner [1979] provides a full translation of the commentary into English. I refer the reader to these publications for greater detail,concentrating here only on the features that are important for my argument.


Fig. 6. A yangma and a bienao form a qiandu (half-parallelepiped).

Fig. 7. Halving the dimensions of the yangma pyramid decomposes the solid into unit blocks.

His �rst step consists in forming the bienao and the yangma with dimensions all equal to 2 chi from blocks ofthe following types: parallelepiped, qiandu, yangma, bienao, all with dimensions of 1 chi.47 Let us describe thecomposition of the yangma and the bienao successively.

The yangma is decomposed into several kinds of pieces, the dimensions of which are all half of the dimensions ofthe original body (see Fig. 7). On top, and in front on the right-hand side, we see two small yangma, similar to the onewe started with. In the back, on the right-hand side, and in front on the left, we see two small qiandu, whereas in theback, below and on the left-hand side, we have a small parallelepiped. This is how, for space geometry, Liu Hui usesblocks to compose solids. The reasoning that follows is typical of Liu Hui’s reasonings of that kind.

As for the bienao, the original body is composed of two kinds of blocks, the dimensions of which are all half ofthe original dimensions (see Fig. 8). On top in the rear, and in front on the right-hand side, we see two small bienaosimilar to the one we started with. In front, on the left-hand side, on top of each other, we see two small qiandu.

If we bring together the solids thus composed, we obtain a global decomposition of the original qiandu, as shownin Fig. 9. I have drawn it with the blocks set apart to make easier the enumeration of the components (see Fig. 10).However, it must be kept in mind that Liu Hui’s commentary refers throughout to the qiandu as a whole.

Liu Hui considers separately two zones in the qiandu.The �rst zone is the space in it that is occupied by pieces similar to the original bienao and yangma, but the

dimensions of which are all half of the original ones. We know that there are two such pieces of each kind. Within

47 The bienao is formed with vermillion blocks, whereas black blocks are used for the yangma. But I will not mention the colors any further here,because they do not matter for the point I want to make. The reader interested is referred to Chemla [1994]. More generally, the reader is referredto the published translations of Liu Hui’s commentary on the yangma to get a more precise idea of the original text (see footnote 46).


Fig. 8. Halving the dimensions of the bienao tetrahedron decomposes the solid into unit blocks.

Fig. 9. The half-parallelepiped decomposed.

the qiandu, they form, two by two, smaller qiandu, one situated in the lower and front part, on the right of Fig. 9 (orFig. 10, indifferently) and the other situated on the upper and rear part of Fig. 9 (or Fig. 10), on the left.

The second zone, in the space within the qiandu, is composed of smaller qiandu and a small parallelepiped. Allthese pieces may be oriented in different ways, but their three dimensions are uniformly half the length, half the width,and half the height of the original qiandu. Liu Hui has already established that half a parallelepiped has a volume thatis half the volume of the corresponding parallelepiped. The volume of these pieces is hence easy to compute. However,this is not the relevant feature in the situation—and Liu Hui does not mention it. The only useful information is thatthe volumes of all these half parallelepipeds are the same.

But there are two other features that are essential.The �rst key feature derives from evaluating the relative occupation, in the second zone, of blocks coming from the

yangma and blocks coming from the bienao. It turns out that in the second zone, there is twice as much space occupiedby pieces coming from the yangma as there is space occupied by pieces coming from the bienao. The property soughtfor (i.e., the yangma occupies twice as much volume in the qiandu than the bienao) is hence established in the secondzone.

As for the �rst zone, it replicates, on a different scale, the situation we started with.The second key feature in the situation concerns the evaluation of the respective proportion of the space in which

the situation is known and the space where it still needs to be clari�ed.To determine this proportion, by making two cuts in the qiandu, Liu Hui removes the upper half-parallelepiped

in the front and brings it closer to the similar half-parallelepiped in the lower rear part of the �gure. In addition, heremoves the rear upper half-parallelepiped, which is composed of a smaller bienao and a smaller yangma, and movesit near the similar one in the lower front part of the �gure (see Fig. 11).


Fig. 10. The half-parallelepiped decomposed, with the pieces set apart.

Fig. 11. The original qiandu rearranged into four smaller qiandu, which in fact compose a parallelepiped.

This rearrangement, which moves only two pieces—two smaller qiandu—transforms the original qiandu into a par-allelepiped, composed of four smaller parallelepipeds, the dimensions of which are all half of the original ones. Seenfrom this perspective, the second zone considered above—the one in which the property sought for is established—clearly appears to occupy 3/4 of the original body, whereas the �rst zone occupies only 1 /4 of it. These relativeproportions ensure that when one repeats the same reasoning, 3/4 of the remaining 1/4 of the original qiandu can beshown to hold the property, and so on.

4.2. The proof is general, but it makes use of speci�c values

To make the points I have in mind about the meaning of the numerical values appearing in the statements ofproblems, we need not discuss the �nal part of Liu Hui’s general proof, in which he establishes the proportion of 2to 1 that he chose as his new goal to prove the correctness of the algorithm computing the volume of the yangma.We have seen enough of this proof to note how different it is from the proof for the special case. Yet both proofs areconstructed using particular values, for which all the dimensions of the solids involved are equal. If Liu Hui was only


(a) (b)

Fig. 12. The qiandu and the geometrical transformation underlying the proof of the correctness of the procedure computing its volume.

aiming at making a proof valid for the special case—precisely the one he introduces by his change of values for thedimensions of the yangma—he would use a simpler argument: this argument is simply the �rst argument he makesand then discards as not general.

Two conclusions can be derived from these remarks. To begin with, when Liu Hui �rst changes the values ofthe dimensions of the yangma, his goal is not to introduce a simpler case for which a straightforward argument canbe made to establish the correctness of the algorithm. Second, the second proof, conducted and presented withinthe framework of the simplest situation possible—that with dimensions all equal—is meant to be general. We arehence led to the same conclusion as we have drawn at the end of the previous section: with respect to the proof,the commentator discusses the general in terms of the particular, and even the most particular possible. The sameconclusion holds true for the various problems and procedures, whether they are those from The Nine Chapters itselfor those used within the context of Liu Hui’s proof.48 In the case with gold, persons, and horses, discussed in Section 3,as well as in the case of the yangma, the extension of the validity of the reasoning is determined on the basis of theoperations put into play. We meet again with a parallel between problems and visual tools, based on the fact that theyare used in similar ways. We shall come back to this remark shortly.

In Section 2, we have shown how the proof of the correctness of the procedure for the trapezoidal prism broughtto light that the geometrical transformation needed could be subsumed under the general formal operation of “usingthe excess to �ll up the void.” In my view, the way of conducting the general proof for the yangma also reveals howit brings into play the same general operation.49 To explain this point, we need to sketch how, after Problem 5.14 andthe procedure for it, Liu Hui comments on the volume of the qiandu. His commentary on the algorithm computingthe volume of this solid offers two ways of accounting for the correctness of the procedure. The �rst one relies onthe fact that two identical qiandu make a parallelepiped. However, there is a second argument, which is not easy tounderstand. I have suggested that it refers to the qiandu as a particular case of the trapezoidal prism Chemla [1992b].As a consequence, the procedure given for the qiandu appears simply to be an application of that for the prism andthe proof of its correctness derives from the general proof for the prism, applied to the speci�c case of the qiandu (seeFig. 12).

In turn, the proof for the correctness of the procedures for the yangma and the bienao, which depends on that forthe qiandu, �ts within this framework, except that the piece to be moved needs to be cut in the middle and the piecesobtained have to be exchanged before they are moved. In this way, again, the operation of “using the excess to �ll upthe void” appears to be ef�cient and in fact to underlie the transformations needed to conclude the proof.

From the previous remarks, we can conclude that in the case of the yangma, as in the other cases examined above,through proving, the commentator also here seems to aim at identifying general and fundamental transformationsunderlying a number of different algorithms. Yet, in contrast with the multiplication of fractions examined in Section 3,for which the change of the situation of the problem was linked to the goal of exhibiting the general formal operationsunderlying the procedure, for the yangma the change of the values is not linked to this question. If we are to understand

48 Indeed, as in Liu Hui’s commentary on the area of the circular segment, it is clear that in order to unfold, the proof needs the original bodies tobe cut into other bodies. These bodies are associated with procedures that compute their volumes, and these procedures are employed in the proof.49 I make this point in greater detail in Chemla [1992b].


the use of the particular values provided by problems, we must therefore answer the following question: Why doesLiu Hui twice change the values of the problem, when clearly any particular values could do?

The answer is clear when we observe the two proofs above: the values of the dimensions used by Liu Hui aredetermined by the visual tools he used, that is, the blocks, which were concrete objects, all dimensions of which wereequal to 1 chi. In fact, the �rst reasoning brings together three identical blocks. In relation to the physical operationcarried out, the values needed are all equal to 1 chi. The second reasoning illustrates the composition of a yangmaand a bienao with blocks. The same blocks being used, the values needed to refer to the composition of the solidsare 2 chi. As for the iteration of the decomposition of the yangma and the bienao in the second step of this proof, itsanalysis is based on the same con�guration as in the �rst step. Blocks with the simplest dimensions have the propertyof being polyvalent for all these uses. Using the same blocks, Liu Hui can develop a proof that is particular (the �rstone) or a proof that is general (the second one).

It is thus the physical features of the objects with which the proof is conducted that dictate the change of values. Thisconclusion implies that the commentaries were written down by reference to objects used in the course of proving.Note that there is no hint that blocks were used at the time when The Nine Chapters itself was compiled.50 Onemay assume that these blocks—or some of their uses—were introduced for the sake of exegesis. This hypothesiswould explain why the commentator had to change the values to refer to the concrete use of objects in relation tohis reasonings, whereas the problems in The Nine Chapters would not mention values relating to material objects. Inany event, the conclusion reached accounts for the fact that, except for the introduction of new values for a problememployed as a counterexample, the only systematic cases in which the commentators change the numerical values ofproblems occur in the context of geometry and in relation to the use of material visual tools. The new values are alldetermined by the objects used as visual tools to compose the body under consideration. The generality of the proofsdeveloped is an issue that is completely dissociated from the fact that the texts could be written as referring to speci�cmaterial objects. The next point will highlight this conclusion from another angle.

4.3. Visual devices and problems as tools to express the “meaning yi” of operations

Although the particular and the general proofs are different on various accounts, they both use blocks to expressthe “meaning yi” of operations in the same way. In the former proof, gathering the three blocks yields a solid, thevolume of which expresses the “meaning” of the multiplication together of length, width and height. The coef�cient 3is interpreted as related to the composition of the cube with three yangma. In the latter proof, like in the commentaryfollowing Problem 1.36 and bearing on the area of the circular segment, the commentator introduces a procedurethat computes iteratively the extension of the space, within the qiandu, in which the volumes of the pieces coming,respectively, from the yangma and the bienao are in the proportion of 2 to 1. In order for its various steps to receivea “meaning,” the interpretation of the procedure requires that the volume of the yangma be divided. Interestingly, thisquestion of interpretation brings us back to the parallel between problems and visual tools alluded to above.

In the case analyzed in Section 3, a problem was changed into another problem, the situation of which was richerin possibilities of interpretation, since this feature was required for making explicit the “meaning” of the speci�coperations required by a proof. The same phenomenon recurs here in relation to the yangma: the change of values

50 Several points should be stressed about the solids used in the earliest surviving documents and the hints about the early use of blocks. First, incontrast to The Nine Chapters, the Book of Mathematical Procedures does not mention the yangma or the bienao, that is, pieces among the blocksessential to establish the correctness of the procedures for other solids. However, the book treats other solids also found in The Nine Chapters inways that require knowing the volumes of the yangma and the bienao. Several algorithms for volumes in both The Nine Chapters and the Bookof Mathematical Procedures point to a geometrical interpretation, which provides reasons for the correctness of the procedures. This is how thecommentators read them in the case of The Nine Chapters: they interpret the procedures step by step with the help of blocks [Chemla, 1990]. Insuch commentaries, Liu Hui also regularly changes the values of the problems and uses blocks in the �rst way described above, that is, in a waythat is not geometrically general. Does this indicate that the commentators knew that objects of the type of blocks were used at the time when ourearliest extant sources were composed, but used only in a certain way? Or did they introduce blocks as a tool for exegesis? It is impossible to give acertain answer to this question. However, the generic terms for “�gure” or “block” do not occur in the earliest sources. The �rst known occurrencesof these terms are found in the commentaries. Moreover, we may assume that if blocks did not exist, geometrical practice may have been similar towhat some commentaries betray within the context of some problems dealing with geometrical topics: instead of referring to speci�c visual aids,the commentators interpret the operations directly in terms of the physical features of the situation of the problem. See my introduction to Chapter 9in Chemla and Guo Shuchun [2004, 673–684].


in the problem of The Nine Chapters is correlated to the introduction of an inner decomposition of the solid, whichcreates further possibilities for the interpretation of the steps needed by the proof. Not only does this conclusion offera similar explanation for the change of a situation and for a change of numerical values. It also leads to an interestingobservation. Seen from this angle, the use of a problem and the use of a visual tool are similar51: On the one hand,they “illustrate” a situation. On the other hand, they are put into play to express the “meaning” of operations, andwhen they are not rich enough to support the needs of interpretation required by a proof, they are replaced. In thiscontext, it is worth recalling how the commentator introduces blocks when he �rst resorts to them: “ ‘Speech cannotexhaust the “meaningyi” ’ (yan bu jin yi),52 hence to dissect/analyze (jie) this (volume), one must use blocks; this isthe only way to get to understanding (the procedure). , , ” We �nd here again, inrelation to visual devices, the combination of terms (meaning, understanding) that in Section 3 was used in relation toproblems.

The previous discussion highlighted why in the context of the commentary on the yangma the values of the problemwere changed into other values, which refer to the dimensions of a block: In fact, the continuity between the two kindsof item—problem and visual tool—is thereby manifested by, and inscribed in, the text.

All that has been established for blocks in fact also accounts for the speci�cities of what is known regarding �gures.This is why, even though all the evidence examined here bears on blocks, we have formulated our conclusions forvisual devices in general. The perspective developed further suggests an interpretation of Liu Hui’s own description ofhis activity, when, in his preface to the book, he writes: “The internal constitutions ( li) are analyzed with statements( ci) and the bodies are dissected with �gures ( tu).53 , ” [Chemla and Guo Shuchun, 2004,126–127].

This concludes what I wanted to establish about Liu Hui’s practice with problems. Until this point in the paper,I have mainly concentrated on the commentaries, since they provide evidence to support conclusions about the practicewith problems. To which extent can these conclusions on the practice with problems obtained thanks to the evidencefound in the commentary be transferred to the mathematical activity at the time of the composition of The NineChapters itself, or even the Book of Mathematical Procedures? This question will be examined in Section 5. Withthe example of visual tools we have already seen reasons to believe that not all mathematical practices were thesame. What about problems? This point is all the more important since in the earliest extant writings problems play aprominent part.

5. Back toThe Nine Chapters: Connecting the evidence from the commentaries and the Canon

How can one determine whether the editors of the Canon also used problems in a way similar to that of Liu Huidescribed above? Let us repeat that, unless new sources are found, we will not be able to answer this question with fullcertainty. The method I will suggest here is to gather hints in The Nine Chapters indicating continuities with respectto the practice evidenced by Liu Hui’s more proli�c writings. I shall allude to some of these hints below. However,another paper would be needed to deal with the question more systematically.

One essential point should be �rst stressed. Much of what I have said bears on the use of problems within thecontext of proof. If we believe, as is often stated, that in sheer contrast to the commentaries The Nine Chapterscontains nothing relating to proof and betrays no interest in this dimension of mathematical activity, this would denyfrom the outset the possibility of a real continuity. However, I have argued several times that, even though The NineChapters includes no fully developed proof, various facts indicate that the authors had an interest in understandingwhy their procedures were correct. We have already indicated some of them. For example, we have noticed that thequali�cation of division as “dividing in return ( baochu)” adheres to the sphere of justifying procedures. It is hence

51 Even though, in his unpublished dissertation, Wang Ling did not analyze in detail how problems and �gures were used in the course of proving,it is interesting that he used the same term of “model” to refer to both problems (he speaks of “‘model’ problems” or “variant models”) and �gures.Moreover, his conclusion regularly stresses that in the mathematics of ancient China, the particular was used to deal with the general [Wang Ling,1956, 211, 282, 287, 295]. However, he did not provide evidence to support his claims, which is my main purpose in this paper.52 The commentator quotes here the “Great commentary” (Xici dazhuan) to the Book of changes (�rst chapter, paragraph 12). Compare Chemlaand Guo Shuchun [2004, 374–375].53 Here, probably, the generic term of “�gure” stands for all the visual tools used in the commentary. On the terms occurring in the statement, seemy glossary.


quite striking that this expression occurs several times in The Nine Chapters itself. Why should one prescribe to“divide in return” instead of “divide,” unless to indicate the reason for carrying out the operation? In addition, we havementioned that Liu Hui interpreted some algorithms for computing the volume of solids as indicating a proof of theircorrectness through the description of the procedure (see footnote 50).

A further hint in support of the claim that the authors of The Nine Chapters had an interest in the correctness ofalgorithms is provided by how Liu Hui appears to interpret the fact that they described procedures in the context ofproblems. Not only does this piece of evidence indicate that he reads a concern for proof in the Canon, but it alsoreveals that he links the problems in the Canon to this concern. Let us therefore examine this evidence in greaterdetail.

In fact, there are cases where procedures in the Canon are described outside of the context of problems. Suchis the case, for instance, for the rule of three (jinyoushu ) [Chemla and Guo Shuchun, 2004, 222–225]. Thisobservation indicates that a problem is not an indispensable component in presenting a procedure. It is interesting,incidentally, that, to establish the correctness of the rule of three, Liu Hui’s commentary introduces a problem al-lowing the interpretation of the “meaning” (yi) of the operations. This fact con�rms from yet another angle the partplayed by problems in proofs in relation to making explicit the “meaning” of the operations of an algorithm. Whatis most interesting in this case, however, lies elsewhere. The commentator states about this procedure for the rule ofthree: “This is a universal procedure ( ci doushu ye).” There is only one other passage in which Liu Huirepeats this statement. There it applies to the procedure for solving systems of simultaneous linear equations (fangchengshu), mentioned in Section 2 above [Chemla and Guo Shuchun, 2004, 616–617]. However, in contrast withthe rule of three, in this case the “universal procedure” is described by The Nine Chapters within the context of aproblem about different types of millet. It reads as follows:

(8.1) Suppose that 3 bing of high-quality millet, 2 bing of medium-quality millet and 1 bing of low-quality millet pro-duce (shi) 39 dou; 2 bing of high-quality millet, 3 bing of medium-quality millet and 1 bing of low-quality milletproduce 34 dou; 1 bing of high-quality millet, 2 bing of medium-quality millet and 3 bing of low-quality millet pro-duce 26 dou. One asks how much is produced respectively by one bing of high-, medium- and low-quality millet.54

, , , ; , , , ; , , ,[Chemla and Guo Shuchun, 2004, 616–617]

The piece of evidence in which I am interested is found immediately after the sentence in which Liu Hui quali�esthe procedure as being “universal.” The statement he adds to this can be interpreted as accounting for why, here,in contrast to the case of the rule of three, the Canon uses the context of problems to present the procedure. Thecommentator writes:

It would be dif�cult to understand (the procedure) with abstract expressions (kongyan), this is why one deliberatelylinked it to (a problem of) millet to eliminate the obstacle. , 55

This statement reveals that, in Liu Hui’s perspective, the purpose of the Canon for presenting a procedure in thecontext of a problem was related to the aim of having the procedure be understood. We have already shown therelationship between “understanding” a procedure and establishing its correctness. In the text that follows, Liu Huiuses the context of the problem on millets to interpret the operations of the procedure and thereby bring to light that

54 Bing designates a unit of capacity from a system of units different from that of the dou. I am grateful to Michel Teboul, who suggestedanother interpretation for this problem: the statement may be understood as referring to distinct units of capacity all named bing and the value ofwhich would depend on the grain measured. I have shown that in The Nine Chapters, the Canon gives evidence of a similar phenomenon withrespect to the unit of capacity hu , which had different values for different grains [Chemla and Guo Shuchun, 2004, introduction to Chapter 2,201–205]. The key term shi in Problem 8.1 could thus be interpreted either as “�ll up” or as “capacity” (the basic sentence would then read:“Suppose that 3 bing of high-quality grain, 2 bing of medium-quality grain, and 1 bing of low-quality grain (correspond to) a capacity (shi) of 39dou”). The meaning of shi as “capacity” is attested in a passage from the Records on the scrutiny of the crafts (Kaogongji , dated to thethird century B.C.E.), which deals with standard vessels and is quoted in Liu Hui’s commentary on Problem 5.25 [Chemla and Guo Shuchun, 2004,450–453]. However, the actual values in the problem still make me prefer the �rst interpretation, which I have therefore inserted here.55 On the interpretation of kongyan, see Chemla [1997a] and the entry for kong “abstract,” in my glossary [Chemla and Guo Shuchun, 2004, 947].Chemla [2000] develops a reading of Chapter 8 of The Nine Chapters along these lines, that is, by assuming that the millets were intended to enablean interpretation of the operations in the procedures.


the pattern of “equalizing” and “homogenizing” underlies it, thus proving its correctness (see Section 2). The piece ofevidence that the above statement constitutes may hence indicate that Liu Hui reads the Canon as providing problemsto be put into play to prove algorithms.

Several additional features of the problems used in the same chapter are quite interesting to support our argumentthat the practice of problems in The Nine Chapters presents continuities with Liu Hui’s commentary [Chemla, 2000].

First, in fact, the statement of Problem 8.1, regardless of whether one interprets it in terms of production of milletor of capacities (cf. footnote 54), can be interpreted in a still completely different way:

(8.1, alternative interpretation) Suppose that 3 bing of high-position millet, 2 bing of medium-position millet and 1 bingof low-position millet correspond to the dividend (shi) 39 dou; 2 bing of high-position millet, 3 bing of medium-positionmillet and 1 bing of low-position millet correspond to the dividend 34 dou; 1 bing of high-position millet, 2 bing ofmedium-position millet and 3 bing of low-position millet correspond to the dividend 26 dou. One asks how much is thedividend corresponding respectively to one bing of high-, medium- and low-position millet.

In addition to interpreting “high,” “middle,” and “low” as positions on the calculating surface, this second readinginterprets the term shi (“produce”/capacity) with its technical meaning in mathematics, “dividend.” From the �rstcentury C.E. till the fourteenth century at least, in ancient China an algebraic equation was conceptualized as anopposition between divisors (the various coef�cients of the unknown) and a dividend (the constant term). In fact, itcan be shown that Liu Hui also reads the statement of Problem 8.1 in this alternative way and understands a linearequation as opposing a dividend (its constant term) to divisors.

This �rst problem of Chapter 8 is followed by �ve others similar to it, all relating to grain. The full procedure forsolving systems of linear equations, with positive and negative numbers, is then unfolded in relation to them.

If Problem 8.6 were interpreted along the same lines as the previous ones, it would correspond to a system ofequations with two negative constant terms. However, all ancient sources give the two “productions/dividends” aspositive, a fact that the commentators do not stress as erroneous. In other terms, it seems that the interpretation ofthe “dividends” in terms of “production” or “capacities” imposes that the shi remains positive. If this is the case, thisconstraint, which derives from the interpretation of the operations in the terms of the problem, poses limits to the fullpresentation of the mathematical topic. What follows in The Nine Chapters supports this hypothesis: Problems 8.7and 8.8 turn to a new situation—buying animals—to discuss new types of systems in which dividends—in this contextno longer “productions” or “capacities”—are either positive, negative or zero.

This seems to indicate that in The Nine Chapters the situation with millet producing grain (or contained in unitsof capacities) was used to deal with systems of equations as long as the interpretation allowed by the situation didnot con�ict with the mathematical requirement. As soon as a divergence arose between the interpretation in the termsof the problems and the mathematical meaning, the situation for discussing the topic was changed to allow furtherdiscussion. If our interpretation is correct, this implies that, in agreement with Liu Hui’s explanation, the use of thesituation of a problem for interpreting the operations of an algorithm, in relation to understanding and proving it, datesto the time of The Nine Chapters.

Such a conclusion suggests that there may have existed a mathematical culture of situations selected for their use-fulness in presenting procedures. Another element appears to con�rm this hypothesis: many mathematical questionsare discussed within the framework of the same kind of situation throughout the centuries. One way of accounting forthis would be that these situations proved particularly suitable in relation to interpreting the operations of the proce-dures. Inquiring further into this question exceeds the framework of this paper. Let me simply claim for now that thisis quite plausible.

6. Conclusion

We have seen that not only does the commentator Liu Hui attest to a practice of problems, described in Sections 1to 4, that is peculiar and differs from the one most common today, but he also seems to assume that his way of usingproblems was in continuity with former practices, especially those contemporary with the making of the Canon. Infact, we have found hints in The Nine Chapters that appear to support his belief. In this tradition, whether problemswere practical in form or abstract, they were read as general statements, the extension of which was determined on thebasis of the procedure relating to them. This �eshes out what the opening sections of The Gnomon of the Zhou intended


when depicting intellectual activity in mathematics or astronomy as aiming at widening “classes (lei)” of problems.Instead of interpreting these statements on the basis of our own experience, it seems to me more appropriate to relyon the evidence we have about mathematical activity in ancient China to interpret this early theoretical descriptionof what mathematics was about. Furthermore, the results obtained in this paper give us elements for establishing amethod to interpret our earliest Chinese sources.

As we have seen, the procedure given for a problem can either solve it or be used for establishing the correctnessof another algorithm. This is why problems can be either questions to be solved or statements describing a situationin order to interpret the meaning of operations. Clearly, ignoring such facts would be quite detrimental when readingmathematical sources from ancient China. This is one of the main errors responsible for the mistaken idea that thesetexts can be adequately interpreted as merely practical. However, the bene�t of promoting a new method of readingis not limited to this aspect. Instead of assuming that mathematical practice has been uniform in space and time,such a way of approaching texts, attempting to establish how they should be read before one sets out to read them,contributes to restoring the diversity of mathematical practice. I hope the above arguments will inspire further researchthat by gathering speci�c evidence will enable us to restore various practices of mathematical problems. Once we haveassembled several similar case studies, we shall be in a position to outline a research program that may create newconditions for interpreting our sources.

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Available online at www.sciencedirect.com


The correspondence between Moritz Pasch and Felix Klein

Dirk Schlimm

Department of Philosophy, McGill University, 855 Sherbrooke St. W., Montreal QC, H3A 2T7, Canada


Abstract

The extant correspondence, consisting of ten letters from the period from 1882 to 1902, from Moritz Pasch to Felix Klein ispresented together with an English translation and a short introduction. These letters provide insights into the views of Pasch andKlein regarding the role of intuition and axioms in mathematics, and also into the hiring practices of mathematics professors in the1880s.© 2013 Elsevier Inc. All rights reserved.

MSC: 01A55

Keywords: Correspondence; Klein; Pasch

1. Introduction

If one is interested in an historically informed understanding of the development of mathematical ideas,it is vital to look beyond the published works and to take other sources, like unpublished manuscripts,drafts, and letters, into account. Correspondence can be a particularly rich source of insights, since, onthe one hand, letters are usually not meant for publication and thus allow for more candid and tentativeformulations. On the other hand, the relationship between the correspondents has a major effect on thecontent of letters and historians can learn about this relation from what is written, how it is written, andfrom what was left out. The information gleaned from letters allows us to get a glimpse into the intellectual,personal, and social lives of their authors.

In the case where a mathematician has been a proli�c letter-writer, like Leibniz or Gödel, and the cor-respondence has been preserved in its entirety, it constitutes a gold mine for historians of mathematics.In many other cases, however, the available source material is scarce, fragmented, or simply non-existent.Circumstances often allowed only a small portion of the correspondence to be preserved. Nevertheless, thelack of evidence for an extensive body of letters cannot immediately be taken as evidence that there neverwas any such in the �rst place, because it could have been destroyed in the course of time or lost, waitingto be rediscovered in the future.

E-mail address: [email protected].

0315-0860/$ – see front matter © 2013 Elsevier Inc. All rights reserved.http://dx.doi.org/10.1016/j.hm.2013.02.001

http://www.sciencedirect.com





184 D. Schlimm / Historia Mathematica 40 (2013) 183–202

Below is presented a German transcription together with an English translation of the known extant cor-respondence between Moritz Pasch (1843–1930) and Felix Klein (1849–1925), which consists of ten lettersfrom Pasch to Klein from the period 1882–1902.1 If considered in the context of the other known corre-spondence of these authors, the relative weight of these letters for the historical scholarship on Klein andPasch is very different. To give a �rst impression of this imbalance, the Kalliope Verbundkatalog Nachlässeund Autographen database2 lists 306 items addressed to Klein and 1785 from Klein, where each item cancontain several manuscripts or letters; in comparison, only 13 items from Pasch are listed (two of these con-tain the letters translated below) and none to him. In other words, for each listed item from Pasch, there areover 160 items to or from Klein! Here are two factors that may have contributed to this discrepancy. First,from 1877 to 1924 Klein was editor of the Mathematische Annalen (from vol. 10 to 92), which became oneof the most prestigious mathematical journals during this time. In this capacity he dealt with submissionsfor publication, referees, and general administrative issues. Indeed, three of the letters from Pasch wereprompted by him submitting articles for publication to Klein. Second, Klein actively cultivated many con-tacts with other mathematicians and scientists and he was very much involved in improving the standingof mathematics and mathematics education at national and international levels. This knowledge about theprofessional and personal situation of many mathematicians is one reason why Pasch approached him forguidance when the second chair in mathematics at the University of Giessen became vacant in 1886. Later,in 1896/97 Klein became the most important consultant for the Prussian Ministry of Education (Kultusmin-isterium) with regard to appointments of mathematics professors [Tobies, 1987, 41]. Klein reported to havedestroyed his correspondence in the fall of 1878 [Tobies and Rowe, 1990, 8], but he seems to have keptmost of it from after this incident and it has been preserved in the archives at the University of Göttingen.Given the availability and the content, parts of Klein’s correspondence have already been published: forexample, selections of the correspondence with Hilbert, consisting of 129 letters from 1886 to 1918 [Frei,1985], and a selection of 186 letters between Klein and Adolph Mayer from the period 1871–1907 [Tobiesand Rowe, 1990].

The state of affairs is very different in the case of Pasch. His Nachlaß at the University of Giessen,where he taught from 1870 until 19113 contains only a handful of short notes. This is consistent with theinformation given by Pasch’s son-in-law, Clemens Thaer, to Heinrich Scholz that Pasch had the habit ofdestroying his correspondence after having replied to it [Frege, 1976, 169]. This makes it extremely dif�cultto gauge the original extent of Pasch’s correspondence. To get a rough idea of it, we have to look for theletters that Pasch wrote in the estates of the recipients. One would expect a promising starting point to bethose mathematicians who shared Pasch’s main research interests, e.g., in axiomatics and geometry. But,unfortunately, the yield is rather thin. In Hilbert’s Nachlaß, for example, we �nd only two postcards fromPasch (from 1888 and 1913), which are thank-you notes for the sending of a publication and for Hilbert’scontribution to the celebration of Pasch’s 70th birthday. Pasch’s correspondence with Gottlob Frege, whichconsists of �ve letters and a postcard from Pasch to Frege, but no extant item from Frege to Pasch [ Frege,1976, 169–174], contains the beginnings of interesting discussions. However, these are not continued dueto the fact that Pasch was often occupied with other duties, as he frequently laments (this is also a commontheme in the letters to Klein). It is also interesting to note that �ve of these items are prompted by Frege’ssending of his writings to Pasch. In the Nachlaß of Max Dehn, who wrote a long article on the historyof geometry that was appended to the second edition of Pasch’s lectures on geometry in 1926, only oneshort letter can be found. Of the colleagues that Pasch mentions in his autobiography [Pasch, 1930], eitherthere seems to be no extant Nachlaß, e.g., of Jakob Rosanes, whom Pasch refers to as ‘his friend’, or nocorrespondence from Pasch could be found, as in the case of Kronecker and Weierstrass. The following

1 For some biographical background, see Pasch [1930], Engel and Dehn [1934], and Tobies [1981].2 Online: http://kalliope.staatsbibliothek-berlin.de.3 He continued to live in Giessen until his death in 1930.

http://kalliope.staatsbibliothek-berlin.de

D. Schlimm / Historia Mathematica 40 (2013) 183–202 185

list shows all recipients of more than one letter from Pasch that I have been able to track down, togetherwith the total number of extant items: Felix Klein (10), Gottlob Frege (6), Mario Pieri (3), Ernst RichardNeumann (3), Otto Behagel (2), David Hilbert (2), Otto Toeplitz (2), Ventura Reyes y Prósper (2). To LuigiCremona, Max Dehn, W. Keller, Adolf Kneser, Alexander Naumann, Bernhard Stade, and Hans Vaihinger,I have been able to locate only one single letter from Pasch addressed to each. Thus, based on all thecurrently available evidence it appears that Pasch indeed was not an avid letter writer and that he did notmaintain an extensive correspondence with this colleagues. The letters to Klein reproduced below form thelargest set of known letters from Pasch to any other mathematician. As such, they might not tell us muchabout Klein, but they are a rare source of information about Pasch.

2. The Pasch–Klein correspondence

2.1. Description of the letters

Below are English translations and German transcriptions of the extant correspondence of Moritz Paschto Felix Klein, which is held in the archives of the Niedersächsische Staats- und UniversitätsbibliothekGöttingen. This correspondence consists of ten letters from Pasch written in the period from 1882 to 1902.From Pasch’s remarks in these letters we can infer that Klein must have written at least seven times toPasch, but, as far as I know, the part of the correspondence addressed to Pasch has not been preserved.

The original letters are handwritten by Pasch in Kurrent script and a few words are extremely dif�cult todecipher. Best guesses for these are reproduced with a small superscripted question mark (?). Punctuationand spelling have not been altered in the German transcriptions; in the English translation underliningshave been rendered in italics and crossed-out words are omitted. Page breaks in the originals are markedwith a vertical line followed by the page number in angle brackets in the transcription. For example, ‘|〈2〉’indicates the beginning of the second page in the original. The numbers in square brackets in the headingsof the German transcriptions indicate the signature numbers from the library of the University of Göttingen.

2.2. Content of the letters

The letters from Pasch to Klein can be divided up into thee chronological phases, which are presentednext. The letters from Klein to Pasch, whose existence can be inferred from the content of Pasch’s letters,are also indicated below.

2.2.1. Empiricism and publications in Mathematische Annalen (1882–1887)

No. Date Content

– (date unknown) Letter from Klein, after the publication of Pasch’s Vorlesungen [1882a],informing Pasch about the views expressed in Klein [1874b].

1. 16 Jun. 1882 Reply from Pasch, noting that he does not possess Klein’s article.– (date unknown) Letter from Klein, including the 1874 article.2. 22 Jun. 1882 Reply from Pasch, returning the sent materials.3. 9 Oct. 1883 Pasch sends Klein articles for publication in the Annalen.4. 31 Mar. 1887 Pasch sends Klein articles for publication in the Annalen.– 23 Apr. 1887 Reply from Klein.

The occasion for the �rst exchange of letters in June 1882 was a letter from Klein. Herein he informsPasch about an article on the general concept of functions [Klein, 1874b], in which he expressed viewssimilar to those that Pasch put forward in his recently published lectures on projective geometry [Pasch,1882a], the preface of which is dated March 1882. In his reply, Pasch brie�y presents his philosophicalviews on the nature of geometry, including a discussion of the differences between our intuitions and the


mathematical concept of function.4 He explains that Klein’s 1874 article was not mentioned in this context,mainly because Pasch misremembered its exact content, and he also reveals some interesting backgroundinformation regarding the development of his own views. In particular, he traces back his empiricist leaningsto the lectures of Kronecker and Weierstrass he attended in 1865/66 and to discussions with his friendJakob Rosanes.5 Pasch remarks that he thought his views to be fairly uncontroversial, especially after�nding similar views expressed by Lipschitz, and he shows genuine surprise after hearing about the negativereactions to Klein’s views. Pasch admits that he has not read much philosophy, and the bits that he hadread quickly disappointed him; he considered the authors to be on the wrong track. This exchange ledto the addition of a brief remark with a reference to Klein’s paper at the end of Pasch’s Einleitung indie Differential- und Integralrechung [Pasch, 1882b], which was published soon afterwards, and also to areference to Pasch’s books in the republication of Klein’s article in the Mathematische Annalen a year later[Klein, 1883], which Pasch had encouraged.

In the letters from 1883 and March 1887 Pasch sent Klein articles meant for publication in the Mathe-matische Annalen, for which Klein was one of the editors. Interestingly, Pasch published more often in theMathematische Annalen than in the Journal für die reine und angewandte Mathematik (Crelle Journal),which was the journal closely associated with the Berlin school of mathematics of Kronecker and Weier-strass [Tobies and Rowe, 1990, 38–44]. In these letters Pasch emphasizes the af�nity of their views, butalso hints at some differences, but without going into any detail.

2.2.2. Filling of vacant position in Giessen (1887–1888)

No. Date Content

5. 3 Dec. 1887 Pasch sends list of possible successors of Baltzer.– (date unknown) Reply from Klein.6. 13 Dec. 1887 More on the succession of Baltzer.– (date unknown) Reply from Klein.7. 22 Dec. 1887 Brief reply from Pasch.8. 17 Feb. 1888 Pasch informs Klein about decision on succession of Baltzer.

The four letters from December 1887 to February 1888 deal exclusively with discussions surroundingthe position at the University of Giessen that had become vacant with the death of Richard Baltzer (1818–1887), who had been professor of mathematics in Giessen for almost two decades.6 Baltzer started thisposition in Giessen on April 28, 1869 as the successor of Alfred Clebsch, who had moved to Göttingen. Atthe time there was only one chaired position (Ordentlicher Professor) in mathematics in Giessen and oneposition without chair (Ausserordentlicher Professor). Baltzer had been instrumental in getting Pasch tosubmit his habilitation in Giessen in 1870, where he then became private docent. As was anticipated, Paschwas appointed to the unchaired position in 1874, which was converted into a chaired position in 1876. In theletters, Pasch asks Klein for advice for determining who to offer the second chair of mathematics in Giessen.In this connection he reveals some of the pertinent considerations, like renumeration, teaching experiencein mechanics, and he also hints at some aspects of the local university politics. Quite interesting with regardto the general attitude at the time is Pasch’s frank remark that hiring another Jewish mathematician (Paschhimself was Jewish) would be out of the question.7 The vacant position was �lled in 1888 by Eugen Netto(1848–1919).

4 For an overview of Pasch’s philosophy of mathematics, see Schlimm [2010].5 This account is repeated 48 years later in Pasch’s autobiography [Pasch, 1930].6 See Baltzer [1888] for a short biography.7 On the situation of Jewish mathematicians during the German Empire (1871–1918), see Bergmann and Epple [2009], and Elon

[2003] for a more general account.


2.2.3. Lecture notes on geometry (1891–1902)

No. Date Content

– (date unknown) Klein sends lecture notes on non-Euclidean geometry (1889/90).9. 19 Oct. 1891 Pasch thanks Klein for the lecture notes and sends articles for

publication in the Annalen.– (date unknown) Klein sends notes of lectures held in the summer 1901.10. 21 Apr. 1902 Pasch replies and discusses some of Klein’s remarks.

In the �nal part of the correspondence, Klein twice sent lithographed notes of his lectures on geometryto Pasch, which were the occasions for Pasch’s replies to Klein in 1891 and 1902. Here Pasch returns to thediscussion of philosophical issues, in particular the role of intuition, diagrams, and axioms in mathemat-ics, in order to clarify the points of agreement and disagreement between him and Klein. Pasch commentsdirectly on some of Klein’s remarks in the lecture notes and hints at having become ‘more critical’ to-wards the approach taken in his two books of 1882, but without divulging any more details about thischange of heart. These comments supplement and provide additional background for similar remarks thatcan be found in Pasch’s and Klein’s publications.8 A concluding remark reveals Pasch’s disappointmentthat his students did not appreciate his foundational investigations, which can be explained by the factthat most students of mathematics in Giessen wanted to become teachers and not pursue a research ca-reer.

2.3. Names mentioned in the Pasch–Klein correspondence

The numbers refer to the letters in which the name is mentioned, according to the numbering schemeintroduced in the previous section.

Baltzer, Richard (1818–1887): 5Betti, Enrico (1823–1892): 8Dyck, Walther (1856–1934): 5 and 6Gordan, Paul (1837–1912): 1Hölder, Otto (1859–1937): 6Henneberg, Lebrecht (1850–1933): 5Hettner, Georg (1854–1914): 5Hurwitz, Adolf (1859–1919): 6Köpcke, Alfred (1852–1932): 4Kiepert, Ludwig (1846–1934): 5Krause, Martin (1851–1920): 5Krazer, Adolf (1858–1926): 6Kronecker, Leopold (1823–1891): 2Lindemann, Ferdinand (1852–1939): 5 and 9Lipschitz, Rudolf (1832–1903): 1Meyer, Franz (1856–1934): 5, 6, and 8Nöther, Max (1844–1921): 6

Netto, Eugen (1848–1919): 5 and 8Peano, Giuseppe (1858–1932): 9Reyes y Prósper, Ventura (1863–1922): 4 and 9Röntgen, Wilhelm (1845–1923): 8Rosanes, Jakob (1842–1922): 2 and 3Runge, Carl (1856–1927): 5Schön�ies , Arthur (1853–1928): 6Schering, Ernst (1824–1897): 5Schottky, Friedrich (1851–1935): 5Schröter, Heinrich (1829–1892): 8Schur, Friedrich (1856–1932): 5, 6, 8, and 9Staude, Otto (1857–1928): 5, 6, and 8Stickelberger, Ludwig (1850–1936): 5 and 6Voss, Aurel (1845–1931): 6 and 7Weierstrass, Karl (1815–1897): 2Wiltheiss, Ernst (1855–1900): 5

8 A more detailed discussion of similarities and differences between Pasch’s and Klein’s views, in particular with regard to therole of intuition in mathematics, is in preparation by the author.


3. Letters from Pasch to Klein (1882–1902)

3.1. Pasch to Klein, June 16, 1882

Giessen, June 16, 1882Highly honoured colleague,

The answer to your kind letter was delayed because I wanted to get the Erlanger Berichte here sothat I could �rst have another look at your article, of which I do not own a copy. 9 I found out that theOberhessische Gesellschaft, which I do not belong to, exchanges publications with the Erlanger Societät;however, the issues have not been collected carefully, and, among others, the one that contains your articleis missing. I read your article at Gordan’s, as he got a copy from you (it was during the semester in whichI taught the �rst course of lectures on the subject matter covered in my book) 10; the content appealed tome all the more, because I entirely agreed with your basic view. However, I still believe that the mainpoint of your discussion was to show how the concept of a function strip leads to the tangent. On the otherhand, I considered the basic view, according to which geometric concepts, like other empirical concepts,are af�icted with imprecision, to be more or less widespread. For example, I saw in the acknowledgmentthat the intuition of a curve and the concept of function do not coincide, the reason why function theorysought to free itself from so-called geometrical proofs. In the meantime, statements about the imprecisionof all determinations of measurement that I found in Lipschitz (Differential- und Integralrechnung, §40)11

further con�rmed my assumption.Thus it came about that I did not mention your article in the book.12 I deeply regret having had this

understanding and will not fail to make up for the omission at the next suitable occasion. Your remark thatthe ideas that you put forward at that time encountered �erce opposition leads me to assume that my workwill also gain little approval. All the more I am pleased to hear that you intend to return to this subjectyourself.

With sincere regards,respectfully Yours,

M. Pasch

3.2. Pasch to Klein, June 22, 1882

Giessen, June 22, 1882Highly honoured colleague,

I owe you many thanks for your kind mail, which arrived on Monday. I have now received the last proofsheet for a small work, which will be published by Teubner under the title Einleitung in die Differential- und

9 Pasch refers here to Klein [1874b]. See also Pasch’s remarks in the letter from October 9, 1883. Klein’s paper was reprinted inthe Mathematische Annalen in 1883, with an additional footnote by Klein referring to Pasch [1882a] and Pasch [1882b]. It seemsthat Klein had informed Pasch about this article in an earlier letter.10 In Pasch [1882a, IV] the author notes that this book is based on lectures that we held �rst in the winter semester 1873/74.Gordan moved from Giessen to join Klein in Erlangen in 1874.11 Lipschitz [1880, 207–213]. On p. 208, Lipschitz writes: ‘Before discussing in detail the nature of equation (2), let me remind youthat all measurements of actual objects lead to equations in which the representation differs from that which is to be represented bya certain quantity and thus that every application of the calculation to actual objects that are subjected to measurements implies theuse of equations, which are af�icted with certain errors.’ (‘Ehe auf das Wesen der Gleichung (2) näher eingegangen wird, sei daranerinnert, dass alle Massbestimmungen von Gegenständen der Wirklichkeit zu Gleichungen führen, bei denen das dargestellte vondem darzustellenden um eine gewisse Grösse abweicht, und dass daher jede Anwendung der Rechnung auf wirkliche Gegenstände,die dem Masse unterworfen sind, den Gebrauch von Gleichungen in sich schliesst, die mit gewissen Fehlern behaftet sind.’)12 Pasch [1882a].


Integralrechnung.13 Like my work on geometry, which has already appeared, this is the result of my effortsto come to terms with myself about the material I have to lecture on. Since I refer to the book on geometryin the new piece, when I discuss the imprecision of measurements, I took the opportunity to add a remarkin the conclusion alluding to your article.14

I really appreciated being able to read your article once again, in particular because I see that already atthe time of the main revision of my Vorlesungen (1877/78), I remembered its content only incompletely.The fact that already in 1865 or 1866 Kronecker had told me about the impossibility of geometrically rep-resenting certain functions, and that already then Weierstraß placed great importance on arithmetical proofsin his lectures, in addition to some conversations I had with Rosanes in the 1860s, may have contributed tomy views on the matter, which I recently described to you. Thus, I was quite surprised by what you relateabout the comments made by some function theorists.

I did not have the opportunity to make such observations myself. Philosophers cannot do much here inmy view; they must wait until things are put in order by mathematicians. I, too, must confess to have hardlyread any of the philosophical writings; after brief attempts at reading them, it always appeared to me asthough these men were on the wrong paths.

If I may be permitted to state my personal opinion, I would like to speak in favour of a wider distributionof your article through the Annalen. On its few pages the relevant issues are expressed and brought tobear. I myself had to settle on a narrow aim, since I already needed very much time and effort for bringingeverything into a strictly logical system, and thus didn’t advance my discussion to curved entities at all.

With the return of your collection15 I remain,

YoursRespectfully

M. Pasch

3.3. Pasch to Klein, October 9, 1883

Giessen, October 9, 1883Highly honoured colleague,

I take the liberty of sending you the three enclosed articles, two by Rosanes and one by myself,and to enquire as to whether you might want to allocate a spot to them in the Annalen.16 If so, wewould like to request that they be published in succession, �rst mine about collinearity and reciprocity,then Rosanes’: ‘Erweiterung etc.’ and ‘Bemerkung zur etc.’ I should also mention that Rosanes, whohas now returned from lengthy travels, put both notes quickly down on paper in order not to hold upthe work that I drafted last year even more, and which is already rather delayed, and that he there-fore would like to reserve the opportunity of �lling in possibly missing information in the bibliogra-phy.

I was happy to see your article from the Erlanger Berichte in print in the next issue of the Annalen.17

Particularly after you mentioned my books in your footnote, I would very much like to return to the matter

13 Pasch [1882b].14 Pasch [1882b, 188]: the ‘Remark to page 13’ contains a reference to Klein [1874b].15 This indicates that Klein must have sent something to Pasch that he is now returning.16 These articles appeared successively in the following order: Rosanes [1884a] and Rosanes [1884b], both dated ‘July 1883,’ andPasch [1884], dated ‘1882.’ This is not the order that Pasch requested in this letter. In Rosanes [1884a, 414] the author mentionsthat his result had been obtained earlier by ‘my friend Pasch.’17 See Footnote 9, above.


and eventually be able to clarify the issues that still separate our views. For this, however, I would need tohave more undisturbed time at my disposal than I can presently hope for.

With highest regards I remainYours sincerely

M. Pasch

3.4. Pasch to Klein, March 31, 1887

Giessen, March 31, 1887Highly honoured colleague,

I take the liberty of sending you two articles and of humbly asking whether you would be inclinedto print them in the Annalen.18 The article on function theory was developed from lectures, which madeapparent the need for an essential simpli�cation of certain sections. Prósper’s article 19 really only providedan external motive for the small note; for, I do not want to give particular weight to the simpli�cation ofthe proofs vis-à-vis Mr. Prósper, but rather I would like to point out again — especially after having readKöpcke’s article20 — the views that you pioneered at the time.

With high regardsYours

M. Pasch

3.5. Pasch to Klein, December 3, 1887

Giessen, December 3, 1887Highly honoured colleague,

Baltzer’s21 sudden death forced our discipline to make preparations for an appropriate appointment.Because I had to take over a replacement lecture as well as take care of a number of other things, I am onlynow able to devote myself to the task I was assigned — a task which, thanks to the great number of namesto be considered, is not easy. Firstly, with as much regard as possible to the circumstances, I made deletions,and now have a list, which I allow myself to share with you along with a request for your discretion. I hopethat I was not unfair with my deletions; on the other hand, there are probably some remaining names thatwill have to go because of �nancial considerations of which I do not have suf�cient knowledge, and othersthat are not suitable because of their specialties, given that it is preferred, for example, for the professor tobe appointed to enjoy and to be successful in teaching those parts of mathematics that are closely related tophysics.

Perhaps I may assume that you, my honored colleague, also take some interest in this matter, afterall quite important to our discipline, and that you will assist us with your extensive knowledge of thecircumstances in question. I would genuinely appreciate it if you did this.

18 Pasch [1887a] and Pasch [1887b].19 Reyes y Prósper [1887].20 Köpcke [1887].21 Richard Baltzer (January 27, 1818 – November 7, 1887) held a chair in mathematics in Giessen; on the history of mathematicsat the University of Giessen, see Scharlau [1990, 111–116].


I also give my belated thanks for your kind letter of April 23. The prospect that you speak of, that theremight be a gathering this summer in Göttingen of the three universities, was regretfully not realized.22

With high regards,Yours

M. Pasch

Dyck, Henneberg, Hettner, Kiepert, Krause, Lindemann, Franz Meyer, Netto, Runge, K. Schering,Schottky, Schur, Staude, Stickelberger, Wiltheiss23


Giessen, 13 December 1887Highly honoured colleague,

Given the great kindness of the extensive and most valuable advice that you gave in response to my stillvaguely focused questions, the delay of my thanks would be doubly unjusti�able if it had not been causedby university issues, which were quite unexpectedly loaded upon me in that past week and, when combinedwith the already present burden, completely distracted me from pursuing the question of hiring. TodayI return to the matter and acknowledge �rst of all that I, in fact, very much appreciated your particularattention to personal characteristics. As to your offer of more advice, I would like to make use of it, �rstof all, regarding the following points. For Voss and Dyck it would be necessary to know what wagesand bene�ts they receive, at which point the Bavarian quinquennial bene�ts also come into consideration.Given that I am lacking other connections, I would much appreciate your intervention, but only if doing sodoes not cause too much trouble. Staude will probably remain out of consideration because of his salary,as he apparently earns 4000 Rubel; perhaps you have by chance more exact knowledge about this. Ofcourse, various suggestions have been received — mostly indirectly. Krazer and Stickelberger, about whosepersonalities I know almost nothing, were named. In particular the physicists will consider mechanics andrelated courses to be of great importance. In this respect, how do things stand with Hölder, F. Meyer,Schön�ies, and Schur? The great dif�culty for me lies in the overabundance of names which could beconsidered in this case. I was of course not allowed to include any Jewish colleagues in the list; otherwisescholars like Nöther and Hurwitz would not have been omitted.24

Once again, my warmest thanks for your kind cooperation and high regards from

YoursM. Pasch


Giessen, December 22, 1887Highly honoured colleague,

I do not want to delay any longer in thanking you for your kind advice, and especially taking the troubleto make the inquiries in Munich yourself.25 Now the prospect of being able to consider Voss is, in thegiven circumstances here, indeed a remote one. The main point, however, is that I must put aside my job ofmaking suggestions until after the end of the holidays and therefore cannot at all forsee what will happen.

22 It is possible that Klein had planned something in connection with the celebration of the 150th anniversary of the founding ofthe University of Göttingen, which took place on August 7–9, 1887, but which Klein did not attend. The celebration is mentionedin a letter from Klein to Mayer, August 21, 1887 [Tobies and Rowe, 1990, 163].23 This list of names in alphabetical order is written in a single column on a separate page.24 Pasch himself was Jewish.25 Aurel Voss and Walther Dyck, who were mentioned in the previous letter, were professors in Munich at the time; Voss had beena postdoctoral student and Dyck a doctoral student of Klein.


This situation is the outcome of the unfortunate way this matter got started, which made it necessary to waitfor the government’s decision on a preliminary question. I reserve the right to be allowed to return to all ofthese things later and remain in the meantime, with the best wishes,

YoursM. Pasch

3.8. Pasch to Klein, February 17, 1888

Giessen, February 17, 1888Highly honoured colleague,

I am only now in a position to let you know the result of the negotiations for the local appointment.Allow me �rst to elaborate on an earlier intimation. We had a chair as well as a professorship without chairfor mathematics; I held the latter as a personal appointment. Generally, people thought that this relationshipwould now have to be appropriately settled. However, our physicist (Röntgen), whose behavior had already,particularly in the last semester, caused my disapproval, used the fact that he was currently dean to preventthe resolution in question. This made necessary negotiations at the faculty level, which the dean did notreally help with, but which �nally led to a resolution of the matter. The conclusion was reached onlylast week, and the proposals could then be made soon. The government approved the retention of twochairs, however — as was expected based on the local circumstances — with the requirement that as littleas possible would be spent on salaries. We thus had to abandon the idea of considering already chairedprofessors, with the exception of Staude, whom Schröter repeatedly recommended strongly, also with theremark that he only earns 2400 Rubel and an insigni�cant amount of extra income, and that he will makeany sacri�ce to return to Germany from Russia, where the conditions are unbearable for him. 26 Moreover,the decisive factor was the request from the natural sciences that the appointed person should have provenhimself as lecturer in mechanics and related �elds. Thus came about the list with 1) Netto, 2) Staude,and 3) Franz Meyer. Netto, as the oldest among the scholars under consideration, had to come �rst. Theattendance is supposed to have gone down considerably in Berlin as well.27 However, if Netto would notaccept, the chances of getting Staude would be certain. The determining factor for Meyer was that hetranslated Betti28 and lectured on mathematical physics, astronomy, and the like. Unfortunately, Schur’sinterests did not meet our needs. The suggestions still need the approval of the whole senate, which weexpect will be given on the 25th of the month. For this reason I must request that this communication betreated as con�dential until then. I once more take the opportunity to give my most heartfelt thanks for yourgenerous assistance, and remain,

with highest regards,Yours

M. Pasch

3.9. Pasch to Klein, October 19, 1891

Giessen, October 19, 1891Highly honoured Colleague,

I am taking the liberty of sending you the enclosed pages for your Annalen29; their occasion is describedin the introduction. The presented view is quite trivial; I nevertheless took such a long time with the �nalversion because I wanted to dispel, at least for myself, the concerns that might be raised.

26 Otto Staude received his habilitation under Schöter, and was professor in Dorpat, Russia (now Tartu, Estonia) from 1886 to1888.27 Netto was professor without chair in Berlin since 1882.28 Betti [1885].29 Pasch [1892].


I am also late in thanking you for sending your essay on non-Euclidean geometry from last year.30

I allowed myself to refer to the last sentences of your essay in the second footnote on p. 7 of my manuscript.In the �rst footnote I do not refer to your comments in vol. 6, p. 136 31 and vol. 7, p. 53232 for the timebeing, since I would rather like to interpret these passages in a different way now.33 I also agree with mostof what you say in the last two pages of your essay. The content of the axioms comes from observations(intuition as an internal activity is based on remembering what has been observed); the concepts used inthe axioms, however, are inexact, and thus so are the axioms themselves. These latter can, however, onlybe used purely logically if they are presented as being exact. By further working on the axioms and seekinggeometric propositions, we commonly make use of �gures, either by drawing them or by ‘imagining’ them.In fact, where one does not calculate, one would not be able to make any progress without this help withregard to somewhat complex phenomena. However — and here our views apparently diverge — the useof �gures is merely a facilitation of the work; otherwise, the work would exceed our powers, or at leastwould progress much too slowly, or would not progress far enough. The consideration must be possibleeven without the �gures, in other words: that which is derived from the �gures must already be containedin the axioms, for otherwise the axioms are not complete. It is precisely this characteristic that I wanted mybasic propositions to have, and I therefore had to admittedly increase their number (see Geometrie34 p. 6bottom, pp. 43–45, pp. 98–100). The ful�llment of this requirement is possible, and thus the requirementis no doubt justi�ed. Proven, in geometry as in analysis, is only that which can be derived from the axiomsalone. However, with regard to external representation, I do not want to go so far as, for example, Peano(I principii di Geometria logicamente esposti, Torino 1889)35, who criticised me a little because of this onpp. 32–33.36

In vol. 39, issue I,37 Mr. Schur presented simpler proofs for the existence of ideal elements. However,this shortening had already been carried out by Prósper and myself in vol. 3238 on the basis of quite simpleprinciples. Mr. Schur wrote to me that he missed these small notes, because at the time he was moving toDorpat.

It has been noticed, not only by me but also by others,39 that Lindemann does not refer to my work ongeometry in his Raumgeometrie.40 I really do not know how this could be explained.

30 Klein [1890].31 Klein [1873].32 Klein [1874a].33 There is only one footnote in the published article in which Pasch refers to Klein [Pasch, 1892, 151]. Here Pasch mentionsKlein [1874a] and Klein [1890].34 Pasch [1882a].35 Peano [1889].36 In these pages Peano refers to Pasch’s valuable (‘pregevole’) book and points out two ambiguities in the formulation of Pasch’s�rst basic principle, noting that such ambiguities are dif�cult to avoid if a natural language is used instead of a logical notation.37 Schur [1891].38 Reyes y Prosper [1888] and Pasch [1888].39 Friedrich Schur, who Pasch mentions to have corresponded with in the previous sentence, is a possible source for this remark.Schur wrote in a letter from Friedrich Engel, January 3, 1892: ‘Another strange thing is that he [Lindemann] doesn’t seem to haveknown at all Pasch’s book on projective geometry, where everything is treated more thoroughly, albeit boringly. It surely is moreconvenient to ignore such works.’ (‘Eine andere komische Sache ist die, dass er das Buch von Pasch über projektive Geometriegarnicht gekannt zu haben scheint, wo alles viel gründlicher, wenn auch langweilig behandelt wird. Es ist freilich bequemer solcheSchriften zu ignorieren.’) Nachlaß Friedrich Engel, Sig. NE110318, Justus-Liebig Universität Giessen (online: http://digibib.ub.uni-giessen.de/cgi-bin/populo/mat.pl). The whereabouts of the correspondence between Pasch and Schur are unknown.40 Clebsch [1891]. An editorial footnote on p. 348 refers to Pasch [1873].

http://digibib.ub.uni-giessen.de/cgi-bin/populo/mat.pl

http://digibib.ub.uni-giessen.de/cgi-bin/populo/mat.pl


Finally, I would like to express my regret for my absence in Halle,41 where so many interesting thingswere presented — my presence was necessary here. I remain, with best regards,

YoursM. Pasch

3.10. Pasch to Klein, April 21, 1902

Giessen, April 21 1902Highly honoured colleague,

I have barely been able to look over the �rst quarter of your latest publication, which is based on thelectures from the summer of 1901.42 However, I did not want to delay any longer in sincerely thankingyou for kindly sending it. You must have known that these expositions would interest me very much. Myhumble self is entirely in agreement with you about emphasizing their importance again and again. Let ushope that this view gets through to the mathematicians and then in�uences the physicists, and maybe alsothe philosophers.

Like you, I cannot ascribe any greater certainty to spatial imaginations (p. 7) than I can to empiricalmeasurements and the like. Imagination allows us only to recall something we have seen, or to createshapes, whose parts we have (or could have) seen; it cannot prove anything. Abstract geometry arguablyarose out of the same need as the wish to assert natural laws as being absolutely precise (pp. 42, 44, etc.).The way that we form the image of a continuum (p. 36) must also have contributed. In fact, practicalgeometry just has to do with what the unaided eye sees, what the unaided senses perceive at all. If one usesa microscope or the like, then things become quite complicated. But, in any case, geometry proper does notknow incommensurable segments (p. 21); rather, it is only the additional, arti�cially constructed one thatdoes. The latter creates a new concept, that of ‘mathematical’ points; according to my understanding of it,this does not require any postulates, but de�nitions are suf�cient. Thus, I am not in a position to say: a pointhas no spatial extension (p. 15); here I don’t know any de�nition of ‘extension’.

I found the amicable way in which you mentioned my older works particularly pleasing.43 Unfortunately,I have become considerably more critical and this has prevented me from completing further investigationsin this direction. It is quite dif�cult to �nd interest and appreciation for this among ordinary students.

Highly honored colleague, please accept once again the greatest thanks and highest regards fromYours

M. Pasch

Acknowledgments

The correspondence from Pasch to Klein is reproduced with kind permission from Dr. Helmut Rohl�ng, Nieder-sächsische Staats- und Universitätsbibliothek Göttingen, Abteilung Spezialsammlungen und Bestandserhaltung. Theoriginal transcriptions were made by Irmgard Pientka. Rachel Rudolph helped in correcting these transcriptions andproduced �rst drafts of the translations. Katrina Sark helped with improving the translations. The author would alsolike to thank June Barrow-Green, Douglas Campbell, and two anonymous reviewers of this journal for their helpfulcomments. Work on this paper was funded by Fonds québécois de la recherche sur la société et la culture (FQRSC).

41 The annual meeting of the newly founded Deutsche Mathematiker-Vereinigung was held on September 22–26, 1891 in Halle.Pasch is listed in the member directory (dated June 1, 1891). See N.N. [1892] and Tobies and Volkert [1998, ch. 4].42 Klein [1902].43 Both of Pasch’s 1882 books and his [1887a] are mentioned on pp. 34–35 of Klein [1902].


Appendix A

A.1. Pasch to Klein, June 16, 1882 [Klein 11, 176]

Giessen, 16. Juni 1882.Hochgeehrter Herr College!

Die Beantwortung Ihres freundlichen Briefes hat sich dadurch verzögert, daß ich mir die ErlangerBerichte hier zu verschaffen suchte, um Ihren Aufsatz, den ich nicht besitze, zuvor nochmals einzuse-hen. Ich erfuhr, daß die Oberhessische Gesellschaft, der ich nicht angehöre, mit der Erlanger Societät dieSchriften austauscht; die Hefte werden jedoch nicht sorgfältig gesammelt, und u. A.44 fehlt auch dasjenige,welches Ihren Aufsatz enthält. Ich habe nun? Ihren Aufsatz bei Gordan, als dieser einen Abzug von Ihnenerhielt (es war gerade in dem Semester, wo ich das erste Colleg über den in meinem Buche behandel-ten Gegenstand vortrug), gelesen; der Inhalt sagte mir umso mehr zu, als ich die Grundanschauung |〈2〉völlig theilte. Doch blieb ich seitdem in dem Glauben, daß der Schwerpunkt Ihrer Ausführungen in demNachweise lag, wie der Begriff eines Functionsstreifens zur Tangente führt. Dagegen hielt ich die Grundan-schauung, wonach die geometrischen Begriffe, wie andere empirische Begriffe, mit Ungenauigkeit behaftetsind, für eine mehr oder weniger verbreitete. Ich erblickte z. B. in der Anerkennung des Umstandes, daßdie Anschauung von der Curve und der Begriff der Function sich nicht decken, den Grund, weshalb dieFunctionentheorie sich von den sog. geometrischen Beweisen frei zu machen gesucht hat. Inzwischen fandich auch bei Lipschitz (Differential= und Integralrechnung § 40) Aeußerungen über die Ungenauigkeit allerMaßbestimmungen und wurde dadurch in jener Annahme bestärkt. |〈3〉

So ist es gekommen, daß ich Ihren Aufsatz in dem Buche nicht erwähnt habe. Ich bedauere außeror-dentlich, mich in einer solchen Auffassung befunden zu haben, und werde nicht verfehlen, das Unterlassenebei einer demnächstigen geeigneten Gelegenheit nachzuholen. Ihre Bemerkung, daß die damals von Ihnenausgesprochene Ideen auf heftigen Widerspruch gestoßen sind, läßt mich annehmen, daß auch meine Schriftwenig Zustimmung �nden wird. Um so mehr freue ich mich, von Ihnen zu hören, daß Sie selbst auf denGegenstand zurückzukommen gedenken.

Mit vorzüglicher HochachtungIhr ergebener

M. Pasch.

A.2. Pasch to Klein, June 22, 1882 [Klein 11, 177]

Giessen, 22. Juni 1882.Hochgeehrter Herr College!

Für Ihre freundliche Sendung, welche am Montag eintraf, bin ich Ihnen sehr zu Dank verp�ichtet. In-zwischen habe ich den letzten Correcturbogen einer kleinen Schrift erhalten, welche bei Teubner unterdem Titel „Einleitung in die Differential= und Integralrechnung“ erscheint und ebenso, wie die bereits er-schienene geometrische, aus dem Bemühen, über die Dinge, welche ich vorzutragen habe, mit mir selbstins Reine zu kommen, hervorgegangen ist. Da in der neuen Schrift an einer Stelle, wo von der Unge-nauigkeit der Messungen die Rede ist, auf das geometrische Buch Bezug genommen wird, so habe ich dieGelegenheit benutzt, um am Schluß eine auf Ihren Aufsatz hinweisende Bemerkung anzubringen. |〈2〉

Es war mir sehr lieb, Ihren Aufsatz noch einmal lesen zu können, zumal ich sehe, dass ich den Inhaltdesselben schon zur Zeit der Hauptredaction meiner „Vorlesungen“ (1877/78) nur noch unvollkommenim Gedächtniß gehabt habe. Der Umstand, daß schon 1865 oder 66 Kronecker mir einmal von der Un-möglichkeit beliebige Functionen geometrisch darzustellen gesprochen und daß Weierstrass schon damals

44 Abbreviation for: unter Anderem.


in seinen Vorlesungen den größten Werth auf eine arithmetische Beweisführung gelegt hatte, mag im Vereinmit manchen Gesprächen zwischen Rosanes und mir in den 60er Jahren hauptsächlich dazu beigetragenhaben, daß ich mir von dem Sachverhalt die Vorstellung bildete, die ich Ihnen neulich beschrieb. Deshalbhat mich auch? das, was Sie über Aeußerungen von Functionentheoretikern mittheilen, nicht wenig über-rascht. |〈3〉

Ich selbst habe keine Gelegenheit gehabt, bezügliche Wahrnehmungen zu machen. Die Philosophen kön-nen hier meines Erachtens wenig thun; sie müssen abwarten, bis von Mathematikern die Sache in Ordnunggebracht sein wird. Auch ich muß Ihnen meinerseits gestehen, daß ich von den philosophischen Schriftenfast keine gelesen habe; nach kurzen Leseversuchen schien es mir immer bald, als ob die Herren sich auffalschen Wegen befänden.

Wenn ich nun meine persönliche Meinung sagen darf, so möchte ich einer allgemeineren VerbreitungIhres Aufsatzes durch die Annalen das Wort reden. Die in Betracht kommenden Momente �nden sich aufden wenigen Seiten zum Ausdruck und zur Anwendung gebracht. Ich selbst mußte mir ein enges Zielstecken, da ich schon ohnehin sehr viel Zeit und Mühe brauchte, um Alles in ein |〈4〉 streng logischesSystem zu bringen, und bin daher zu krummen Gebilden gar nicht vorgedrungen.

Unter Rücksendung Ihrer Sammlung verbleibe ich

IhrHochachtungsvoll ergebener

M. Pasch.

A.3. Pasch to Klein, October 9, 1883 [Klein 11, 178]

Giessen, 9. October 1883.Hochgeehrter Herr College!

Die drei beiliegenden Aufsätze, zwei von Rosanes, einen von mir, erlaube ich mir mit der Anfrage Ihnenzu übersenden, ob Sie denselben einen Platz in den Annalen anweisen wollen. Wir würden dann darumbitten, daß dieselben hintereinander abgedruckt werden, und zwar zuerst der meinige über Collineation undReciprocität, dann die von Rosanes: „Erweiterung etc.“ und „Bemerkung zur etc.“ Auch soll ich erwähnen,daß Rosanes, von längerer Reise zurück gekehrt, die beiden Noten schnell zu Papier gebracht hat, um meineim vorigen Jahre entworfene und bereits ziemlich verzögerte Arbeit nicht länger aufzuhalten, und daß ersich deshalb vorbehalten möchte, was etwa an Literaturnachweisungen fehlen sollte, noch nachzuholen.|〈2〉

Ihren Aufsatz aus den Erlanger Berichten freute ich mich im nächsten Heft der Annalen abgedruckt zu�nden. Ich wünschte sehr, zumal nach dem in Ihrer Fußnote enthaltenen Hinweis auf meine Bücher, auf denGegenstand zurückkommen und gelegentlich auch die Dinge, in denen Ihre und meine Auffassung nochauseinandergehen möchten, präcisieren zu können. Doch müßte ich zu diesem Zweck mehr ungestörte Zeitzur Verfügung haben, als ich vorläu�g hoffen darf.

Mit vorzüglicher Hochachtung verbleibe ichIhr ergebener

M. Pasch.

A.4. Pasch to Klein, March 31, 1887 [Klein 11, 179]

Giessen, 31. März 1887.Hochgeehrter Herr College!

Ich nehme mir die Freiheit, Ihnen zwei Aufsätze zu übersenden und ergebenst anzufragen, ob Sie diesel-ben in den Annalen abzudrucken geneigt wären. Der functionentheoretische Aufsatz ist gelegentlich vonVorlesungen entstanden, welche das Bedürfniß wesentlicher Vereinfachung gewisser Partien emp�nden


ließen. Zu der |〈2〉 kleinen Note hat der Artikel von Prósper mehr die äußere Veranlassung gegeben,denn ich will nicht etwa Herrn Prósper gegenüber auf die Beweisvereinfachung besonders Gewicht legen,möchte aber doch — auch nach der Lektüre der Köpcke’schen Arbeit — auf die Anschauungen, mit denenSie s. Z.45 vorangegangen sind, von Neuem hinweisen.


M. Pasch

A.5. Pasch to Klein, December 3, 1887 [Klein 11, 180]

Giessen, 3. December 1887.Hochgeehrter Herr College!

Der plötzliche Tod Baltzer’s hat unsere Fachwelt in die Lage versetzt, Vorbereitungen zu einer geeignetenBerufung treffen zu müssen. Durch Uebernahme einer Ersatzvorlesung und andrer Umstände vollauf inAnspruch genommen, kann ich mich erst jetzt der mir zufallenden Aufgabe voll widmen, einer Aufgabe,welche nicht leicht ist gegenüber der großen Anzahl der in Betracht zu ziehenden Namen. Ich habe nunzunächst unter möglichster Berücksichtigung der Verhältnisse gestrichen und stehe jetzt vor der Liste,welche ich mir erlaube, Ihnen mit der Bitte um Discretion mitzutheilen. Ich will hoffen, daß ich bei demStreichen nicht ungerecht gewesen bin; andererseits wird von dem Zurückbehaltenen wohl Mancher schonaus pekuniären Rücksichten, die ich nicht hinreichend übersehe, fortfallen müssen, Mancher wegen seinerRichtung ungeeignet sein, daß es z. B. wünschenswerth ist, daß der zu berufende Ordinarius die der |〈2〉Physik nahestehenden Theile der Mathematik gern und mit Erfolg vorträgt.

Vielleicht darf ich voraussetzen, daß auch Sie, verehrter Herr College, an dieser für unser Fach im-merhin wichtigen Angelegenheit einigen Antheil nehmen und uns mit Ihrer ausgedehnten Kenntniß derbetreffenden Verhältnisse zu unterstützen sich bereit �nden lassen. Sie würden dadurch mich in erster Liniezu aufrichtigem Dank verp�ichten.

Nachträglich noch meinen besten Dank für Ihren freundlichen Brief vom 23. April. Die Aussicht, von derSie sprechen, daß in Göttingen im Sommer eine Zusammenkunft der drei Universitäten statt�nden würde,hat sich zu meinem Bedauern nicht verwirklicht.


M. Pasch.

Dyck, Henneberg, Hettner, Kiepert, Krause, Lindemann, Franz Meyer, Netto, Runge, K. Schering,Schottky, Schur, Staude, Stickelberger, Wiltheiss


Giessen, 13. Dezember 1887.Hochgeehrter Herr Kollege!

Gegenüber der großen Freundlichkeit, mit der Sie auf meine in der That noch wenig �xierten Fragen mirausführliche und höchst werthvolle Auskunft gegeben haben, wäre die Verzögerung meines Dankes dop-pelt unverantwortlich, wenn Sie nicht durch Universitätsgeschäfte, welche ganz unerwartet in der vorigenWoche mir aufgeladen wurden und im Verein mit der schon an sich vorhandenen Belastung mich vonder Verfolgung der Berufungsfrage gänzlich abzogen, verschuldet gewesen wären. Ich komme nun heutnochmals auf den Gegenstand zurück und bestätige vor Allem, daß Ihre besondere Berücksichtigung der

45 Abbreviation for: seinerzeit.


persönlichen Charakteristiken mir in der That sehr erwünscht gewesen ist. Von |〈2〉 Ihrem Anerbieten wei-terer Auskunft möchte ich zunächst bezüglich folgender Punkte Gebrauch machen. Bei Voss und Dyck wärees nöthig, zu wissen, welche Gehälter und Nebeneinnahmen sie beziehen, wobei wohl noch die bayerischenQuinquenalzulagen in Betracht kommen. Ich würde mich, da mir andere Verbindungen fehlen, Ihrer Vermit-telung gern bedienen, aber nur dann, wenn Sie nicht dadurch besonders bemüht werden. Staude wird wohlschon des Gehalts wegen außer Betracht bleiben, er hat angeblich 4000 Rubel; vielleicht ist Ihnen zufälligGenaueres bekannt geworden. Natürlich sind mancherlei Empfehlungen — meist indirekt — eingegangen.So hat man Krazer und Stickelberger genannt, über deren Persönlichkeiten mir eben nichts Näheres bekanntist. Auf Mechanik und verwandte Vorlesungen wird namentlich der Physiker Gewicht legen. Wie mag esin dieser Hinsicht mit Hölder, F. Meyer, |〈3〉 Schön�ies und Schur stehen? Die große Schwierigkeit bestehtfür mich in der Ueberfülle von Namen, die in dem hiesigen Falle genannt werden können. Nur habe ichnatürlich keinen jüdischen Fachgenossen in das Verzeichniß aufnehmen dürfen, sonst hätten Gelehrte wieNöther und Hurwitz darin nicht gefehlt.

Nochmals meinen wärmsten Dank für Ihr liebenswürdiges Entgegenkommen und ergebenste Grüße von

IhremM. Pasch.


Giessen, 22. December 1887.Hochgeehrter Herr Kollege!

Ich will nicht länger zögern, Ihnen für Ihre erneute freundliche Auskunft und ganz besonders dafür sehrzu danken, daß Sie sich sogar die Mühe genommen haben, die Erkundigung in München selbst einzuziehen.Freilich ist danach die Aussicht, Voss in Betracht ziehen zu können, unter den hiesigen Verhältnissen einegeringe. Die Hauptsache aber ist, daß ich mein Geschäft, Vorschläge zu machen, bis nach dem Ende |〈2〉 derFerien ruhen lassen muß und daher noch gar nichts abzusehen vermag. Es ist dieser Zustand die Folge derArt und Weise, wie die Angelegenheit hier leider eingeleitet worden ist, wodurch es nöthig geworden ist,die Entscheidung der Regierung über eine Vorfrage abzuwarten. Ich behalte mir vor, auf alle diese Dingespäter zurückkommen zu dürfen, und bleibe inzwischen mit bestem Gruß

Ihr sehr ergebenerM. Pasch.

A.8. Pasch to Klein, February 17, 1888 [Klein 11, 183]

Giessen, 17. Februar 1888.Hochgeehrter Herr Kollege!

Erst jetzt bin ich in der Lage, ein Ergebniß der hiesigen Berufungsverhandlungen mitzutheilen. GestattenSie mir zunächst, eine frühere Andeutung näher auszuführen. Wir besaßen für Mathematik ein Ordinariatund ein Extraordinariat, das letztere hatte ich als persönlicher Ordinarius inne. Allgemein war man derAnsicht, daß dieses Verhältniß jetzt angemessen geregelt werden müsse. Unser Physiker jedoch (Röntgen),dessen Verhalten mich allerdings schon früher und insbesondere im vorigen Semester zu einer bestimmtenZurückweisung genöthigt hatte, benutzte den Umstand, daß er gerade Dekan war, |〈2〉 um die betreffendeEntschließung zu verhindern. Dadurch wurden Fakultätsverhandlungen nöthig, die durch den Dekan nichteben gefördert wurden, schließlich jedoch zu einer Regelung der Sache geführt haben. Der Abschluß isteigentlich erst in der vorigen Woche erfolgt, und die Vorschläge konnten dann bald gemacht werden. DieBeibehaltung zweier Ordinarien hat die Regierung bewilligt, jedoch — wie den hiesigen Verhältnissennach zu erwarten war — mit der Maßgabe, daß möglichst wenig Gehalt in Anspruch genommen werde.


Von angestellten Ordinarien mußten wir sonach ganz abgesehen, mit Ausnahme von Staude, den Schröterwiederholt dringend empfohlen hatte, zuletzt mit dem Bemerken, daß er nur 2400 Rubel Gehalt und ge-ringfügige Nebeneinnahmen bezieht und für die Rückkehr nach Deutschland aus den ihm unerträglichen?

russischen Verhältnissen jedes Opfer bringen will. Maßgebend war ferner der von naturwissenschaftlicherSeite geltend gemachte Wunsch, der zu Berufende solle sich auf dem Gebiet der Mechanik und benach-barter Theile als Docent bewährt haben. So kam die Liste 1) Netto, 2) Staude, 3) Franz Meyer zu Stande.Netto als ältester unter den in Betracht kommenden Gelehrten mußte an die erste Stelle. Die Zuhör-erzahl soll auch in Berlin stark abgenommen haben. Sollte Netto dennoch nicht kommen, so ist wohl dieAussicht auf Staude eine sichere. Für Meyer war ausschlaggebend, daß er Betti übersetzt, math. Physik, As-tronomie u. dgl.gelesen hat. Leider war die Richtung von Schur unseren Bedürfnissen nicht entsprechend.Die Vorschläge bedürfen noch der Zustimmung des Gesammtsenats, welche voraussichtlich am 25. d. M.erfolgen wird. Aus diesem Grunde erlaube |〈3〉 ich mir die Bitte, diese Mittheilung bis dahin als vertraulicheansehen zu wollen. Indem ich die Gelegenheit benutze, um nochmals meinen herzlichen Dank für Ihre be-reitwillige Unterstützung auszusprechen, bleibe ich

mit vorzüglicher HochachtungIhr ergebener

M. Pasch.

A.9. Pasch to Klein, October 19, 1891 [Klein 11, 184]

Gießen, 19. October 1891.Hochgeehrter Herr Kollege!

Die beifolgenden Blätter, deren Veranlassung aus der Einleitung hervorgeht, erlaube ich mir, Ihnen fürIhre Annalen zur Verfügung zu stellen. Die darin dargelegte Auffassung ist ganz trivial; trotzdem habe ichmit der endgültigen Niederschrift so lange gewartet, weil ich die etwa zu erhebenden Bedenken wenigstensvor mir selbst zerstreuen wollte.

Ebenfalls spät kommt der Dank, welchen ich Ihnen gleichzeitig für die freundliche Zusendung IhrerAbhandlung zur Nicht- Euklidischen Geometrie vom vorigen Jahre abstatten wollte. Auf die letzten SätzeIhrer Abhandlung habe ich geglaubt, in meinem Manuscript S. 7, zweite Fußnote, Bezug nehmen zu dürfen,während ich in der ersten Fußnote einen Hinweis auf Ihre Aeußerungen in Bd. 6 S. 136 und Bd. 7 S. 532zunächst nicht gebracht habe, vielmehr |〈2〉 diese Stellen jetzt in anderem Sinne auffassen möchte. Auchsonst schließe ich mich dem, was Sie auf den beiden letzten Seiten der Abhandlung sagen, in den meistenPunkten an. Der Inhalt der Axiome wird aus der Beobachtung (Anschauung als innerer Vorgang beruht aufErinnerung an Beobachtetes) entnommen, die zu den Axiomen verwendeten Begriffe sind aber ungenau undmithin auch die Axiome selbst, diese lassen sich jedoch rein logisch nur dann verwenden, wenn sie als exakthingestellt werden. Indem wir nun an den Axiomen weiter arbeiten und geometrische Sätze suchen, p�egenwir allerdings die Figuren zu Hülfe zu nehmen, sie entweder zu entwerfen oder innerlich „vorzustellen“.In der That wird man, wo nicht gerechnet wird, bei einigermaßen zusammengesetzten Erscheinungen ohnesolche Hülfe nicht vorwärts kommen. Aber — und hierin gehen offenbar unsere Ansichten auseinander —die Benutzung der Figur ist bloß eine Erleichterung der Arbeit, die Arbeit würde sonst unsere Kräfte über-steigen oder wenigstens allzu langsam |〈3〉 vorwärts gehen, nicht weit genug vorwärts gehen. Möglich mußdie Betrachtung auch ohne die Figuren sein, m. a. W.: das, was aus den Figuren entnommen wird, mußbereits in den Axiomen niedergelegt sein, andernfalls sind die Axiome nicht vollständig. Gerade diesenCharakter habe ich meinen Grundsätzen zu geben versucht und habe deshalb allerdings ihre Zahl ver-mehren müssen (s. Geometrie S. 6 unten, S. 43–45, S. 98–100). Die Erfüllung jener Forderung ist möglich,und darum wohl die Forderung berechtigt. Bewiesen ist eben in der Geometrie wie in der Analysis nurdas, was aus den Axiomen allein hergeleitet werden kann. Ich möchte allerdings in Hinsicht der äußeren


Darstellung nicht soweit gehen, wie z. B. Peano (I principii di Geometria logicamente esposti, Torino 1889),der mir deshalb auf S. 32–33 kleine Ausstellungen macht.

In Bd. 39 Heft I hat Herr Schur einfachere Beweise für das Bestehen der idealen Elemente gebracht.Indeß war schon, auf recht einfacher Grundlage, von Prosper und mir in Bd. 32 |〈4〉 die Kürzung ausgeführtworden. Wie Herr Schur mir schreibt, waren ihm diese kleinen Noten entgangen, da er damals nach Dorpatübersiedelte.

Nicht bloß mir selbst, sondern auch Anderen, ist es aufgefallen, daß Lindemann in der Raumgeometriemeine Schrift über Geometrie nicht erwähnt hat. Ich weiß in der That nicht, wie dies zu erklären sein mag.

Indem ich zum Schluß noch mein Bedauern ausspreche, in Halle, wo soviel Interessantes zu hörenwar, gefehlt zu haben — meine Anwesenheit war hier nöthig gewesen —, bleibe ich mit ausgezeichneterHochachtung

Ihr ergebenerM. Pasch.

A.10. Pasch to Klein, April 21, 1902 [F. Klein 22 F, Bl. 127–128]

Gießen, 21. April 1902.Hochgeehrter Herr Kollege!

Von Ihrer neuesten Veröffentlichung, nach der im Sommer 1901 gehaltenen Vorlesung, habe ich zwarkaum das erste Viertel durchsehen können, will aber doch den Ausdruck meines herzlichen Dankes fürdie liebenswürdige Übersendung nicht noch länger verzögern. Sie durften ja mit Sicherheit annehmen, daßmich diese Darlegungen in hohem Maße interessieren würden. Wenn Sie deren Wichtigkeit immer undimmer wieder betonen, so haben Sie meine |〈2〉 Wenigkeit ganz auf Ihrer Seite. Hoffen wir, daß dieseAnschauung bei den Mathematikern durchdringt und die Physiker dann beein�ußt, vielleicht auch diePhilosophen.

Mit Ihnen kann ich dem räumlichen Vorstellen (Seite 7) keine größere Genauigkeit beimessen, als demempirischen Messen u. dgl. Die Vorstellung vermag uns nur an Gesehenes zu erinnern oder Gestalten,deren Theile wir gesehen haben (oder haben können), in uns zu erzeugen; beweisen kann sie nichts. Dieabstrakte Geometrie ist wohl aus demselben Bedürfnis hervorgegangen, wie der Wunsch, Naturgesetzeals absolut genau hinzustellen (Seite 42, 44 u. s. w.). Mitgewirkt hat wohl auch die Art (S. 36), wie sichin uns die Vorstellung eines Con|〈3〉tinuums bildet. Eigentlich hat die praktische Geometrie es wohl nurmit dem zu thun, was das unbewaffnete Auge sieht, der unbewaffnete Sinn überhaupt wahrnimmt. Nimmtman Mikroskop u. dgl.zu Hülfe, so verwickelt sich die Sache schon sehr. Aber inkommensurable Strecken(S. 21) kennt die eigentliche Geometrie jedenfalls nicht, sondern nur die künstlich dazu geschaffene. Letz-tere schafft einen neuen Begriff, den des „mathematischen“ Punktes; nach meiner Auffassung hierüberwürde es keines Postulats bedürfen, De�nitionen genügen vielmehr. Ich komme dann nicht in die Lage,zu sagen: Ein Punkt hat keine räumliche Ausdehnung (S. 15); hier weiß ich nämlich keine Erklärung für„Ausdehnung“.

Die liebenswürdige Art, wie Sie nun |〈3〉 alt gewordene Arbeiten von mir erwähnen, ist mir besonderserfreulich gewesen. Leider bin ich erheblich kritischer geworden, und das hat mich an Vollendung weitererÜberlegungen in diese Richtung gehindert. Bei den gewöhnlichen Studierenden Interesse und Verständnißdafür zu �nden, ist gar schwer.

Empfangen Sie, hochgeehrter Herr Kollege, nochmals die Versicherung besten Dankes und vorzüglicherHochschätzung von

Ihrem ergebenenM. Pasch


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Sitzungsberichte der physikalisch-medicinischen. Societät zu Erlangen 6, 52–64. Dated December 8, 1873.Reprinted in Klein [1883] and in Klein [1922, ch. 45, pp. 214–224].

Klein, F., 1883. Ueber den allgemeinen Functionsbegriff und dessen Darstellung durch eine willkürliche Curve. Ma-thematische Annalen 22 (2), 249–259. Reprint of Klein [1874b]. Reprinted in Klein [1922, ch. 45, pp. 214–224].

Klein, F., 1890. Zur Nicht-Euklidischen Geometrie. Mathematische Annalen 37 (4), 544–572. Reprinted in Klein[1921, ch. 21, pp. 353–383]. Discussion of the lectures held in the Winter Semester 1889/90, published as Klein[1892].

Klein, F., 1892. Nicht-Euklidische Geometrie. Göttingen. Lithographed notes (autograph) worked out by FriedrichSchilling of lectures held in Winter Semester 1889/90 in Göttingen.

Klein, F., 1902. Anwendung der Differential- und Intergralrechnung auf Geometrie, eine Revision der Principien.B.G. Teubner, Leipzig. Lithographed notes (autograph) worked out by Conrad Müller of lectures held in SummerSemester 1901 in Göttingen. Second edition, 1907. Third edition, Präzisions- und Approximationsmathematik,vol. 3 of Elementarmathematik vom höheren Standpunkte aus, worked out by Conrad H. Müller, edited by FritzSeyfarth, Julius Springer Verlag, Berlin, 1928.

Klein, F., 1921. Gesammelte mathematische Abhandlungen, vol. 1. Springer, Berlin. Edited by R. Fricke and A. Os-trowski, with additions by F. Klein.

Klein, F., 1922. Gesammelte mathematische Abhandlungen, vol. 2. Springer, Berlin. Edited by R. Fricke and H. Ver-meil, with additions by F. Klein.

Köpcke, A., 1887. Ueber Differentiirbarkeit und Anschaulichkeit der stetigen Functionen. Mathematische Annalen 29(1), 123–140.

Lipschitz, R., 1880. Lehrbuch der Analysis, vol. 2: Differential- und Integralrechnung. Max Cohen & Sohn (Fr. Co-hen), Bonn.

N.N., 1892. Chronik der Deutschen Mathematiker-Vereinigung. Jahresbericht der Deutschen Mathematiker Vereini-gung (JDMV) 1 (1), 1–14.

Pasch, M., 1873. Zur Theorie der linearen Complexe. Journal für die reine und angewandte Mathematik 75 (2), 106–152.

Pasch, M., 1882a. Vorlesungen über Neuere Geometrie. B.G. Teubner, Leipzig. Reprinted as Pasch [1912].Pasch, M., 1882b. Einleitung in die Differential- und Integralrechnung. B.G. Teubner, Leipzig.Pasch, M., 1884. Zur Theorie der Collineation und der Reciprocität. Mathematische Annalen 23 (3), 419–436.


Pasch, M., 1887a. Ueber die projective Geometrie und die analytische Darstellung der geometrischen Gebilde. Ma-thematische Annalen 30, 127–131.

Pasch, M., 1887b. Ueber einige Punkte der Functionentheorie. Mathematische Annalen 30, 132–154.Pasch, M., 1888. Ueber die uneigentlichen Geraden und Ebenen. (Auszug aus einem Schreiben an Herrn V. Reyes y

Prosper). Mathematische Annalen 32, 159–160.Pasch, M., 1892. Ueber die Einführung der irrationalen Zahlen. Mathematische Annalen 40, 149–152.Pasch, M., 1912. Vorlesungen über Neuere Geometrie, second edition, with additions. B.G. Teubner, Leipzig, Berlin.Pasch, M., 1930. Eine Selbstschilderung. von Münchowsche Universitäts-Druckerei, Gießen. Reprinted in Mitteilun-

gen des mathematischen Seminars Giessen 146, 1–19.Peano, G., 1889. I principi di geometria logicamente esposti. Fratelli Bocca, Torino. Reprinted in Peano [1957–1959,

vol. 2, pp. 56–91].Peano, G., 1957–1959. Opere scelte. Edizioni Cremonese, Roma. 3 volumes, edited by Ugo Cassina.Reyes y Prósper, V., 1887. Sur la géométrie non-Euclidienne. Mathematische Annalen 29 (1), 154–156.Reyes y Prosper, V., 1888. Sur les propriétés graphiques des �gures centriques (Extrait d’une lettre adressée à

Mr. Pasch). Mathematische Annalen 32, 157–158.Rosanes, J., 1884a. Erweiterung eines bekannten Satzes auf Formen von beliebig vielen Veränderlichen. Mathemati-

sche Annalen 23 (3), 412–415.Rosanes, J., 1884b. Bemerkung zur Theorie der Flächen zweiter Ordnung. Mathematische Annalen 23 (3), 416–418.Scharlau, W. (Ed.), 1990. Mathematische Institute in Deutschland, 1800–1945. Vieweg, Braunschweig. Number 5 in

Dokumente zur Geschichte der Mathematik.Schlimm, D., 2010. Pasch’s philosophy of mathematics. Review of Symbolic Logic 3 (1), 93–118.Schur, F., 1891. Ueber die Einführung der sogenannten idealen Elemente in die projective Geometrie. Mathematische

Annalen 39 (1), 113–124.Tobies, R., 1981. Felix Klein. Teubner Verlag, Leipzig.Tobies, R., 1987. Zur Berufungspolitik Felix Kleins. NTM Schriftenreihe für Geschichte der Naturwissenschaften,

Technik und Medizin 24 (2), 43–52.Tobies, R., Rowe, D.E. (Eds.), 1990. Korrespondenz Felix Klein – Adolph Mayer. Auswahl aus den Jahren 1871 bis

1907. B.G. Teubner, Leipzig.Tobies, R., Volkert, K., 1998. Mathematik auf den Versammlungen der Gesellschaft Deutscher Naturforscher und

Ärzte 1843–1890. Wissenschaftliche Verlagsgesellschaft, Stuttgart.

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