Abstract
The earliest extant mathematical books from China contain a lot of problems and data about grains. They also betray a close relationship with imperial bureaucracy in this respect. Indeed, these texts quote administrative regulations about grains. For instance, the Book on mathematical procedures 筭數書, found in a tomb sealed ca. 186 BCE, has a section in common with the “regulations on granaries” from the Qin statutes in eighteen domains, known thanks to slips excavated at Shuihudi. Mathematical writings also deal with official vessels used to measure grains. They cast light on statements from, and practices evidenced by, official histories and administrative documents. This article addresses the following issues. Which information about the concrete management of grains can we derive from mathematical writings in relation to administrative documents? Which data can we find in these writings about continuities and changes in the management of grains in the time span between the Qin and Han dynasties? In particular, how can we account for the fact that in a later mathematical text, namely, The Nine Chapters, probably completed in the first century CE, there was a change in the form in which the data about grain equivalences were given, by comparison with the “regulations on granaries”? Finally, what do our conclusions imply with respect to the nature of the earliest extant mathematical writings. In this article, we gather the various types of statement that mathematical writings contain about grains and offer several elements of interpretation for the “regulations on granaries” and the related text in The Nine Chapters. From this perspective, we offer several hypotheses about the management of grain in the Qin and Han dynasties.
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Notes
When we speak of early China in this article, this refers mainly to the period between the fourth century B.C.E. and the third century C.E. During this time period, the state of Qin (one of the “Warring States”) defeated the other major states and unified the Chinese empire, establishing in 221 B.C.E. the so-called Qin dynasty. This dynasty was short-lived—it lasted between 221 B.C.E. and 206 B.C.E.—and was quickly overturned, giving way to the Han dynasty (206 B.C.E.–220 C.E.). The regent Wang Mang 王莽 (45 B.C.E.–23 C.E.) attempted to overthrow the Han dynasty and establish a new dynastic rule, the Xin 新 dynasty. The latter was short-lived (9–23 C.E.), and Han rule was reestablished in 25 C.E. This event divides the Han dynasty into two periods: the “Western Han dynasty” (206 B.C.E.–9 C.E.) and the Eastern Han dynasty (25–220 C.E.).
Let us recall the fact that the mathematical manuscripts that were excavated, including those discussed in this article, were all written on bamboo slips. Originally, the slips were interlinked by vegetable fiber cords to make a book. However, the cords decomposed during the two thousand years they remained buried underground. A key operation that archeologists face in order to publish critical editions of these documents is to attempt to recover the order in which the slips were originally arranged.
Qin statutes, slips 41–43, see (Peng Hao 彭浩 2001, 80), note 1.
When we present the various manuscripts below, we shall provide reference to publications that provide the basis for making this point.
For a critical edition and a translation into French of The Nine Chapters, as well as the commentaries handed down with the book through the written tradition, (see Chemla and Guo Shuchun 郭書春 2004).
On these points, see K. Chemla, introduction to chapter 2, in Chemla and Guo Shuchun (2004, 201–205).
(Ban Gu et al. 1962, 966–967) deals with measuring units of length; the whole system of measuring units is presented in chapter 21a. In the present article, we shall number all the passages quoted to enable us to refer easily to documents under discussion.
In what follows, when we quote The Nine Chapters, we use small capitals to distinguish the book from the commentaries, which we render in lower case letters.
This identification was provided by Wang Guowei 王國維, “Shi er 釋二” in (Luo Zhenyu 羅振玉 and Wang Guowei 1993, 164). Wang Guowei offered this interpretation on the basis of occurrences in bamboo slips from the Han dynasty (Juyan). Yang Lien-sheng 楊聯陞 (1950), quoted from (Yang Lien-sheng 2006, 8), confirmed this interpretation and established its relationship to this passage of The Nine Chapters.
The Grand Dictionary of Chinese characters 漢語大字典, p. 3163 (Sichuan cishu chubanshe and Hubei cishu chubanshe, 1984–1990) gives 糵 nie as equivalent to 糱 nie, relying on the Shuowen jiezi 說文解字 dictionary and its 1815 commentary by Duan Yucai 段玉裁. This suggests the interpretation that the character refers to sprouts of either barley or beans. The value with which the state of the grain is associated in the table seems to support this interpretation, as we can state on the basis of the meaning of the table expounded below.
For a study of the evolution of crops in northern China, see (Li Qiufang 李秋芳 2012). Although millet was the most common plant in the north, note that both millet and paddy could be found in the north from a very early period onwards.
In contexts in which we believe that su also designates the unhusked state of millet in particular, we shall translate as follows “unhusked grain (millet).” Whether li mi, bai mi and zuo mi were states of grain specifically attached to millet or could be attached to other types of grain is a topic awaiting further research. We shall thus translate them as states of grain, as is done in the table, adding “(millet)” in brackets. Zou Dahai (2005, 538–540) discusses the various terms used to designate these states of grain in different documents and attempts to account for why The Nine Chapters differs from several other documents in this respect.
Peng Hao (2012, 196) obtained a similar conclusion by studying the occurrence of the same term in various contexts in the mathematical manuscripts. We return to this question below.
See, for instance, (Shen Kangshen et al. 1999, 141–142), in particular note 1.
On these statutes, see the introduction of (Zhangjiashan er si qi hao Han mu zhujian zhengli xiaozu 張家山二四七號漢墓竹簡整理小組 Group of editors of the bamboo strips from the Han tomb 247 at Zhangjiashan 2001, 1), a book containing an annotated critical edition of the text (pp. 131–210).
As mentioned above, the parallel was first noticed in (Peng Hao 2001). See the edition of Edicts and statutes of the Qin dynasty, especially the slips 41–43, in (Shuihudi Qin mu zhujian zhengli xiaozu 睡虎地秦墓竹簡整理小組 Group of editors of the bamboo strips from Qin tombs at Shuihudi 1990, 29–30). Hulsewé (1985) contains a complete English translation of these statutes. (Ikeda Yûichi 池田雄一 2008, 147–195) (chapter “Kohoku Unbō Suikochi shinbo kanken 湖北雲夢睡虎地秦墓管見) (Personal ideas about the Qin tombs at Shuihudi (Yunmeng, Hubei)”) suggested that Edicts and statutes of the Qin dynasty deals with regulations related to officials’ management of several types of activities as well as to duties they had to carry out. Ma Biao 馬彪 (2013, 88–102) (chapter 6 "Unbô Soôjô no nijyû no seikaku 雲夢楚王城の二重の性格 (The double nature of the Chu royal city at Yunmeng") further puts forward the idea that the Edicts and statutes of the Qin dynasty are almost all about statutes and regulations for local administrative offices dispatched by the court.
Peng Hao (2001, 80), note 3, comments that the repetition mark was omitted. Evidence for this is provided by the parallel text in the Qin statute. Unless otherwise stated, here we follow Peng Hao’s edition of the text.
Peng Hao (2001, 81), note 7, explains that he reads here on the slip another character (
), which was erroneously copied instead of zuo, which we use here. On p. 5, however, he seems to suggest that the former is borrowed to write down the latter. Cullen (2004, 130) takes up the former character, without mentioning a difference. Chôka zan kankan Sansûsho kenkyûkai 張家山漢簡『算數書』研究会編 The research group on the Han bamboo strips from Zhangjiashan Book of Mathematical Procedures (2006, 52) suggests reading the character on the slip as 糳. Dauben (2008, 137) quotes the same text for this section of the slip. We also follow the reading of the Japanese edition. The latter three publications provide translations. Note that we do not punctuate in the same way as previous authors, even though the Japanese translation, the Chinese one, and the two English translations follow this punctuation. All these authors discuss the meaning of the grains and compare these slips, on the one hand, with the text of the Qin regulations and, on the other hand, with The Nine Chapters (Chôka zan kankan Sansûsho kenkyûkai 2006, 52–56, Cullen 2004, 66–68, Dauben 2008, 137–139). In particular, they quote the corresponding text of the Qin regulation for the last sentence here. It reads “稟毇粺者, 以十斗為石 bing hui bai zhe, yi shi dou wei dan.” In this parallel text, the character of the Book of Mathematical Procedures under discussion is, in fact, replaced by bai 粺. Peng Hao (2001, 80–81), note 3, observes that, except for this set of slips quoted here, zuo never recurs in the Book of Mathematical Procedures. Elsewhere the state of millet corresponding to the same degree of fineness is always referred to as bai, as in the version of the sentence under discussion found in the Qin regulation.Peng Hao (2001, 80), note 2, rightly suggests that this measuring unit dan, as well as the corresponding one below, designate a measuring unit for weight.
We see here that the relationship between three different degrees of fineness in the husking, namely, li mi, zuo mi, hui mi, is defined by the ratios between, respectively, 10, 9, and 8 dou. The table included in The Nine Chapters contains numerical values with the same ratios for, respectively, li mi, bai mi, and zuo mi (namely, 30, 27, 24). It thus seems that the definition of zuo mi changed between the time of the Qin statute and the moment when The Nine Chapters was composed (Peng Hao 2001, 80–81, note 3, and Zou Dahai 2005). However, it is worth noticing that these occurrences of the expression zuo mi are the only occurrences in all the manuscripts. By contrast, in all statements that assert a ratio of 10 to 9 between two degrees of husking in the mathematical manuscripts, from the oldest one known, these states of grain are referred to as, respectively, li mi and bai mi, that is, by means of the same expressions as those found in The Nine Chapters. See for instance, the Book of Mathematical Procedures, slip 98 (Peng Hao 2001, 84), or the Qin manuscript discussed below, Mathematics, (Zhu Hanmin 朱漢民 and Chen Songchang 陳松長 zhubian 主編 (gen. ed.) 2011), slip 0822 (editors’ number 94), in which one reads: “one sheng of li (mi) makes nine tenths sheng of bai (mi)
” (see also Xiao Can 肖燦 2011, 56). These facts indicate that further research is required on the history of the use of the term zuo mi and the state of grain it designated at different time periods.This “unhusked grain su” must be “rice.” We encounter evidence below that the expressions “unhusked rice 稻粟” and “husked rice 稻米” were used.
Peng Hao (2012, 195) shows that the text as preserved in the Book of Mathematical Procedures is correct in this respect. He points out further that in contrast to the system of states of grain deriving from millet, the system of states of grain deriving from paddy had only two degrees: one called mi (Mathematics, slip 0756, calls this state 稻米 dao mi “husked rice”) and a second one called hui can mi or simply hui mi.
Peng Hao (2001, 80–81) provides an annotated edition of the text quoted above, and as mentioned, in his note 1, he notes the parallel between the Qin statute contained in slips 41–43 in question and the Book of Mathematical Procedures. Peng Hao (2012, 194–196) summarizes different scholars’ contributions to the establishment of the text of this Qin statute and shows how one can rely on the Book of Mathematical Procedures to restore the text as transmitted in the extant copy of Qin statutes. See (Hulsewé 1985, 42) for a translation into English of the Qin statute parallel to the paragraph of the Book of Mathematical Procedures under discussion. Regarding the final sentence of the text, (Peng Hao 2012, 199), in conformity with his punctuation of the text, suggests following the editors of the Qin regulations in reading huizuo as a single word, designating states of millet with finer degrees of husking. He interprets this sentence in the context of Qin regulations, as referring to special ways of measuring rations of grains for officials in mission, in relation to their degrees in the hierarchy. Chôka zan kankan Sansûsho kenkyûkai (2006, 52), Cullen (2004, 67–68), and Dauben (2008, 139) interpret the text as referring to two types of grain. Cullen admits that the text is difficult to understand, whereas Dauben suggests that the two types of grain could refer more specifically to millet. Peng Hao (2012, 195–196) establishes that the finer state of rice can be designated as hui, that is, with the same term as the finer state for millet (see previous footnote). The two terms zuo and hui could thus either designate all states of rice and millet finer than the coarsely husked state, or alternatively, in each of the series deriving one from rice and the other from millet, respectively, the second state of grain after the state of “coarsely husked grain.” For the interpretation of dan, see below.
This question was raised by K. Chemla in the introduction to chapter 6 (Chemla and Guo Shuchun 2004, 477–478).
The structure of the text is interesting, but this is the topic of another publication.
Peng Hao (2001, 89), note 1, thinks that after the character “part fen 分,” used to express the fraction, the measuring unit of capacity 升 sheng, which occurs after the integral part of the result, was omitted.
See the whole text in Peng Hao (2001, 89), slips 113–114.
Peng Hao (2001, 88–89), slips 109–110. We use the same convention as above for the translation of names of grains.
One key reason for this is that it clearly uses the character 券 quan in a way quite similar to the use of quan by another mathematical manuscript, and only this one. However, the slips of this other manuscript showing that use of quan were only published in 2008. See (Chemla and MA 2011).
Several papers devoted to different aspects of the book Mathematics appeared in 2009: (Xiao Can and Zhu Hanmin 2009a, b, Zhu Hanmin and Xiao Can 2009). In 2011, the contents of the book were made public through (Xiao Can 2011) and the critical edition of Mathematics, with reproductions and annotations, published in (Zhu Hanmin and Chen Songchang (gen. ed.) 2011). Xiao Can and Zhu Hanmin (2009b) focus in particular on the units of measurement for quantities of grain in Qin China. Since the topic is close to that of our article, let us sketch their relevant results. In their terms, they mainly discuss the conversion, for grains, between measuring units of volumes and weights. Their conclusion is that Mathematics records many coefficients expressing the ratios between volume and weight for grains. Moreover, they suggest that in the third century B.C.E., before the unification of the Chinese empire, people usually measured grains by means of measuring units of volume and then used these coefficients to compute the related weight. We show below why we have come to different conclusions.
See (Chen Songchang 2009, 86–87). According to the description given there, the statutes seem to be comparable to those found in Shuihudi and mentioned above.
See, for example, Mathematics, slip 0939 (editors’ number 11) (Zhu Hanmin and Chen Songchang (gen. ed.) 2011, 4). This slip also confirms that in this system one dan is equal to ten dou.
This article does not provide the scope to analyze extensively the various formulations of the rules of three contained in these books (several different formulations in the same book, as well as similarities and differences across books), their relationships with each other, and what these facts reveal in terms of conceptual history. Such a systematic study is interesting and important. It will be the topic of another publication.
Zhu Hanmin and Chen Songchang (gen. ed.) (2011, 13).
The expression by means of which one refers to unhusked millet here is different from that used in the Book of Mathematical Procedures (Text 9). Note that the manuscript writes down these tables by means of registers, which we indicate with spaces. In the translation, each clause is translated in a separate paragraph.
It is this development that (Xiao Can and Zhu Hanmin 2009b) interpreted as the introduction in third century B.C.E. China of the concept of specific gravity.
The term tong 童 that we translate as “heap” and interpret as “granary” occurs in The Nine Chapters as well as in the Book of Mathematical Procedures. Xiao Can (2011, 85), note 1, and Zhu Hanmin and Chen Songchang (gen. ed.) (2011, 127), note 1, suggest relying on the third century commentator Liu Hui’s gloss on this character in The Nine Chapters to interpret it. Liu Hui writes: “凡積芻有上下廣曰童 Every time when, piling up forage grass, in the (shape constructed) there is a higher width and a lower width, one calls (the shape) tong.” We suggest that the character tong could be a character borrowed to write chong 重 “heap.” The meaning could thus have been extended to designate the “height of the granary” itself.
For the moment, we translate rong loosely. We shall return to this concept below, as its interpretation demands revision. Note that we also translate “jichi the chi of the volume” loosely. The precise interpretation is the topic of another publication.
Zhu Hanmin and Chen Songchang (gen. ed.) (2011, 25). The procedure solving the problem is not complete, since the editors could not identify the following slip among the extant slips. A similar mathematical problem can be found on slips 175 and 176 (editors’ numbers), Zhu Hanmin and Chen Songchang (gen. ed.) (2011, 24, 126).
The explanation of the system was first published in (Li Jimin 李繼閔 1998, 768–778). We are grateful to Zhu Yiwen for having called our attention to the first publication explaining a feature now universally accepted.
Peng Hao (2012, 199–201) endorses this interpretation, and this is a key point where our explanations of the measuring units used for grains diverge. In fact, the weights he believes these statements refer to are not coherent. On p. 200, he suggests at the same time that the expression in Text 14 refers to the quantity of unhusked millet corresponding to a weight of the plant with stalks and leaves (he establishes that heshu 禾黍 refers to this on p. 195) and also that the weight is that of the unhusked grain (1 dan would be the weight of 2 chi 7 cun or 16 dou 2/3 dou of unhusked millet). However, on p. 196, he showed these weights were different (a weight of 1 dan of harvested plant yields 16 dou 2/3 dou of unhusked millet, after deletion of stalks and leaves). As a result, Peng is compelled to conclude that the similar clauses found in The Nine Chapters either derive from compilers’ mistakes or refer to artificial problems. Further, he is compelled to think that the commentaries on The Nine Chapters also made the same mistake. In the interpretation we suggest, all these sources are perfectly clear and coherent.
Zou Dahai (2009, 513–515) is the publication that went farthest in the direction explored in this article. As we suggest here, Zou Dahai understood that in addition to being a measuring unit for weight and capacity, dan had a third meaning in relation to the measurement of cereals and on the basis of capacity. However, he did not address the question of the meaning of this measuring unit. Nor did he establish the link with the similar phenomenon related to hu. Further, Zou Dahai rightly perceived that a state of husked millet and husked rice allowed this third meaning of dan to correspond with the capacity amount of 1 dan, without, however, bringing to light the specific part played by “standard husked grain” in structuring the system of grains (see below). Zou seems also not to understand the use of this measuring unit as we do for states of grain finer than the “standard husked grain.” Finally, because Zou treats volume and capacity together, he did not address the question of the part played by volume, as opposed to capacity, in the statement of norms relating to grains. Various clues indicate that the management of grain in early imperial China made use of distinct types of measuring units in various instances. Zou Dahai seems to think that capacity measurement played a central role, a point we question below.
Many states of the grain that appear in the table contained in The Nine Chapters (Text 4) are not mentioned in the Book of Mathematical Procedures. On the other hand, the state of “highly refined rice hui can mi” seems not to occur in The Nine Chapters.
As the commentator explains later on, the circumference here designates the half-circumference, which is the lower dimension of the heap, which has the shape of a half-cone.
As the commentator explains later on, the circumference here designates a quarter of the circumference of a cone. It is the lower dimension of the heap, which now has the shape of a quarter of a cone.
A critical edition with annotations can be found in (Chemla and Guo Shuchun 2004, 446–453). Here, we simply summarize the main point of the argument, since it provides a foundation for the following part of the article. See the discussion in relation to the problem of hu in K. Chemla, introduction to chapter 2, in (Chemla and Guo Shuchun 2004, 201–205).
On the analysis of Jia Gongyan’s computation in the context of the subcommentary on Zheng Xuan’s commentary on the Rites of Zhou, see Zhu Yiwen, forthcoming. We are grateful to Zhu Yiwen for drawing our attention to this piece of evidence. We shall see below that in the case of “coarsely husked millet,” the unit of capacity hu coincides with the unit of value hu.
More precisely, the values used in the rules of three have exactly the same ratio in all the books. However, as we stressed above, the volume that Mathematics associates with one dan of barley slightly differs from the one The Nine Chapters mentions.
Peng Hao (2012, 196) noticed the relationship between these two data. He emphasized the simplicity in the management of grains that such a relationship entailed.
On Shang Yang, see, for instance, (Vandermeersch 1965), in particular chapter 1.
For a discussion of Shang Yang’s vessel with a photograph of the vessel and its inscriptions, as well as a transcription of the inscribed text, see (Qiu Guangming et al. 2001, 165–166).
See the translation and annotation of passages in (Chemla and Guo Shuchun 2004, 182–185, 452–453, 456–457).
This translates the term tiaopang, which commentators interpreted as the distance between the inner circumference of the vessel and the vertices of the fictitious square inscribed in its center (“what exceeds/goes beyond on the side”). Note that the hao is about 0.02 mm, which implies that the order of magnitude of accuracy probably has no relation to the actual design of the vessel.
For a photograph of the vessel and a discussion of its inscription, see (Qiu Guangming et al. 2001, 221–222). We return below to the term rong used here to designate the capacity.
Peng Hao (2012, 199–200, 201) also uses the tool provided by volumes to analyze the text. In Peng Hao (2013, 30-01-2013, at 10:42), he comes back to this point. For this, in both publications, he relies on Shang Yang’s parallelepipedic sheng. However, our conclusions are different. With respect to our conclusions, see the explanations above and below. Peng Hao, for his part, uses the relationship between the first measuring unit for weight dan and the corresponding volume of unhusked grain to propose a hypothesis: In his view, unhusked grain was the foundation from which the whole system of grain was derived on the basis of determining conversions between unhusked grain and all the other grains and states of grain. One key difference, as will emerge below, is that he does not establish any connection between Shang Yang’s parallelepipedic sheng and coarsely husked grain.
This statement is generally true, with the exceptions mentioned earlier. First, the state the text calls zuo mi (finely husked grain (millet)) is called bai mi in the rules of three inserted in all three texts. However, in The Nine Chapters, the term zuo mi is used to refer to what is in this text called hui mi. Secondly, in fact the volume associated with barley, or soy, mung, and hemp beans in Mathematics is 2,400 cun and not 2,430 cun, as it is the case in Text 7 and Text 16 from The Nine Chapters. Finally, for the moment, the final statement of Text 7 cannot be explained in these terms.
The same conclusion holds true in The Nine Chapters. If we observe the problem similar to this one, which is included in Chapter 5 (problem 5.27), the problem deals with a granary and relates volume and measuring units of value. It is again the term rong that designates the amount in measuring units of value, this time designated as hu.
Peng Hao (2001, 105–106). We opt for the interpretation of xuan suggested by Peng Hao in first page note.
This expression refers to the volume of the parallelepiped, which is to be divided by 36 to yield the volume of the cone sought.
This is one of the main points where our interpretation differs from that presented in Peng Hao (2012, 199–200). In that passage, Peng Hao explains why he believes that these dan must be interpreted as measuring units for weight. As we explained above (see note 47), we think this interpretation is not feasible. We do not understand how this interpretation fits with Peng Hao’s interpretation of dan as referring to “capacity” in assessing millet (p. 197).
See some samples of evidence in slips 13 or 16, in document W2 from Wayen-torei (Loewe 1967, 334–335) (volume 1, plate 41, and see pp. 317–331 for a general introduction on this set of slips). Let us summarize the discussion that has been developed since the 1950s about this vexed issue. Lao Gan mentioned the existence of the “small dan” and the “large dan” as well as the problem raised by their interpretation as early as 1950 (Lao Gan 勞幹 1950). He put forward two ideas. First, he suggested that “small dan” was a measuring unit used by officials, in contrast to “large dan,” which was used by ordinary people. Secondly, he offered several examples taken from Han bamboo slips, which evidenced conversions at the time between “small dan” and “large dan.” At the same time, (Yang Lien-sheng 1950) (quoted here on the basis of the reprint in Yang Lien-sheng 2006) discussed this problem and made three further points. First, in his view, “small dan” and “large dan” did not correspond to different vessels but were only nominal units used in computations, being both equal to a single actual measuring unit of capacity: the dan. Secondly, “small dan” referred to the use of dan when assessing unhusked grain (millet), whereas “large dan” designated the use of dan in relation to husked grain (millet). Thirdly, he pointed out the ratio of 5 to 3 as that between “large dan” and “small dan,” respectively. Later authors criticized the association he made between types of grain and types of dan. In 1951, Utsunomiya Kiyoyoshi 宇都宮清吉 published “Zoku Kanshi hyakkan jyuhôrei-kô sairon 續漢志百官受奉例考再論 Resuming discussion on the Salary of Government Officials according to the Sequel to the Treatises of (the History of Later) Han (Dynasty),” which we quote from the reprint (Utsunomiya 1955). He dealt with this issue in a passage there (Utsunomiya 1955, 221). To begin with, he suggested that different vessels were used during the Han dynasty to measure different species and states of grains. Accordingly, the same name (dan) could refer to different actual measurement units. “Small dan” and “large dan” are an example thereof, but in his view, this phenomenon also affected dou and sheng. Moreover, he argued that to measure unhusked grain (millet), one would use “large dan,” whereas for husked grain (millet), one would use the “small dan.” Later on, Yang Lien-sheng accepted the view that there may have been two distinct measuring units. Finally, Utsunomiya put forward the hypothesis that using such vessels would enable officials to carry out conversions. Subsequently, the problem was actively debated on these bases, without reaching a conclusion generally accepted. The reader can find a summary of the debate in (Loewe 1961/1962) and (Loewe 1967, vol. 1, 330–331).
For example, a vessel bearing an inscription from the Qin dynasty and measuring 1 dan 6 dou in Qin measuring units is mentioned in (Qiu Guangming et al. 2001, 183–184). Qiu Guangming et al. suggest that this vessel could embody the da dan “large dan.”
(Chen Shou 陳壽 1959, 55), Chapter 1, Book of Wei. Annals of Emperor Wu.
See above, note 17.
At the beginning of the “Monograph on pitch-pipes and the calendar” of the History of the Han, Ban Gu, its author, makes clear that he has used texts by the scholar who helped Wang Mang to implement his reforms, namely, Liu Xin 劉歆 (Ban Gu et al. 1962, 955).
Peng Hao (2001, 83). Mathematics also provides evidence that measuring units of weight are used to express amounts to be paid for tax. However, this does not occur in relation to grains, but in relation to rather coarse stuff. See, for instance, Zhu Hanmin and Chen Songchang (gen. ed.) (2011, 41). This use of weight measuring units can be compared to their use to measure amounts of cereal crops that were just harvested, as evidenced in Text 7 (“Regulation on cereal crops”). It must be distinguished from the use of weight measuring units explained below, in relation to grains.
), which was erroneously copied instead of zuo, which we use here. On p. 5, however, he seems to suggest that the former is borrowed to write down the latter. Cullen (
” (see also Xiao Can 肖燦 References
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Communicated by: Han Qi.
The research leading to these results received funding from the European Research Council under the European Union's Seventh Framework Programme (FP7/2007–2013)/ERC Grant Agreement No. 269804 “Mathematical Sciences in the Ancient World (SAW).” The results presented in this article were obtained through a discussion between the two authors that started in the summer of 2011. These results were presented first on March 9, 2012, and subsequently on March 20 and 26, 2013, within the framework of the seminars and conferences of the SAW Project (http://sawerc.hypotheses.org/workshop-cultures-of-computation-and-quantification/conference-cultures-of-computation-and-quantification-in-the-ancient-world). We would like to thank the audience for their questions, which helped us refine our argument. Most of all, we are grateful to the Director of the SPHERE research group, who graciously lent us his office for several weeks, as 18 months after the beginning of the ERC project the University had still not provided any space for its activities. We would also like to thank Annick Horiuchi and Karen Margolis for kindly helping us prepare the final version of this article. Han Qi’s and Jeremy Gray's editorial advice was invaluable, and we have pleasure in expressing our gratitude for their careful reading.
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Chemla, K., Ma, B. How do the earliest known mathematical writings highlight the state's management of grains in early imperial China?. Arch. Hist. Exact Sci. 69, 1–53 (2015). https://doi.org/10.1007/s00407-014-0139-3
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DOI: https://doi.org/10.1007/s00407-014-0139-3