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

The rst Chinese translation of the last nine books of 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 Hypsicles of Alexandria) were rst translated into Chinese by the British missionary Alexander Wylie and the Chinese mathematician 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 specic original source. Accordingly, historians have pondered this question ever since. Some took a bold guess that its source was Isaac Barrow’s English translation. This article provides solid evidence to show that the guess is wrong, and argues that 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 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 as early as the Yuan dynasty, sometime between 1250 and 1270. This has at least been the case ever since the historian of Chinese mathematics Yan Dunjie pointed out in 1943 that a book mentioned in the “Catalogue of the Muslim Books (Huihui shuji )” included in chapter seven of the History of the 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, in the 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 the intelligence of Möngke Khan (Mengge Han , reigned 1251–1259) for “solving some propositions of Euclid,” which clearly suggests that Euclid’s Elements was known in the Mongolian court by the early 1250s [Yan, 1943, 36].3 One year before Yan’s publication, the Japanese scholar Tasaka Kod¯ o,¯ in his study of the introduction of Islamic culture into China during the Yuan dynasty, also discussed the possible meanings of the recorded book title in the Catalogue of the Muslim Books. But Tasaka held that it 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,” Tasaka changed his view, and also concluded that the title refers to an Arabic version of Euclid’s Elements,by using 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 whether any Chinese scholar of that time was familiar with any of the contents of the book. It is also unknown whether there may have been a Chinese translation of all or only part of the copy of Euclid’s Elements which was in the library of the Northern, or Islamic, Observatory (Bei sitiantai ) built in 1271 by the Mongolians in the upper capital (Shangdu ), situated near Dolon Nor (Zhenglanqi )in Inner Mongolia. Moreover, Rash¯d’s brief and ambiguous description is really not sufcient to prove that Möngke Khan had studied Euclid’s Elements as Yan suggested. Nevertheless, the Elements, in either an Arabic or a Persian version, certainly existed in China no later than 1273, the year in which the Islamic Observatory 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 academic world. The Mishujian is rendered here as “imperial archive and library,” but it functioned much more broadly than an archive or 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 13th centuries, see Hucker [1985, 376b–377b], as well as the introduction to the latest Chinese edition of the Mishujian zhi by Gao Rongsheng 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, sa prudence, 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 was distinguished among Mongol rulers for his intelligence, perspicacity, sharp wit and mind, and he even solved some problems left 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 of a Latin version by Christopher Clavius, professor of mathematical sciences at the Collegio Romano (the Jesuit seminary founded in Rome by Ignatius Loyola in 1551). It was the Jesuit Matteo Ricci, a student of 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 Western works, in hopes of gaining the support that he and his fellow Jesuits sought from the Chinese literati.In doing so the Jesuits expected to secure a footing in pagan China, and eventually to convert the Chinese to Christianity. Not long after his arrival in China, Ricci sought help from one of his early converts and together they began to translate the Elements. But at the time he was only able to nish the rst book, the manuscript of which was later lost [Engelfriet, 1998, 59–66]. Ricci never lost interest in translating the Elements. In 1604, when he eventually settled in Beijing after 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 of the literati who was on the verge of getting the highest degree the Ming dynasty could offer.5 Ricci proposed they collaborate in translating the Elements. Fascinated by its style and the entirely different approach to geometry it represented in contrast to traditional Chinese geometry, Xu Guangqi devoted himself wholeheartedly to assisting Ricci in translating Euclid’s Elements.6 After three years of hard work, they had nished the rst six books. Although Xu Guangqi wanted to continue to translate the rest of the book, Ricci thought it would be better to publish their partial translation rst. Depending upon its reception, they could then decide whether to continue the project or not. Accordingly, they published their 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 exotic style and formality (not its axiomatic–deductive system) was warmly accepted, and the newly invented term Jihe for geometry was widely adopted in titles of Chinese mathematical books published soon thereafter. Moreover, a number of books were written to elucidate parts of the Ricci/Xu translation. On the other hand, some mathematicians, including Mei Wending , who was one of the most prominent mathematicians in the early Qing period, held that traditional Chinese mathematics, represented by the Nine Chapters on Mathematical Procedures (Jiuzhang suanshu ), included all of the methods to be found in the translation, with but one possible, if notable, exception—namely, Euclid’s introduction of extreme and mean ratios.7 Moreover, Chinese mathematicians regretted that the translation was only partial, and some, notably Mei, even suspected that Ricci was trying to hide something important from the Chinese in the later books of the Elements [Liu, 1989, 57]. As we shall see below, Mei’s false accusation was to become a major factor in the decision, made early in the 1850s by a later missionary, Alexander Wylie, to complete the work left undone by Ricci.

4 For recent studies of scientic 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 in China. 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]. Y. Xu / Historia Mathematica 32 (2005) 4–32 7

During Mei’s time, the Emperor Kangxi (reigned 1662–1722) was especially interested in Western science in general and mathematics in particular. In order to learn Euclidean geometry, sometime around 1690 he requested two French Jesuits, Joachim Bouvet and Jean-François Gerbillon, who had recently arrived in China, to lecture at his court. Bouvet and Gerbillon, who both held the title “King’s Mathematician” granted by the Sun King, Louis XIV, compiled their lessons for the Emperor in Manchu from a short textbook by their countryman Ignace-Gaston Pardies: Elémens de Géométrie (rst published in Paris in 1671).8 The resulting lecture notes for Emperor Kangxi were subsequently translated into Chinese and later incorporated into a mathematical encyclopedia, the Essence of Mathematics (Shuli jingyun ), which was compiled under the auspices of Kangxi, but not published until 1723, the year immediately after his death.9 The Chinese translation (with some additions and deletions) of Pardies’ Elémens, when it appeared in the Essence of Mathematics, bore the same title as the one used by Ricci and Xu, but its contents and style were very different. The Pardies translation was divided into 12 chapters (juan ), the rst ve of which covered, respectively, lines and angles, triangles, quadrilateral gures and polygons, circles, and solids. 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, and logarithms, were left out, presumably because of their difculty. The last book of the original text, Book IX, which included 48 practical geometrical problems, provided the contents for chapters 11 and 12 of the translation. The contents of chapter 10 were devised by Bouvet and Gerbillon themselves or other missionaries [Han, 1994, 3–4]. The Essence of Mathematics version of the Elements was not, in reality, even a close translation of Euclid’s Elements, nor was it complete. But the drastic negative change in Qing dynasty policy toward the missionaries certainly did nothing to encourage either the Chinese or the Jesuits—or indeed any other missionaries—to complete Ricci’s unnished project for another 130 years. The Emperor Yongzheng , Kangxi’s immediate successor, adopted harsh measures against the missionaries in China as a direct result of the rites controversy that had started during the Kangxi reign. The controversy was mainly about whether ancestor worship should be an allowable practice for Chinese converts to Catholicism or whether the 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 the Qing court to close China’s doors to all foreigners, but especially to missionaries, and this situation was only 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 from Great 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,threevolumes(san ce ) in bulk, which undoubtedly came from the lecture 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, one preserved in the National Library in Taipei and the other two in the Palace Museum in Beijing. Each of these three Chinese manuscripts 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 Chinese versionintheEssence of Mathematics,seeChen [1931]; Li [1984]; Liu [1991]; and Martzloff [1993a]. 10 For details and historical meanings of the Chinese rites controversy, see, for example, Mungello [1994]. 8 Y. Xu / Historia Mathematica 32 (2005) 4–32

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 unnished by Ricci 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 Ming dynasty. 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 difculty of translating. The nature of knowledge is to provide means to all people. Why hide it and not reveal it?... My translation has just been completed. It has not only fullled Ricci’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 less important, 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]. Y. Xu / Historia Mathematica 32 (2005) 4–32 9

Fig. 2. Li Shanlan (1811–1882). From The Chinese Scientic and Industrial Magazine (its Chinese title is Gezhi huibian), 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 Elements in 1852, and took about four years to complete their work. The translation was published under the patronage of Han Yingbi in Shanghai in 1857, and later was reprinted along with the rst six books by Ricci and Xu Guangqi in 1865. Over the next half a century, Chinese interest in the last nine books of the Elements was exclusively technical. As time passed by, historical interest in the translation gradually arose. Historians quite naturally wanted to know about the original source, or sources, that Wylie and Li had used to prepare their translation in order to evaluate the Chinese translation itself, assess the transmission of Western mathematics into China in the second half of the 19th century, and study the inuence of Euclid’s Elements upon Chinese mathematicians, especially the translator Li Shanlan, among other historical interests. However, neither Wylie nor Li had said anything, specically, in their prefaces to the translation about 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 completed book 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 mathematical education in China, see, for instance, Fang [1943]; and Horng [1991]. 10 Y. Xu / Historia Mathematica 32 (2005) 4–32

a translation from Greek into my native tongue. It has not been reprinted recently. The copy I have is an old version, but not well proofread and has mistakes. Such mistakes, even minor, would cause great difculties 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 an old English translation of a complete Greek version of the Elements), the just-quoted passage does not mention 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 original source in vain [Li, 1940, 801; Yan, 1943, 35]. However, another historian, Qian Baocong ,ventured the 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’s English translation seems to be the original source of the Chinese translation” [Qian, 1981, 324]. Qian’s hypothesis has been widely cited ever since, but the present paper will show that Qian’s hypothesis is wrong. 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 Euclid by Henry Billingsley of 1570. A conjecture about the actual copy of Billingsley’s book that Alexander Wylie 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 French geometer, André Tacquet, Barrow decided to produce his own Latin version of the Elements. This resulted in a translation in 15 books, which was printed in London in 1655. Meanwhile, having been forced to leave the University because of his family’s Royalist connections, with the Restoration of the monarchy Barrow was appointed to the professorship of Greek at Trinity College four years later. It was then that he prepared the second edition of his Latin version of Euclid, Euclidis Elementorum libri xv (1659), and then 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 achieve his goal of “[conjoining] the greatest compendiousness of demonstration with as much perspicuity as the quality 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 of Book X of the Elements [Barrow, 1660/1983, preface; Murdoch,√ 1971, 451–452]. For instance, Barrow used × for multiplication; + for addition; − for subtraction; for a root of any degree; −: for the difference of the two quantities or magnitudes on the two sides of the symbol; Q&q for a square; C&c for 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 the propositions 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 or Characters,” which immediately precedes Book I. Y. Xu / Historia Mathematica 32 (2005) 4–32 11

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 less necessary, 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 demonstrate Euclide 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 in the second and thirteenth, & sparingly in the seventh, eighth, and ninth Books, where it seem’d convenient to vary something from him. [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 priority controversy. In the preface to his own treatment of the Elements (containing Books I–VI, XI, and XII only), 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 and difcult to one [who is] not versed in geometry. There, many propositions which appear conspicuous in reading Euclid himself, are made knotty, and scarcely intelligible to learners, by his algebraical way of demonstration; as is, for example, prop. 13 Book I. And the demonstrations which he lays down in Book II, are still more difcult: Euclid himself has done much better, in shewing their evidence by the contemplations of gures, as in geometry should always be done. The Elements of all science ought to be handled after 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. Over the course of the century since it rst appeared in 1660, the book was reproduced (reprinted or reset) at least six times, rst in 1686, and then in 1696, 1705, 1714, 1722, and 1732. The last edition appeared in 1751. The editions of 1705 (and after) also included as an appendix Archimedes’ Theorems on the Sphere and Cylinder. The 1714 edition added Euclid’s Data, as well as François Foix-Candale’s Brief Treatise of Regular Solids.14 The next edition of 1722 also included additional supplements, namely some corollaries Barrow had deduced himself from the propositions of Euclid’s Elements. The 1732 edition was issued after 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 the layouts of these editions, the content of the Euclidean geometry in all editions remained virtually the same. Consequently, in what follows, Barrow’s rst English edition is used for comparison with Wylie and 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 source of the latter, it is necessary to describe the format and major characteristics of the latter. As already mentioned above, Alexander Wylie considered it his mission to fulll the unnished project left behind

14 François Foix-Candale’s full French name is François de Foix, Comte de Candale, known in Latin as Franciscus Flussatus and 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]. 12 Y. Xu / Historia Mathematica 32 (2005) 4–32 by Matteo Ricci. Thus it was natural for him to adopt the terminology and format created by Ricci and Xu 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 Ricci and Xu’s translation by Peter M. Engelfriet [1998, 142–145], were followed in the arrangement of propositions 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 to include “another explanation” (you jie yue ) and “another proof” (you lun yue ). The other basic 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 heading xi (literally, “connected” or “subordinate”), which usually came at the end of a proposition. Some propositions also include a “note” (an ), an “example” (li ), or a “comment” (zhu yue ), and sometimes all of these. In WLC, everything was articulated in words without use of mathematical symbols or formulas, and most of the propositions were illustrated with diagrams. The Roman and Greek letters used in the diagrams were replaced by Chinese characters borrowed from the 10 heavenly stems and 12 earthly branches.15 With the above brief description of the basic format and characteristics of WLC in mind, it is now possible to compare it with IBE. The most obvious contrasts between the two books are that (1) IBE makes 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 of such additional explanatory commentary. For example, consider Book X, Proposition 42, which plainly illustrates these differences (see Figs. 3 and 4). In order to facilitate this comparison, and make what follows below clearer, the following is a literal translation, back into English, of Proposition 42 as it was rendered 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 lines is also medial. The rectangle is incommensurable with the sum of the two squares. Thus, the sum of the two lines is irrational, naming it 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. The Problem 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 the square 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 a medial. But it is applied to the rational line DE, so DG is rational and incommensurable with DE (this Book, 23). GI is also rational and is incommensurable with FG, that is, DE. Same reason as above. Since the sum of the squares AB, BC, is incommensurable with twice the rectangle AB, BC, the two rectangles DF, GH are incommensurable (Book VI,1;this Book, 10). The two lines [DF and GH] are rational. Therefore, DF and GI are rational lines commensurable only in square. So, DI is irrational and called binomial (this Book, 37). While DE is rational, so the rectangle DH is irrational (this Book, a note in Proposition 39), therefore the side of a square which has the same area is irrational. The rectangle DH is equal to the square AC, therefore, AC is irrational, and is called the line of the two medials.

15 For a list of the correspondence between English and Greek letters and their associated Chinese characters, see Martzloff [1997, 121]. Y. Xu / Historia Mathematica 32 (2005) 4–32 13

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 two medials. 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 is greater than CB. Since AD is equal to DB, therefore ED is less than DC. The distances between the midpoint D and the points C and E 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 rectangle AC, 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 equal to that of the rectangle AE, EB, and the square ED. Since the square ED is less than the square CD, the remaining rectangle AC, CB is less than AE, EB. Therefore, twice the rectangle AC, CB is less than twice AE, EB. Their difference is greater than the difference between 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 following discrepancies:

(1) IBE’s description is concise and replete with symbols such as =, +,and , whereas WLC’s description 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, none of 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 symbols of 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, for many other propositions throughout WLC. Doing all of this would have required tremendous effort and considerable ingenuity. Further comparisons of the two books reveal other major discrepancies. For instance,

(1) Book VII of IBE includes 23 denitions, the last of which states: “One number is said to measure another, by that number, which, when it multiplies, or is multiplied by it, it produces” [Barrow, 1660/1983, 143]. This denition cannot be found in WLC, and the three postulates and twelve axioms immediately following the denition 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 the same proposition in WLC has two [Wylie and Li, 1857/1865, Book VII, 4a]; moreover, Propositions 18, 19, and 36 all have a scholium or corollary, but there are none for the corresponding propositions in WLC [Barrow, 1660/1983, 154, 159, and 161; Wylie and Li, 1857/1865, Book VII, 13b–15a, and 25a–26a]. (3) The denitions for a parallelepiped and inscribed and circumscribed solid gures (Denitions XXX, XXXI, and XXXII) in Book XI of IBE are not to be found in WLC [Barrow, 1660/1983, 270; Wylie and Li, 1857/1865, Book XI, 4]. 14 Y. Xu / Historia Mathematica 32 (2005) 4–32

Fig. 3. Alexander Wylie and Li Shanlan’s Chinese translation of Euclid’s Elements, Book X, 42. From the 1865 edition of Jihe yuanben , Book X, part I, pp. 47b–49a. Y. Xu / Historia Mathematica 32 (2005) 4–32 15

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 Microlm International, 1981–1982).

(4) At the beginning of Book XIV of WLC, the translators state clearly that neither Book XIV nor XV was written by Euclid, but both were added to the Elements by Hypsicles of Alexandria. Wylie and Li also rendered the preface by Hypsicles into Chinese [Wylie and Li, 1857/1865, Book XIV, 1] but IBE does not include the Hypsicles preface [Barrow, 1660/1983, 345]. 16 Y. Xu / Historia Mathematica 32 (2005) 4–32

Many more discrepancies, major and minor, between IBE and WLC could easily be identied. But the comparisons made above sufce to cast considerable doubt on Qian Baocong’s hypothesis that Isaac Barrow’s English version of Euclid’s Elements served as the basis for the Wylie/Li translation of Books VII–XV into Chinese. The question that now arises concerns the book Alexander Wylie and Li Shanlan actually used in preparing their Chinese translation of the last nine books of Euclid’s Elements, and whether the exact edition of Euclid’s Elements they used can be identied. The next section establishes beyond doubt that the source of the Wylie/Li translation must have been the rst English 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 to translate the last nine books into Chinese. Nevertheless, most included only the rst six books, although some added XI and XII as well [Heath, 1956, 109–112]. Among English versions with 15 or even 16 books of the Elements are the rst English translation in 15 books by Henry Billingsley of 1570 (hereafter referred to as HBE) and a translation by John Leeke and George Serle of 1661 (hereafter referred to as LSE). Of these, the New York Public Library’s Rare Book Division has an original copy of HBE,and Columbia University’s Butler Library has a microlm copy of LSE. Comparing both of these with the Chinese translation leads to some interesting conclusions, but rst it is necessary to introduce briey these two English editions. The rst English translation of Euclid’s Elements, by Billingsley, was made from the 1558 Latin edition: Euclidis Megarensis mathematici clarissimi Elementorum geometricorum libri XV (Fig. 5). This in turn was the rst Latin translation based upon a Greek text, and was made by Bartolomeo Zamberti who published it in Venice in 1505 [Archibald, 1950, 445–448]. A brief description of HBE can be found in Charles Thomas-Stanford’s Early Editions of Euclid’s Elements, which provides a facsimile of the title page [Thomas-Stanford, 1977, 13–16, 43–44, Plate X]. More detailed information may also be found in Walter F. Shenton’s “The First English Euclid” [Shenton, 1928]. In addition to including a facsimile of the 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 out of folded pieces of paper serve to illustrate the text. Shenton also reprints the preface to the book in its entirety [Shenton, 1928, 505–512].16 Following “The Translator to the Reader,” HBE has a lengthy introduction, 24 folios in all, by the Renaissance polymath John Dee, who was misattributed as the “true” translator of the book by August De Morgan [De Morgan, 1837, 38].17 The translation of the 13 books by Euclid cover 414 folios and are followed by the 14th and 15th books by Hypsicles (29½ folios). The book ends with an appendix as Book XVI (13 folios) and a “Brief Treatise of Mixed and Composed Regular Solids” (5 folios) by François 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 York Public 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 History Publications in New York. See Dee [1975]. Y. Xu / Historia Mathematica 32 (2005) 4–32 17

Fig. 5. Title page of the Henry Billingsley Elements. Stuart Collection, Rare Books Division, The New York Public Library, Astor, Lenox, and Tilden Foundations. 18 Y. Xu / Historia Mathematica 32 (2005) 4–32 argument” of the main contents of the book in question. For Books VII, X, and XI, explanations are also provided after every denition. A double numbering system, for both theorems and propositions, is used in the book. The translation is not a mere word-for-word translation, but contains “all the earlier and current commentary of importance” [Archibald, 1950, 449]. Many corollaries, assumptions, and notes by Apollonius, Archimedes, Campanus, John Dee, Eudoxus, François Foix-Candale, Hypsicles, Montaureus, Pappus, Plato, Proclus, Pythagoras, Regiomontanus, and Theon of Alexandria, among others, 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 denition. A dual number system is adopted and descriptions of the 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 by Robert Potts, a 19th-century expert on Euclid [Heath, 1956, 110: 22, 233]. Potts’ view was widely ac- cepted by many inuential 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, by Raymond 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 model for WLC. Nevertheless, there is direct evidence that shows that LSE could not have been the basis for WLC, and at the same time establishes the link between HBE and WLC. Book X, Proposition 42, which was 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 the proposition in LSE (Fig. 6) with that in WLC as translated above (but see Fig. 3 as well), the following differences can be observed: (1) The letterings G, F ,andH in the diagrams do not correspond to one another; (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 an and 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 explanations accompanying the statement of each proposition. Moreover, the contents of the an and li in other places of WLC, some of which have been pointed out above, would have been added by Alexander Wylie and Li Shanlan themselves if LSE were the book they used for preparing WLC. But when we turn to compare Book X, Proposition 42 in WLC with its counterpart in HBE, we shall see there is no such possibility. As the illustration (Fig. 7) shows, the contents of “The 29. Theoreme. The 41. Proposition.” correspond very 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 jump from folio 203 to folio 205, and in some cases the pages are wrongly numbered, for example, the three folios after 319 were numbered as 340, 341, and 317, respectively. But following these errors, the correct pagination is restored. Y. Xu / Historia Mathematica 32 (2005) 4–32 19 , 1641–1700, 1283: 17, Early English Books translated by John Leeke and George Serle and published in 1661. From Elements pp. 288–289 (Ann Arbor, Michigan: University Microlm International, 1981–1982). Fig. 6. Book X, 41 of the 20 Y. Xu / Historia Mathematica 32 (2005) 4–32 . recto ,verso 261 . From Henry Billingsley’s English translation (printed by John Daye in 1570), folios 260 Elements Stuart Collection, Rare Books Division, The New York Public Library, Astor, Lenox, and Tilden Foundations. Fig. 7. Book X, 41 of the Henry Billingsley Y. Xu / Historia Mathematica 32 (2005) 4–32 21 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 same meaning. Just as “jieyue ” is part of the standard format of WLC, “Suppose that... Then I say,” or less 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 squares of which added together, make a medial supercies, and that which is contained under them is also mediall, and moreover it is incommensurable 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 two assumptes here following. [Billingsley, 1570, folio 261 recto]

These comparisons clearly show that the two notes in WLC were doubtlessly translated from these two passages. Moreover, the last sentence of the previous quotation indicates that two “assumptions” are provided immediately after. Having compared the rst “assumption” with the li, we must conclude that the contents accord directly with each other. The above example is not an isolated case. Compare the last proposition of Book X, which is numbered as 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’s Elements was not the basis for WLC.) At the end of the problem, WLC includes three notes (one an and two 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 A and 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 incommensurable lines, 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 two solids.

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 the circles. 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 these irrational 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 be founde out many other magnitudes having length and breath (such as are playne supercieces) which shall be incommensurable the one 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 22 Y. Xu / Historia Mathematica 32 (2005) 4–32

Fig. 8. Alexander Wylie and Li Shanlan’s Chinese translation of Euclid’s Elements, Book X, 117. From the 1865 edition of Jihe yuanben , 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 gure described upon the line C, being both like and in like sort described, that is, whether they be squares (which are alwayes like the one to the other), or whether they be any other like rectiline gures, or whether they be circles aboute the diameters A and C. For circles have that proportion the one to the other, that the squares of their diameters have (by the 2. of the twelfth). Wherfore (by the second part of the 10. of the tenth) the gures described upon the lines A and C being like and in like sort described are incommensurable the one to the other, wherfore by this means there are founde out supercieces incommensurable the one to the other. In like sort there may 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 to the 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 to the 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 any other rectiling gures equal to these square be erected solids of equall altitude, whether those solids be composed of equidistance supercieces, or whether they be pyramids, or prismes, those solids so erected shal be in that proportion the one to the other that theyr 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 solids also shall be commensurable the one to the other, and if the bases be incommensurable the one to the other, the solids also shall be incommensurable 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 be erected Cones or Cylinders of equal altitude, those Cones & Cilinders shall be in that proportion the one to the other that the circles are, which are their bases (by the 11. of the twelfth): and so if the circles be commensurable the one to the other, the Cones and Cilinders also shall be commensurable the one to the other. But if the circles be in commensurable the one to another, the Cones also and Cilinders shall be incommensurable the one to the other (by the 10. of the tenth). Wherfore it is manifest that not only in lines and supercieces, but also in solids or bodies is found commensurable or incommensurability. [Billingsley, 1570, folio 310 verso and folio 311 recto] Y. Xu / Historia Mathematica 32 (2005) 4–32 23 ,verso 311 . From Henry Billingsley’s English translation (printed by John Daye in 1570), folios 310 Elements .recto Stuart Collection, Rare Books Division, The New York Public Library, Astor, Lenox, and Tilden Foundations. Fig. 9. Book X, 116 of the Henry Billingsley 24 Y. Xu / Historia Mathematica 32 (2005) 4–32

The three notes in WLC are all apparently translations from the “instruction” added by the prominent Greek mathematician Theon of Alexandria.19 The above lengthy quotation not only provides further evidence to establish that HBE is indeed the source of WLC, but it also makes clear that an and you an are translations of the notes, comments, assumptions, and other supplements that Billingsley inserted in his 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 wrongly praised 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 author of Books XIV and XV, before Book XIV in WLC. Although missing from IBE, it is to be found in HBE [Billingsley, 1570, folio 416 recto–verso]. Nevertheless, WLC is not a cover-to-cover translation of HBE. In order to follow in general the format created by Ricci and Xu Guangqi, Wylie and Li Shanlan did not translate the outlines of each book provided in HBE, nor the lengthy explanation for every denition. Even the two denitions of Book XV, and many propositions, theorems, corollaries, assumptions, comments, and notes by scholars other than Euclid were omitted. Twenty propositions after Hypsicles’ work, the entire Book XVI, and the appendix by François Foix-Candale in HBE are also missing from WLC [Billingsley, 1570, folios 437 recto–463 recto]. In addition, the total number of propositions in Book X, and of the denitions in Book XI, is not identical. Book XI of WLC has 29 denitions, but HBE only 25. This difference was caused by breaking down certain original denitions into separate parts. For instance, the rst denition in HBE was translated as the rst two denitions in WLC [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 was caused by “An Assumption” added after Proposition 12 in HBE: “If there be two magnitudes compared to one and the self same magnitude, and if the one of them be commensurable unto it, and the other incommensurable; 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 source of 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 for the rst Chinese translation of the last nine books of Euclid’s Elements. We may ask ourselves one nal question related to the Chinese translation, namely, where is the book Wylie and Li used? We suspect that the copy preserved at the Rare Book Division of the New York Public Library may very well be the copy in question, strange and unlikely as this may seem at rst. But this surmise is not the product of sheer imagination, but the result of several factors based upon certain specic concrete evidence. Wylie in his preface briey 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]. Y. Xu / Historia Mathematica 32 (2005) 4–32 25

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 actually owned the copy of the book that he and Li Shanlan were translating. The Catalogue of the London Mission Library, Shanghai, which Wylie edited himself and printed in 1857, the year in which his translation was published, conrms that the Society did not itself have a copy of the Billingsley Euclid.20 In the section of the Catalogue devoted to “Books Belonging to the London Missionary Society,” Wylie recorded two copies of “Euclid’s Elements, Cambridge edition.” These were copies of Robert Pott’s edition, but there 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 London Missionary Society in Shanghai did not at the time have its own library, the books were distributed among its missionaries, who took responsibility for their safe keeping. The missionaries Alexander Williamson and Grifth John each had a copy of Pott’s Elements, while the Lardner edition was in the care of William Lockhart. Under Wylie’s own name in this section, only four books are listed, namely, Robert Morrison’s A Dictionary of the Chinese Language, A View of China for Philological Purposes, Joshua Marshman’s Clavis Sinica, and Matthew Henry’s Commentary for the NIV: Genesis to Revelation [London Mission Library, 1857, 15]. No edition of Euclid’s Elements is listed under Wylie’s name, conrming that the copy of Billingsley’s Euclid was his own possession. Soon after his translation was published, Shanghai was engulfed in the political and social turmoil caused 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 he left, Wylie sold his collections of Chinese books to the Asiatic Society, although it is reasonable to suspect that 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 have sold the Billingsley Euclid for nancial reasons. There is some evidence to link Wylie’s copy with a copy of the Billingsley Euclid currently in the possession of the New York Public Library. This was a copy donated to the Lenox Library in 1892 by the widow of the millionaire Robert L. Stuart (1806–1882). Stuart was an important philanthropist in New York City and president of the American Museum of Natural History from 1872 to 1881. Stuart had 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 up his collections, and in 1867 he purchased Billingsley’s book in London, only a few years after Wylie mayhavesoldhiscopy[Stuart, 1860, 88]. According to a note under the entry of the book by a person who identied himself as “Thomas” in the published Catalogue of the Library of Robert L. Stuart,the book “belonged to Schumacher” [Stuart, 1884, 176]. Further information is revealed in a clipping of the 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 ENGLISH EDITION... with Latin MSS. Notes in a contemporary hand, in the ORIGINAL BINDING, ... £L12 6s. 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 Mission Library [1857]. 26 Y. Xu / Historia Mathematica 32 (2005) 4–32

Fig. 10. Detail of the recto of the front yleaf of the Henry Billingsley English Elements (printed by John Daye in 1570). Stuart Collection, 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 and of the Differential and Integral Calculus (London Mission Society Press, 1859), preserved at the Prentis Library of Columbia University. 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 in London, or was he a previous owner of the book from whom Wylie purchased it, or someone else? What is 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 book entitled Daiweiji shiji (The Elements of Analytical Geometry and of the Differential and Integral Calculus), which is in the Prentis Library of Columbia University. This was the rst Chinese translation, also by Wylie and Li Shanlan, of a calculus book by the American mathematician Elias Loomis. The mark “A No. 15” looks very similar, and may well have been written by Wylie’s hand Y. Xu / Historia Mathematica 32 (2005) 4–32 27

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

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 was available 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 the various attempts to provide versions in Chinese of Euclid’s Elements. One interesting fact is that each translator chose for his original source a version of Euclid as close to his own culture as possible. Matteo Ricci 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 his translation upon an English version of the Elements. The specic English version of Euclid’s Elements that Wylie used to prepare the rst Chinese translation of Books VII to XV of the Elements was not the one by Isaac Barrow as some historians have speculated, but the one published in 1570 by Henry Billingsley, as this paper has argued. This conclusion also coincides with Wylie’s own brief description of the source he used. As he noted in his preface, “The book is a translation from Greek into my native tongue. It has not been reprinted recently. The copy is an old version, but it was not well proofread and has mistakes” [Wylie and Li, 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 Billingsley translated 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] of Euclid was either made from the Greek, or corrected by the Greek” [De Morgan, 1837, 39], and this widely held view may have led Wylie to believe that the source of the English Euclid he was using was based directly upon a Greek version. Wylie did not mention the name of the translator of the book he used, which may be explained by the fact that most writers of the 19th century did not bother to identify the sources they used, and also by the fact that De Morgan doubted that Billingsley was the true 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 nothing about the authorship of the English Euclid he was using. Billingsley’s translation was beautifully printed by John Daye, but like any book, it was not free of printing errors. By saying that “it was not well proofread and has mistakes,” Wylie may have intended to exaggerate the difculties he and Li Shanlan had faced in translating Books VII through XV of Billingsley’s Euclid for the rst time into Chinese.

21 The full name of I.J. Roberts is Issachar Jacob Roberts (1802–1871). He was sent to China in 1837 by the American Baptist Missionary Union and in 1847 taught Hong Xiuquan , who was to become the leader of the Taiping Rebellion (1851–1864), the essence of Christianity. 28 Y. Xu / Historia Mathematica 32 (2005) 4–32

7. Epilogue

In closing, having argued that it was the Billingsley version of Euclid and not the Barrow translation as others have suggested that served as the basis for the Chinese translation completed by Wylie and Li in the 1850s, several further questions remain to be addressed. Did it make a difference that Wylie and Li used the Billingsley translation, rather than the Barrow or possibly the Leeke–Serle translation? And why should they have focused on translating Euclid rather than more cutting-edge geometry that from a Western perspective would surely have been of greater mathematical interest, for example, the works of Carl F. Gauss, Jakob Steiner, Nikolai I. Lobachevsky, Julius Plücker, and others that were transforming European geometry in the 19th century? As for the signicance of the Billingsley translation, as already noted, the value of this particular version of Euclid lay in its notes and explanatory comments, which Wylie and Li duly included in their Chinese version. Not only was the notation of the Barrow edition at odds with the translation already available in Chinese of the rst six books, which would have made it an inappropriate choice for Wylie and Li, but the Leeke–Serle version, lacking the more extensive commentary of the Billingsley translation, would not have been as useful for Chinese readers, nor perhaps as easy to penetrate. This, of course, is speculation, and it may simply have been that Wylie used the Billingsley translation because this is the one that most easily came to hand; but even if it was a choice due to expedience, it was a fortuitous one. As for why Wylie and Li chose to translate Euclid rst rather than other, more modern geometers of the 19th century, there may be several factors worth considering. First and foremost, the translation of Euclid into Chinese was unnished business. Chinese familiar with the Ricci/Xu translation knew only the rst six books, and completing the translation indeed made available for the rst time in Chinese the most fundamental work in the history of geometry, a prerequisite for moving on to the more sophisticated geometries of the 19th century, including non-Euclidean geometry. As the vocabulary and methods of Euclidean geometry became increasingly familiar to Chinese studying Western 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 appearance of the Wylie/Li completion of the Euclidean Elements. But as Jean-Claude Martzloff has observed, in commenting on European mathematical works that had been translated into Chinese by the end of the 19th century:

...this literature does not contain echoes of the philosophical controversies about innitesimal calculus, imaginary numbers and parallelism. It does not contain any information whatsoever about the theories of Cauchy, Gauss, Riemann or Dirichlet and algebraic symbolism similar to that of Descartes did not appear in China until the second half of the 19th century. Similarly, differential equations, 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 overly theoretical 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’s Algebra was more acceptable. [Horng, 1991, 383] Y. Xu / Historia Mathematica 32 (2005) 4–32 29

Horng then goes on to cite Chang Hao, who also explains that “To the extent that they [Chinese scholar–ofcials] responded to the West at all, their responses reected basically the same pragmatic approach and technically rational mode of thinking they had long applied to tackling domestic problems of statecraft” [Chang, 1976, quoted from Horng, 1991, 382]. Whether the subject is algebra, calculus, geometry, or any other branch of mathematics, the increasingly powerful and abstract concerns of European theoretical mathematicians would not become prominent 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 had studied rst in Europe or America, and then brought back to China. But this is a more complex story about the transmission of mathematics from West to East, and of the emergence of an indigenous mathematical community in China, than can be recounted here.22 Nevertheless, it is certainly an important feature of the assimilation of Western mathematics that one of its essential works, the Euclidean Elements, was at last available to Chinese mathematicians in its entirety, thanks to Alexander Wylie and Li Shanlan, when they 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 and suggestions made on earlier drafts of this article. He is equally grateful to Adrian C. Rice for his help in locating Auguste De Morgan’s article on Euclid. He thanks the New York Public Library for allowing him 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. Rathe and Elie Weitsman of the Rare Rook Division of the New York Public Library, for their assistance. An earlier version of this paper was presented in the Special Sessions for History of Mathematics at the joint annual 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 special session, Joseph W. Dauben and David Zitarelli, for inviting him to participate, and to the audience for its comments and questions. Last but not least, the author thanks the anonymous referees and the editors of this journal for their comments and criticism, which have also helped to improve the published version of this paper.

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22 For transmission of modern mathematics from the West to China and the emergence of mathematical institutions in China, see Zhang and Dauben [1994]; Li and Martzloff [1998]; and Dauben [2002]. 30 Y. Xu / Historia Mathematica 32 (2005) 4–32

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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 is not a single coherent book, but is made up of approximately 69 independent sections of text, which appear to have been assembled from a variety of sources. Problems treated range from elementary calculations with fractions to applications of the Rule of False Position and nding the volumes of various solid shapes. The Suàn shù shu¯ is now the earliest datable extensive Chinese material on mathematics. This paper discusses its relation to ancient works known through scribal transmission, such as the so-called “Nine Chapters,” Jiuzh ang¯ suàn shù , which is rst mentioned in connection with events around A.D. 100, but may have been compiled about a century earlier. It is proposed that the evolution of Chinese mathematical literature in the centuries that separate these two texts may be understood through comparison with what is known to have taken place during that time in another area of Chinese technical literature, that of medicine.

© 2005 Elsevier Inc. All rights reserved.

MSC: 01A25

Keywords: China; Hàn dynasty; Jiuzh ang¯ suàn shù; Suàn shù shu¯

✩ I explain my use of tone-marked romanization in the appendix, in which I also discuss the status and signicance 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 C. Cullen / Historia Mathematica 34 (2007) 10–44 11

Contents

0. Chronology ...... 11 1. Introduction ...... 12 2. Background and context ...... 13 3. The physical and formal structure of the Suàn shù shu¯ ...... 13 3.1. Physical form, reconstitution, and transcription ...... 13 3.2. The formal structure of the Suàn shù shu¯ ...... 16 3.2.1. The individual characters ...... 16 3.2.2. The clause and the sentence ...... 16 3.2.3. Thesection...... 16 3.2.4. Sequence and grouping of sections ...... 19 4. Mathematical content of the Suàn shù shu¯ ...... 19 4.1. Mathematical language ...... 19 4.2. Topicsandtechniques...... 20 5. The origins of the Suàn shù shu¯ ...... 26 5.1. The received tradition of ancient Chinese mathematics ...... 27 5.2. MathematiciansoftheearlyChineseempireandtheirtexts...... 30 5.3. Liú Hu¯ and the history of the Nine Chapters ...... 32 5.4. The commentators and the “Nine Reckonings” ...... 33 5.5. The Nine Chapters and the Suàn shù shu¯ ...... 35 5.5.1. Problems in constructing a narrative ...... 35 5.5.2. A parallel: Medicine in the Western Hàn and the format of the Suàn shù shu¯ ...... 35 5.5.3. The role of Wáng Mang...... 37 5.5.4. The Suàn shù shu¯ and the purpose of mathematical study in the Western Hàn ...... 38 6. Conclusionandsummary...... 40 Acknowledgments...... 41 Appendix. Suàn shù shu¯—title or label? ...... 41 References...... 42 Pre-modernworks ...... 42 Modernworks...... 43

0. Chronology

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

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 powerful vassals until 256 B.C. The period from 771 to 256 B.C. is often referred to as “Eastern Zhou.”¯ The Eastern Zhou¯ includes the “Spring and Autumn” period (722–481 B.C.), and most of the “Warring States” period (481–222 B.C.). 221 B.C. State of Qín destroys all rivals and founds a unied centralized empire. 206 B.C.–A.D. 9 Western Hàn dynasty. A.D. 9–23 Reign of Wáng Mang . 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 the tendency 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 mathematics approximately 7000 characters in length, written on 190 bamboo strips, which were originally bound side by side with string and rolled up into a scroll-like bundle. It was discovered in 1983 when archaeologists opened a tomb at Zhangji¯ ash¯ an¯ in Húbei province [Zhangji¯ ash¯ an,¯ 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 ofcial who had begun his career in the service of the Qín dynasty but started work for its successor, the Hàn, in 202 B.C. [Péng, 2001, 11–12]. The work discussed here was not the only one deposited in this tomb; in addition to material containing administrative regulations there were also 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 explanatory commentary 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 Jiuzh ang¯ suàn shù . The title of this book has been variously rendered. While I would prefer something like “Mathematical methods [or procedures]1 under a ninefold classication,” it is more common to call it “Nine chapters on mathematical procedures,” “Nine chapters on the mathematical arts” or “The Nine Chapters” for short. For simplicity I shall follow the 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 major mathematical 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 to what would follow from the discovery of an ancient Greek manuscript with substantial material by such pre-Euclidean mathematicians as Hippocrates (c. 485 B.C.) and Archytas (c. 385 B.C.). The importance of the Suàn shù shu¯ for the history of world mathematics is indisputable. This article outlines the nature and signicance of this text and indicates some of the ways in which its discovery changes our views of the beginning of the ancient Chinese mathematical tradition, especially in relation to the place of 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 shù shu¯ belonged. Early imperial China was a world very different from the ancient cultures more familiar to most English speaking historians of mathematics, and some preliminary orientation is indispensable to understanding what follows. (b) The physical and formal structure of the text (Section 3). Here I discuss the implications of the fact that the Suàn shù shu¯ took the uniquely Chinese form of a bundle of bamboo strips, and I outline the way that the writings it contains are structured and divided. Such issues will be of major importance in analyzing the historical place of this 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 in the Suàn shù shu¯ and give samples to illustrate the way they are applied. Readers who require more details can of 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 now available 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 dominated by the presence of the Nine Chapters. It was taken for granted that the Nine Chapters represented a natural or at least unproblematic form for mathematical writing in ancient China. No other early examples of extensive mathematical writing were known, and it was claimed that the ninefold classication of mathematics on which the book 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 preference for using less technical terms unless the original text renders it impossible. C. Cullen / Historia Mathematica 34 (2007) 10–44 13

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 and creativity, in which we do not simply see texts like the Nine Chapters as intellectual monuments, but can analyze the processes which created them. In the light of the evidence provided by the Suàn shù shu¯ I suggest that the Nine Chapters is a more recent text than has commonly been supposed, and I identify a plausible set of historical and 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 in this collection was no doubt composed considerably earlier, the consensus among all scholars who have studied the Suàn shù shu¯ so far is that in broad terms it belongs in the cultural milieu of around 200 B.C. Unlike the Rhind/Ahmose papyrus, it is probably not a copy of a much more ancient document. If we want to understand the nature and purpose of the Suàn shù shu¯, it will be helpful to say something about what kind of place China was at the time this collection was placed in the tomb. China as a unied and centrally governed imperial state is not very old compared to (say) the cultures of ancient Mesopotamia. The rst emperor who unied the Chinese world under one centralized government took the throne of his short-lived Qín dynasty in 221 B.C., only 35 years before the Zhangji¯ ash¯ an¯ tomb was closed. For the rst time 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 of aristocratically governed states into which China was divided. When we do go back far enough to nd such a claim being maintained, the central power of the rulers of the Western Zhou¯ kingdom (1046–771 B.C.) was nothing like as strong and ne-meshed as the imperial system which the Qín inaugurated. The Suàn shù shu¯ is a product of the new 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 by some writers in the early imperial age. As we shall see, some scholars of that period who discussed the origins of mathematics believed that the mathematics of their own age could be traced back to Zhou¯ models. This view will require 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 from the break under Wáng Mang from A.D. 9–23), and its government was largely founded on a continuation of Qín institutions. Under the Hàn emperors a new and self-conscious social group took deep root and ourished: this was the scholar-ofcial gentry who staffed the new empire. The rich grave-goods in their tombs are witness to the prosperity brought to them by service to the emperor. For our present purposes, the most signicant point about Hàn funerary practices is that for at least the rst two centuries of the dynasty ofcials were sometimes buried with collections of books to take into the next world [Tsien, 2004, 207–232]. Now before such collections began to be excavated in recent decades, 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 scribal copying, and then after about 1000 A.D. preserved through printing. The Nine Chapters is the oldest mathematical literature to reach us by that route, but our earliest surviving (and incomplete) copy of this book represents an early 13th century reprint of an edition of 1084 [Chemla and Guo,¯ 2004, 71–72]. The other method, less dependent on the accidents 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-ofcials read 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 14 C. Cullen / Historia Mathematica 34 (2007) 10–44 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 from top to bottom with the strip held vertically: see Fig. 1 for some examples. Each strip was cut from the original stalk so that there is a “node” or thickening marking the division between sections of growing bamboo about 15 mm below the top of the strip, and about 20 mm above the lower end. The main body of writing on any given strip lies between the two nodes. To make the roll, the strips of bamboo were laid side by side, and strings were then knotted round the strips near the top and bottom to hold them in place so that they could be rolled up like a mat. When the bundle was unrolled, the columns of characters on successive strips were read from top to bottom and right to left, as was also the case with all Chinese texts on silk or paper until the horizontal and left to right arrangement for writing was adopted under Western inuence in recent times. Since there are 190 strips in the reconstituted collection, allowing a few millimeters for the knotted cord between strips, it seems that the text would have been somewhat less than 2 m long when fully unrolled. When the Suàn shù shu¯ was found, its strings had long since perished and its strips had spread out in disorder through the action of soil and water movement on the original bundle during two millennia. A drawing showing the widely dispersed layout of the strips as found was prepared by the team who excavated them [Péng, 2001, fold-out following p. 134]. The rst scholars who studied the text could only attempt to restore the correct order of the strips using such clues as continuity of text and proximity when excavated [Péng, 2001, 1–4]. The difculty of this task may be judged from the comparative table of excavation numbers of the strips (which roughly reect proximity as found) and the numbers used for the strips in the order of the published transcription, which reect the editors’ judgments of the original order of the strips in the bundle [Péng, 2001, 129–130]. While the dispersion of the original bundle of strips had clearly not produced a completely random order on the ground, it was rare for the excavators to nd a sequence of strips lying together in just the right order. Added to that is the fact that although most of the strips were quite legible a number were damaged or partly obliterated. Transcribing and ordering the strips was therefore no easy task, although that in itself seems insufcient to explain why it took 17 years before the rst transcription of the text appeared in published form. It seems that several different arrangements of the strips were made before the version now before us was decided on [Péng, 2001, 133]. Slight differences between the successive transcriptions published show that revisions were still being made after the rst version appeared. During the long period when the work of transcription was in hand, only brief references to the contents of the text appeared in books and periodicals. The delay that has already occurred makes it all the more urgent to make this text accessible to historians of mathematics in general as well 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¯ Zhangji¯ ash¯ an¯ Hànjian ‘Suàn shù shu¯’ shìwén” (English title given in original publication as “Transcription of bamboo Suàn shù shu¯ or a book of arithmetic from Jianglíng”).¯ This work [Jianglíng,¯ 2000] was ascribed to the Study Group of the Jianglíng¯ Documents. The transcription from early Hàn script in this publication was made using modern simplied characters 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 therefore limited. A further publication appeared the following year [Péng, 2001]. This gave a transcription using full form modern characters, indicated the start and end of strips, and included editorial notes on the transcription and a commentary on the 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 the editors. Later in the same year a full set of full-sized photographs of all the Zhangji¯ ash¯ an¯ strips was nally published [Zhangji¯ ash¯ an,¯ 2001]. This included a transcription of the Suàn shù shu¯ text with editorial notes, substantially identical to 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 suggest emendations to the text than others to date. In part this is perhaps linked to the fact that I have not felt justied in assuming that where the text is irregular or elliptic, this is because a scribe has failed to reproduce fully a hypothetical source text that did not have such features. The Suàn shù shu¯ is what it is: it does not deserve to be treated as an imperfect version of some other shadowy document now lost. C. Cullen / Historia Mathematica 34 (2007) 10–44 15

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. The ninth 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 with permission from [Péng, 2001]. 16 C. Cullen / Historia Mathematica 34 (2007) 10–44

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 elements of the text, and moving up from that level step by step to the overall structure of the entire collection.

3.2.1. The individual characters These 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 supercially 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 mean that a single scribe copied all the text we see, or it may mean that a number of scribes working on the text had been trained in ways that produced a fairly standardized style. It may also mean that I am not sufciently familiar with early Hàn documents to detect differences that may be obvious to the more expert. It is signicant, however, that the group of Chinese scholars who examined the text after its excavation also concluded that the uniformity of the writing suggested that it was all the work of a single copyist [Péng, 2001, 3]. A small number of characters found in this text are unknown to dictionaries based on the printed literary tradition as it existed in late imperial times, such as in strip 80 for what appears to be a type of fat (here as in [Cullen, 2004] I follow the strip numbering in the full photographic reproduction [Zhangji¯ ash¯ an,¯ 2001]). In a few other cases characters of standard form are used in ways that later ceased to be common; for instance, is used frequently for shù “procedure, method” where later mathematical 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 we have zeng¯ in the title, followed two characters later by zeng¯ in the same sense of “increase.” An indication of the attitude of the scribe or scribes to the effort required to write down characters is shown by the frequent use of the ditto mark = indicating that the preceding character is to be repeated. No consideration of elegance or reader-friendliness seems to stand in the way of saving ink and reducing the labor of copying wherever possible.

3.2.2. The clause and the sentence Punctuation 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 order to 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 by obvious units of sense, by introductory formulae such as shù yue¯ , “the method says,” and by familiar particles in classical Chinese such as zhe (which mainly functions to nominalize a preceding verbal expression) and ye (which roughly speaking marks the end of an assertion). However, in parts of the Suàn shù shu¯ one frequently nds that the scribe has already provided the punctuation, in the form of the “hook-mark” , sometimes written clearly as shown here, but sometimes little more than a slightly bent streak. The usage of this mark does not seem to follow any obvious pattern. Sometimes it seems quite superuous, as in several cases where it occurs only after ye has already made the clause division clear. Elsewhere its role in dividing groups of gures from one another does sometimes make the task of grasping where separate numbers begin and end a little easier. A less frequent punctuation mark is the round blob •, which may perhaps mark a more signicant pause than the hook-mark. But for large portions of the text punctuation does not occur at all.

3.2.3. The section In most cases, any text forming part of the Suàn shù shu¯ can be identied as part of what I have called a “section,” a self-sufcient piece of writing possibly extending over several strips, beginning with a title written by the scribe above the upper node of a strip. The end of a section may coincide with the end of a strip, and the scribe may squeeze characters together to avoid running on to the next strip. Often, however, there is an obvious gap in the form of a length of blank bamboo between the end of the text and the lower node of the strip on which the section ends. We must not forget that such a section as we see it today in a published transcription has been reconstituted from scattered strips by the original Chinese editors. However, the continuity of text recording calculations or specifying methods is often so strong from one strip to the next that one can feel considerable condence that the editors have made the right decision in ordering strips to reconstitute a section. In some cases the continuous text of a section can be seen C. Cullen / Historia Mathematica 34 (2007) 10–44 17 to consist of a number of subsections, which may be variants on the main theme apparently from the same source, or may 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 1/4is 1/16,” or tell us what amount of a given commodity has the same exchange value as a given amount of some other commodity. The majority, however, follow some variant of the following form, in which a problem is stated and solved:

(a) a description of a situation, sometimes including the expressions j¯n you , “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 jhé X , “It is asked: how much is X?” (26 in- stances), or simply X jhéX , “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 rather weak word that often functions in classical Chinese as no more than a divider between one block of direct speech and the next. In translating the above expressions as part of continuous prose it seems to me that it is usually better rendered 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 to see any consistent pattern even within individual sections, which were presumably likely to have been copied from a single 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 of the 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 to end of strip) (12) The fox goes through a customs-post A 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 paid out 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 18 C. Cullen / Historia Mathematica 34 (2007) 10–44

out 63 cash and 3 parts. Method: let them be double one another, and combine them [into] 7 to make the divisor; multiply each 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, who will 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 signicant divisions within a section, as in the following example. The letters (a), (b), and (c) have been 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 go on to simplify. (b) Further, the method for simplifying parts: What can be halved, halve it. Where one [can be counted for each multiple] 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 from the numerator. When [the numbers on the sides of] the numerator and denominator are equal, take that [number] as the divisor. 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 title for this section. The long gap at the end of strip 18 suggests that at this point the scribe felt that he had nished a 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 and denominator of a fraction to be simplied) similar to that found in Euclid, Book 7, Proposition 2. Subsection (b) then gives another method for simplication, 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 from a 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 signicant (though somewhat cryptic) evidence as to what the aims of early Chinese mathematical writing may have been. I believe that the hypothesis I introduce later in this article goes some 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-dened section in the form of the announcement Wáng y chóu , “Checked by Wáng.” In the other two instances the name alone appears. As for Yáng, his name is found 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 ends Section 21, which is on a theme (rates of weaving in geometrical progression) similar to Section 15. We read here Yáng y chóu , “Checked by Yáng,” a clear parallel to the earlier Wáng y chóu . In this case the example is incorrectly worked. Apart from Sections 15 and 21, the rule seems to be that the name is given on the rst strip of a section of material associated with the person in question. It is only in Section 47, which divides naturally into subsections, that both names occur. The names Wáng and Yáng are never found on the same strip, nor do they ever occur on consecutive strips that are linked by continuous text. Of course such named strips are a minority: 14 bear names, but that leaves 176 out of 190 unaccounted for. Sections bearing the names of Wáng and Yáng seem to differ C. Cullen / Historia Mathematica 34 (2007) 10–44 19 from one another in their names for grains, and also in their terminology for fractions (see below). Who this Wáng and 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 the last copyist bother to preserve their names, and why only label 14 strips out of 190? And if they were no more than scribes, it is hard to see why their names should be associated in a distinctive way (“checked by ...”)withthetwo weaving problems. On the whole the most probable solution seems to be that Wáng and Yáng were in some sense mathematical teachers and (to a limited extent at least) authors of mathematical problems, and that material by them has been copied into the present collection.

3.2.4. Sequence and grouping of sections Even 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 actually represent an original feature of the collection as deposited in the tomb. The same cannot, however, be said of the order of sections within the transcription. Looking again at the table of excavation numbers and transcription numbers for slips, together with the diagram showing the strips as found, we can see straight away that the order in which the sections are placed in the transcription does not relate in any obvious way to the positions and sequence of the strips as 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 similar topics 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 have come early enough in the series for it to be visible when the bundle was wrapped up with the text facing inward and the 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 conrming to some extent the notion that elementary topics preceded the rest. Under these circumstances, however, we cannot deduce much about the order of the sections in the original text from the order in which the editors have chosen to place them here. When therefore in my translation I propose the 14 rough groupings of sections under broad topic headings listed below, I merely claim that these groupings give a useful idea of the topics that are covered within the collection as a whole. I certainly cannot prove that the relevant sections were so grouped in the undamaged roll of strips, or that even if they were so grouped the groups stood in the order in which they are currently found. On the other hand it does seem unlikely that a collection of mathematical material would have been compiled in a completely random order, and as we have already seen, material on similar topics was deliberately grouped together within sections. It seems likely, therefore, that sections bearing related material would also 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 have been a systematic and ordered treatise. It is, even at the level of the individual section, a patchwork compilation of material from different sources rather than something cut from whole cloth by an individual with a didactic plan in mind.

4. Mathematical content of the Suàn shù shu¯

4.1. Mathematical language

The ways in which the Suàn shù shu¯ expresses numbers and species mathematical processes are not markedly different from what we know of early Chinese mathematics through the received tradition. One striking feature of the collection is, however, its lack of standardization of mathematical vocabulary and syntax compared with later texts. To take one obvious example, the concluding phrase of a problem frequently takes the form of an instruction to perform an operation in which we nd the number of times one number (the fa , 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 in the opening sections alone. I translate as literally as possible for the sake of clarity; the small roman letters after each section number designate subsections. 20 C. Cullen / Historia Mathematica 34 (2007) 10–44

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 various parts 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 second pattern. In general, a section that uses one of these patterns uses it consistently. In the two instances where a single section contains examples of both, this happens in different subsections. While Section 33 (a) and (b) use the rst form, 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 of Wáng and Yáng. It is striking that of the sections using the second form, none of them bears Wáng’s name and three of them (Sections 21, 40, 41) bear Yáng’s. If we turn to Section 47, which has both usages, we nd that (b) with the second form bears Yáng’s name, while (a) with the rst form bears Wáng’s. We cannot simply say that Yáng never uses 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 may note that Yáng prefers the second fractional form for his (incorrect) weaving problem in Section 21, while Wáng uses the rst form for his (correct) weaving problem in Section 15. All this certainly strengthens the impression that the material in the collection comes from several different hands, even if a single scribe wrote out the material as we have it 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 fourteen groupings of sections that I have proposed. It will be evident that in a few cases I have changed the published order of strips 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 prot or liability in proportion to the amount contributed by a number of persons; the case of contributions in geometrical progression; nding the original amount that has been repeatedly diminished in a given proportion to produce a given result; value of a given amount of some commodity, given a unit price.

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) C. Cullen / Historia Mathematica 34 (2007) 10–44 21

Sharing; reaching a total amount through contributions at different rates; nding the cost of some quantity of a commodity from the price of a given amount; nding interest payable on a loan for a given time on the basis of a given 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 few translated 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 the text—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. 22 C. Cullen / Historia Mathematica 34 (2007) 10–44

The labeling convention that I use here encodes the fact that the material translated runs continuously from strip number 1 to strip number 2. The name Yáng appears below the lower node of the rst strip, and there is a long gap after the end of the material on strip 2, suggesting that what we have here is a complete unit of text. The title translated here as “Multiplying together” is written above the upper node of strip 1. Like the section numbers, the lower case letters identifying subsections are my own, and are not paralleled in the original text. The cùn and chí are units of length 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 be described 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 weaving In 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 the subsequent ones, how much [was produced] in each case? Reply: At the start she wove 1 cùn and 38/62 cùn;next3cùn and 14/62 cùn;next6cùn and 28/62 cùn;next1chí 2 cùn and 56/62 cùn;next2chí 5 cùn and 50/62 cùn. Method: set out 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 a chí, 10-fold it; [count] 1 cùn for [each time the result] accommodates the divisor; for what does not ll a cùn, designate the 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 the characters Wáng y chòu , “checked by Wáng,” below the lower node of the third strip, while Section 21 has a similar problem which mentions the name Yáng in the same phrasing used here. The solution starts from the notion that the production of successive days can be said (in modern terms) to be in the ratio 2 : 4 : 8 : 16 : 32. The present text 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 are allocated to days in accordance with the ratios listed. It is not clear why the ratios start from 2 rather than 1, nor why the 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 weaving There 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 the dividends; one chí results from [each time the dividends] accommodate the divisor; for what does not ll a chí, designate the parts according to the divisor. 3 is the dividend for the eldest one; 2 is [the dividend] for the middle one; 1 is [the dividend] for the youngest one.

It seems possible that Yáng may have set out to construct a distinctive weaving problem of his own involving a series of numbers based on rates of work, leading to a division of a total length of cloth, 50 chí in both cases. There is, 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 a known total at different rates, in which a pool is lled by ve separate streams, and in each case we are told how many days each would take to ll the pool on its own. The correct step is then to take the reciprocals of these rates to nd how many times each day the pool would be lled by each stream alone. These are then amalgamated and divided into one day to nd how long all the streams together take to ll the pool. Clearly Yáng’s calculation would have worked in the same way if he had taken the number of times each woman wove 50 chí in one day as his basic rates, rather C. Cullen / Historia Mathematica 34 (2007) 10–44 23 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 = 91/11 chí.

It seems that the author’s check of his answer must have been limited to seeing whether the total was 50 chí—which it 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 of the 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 the context 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 ofcial documents [ Tsien, 2004, 115–118]. The heading of this section contains the rst occurrence so far in this text of the term chéng , familiar from Qín administrative documents with the sense of “regulation,” “standard,” or (as suggested here) “norm” as a standard for productivity [Hulsewé, 1985, 61]. Thus we nd three items in the Shuìh udì regulations labeled gong¯ rén chéng ,“chéng for workmen,” detailing standards for production by workers [Shuìhudì, 2001, 45–46]. An editors’ note to Shuìhudì strip 108 interprets chéng in this context as the amount of work done per day. This word is found 20 times in the present text, usually in circumstances suggesting that the writer is quoting some ofcial standard. It is therefore tting that the rst instance of its use should refer to the production of the standard ofce stationery of the early imperial bureaucrat. In mathematical terms, this problem simply looks at the consequences of changing the circumference of the bamboo to 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. Naturally we 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 diminished half 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 the numerator, [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. 24 C. Cullen / Historia Mathematica 34 (2007) 10–44

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 food grain in ancient northern China. The volume measure sheng¯ in ancient China was about 1/5 litre, which is about the amount of grain one needs to cook a meal for two persons. The expression “diminished half,” shao bàn , simply means one-third; elsewhere in the text we also nd “augmented half,” dà bàn , meaning two thirds. So the rst sentence of the passage given here is simply saying that 1 /3 sheng¯ of hulled millet is the same value as 3/10 sheng¯ of bài grain, which is a slightly more highly milled form of millet. And clearly 1/3 × 9/10 = 3/10, as stated here. The form described as hu takes the milling a step further. Once again, the material presented here does not seem homogeneous. Much of Group 9 is taken up with lists of conversion ratios of the type sampled here. One problem in this group runs as follows:

Strips 117, 118 (nine character gap after end of text) (46) Hulled and unhulled grain combined There 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 dou 8/16 dou ; the owner of the unhulled grain takes 7 dou 8/16 dou . Method; Set out hulled grain 10 dou and 6 dou , 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 dou is the number of the unhulled grain in hulled [terms].

Measures of grain volume are related as follows:

1 shí = 10 dou , and 1 dou = 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 dou of unhulled grain produces 6 dou of hulled grain (as indeed the last sentence notes). Clearly then, the whole 2 shí of mixed grain is worth 16 shí of hulled grain, to which the contributors are entitled to shares in the proportions of 10 to 6. The 2 shí is 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 a typical example:

Strip 131 (10 character gap after end of text) (50) Feathering arrows The 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 and feathering to make the divisor; take the arrows and the feathering multiplied together to make the dividend.

The calculation specied is (30 × 20)/(30 + 20) = 12. It is interesting to think how this result could have been arrived at without using modern algebra. I suggest that the multiplication 30×20 indicates that we begin by imagining the man being told to make 30 × 20 feathered arrows. How long will he take to do this? Clearly, since he can make 30 arrows a day he will take 20 days to make the arrows, and since he can feather 20 arrows a day he will take 30 days to feather these arrows. Hence the daily rate of production of the nished product will indeed be (30 × 20)/(30 + 20) as specied. In Group 11 we nd four problems all using the method of Yíng bù zú , “excess and decit,” which is equivalent to the method known much later in the West as the “Rule of False Position.” The last of these is particularly interesting, since the method is there applied to a problem which involves nonlinear functions of the unknown: C. Cullen / Historia Mathematica 34 (2007) 10–44 25

Strips 185, 186 (long gap after end of text) (54) Squaring a eld There is a eld of one mu :howmanybù is it square? Reply: it is square 15 bù and 15/31 bù. Method: If it is square 15 bù it is in decit by 15 bù; if it is square 16 bù there is a remainder of 16 bù. Reply: combine the excess and the decit to make the divisor. Let the numerator of the decit multiply the denominator of the excess, and the numerator of the excess multiply the denominator of the decit; combine to make the dividend. Reverse this like the method for revealing the width.

The mu (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 contexts the “bù” referred to is a square bù as was the case with the cùn and the chí; there are 240 square bù in a mu .Itis intriguing to nd the “false position” method used here to nd a good approximation to a square root, thereby perhaps suggesting 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/decit (“denominator”) 15 (decit) 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 an inverted 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 of the usual approximation π = 3. For all but one of the sections in this group, the language used is consistent enough to suggest that the problems come from a common source. In the case of the cone, however, we have two different sections 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 grain A whirl of hulled grain has a height of 5 chí; its lower circumference is 3 zhang¯ ; the volume is 125 chí.[Thereis] 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 be multiplied by itself; multiply it by the height; make 1 [from] 36. The greater volume is 4500 chí.

Strip 148 (59) A granary cover A 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. Further multiply 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 to calculate the amount of grain in a given volume. But the real interest of that section is that the thinking behind the algorithm used in the Suàn shù shu¯ is revealed in the reference to the “greater volume.” This quantity is the volume of a cuboid of the same height as the cone, but with a side equal in length to the circumference. As Liú Hu¯, the third century A.D. commentator on the Nine Chapters, points out [Chemla and Guo,¯ 2004, 429; Shen, 1999, 267], the factor of 1/36 follows from the fact that the volume of a pyramid on that square base would be 1/3ofthevolume 26 C. Cullen / Historia Mathematica 34 (2007) 10–44 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 volume of the rectangular hopper (Section 57), the Suàn shù shu¯ problem clearly bases itself on the system of dissecting complex forms into elementary units such as cubes, wedges and pyramids which also underpins the methodology of the Nine Chapters in such cases [Chemla and Guo,¯ 2004, 391–405; Shen, 1999, 251–306]. Interestingly the way the algorithm is presented suggests strongly that the writer of the Suàn shù shu¯ “hopper” problem was thinking of a different 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 last substantial part of the text as edited today is Group 14, which is mainly concerned with problems involving the elds where the area is given as 1 mu and one side is given whose length has a fractional part. The problem is then to nd the 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 way the problems are posed and solved is it clear that the point is to nd a fraction by which one can multiply a group of fractions with different denominators to produce a whole number for convenience of calculation. We may illustrate this 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 and 6939/7381 bù; multiply it to form a eld of 1 mu .

Now there are 240 square bù in a mu . 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 also proceed to “take 1 as 2520” in the case of the dividend, which becomes 240 × 2520 = 604,800. Finally we calculate 604,800/7381 = 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 a myriad is 10 myriad, and so on [Cullen, 2004, 37]), it never actually tells us how to carry out the four basic operations of arithmetic—addition, subtraction, multiplication, and division. Nor is there any direct reference to a physical means for 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 to qualify 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 imperial economy [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 out using small wooden, metal, or ivory counting-rods (variously named chóu , cè , suàn , etc.) arranged on a horizontal surface to represent decimal digits, and manipulated according to rules to represent the new numbers generated as calculation proceeds. Descriptions of the way this was done are available from the rst few centuries of the Christian era [Chemla and Guo,¯ 2004, 15–20; Lam and Ang, 1992, 20–73], but the details need not concern us here.

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 we have seen, the known date of closure of the tomb in the early second century B.C. gives us a rm chronological context 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 C. Cullen / Historia Mathematica 34 (2007) 10–44 27

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 has been, written by real people within particular mathematical cultures which are themselves the products of the society in 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 supercial account 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 supercial level, we need to situate it within the mathematical culture of early imperial China. As I hope to show, the attempt to t the Suàn shù shu¯ into what we already know about that culture poses problems that cannot be solved unless we widen our perspective to include the whole 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 age which concerns us here. Then I shall go on to say a little about the few people known to us who were renowned for mathematical skills in the early imperial age. That will lead us back to the received textual tradition, and at that point we shall be in a position to evaluate the earliest attempt to write a history of that tradition, which dates from the third century 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 point to be able to refer the reader to a wider synthetic discussion of the role and content of mathematical activity in early imperial China, but unfortunately no one has yet attempted successfully to write generally on this topic in English. It would be an invidious task to list books that might have been expected to succeed in this aim, but do not. Commonly read texts such as [Martzloff, 1997; Needham and Wang, 1959] fall into this category. There is, however, a succinct and helpful introduction to ancient Chinese mathematics in French, translated from the writing of one of the 20th century’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 through a continuous literary transmission are the “The gnomon of Zhou,”¯ Zhou¯ bì (in which “Zhou”¯ is probably a reference to the capital of the Western Zhou¯ dynasty) and the “Nine Chapters,” Jiuzh ang¯ suàn shù . There is a full translation of the Zhou¯ bì into English [Cullen, 1996], and translations of the Nine Chapters have been produced in English [Shen, 1999] and French [Chemla and Guo,¯ 2004]. Both these books are anonymous, and we have no direct accounts of the circumstances of their composition. Neither of them is mentioned in any source from before the 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 have today 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 at dates which may extend from as early as the second century B.C. to the early rst century A.D. Although the Zhou¯ bì has been listed as the earliest of the “Ten Mathematical Classics” since the seventh century A.D., the primary aim of those whose writing is included in the book does not appear to have been to expound mathematical methods systematically, but rather to apply available methods to calendrical calculation and to working out the dimensions of the cosmos. The mathematical content of the Zhou¯ bì may be described quite briey. In general it is simply assumed that the reader can add, subtract, multiply, divide, square, and extract square roots using fairly large numbers with mixed 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 objects using a vertical gnomon. The second is equivalent to the Pythagorean relation between the lengths of the side of a right triangle. There is also a certain amount of what may be called “metamathematical discourse,” most of it in a dialogue between a (possibly ctional) master and pupil who discuss the difculties of mathematical study, and the ways 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 shù shu¯. In the case of the Nine Chapters, however, there is a clear family resemblance that demands further analysis. Like the Suàn shù shu¯, the Nine Chapters consists largely of problems, for each of which a solution is provided together with an explanation of the method to be used. However, unlike the looseness and variety we found in the Suàn shù 28 C. Cullen / Historia Mathematica 34 (2007) 10–44 shu¯, the format of the Nine Chapters is highly regular, though not rigid. Every chapter consists mainly of a series of problems, and each problem is introduced by “Now we have ...,”j¯n you (or yòu you , “Further we have” if more than one problem is enunciated). The result is always given as dayu¯ e¯ . “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 problems are 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 related family of problems. The problems treated range from the handling of elementary fractions up to more advanced procedures involving quantities such as the surface area and volumes of various geometrical gures, the method named in English “The Rule of False Position,” the solution of problems involving (in modern terms) simultaneous linear 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: Shao guang , “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 decit” 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¯ gu , “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 Nine Chapters. 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]. C. Cullen / Historia Mathematica 34 (2007) 10–44 29

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 weaves 6 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 the fractions 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 material present 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. Shao guang , “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 yángma and bienáo¯ (special forms required for volume dissection techniques); volume of square pyramid (although the more complex chútong¯ , a frustum of a rectangular pyramid, is treated, as is the 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 the delivery of tax has to be made. Large-scale arrangements of this kind probably date back no further than about 115 B.C. under Hàn Wudì [Chemla and Guo,¯ 2004, 475–478; Twitchett and Loewe, 1986, 602–607], although another document from the Zhangji¯ ash¯ an¯ tomb may possibly bear the words jun¯ shu¯ on one rather obscure strip [Zhangji¯ ash¯ an,¯ 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 decit” No major omissions. 8. Fang¯ cheng¯ , “The rectangular array” Absent. 9. Gou¯ gu , “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 are the only other signs of mathematical activity in China in the early centuries of the empire. For instance such obvious sources as the rst two standard Chinese histories, the Shjì , “Records of the Historian” by S¯maQi an¯ (completed c. 90 B.C.) and the Hàn shu¯ , “History of the [Western] Hàn Dynasty” by Ban¯ Gù (largely complete by his death in A.D. 92) both contain substantial discussions of such topics as astro-calendrical calculation and the mathematics of the standard sets of musical pitch-pipes. Sources of this kind are, however, concerned with applications of fairly elementary arithmetic rather than with discussing methods of calculation and problem-solving in 30 C. Cullen / Historia Mathematica 34 (2007) 10–44 their own right. The same may be said of the excavated fragments of Western Hàn administrative records on wooden strips 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 have performed addition, subtraction, multiplication, and division using whole numbers and some mixed numbers, with one example of the calculation of the area of a rectangular eld. However, English-speaking readers may also refer to 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 to soldiers’ families (I modify Loewe’s translations to be consistent with the practices adopted in this article, and follow the Chinese text given by LDí):

Fù Fèng, private, “Grasping the Barbarians” beacon post: wife Juny¯ , adult, aged 28, grain consumed 2 shí 1 dou 6and2/3 sheng¯ ; daughter Sh, child age 7, grain consumed 1 shí 6 dou 6and2/3 sheng¯ ; daughter Jì, infant age 3 grain consumed 1 shí 1 dou 6and2/3 sheng¯ . Total of grain consumed 5 shí.[Loewe, 1967, 86–87]

Here the scribe had to do nothing more difcult than simple addition. In the next example division is required to produce the result 510 bundles/17 men = 30 bundles/man. The signicance 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 men piled 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 ofcial data and their processing, with no discussion of method, but they certainly indicate how much accounting work went on at the lowest levels of the Hàn state machine. It is not surprising to nd that in three examples of what appear to be prociency certicates issued to military ofcers, it is said of all three néng shu¯ kuài jì , “he is able to write and keep accounts” [Loewe, 1967, 178–179]. So far I know of only one example of a bamboo strip from a government site of Hàn date that shows signs of going beyond the mere recording of accounts to indicate how particular problems are to be tackled. The fragment in question is one of many unearthed 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:

...5dou 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 ofcial had by him a collection of 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 ofcials. The problems in the Suàn shù shu¯ would have been comprehensible to many of them, even if they did not often meet them in daily work or could not have tackled them unaided. But so far as named individuals famous for their outstanding mathematical skills 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. Cangalsoexcelledintheuse¯ of calculating rods, and in mathematical harmonics and the calendar. [Shjì, 96, 2676]

With the change of dynasties, Zhang¯ Cang¯ went on to become a high ofcial under the Hàn, with overall respon- sibility for aspects of government activities involving calculation. One of his successors was the economic reformer Sang¯ Hóngyáng, already mentioned above for his skills in mental calculation. For another high Western Hàn ofcial, C. Cullen / Historia Mathematica 34 (2007) 10–44 31

Geng Shòuchang¯ (. c. 57–52 B.C.), we have good evidence of his skill in mathematical questions relating to civil engineering and administration:

He excelled in calculation, and was able to evaluate the benets of works . . . [It was said of him] “He is practiced in the 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 his time Geng seemed to have enjoyed a role as a mathematical expert only a few degrees below the eminence accorded to Zhang¯ Cang¯ in the previous century. Shortly after the words quoted here, Geng is credited with the innovation of the institution of cháng píng cang¯ , “Ever normal granaries,” to even out variations in the price of grain by state purchase and resale. Geng Shòuchang¯ was also an observational astronomer: his work on the motions of the moon is discussed by JiaKuí in A.D. 92 [Hòu Hàn shu¯, zhi 2, 3029]. The Hàn shu¯ bibliography, based on a listing of the holdings of the imperial library as they were c. 10 B.C., records the presence in the library of 32 juàn, “rolls,” of charts and 2 juàn, of numerical data by him on this topic [Hàn shu¯, 30, 1766]. Another ofcial said to have been skilled in reckoning generally, as well as in “evaluation of works” was X uSh ang¯ (. c. 30 B.C.) [ Hàn shu¯, 29, 1688–1689]. The Hàn shu¯ bibliography records a book (long lost) under his name:

XuSh ang’s¯ mathematical methods,36juàn.[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 Geng Shòuchang¯ mentioned earlier, are found in the section of the bibliography labeled lì pu , “[astronomical] systems and listings.” The titles of other books in that section suggest that they are concerned with mathematical astronomy, an impression which is in line with the editorial note that follows the section. Indeed it is striking that nowhere in the whole Hàn shu¯ bibliography is there any group of books described by the editors as being specically related to practical arithmetical calculations of the kind that currently concern us. The genre of which the Nine Chapters was the rst example in the received tradition appears not to have been recognized as a distinct form of writing at the time the original imperial bibliography was compiled. The earliest rm evidence of a scholar having studied any mathematical book known today is found in a bio- graphical note on MaXù ,thesonofMaYán (A.D. 17–98) and elder brother of the famous scholar and commentator on the classics Ma Róng (A.D. 79–166). Hence he presumably ourished c. A.D. 100. According to the Hòu Hàn shu¯,

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

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

He began by going through . . . the Nine Chapters on Mathematical Methods. [Hòu Hàn shu¯, 35, 1207] 32 C. Cullen / Historia Mathematica 34 (2007) 10–44

It seems very likely, therefore, that the Nine Chapters was a well-known book by the beginning of the second century A.D. Later we hear brief mentions of other students of this book, such as Xú Yuè (. c. 210?) and K an Zé (?–243) [Chemla and Guo,¯ 2004, 59]. Zhào Shuang , writing in his commentary on the Zhou¯ bì in the third century, says of a particular mathematical principle shù zài jiuzh ang¯ , “The method is in the Nine Chapters” [Guo¯ and Liú, 2001, 36]. But none of these records takes us back before around A.D. 100, when MaXù “excelled” in its study. Even if we allow a whole century after its presumed compilation for the Nine Chapters to become widely enough known 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 Nine Chapters in the Hàn shu¯ bibliography, which reects the state of the imperial collection around 10 B.C. There are thus no 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 later included in the Nine Chapters was known about three centuries before the Nine Chapters appears in the historical record. It is therefore clear that mathematical writing of a very different kind from the Nine Chapters did exist in the Western 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 position on 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 that accompanies all extant editions of the main text [Chemla and Guo,¯ 2004, 57]. Liú’s commentary is a major contribution to the development of Chinese mathematics, and he is undoubtedly the earliest identiable Chinese mathematician of distinction from whom we have a substantial extant body of writing. In his preface to the Nine Chapters, he gives his views on the history of mathematics before his own day, and one obviously turns to his account with considerable interest, in the hope that it will tell us something about the milieu in which both the Suàn shù shu¯ and the Nine Chapters originated. Liú Hu¯ begins his preface by claiming that the arts of reckoning go back to the time of the legendary sage Bao¯ X¯ (more commonly known as Fú X¯ ), who rst drew the eight trigrams of the “Book of Change,” Yì j¯ng , and made the jiuji u , “Nine nines”—the multiplication table. But after a few more such gestures at remote antiquity, he tells us disarmingly enough—and quite convincingly—qí xiáng wèi zh¯wénye , “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 Chapters that 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. After that time, Zhang¯ Cang¯ the Beipíng Marquis, and the Grand Supervisor of Agriculture Geng Shòuchang,¯ [men of the] Hàn, both won fame in their age through excellence in reckoning. [Zhang]¯ Cang¯ and others took the remaining fragments of the old texts, and each was equal to [the task of ] making deletions and additions. So if one compares the headings, there are 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]aspart of 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.). C. Cullen / Historia Mathematica 34 (2007) 10–44 33

(b) The Nine Chapters came into existence on the basis of the “Nine Reckonings” tradition (how close this was to the time of the Duke of Zhou¯ he does not indicate), but the transmission of the text was interrupted by the destruction of books under the Qín, and Western Hàn scholars (including Zhang¯ Cang¯ and Geng Shòuchang)¯ had therefore been 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 Nine Chapters 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 I suggest that as well as being dubious in itself, Liú’s view is cast into further doubt by the discovery of the Suàn shù shu¯. My reasons for this may be stated briey 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 of Geng Shòuchang¯ and Zhang¯ Cang.¯ That is comparable to a reader in the year 2000 contemplating the activities of persons in 1700 and 1600. Between Liú Hu¯ and the events he is describing lie the major losses of documents in the destruction with which both the Western and Eastern Hàn ended. If he had named as the reconstituters of the Nine Chapters a person or persons unknown to the historical sources we have today, we might have been forced to conclude provisionally that Liú Hu¯ had access to sources now lost to us. But the fact that he names the only two Western Hàn gures said in extant records to have enjoyed real eminence as mathematicians suggests his information may have been no better than ours, and he is simply looking for likely candidates to t into a story based on conjecture. Other extant sources on the mathematical activity of Zhang¯ Cang¯ and Geng Shòuchang¯ say nothing 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 the habit of lamenting, and indeed exaggerating when possible. But it is exceedingly unlikely that the Nine Chapters would have been among the books destroyed by the Qín, had it existed at the time. Such destruction as there was only targeted on ideologically dangerous works that “used the past to criticize the present,” and books on useful subjects were explicitly exempted; moreover, books held by high-ranking state advisers like Zhang¯ Cang¯ were also exempted [Shjì, 6, 255]. If the Nine Chapters had existed in Zhang¯ Cang’s¯ day he would undoubtedly have had a copy. If Liú Hu¯ was anything of a historian he could not have missed the very well-known source in which all this is made clear. The fact that despite this he believes the Qín could have burned the Nine Chapters does not say 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 it was by calculation and accountancy—destroying a text such as the Nine Chapters, or for that matter the Suàn shù shu¯, some of whose material bears a close resemblance to what we know to have been the content of Qín administrative regulations [Cullen, 2004, 67].

5.4. The commentators and the “Nine Reckonings”

But what grounds did Liú Hu¯ have for claiming the Jiushù , “Nine Reckonings,” had any connection with the 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 of Liú Hu¯ in the late third century A.D., the Zhou¯ l was a famous and widely studied text, which was assumed to give a true account of the most ancient institutions of Chinese government. The modern view is, however, that it is probably a work of the later Warring States period, perhaps of the fourth century B.C. The Zhou¯ l is not mentioned before the middle of the second century B.C., when it is said to have been presented to a younger brother of the emperor Wudì by a scholar who had rediscovered it. It was not then highly regarded. However, in A.D. 9, after having been the effective ruler for several years, the powerful minister Wáng Mang received the abdication of a boy emperor, and began to 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 Zhou¯ l was given an honored place as the basis of these innovations—which were of course presented as the restoration of antique purity. From that time onward this book was given much attention by scholars [Loewe, 1993, 24–32]. In the Zhou¯ l, the “Nine Reckonings” are prescribed as part of the “Six Arts,” liù yì , which form the educational 34 C. Cullen / Historia Mathematica 34 (2007) 10–44 curriculum of young aristocrats. The other ve subjects are the Five Rituals, the Six Musics, the Five Archeries, the Five Horsemanships, and the Six Writings [Zhou¯ l, 212–213]. The Zhou¯ l itself does not give us any idea of what the Nine Reckonings (or the Five Rituals, etc.) actually were. No ancient text of any kind currently known mentions the Nine Reckonings before the Eastern Hàn. If we look for other ancient examples of expressions involving the number nine and something to do with calculation, possibly the earliest comes from the Guan Z book, an eclectic text containing material dating from the late Warring States to the early Western Hàn [Loewe, 1993, 246–249]. Here the multiplication table is referred to as jiuji uzh ¯shù , “the nine nines reckoning” [Rickett, 1998, 499]. If that was all we had to go on, it would seem quite plausible that the expression “Nine Reckonings” of the Zhou¯ l was simply an abbreviated version of this phrase; the multiplication table is about the limit of mathematical knowledge one can imagine being inicted 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 problem posed by the fact that many ideologically important ancient texts such as the Zhou¯ l contained terms and expressions that were no longer comprehensible. In response they wrote voluminous commentaries which aimed to provide the student with solutions—sometimes based on evidence, sometimes clearly conjectural—to (almost) all such difculties. Perhaps the most renowned of all such commentators was Zhèng Xuán—who as we have seen was the second person in Chinese history known to have studied the Nine Chapters. Zhèng Xuán wrote a commentary on the Zhou¯ l, and Liú Hu¯’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, Chaf¯ en,¯ Shao guang, Shang¯ gong,¯ Jun¯ shu,¯ Fang¯ chéng, Yíng bù zú, Páng yào; nowadays we have the Chóng cha,¯ Xì jié and Gou¯ gu .[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 Nine Chapters. Apart from trivial differences of wording and the fact that the seventh and eight topics are interchanged, the only noteworthy difference is that the present text has Gou¯ gu instead of Páng yào (for brevity, I leave to one side what the topics Páng yào, Chóng cha¯, and Xì jié might refer to). It is clear, therefore, that Liú Hu¯’s claim that the Nine Chapters originates from the Nine Reckonings is based on Zhèng Xuán’s statement. It appears from adjacent text that Zhèng Xuán is himself quoting from Zhèng Zhòng (?–A.D. 83), who is roughly of the same generation as Ma Xù, the rst known student of the Nine Chapters at a time when it was apparently already well known. The critical point in evaluating Liú Hu¯’s view on the origins of the Nine Chapters is therefore what Zhèng Xuán’s (or rather Zhèng Zhòng’s) basis might be for their identication of the Nine Reckonings with the headings of the Nine Chapters. It is clearly highly signicant that the rst reference to the Nine Chapters as a well-known book appears simultaneously with the rst attempt to assimilate the Nine Reckonings to the Nine Chapters, and that both events take place in the rst century of the Eastern Hàn. The coincidence of the appearance of the book at the same time as the explanation is striking enough. I shall shortly suggest a possible reason for this coincidence. For the moment, we may simply note that the existence in their own day of a well-known book on mathematics in nine sections would seem to have been sufcient reason for the two Zhèngs to explain the Nine Reckonings in the way they did, without any need to 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 a claim 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 the reliability of the explanations he quotes, which will turn on our analysis of the period in which the original writers worked. From the point of view of the Suàn shù shu¯ Liú Hu¯’s writing does nothing to convince us that the Nine Chapters were (in whatever form) in existence at the time the Suàn shù shu¯ was placed in the tomb, nor does it make it seem likely that the Nine Reckonings were an accepted way of organizing mathematics at that period. Under these circumstances we can treat the Suàn shù shu¯ as a collection in its own right, rather than as an imperfect attempt to do the same job as the Nine Chapters. C. Cullen / Historia Mathematica 34 (2007) 10–44 35

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 from the account of the Suàn shù shu¯ and its context given above. The task is essentially to construct an evidence-based narrative that can deal with the fact that the two earliest texts of Chinese mathematics can be securely dated no closer than two or three centuries from each other, and while clearly belonging to the same tradition differ very signicantly in format as well as in detailed content. What follows is directed toward those who are not satised by the simple assumption that mathematical progress is inevitable, but who are interested in seeing how far mathematical activity and mathematical innovation can be seen as part of a wider context of social and intellectual change.

5.5.1. Problems in constructing a narrative The 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 in the early Western Hàn and early Eastern Hàn, we have to explain how the practice of mathematics and the modes of transmission of mathematical knowledge can have changed sufciently to move us from a world in which the Suàn shù shu¯ was seen as normal to one in which the Nine Chapters were seen as normal. We have to explain how mathematical writing changed from being a matter of generating short problem-sized passages to being a matter of writing a lengthy and well-organized book—and a book whose organization depended crucially on the number nine. (2) However, as we have seen, we have rather little direct information on the activities of those who used mathematics in the intervening centuries, or on the social framework within which they operated. The only ancient attempt at writing 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 to attempt a systematic review of the opinions of Chinese scholars on the origins of the Suàn shù shu¯ and its connections with the Nine Chapters. Briey, 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 GuoSh¯ uch¯ un¯ , takes Liú Hu¯’s account of the origins of the Nine Chapters fairly literally [Chemla and Guo,¯ 2004, 54–56; Guo,¯ 2003]. It is thus assumed that the Nine Chapters actually did exist in some form as a book before the Qín, and that it was damaged or scattered and later reconstituted as Liú Hu¯ tells us. The question therefore arises whether the Suàn shù shu¯ is in some way in the true line of descent that leads to the Nine Chapters as we have it today, and the answer is negative. The polymath historian of science LDí , on the other hand, discounts Liú Hu¯ as a reliable historian so far as the story of the “res of Qín” is concerned, and believes that mathematical knowledge in the Western Hàn circulated in the form of what he calls guan¯ jian , “ofcial bamboo strips.” It was from such material that the Nine Chapters was assembled and edited [L, 1997, 88–138]. L’s views have developed from conjectures expressed before the text of Suàn shù shu¯ was fully published and widely discussed, but he has recently stated that they remain basically unchanged [private communication, 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 shu¯ These 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 already have the main outlines of the history of another technical eld that showed major changes in the writing down and transmitting of knowledge between Western and Eastern Hàn—changes similar to those that would take us from the Suàn shù shu¯ to the Nine Chapters. The eld in question is that of medicine. In medicine, as in mathematics, we nd ourselves 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 what happened in another such eld. 36 C. Cullen / Historia Mathematica 34 (2007) 10–44

For a description and evaluation of the major portions of the Western Hàn medical material, the reader may turn rst to the work of Harper [1998] to which may be added the research results of Vivienne Lo, embodied in her unpublished Ph.D. thesis [1998]. A pioneering and original discussion was contributed by Yamada [1979] and there is also an important 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 literature was like under the Western Hàn, and to elucidate some of the ways the Western Hàn heritage was transformed in succeeding 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 Mawángduì as well as from Zhangji¯ ash¯ an,¯ led him to the conclusion that the elementary unit of that literature was not to be seen as the “text” in its usual meaning of an extensive piece of writing equivalent to what we would call a book, but rather what we may call a “textlet,” a shorter piece of writing capable of being transmitted on its own. Different extended texts might contain overlapping but not identical collections of textlets, and one text might separate textlets contiguous in another text, while bringing together textlets separated in other collections. In studying medicine, one increased one’s knowledge base in part by receiving more material from one’s teachers, who might sometimes pass on textlets only after an impressive ritual requiring a commitment not to transmit them to the unworthy. Such rituals have been reconstructed and evidenced from historical texts by Sivin. Clearly different doctors within a given tradition of medicine would tend to have overlapping but often differently ordered collections of material at their disposal. By the Eastern Hàn, however, Keegan claims that this process had led to the formation of more than one large ossied collection of material that was no longer subject to “textlet” transmission in the old way. This is his explanation of the origin of the different recensions of the so-called Huángdì , “Yellow Emperor,” corpus of which signs appear for the rst time in the Hàn shu¯ bibliography. An examination of three of these early medical compilations shows that they do indeed embody much material common to one another, and that some of their contents closely resemble “textlets” from the Mawángduì and Zhangji¯ ash¯ an¯ material, but in an order and arrangement different enough to 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 read them as deliberately composed and systematic treatises will lead rapidly to a sense of confusion on the part of the conscientious 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 jiay j¯ng , “Huángdì’s ABC canon,” was composed by HuángfuMì around A.D. 256–282, so that he might have been a contemporary of Liú 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 Nán j¯ng , “Canon of Difculties”: this is a highly formalized and ordered book in which 81 sections each raise a question in the form of a “difculty” which is then answered. With the Nine Chapters in mind, we may note that 9 × 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 is quoted shortly thereafter. Turning back indeed to the Nine Chapters and its relation to the Suàn shù shu¯, it does seem that a pattern similar to the one sketched above for the case of medicine can be detected. Even without the work of the scholars mentioned above, an inspection of the Suàn shù shu¯ suggests that in one technical eld, that of mathematics, the independently circulating unit of knowledge in the early Western Hàn—the “molecule of written information,” so to speak—was a textlet rather than an extended and orderly treatise. In many cases the textlets of the Suàn shù shu¯ take the form of a complete section beginning with a title. In other cases, a section with a title contains what is clearly a deliberately collected group of related textlets, which make it plain that an individual scribe has felt that gathering textlets in this way was just what his reader would expect him to have done. The frequent duplications and repetitions that result from this practice make it plain how much the accumulation of transmitted textlets was valued for its own sake, even if 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 hand to 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 to look back a little further than the manuscript itself, though our ability to draw any reliable conclusions is reduced by our inability to say just what sort of people they were, and what roles they played. But we can note that the way their names appear does not suggest any association with long passages of text: the longest continuous passages marked C. Cullen / Historia Mathematica 34 (2007) 10–44 37 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 a mathematically active individual was more likely to generate mathematics in the form of textlet-sized packages than to write a discursive treatise. In other words, Western Hàn scholars who wrote mathematics managed their knowledge bases 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 he said there were only “remaining fragments of the old texts,” jiù wén zh¯yícán , available to scholars. This certainly seems like a reection, perhaps in distorted form, of the fact that Western Hàn mathematical writing really did 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 during the 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 Mang The 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 jiay j¯ng at the same period or a little later. To get from the Suàn shù shu¯ to the Nine Chapters we need to apply the same kinds of transformations that take us from the somewhat disorderly Western Hàn medical material to the more systematic medical texts of the Eastern Hàn—one of which, as we have seen, also used a division of its subject based on the number 9. Reversing the argument, we may say that given the existence of the Nine Chapters in the Eastern Hàn, a collection such as the Suàn shù shu¯ is (given the evidence from medicine) just the kind of mathematical text we would expect to nd from the early Western Hàn. We may now return to the fact that the rst evidence of the existence of the Nine Chapters appears at the same time as an explanation of the Zhou¯ l’s Nine Reckonings that ties them rmly to 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 a revered guide to government. As mentioned above, this was Wáng Mang, who during his reign as emperor from A.D. 9 to 23 consciously presented himself as a new Duke of Zhou,¯ and modeled his governmental structure on the Zhou¯ l.It is well known that Wáng Mang wanted his reign to be signalized by new learning in the numerical disciplines, ranging from calendrical astronomy to divination of all kinds. We know that in A.D. 5, when he was still regent, Wáng Mang held a great gathering of those interested in such topics and learned in ancient texts [Hàn shu¯, 1962, 21A, 955], which was attended by several thousand scholars [Hàn shu¯, 1962, 12, 359]. There is persuasive evidence that Wáng Mang’s patronage may have led to the compilation of the Zhou¯ bì, the text dealing with astronomical calculation mentioned above, which begins with a dialogue involving the Duke of Zhou¯ [Cullen, 1996, 153–156]. That dialogue also stresses the importance of the number nine as the basis of numerical data about the cosmos:

...Thepatternsforthesenumberscomefromthecircleandthesquare.Thecirclecomesfromthesquare,thesquare comes from the trysquare, and the trysquare comes from [the fact that] nine nines are eighty-one. [Cullen, 1996, 174]

If Wáng Mang had sponsored the compilation of a treatise on methods of calculation, it would not have been surprising to nd that it based its structure on the only phrase in the Zhou¯ l which could be interpreted as a reference to the divisions of mathematics, which is of course the cryptic reference to the Nine Reckonings. It would thus have been likely to have used a ninefold division of its subject. Another famous work by one of Wáng Mang’s supporters is 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 Difculties,” mentioned above, likewise divided into 81 sections, may also originate from the same period [Unschuld, 1986, 424]. Another feature of the Nine Chapters that points to the time of Wáng Mang is the widespread use of the hú measure for grain in that text, a unit entirely absent from the Suàn shù shu¯. According to Michael Loewe’s careful examination of datable excavated texts bearing government accounting records, these point to the conclusion that “the term hu [as a unit of capacity] was introduced ofcially from the Wang Mang period” [ Chemla and Guo,¯ 2004, 202; Loewe, 1961, 73]. If the Nine Chapters originated in the time of Wáng Mang, that would t perfectly with its rst appearance in the historical record as a well-known book studied by Ma Xù about eighty years after the fall of Wáng Mang’s regime. The opprobrium later heaped on Wáng and on those scholars associated with him would have ensured that even if Eastern Hàn scholars found such a book as the Nine Chapters too useful to ignore, they might not have wished to advertise its 38 C. Cullen / Historia Mathematica 34 (2007) 10–44 origins. The fact that Zhèng Zhòng gave an explanation of the Nine Reckonings close to the chapter headings of the Nine Chapters would then follow from the fact that, like Liú Hu¯, he simply assumed that the Nine Chapters were an authentic descendant of the Nine Reckonings tradition—whereas in fact they were merely an attempt to reconstruct it by dividing the content of mathematics into nine articial categories based on the fact that the Zhou¯ l specied nine “reckonings.” Indeed it is noticeable that after Zhèng has listed the Nine Reckonings, he records the existence of three other mathematical topics known in his own day that were not then counted in the Nine Reckonings—which would suggest that the ninefold division had not had very long to take hold, and that it was by no means natural for mathematicians at that time to think of mathematical methods as subject to a ninefold classication. We may recall that the only books with titles referring to “mathematical methods” in the ofcial listing of c. 10 B.C. were in 36 and 16 sections, respectively.

5.5.4. The Suàn shù shu¯ and the purpose of mathematical study in the Western Hàn On 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 “Nine Reckonings.” But what then can we say about the aims of those who wrote the material copied and assembled in the Suàn 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: is this material simply a practical handbook of reckoning, or is it written in part by or for people whose interest in mathematics goes beyond its use for administrative purposes? That question is somewhat crudely put, but rather than trying to add in all the possible qualications, 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 same could 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 ofcial, it is in any case difcult to be sure whether the more complex problems about amounts of grain, and elaborate arrangements about sharing and mixtures requiring treatment by the Rule of False Position, really do reect 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, we are faced by texts where the main interest is in problem-solving structures rather than any conceivable administrative reality. And since it is notable that these are the two cases where named persons are linked with problems in the clearest and most formal manner, it does seem probable that at least two persons near the beginnings of imperial China were for at least part of the time interested in displaying their ability to create (or at least to pass on) problems and methods of solution whose interest lay in their structure and ingenuity rather than in their practical value. The fact that one of the methods is faulty shows that we are not just reading a recital of standard methods, but are in the presence of real attempts at mathematical creativity—which can sometimes go wrong. It therefore seems safe to conclude that some early Western Hàn scholars were to some extent interested in clever methods of calculation for reasons other than their straightforward usefulness in administration. The intriguing question that then presents itself is who the audience for such mathematical virtuosity might have been, what criteria that audience operated in deciding what counted as good mathematics, and what rewards followed from a reputation for mathematical skill beyond the call of ofcial duty. For the rst and third parts of that question we are currently far from being able to give a well-based answer. But for the second part of the question, we may be able to hear some Western Hàn mathematicians telling us what they thought good mathematics was about, and that evidence will prove directly relevant to the Suàn shù shu¯ itself, and to its relation to the Nine Chapters. This is possible because we have a piece of metamathematical writing that probably dates from the Western Hàn dynasty, and that sums up the essentials of the work in which Chinese mathematicians were then engaged on the basis of collections such 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 from the rst century B.C., a teacher, Chén Z , is represented as telling his student Róng Fang¯ how to learn mathematics. Neither of these characters is known to history. It is notable that as in the case of the dialogue between Socrates and the slave-boy in Plato’s Meno, the rst piece of Chinese writing about mathematics is in the form of a conversation between master and student. C. Cullen / Historia Mathematica 34 (2007) 10–44 39

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 [ ...] In relation to numbers, you are not as yet able to generalize categories [it is categories of problems that are meant here]. This shows 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 but their 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 difculties, which shows that your understanding of the categories is too simple. The difcult part about understanding the Way, is that when one has studied it, one has to worry about broad application of it. Once it has been broadly applied, one has to worry about not [being able to] put it into practice. Once one has put it into practice, one worries about not being able to understand it. So similar methods are studied comparatively, and similar problems are comparatively considered. This is what sorts the stupid scholar from the clever one, and the worthy from the worthless. So being able to categorize in order to unite categories—this is the substance of how the worthy will devote themselves to rening practice and understanding. (See [Cullen, 1996, 175–178], from which this translation is slightly modied.)

We may note that in telling his student how to study mathematics, Chén Z says nothing about the Nine Reckonings as a means of organizing mathematical knowledge, nor about the Nine Chapters or indeed about any kind of standard mathematical book to be studied. That is consistent with the view suggested above, according to which such things were not yet part of the intellectual repertoire of Chinese mathematics before the rst century A.D. The key to this passage 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 unication of mathematical methods:

The categories under which the matters [treated herein fall] extend each other [when compared], so that each benets [from the comparison]. So even though the branches are separate they come from the same root, and one may know that they 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 solved by one and the same method. In his actual commentary, Liú Hu¯ gets nowhere near being able to realize this ideal program. But the overall ambition to connect and to unify mathematical methods is certainly an aspiration that Liú Hu¯ and Chén Z have in common. I have quoted the passage from the Zhou¯ bì at length because I believe it sums up the spirit of early Hàn dynasty mathematical learning that eventually owed into a systematic and powerful book as the Nine Chapters, in which “words are simple but their application is wide-ranging [and] when you ask about one category [you] are thus able to comprehend a myriad matters.” That is what I believed when I rst translated those words of Chén Z, at a time when the Suàn shù shu¯ was just being unearthed. But now that I have seen the Suàn shù shu¯, I am all the more sure that in Chén Z’s advice to his student we can hear the authentic voice of the early Hàn mathematicians who assembled collections such as the Suàn shù shu¯. It is obvious that the aim of those who gathered the material in each section of that collection was precisely to bring together related problems and methods from various sources so that as Chén Z says “similar methods are studied comparatively, and similar problems are comparatively considered.” To underline the distinctive nature of the program of algorithmic generalization that underlies the mathematical texts we are discussing, an East–West comparison will be helpful. It was the fate of the Nine Chapters to be spoken of in later centuries as a paradigmatic work that summed up the essential spirit and content of mathematics in China. The same thing, more or less, happened to Euclid’s Elements in the West, and with about as little justication, as can easily be seen by considering the contrast between Euclid and (for instance) Diophantos or Heron. But even stereotypes may tell us something interesting, and there is an illuminating contrast here. 40 C. Cullen / Historia Mathematica 34 (2007) 10–44

The Elements, as we know, treat mathematics with a well-dened program in mind, which may in part be described as 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 this program executed that it is not surprising that some of Euclid’s later readers were tempted to think that this was what all “real” or “true” mathematics should be like, and that anything else was in some sense a falling-short. Now judged in that way, much of early Chinese mathematics including the Nine Chapters is a lamentable failure to do real mathematics at all. However, on the view set out above, the Nine Chapters had an aim that was in some sense orthogonal to that of Euclid: 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 innite variety of mathematical problems to the smallest number of general algorithms that could solve them all, grouped under the nine main headings of its chapters. And it seems clear that some part of that program was already implicit in the way the writers of the Suàn shù shu¯ organized their material, and in the way their approximate contemporaries thought of the way one should study mathematics.

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 Chapters represented the only substantial example of a mathematical text from the early imperial age. The history of early mathematics in China largely consisted of attempts to nd some way of dating the Nine Chapters, a process conducted by searching for the earliest dates at which the techniques set out in that book could have existed. The history of early Chinese mathematics was, essentially, the history of the Nine Chapters, and writing that history was not so much a matter of analyzing how Chinese mathematics was created, but rather concerned itself with trying to nd where a unique mathematical monument should be set in place on a timeline. After the Suàn shù shu¯ things are different: we are now able to sketch a vision of the early imperial age as an epoch of change and creativity, in which we do not simply see mathematical texts as intellectual monuments, but can begin to analyze the processes and inuences 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 least as far back as c. 200 B.C., given that the tomb in which it was found was closed not long after that date. These techniques 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, note the absence of certain other techniques, such as the extraction of square roots, the right angled triangle theorem and the solution of linear equations in several unknowns, all of which are found in the Nine Chapters. Nor is there any evidence, from the Suàn shù shu¯ or elsewhere, that Western Hàn (or earlier) scholars regarded the “Nine Reckonings” 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 style and format are not mere matters of the idiosyncrasies of different writers, but can be shown to relate closely to the changing nature of the creation and transmission of technical knowledge in ancient China, and to the inuence of political changes at the highest levels of the Chinese imperial state. Despite these differences, both texts show clear signs of a commitment to a particular style of mathematical learning that is advocated in more than one ancient 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 fact that the Nine Chapters was the earliest known text on mathematics, it may have been natural to assume that the book must have been in existence in some form or other from the earliest period when the techniques it contains could reasonably have been thought to be in use, certainly as far back as the Warring States. The existence of the Suàn shù shu¯ early in the Western Hàn with its comparatively rich content in a format quite different from the Nine Chapters liberates us from that assumption, and allows us to confront the historical evidence without presuppositions. Given that before the early Eastern Hàn there is no sign whatsoever of the Nine Chapters or indeed of a tradition of dividing mathematical methods into nine sections, our working hypothesis has to be that C. Cullen / Historia Mathematica 34 (2007) 10–44 41

the Nine Chapters originated early in the rst century A.D. And given that we can demonstrate the existence of special 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 could currently be based.

The views I outline here are intended to make the most economic and cautious use of the historical and textual evidence actually before us. Nevertheless, no conclusions in this eld can ever be nal. Exciting new nds from Chinese archaeological excavations are now reported at monthly intervals, and we may hope that it is only a matter of time before more mathematical material from the early imperial age comes to light. When that happens, it may be possible 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 the history of mathematics, that the Suàn shù shu¯ existed, and that it was an early Hàn text on mathematics, but that was all 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 mainland Chinese journal of archeology Wénwù [Jianglíng,¯ 2000]. When I began to read it, I found that the only way to understand the harder sections was to write down a rough version in English. The process continued until by the spring of 2001 I had completed a draft translation of the whole text, with a methodological commentary and citations of parallels from other mathematical works, principally the Nine Chapters. I also drafted an essay on the origins and signicance of the Suàn shù shu¯. Professor Sir Geoffrey Lloyd and Professor Nathan Sivin very kindly read this draft material in its earliest form and gave me detailed comments. It was not, however, possible to make further progress without 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 [Zhangji¯ ash¯ an,¯ 2001], which enabled me to resolve a number of outstanding points. The substance of my work was therefore largely complete in draft by early 2002. In the summer of that year I also had the chance to discuss my views with Karine Chemla during a visit to Paris, and I recall nding her suggestions on the dissection of solid forms particularly helpful. My work was however in no t state for publication, because it represented no more than the results of my own obsession with the text, still largely based on single combat with the Wénwù transcription, translated and annotated in the light of what I felt I already knew about early Chinese mathematics. 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 more full of major distractions from scholarship than 2001, and I was able to do little more than ddle with minor details of what I had already written. It was not until the autumn of 2003 that I began the nal efforts that led the project to its present state. The results of my translation work so far have now been published [Cullen, 2004]. I should like to acknowledge the great help I have derived from the generous practical assistance and critical input given by Catherine Jami in the preparation of that work, as well as in the present article. I am also grateful to the editors of Historia Mathematica 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 article title I have added the gloss “Writings on Reckoning.” In this appendix I should like to discuss the signicance of these decisions. Firstly we may turn to the relatively trivial point of the pronunciation of the name itself. “Suàn shù shu¯” is a representation in Roman script of the sounds of the characters in modern standard Chinese, which is an idealized version of the northern Chinese language sometimes known as “Mandarin.” The accents mark the tones which are an integral part of the language. The character is simply an archaic form of the more common , with identical meaning and pronunciation. However, since the material we are studying is over 2000 years old, one would not expect the original owner to have pronounced the characters in the modern fashion. Various attempts have been 42 C. Cullen / Historia Mathematica 34 (2007) 10–44 made to reconstruct the ways in which Chinese pronunciation has evolved in past millennia: one of the best-known and most accessible is still that of Karlgren [1964]. Karlgren reconstructs two pronunciations, an “archaic” scheme representing the rst millennium B.C., and an “ancient” scheme centering around A.D. 600. In the “archaic” scheme and using Karlgren’s diacritic markings, would be read . For interest, we may note that in the same scheme the title of the Nine Chapters, Jiuzh ang¯ suàn shù , would have been read .The useful point of such reconstructions is when, as here, we note that two words with the same modern pronunciation such as and , both shù today, were formerly quite distinct as and respectively. But no argument based on such considerations is central to my present discussion. There are, however, some signicant 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 expression rather than just a pair of separate characters that happen to follow one another: an electronic search of the entire texts of the 25 standard histories gives 51 separate examples of this pairing. It also, as we have seen above, occurs in the Zhou¯ bì. Following the entry for in the now standard Hànyudàcídi an [Luò, 1997, 5231]wemay agree that a broad characterization of this usage over Chinese literature as a whole would be that it means about the same as the now identically pronounced but differently written expression suàn shù , “methods/procedures of calculation/mathematics” (leaving to one side the few instances in which should be read as suàn shu , with the meaning “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 inuenced by the elementary nature of much of the material 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¯ into western languages so far appear to have used “book” or in the case of the one French example known to me the equivalent “livre.” While not denying that shu¯ can be used in Chinese in ways equivalent to English “book,” I feel very uncomfortable with this rendering. As I have argued above there is no way that the material written on the original scroll of 190 bamboo strips from the Zhangji¯ ash¯ an¯ tomb can be thought of as once having constituted a continuous piece of writing with some kind of large-scale order to it—which is surely a minimum requirement for something to be called a “book” in English. A mere collocation of short pieces of textual material, such as a collection of papers in a binder, or an electronic le holding pages downloaded from the Internet, would not, I suggest, usually qualify as a “book” in the view of most native English speakers. Or if I was to copy notes on (say) French grammar from various sources onto a collection of sheets permanently bound together in hard covers, no one would normally say I had composed a book on that subject. However, as everybody familiar with classical Chinese will acknowledge, shu¯ means much more than just “book.” As a verb it means “to write,” and its basic sense as a noun is indeed the broader one of “things written, writings, the act of writing.” I have therefore adopted “writings” as my preferred sense of this word 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 shu¯ should be described as a book title [L, 1997, 91]. Like him, I feel that these words are best viewed as a descriptive label 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 feel that it could well have been. I suggest that the same applies to the characters written on the back of strips from four other rolls from the same tomb, such as Èr nián lü` lìng , “Statutes and ordinances of the second year,” Mài shu¯ , “Writings on the vessels,” Gài Lú , “[Questions by] Gài Lú,” and Yn 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 substantial discussions of content, text history, and associated questions in [Loewe, 1993]. C. Cullen / Historia Mathematica 34 (2007) 10–44 43

Hòu Hàn shu¯ , “History of Later [i.e. Eastern] Hàn dynasty” by Fàn Yè , c. A.D. 450, edn. 1963, Beijing, Zhonghua press. Jiuzh ang¯ 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]. Shjì , “Records of the Historian” by S¯maQi an¯ , 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,¯ Shuch¯ un,¯ 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 Mathematical Collection of the Second Century B.C., with Explanatory Commentary. Needham Research Institute Working Papers 1, Needham Research Institute, Cambridge. This publication has been made available in both hard copy and electronic form: See the details available on the Website of the Needham Research Institute, http://www.nri.org.uk. Guan,¯ X¯hua¯ , 2002. Zhongguó¯ gudài biaodi¯ an fúhào fazh¯ an (A History of the Devel- opment of Punctuation Marks in Pre-modern China). Bashu, Chengdu. Guo,¯ Shuch¯ un¯ , 1990. Jiuzh ang¯ suàn shù ([Critical Edition of the] Nine Chapters). Liaoning Education Press, Shenyang. Guo,¯ Shuch¯ un¯ , 2003. ‘Suàn shù shu’¯ chut¯ an¯ (A preliminary investigation of the Suàn shù shu¯). Guóxué yánjiu¯ 11, 307–349. Guo,¯ Shuch¯ un¯ ,Liú,Dùn , 2001. Suànj¯ng shíshu¯ ([Critical Edition of the] Ten Mathematical Classics). Jiuzhang Publishers, Taipei. Harper, D.J., 1998. Early Chinese Medical Literature: The Mawangdui Medical Manuscripts. Sir Henry Wellcome Asian Series. Royal Asiatic Society, London. Hulsewé, A.F.P., 1985. Remnants of Ch’in Law: An Annotated Translation of the Ch’in Legal and Administrative Rules 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¯ Zhangji¯ ash¯ an¯ Hànjian ‘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 Signicance 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 Scientic, Singapore. Reprinted and revised in 2004. L, Dí , 1982. ‘Jiuzh ang¯ 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¯ : Shànggu dào Wudài juán (General History of Chinese Mathematics: Volume on Early Times to the Five Dynasties). Jiangsu Educational Press, Nanjing.

3 The use of full or simplied Chinese characters in this listing follows the style of the original publication. Names of Chinese authors in western language publications are given as in those publications. 44 C. Cullen / Historia Mathematica 34 (2007) 10–44

Lo, V., 1998. The inuence of Yangsheng culture on early Chinese medical theory. Ph.D. thesis in history. University of 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, Zhuf¯ eng¯ (Ed.), 1997. Hànyudàcídi 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. Zhangji¯ ash¯ an¯ Hànjian ‘Suàn shù shu’¯ zhùshì (The Hàn Dynasty Book on Wooden Strips Suàn shù shu¯ Found at Zhangji¯ ash¯ an¯ 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. Robson, E., 2001. Neither Sherlock Holmes nor Babylon: A Reassessment of Plimpton 322. Historia Mathematica 28, 167–206. 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. Shuìhudì [anonymous collective work by team in charge of Shuìhudì documents], 2001. Shuìhudì Qínmù zhuji¯ an (Bamboo Strips from the Qín Tomb at Shuìhudì). 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, second edition. 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 Difcult 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. ‘Jiuzh ang¯ suàn shù’ yú Liú Hu¯ (Liú Hu¯ and the Nine Chapters). Beijing Shifan Daxue Press, Beijing. Xiè, Guìhua¯ et al. (Eds.), 1987. Juyán¯ Hànjian shìwén héjiào (Complete Transcriptions of Hàn 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. Zhangji¯ ash¯ an¯ [anonymous collective work by team in charge of Zhangji¯ ash¯ an¯ documents], 2001. Zhangji¯ ash¯ an¯ Hànmù zhuji¯ an: Èr sì q¯ hào mù (Bamboo Strips from the Hàn Tombs at Zhangji¯ ash¯ an:¯ Tomb 247). Wénwù Publishing, Beijing. Historia Mathematica 36 (2009) 213–246 www.elsevier.com/locate/yhmat

On mathematical problems as historically determined artifacts: Reections inspired by sources from ancient China

Karine Chemla ∗

CNRS, UMR 7596, REHSEIS, Paris, France Université Paris 7, 3 square Bolivar, 75019 Paris, France Available online 4 March 2009 For Menso Folkerts On 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 social groups considered? This paper discusses evidence found in the oldest Chinese mathematical text handed down by the written tradition, the canonical work The Nine Chapters on Mathematical Procedures and its commentaries, to answer this question in the 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 of algorithms. © 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 social considéré ? Cet article analyse des témoignages prélevés dans les sources chinoises les plus anciennes à avoir été transmises par la tradition écrite, l’ouvrage canonique Les neuf chapitres sur les procédures mathématiques et ses commentaires, pour répondre par la négative à cette question. Ces documents attestent que, dans la tradition dont ils témoignent, les problèmes n’étaient pas seulement des 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 of algorithms

* Fax:+33144278647. 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 often worked under the assumption that the main components of scientic texts—problems, algorithms and so on—are essentially 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 reect tax systems or civil works, historians have failed to recognize that their statements adhere to a given historical context. Rather, I claim that they have mainly viewed problems in the way we commonly use them today—or as we commonly think we do—that is, as formulations of questions that require something to be determined and call for the execution of this task. In contrast, by focusing on mathematical problems found in ancient Chinese sources, this paper aims at establishing that we cannot take for granted that we know a priori what, in the contexts in which our sources were produced or used, a problem was. Moreover, besides providing evidence to support this claim, it suggests a method that can be used to describe the nature of mathematical problems in a given historical tradition. My conclusion is that developing such descriptions should be a prerequisite to setting out to read sources of the past. To substantiate these statements and illustrate this method, I shall concentrate on the earliest extant mathematical documents 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 this document, which was not handed down, the earliest book that has come down to us through the written tradition, The Nine 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 books are 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 echo the concrete problems that bureaucrats or merchants of that time might have confronted in their daily practice. It has hence 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, specically attached to sources such as the Chinese ones, is that the situation used to set a problem should be taken at face value: learning how to handle the concrete task presented by the problem would precisely be what motivated its inclusion in a text. The second assumption, much more generally held, is that the sole aim of a problem is to present a mathematical task to be executed and to provide means to do so. In this paper, I shall cast doubt on the second assumption, showing that it distorts our view of the way in which problems were actually used in ancient China. As a side result, the rst assumption will also be undermined. This conclusion implies that we should be 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 audience for 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 she be thanked for her generosity and encouragement. Markus Giaquinto, the referees, and the editors of Historia Mathematica spent much time trying to 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, such as Guo Shirong [2001], Guo Shuchun [2001]. Two translations into English are forthcoming: [Cullen, 2004], the rst version of which 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 Book of Mathematical Procedures, 2006]. Recently, Cullen [2007] presented his rewriting of the mathematics of ancient China based on the discovery of the 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], Guo Shuchun 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 Nine Chapters was completed in the rst century B.C.E. In what follows, I shall regularly rely on the glossary of mathematical terms I composed to back 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 as The Nine Chapters considered to be “Canons.” In Chemla [2001 (forthcoming), 2003b], on the basis of the extant evidence regarding The Nine Chapters, 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 written only for practical purposes. How can we devise a method for determining what a mathematical problem was in a given historical context? The suggestion I develop in this paper is to look for readers of the past who left evidence regarding how they read, or reacted to, our sources. This is why the Chinese case is most helpful: in the form of commentaries, it yields source material documenting early readings of some mathematical sources. This is also where the difference in type between the two Chinese documents mentioned above is essential. In relation to its status as a Canon, The Nine Chapters was handed down and several commentaries were composed on it. Among these commentaries, two were selected by the written tradition to be handed down together with the text of the Canon: the one completed by Liu Hui in 263 and the one presented to the throne by Li Chunfeng in 656. As we have already stressed above, The Nine Chapters is for the most part composed of problems and algorithms solving them. The commentaries bear on the algorithms and, less frequently, on the problems. They are placed at the end of the piece of text on which they comment or between its statements. In this paper, I shall gather evidence from Liu Hui’s commentary with respect to how he used problems and also how he read those included in The Nine Chapters. This will provide us with source material to reconsider in a critical way the nature of a mathematical problem in ancient China and to establish that problems were submitted to a mathematical practice differing, on many points, from the one we spontaneously attach to them. What does such evidence tell us? To start with, it shows us why the questions we raise on problems are essential and cannot be dismissed if we are to read our sources in a rational way. We shall select two examples of the difculties which face the historian and which are illustrated by the commentary. First, The Nine Chapters describes a procedure for 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 century commentator 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 fractions by one another. Obviously, from Liu Hui’s perspective, they differ. Otherwise, he would not need to change one for the other. Inquiring into their difference as perceived by Liu Hui should hence disclose one respect in which the practice of problems in ancient China differs from the one we would be spontaneously tempted to assume. This will be our rst puzzle. In the example above, the numerical values remain the same. The rst difculty hence regards the interpretation of the situation chosen to set a problem. In addition, the commentator also regularly changes the numerical values with which a problem is stated, without changing the situation, before he comments on the procedure attached to it. For instance, 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–west direction) 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 seem concrete. 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 formulates this problem. Note that procedures for multiplying fractions similar to the one in The Nine Chapters occur in several contexts in the Book of Mathematical 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 an arrow and getting cash in return, bamboo slips 57–58 [Peng Hao , 2001, 65]). In yet other cases, the procedure is formulated in relation to computing the area of a rectangular eld and showing how it inverts procedures for dividing (bamboo slips 160–163 [ Peng Hao , 2001, 114]). 216 K. Chemla / Historia Mathematica 36 (2009) 213–246

(5.15) Suppose one has a yangma,whichis5chi 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 the numerical values here, as he also regularly does when he discusses other surfaces and solids? This will be our second puzzle. My claim is that it is only when we are in a position to argue for an interpretation of the differences between, on the one hand, the situations and, on the other hand, the numerical values that we may think we have devised a non-naive reading 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 commentators read a particular problem as standing for a class (lei) of problems. Moreover, they determined the extension of the class on the basis of an analysis of the procedure for solving the particular problem rather than simply by a variation of its numerical values. An even stronger statement can be substantiated: one can prove that these readers expected that an algorithm given in The Nine Chapters after a particular problem should allow solving as large a class of problems as possible [Chemla, 2003a]. In making these points, we shall thereby establish the rst elements of a description of the 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 by commentators 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 being reduced to questions to be solved, the statements of problems, and more specically the situations with which they are formulated, were an essential component in the practice of proof as exemplied in the commentaries. Viewing the situations from this angle allows accounting for why the problem with cash was substituted for that about the area of a eld. Section 4 focuses on the numerical values given in the problems and suggests that, if we set aside cases in which a change of values aims at exposing the lack of generality of a procedure, it is only within the framework of geometry that Liu Hui changed the values in the statements of problems. In Section 4, I argue that the reason for this is that the commentators introduce material visual tools to support their proofs and that the numerical values given in the problems refer to these tools. This leads me to suggest a parallel between the role played by visual tools in the commentaries and the part devoted to the situations described by problems. Sections 1to4concentrate on how the evidence provided by the commentary allows solving the two puzzles put forward. To be sure, these puzzles could also be grasped only thanks to the commentaries. As a result, my argument establishes how the practice with problems attested to by the commentators differs from our own and how this de- scription accounts for the difculties presented. Even if this is to be considered as the only outcome of the paper, we would have fullled our aim of providing an example of a practice with problems that does not conform to our expectations. Now, the question is: can we go one step further and transfer the results established with respect to the use of problems by the commentators to The Nine Chapters itself, even though the earliest extant commentary on The

6 The yangma designates a specic pyramid, with a rectangular base. Its shape is dened by the fact that its apex is above a vertex of the rectangular 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 of pyramid, 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 object the Chinese term designates the pyramid. Nor do we know whether, at the time when The Nine Chapters was composed, the term had acquired a technical meaning or was only designating a specic object with the geometrical shape required. Note also that, at that time, 1 chi was between 0.23 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),”whichinthe commentaries as well as in some later treatises is more frequently used to introduce a problem. K. Chemla / Historia Mathematica 36 (2009) 213–246 217

Nine Chapters was composed probably some two or three centuries after the completion of the Canon? This question is addressed somewhat briey in Section 5. The reason that we must proceed in this indirect way relates to the fact that commentators wrote in a style radically different from that of the Canons. More precisely, commentaries express expectations, motivations, and second-order remarks, 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 difcult to approach on the basis of the Canon itself, at least when one demands that a reading of ancient sources be based on arguments. One point must be emphasized: Unless we nd new sources, there is no way to reach full certainty about whether what was established 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 to be a less inadequate method than relying on one’s personal experience of a mathematical problem. It seems to me to be more plausible that the practice of problems contemporary with the compilation of The Nine Chapters is related to Liu 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 many hints indicating that some conclusions probably hold true for The Nine Chapters itself and even for the Book of Mathematical Procedures. Section 5 is devoted to discussing such hints. With these warnings in mind, let us turn to examining 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 particular in two respects. The statement of the problem refers to a particular and most often apparently concrete situation, such as, in this case, computing the area of a eld. Moreover, it mentions a particular numerical value for each of the data involved—in this case, 3/5 bu and 4/7 bu for the data “length” and “width,” respectively.8 However, some problems are only particular in this latter respect. An example of this is Problem 1.7, one of three that precede the procedure for the 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 gathers them. , ,

Although numerical values are given, the fractions to be added, or in other terms, the units out of which parts are taken, are abstract.9 All problems in The Nine Chapters are of one of the two types exemplied by 1.19 and 1.7. The procedures associated with the problems in The Nine Chapters also show some variation. Problem 1.19 is a · c = ac followed by a procedure for the multiplication of parts, which amounts to b d 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 parts Procedure: 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 e Chemla and Guo Shuchun [2004, Glossary, 992–993]. It is hence difcult to determine whether the statement of the problem in The Nine Chapters still uses the term with a concrete sense or already with a technical meaning. The same problem was raised above regarding the interpretation of yangma. More generally, the qualication 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 dealing with problems of daily life and those that can be considered as recreational problems. In my view, this opposition is anachronistic and does not t the evidence we have from ancient China (see the following discussion). Moreover, Vogel fails to point out that there are problems formulated in abstract 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. 218 K. Chemla / Historia Mathematica 36 (2009) 213–246

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 concrete situation and values described by the problem. An example of this is the following problem from Chapter 9, “Basis and 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 saw at 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 (the result of the previous operation) by the depth of 1 cun, which gives the diameter of the log. , , ,

, , , 10

11 AC2 + = In modern terms, the procedure amounts to the formula CD CD 2AO. Since both the problem and the procedure are formulated in the same concrete terms, we might, for such cases at least, be tempted to assume that they are to be read as standing only for themselves. Interestingly, as we will see below, we can nd evidence allowing us to determine 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 comes from 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 the problem of the log stuck in a wall from the Canon. The Problems 1.35 and 1.36 are similar. The second one reads as follows (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 gure drawnbymyself(Fig. 1). Needless to say, neither the gure nor the references are to be found in The Nine Chapters. More generally, the Canon does 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 the hypotenuse 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 the translation of the original text references to the gure between brackets. K. Chemla / Historia Mathematica 36 (2009) 213–246 219

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 the circle, in fact the algorithm computes the area of the half-dodecagon inscribed in the circle. Moreover, he stresses that the imprecision increases when the circular segment is smaller than the half-circle. This imprecision motivates him to establish a new procedure, which derives from tiling the circular segment with triangles and computing its area as the sum of their areas.14 It is within this context that Liu Hui rst needs to compute the diameter of the circle containing the circular segment from the two data of Problem 1.36, the chord (AB) and the arrow (CD). For this, he refers to Problem 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 of the circular segment as the length of the path of the saw,andthe arrow as the depth of the piece sawn. Once one knows the 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 procedure attached to it as merely standing for themselves, but as expressing something more general. It thus reveals that, even in a case like Problem 9.9, where the procedure is expressed with reference to the concrete situation of a log stuck in a wall and particular values, the third-century commentator reads its meaning as exceeding this particular case. This holds true in the entire commentary, in which one can nd other pieces of evidence conrming this conclusion, as we will see below.

1.3. The procedure denes 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 equal to 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 innite number of tiles and it has been the topic of much discussion in the literature. It falls outside 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 dened by the integers 5 to 1 and that the share they obtain depends on their rank. We may feel that the statement of the problem is incomplete, but this would mean that we project our own expectations of how a problem should be formulated. Actual description of how problems recorded in historical documents were formulated should replace this anachronistic approach. However, dealing with this topic would exceed the scope of this paper. 220 K. Chemla / Historia Mathematica 36 (2009) 213–246

The procedure following the answers to the problem is also expressed with reference to the particular situation and values mentioned in the statement. However, in this case, through the analysis of the procedure that he then develops, Liu Hui brings to light that the procedure is not general. To be more precise, the procedure adequately solves the problem, but cannot solve all similar problems, because it uses features specic to the situation described in Problem 6.18. Note that this is the only case when this happens for a procedure given in The Nine Chapters and that the commentator immediately exposes the lack of generality. This reaction betrays his expectation that a problem stands 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 particular problem 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) is equal 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 to this problem.18 Despite the appearances, on the basis of the procedure stated, the two problems do not belong to the same class. In other terms, the category of problems for which a problem stands is not determined by a variation of its numerical 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 from another angle, Liu Hui aims at stating a procedure for which all similar problems belong to the same class. We see that the commentary on a procedure analyzes it in such a way as to inquire into the extension of its validity and modies it to extend the class of problems that can be solved by it (and for which a particular problem stands). As a result, on the basis of the sections of Liu Hui’s commentary examined so far, and in fact of others, we can thus state that, in his view, a problem stands for a class (lei) of problems that is determined on the basis of the procedure described after it. It is not so much the similarity of structure between the situations described by different problems that allow considering them as sharing the same category, but, most importantly, the fact that they are solved by the same procedure.19 We hence reach the conclusion that far from being only the sequence of operations allowing a given problem to be solved, the procedure is read beyond the specic context within which it is formulated, and, further, it even 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 general procedure for dealing with Problem 6.18. In fact, the commentator does this simply by suggesting “to imitate the procedure” 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 the coefcients 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 numerical values chosen to set a problem. However, clearly they call for comment. The gures used in Problem 6.18 in The Nine Chapters are the simplest possible with which the mathematical question can be formulated. In his commentary, Liu Hui introduces values that are the simplest possible to make his point. Also, the gures occurring in Problem 1.19 (3, 4, 5, 7; see above) are probably chosen on purpose. Compare my introduction to Chapter 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 coefcients 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. As rightly 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 practice of “categories” in mathematics are central [Cullen, 1996, 74–75, 177]. In my glossary of mathematical terms, I discuss more generally the various uses of the term lei in the commentaries on The Nine Chapters and in philosophical texts of antiquity [Chemla and Guo Shuchun, 2004, 48–949]. It is interesting that the Book of Mathematical Procedures also attests to the use of the term lei in the description of mathematical procedures as early as the second century B.C.E. (see bamboo slip 21 [Peng Hao , 2001, 45]). K. Chemla / Historia Mathematica 36 (2009) 213–246 221

(6.19) Suppose that a bamboo has 9 internodes20 and that the 3 inferior internodes have a capacity of 4 sheng whereas the 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 values displayed in the statement. However, despite the differences on both counts between Problems 6.18 and 6.19, Liu Hui directly imitates the procedure solving 6.19 (that is, transfers it step by step) to solve Problem 6.18, and, beyond, the problems that now belong to the same class. As a result, the new procedure shapes Problem 6.18 as standing for a much 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 of the commentary on the circular segment (Problem 1.36) what we mean by the “circulation” of a problem. Liu Hui does not feel the need to express a more abstract statement or procedure that would capture the “essence” of Problem 9.9 and could be applied to similar cases such as Problem 1.36. On the contrary, he directly makes use of the procedure given after 9.9, with its own terms, in the context of 1.36, by establishing a term-to-term correspondence, “taking the chord of the circular segment as the length of the path of the saw, and the arrow as the depth of the piece sawn.” This seems to indicate that the situation described in Problem 9.9 can be directly put into play in other concretely different situations. 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 procedure for 6.19 within the context of the most singular of all problems (6.18) and not that of the more “generic” one, which he introduced as a counterexample. More importantly, for all problems such as 6.18, he could simplify the procedure given for 6.19 to make it t certain specic features that these problems all share—for problems like 6.18, in contrast to the bamboo problem and its middle internodes, there are no persons who do not belong to either the group of inferiors or that of superiors. Instead, Liu Hui prefers to keep the procedure with the higher generality that characterizes it. The conclusion of the previous paragraph can be stated in a stronger way: the most particular of all paradigms is used to formulate 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 useful below. Liu Hui uses the procedure solving Problem 9.9 in a commentary in which he describes a new procedure for computing 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 systematically establish that the procedures in The Nine Chapters are correct. Problems play a key role in achieving this goal. In the context examined, the use of Problem 9.9 is signaled by the verb “to look for qiu .”23 More generally, this term signals the use of a problem in a proof. The commentary on the area of a circular segment illustrates the following use of a problem in a proof: The task of establishing the new procedure is divided into subtasks, which are identied with problems known—to start with, Problem 9.9. On the one hand, using the procedure that The Nine Chapters gives to solve Problem 9.9 yields the rst segment of the procedure now sought for. On the other hand, since the correctness of the procedure solving 9.9 has already been established, the commentator can rely on the fact that it yields the magnitude needed at this point of the reasoning, that is, the diameter of the circle. In the terms Liu Hui uses to speak of the proof of the correctness of a procedure, the “meaning yi ” of the result of the rst segment of the procedure has been ascertained. More generally, let us stress the fact that, within the proof of the correctness of an algorithm he is shaping, Liu Hui uses problems and procedures solving them to determine step by step the “meaning” of the whole sequence 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 volume thus 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 [Peng Hao , 2001, 114]. 222 K. Chemla / Historia Mathematica 36 (2009) 213–246

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 for the use of the procedure solving Problem 9.9 for dealing with the area of a circular segment, was not specic to Liu Hui. In fact, one can nd a similar piece of evidence four centuries later, in a seventh-century commentary on the Mathematical Canon Continuing the Ancients (Qigu suanjing), written by Wang Xiaotong in the rst half of the seventh century.24 The commentary relates the rst problem of the book, which is devoted to astronomical matters, to a problem dealing with a dog pursuing a rabbit, indicating that the latter problem is included in The Nine Chapters.25 In this case, too, the latter problem is not reformulated in astronomical terms, nor is a third and abstract description of it introduced as a middle term, to allow the result concerning the dog and the rabbit to be applied in astronomy. In exactly the same way as described above, although The Nine Chapters presents the problem within a particular concrete context, the rst reader that we can observe, namely the seventh-century commentator, reads it as exemplifying a set of problems sharing a similar structure and solved by the same algorithm. Furthermore, as in the previous example, the commentator feels free to make the problem and procedure, which apparently do not relate to astronomy, “circulate” as such into a different, astronomical context. This seems to indicate that there was an ongoing tradition in ancient China that did not mind discussing general mathematical procedures in the particular terms of the problems in which they had been formulated, although the questions discussed exceeded the case illustrated by the 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 Nine Chapters conceived of the problems that they included in the Canon. This seems all the more reasonable because, as we 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 Hui regularly comments on a given problem and procedure on the basis of particular numerical values, he understands the meaning of his discussion as extending beyond this particular set and as, in fact, general. Again, in this respect, the commentator 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 may be perceived as recreational by some readers of today, because of the terms in which they are cast. The evidence examined proves that things are not so simple. The historian is thus warned against the assumption that the category of “mathematical problem” remained invariant in time. Such a historical reconstruction guards us against mistaking a problem as merely particular or practical, when Chinese scholars read it as general and meaningful beyond its own context, 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, and how it did so, we have discarded the simple solution that could have accounted for our puzzles. We are hence left with the question: Why is it that, within the context of his commentary on the procedure for “multiplying parts,” or on that for the volume of the yangma, Liu Hui feels it necessary to substitute one situation for another, or one set of values for another, 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 are to be found in Qian Baocong [1963, 2, 495–496]. Eberhard (Bréard) [1997] discusses this example. Part of her discussion and her translation are published 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 interested here 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 of The Nine Chapters. Another hint is provided by how the “chapters zhang ” are composed. They each embody a part of mathematics that derives from 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 Computational Procedures for Nine Categories [of Mathematical Problems][Volkov, 1986, 2001]. Compare the translations of the title as Nine Chapters on the Mathematical Art [Needham and Wang Ling, 1959, 19], Nine Books on Arithmetical Techniques [Vogel, 1968], Nine Chapters of Calculation [Wang Ling, 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 process of development, as well as, more generally, a distinction. Whatever the interpretation of the title may be, the division of the book into nine chapters manifests the inuence of the value of generality. K. Chemla / Historia Mathematica 36 (2009) 213–246 223 say that after examining the evidence presented so far, our puzzles look even more intriguing. Elaborating a solution for these puzzles will compel us to enter more deeply into the practice of mathematical problems as exemplied in Liu 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 of the yangma requires that we recall some basic information regarding the mathematical practices linked to the exegesis of 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 a set of problems, the commentators systematically establish its correctness. However, the way in which they deal with the issue of correctness manifests a specic practice of proof, which can be linked to the context of exegesis within which 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 up fractions, 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 to a + c = ad+cb b d bd :

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

, , [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 this procedure. The Canon and the commentaries approach the object “fraction,” or in Chinese terms: “parts,” in two ways. The expression for m/n used in The Nine Chapters,“m of n parts”(n fen zhi mn m), gives the fraction as being composed of “parts.” This dimension is the one emphasized in what I call the “material” approach to fractions. The expression also displays a numerator and a denominator (“m of n parts”), which are the basis for what I designate as the “numerical” approach to fractions. These two approaches to fractions appear to have been combined by the mathematicians of ancient China. On the one hand, the problem asking us to add up fractions requires gathering vari- a + c = ous disparate parts together to form a single quantity, which must hence be evaluated ( b d ?). On the other hand, the algorithm prescribes computations on numerators and denominators—the numerical dimension of the fractions ad+cb + involved—to yield a value as the result of a division ( bd —where ad cb is the dividend and bd the divisor). ad+cb Establishing the correctness of the algorithm requires proof that the value obtained ( bd ) measures the quantity a + c formed by bringing together the various parts ( b d ). In his commentary on the simplication 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 consisting of 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 specic practice of proof, on the other hand, is addressed in Chemla [2001 (forthcoming)]. In Chapter A of Chemla and Guo Shuchun [2004, 27–39], I discuss the fundamental operations 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 this section, the reader is referred to this paper. 224 K. Chemla / Historia Mathematica 36 (2009) 213–246 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 needed only 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: “simplied” fractions correspond to “coarser parts” (halves instead of fourths), “complicated” ones to “ner” parts (eighths instead of fourths). At this level, Liu Hui 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 c ad + bc + = , b d bd the commentator shows that the algorithm carries out the following program: by multiplication of both the numerator and the denominator, it renes 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 bdths). Quoting the Canon, Liu Hui expounds the actual “meaning” of each step, in terms of both parts and numerators/denominators, making clear how they combine to fulll the program outlined.29 When “the denominators are multiplied by one another”—an operation that, in the course of the proof, Liu Hui names “to equalize”—this computes the denominator common to all fractions (bd) and denes the size that the different parts can have in common: when parts have a common size, the fractions can be added. Moreover, when “the denominators 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 correct measure 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 are highlighted 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 and equalize 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 previous proof 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 parts into coarser parts, equalizing denominators, homogenizing numerators. These operations constitute the topic of the subsequent considerations: exhibiting them appears to be one motivation for carrying out the proof. Moreover, all these operations were introduced in relation to fractions, for which they referred to precise operations on numerators and denominators. However—and this is the point difcult 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 the situation 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 their importance 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. Are they 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 completely missed. The consequences should appear clearly with what is explained below, in this section and in the following one. K. Chemla / Historia Mathematica 36 (2009) 213–246 225 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 other proofs 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 value of x is easily obtained. When Liu Hui accounts for the correctness of the algorithm, he brings to light that one can multiply and divide the coefcients of an equation by the same number without altering the relationship it expresses between the unknowns. Moreover, he points out that the algorithm “equalizes” the values of the coefcients of x, whereas it “homogenizes” the values of the other coefcients. This is how all the operations included in the key declaration quoted above occur again in the proof of the correctness of another algorithm. The same holds true for other 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 which goes beyond the context of fractions. However, if we compare the two situations alluded to, in which Liu Hui identies “equalization,” clearly in the context of equations we do not have an “equalization of denominators,” since what is “equalized” is the coefcients of x. In other words, what is common between the two contexts is not the concrete meaning that “equalization” takes, although in each context the specic concrete meaning of “equalization” is what is required for the proof to work. The declaration invites us to nd something else that is common to the various contexts in which the operations identied occur and which would justify its validity. This leads us to note that in each proof in which they occur, the terms 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. In that of equations, it was interpreted as the operation that made the coefcients of x equal. Similarly, in all the other contexts in which “equalizing” occurs, it can be interpreted in terms of its effect with respect to the particular situation to which it is being applied. This material effect constitutes the rst meaning of “equalization,” one that changes according to the context. However, the fact that the operation recurs in different contexts reveals that the term takes a second meaning, a formal one that is common to all contexts: the term “equalizing” points to how the algorithms work. All algorithms for which the proof of correctness highlights that “equalizing” and “homogenizing” are at play proceed by “equalizing” some quantities while “homogenizing” others. The second meaning of the two terms captures and expresses the strategy followed by the procedure. And the parallel between the proofs discloses that, in fact, the algorithms follow the same formal strategy of equalizing and homogenizing. Even though the concrete meanings of equalizing and homogenizing vary according to the contexts in which they are at play, formally, they operate in the same way. These remarks reveal a key feature shared by the proofs: They bring to light that the same fundamental algorithm underlies various procedures. It is on the basis of the actual reasons accounting for the correctness of the 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 simplied in its detail: cf. [ Chemla, 2000]. In Section 3, I shall describe with greater detail how “equalizing” and “homogenizing” occur in another context. 226 K. Chemla / Historia Mathematica 36 (2009) 213–246

Fig. 3. The transformation of the trapezoid prism. reveals that while proving the correctness of an algorithm, the commentator concentrates on formal dimensions in the procedure.32 This concern relates to one of the reasons for Liu Hui to carry out proofs, i.e., bringing to light such fundamental formal 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 number of strategies systematically used in designing all its procedures. Multiplying or dividing all numbers in an adequate set, as well as equalizing and homogenizing, appear to underlie many of the algorithms of the Canon in domains that 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 the most fundamental algorithms underlying the procedures of The Nine Chapters, those procedures being brought to light 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 the proofs that Liu Hui develops in the context of geometry. This can be deduced from the fact that the proofs establishing the correctness of the most important algorithms related to geometrical shapes all bring to light that these algorithms use 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 this problem either in the commentary or in The Nine Chapters). This solid is the rst one considered in Chapter 5, in which 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 height or 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 also considered 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. What correspond to the “lower” and “upper width” are shown on the gure as the front and rear width. The following argument is not affected by this rotation. K. Chemla / Historia Mathematica 36 (2009) 213–246 227

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 excess to ll up the void , this yields the average width. “Multiplying this (the result) by the height or the depth” yields the erected surface of a front. The reason that “again, one multiplies this, (the result) by the length” is that it yields the volume corresponding to the solid; this is why this makes “the chi of the volume” , , “ ”, “ ” , , (My emphasis) [Chemla and Guo Shuchun, 2004, 412–413]

Given the position in which I have drawn the trapezoid prism, what Liu Hui calls a front is represented in the upper and lower planes. The rst steps of the procedure are interpreted as computing the area of a face, by means of its transformation into a rectangle. With the expression “one uses the excess to ll up the void,” Liu Hui indicates the concrete transformation of the solid that is at the basis of the proof (it is illustrated in Fig. 3) and that allows him 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 concrete transformations that make the proof work in other geometrical contexts. In each context, the actual transformations differ. However, the recurrence of the same formula to refer to them reveals that viewed from a certain angle, they are formally the same. This conclusion conrms 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 despite the apparent difference between them that derives from the fact that they prescribe different computations.36 How does the proof fulll 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 enabling the proof to fulll this function and, in this case, disclose the hidden action of “equalizing” and “homogenizing” in its process. 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 “multiplying parts,” requires computing the area of a rectangular eld, 3/5 bu long and 4/7 bu wide. However, the procedure itself is formulated without reference to any concrete situation. Liu Hui’s commentary on the procedure provides key evidence for understanding the part played by problems for proofs to fulll 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. Its opening 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 uses the excess to ll up the void,” which occurs only in relation to geometrical shapes. Most probably, this type of transformation was conceived of as one of the general patterns with which the operations identied 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 a position 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 step of 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 operation yielding the fraction. The commentary alternately uses the two sets of terms, with the greatest precision. 228 K. Chemla / Historia Mathematica 36 (2009) 213–246

correlatively, it (i.e., the dividend produced by the multiplication) lls up the divisor, the (division) hence only yields an integer.39 If, furthermore, one multiplies something by the numerator, the denominator must consequently divide (the product) in return (baochu). “Dividing in return,” this is dividing the dividend by the divisor. , , , , , , [Chemla and Guo 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. The expression of “dividing in return” (baochu) is particularly important to note. The commentator introduces it here for the rst time in the whole text (Canon and commentaries). Seen as an operation, as Liu Hui makes it clear, it consists in a division. However, the qualication “in return” adds something to the prescription of a division: it makes explicit the reason for dividing. Using “dividing in return” means that it was needed, for some reason, earlier in the procedure, to carry out a multiplication, which was superuous with regard to the sought-for result: this division compensates for the earlier multiplication, deleting its effect. This general idea clearly makes sense in the passage of Liu Hui’s commentary translated above. The use of the technical term indicates that multiplying by the numerator is to be interpreted as follows: instead of multiplying “something” by a fraction a/b, one multiplies the “something” by its numerator a. Multiplying the fraction a/b by b, one obtains the numerator a. Having multiplied the “something” by the numerator a, instead of a/b, one has multiplied by a value that was b times what was desired. Consequently, one has to “divide in return” by that with which one had superuously multiplied, that is, by b. Multiplying something by a/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 qualications of division that one can nd in The Nine Chapters itself too. This fact calls for two remarks. First, the expression clearly adheres to the sphere of justication, since the very prescription of the division indicates a reason for carrying it out. In other words, such hints prove that The Nine Chapters does refer to arguments supporting the correctness of algorithms in some specic ways. Second, we can nd the expression both in The Nine Chapters and in the commentaries. This fact points to continuities 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 the meaning 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 multiplies the numerators instead of multiplying the fractions, one must divide the product by each of the denominators. The last sentence of Liu Hui’s rst proof transforms the sequence of operations just established to carry out the multiplication of fractions (multiplication of the numerators, division by a denominator, division by the other denominator) into the procedure 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 the numerator is multiplied, the product should be divided by the denominator. Since the product of [all the] numerators is taken as dividend, it should be 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 second sentence, “multiplying the numerator by the dividend” may mean. Moreover, there is no word for “numerator” in the text at this point. Later, “if the 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 divisions into 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 mathematical explanation 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 various cases in the multiplication and fails to read the argument that Liu Hui makes in it. K. Chemla / Historia Mathematica 36 (2009) 213–246 229

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, that is, between the validity of the transformation and the exactness of the results of division [Chemla, 1997/1998]. As above, the formulation “to divide altogether” prescribes a division in a way that indicates the reason why the division prescribed is to be carried out in this particular way. This expression recurs regularly in Liu Hui’s commentary. The qualication adheres to the sphere of justication. In contrast to the “division in return,” however, the expression lianchu 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 proof discloses that multiplication and division, two of the fundamental operations listed in Liu Hui’s declaration, are put into play, as opposed operations, for designing the procedure as it stands. For our purpose, it is interesting to note that this proof of the correctness of the algorithm develops independently of the framework of the problems in the context of which the “procedure for multiplying parts” was formulated, that is, the problems about rectangular elds. The arguments only make use of general properties of dividend and divisor, numerator and denominator. Moreover, they bring 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 difcult to have (the procedure) be understood with a greater generality nan yi guang yu , [Chemla and Guo Shuchun, 2004, 170– 171]

In fact, this sentence justies why the commentator discards Problem 1.19 in the context of which the procedure for “multiplying parts” is described in The Nine Chapters and why he introduces, immediately thereafter, another problem equivalent to the original one: 1 horse is worth 3/5 jin of gold. If a person sells 4/7 horse, how much does the person get? Let us analyze more closely the assertion introducing the alternative proof that the commentator develops in the sequel. Since it is a key assertion for my argument, I shall insist on it and distinguish all the facts that it reveals. First, Liu Hui’s above statement establishes a link between the context of a problem—here the problem provided by The Nine Chapters, i.e., that of a rectangular eld—and the aim of understanding the procedure. It appears here that a given problem can prevent one from gaining a wider understanding of the procedure. Second, this seems to indicate that Liu Hui also perceives the preceding passage as contributing to an understanding of the procedure. This link between establishing the correctness of a procedure and understanding it is not fortuitous. It will recur again in the passage of the commentary following the above statement. Now, the change of problem is justied 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 context of a problem and the proof of the correctness of a procedure. Again, this link will be conrmed in what follows. In fact, Liu Hui next introduces a sequence of three new problems, all formulated with respect to a single kind of situation, which make it possible to develop the second proof of the correctness of the procedure conforming to the practice of demonstration sketched in Section 2. Let us stress right at the outset the essential consequence that can be derived from both the above statement and the analysis developed below: problems appear here not to be reduced to questions that require a solution—as we would readily assume—but they also play a part in proofs. Observing Liu Hui’s next development should hence allow us to progress on two fronts. It should provide evidence showing how problems intervene in proving the correctness of a procedure. And this is where the difference between the problem with the eld and those with the persons, the horses, and the gold should become apparent. Moreover, it should help us grasp how Liu Hui conceives of “understanding a procedure.” 230 K. Chemla / Historia Mathematica 36 (2009) 213–246

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

a c ac bc ac · = · = . b d bc bd bd The second proof brings to light that the computations of ac and bd, with which the procedure for “multiplying parts” 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 shall examine 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 to proofs: 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 us to infer why the rst problem cannot be used here. The situation of the eld with length and width is unsuitable from a semantic 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 eld does not allow bringing to light the “equalization” that underlies the procedure for “multiplying parts.” The scheme of “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 components constituted 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 the procedure for “multiplying parts.” Consequently, the new situation allows revealing in another way how the procedure for “multiplying parts” relates to the fundamental operations identied by Liu Hui and discussed in Section 2.Thisis what 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” belongs only to the sphere of proving. The equalization plays no role in the actual computation of the result. This is why the problem in the context of which the algorithm is described differs from the problems in relation to which the proof is carried out. A link is thereby established between bringing to light a formal strategy accounting for the correctness of the procedure and interpreting the “intentions,” the “meanings” (yi) of the operations in the eld of interpretation offered by the situation of a problem. The example of the multiplication of fractions, in which the description of the algorithm and the development of its proof require a sequence of different problems, reveals the essential part played by problems in establishing the correctness of algorithms. Note, however, that the numerical values are common to all problems: Liu Hui introduced the new problem mentioned above in such a way that the task to be carried out is still to multiply 3/5by4/7. As a consequence, the computations leading to a solution of the problem included in The Nine Chapters (3 times 4, 5 times 7, dividing the former result by the latter) are an actual subset of the operations involved in 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 an adequate way, a sequence of equivalent problems that are transformed one into the other. What would be done in modern mathematics by formal computations is here carried out by interpreting the results of operations with respect to a succession of problems. In the alternative proof, Liu Hui’s rst step consists in formulating a rst problem, the solution of which requires use of the last operation of the procedure for “multiplying parts,” i.e., dividing 12 (ac)by35(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 share this [the gain], how much does a person get? Answer: 12/35 jin. To solve it, one must follow the procedure for “directly sharing (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 division involving integers and fractions [Chemla, 1992a]. Note that division is dealt with before multiplication and that, in correlation with this fact, the K. Chemla / Historia Mathematica 36 (2009) 213–246 231

, , , , , [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 commentary on the preceding section of The Nine Chapters, devoted to “sharing parts.” This problem is immediately followed by another 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 if 7(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 homogenize these quantities of gold (ac) and persons (bd). They then all conform to the rst problem and this is solved by the (procedure for) “directly sharing.” , , , [Chemla and Guo Shuchun, 2004, 170–171]43

The use of the term “homogenize” implies that, in parallel, an equalization is carried out. Only the operation of equalization can confer the meaning of “homogenization” on the other operations. Liu Hui will make this point explicitly in one of the subsequent sentences of his proof. “Homogenizing” quantities of gold and persons goes along with “equalizing” quantities of horses. This is why, as Liu Hui states, the homogenizations prescribed lead to the “rst problem”: 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 of the relationship between the values considered. The procedure solving the rst problem can then conclude the solution of the second one. Now, if one considers the whole procedure that correctly solves the second problem, through transforming it into the rst one, one realizes that the procedure for “multiplying parts” is embedded in it. What makes the difference between the two is that the procedure for “multiplying parts” does not prescribe the computation of bc. However, if we pay closer attention to the way in which Liu Hui formulates the transformation of the second problem into the rst one, we observe that there, too, he evokes the computation of bc only in an allusive way, by referring to the computation of ac and bd as “homogenizations,” and by stating that the second problem is reduced to the rst one. In fact, the sequence of operations that is concretely given to solve the second problem corresponds exactly to the procedure given by The Nine Chapters for “multiplying parts.” To recapitulate, Liu Hui describes the procedure 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 procedure for “multiplying parts.” He writes:

If this is so, “multiplying the numerators by one another to make the dividend (ac)” is like homogenizing the correspond- ing gold. “Multiplying the denominators by one another to make the divisor (bd)” is like homogenizing the corresponding persons. 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 translation modern 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 by the rule of division.” [Shen Kangshen et al., 1999, 83], which does not t with the Chinese. Again, the argument developed by Liu Hui cannot be understood from the translation. 232 K. Chemla / Historia Mathematica 36 (2009) 213–246

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 second problem correctly and, on the other hand, the procedure involves only homogenizations. Liu Hui devotes the following statement to considering the ambiguous status of “equalization.” The computation of bc as the equal quantity in the statement of the rst problem is essential to ascertain that ac and bd correctly correspond to each other in a pattern similar to that of the rst problem. In other words, it is essential for the proof. But this computation is useless in obtaining the result: once one knows the reason why ac and bd correspond to each other, it sufces to divide one by the other to yield the solution. This explains why Liu Hui emphasizes that the “equalization” plays no part in the procedure itself, except for allowing an interpretation of its rst steps as “homogenizations.” In turn, the interpretation of 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 the framework of the new situation with persons, gold, and horses, in which the equalization can be interpreted and thereby 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 a third 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 equivalence between 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 makes 1 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 possibly multiplied 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 equivalent problem, 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 to analyze to solve our rst puzzle and to account for why Liu Hui substituted the third problem for Problem 1.19. We now 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 the procedure 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 to the fact that a denominator and a numerator are made equal so that the procedure can fulll its task. Even though the concrete meanings differ, the procedures use the same strategy formally: in both contexts, it is by making some quantities equal and then making other quantities homogeneous that one can obtain the solution. This is what the proofs disclose. This remark indicates yet again the sense in which, in Lu Hui’s eyes, the fundamental operations he listed can be deemed fundamental for mathematics. Each proof sheds a different light on the procedure. Through highlighting how the rst operations prescribed by the procedure for “multiplying parts” amount to homogenizing some quantities, the second proof discloses a new meaning for 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]. K. Chemla / Historia Mathematica 36 (2009) 213–246 233 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 the meaning of the operation. It is in this way that we can interpret Liu Hui’s introductory statement, where he claims that he needs to discard the problem of the area of the rectangular eld in order to “have (the procedure) be understood with a greater generality.” The second proof requires introducing a procedure—the procedure of homogenization and equalization—in which the procedure for “multiplying parts” is embedded. This brings us back to the remark I made in Section 1,onthe commentary Liu Hui devoted to Problem 1.36 and the procedure for the area of the circular segment: proofs of the correctness 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 a new problem. In the case of the commentary on the topic of Problem 1.36, in order to formulate his new procedure, Liu Hui introduces new elements in the gure of the circular segment: the circle from which it derives, and then triangles tiling its surface. These elements are in fact the topics of other problems in The Nine Chapters and are thus associated with procedures that the commentator uses to establish his new procedure for the area of the circular segment. In particular, Liu Hui employs these new elements to express the “meaning” of the steps of the new procedure. We hence 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 important for explaining the substitution of one problem for the other in the case of the multiplication of fractions. The difference between the new problem and the one described in The Nine Chapters now appears precisely to lie in the fact that the procedure developed within the context of the proof can be “interpreted” only with respect to the new problem. The key point is that the operation of “equalization,” which is part of the procedure needed by the proof, is not needed by the procedure that computes the result. In correlation with this, the new problem, and not the one in The Nine Chapters, allows an interpretation of the “meaning yi” of the procedure. These remarks highlight how problems play a key part in Liu Hui’s practice of proof. They account for the fact that problems that appear to be the same to us are in 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 a solution, but were also used as providing a situation in which the semantics of the operations used by an algorithm could be formulated in order to establish its correctness. This second function becomes apparent when the problem fullling the rst function cannot fulll the second. This aspect appears to be an essential component of the practice of 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 to carry 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 for which a particular problem stands. In Section 1, we have shown that a problem stands for a class of problems sharing the same category. But this conclusion seemed to be contradicted by the need to replace the problem about the area of a rectangular eld by a problem with horses, gold, and persons, both problems sharing exactly the same particular values: 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 dened by examining the procedure attached to it. With respect to the procedure for “multiplying parts,” these problems belong to the same category, but, with respect to the procedure developed for the sake of the proof, they can no longer be substituted one for the other. However, what is essential is that all problems similar to the one with horses, gold, and persons, in which the procedure of the proof, i.e., equalization and homogenization, can be unfolded share the same category. Consequently, even though the proof is discussed in terms of 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 the problem 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 the context of a problem is worth quoting (p. 181): “Evidently, he [Liu Hui] chose these specic 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.” 234 K. Chemla / Historia Mathematica 36 (2009) 213–246

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 cannot be underestimated. Proofs developed by the commentators were regularly written down in ways that conict with our expectations, since they seem to be formulated only with respect to particular cases. As a result, they were interpreted as lacking generality. Such an interpretation is in my view anachronistic, because it reads our sources in ways that project our modes of interpretation onto documents that require other readings. I think that the previous discussion establishes 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 the particular values it involves. Proofs relying on problems with particular values are also meant to be general in this sense: the example of the pyramid with a square basis (yangma) provides a clear illustration. We shall now use this example 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 method as 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 modies the numerical values of a problem presented in The Nine 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 to expose 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 the volume of the pyramid yangma, already mentioned in the Introduction, is more difcult to account for. We will now examine 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 type of pyramid called yangma , in Problem 5.15, is represented in Fig. 4. Also in this case, there is no diagram in the sources. However, Liu Hui’s commentary does refer to a visual tool of a different kind, namely “blocks qi .” This is the 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 modern ways of dealing with visual aids. In particular, I have not added letters by which one could refer to the vertices of a block. Moreover, I have drawn some of the diagrams in such a way as to give a sense that the actual visual tools used were 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 K. Chemla / Historia Mathematica 36 (2009) 213–246 235

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 of his 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 the problem by considering a yangma with dimensions (width, length, height) equal to 1 chi. He then remarks that three such 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 the general case, when the dimensions are not all equal. The reason for this is that when the dimensions are different, the three pyramids into which the parallelepiped with the same dimensions can be decomposed are not identical, as was the case for the cube. Thus from this decomposition, one cannot conclude that the volumes of the three pyramids are equal before the procedure under study has been proved correct. As a result, Liu Hui discards the rst reasoning and starts 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 focus the reader’s attention on a specic preliminary case. Surprisingly, in the general reasoning, instead of making use of the 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 does he change the values he has just introduced to new values that are also equal to each other? We shall sketch the rst and main part of Liu Hui’s general proof—the only one important for the points we want to make—to nd answers to these questions. For the second reasoning, Liu Hui considers simultaneously a yangma and a specic 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 the qiandu thus formed, Liu Hui sets himself a new goal: establishing that the yangma occupies twice as much volume in the qiandu as the bienao. From this property, the correctness of the procedure for computing the volume of the pyramid can be derived immediately. However, we do not need to examine this part of the reasoning here, since to make our points, it sufces to evoke the 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 this book, but the second sketches the essential points of the proof), Wagner [1979], Guo Shuchun [1984a, 51–53],andLi 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. 236 K. Chemla / Historia Mathematica 36 (2009) 213–246

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 of the following types: parallelepiped, qiandu, yangma, bienao, all with dimensions of 1 chi.47 Let us describe the composition 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 of the original body (see Fig. 7). On top, and in front on the right-hand side, we see two small yangma, similar to the one we started with. In the back, on the right-hand side, and in front on the left, we see two small qiandu, whereas in the back, below and on the left-hand side, we have a small parallelepiped. This is how, for space geometry, Liu Hui uses blocks 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 of the original dimensions (see Fig. 8). On top in the rear, and in front on the right-hand side, we see two small bienao similar 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,asshown in 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,butthe 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 referred to the published translations of Liu Hui’s commentary on the yangma to get a more precise idea of the original text (see footnote 46). K. Chemla / Historia Mathematica 36 (2009) 213–246 237

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 (or Fig. 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. All these 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 that is 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 that the 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 occupied by pieces coming from the yangma as there is space occupied by pieces coming from the bienao. The property sought for (i.e., the yangma occupies twice as much volume in the qiandu than the bienao) is hence established in the second zone. 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 claried. 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, he removes the rear upper half-parallelepiped, which is composed of a smaller bienao and a smaller yangma, and moves it near the similar one in the lower front part of the gure (see Fig. 11). 238 K. Chemla / Historia Mathematica 36 (2009) 213–246

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. Seen from 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 relative proportions ensure that when one repeats the same reasoning, 3/4 of the remaining 1/4 of the original qiandu can be shown to hold the property, and so on.

4.2. The proof is general, but it makes use of specic values

To make the points I have in mind about the meaning of the numerical values appearing in the statements of problems, we need not discuss the nal part of Liu Hui’s general proof, in which he establishes the proportion of 2 to 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 are constructed using particular values, for which all the dimensions of the solids involved are equal. If Liu Hui was only K. Chemla / Historia Mathematica 36 (2009) 213–246 239

(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 the dimensions of the yangma—he would use a simpler argument: this argument is simply the rst argument he makes and then discards as not general. Two conclusions can be derived from these remarks. To begin with, when Liu Hui rst changes the values of the dimensions of the yangma, his goal is not to introduce a simpler case for which a straightforward argument can be made to establish the correctness of the algorithm. Second, the second proof, conducted and presented within the framework of the simplest situation possible—that with dimensions all equal—is meant to be general. We are hence 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 same conclusion holds true for the various problems and procedures, whether they are those from The Nine Chapters itself or 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 the operations put into play. We meet again with a parallel between problems and visual tools, based on the fact that they are 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 brought to light that the geometrical transformation needed could be subsumed under the general formal operation of “using the excess to ll up the void.” In my view, the way of conducting the general proof for the yangma also reveals how it brings into play the same general operation.49 To explain this point, we need to sketch how, after Problem 5.14 and the procedure for it, Liu Hui comments on the volume of the qiandu. His commentary on the algorithm computing the volume of this solid offers two ways of accounting for the correctness of the procedure. The rst one relies on the fact that two identical qiandu make a parallelepiped. However, there is a second argument, which is not easy to understand. 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 and the proof of its correctness derives from the general proof for the prism, applied to the specic case of the qiandu (see Fig. 12). In turn, the proof for the correctness of the procedures for the yangma and the bienao, which depends on that for the qiandu, ts within this framework, except that the piece to be moved needs to be cut in the middle and the pieces obtained have to be exchanged before they are moved. In this way, again, the operation of “using the excess to ll up the void” appears to be efcient 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 transformations underlying 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 operations underlying 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 to be 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]. 240 K. Chemla / Historia Mathematica 36 (2009) 213–246 the use of the particular values provided by problems, we must therefore answer the following question: Why does Liu 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 are determined by the visual tools he used, that is, the blocks, which were concrete objects, all dimensions of which were equal to 1 chi. In fact, the rst reasoning brings together three identical blocks. In relation to the physical operation carried out, the values needed are all equal to 1 chi. The second reasoning illustrates the composition of a yangma and a bienao with blocks. The same blocks being used, the values needed to refer to the composition of the solids are 2 chi. As for the iteration of the decomposition of the yangma and the bienao in the second step of this proof, its analysis is based on the same conguration as in the rst step. Blocks with the simplest dimensions have the property of being polyvalent for all these uses. Using the same blocks, Liu Hui can develop a proof that is particular (the rst one) 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. This conclusion 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 One may assume that these blocks—or some of their uses—were introduced for the sake of exegesis. This hypothesis would explain why the commentator had to change the values to refer to the concrete use of objects in relation to his reasonings, whereas the problems in The Nine Chapters would not mention values relating to material objects. In any event, the conclusion reached accounts for the fact that, except for the introduction of new values for a problem employed as a counterexample, the only systematic cases in which the commentators change the numerical values of problems occur in the context of geometry and in relation to the use of material visual tools. The new values are all determined by the objects used as visual tools to compose the body under consideration. The generality of the proofs developed is an issue that is completely dissociated from the fact that the texts could be written as referring to specic material 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 express the “meaning yi” of operations in the same way. In the former proof, gathering the three blocks yields a solid, the volume of which expresses the “meaning” of the multiplication together of length, width and height. The coefcient 3 is interpreted as related to the composition of the cube with three yangma. In the latter proof, like in the commentary following Problem 1.36 and bearing on the area of the circular segment, the commentator introduces a procedure that 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 receive a “meaning,” the interpretation of the procedure requires that the volume of the yangma be divided. Interestingly, this question 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 richer in possibilities of interpretation, since this feature was required for making explicit the “meaning” of the specic operations 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, in contrast to The Nine Chapters,theBook of Mathematical Procedures does not mention the yangma or the bienao, that is, pieces among the blocks essential to establish the correctness of the procedures for other solids. However, the book treats other solids also found in The Nine Chapters in ways that require knowing the volumes of the yangma and the bienao. Several algorithms for volumes in both The Nine Chapters and the Book of Mathematical Procedures point to a geometrical interpretation, which provides reasons for the correctness of the procedures. This is how the commentators read them in the case of The Nine Chapters: they interpret the procedures step by step with the help of blocks [Chemla, 1990]. In such commentaries, Liu Hui also regularly changes the values of the problems and uses blocks in the rst way described above, that is, in a way that is not geometrically general. Does this indicate that the commentators knew that objects of the type of blocks were used at the time when our earliest 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 a certain answer to this question. However, the generic terms for “gure” or “block” do not occur in the earliest sources. The rst known occurrences of these terms are found in the commentaries. Moreover, we may assume that if blocks did not exist, geometrical practice may have been similar to what some commentaries betray within the context of some problems dealing with geometrical topics: instead of referring to specic 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 9 in Chemla and Guo Shuchun [2004, 673–684]. K. Chemla / Historia Mathematica 36 (2009) 213–246 241 in the problem of The Nine Chapters is correlated to the introduction of an inner decomposition of the solid, which creates further possibilities for the interpretation of the steps needed by the proof. Not only does this conclusion offer a similar explanation for the change of a situation and for a change of numerical values. It also leads to an interesting observation. 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, and when they are not rich enough to support the needs of interpretation required by a proof, they are replaced. In this context, it is worth recalling how the commentator introduces blocks when he rst resorts to them: “ ‘Speech cannot exhaust the “meaning yi”’(yan bu jin yi),52 hence to dissect/analyze (jie) this (volume), one must use blocks; this is the only way to get to understanding (the procedure). , , ” We nd here again, in relation to visual devices, the combination of terms (meaning, understanding) that in Section 3 wasusedinrelationto problems. The previous discussion highlighted why in the context of the commentary on the yangma the values of the problem were changed into other values, which refer to the dimensions of a block: In fact, the continuity between the two kinds of 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 specicities of what is known regarding gures. This is why, even though all the evidence examined here bears on blocks, we have formulated our conclusions for visual devices in general. The perspective developed further suggests an interpretation of Liu Hui’s own description of his 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 practice with problems. To which extent can these conclusions on the practice with problems obtained thanks to the evidence found in the commentary be transferred to the mathematical activity at the time of the composition of The Nine Chapters itself, or even the Book of Mathematical Procedures? This question will be examined in Section 5. With the example of visual tools we have already seen reasons to believe that not all mathematical practices were the same. What about problems? This point is all the more important since in the earliest extant writings problems play a prominent part.

5. Back to The 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 Hui described above? Let us repeat that, unless new sources are found, we will not be able to answer this question with full certainty. The method I will suggest here is to gather hints in The Nine Chapters indicating continuities with respect to the practice evidenced by Liu Hui’s more prolic 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 the context of proof. If we believe, as is often stated, that in sheer contrast to the commentaries The Nine Chapters contains nothing relating to proof and betrays no interest in this dimension of mathematical activity, this would deny from the outset the possibility of a real continuity. However, I have argued several times that, even though The Nine Chapters includes no fully developed proof, various facts indicate that the authors had an interest in understanding why their procedures were correct. We have already indicated some of them. For example, we have noticed that the qualication 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)totheBook of changes (rst chapter, paragraph 12). Compare Chemla and 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, see my glossary. 242 K. Chemla / Historia Mathematica 36 (2009) 213–246 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 have mentioned that Liu Hui interpreted some algorithms for computing the volume of solids as indicating a proof of their correctness 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 of algorithms is provided by how Liu Hui appears to interpret the fact that they described procedures in the context of problems. Not only does this piece of evidence indicate that he reads a concern for proof in the Canon, but it also reveals that he links the problems in the Canon to this concern. Let us therefore examine this evidence in greater detail. In fact, there are cases where procedures in the Canon are described outside of the context of problems. Such is the case, for instance, for the rule of three (jinyoushu )[Chemla and Guo Shuchun, 2004, 222–225]. This observation 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 conrms from yet another angle the part played by problems in proofs in relation to making explicit the “meaning” of the operations of an algorithm. What is most interesting in this case, however, lies elsewhere. The commentator states about this procedure for the rule of three: “This is a universal procedure ( ci doushu ye).” There is only one other passage in which Liu Hui repeats 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 with the rule of three, in this case the “universal procedure” is described by The Nine Chapters within the context of a problem 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)39dou;2bing of high-quality millet, 3 bing of medium-quality millet and 1 bing of low-quality millet produce 34 dou;1bing 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 qualies the 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. The commentator writes:

It would be difcult to understand (the procedure) with abstract expressions (kongyan), this is why one deliberately linked 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 the context of a problem was related to the aim of having the procedure be understood. We have already shown the relationship between “understanding” a procedure and establishing its correctness. In the text that follows, Liu Hui uses 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 suggested another interpretation for this problem: the statement may be understood as referring to distinct units of capacity all named bing and the value of which would depend on the grain measured. I have shown that in The Nine Chapters, the Canon gives evidence of a similar phenomenon with respect 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)of39 dou”). The meaning of shi as “capacity” is attested in a passage from the Records on the scrutiny of the crafts (Kaogongji , dated to the third 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,seeChemla [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 enable an interpretation of the operations in the procedures. K. Chemla / Historia Mathematica 36 (2009) 213–246 243 the pattern of “equalizing” and “homogenizing” underlies it, thus proving its correctness (see Section 2). The piece of evidence that the above statement constitutes may hence indicate that Liu Hui reads the Canon as providing problems to be put into play to prove algorithms. Several additional features of the problems used in the same chapter are quite interesting to support our argument that 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 millet or 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 bing of low-position millet correspond to the dividend (shi)39dou;2bing of high-position millet, 3 bing of medium-position millet and 1 bing of low-position millet correspond to the dividend 34 dou;1bing of high-position millet, 2 bing of medium-position millet and 3 bing of low-position millet correspond to the dividend 26 dou. One asks how much is the dividend 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 reading interprets the term shi (“produce”/capacity) with its technical meaning in mathematics, “dividend.” From the rst century C.E. till the fourteenth century at least, in ancient China an algebraic equation was conceptualized as an opposition between divisors (the various coefcients of the unknown) and a dividend (the constant term). In fact, it can be shown that Liu Hui also reads the statement of Problem 8.1 in this alternative way and understands a linear equation 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 for solving 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 of equations with two negative constant terms. However, all ancient sources give the two “productions/dividends” as positive, a fact that the commentators do not stress as erroneous. In other terms, it seems that the interpretation of the “dividends” in terms of “production” or “capacities” imposes that the shi remains positive. If this is the case, this constraint, which derives from the interpretation of the operations in the terms of the problem, poses limits to the full presentation of the mathematical topic. What follows in The Nine Chapters supports this hypothesis: Problems 8.7 and 8.8 turn to a new situation—buying animals—to discuss new types of systems in which dividends—in this context no 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 units of capacities) was used to deal with systems of equations as long as the interpretation allowed by the situation did not conict with the mathematical requirement. As soon as a divergence arose between the interpretation in the terms of the problems and the mathematical meaning, the situation for discussing the topic was changed to allow further discussion. If our interpretation is correct, this implies that, in agreement with Liu Hui’s explanation, the use of the situation of a problem for interpreting the operations of an algorithm, in relation to understanding and proving it, dates to 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 conrm this hypothesis: many mathematical questions are discussed within the framework of the same kind of situation throughout the centuries. One way of accounting for this 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 this is quite plausible.

6. Conclusion

We have seen that not only does the commentator Liu Hui attest to a practice of problems, described in Sections 1 to 4, that is peculiar and differs from the one most common today, but he also seems to assume that his way of using problems was in continuity with former practices, especially those contemporary with the making of the Canon. In fact, we have found hints in The Nine Chapters that appear to support his belief. In this tradition, whether problems were practical in form or abstract, they were read as general statements, the extension of which was determined on the basis of the procedure relating to them. This eshes out what the opening sections of The Gnomon of the Zhou intended 244 K. Chemla / Historia Mathematica 36 (2009) 213–246 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 rely on the evidence we have about mathematical activity in ancient China to interpret this early theoretical description of what mathematics was about. Furthermore, the results obtained in this paper give us elements for establishing a method 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 correctness of another algorithm. This is why problems can be either questions to be solved or statements describing a situation in order to interpret the meaning of operations. Clearly, ignoring such facts would be quite detrimental when reading mathematical sources from ancient China. This is one of the main errors responsible for the mistaken idea that these texts can be adequately interpreted as merely practical. However, the benet of promoting a new method of reading is 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 research that by gathering specic evidence will enable us to restore various practices of mathematical problems. Once we have assembled several similar case studies, we shall be in a position to outline a research program that may create new conditions for interpreting our sources.

References

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Historia Mathematica 40 (2013) 183–202 www.elsevier.com/locate/yhmat

The correspondence between Moritz Pasch and Felix Klein

Dirk Schlimm

Department of Philosophy, McGill University, 855 Sherbrooke St. W., Montreal QC, H3A 2T7, Canada Available online 14 March 2013

Abstract The extant correspondence, consisting of ten letters from the period from 1882 to 1902, from Moritz Pasch to Felix Klein is presented together with an English translation and a short introduction. These letters provide insights into the views of Pasch and Klein regarding the role of intuition and axioms in mathematics, and also into the hiring practices of mathematics professors in the 1880s. © 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, on the one hand, letters are usually not meant for publication and thus allow for more candid and tentative formulations. On the other hand, the relationship between the correspondents has a major effect on the content of letters and historians can learn about this relation from what is written, how it is written, and from 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 prolic 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, the lack of evidence for an extensive body of letters cannot immediately be taken as evidence that there never was any such in the rst place, because it could have been destroyed in the course of time or lost, waiting to 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.2013.02.001 184 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 letters from 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 and Pasch is very different. To give a rst impression of this imbalance, the Kalliope Verbundkatalog Nachlässe und Autographen database2 lists 306 items addressed to Klein and 1785 from Klein, where each item can contain 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 are over 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 one of the most prestigious mathematical journals during this time. In this capacity he dealt with submissions for publication, referees, and general administrative issues. Indeed, three of the letters from Pasch were prompted 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 standing of mathematics and mathematics education at national and international levels. This knowledge about the professional and personal situation of many mathematicians is one reason why Pasch approached him for guidance 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].Kleinreportedtohave destroyed his correspondence in the fall of 1878 [Tobies and Rowe, 1990, 8], but he seems to have kept most 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: for example, 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 [Tobies and 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 the information given by Pasch’s son-in-law, Clemens Thaer, to Heinrich Scholz that Pasch had the habit of destroying his correspondence after having replied to it [Frege, 1976, 169]. This makes it extremely difcult to gauge the original extent of Pasch’s correspondence. To get a rough idea of it, we have to look for the letters that Pasch wrote in the estates of the recipients. One would expect a promising starting point to be those 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 from Pasch (from 1888 and 1913), which are thank-you notes for the sending of a publication and for Hilbert’s contribution to the celebration of Pasch’s 70th birthday. Pasch’s correspondence with Gottlob Frege, which consists 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 due to the fact that Pasch was often occupied with other duties, as he frequently laments (this is also a common theme in the letters to Klein). It is also interesting to note that ve of these items are prompted by Frege’s sending of his writings to Pasch. In the Nachlaß of Max Dehn, who wrote a long article on the history of geometry that was appended to the second edition of Pasch’s lectures on geometry in 1926, only one short letter can be found. Of the colleagues that Pasch mentions in his autobiography [Pasch, 1930], either there seems to be no extant Nachlaß, e.g., of Jakob Rosanes, whom Pasch refers to as ‘his friend’, or no correspondence 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],andTobies [1981]. 2 Online: http://kalliope.staatsbibliothek-berlin.de. 3 He continued to live in Giessen until his death in 1930. 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, together with the total number of extant items: Felix Klein (10), Gottlob Frege (6), Mario Pieri (3), Ernst Richard Neumann (3), Otto Behagel (2), David Hilbert (2), Otto Toeplitz (2), Ventura Reyes y Prósper (2). To Luigi Cremona, 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 the currently available evidence it appears that Pasch indeed was not an avid letter writer and that he did not maintain an extensive correspondence with this colleagues. The letters to Klein reproduced below form the largest set of known letters from Pasch to any other mathematician. As such, they might not tell us much about 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 Pasch to Felix Klein, which is held in the archives of the Niedersächsische Staats- und Universitätsbibliothek Göttingen. 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 to Pasch, 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 difcult to decipher. Best guesses for these are reproduced with a small superscripted question mark (?). Punctuation and spelling have not been altered in the German transcriptions; in the English translation underlinings have been rendered in italics and crossed-out words are omitted. Page breaks in the originals are marked with 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 headings of 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 presented next. 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 informs Pasch about an article on the general concept of functions [Klein, 1874b], in which he expressed views similar 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 briey presents his philosophical views on the nature of geometry, including a discussion of the differences between our intuitions and the 186 D. Schlimm / Historia Mathematica 40 (2013) 183–202 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 background information regarding the development of his own views. In particular, he traces back his empiricist leanings to the lectures of Kronecker and Weierstrass he attended in 1865/66 and to discussions with his friend Jakob 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 negative reactions to Klein’s views. Pasch admits that he has not read much philosophy, and the bits that he had read quickly disappointed him; he considered the authors to be on the wrong track. This exchange led to the addition of a brief remark with a reference to Klein’s paper at the end of Pasch’s Einleitung in die Differential- und Integralrechung [Pasch, 1882b], which was published soon afterwards, and also to a reference 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 the Mathematische 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 afnity of their views, but also 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 surrounding the 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 this position in Giessen on April 28, 1869 as the successor of Alfred Clebsch, who had moved to Göttingen. At the time there was only one chaired position (Ordentlicher Professor) in mathematics in Giessen and one position without chair (Ausserordentlicher Professor). Baltzer had been instrumental in getting Pasch to submit his habilitation in Giessen in 1870, where he then became private docent. As was anticipated, Pasch was appointed to the unchaired position in 1874, which was converted into a chaired position in 1876. In the letters, 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 experience in mechanics, and he also hints at some aspects of the local university politics. Quite interesting with regard to the general attitude at the time is Pasch’s frank remark that hiring another Jewish mathematician (Pasch himself 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],andElon [2003] for a more general account. D. Schlimm / Historia Mathematica 40 (2013) 183–202 187

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 geometry to Pasch, which were the occasions for Pasch’s replies to Klein in 1891 and 1902. Here Pasch returns to the discussion 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 comments directly 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 this change of heart. These comments supplement and provide additional background for similar remarks that can be found in Pasch’s and Klein’s publications.8 A concluding remark reveals Pasch’s disappointment that his students did not appreciate his foundational investigations, which can be explained by the fact that 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 scheme introduced in the previous section.

Baltzer, Richard (1818–1887): 5 Netto, Eugen (1848–1919): 5 and 8 Betti, Enrico (1823–1892): 8 Peano, Giuseppe (1858–1932): 9 Dyck, Walther (1856–1934): 5 and 6 Reyes y Prósper, Ventura (1863–1922): 4 and 9 Gordan, Paul (1837–1912): 1 Röntgen, Wilhelm (1845–1923): 8 Hölder, Otto (1859–1937): 6 Rosanes, Jakob (1842–1922): 2 and 3 Henneberg, Lebrecht (1850–1933): 5 Runge, Carl (1856–1927): 5 Hettner, Georg (1854–1914): 5 Schönies , Arthur (1853–1928): 6 Hurwitz, Adolf (1859–1919): 6 Schering, Ernst (1824–1897): 5 Köpcke, Alfred (1852–1932): 4 Schottky, Friedrich (1851–1935): 5 Kiepert, Ludwig (1846–1934): 5 Krause, Martin (1851–1920): 5 Schröter, Heinrich (1829–1892): 8 Krazer, Adolf (1858–1926): 6 Schur, Friedrich (1856–1932): 5, 6, 8, and 9 Kronecker, Leopold (1823–1891): 2 Staude, Otto (1857–1928): 5, 6, and 8 Lindemann, Ferdinand (1852–1939): 5 and 9 Stickelberger, Ludwig (1850–1936): 5 and 6 Lipschitz, Rudolf (1832–1903): 1 Voss, Aurel (1845–1931): 6 and 7 Meyer, Franz (1856–1934): 5, 6, and 8 Weierstrass, Karl (1815–1897): 2 Nöther, Max (1844–1921): 6 Wiltheiss, 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 the role of intuition in mathematics, is in preparation by the author. 188 D. Schlimm / Historia Mathematica 40 (2013) 183–202

3. Letters from Pasch to Klein (1882–1902)

3.1. Pasch to Klein, June 16, 1882

Giessen, June 16, 1882 Highly honoured colleague, The answer to your kind letter was delayed because I wanted to get the Erlanger Berichte here so that I could rst have another look at your article, of which I do not own a copy. 9 I found out that the Oberhessische 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 article is missing. I read your article at Gordan’s, as he got a copy from you (it was during the semester in which I taught the rst course of lectures on the subject matter covered in my book) 10; the content appealed to me all the more, because I entirely agreed with your basic view. However, I still believe that the main point of your discussion was to show how the concept of a function strip leads to the tangent. On the other hand, I considered the basic view, according to which geometric concepts, like other empirical concepts, are aficted with imprecision, to be more or less widespread. For example, I saw in the acknowledgment that the intuition of a curve and the concept of function do not coincide, the reason why function theory sought to free itself from so-called geometrical proofs. In the meantime, statements about the imprecision of all determinations of measurement that I found in Lipschitz (Differential- und Integralrechnung, §40)11 further conrmed 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 that the ideas that you put forward at that time encountered erce opposition leads me to assume that my work will also gain little approval. All the more I am pleased to hear that you intend to return to this subject yourself. With sincere regards, respectfully Yours, M. Pasch

3.2. Pasch to Klein, June 22, 1882

Giessen, June 22, 1882 Highly honoured colleague, I owe you many thanks for your kind mail, which arrived on Monday. I have now received the last proof sheet 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 in the Mathematische Annalen in 1883, with an additional footnote by Klein referring to Pasch [1882a] and Pasch [1882b]. It seems that 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 you that all measurements of actual objects lead to equations in which the representation differs from that which is to be represented by a certain quantity and thus that every application of the calculation to actual objects that are subjected to measurements implies the use of equations, which are aficted with certain errors.’ (‘Ehe auf das Wesen der Gleichung (2) näher eingegangen wird, sei daran erinnert, dass alle Massbestimmungen von Gegenständen der Wirklichkeit zu Gleichungen führen, bei denen das dargestellte von dem 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]. D. Schlimm / Historia Mathematica 40 (2013) 183–202 189

Integralrechnung.13 Like my work on geometry, which has already appeared, this is the result of my efforts to come to terms with myself about the material I have to lecture on. Since I refer to the book on geometry in the new piece, when I discuss the imprecision of measurements, I took the opportunity to add a remark in 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 at the 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 proofs in his lectures, in addition to some conversations I had with Rosanes in the 1860s, may have contributed to my views on the matter, which I recently described to you. Thus, I was quite surprised by what you relate about the comments made by some function theorists. I did not have the opportunity to make such observations myself. Philosophers cannot do much here in my view; they must wait until things are put in order by mathematicians. I, too, must confess to have hardly read any of the philosophical writings; after brief attempts at reading them, it always appeared to me as though 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 distribution of your article through the Annalen. On its few pages the relevant issues are expressed and brought to bear. I myself had to settle on a narrow aim, since I already needed very much time and effort for bringing everything 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, Yours Respectfully M. Pasch

3.3. Pasch to Klein, October 9, 1883

Giessen, October 9, 1883 Highly 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, we would 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, who has now returned from lengthy travels, put both notes quickly down on paper in order not to hold up the 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,’ and Pasch [1884], dated ‘1882.’ This is not the order that Pasch requested in this letter. In Rosanes [1884a, 414] the author mentions that his result had been obtained earlier by ‘my friend Pasch.’ 17 See Footnote 9, above. 190 D. Schlimm / Historia Mathematica 40 (2013) 183–202 and eventually be able to clarify the issues that still separate our views. For this, however, I would need to have more undisturbed time at my disposal than I can presently hope for. With highest regards I remain Yours sincerely M. Pasch

3.4. Pasch to Klein, March 31, 1887

Giessen, March 31, 1887 Highly honoured colleague, I take the liberty of sending you two articles and of humbly asking whether you would be inclined to print them in the Annalen.18 The article on function theory was developed from lectures, which made apparent the need for an essential simplication of certain sections. Prósper’s article 19 really only provided an external motive for the small note; for, I do not want to give particular weight to the simplication of the proofs vis-à-vis Mr. Prósper, but rather I would like to point out again — especially after having read Köpcke’s article20 — the views that you pioneered at the time. With high regards Yours M. Pasch

3.5. Pasch to Klein, December 3, 1887

Giessen, December 3, 1887 Highly 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 only now able to devote myself to the task I was assigned — a task which, thanks to the great number of names to 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 hope that I was not unfair with my deletions; on the other hand, there are probably some remaining names that will have to go because of nancial considerations of which I do not have sufcient knowledge, and others that are not suitable because of their specialties, given that it is preferred, for example, for the professor to be appointed to enjoy and to be successful in teaching those parts of mathematics that are closely related to physics. Perhaps I may assume that you, my honored colleague, also take some interest in this matter, after all quite important to our discipline, and that you will assist us with your extensive knowledge of the circumstances 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 mathematics at the University of Giessen, see Scharlau [1990, 111–116]. D. Schlimm / Historia Mathematica 40 (2013) 183–202 191

I also give my belated thanks for your kind letter of April 23. The prospect that you speak of, that there might 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 3.6. Pasch to Klein, December 13, 1887 Giessen, 13 December 1887 Highly honoured colleague, Given the great kindness of the extensive and most valuable advice that you gave in response to my still vaguely focused questions, the delay of my thanks would be doubly unjustiable if it had not been caused by university issues, which were quite unexpectedly loaded upon me in that past week and, when combined with the already present burden, completely distracted me from pursuing the question of hiring. Today I return to the matter and acknowledge rst of all that I, in fact, very much appreciated your particular attention to personal characteristics. As to your offer of more advice, I would like to make use of it, rst of all, regarding the following points. For Voss and Dyck it would be necessary to know what wages and benets they receive, at which point the Bavarian quinquennial benets also come into consideration. Given that I am lacking other connections, I would much appreciate your intervention, but only if doing so does 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. Of course, various suggestions have been received — mostly indirectly. Krazer and Stickelberger, about whose personalities I know almost nothing, were named. In particular the physicists will consider mechanics and related courses to be of great importance. In this respect, how do things stand with Hölder, F. Meyer, Schönies, and Schur? The great difculty for me lies in the overabundance of names which could be considered in this case. I was of course not allowed to include any Jewish colleagues in the list; otherwise scholars like Nöther and Hurwitz would not have been omitted.24 Once again, my warmest thanks for your kind cooperation and high regards from Yours M. Pasch 3.7. Pasch to Klein, December 22, 1887 Giessen, December 22, 1887 Highly honoured colleague, I do not want to delay any longer in thanking you for your kind advice, and especially taking the trouble to make the inquiries in Munich yourself.25 Now the prospect of being able to consider Voss is, in the given circumstances here, indeed a remote one. The main point, however, is that I must put aside my job of making 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 of the University of Göttingen, which took place on August 7–9, 1887, but which Klein did not attend. The celebration is mentioned in 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 been a postdoctoral student and Dyck a doctoral student of Klein. 192 D. Schlimm / Historia Mathematica 40 (2013) 183–202

This situation is the outcome of the unfortunate way this matter got started, which made it necessary to wait for the government’s decision on a preliminary question. I reserve the right to be allowed to return to all of these things later and remain in the meantime, with the best wishes, Yours M. Pasch 3.8. Pasch to Klein, February 17, 1888 Giessen, February 17, 1888 Highly 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 chair for mathematics; I held the latter as a personal appointment. Generally, people thought that this relationship would 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 prevent the resolution in question. This made necessary negotiations at the faculty level, which the dean did not really help with, but which nally led to a resolution of the matter. The conclusion was reached only last week, and the proposals could then be made soon. The government approved the retention of two chairs, however — as was expected based on the local circumstances — with the requirement that as little as possible would be spent on salaries. We thus had to abandon the idea of considering already chaired professors, with the exception of Staude, whom Schröter repeatedly recommended strongly, also with the remark that he only earns 2400 Rubel and an insignicant amount of extra income, and that he will make any sacrice 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 proven himself 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. The attendance is supposed to have gone down considerably in Berlin as well.27 However, if Netto would not accept, the chances of getting Staude would be certain. The determining factor for Meyer was that he translated Betti28 and lectured on mathematical physics, astronomy, and the like. Unfortunately, Schur’s interests did not meet our needs. The suggestions still need the approval of the whole senate, which we expect will be given on the 25th of the month. For this reason I must request that this communication be treated as condential until then. I once more take the opportunity to give my most heartfelt thanks for your generous assistance, and remain, with highest regards, Yours M. Pasch 3.9. Pasch to Klein, October 19, 1891 Giessen, October 19, 1891 Highly honoured Colleague, I am taking the liberty of sending you the enclosed pages for your Annalen29; their occasion is described in the introduction. The presented view is quite trivial; I nevertheless took such a long time with the nal version 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 to 1888. 27 Netto was professor without chair in Berlin since 1882. 28 Betti [1885]. 29 Pasch [1892]. D. Schlimm / Historia Mathematica 40 (2013) 183–202 193

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 andvol.7,p.53232 for the time being, since I would rather like to interpret these passages in a different way now.33 I also agree with most of 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 in the axioms, however, are inexact, and thus so are the axioms themselves. These latter can, however, only be used purely logically if they are presented as being exact. By further working on the axioms and seeking geometric 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 with regard to somewhat complex phenomena. However — and here our views apparently diverge — the use of gures is merely a facilitation of the work; otherwise, the work would exceed our powers, or at least would progress much too slowly, or would not progress far enough. The consideration must be possible even without the gures, in other words: that which is derived from the gures must already be contained in the axioms, for otherwise the axioms are not complete. It is precisely this characteristic that I wanted my basic propositions to have, and I therefore had to admittedly increase their number (see Geometrie34 p. 6 bottom, pp. 43–45, pp. 98–100). The fulllment of this requirement is possible, and thus the requirement is no doubt justied. Proven, in geometry as in analysis, is only that which can be derived from the axioms alone. 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 on pp. 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 simple principles. Mr. Schur wrote to me that he missed these small notes, because at the time he was moving to Dorpat. It has been noticed, not only by me but also by others,39 that Lindemann does not refer to my work on geometry 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 mentions Klein [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 difcult 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 have known at all Pasch’s book on projective geometry, where everything is treated more thoroughly, albeit boringly. It surely is more convenient to ignore such works.’ (‘Eine andere komische Sache ist die, dass er das Buch von Pasch über projektive Geometrie garnicht gekannt zu haben scheint, wo alles viel gründlicher, wenn auch langweilig behandelt wird. Es ist freilich bequemer solche Schriften 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]. 194 D. Schlimm / Historia Mathematica 40 (2013) 183–202

Finally, I would like to express my regret for my absence in Halle,41 where so many interesting things were presented — my presence was necessary here. I remain, with best regards, Yours M. Pasch

3.10. Pasch to Klein, April 21, 1902

Giessen, April 21 1902 Highly honoured colleague, I have barely been able to look over the rst quarter of your latest publication, which is based on the lectures from the summer of 1901.42 However, I did not want to delay any longer in sincerely thanking you for kindly sending it. You must have known that these expositions would interest me very much. My humble self is entirely in agreement with you about emphasizing their importance again and again. Let us hope that this view gets through to the mathematicians and then inuences the physicists, and maybe also the philosophers. Like you, I cannot ascribe any greater certainty to spatial imaginations (p. 7) than I can to empirical measurements and the like. Imagination allows us only to recall something we have seen, or to create shapes, whose parts we have (or could have) seen; it cannot prove anything. Abstract geometry arguably arose 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, practical geometry just has to do with what the unaided eye sees, what the unaided senses perceive at all. If one uses a microscope or the like, then things become quite complicated. But, in any case, geometry proper does not know incommensurable segments (p. 21); rather, it is only the additional, articially constructed one that does. The latter creates a new concept, that of ‘mathematical’ points; according to my understanding of it, this does not require any postulates, but denitions are sufcient. Thus, I am not in a position to say: a point has no spatial extension (p. 15); here I don’t know any denition 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 investigations in this direction. It is quite difcult to nd interest and appreciation for this among ordinary students. Highly honored colleague, please accept once again the greatest thanks and highest regards from Yours M. Pasch

Acknowledgments

The correspondence from Pasch to Klein is reproduced with kind permission from Dr. Helmut Rohlng, Nieder- sächsische Staats- und Universitätsbibliothek Göttingen, Abteilung Spezialsammlungen und Bestandserhaltung. The original transcriptions were made by Irmgard Pientka. Rachel Rudolph helped in correcting these transcriptions and produced rst drafts of the translations. Katrina Sark helped with improving the translations. The author would also like to thank June Barrow-Green, Douglas Campbell, and two anonymous reviewers of this journal for their helpful comments. 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]. D. Schlimm / Historia Mathematica 40 (2013) 183–202 195

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 Erlanger Berichte 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 die Schriften 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 Ihnen erhielt (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 dem Nachweise 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 behaftet sind, für eine mehr oder weniger verbreitete. Ich erblickte z. B. in der Anerkennung des Umstandes, daß die Anschauung von der Curve und der Begriff der Function sich nicht decken, den Grund, weshalb die Functionentheorie sich von den sog. geometrischen Beweisen frei zu machen gesucht hat. Inzwischen fand ich auch bei Lipschitz (Differential= und Integralrechnung § 40) Aeußerungen über die Ungenauigkeit aller Maßbestimmungen 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 Unterlassene bei einer demnächstigen geeigneten Gelegenheit nachzuholen. Ihre Bemerkung, daß die damals von Ihnen ausgesprochene Ideen auf heftigen Widerspruch gestoßen sind, läßt mich annehmen, daß auch meine Schrift wenig Zustimmung nden wird. Um so mehr freue ich mich, von Ihnen zu hören, daß Sie selbst auf den Gegenstand zurückzukommen gedenken. Mit vorzüglicher Hochachtung Ihr 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 verpichtet. In- zwischen habe ich den letzten Correcturbogen einer kleinen Schrift erhalten, welche bei Teubner unter dem 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 selbst ins 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 die Gelegenheit 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 Inhalt desselben schon zur Zeit der Hauptredaction meiner „Vorlesungen“ (1877/78) nur noch unvollkommen im 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. 196 D. Schlimm / Historia Mathematica 40 (2013) 183–202 in seinen Vorlesungen den größten Werth auf eine arithmetische Beweisführung gelegt hatte, mag im Verein mit manchen Gesprächen zwischen Rosanes und mir in den 60er Jahren hauptsächlich dazu beigetragen haben, daß ich mir von dem Sachverhalt die Vorstellung bildete, die ich Ihnen neulich beschrieb. Deshalb hat 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 Ordnung gebracht sein wird. Auch ich muß Ihnen meinerseits gestehen, daß ich von den philosophischen Schriften fast keine gelesen habe; nach kurzen Leseversuchen schien es mir immer bald, als ob die Herren sich auf falschen Wegen befänden. Wenn ich nun meine persönliche Meinung sagen darf, so möchte ich einer allgemeineren Verbreitung Ihres Aufsatzes durch die Annalen das Wort reden. Die in Betracht kommenden Momente nden sich auf den wenigen Seiten zum Ausdruck und zur Anwendung gebracht. Ich selbst mußte mir ein enges Ziel stecken, da ich schon ohnehin sehr viel Zeit und Mühe brauchte, um Alles in ein |4 streng logisches System zu bringen, und bin daher zu krummen Gebilden gar nicht vorgedrungen. Unter Rücksendung Ihrer Sammlung verbleibe ich Ihr Hochachtungsvoll 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 Ihnen zu übersenden, ob Sie denselben einen Platz in den Annalen anweisen wollen. Wir würden dann darum bitten, daß dieselben hintereinander abgedruckt werden, und zwar zuerst der meinige über Collineation und Reciprocität, 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 meine im vorigen Jahre entworfene und bereits ziemlich verzögerte Arbeit nicht länger aufzuhalten, und daß er sich 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 den Gegenstand zurückkommen und gelegentlich auch die Dinge, in denen Ihre und meine Auffassung noch auseinandergehen möchten, präcisieren zu können. Doch müßte ich zu diesem Zweck mehr ungestörte Zeit zur Verfügung haben, als ich vorläug hoffen darf. Mit vorzüglicher Hochachtung verbleibe ich Ihr 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 von Vorlesungen entstanden, welche das Bedürfniß wesentlicher Vereinfachung gewisser Partien empnden D. Schlimm / Historia Mathematica 40 (2013) 183–202 197 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 denen Sie s. Z.45 vorangegangen sind, von Neuem hinweisen. Mit vorzüglicher Hochachtung Ihr ergebener 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 geeigneten Berufung treffen zu müssen. Durch Uebernahme einer Ersatzvorlesung und andrer Umstände vollauf in Anspruch 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 nun zunächst 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 dem Streichen nicht ungerecht gewesen bin; andererseits wird von dem Zurückbehaltenen wohl Mancher schon aus pekuniären Rücksichten, die ich nicht hinreichend übersehe, fortfallen müssen, Mancher wegen seiner Richtung 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 der betreffenden Verhältnisse zu unterstützen sich bereit nden lassen. Sie würden dadurch mich in erster Linie zu aufrichtigem Dank verpichten. Nachträglich noch meinen besten Dank für Ihren freundlichen Brief vom 23. April. Die Aussicht, von der Sie sprechen, daß in Göttingen im Sommer eine Zusammenkunft der drei Universitäten stattnden würde, hat sich zu meinem Bedauern nicht verwirklicht. Mit vorzüglicher Hochachtung Ihr ergebener M. Pasch. Dyck, Henneberg, Hettner, Kiepert, Krause, Lindemann, Franz Meyer, Netto, Runge, K. Schering, Schottky, Schur, Staude, Stickelberger, Wiltheiss

A.6. Pasch to Klein, December 13, 1887 [Klein 11, 181]

Giessen, 13. Dezember 1887. Hochgeehrter Herr Kollege! Gegenüber der großen Freundlichkeit, mit der Sie auf meine in der That noch wenig xierten Fragen mir ausführliche und höchst werthvolle Auskunft gegeben haben, wäre die Verzögerung meines Dankes dop- pelt unverantwortlich, wenn Sie nicht durch Universitätsgeschäfte, welche ganz unerwartet in der vorigen Woche mir aufgeladen wurden und im Verein mit der schon an sich vorhandenen Belastung mich von der Verfolgung der Berufungsfrage gänzlich abzogen, verschuldet gewesen wären. Ich komme nun heut nochmals auf den Gegenstand zurück und bestätige vor Allem, daß Ihre besondere Berücksichtigung der

45 Abbreviation for: seinerzeit. 198 D. Schlimm / Historia Mathematica 40 (2013) 183–202 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 wäre es nöthig, zu wissen, welche Gehälter und Nebeneinnahmen sie beziehen, wobei wohl noch die bayerischen Quinquenalzulagen 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 wohl schon des Gehalts wegen außer Betracht bleiben, er hat angeblich 4000 Rubel; vielleicht ist Ihnen zufällig Genaueres bekannt geworden. Natürlich sind mancherlei Empfehlungen — meist indirekt — eingegangen. So hat man Krazer und Stickelberger genannt, über deren Persönlichkeiten mir eben nichts Näheres bekannt ist. Auf Mechanik und verwandte Vorlesungen wird namentlich der Physiker Gewicht legen. Wie mag es in dieser Hinsicht mit Hölder, F. Meyer, |3 Schönies und Schur stehen? Die große Schwierigkeit besteht für mich in der Ueberfülle von Namen, die in dem hiesigen Falle genannt werden können. Nur habe ich natürlich keinen jüdischen Fachgenossen in das Verzeichniß aufnehmen dürfen, sonst hätten Gelehrte wie Nöther und Hurwitz darin nicht gefehlt. Nochmals meinen wärmsten Dank für Ihr liebenswürdiges Entgegenkommen und ergebenste Grüße von Ihrem M. Pasch.

A.7. Pasch to Klein, December 22, 1887 [Klein 11, 182]

Giessen, 22. December 1887. Hochgeehrter Herr Kollege! Ich will nicht länger zögern, Ihnen für Ihre erneute freundliche Auskunft und ganz besonders dafür sehr zu 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 eine geringe. Die Hauptsache aber ist, daß ich mein Geschäft, Vorschläge zu machen, bis nach dem Ende |2 der Ferien ruhen lassen muß und daher noch gar nichts abzusehen vermag. Es ist dieser Zustand die Folge der Art 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 Dinge später zurückkommen zu dürfen, und bleibe inzwischen mit bestem Gruß Ihr sehr ergebener M. 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. Gestatten Sie mir zunächst, eine frühere Andeutung näher auszuführen. Wir besaßen für Mathematik ein Ordinariat und ein Extraordinariat, das letztere hatte ich als persönlicher Ordinarius inne. Allgemein war man der Ansicht, 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 bestimmten Zurückweisung genöthigt hatte, benutzte den Umstand, daß er gerade Dekan war, |2 um die betreffende Entschließung zu verhindern. Dadurch wurden Fakultätsverhandlungen nöthig, die durch den Dekan nicht eben gefördert wurden, schließlich jedoch zu einer Regelung der Sache geführt haben. Der Abschluß ist eigentlich erst in der vorigen Woche erfolgt, und die Vorschläge konnten dann bald gemacht werden. Die Beibehaltung zweier Ordinarien hat die Regierung bewilligt, jedoch — wie den hiesigen Verhältnissen nach zu erwarten war — mit der Maßgabe, daß möglichst wenig Gehalt in Anspruch genommen werde. D. Schlimm / Historia Mathematica 40 (2013) 183–202 199

Von angestellten Ordinarien mußten wir sonach ganz abgesehen, mit Ausnahme von Staude, den Schröter wiederholt 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 naturwissenschaftlicher Seite 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 die Aussicht 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 vertrauliche ansehen zu wollen. Indem ich die Gelegenheit benutze, um nochmals meinen herzlichen Dank für Ihre be- reitwillige Unterstützung auszusprechen, bleibe ich mit vorzüglicher Hochachtung Ihr 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 für Ihre Annalen zur Verfügung zu stellen. Die darin dargelegte Auffassung ist ganz trivial; trotzdem habe ich mit der endgültigen Niederschrift so lange gewartet, weil ich die etwa zu erhebenden Bedenken wenigstens vor mir selbst zerstreuen wollte. Ebenfalls spät kommt der Dank, welchen ich Ihnen gleichzeitig für die freundliche Zusendung Ihrer Abhandlung zur Nicht- Euklidischen Geometrie vom vorigen Jahre abstatten wollte. Auf die letzten Sätze Ihrer 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. 532 zunächst nicht gebracht habe, vielmehr |2 diese Stellen jetzt in anderem Sinne auffassen möchte. Auch sonst schließe ich mich dem, was Sie auf den beiden letzten Seiten der Abhandlung sagen, in den meisten Punkten an. Der Inhalt der Axiome wird aus der Beobachtung (Anschauung als innerer Vorgang beruht auf Erinnerung an Beobachtetes) entnommen, die zu den Axiomen verwendeten Begriffe sind aber ungenau und mithin auch die Axiome selbst, diese lassen sich jedoch rein logisch nur dann verwenden, wenn sie als exakt hingestellt werden. Indem wir nun an den Axiomen weiter arbeiten und geometrische Sätze suchen, pegen wir 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 ohne solche 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 muß die Betrachtung auch ohne die Figuren sein, m. a. W.: das, was aus den Figuren entnommen wird, muß bereits in den Axiomen niedergelegt sein, andernfalls sind die Axiome nicht vollständig. Gerade diesen Charakter 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 nur das, was aus den Axiomen allein hergeleitet werden kann. Ich möchte allerdings in Hinsicht der äußeren 200 D. Schlimm / Historia Mathematica 40 (2013) 183–202

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 ausgeführt worden. Wie Herr Schur mir schreibt, waren ihm diese kleinen Noten entgangen, da er damals nach Dorpat übersiedelte. Nicht bloß mir selbst, sondern auch Anderen, ist es aufgefallen, daß Lindemann in der Raumgeometrie meine 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 hören war, gefehlt zu haben — meine Anwesenheit war hier nöthig gewesen —, bleibe ich mit ausgezeichneter Hochachtung Ihr ergebener M. 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 zwar kaum das erste Viertel durchsehen können, will aber doch den Ausdruck meines herzlichen Dankes für die liebenswürdige Übersendung nicht noch länger verzögern. Sie durften ja mit Sicherheit annehmen, daß mich diese Darlegungen in hohem Maße interessieren würden. Wenn Sie deren Wichtigkeit immer und immer wieder betonen, so haben Sie meine |2 Wenigkeit ganz auf Ihrer Seite. Hoffen wir, daß diese Anschauung bei den Mathematikern durchdringt und die Physiker dann beeinußt, vielleicht auch die Philosophen. Mit Ihnen kann ich dem räumlichen Vorstellen (Seite 7) keine größere Genauigkeit beimessen, als dem empirischen 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. Die abstrakte Geometrie ist wohl aus demselben Bedürfnis hervorgegangen, wie der Wunsch, Naturgesetze als absolut genau hinzustellen (Seite 42, 44 u. s. w.). Mitgewirkt hat wohl auch die Art (S. 36), wie sich in uns die Vorstellung eines Con|3tinuums bildet. Eigentlich hat die praktische Geometrie es wohl nur mit dem zu thun, was das unbewaffnete Auge sieht, der unbewaffnete Sinn überhaupt wahrnimmt. Nimmt man 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 hierüber würde es keines Postulats bedürfen, Denitionen 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 besonders erfreulich gewesen. Leider bin ich erheblich kritischer geworden, und das hat mich an Vollendung weiterer Überlegungen in diese Richtung gehindert. Bei den gewöhnlichen Studierenden Interesse und Verständniß dafür zu nden, ist gar schwer. Empfangen Sie, hochgeehrter Herr Kollege, nochmals die Versicherung besten Dankes und vorzüglicher Hochschätzung von Ihrem ergebenen M. Pasch D. Schlimm / Historia Mathematica 40 (2013) 183–202 201

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