Abstract
From the third century to the seventh century CE, mathematics in China developed rapidly. However, this development is not reflected in the Wu cao suan**g (五曹算經 Mathematical Classic of the Five Government Departments, hereafter abbreviated to Wu cao), which was compiled in the sixth century. The advanced and complicated methods contained in earlier writings appear to have been excluded from this book. The most obvious mark of this is that the Wu cao deals with a greater number of geometric shapes and introduces more names of fields with corresponding formulae for their areas than earlier mathematical works, but adopts less advanced methods with lower precision. It deliberately avoids fractions and prefers methods expressed using specific data rather than methods expressed in an abstract or a general way. When the problems require the execution of divisions that would have remainders to express the results, the book uses smaller decimal measurement units rather than fractions, which occur in most other mathematical documents. All these features of the Wu cao can be accounted for by social factors, such as the need to train lower-grade officials in local governments who had to deal with complex managerial questions involving many calculations. The chapter focuses, in particular, on the issue of the enforcement the Equal-Field System, which was enacted in 485 CE. The choice of methods in the Wu cao and the way in which the Wu cao expresses mathematical knowledge definitely betray the influence of social factors, which provides an example of how the transmission of mathematical knowledge can be influenced by factors beyond the authors’ ability in mathematics and their individual interests.
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Notes
- 1.
An outline of the development of Chinese mathematics in this period can be seen in Guo (2010: iii–xiii). Here we substitute the seventh century of the duration for the end of the sixth century in the book.
- 2.
The concept ‘zhonggu’ 中古 originates from the concept ‘zhongshi’ 中世 (Medieval Age) which was put forward by Japanese historian Naitō Torajirō 内藤湖南 (1866–1934) (Naitō Torajirō 2004: 5–6). Most historians have accepted ‘zhonggu’ as referring to the period beginning from the Three Kingdoms (220–280 CE) or Han dynasties (206 BCE-220 CE) and ending with the turn between the Tang (618–907 CE) and Song (960–1279 CE) dynasties. Here we adopt the third century as the beginning of this period.
- 3.
- 4.
The catalogue of the Book of the Sui (Wei Zheng and Linghu Defen 1982: 1025) records a book called the Jiuzhang liu cao suan**g 九章六曹算經 (Mathematical Classic of Six Government Departments in Nine Chapters), which is lost. The title possibly implies it contained methods from the Nine Chapters to solve mathematical problems of local governments. The catalogue of the Old Book of the Tang mentions Wu cao suan**g twice. The first reference is Wu cao suan**g wu juan, Zhen Luan zhuan 五曹算經五卷, 甄鸞撰 (Wu cao suan**g, five volumes, by Zhen Luan); the second is Wu cao suan**g san juan, Zhen Luan zhuan 五曹算經三卷, 甄鸞撰 (Wu cao suan**g, three volumes, by Zhen Luan) (Liu 1975 [945]: 2039). The Chongwen zongmu 崇文總目 (General Catalogue of Chongwen, which was first compiled in 1041 CE, and whose current edition was recompiled in a later period) records: Wu cao suan**g yi juan, Zhen Luan zhu 五曹算經一卷 甄鸞注 (Wu cao suan**g, one volume, annotated by Zhen Luan) (Wang Yaocheng and Ouyang **u: Vol. III, 60A). The catalogue of the New Book of the Tang records: Zhen Luan Jiuzhang suanshu jiu juan, you Wu cao suan**g wu juan 甄鸞九章算術九卷又五曹算經五卷 (Zhen Luan [compiled] Jiuzhang suanshu in nine volumes, and also Wu cao suan**g in five volumes). It also records Han Yan **ahou Yang suan**g yi juan, you Wu cao suan**g wu juan 韓延夏侯陽算經一卷又五曹算經五卷 (Han Yan’s **ahou Yang suan**g in one volume, and also Wu cao suan**g in five volumes) (Ouyang and Song 1975 [1060]: 1545–1546). The sources not only suggest that Zhen Luan compiled a book entitled Wu cao suan**g in five volumes, but also suggest the possibility that before Zhen Luan, there were mathematical documents relating to the affairs of local governments. They further suggest the possibility that such documents might have been recompiled by different authors, with possible influences from each other. Yan Dunjie 嚴敦傑 pointed out that the last problem of the Wu cao uses the standard of ninety-two wen for one mo 陌 (which usually refers to a money unit of one hundred wen) and that this is consistent with the situation in the Changqing 長慶 (821–824 CE) period of the Tang dynasty (Yan Dunjie 1940: 207, 209). He suggested that the Wu cao handed down to us was compiled by authors in the Tang dynasty. We think the evidence to demonstrate the Wu cao was compiled in the Tang dynasty is not sufficient, but we accept the possibility that the book might have been slightly changed during the period from the Tang to the Northern Song.
- 5.
Qian (1963: 409–410) uses ‘approximate’ (**si **似) and ‘mistaken’ (cuowu 錯誤) to express his evaluation of formulae that might not give exact areas. He adds to ‘mistaken’ the qualifier ‘inconsistent with the theory’ (buhe lilun 不合理論) but he does not relate ‘approximate’ to any other word. We will use these two terms to describe the formulae in the following situations: we use ‘mistaken’ when exact formulae exist in other mathematical works from ancient China, but the Wu cao did not use them. Otherwise, we only use the term ‘approximate’.
- 6.
- 7.
Here, expressions like ‘conversion from fractional hu into dou’ have the following meaning: if, as in this case, the operation of a division results in a remainder, many mathematical documents usually express the result using a fraction of hu. However, the Wu cao prefers to give the result as measurement values expressed with integral numbers of finer measurement units. We will refer to similar situations in the Wu cao in the same way, using expressions like ‘conversion from fractional chi into cun, fen’, ‘conversion from fractional zhu into lei’.
- 8.
In this problem, there are three occurrences of shang shi zhi 上十之 (literally meaning ‘decuple it (by moving it) upwards’), which here we express as ‘ × 10’. In order to convert the amounts expressed in the pattern of ‘a zhang b chi’ into the pattern of ‘(100a + 10b) cun’ which only contains a single unit, the author used this phrase to express twice multiplying 10 to convert zhang into cun and once multiplying 10 to convert chi to cun. Karine Chemla and Ma Biao have discussed similar expressions in the **ahou Yang suan**g 夏侯陽算經 (Mathematical Classic by **ahou Yang, hereafter abbreviated to **ahou Yang). See Chemla and Ma (2011).
- 9.
It is more than possible that there are a few words missing at the beginning of the extant text of the procedure, and the original text contains the words yi yi bai cheng zhi 以一百乘之 (multiply it by 100).
- 10.
The original text is zai shang shi zhi 再上十之, which literally means ‘twice decuple (by moving it) upwards’.
- 11.
Among the 96 multiplications, ‘twice decuple it (by moving it) upwards’ (zai shang shi zhi 再上十之) in the thirteenth problem in Chapter 3 is counted as two operations, while each of the two instances of ‘decuple it (by moving it) upwards’ in the fourth problem in Chapter 4 is counted as only one operation just because the text does not mention how many times such an operation should be used. If we count the number of times the operation should actually be used, then there is a total of 100 multiplications. The original meaning of the text is to make a number become ten times itself.
- 12.
In the extant sources, the Sunzi suan**g (probably completed in the fourth century) and **ahou Yang suan**g (probably completed at the end of the eighth century) record how the ancient Chinese manipulated counting rods to do multiplication and division. See (Sunzi 2001: 262), (**ahou 2001: 463–464).
- 13.
We list the systems of measurement units as an appendix at the end of this chapter.
- 14.
The ancient Chinese usually used counting rods to express numbers and to calculate. The numerical signs represented with counting rods have two forms (see Table 11.2).
The vertical form is used to express the digits in the positions of units, hundreds, ten thousands and so on, while the horizontal form is used to express the digits on the positions of tens, thousands, hundred thousands and so on. A blank space is used for 0. See Qian (1981: 7–9). In the early period, the Chinese usually knelt on a mat placed on the ground, supporting their upper body by sitting on the heels and ankles. Thus, according to the Dunhuang manuscript P.3349 (Li Yan 1963: 411), the position in front of the right knee was defined as the position of units. Such a rule might avoid confusions, such as that arising from the fact that both 600 and 6 are represented as 丅, which does not indicate whether there are empty spaces or not (Zou 2015).
- 15.
See the units of volume/capacity in the appendix.
- 16.
The Wu cao does not include any multiplication tables. Nor does it describe the decimal system used to note integers and the system of measurement units. It assumes that the local government officials were familiar with these kinds of knowledge.
- 17.
Here we express the chapter number and the problem number, respectively, in Roman numbers and Hindu-Arabic numbers.
- 18.
pi 疋 and pi 匹 are two variants of the same word.
- 19.
Zhen (2001: 356).
- 20.
Guo (2010: 189–190).
- 21.
甲得二千五百六十三錢四分錢之二。乙得六千四百八錢四分錢之三。丙得五千一百二十七錢。丁得三千八百四十五錢四分錢之一。戊得七千六百九十錢四分錢之二。 Zhang (2001: 332–333).
- 22.
Zhen (2001: 364).
- 23.
For amounts of grain much smaller than the capacity of granaries, they often took sheng as the smallest measurement unit.
- 24.
This viewpoint was systematically discussed in Wang and Li (1995).
- 25.
Qian Baocong had pointed this out (Qian 1963: 409).
- 26.
- 27.
Sunzi (2001: 273).
- 28.
Zhen (2001: 353).
- 29.
Guo (2004: 25).
- 30.
Zhen (2001: 352).
- 31.
The formula omits the step in the original text for converting the area obtained as a number of bu into the area expressed with respect to mu (240 bu = 1 mu). We will also omit this conversion when, in similar situations, we express the methods using modern formulae.
- 32.
The method of Zhang Qiujian is equivalent to solving the equation: x2 + cx = 2S.
- 33.
A waist drum (yaogu 腰鼓) was a percussion instrument, whose shape in ancient times was approximately two connected systematical cone frustums, and whose cross section (Fig. 11.5) was approximately two congruent isosceles trapezoids with the same small base.
- 34.
The standard of the mu is 240 (square) bu.
- 35.
Zhen (2001: 351–352).
- 36.
Zhen (2001: 352).
- 37.
Zhen (2001: 352).
- 38.
Yang Hui 楊輝, Tianmu bilei chengchu jiefa 田畝比類乘除捷法 (Simplified Methods of Multiplication and Division and Analogical Methods of Field Area), in Guo (1993: 1081–1082).
- 39.
Qian (1981: 91).
- 40.
Zou (2001: 375–376, 387–389).
- 41.
Chen and Zou (2009).
- 42.
As we will mention later, Zhen Luan wrote commentaries on many mathematical classics, including the Nine Chapters and the Zhoubi which had been annotated by Liu Hui and Zhao Shuang, respectively. In fact, for a figure that could be divided into two pieces, the area formulae of which were known, one only needed to calculate the areas of two pieces and add them together. This is much easier than many cases where Liu and Zhao needed to divide a figure into several pieces and move some of them to make a new figure.
- 43.
Guo (2010: 77–78).
- 44.
Guo (2004: 18–23).
- 45.
Zhang (2001: 332–333).
- 46.
Zhen (2001: 353–354).
- 47.
Zhen (2001: 363).
- 48.
We do not include the names of fields when the problems in which they are mentioned do not deal with the computation of their areas on the basis of the lengths of key dimensions. For example, we exclude shui tian 稅田 (tax field) because it just means the field, the harvest of which was for tax.
- 49.
Mathematics (Shu數) is a mathematical book written on the bamboo strips dated to Qin Dynasty (221–206 BCE) preserved at Yuelu Academy of Hunan University (Zhu and Chen 2011).
- 50.
The Book on Mathematics (Suanshu shu 筭數書) is a mathematical book written on bamboo strips and excavated from a tomb probably sealed in ca. 186 (Zhangjiashan Er-si-qi Hao Hanmu Zhujian Zhengli **aozu 2006).
- 51.
Guo (2004).
- 52.
Sunzi (2001).
- 53.
Zhang (2001).
- 54.
Zhen (2001).
- 55.
The title Dunhuang suanshu 敦煌算書 (Mathematical Manuscripts of Dunhuang, hereafter abbreviated to Dunhuang) usually refers to many mathematical manuscripts discovered together with other manuscripts in the Mogao Cave (Mogao Ku莫高窟) in Dunhuang, Gansu province. By this title, here we only refer to the manuscript P.3349 complemented by the two sheets of paper S.5779 and S.930 (Li Yan 1963: 23–39).
- 56.
The current edition of **ahou Yang suan**g 夏侯陽算經 (Mathematical Classic by **ahou Yang, hereafter abbreviated to **ahou Yang) features a mathematical book probably compiled by Han Yan韓延 in the Tang Dynasty (**ahou 2001).
- 57.
Not for area calculation, but for square root extraction.
- 58.
In the Nine Chapters, both square fields and rectangular fields are designated by the expression ‘fang tian’, in which fang may mean square or rectangular. The expression fang tian also refers to a branch of mathematics dealing with the calculation of areas. It usually includes operations with fractions as well. The Nine Chapters does not have the name zhi tian 直田, which appears in Liu Hui’s commentary (Guo 2004: 17).
- 59.
Not for area calculation, but for seeking the arrow from the area and chord, based on the Nine Chapters’ formula.
- 60.
Not for area calculation, but for square root extraction.
- 61.
Li Yan (1958: 75).
- 62.
The Zhoubi provides evidence about how astronomical problems were solved with mathematical methods in early China.
- 63.
The Mathematics for the Five Classics deals with problems relating to mathematics in the Confucian classics.
- 64.
For details, see Yan Gengwang (2007: 141–142, 197–214, 297–283, 340–342, 554–560, 566–572, 579–594, 602, 612–624, 630–637 and 789).
- 65.
This translation for ji cao is based on the discussion below.
- 66.
Zizhi tongjian 資治通鑑, a historical book recording events during the period from 403 BCE to 959 CE.
- 67.
集曹主安集流散, 犹汉之安集掾. Sima (1958: 3938).
- 68.
墾租二石, 義租五斗. Du (1988: 95).
- 69.
Here, shi石 is a measurement unit of volume/capacity equivalent to hu (10 dou).
- 70.
The Hongfan, or Great Rules is a treatise of the Shangshu 尙書, a collection of the oldest Chinese historical documents. It mentions the word wuxing五行 (five elements). During the fourth and the third centuries BCE, wuxing developed into a philosophy which took metal, wood, earth, water and fire as five elements to explain the world. The Explanation on Five Elements of Great Rules, whose author and date are not clearly known, is a book explaining this ancient philosophical theory.
- 71.
**ao (1981: 288).
- 72.
金曹共錢布. The character 共 has two pronunciations gòng and gōng. The latter is used in the above quotation in which gōng共 is used as gōng供 and means ‘to serve’, ‘to take charge of’. Money (**金) and cloth (bu布) were closely related in ancient China. For example, in the laws recorded on the Qin bamboo strips, there is a statute named ‘money and cloth statute’ in which 11 qian (a unit of cash) are equivalent to 1 pi of cloth (Shuihudi Qinmu Zhujian Zhengli **aozu 2001: 35–42).
- 73.
以畝法除之..
- 74.
以斛法一千六百二十寸除之..
- 75.
以二十四銖除之..
- 76.
Li Yan (1958: 75).
- 77.
- 78.
This paragraph is mainly based on Fan (1978: 182–494).
- 79.
This paragraph is mainly based on Gao (2007: 15–18, 70–110, 218–235).
- 80.
From 140 BCE, when Emperor Wu of Han dynasty came into power, the timeline was divided in ancient China using reign titles, usually of a monarch. One monarch might issue one but more usually several reign titles. Taihe (477–500 CE) was the third reign title of Emperor **aowen.
- 81.
- 82.
For example, the formula of the bow-shaped field in the Wu cao would cause errors that were beneficial to the receivers.
- 83.
Although it is difficult to find evidence showing that government officials had practiced this method in land surveying before the sixteenth century, there are pieces of evidence indicating that this method was practiced in the sixteenth century. See Zhao (2006: 39–41). We can also observe the effort to calculate the area of an irregular shape accurately by cutting it apart in Liu Hui’s Commentary on the Nine Chapters (Guo 2004: 24–26). Moreover, Yang Hui used this method to correct the Wu Cao’s methods (Guo 1993: 1081–1082). At the end of the 1970s, China enforced a reform of the economic structure, and the government gave farmers managerial authority on land. In Zou Dahai’s homeland at **nhua county in Hunan province, when the lower grade local officials distributed land to rural families, such surveying methods were used.
- 84.
Li Yan (1963: 23–39).
References
Primary Sources
Du You 杜佑. 801. (1988). Tong dian 通典 (Cyclopaedia of institutions). Bei**g: Zhonghua shuju.
Liu Xu 劉昫. 945. [1975]. Jiu Tang shu 舊唐書 (Old Book of the Tang). Bei**g: Zhonghua shuju.
Ouyang **u 歐陽修 and Song Qi 宋祁. 1060 [1975]. **n Tang shu 新唐書 (New Book of Tang). Bei**g: Zhonghua shuju.
Sima Guang 司馬光. 1084. [1958]. Zizhi tongjian 資治通鑑 (Historical Events Retold as a Mirror for Government). Bei**g: Zhonghua shuju.
Suan**g shishu 算經十書 (Ten Mathematical Classics). [2001], ed. Guo Shuchun. Taipei: Jiuzhang Chubanshe.
Sunzi 孫子. n.d Fourth century CE. [2001]. Sunzi suan**g 孫子算經 (Mathematical Classic by Master Sun). In Suan**g shishu 算經十書, ed. Guo Shuchun. Taipei: Jiuzhang Chubanshe.
Wang Yaocheng 王堯臣 and Ouyang **u 歐陽修. 1041. [1882] Chongwen zongmu 崇文總目 (General Catalogue of Chongwen). Changshu: Hou zhi buzu zhai.
Wei Zheng 魏徵 and Linghu Defen 令狐德棻. 636 (first part), 656 (second part), 936–947 (the two parts are put together). [1982]. Sui shu 隋書 (Book of the Sui). Bei**g: Zhonghua shuju.
Wei Shou 魏收. 554. [1974]. Wei shu魏書 (Book of the Wei). Bei**g: Zhonghua shuju.
**ahou Yang 夏侯陽. End of the eighth century. [2001]. **ahou Yang suan**g 夏侯陽算經 (Mathematical Classic by **ahou Yang). In Suan**g shishu 算經十書, ed. Guo Shuchun. Taipei: Jiuzhang chubanshe.
**ao Ji 蕭吉. n.d [1981]. Wuxing Dayi 五行大義 (Deep Meaning of Five Elements). Taipei: Shangwu yinshuguan.
Zhang Qiujian 張丘建. ca. mid fifth century. [2001]. Zhang Qiujian suan**g 張丘建算經 (Mathematical Classic by Zhang Qiujian). In Suan**g shishu 算經十書, ed. Guo Shuchun, 295–348. Taipei: Jiuzhang chubanshe.
Zhen Luan 甄鸞. ca. mid-sixth century. [2001]. Wu cao suan**g 五曹算經 (Mathematical Classic of the Five Government Departments). In Suan**g shishu 算經十書, ed. Guo Shuchun, 351–372. Taipei: Jiuzhang chubanshe.
Secondary Literature
Chemla, Karine, and Ma Biao 馬彪. 2011. Interpreting a newly discovered mathematical document written at the beginning of Han dynasty in China (before 157.B.C.E.) and excavated from tomb M77 at Shuihudi (睡虎地). SCIAMVS. Sources and Commentaries in Exact Sciences 12: 159–191.
Chen Wei 陳巍 and Zou Dahai 鄒大海. 2009. Zhonggu suanshu zhong de tiandi mianji suanfa yu tudi zhidu 中古算書中的田地面積算法與土地制度 (The methods of area calculation in the mathematics books and the land system in the Mid-Ancient China). Ziran Kexue Shi Yanjiu 自然科學史研究 28 (4): 426–436.
Chen Wei 陳巍. 2010. ‘Wu cao suan**g jiqi lishi yu**g yanjiu 五曹算經及其歷史與境硏究 (Study on the Mathematical Classic of Five Government Departments and its historical context)’. Master Thesis (Chinese Academy of Sciences).
Fan Wenlan 范文瀾. 1978. Zhongguo tongshi [dier ce] 中國通史 (第二册) (A General History of China, Vol. 2). Bei**g: Renmin chubanshe.
Gao Min 高敏. 1987. Wei ** Nanbeichao shehui **gji shi tantao 魏晋南北朝社會經濟史探討 (Discussion on History of Society and Economy in the Period of Wei, **, Southern and Northern Dynasties). Bei**g: Renmin chubanshe.
Gao Min 高敏. 2007. Zhongguo **gji shi. Wei** Nanbeichao - shang 中國經濟史.魏晋南北朝 - 上 (History of Economy in China: Wei, **, Southern and Northern Dynasties, Vol. I). Bei**g: **gji ribao chubanshe.
Guo Shuchun 郭書春 (ed.). 1993. Zhongguo kexue jishu dianji tonghui, shuxue juan 中國科學技術典籍通匯•數學卷 (Collection of Scientific and Technological Records. Volume of Mathematics). Zhengzhou: Henan jiaoyu chubanshe.
Guo Shuchun 郭書春. 2004 (2nd ed.). Huijiao jiuzhang suanshu 匯校九章筭術 (A Critical Textual Study of the Nine Chapters on Mathematical Procedures). Shenyang: Liaoning jiaoyu chubanshe.
Guo Shuchun 郭書春 (ed.). 2010. Zhongguo kexue jishu shi: Shuxue juan 中國科學技術史·數學卷 (History of Science and Technology in China. Volume of Mathematics). Bei**g: Kexue chubanshe.
Li Yan **儼. 1958. Zhongguo shuxue dagang [**uding ben] 中國數學大綱 (修訂本) (An Outline of Mathematics in China) (Revised edition). Vol. I. Bei**g: Kexue chubanshe.
Li Yan **儼. 1963 (2nd ed.). Zhongguo gudai shuxue shiliao 中國古代數學史料 (Historical Sources of Mathematics in Ancient China). Shanghai kexue jishu chubanshe.
Naitō Torajirō 内藤湖南. 2004. Zhongguo shi tonglun——Neiteng Hunan boshi zhongguo shixue zhuzuo xuanyi. 中國史通論——内藤湖南博士中國史學著作选譯 (Chinese translation of selected works on the history of China by Dr. Naitō Torajirō) (Chinese Edition translated by **a Yingyuan 夏應元, Liu Wenzhu 劉文柱, Xu Shihong 徐世虹, Zheng **anwen 鄭顯文 & Xu Jianxin 徐建新). Bei**g: Shehui kexue chubanshe.
Qian Baocong 錢寳琮. 1963. Suan**g shishu 算經十書 (Ten Books of Mathematical Classics). Bei**g: Zhonghua shuju.
Qian Baocong 錢寳琮. 1981. Zhongguo shuxue shi 中國數學史 (History of Mathematics in China). Bei**g: Kexue chubanshe.
Shen Kangshen 沈康身 (ed.). 1999. Zhongguo shuxueshi daxi 中國數學史大繫 (A Great Series of History of Mathematics in China). Vol. IV. Bei**g: Bei**g shifan daxue chubanshe.
Shuihudi qinmu zhujian zhengli xiaozu 睡虎地秦墓竹簡整理小組. 2001. Shuihudi qinmu zhujian 睡虎地秦墓竹簡 (The Bamboo Strips Excavated from the Tomb of Qin Dynasty at Shuihudi). Bei**g: Wenwu chubanshe.
Wang Rongbin 王榮彬 and Li Jimin **繼閔. 1995. Zhongguo gudai mianji, tiji duliang zhidu kao 中國古代面積、體積度量制度考 (Critical study on the system of measurement of area and volume in ancient China). Hanxue Yanjiu 13 (2): 159–167.
Yan Dunjie 嚴敦傑. 1940. Nanbeichao suanxue shu zhi 南北朝算學書志 (Records of mathematical books of the Southern and Northern dynasties). Tushu Jikan 圖書季刊 2 (2): 196–211.
Yan Dunjie 嚴敦傑.1937. Sunzi suan**g yanjiu 孫子算經硏究 (Study on the Mathematical Classic by Master Sun). Xueyi 學藝 13.3: 15–31.
Yan Gengwang 嚴耕望. 2007. Zhongguo difang xingzheng zhidu shi: Wei ** Nanbeichao difang xingzheng zhidu 中國地方行政制度史:魏晋南北朝地方行政制度 (History of System of Local Government in China: The System of Local Government in Wei, ** and Southern and Northern Dynasties). Shanghai: Shanghai guji chubanshe.
Zhangjiashan Er-si-qi Hao Hanmu Zhujian Zhengli **aozu 張家山二四七號漢墓竹簡整理小組. 2006. Zhangjiashan hanmu zhujian [Er-si-qi hao mu] Shiwen xiuding ben 張家山漢墓竹簡[二四七號]釋文修訂本 (Bamboo Strips from Tomb of Han Dynasty at Zhangjiashan [Tomb no. 247]. Revised edition of transcription). Bei**g: Wenwu chubanshe.
Zhao Gang 趙岡. 2006. Zhongguo chuantong nongcun de diquan fenpei 中國傳統農村的地權分配 (The Distribution of Land Right in Traditional Chinese Rural Area). Bei**g: **nxing chubanshe.
Zhu Hanmin 朱漢民 and Chen Songchang 陳松長 (eds.). Yuelu shuyuan cang qinjian [Er] 嶽麓書院藏秦簡[貳] (Qin Bamboo Strips Preserved at Yuelu Academy, Vol. II.) Shanghai: Shanghai cishu chubanshe.
Zou Dahai 鄒大海. 2001. Zhongguo shuxue de xingqi yu xianqin shuxue 中國數學的興起與先秦數學 (The Emergence of Mathematics in China and the Mathematics in Pre-Qin Period). Shijiazhuang: Hebei kexue jishu chubanshe.
Zou Dahai 鄒大海. 2015. Whole number in ancient Chinese civilization: A survey based on the system of counting-units and expressions. In: ICMI Study 23: Primary Mathematics Study on Whole Numbers: Proceedings, eds. sun Xuhua, Berinderjeet Kaur, and Jarmila Novotná, 157–164. Macao: International Commission in Mathematical Instruction, University of Macau, the Education and Youth Affairs Bureau, Macau SAR.
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Appendix of Chapter 11: Units
Appendix of Chapter 11: Units
Units of Length:
-
(1)
hao 毫, li 釐, fen 分, cun 寸, chi 尺, zhang 丈, bu 步, li里
10 hao = 1 li, 10 li = 1 fen, 10 fen = 1 cun, 10 cun = 1 chi, 10 chi = 1 zhang.
6 chi = 1 bu, 300 bu = 1 li.
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(2)
duan 端, pi 疋/匹, only for cloth. 50 chi = 1 duan, 40 chi = 1 pi
Units of Area:
bu 步, mu 畝, qing 頃.
240 bu = 1 mu, 100 mu = 1 qing.
Units of Volume/Capacity:
ge 合, sheng 升, dou 斗, hu 斛, shi 石.
10 ge = 1 sheng, 10 sheng = 1 dou, 10 dou = 1 hu; 1 hu = 1 shi (in most cases)
As a unit of volume/capacity, hu was used more frequently than shi, but shi is still often used for volume/capacity.
Units of Weight:
shi 石, jun 鈞, ** 斤, liang 兩, zhu 銖, lei 絫, shu 黍.
10 shu = 1 lei, 10 lei = 1 zhu, 24 zhu = 1 liang, 16 liang = 1 **, 30 ** = 1 jun,
4 jun = 1 shi.
Units of Money:
guan 貫, wen 文 or qian 錢, fen 分, li 釐.
10 li = 1 fen, 10 fen = 1 wen or 1 qian, 1000 wen = 1 guan
The lowest unit of money in real-life was the wen or qian. The units li and fen for money were borrowed from the units of length.
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Zou, D., Chen, W. (2022). The Characteristics of Mathematical Methods in the Wu Cao Suan**g and Its Social Background. In: Chemla, K., Keller, A., Proust, C. (eds) Cultures of Computation and Quantification in the Ancient World. Why the Sciences of the Ancient World Matter, vol 6. Springer, Cham. https://doi.org/10.1007/978-3-030-98361-1_11
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