Afterthoughts

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AFTERTHOUGHTS

The present research into the other Chinese mathematics can be ex- tended in various directions. Firstly, further comparative aspects which are so important for the history of mathematics and astronomy can be considered by taking avail not only of Chinese sources – which are of course fundamental – but also of works not often, or never, associated with Chinese studies. For instance, concerning the logic of divination, cycles of times and number systems, one might notably start from M. Ascher 2002, S. Chrisomalis 2010, T. Sugiki 2005. For an appraisal of the occult vision of numbers and numerology in China and Europe, at least in the case of the sixteenth century, P. Béhar 1996 is certainly important. For a better appraisal of what the Chinese notion of Superior Epoch implies in the utterly different historical context of ancient Greece and medieval Islam, one might rely on G. de Callataÿ 1996a and 1996b. For Indian questions of the history of mathematics, K. Plofker 2009 would also be a good starting point and, more generally, for a comparison between Chinese and other East Asian calendrical systems J.-C. Eade 1995 and D. Schuh 1973 (Tibetan calendar) would certainly be beneficial. In addition, a renewed appraisal of the problem of indirect influences between Ancient Greece and China would be certainly rewarding: the recent and outstanding work Bill M. Mak 2014 in such a direction opens the way to a completely new understanding of the nature of Chinese science since it offers a convincing proof of an indirect link between Dorotheus of Sidon’s Carmen Astro- logicum (late first century AD) and a Chinese translation – the Yusi jing úB – of a Greco-Persian astral text present in Central Asia some time prior to the seventh century AD. Of course, these references cannot but cover a small fraction of an immense and ever increasing domain. At least, however, their bibliographies certainly contain lists of recent other works of interest with respect to these various topics. Moreover, ancient outstanding works, such as Ginzel, F.K., 1906–1911–1014, are

© Springer-Verlag Berlin Heidelberg 2016 333 J.-C. Martzloff, Astronomy and Calendars – The Other Chinese Mathematics, DOI 10.1007/978-3-662-49718-0 334 AFTERTHOUGHTS likewise precious for everything concerning the calendar, particularly in the European case, even though its chapters concerning China are wholly obsolete. Secondly, from the foregoing technical developments, all calculations of the Chinese calendar based on mean elements are readily accessible. Likewise, but with some supplementary and sometimes difﬁ- cult work concerning mathematical procedures which have been handed down to us in a very corrupted state, those based on true elements are also attainable (for examples of research into this latter direction, see, for instance, Qian Baocong 1983b and Yan Dunjie 1984a). However, as other articles show, interpretations of interpolation procedures, either identical with ours or not, are of course possible (see for instance: Wang Rongbin 1994, Qu Anjing 1996), Wu Jiabi 2008) whereas more ancient studies (Yan Dunjie 1955a, Li Yan 1957, Ang Tian Se 1976) have often been superseded.5 Thirdly, fully operational and explicit descriptions of Chinese procedures for eclipse predictions,6 positional astronomy and astrology are certainly key issues for future developments. If research into these directions were done, then, at least, a distinction between observed and calculated phenomena will became possible and comparisons between Chinese and non-Chinese procedures could be undertaken. Fourthly, in a very difﬁcult but also quite important direction, the study of Chinese chronology would have to be reevaluated to some extant because available tables of the Chinese calendar are in no way direct recordings of authentic dates but, on the contrary, the result of conjec- tural reconstructions, derived, at best, from critical evaluations of all sorts of historical sources. Moreover, given that, in general, such tables do not establish any link between astronomical canons and their lists of dates, any study of this domain is necessarily confronted with the following issues:

5These ancient interpretations are fundamentally identical, they only differ from the viewpoint of their operational character. (Ancient articles only describe the general aspect of interpolation procedures but not everything needed in order to really perform the corresponding calculations. For instance, their piecewise aspect is often overlooked.) 6Research articles on this subject exist but they only provide synthetic overviews in this respect. AFTERTHOUGHTS 335

1. the correctness of the admitted intervals of validity of astronomical canons;

2. the distinction between authentic and non-authentic dates (spuri- ous, erroneous and non-ofﬁcial dates) and the ranks of intercalary months.

The first issue is a consequence of the fact that historical documents are imperfect: the exact dates of validity of official astronomical canons have not always been properly recorded by ancient historians and their works have not been transmitted to us from century to century without distortions. Theoretically, this difficulty can be tackled by using the powerful arsenal of critical methods elaborated by historical research. More originally, the method of deviations of R. Billard7 could also be put to profit (this method depends on plotting ‘deviation curves’, i.e. the graph of the ancient mean longitudes minus the modern, as a function of time). However, most Chinese chronological uncertainties are typically limited to a small number of years and when that is not the case, the technical aspect of the concerned astronomical canons is generally utterly wanting. Hence a probable difficulty of using such a method in a significant number of cases.8 Moreover, a previous study of Chinese astronomical canons well beyond the case of calendrical calculations would be a prerequisite. Still, even when limiting oneself to their luni-solar components a previous statistical analysis of their quantitative data must certainly also be taken into consideration (see Y. Maeyama 1975 to 1979). The second issue about the authenticity of dates, in its turn, concerns uncertainties of limited amplitude. During our period of study, authentic calendrical dates sometimes occur one day earlier or later than those recorded in tables of the Chinese calendar. Moreover, in the case of non- official calendars, the deviations often reach one or two days. The Dun- huang manuscript calendars are typical in this respect but it is a fact that most of them are not official calendars. Yet, they also represent an important aspect of the history of the Chinese calendar and, moreover, we

7See R. Mercier 2002b. 8Initially, this method has been devised for the Indian case where chronological uncertainties are often considerable but the Chinese case is quite different in this respect. 336 AFTERTHOUGHTS may observe in passing that we absolutely ignore how these atypical calendars were elaborated (from calculations or not? The question remains open). In other words, theses dates are subject to micro-uncertainties. In addition, these uncertainties are not significantly increased by the reforms of astronomical canons because the amplitude of their effects is always limited. Keeping in mind these micro-uncertainties, recent research has established the limits of reliability of Chinese calendar tables in a few cases. For instance, the Taiwanese historian of Chinese astronomy, Huang Yi-long ?×¹, has recently pinpointed and corrected 162 errors of dates in the Shiji and Hanshu.9 For the years comprised between 665 and 728, the same historian has evidenced a number of deviations, albeit not exceeding one day, between new moons listed in usual tables and those derived from a reconstitution of the Linde li ¡ÆK calendrical calculations, the astronomical canon then in force.10 Moreover, he has shown that the first year of its reform is not the year 663, as generally believed, but the year 665. For the Liao, Song, Xia, Jin and Yuan (907–1367) dynasties, about twenty similar examples of dating errors and fifty dates of new moons differing by one day from those of the aforementioned Lidai changshu jiyao have been pinpointed in Hong Jinfu 2004, an important work briefly presented on p. 375 below. For the seventy years comprised between 822 and 892, a team of astronomers from Nanjing Observatory has also obtained a puzzling result from a reconstitution of the Xuanming li K calculations: they have discovered one day of difference between their calculated dates and those listed in the Lidai changshu jiyao in eleven cases.11 Consequently, it is impossible to be absolutely certain of the correctness of a number of dates (essentially new moon dates) provided in available calendar tables. The problem, however, is of limited importance because the few uncertainties which have been discovered never exceed one day. Nevertheless, even so, exact calendar dates are essen-

9Huang Yilong 2001a. 10Huang Yi-long 1992a. 11Zhang Peiyu, Wang Guifen et al. 1992, p. 127. AFTERTHOUGHTS 337 tial in order to distinguish authentic official calendars from non-official calendars. In a different order of ideas, it would also be highly desirable to distinguish dates obtained from calendrical calculations from those derived from arbitrary political decisions: it is certain that not everything contained in the Chinese calendar only depends of calculations, but it seems difficult to detect specific instances of the phenomenon beyond those noted on p. 99 f. below.12 In spite of the enormous difficulty of these questions, advances are already possible in the case of limited objectives. For example, all authentic calendars have still not been sufficiently examined in order to distinguish correct dates from incorrect ones while they sometimes al- low us to correct some punctual errors or uncertainties. For example, the same Huang Yi-long has noted that the first day of the eleventh month of an incomplete but authentic official calendar, preserved at the Tai- wanese National Central Library in Taipei (no. 6294), and concerning the year Tianshun 6 (1462), is associated with the sexagenary binomial #28 whereas all modern chronological tables indicate #29 instead.13 From these remarks, it is obvious that the presently available chronological tables have some micro defects, a fact that we could have sus- pected in advance by merely noting that most are devoid of any critical apparatus. Nevertheless, with the possibility of processing large volumes of data offered by computers, systematic comparisons between calendar tables and authentic calendars can be seriously considered, even though an electronic transcription of numerous and complex original sources would first be required. Lastly, despite their importance, these questions lead us somewhat far away from the history of mathematics. Beyond the numerous questions of interest in this latter respect, we note, among many others, a generalized analysis of Chinese conceptions concerning the nature and function of mathematics, taking into account, for instance, what ancient Chinese philosophers and scholars have to say about this issue (see for instance J. Gernet 2005, p. 70 f. and notably p. 79 on the artificial char-

12See also Huang Yi-long 1992b. 13Huang Yi-long 1992a, p. 280. 338 AFTERTHOUGHTS acter of mathematics). In a quite different order of ideas too, problems such as the history the replacement of the kong è zero by the ling ë zero – a problem curiously never previously studied to my knowledge – or a minute analysis of Chinese positive or negative quantities would probably also be of great interest. Appendices APPENDIX A

THE SEXAGENARY CYCLE

The following table successively gives the correspondence between the ranks of the sixty sexagenary binomials, numbered from 1 to 60 rather than from 0 to 59 – as reductions modulo 60 would imply – their phonetic transliterations, Chinese characters and binomial representations. Without such a table, the binomial corresponding to a given integer from 1 to 60 is also easily obtained: let n be such an integer. Then, the remainders x and y obtained by dividing n successively by 10 and 12 give rise to a binomial (x,y) and, when x or y is equal to 0, they are respectively replaced by 10 or 12. Hence the sought binomial. For example, when 30 is divided both by 10 and 12, the respective remainders are 0 and 6. The zero should thus be replaced by 10 and the 6 left unchanged. Consequently, #30 corresponds to (10,6).1 Conversely, the rank #n of the binomial (x,y) with respect to the sexagenary cycle is obtained as follows:

n = (6x − 5y) mod 60. For example, with (4,6), we have n = (6 × 4 − 5 × 6) mod 60 = 54. Hence the rank of this binomial, #54. These procedures are apparently not recorded in any original Chinese source. In all likelihood, the familiarity of the Chinese with the sexagenary cycle was such that they probably required nothing particular in this respect.

1This calculation technique corresponds to the notion of adjusted modulo. See N. Dershowitz and E.M. Reingold 1997, pp. 15,16, 19 and 20.

# 1 jiazi ä (1,1) # 31 jiawu ä= (1,7) # 2 yichou Ø (2,2) # 32 yiwei ØÎ (2,8) # 3 bingyin s× (3,3) # 33 bingshen så (3,9) # 4 dingmao Ù (4,4) # 34 dingyou Ù¤ (4,10) # 5 wuchen Å (5,5) # 35 wuxu ÅU (5,11) # 6 jisi (6,6) # 36 jihai ú (6,12) # 7 gengwu 6= (7,7) # 37 gengzi 6 (7,1) # 8 xinwei Î (8,8) # 38 xinchou (8,2) # 9 renshen Eå (9,9) # 39 renyin E× (9,3) # 10 guiyou .¤ (10,10) # 40 guimao . (10,4) # 11 jiaxu äU (1,11) # 41 jiachen ä (1,5) # 12 yihai Øú (2,12) # 42 yisi Ø (2,6) # 13 bingzi s (3,1) # 43 bingwu s= (3,7) # 14 dingchou Ù (4,2) # 44 dingwei ÙÎ (4,8) # 15 wuyin Å× (5,3) # 45 wushen Åå (5,9) # 16 jimao (6,4) # 46 jiyou ¤ (6,10) # 17 gengchen 6 (7,5) # 47 gengxu 6U (7,11) # 18 xinsi (8,6) # 48 xinhai ú (8,12) # 19 renwu E= (9,7) # 49 renzi E (9,1) # 20 guiwei .Î (10,8) # 50 guichou . (10,2) # 21 jiashen äå (1,9) # 51 jiayin ä× (1,3) # 22 yiyou Ø¤ (2,10) # 52 yimao Ø (2,4) # 23 bingxu sU (3,11) # 53 bingchen s (3,5) # 24 dinghai Ùú (4,12) # 54 dingsi Ù (4,6) # 25 wuzi Å (5,1) # 55 wuwu Å= (5,7) # 26 jichou (6,2) # 56 jiwei Î (6,8) # 27 gengyin 6× (7,3) # 57 gengshen 6å (7,9) # 28 xinmao (8,4) # 58 xinyou ¤ (8,10) # 29 renchen E (9,5) # 59 renxu EU (9,11) # 30 guisi . (10,6) # 60 guihai .ú (10,12)

The sexagenary cycle. APPENDIX B

THE TWENTY-FOUR SOLAR BREATHS

The approximate dates indicated in the two last columns of the following table (next page) correspond to the solar breaths of the initial and final years (104 BC and AD 1644, respectively) of the year span studied in this book. The first is a Julian date and the second a Gregorian date. Both have been directly obtained from Zhang Peiyu’s table of the Chinese calendar (Zhang Peiyu 1990*/1997* (the 1997* edition of this work has been exclusively used here). Of course, for other years in the same interval, these dates should be modified accordingly with the help of any table also listing the Julian or Gregorian dates of all solar breaths. In the case of the year 877, for example, the Spring Equinox q7, the Summer Solstice q13, the Autumn Equinox q19 and the Winter Solstice q1, respectively, occur on Mar. 18, June 17, Sept. 17 and Dec. 17 (see p. 290 f. above). From 104 BC to AD 1644, the dates of all solar breaths have always been obtained from mean motion patterns typical of Chinese traditional calendars from this year span. By contrast, they have been determined from true sun calculations from 1645 onwards. However, the tables precisely used to this end and the exact details of calculations for a given year are not often exactly known.

The Twenty-Four Solar Breaths (104 BC – AD 1644)

zhong (Odd Breaths) Approximate Dates jie ; (Even Breaths) (Limit Values)

q1 Dongzhi Winter Solstice Dec. 25 – Dec. 21

q2 ¯ Xiaohan Slight Cold Jan. 9 – Jan. 5

q3 ¯ Dahan Great Cold Jan. 24 – Jan 20

q4 ñÊ Lichun Beginning of Spring Feb. 8 – Feb. 5

q5 $i Yushui Rain Water Feb. 23 – Feb. 20

q6 a Jingzhe Waking of Insects Mar. 11 – Mar. 7

q7 Ê5 Chunfen Spring Equinox Mar. 26 – Mar. 22

q8 z Qingming Pure Brightness Apr. 10 – Apr. 6

q9 ß$ Guyu Grain Rain Apr. 26 – Apr. 22

q10 ñ Lixia Beginning of Summer May 10 – May 7

q11 Xiaoman Grain Full May 26 – May 22

q12 Ë mangzhong Bearded Grain 10 June – 6 June

q13 Xiazhi Summer Solstice June 26 – June 21

q14 Xiaoshu Slight Heat July 10 – July 7

q15 Dashu Great Heat July 26 – July 22

q16 ñK Liqiu Beginning of Autumn Aug. 10 – Aug. 6

q17 Chushu Limit of Heat Aug. 25 – Aug. 21

q18 ç' Bailu White Dew Sept. 9 – Sept. 6

q19 K5 Qiufen Autumn Equinox Sept. 24 – Sept 21

q20 ¯' Hanlu Cold Dew Oct. 10 – Oct. 6

q21 5ª Shuangjiang Descent of Frost Oct. 25 – Oct. 21

q22 ñ Lidong Beginning of Winter Nov. 9 – Nov. 6

q23 _ Xiaoxue Slight Snow Nov. 24 – Nov. 22

q24 _ Daxue Great Snow Dec. 10 – Dec. 7 THE LUNISOLAR COUPLING 345

The Lunisolar Coupling

Solar Appellations of the Solar Breaths Month Breaths in Relation to the Lunisolar Coupling Membership th th th q24 _ shiyiyue jie è× `; even Breath of the 11 month 10 or 11 th th q1 shiyiyue zhong è× ` odd Breath of the 11 month 11 th th th q2 ¯ shi’eryue jie èÞ `; even Breath of the 12 month 11 or 12 th th q3 ¯ shi’eryue zhong èÞ ` odd Breath of the 12 month 12 st th st q4 ñÊ zhengyue jie Ñ`; even Breath of the 1 month 12 or 1 st st q5 $i zhengyue zhong Ñ` odd Breath of the 1 month 1 nd st nd q6 a eryue jie Þ`; even Breath of the 2 month 1 or 2 nd nd q7 Ê5 eryue zhong Þ` odd Breath of the 2 month 2 rd nd rd q8 z sanyue jie ë`; even Breath of the 3 month 2 or 3 rd rd q9 ß$ sanyue zhong ë` odd Breath of the 3 month 3 th rd th q10 ñ siyue jie °`; even Breath of the 4 month 3 or 4 th th q11 siyue zhong °` odd Breath of the 4 month 4 th th th q12 Ë wuyue jie "`; even Breath of the 5 month 4 or 5 th th q13 wuyue zhong "` odd Breath of the 5 month 5 th th th q14 liuyue jie 0`; even Breath of the 6 month 5 or 6 th th q15 liuyue zhong 0` odd Breath of the 6 month 6 th th th q16 ñK qiyue jie Ú`; even Breath of the 7 month 6 or 7 th th q17 qiyue zhong Ú` odd Breath of the 7 month 7 th th th q18 ç' bayue jie â`; even Breath of the 8 month 7 or 8 th th q19 K5 bayue zhong â` odd Breath of the 8 month 8 th th th q20 ¯' jiuyue jie Ü`; even Breath of the 9 month 8 or 9 th th q21 5ª jiuyue zhong Ü` odd Breath of the 9 month 9 th th th q22 ñ shiyue jie è`; even Breath of the 10 month 9 or 10 th th q23 _ shiyue zhong è` odd Breath of the 10 month 10 APPENDIX C

THE SEVENTY-TWO SEASONAL INDICATORS

The 72 seasonal indicators (hou Î) have been studied in several always useful historical, philosophical or philological works (see Fung Yulan 1952–1953, vol. 2, p. 114–118; Ngo Van Xuyet 1976, p. 172– 177; Huang Yi-long 1992b, p. 30 passim, Sivin 2009, p. 81 and 401– 405, notably). Intended to highlight the correspondence between the 24 solar breaths (q1,q2. . . ) with them, the following table is restricted to essentials.

The Seventy-two Seasonal Indicators qi Initial Indicator Next Indicator Final Indicator chu hou Î ci hou gÎ mo hou ÏÎ

6 (h1) (h2) iô (h3) qiu yin jie mi jiao jie shui quan dong q1 Earth-worms Curl Up Elaphure Shed Antlers Springs and Streams Stir

(h4) ï (h5) MÖç (h6) yan bei xiang que shi chao yeji shi gou q2 Wild Geese Head Magpie Nests Pheasant Begin Northwards to Crow

Ö® (h7) _f{» (h8) igZ¾ (h9) ji shi ru zhi niao li ji shui ze fu jian q3 Hens Begin to Brood Birds of Prey Fierce Rivers and Lakes and Quick Frozen Thick

The Seventy-two Seasonal Indicators qi Initial Indicator Next Indicator Final Indicator chu hou Î ci hou gÎ mo hou ÏÎ

±Û (h10) a©M (h11) eî (h12) dong feng jie dong zhi chong shi zhen yu shang bing q4 East Wind Dissipates Hibernating Creatures Fish Ascend to the Ice the Cold Begin to Stir

B¼ (h14) 'aÀ (h15) Èe (h13) q5 ta ji yu hong yan lai caomu meng dong Otter Sacriﬁces Fish Wild Geese Appear Plants Bud and Grow

~Öº (h16) Ö6 (h17) ¿; ý (h18) tao shi hua cang geng ming ying hua wei jiu q6 Peach Trees Begin to Oriole Sings Hawks Transformed Blossom into Doves

Úf (h19) èÛsÎ (h20) é (h21) xuanniao zhi lei nai fasheng shi dian q7 Black Bird (i.e. Swal- Thunder Sounds First Lightning low) Arrives

{º (h22) â; (h23) (h24) tong shi hua tianshu hua wei ru hong shi jian q8 Pawlownia Begins to Moles Transformed Rainbows begin to Flower into Button-quail Appear

µß (h25) ý^Í (h26) Z{ªöx (h27) ping shi sheng mingjiu fu qi yu dai sheng jiang yu sang q9 Duckweed Begins to Cooing-dove Preens Hoopoe Alights on Grow Mulberry Trees

ðò (h28) 6 (h29) rÜß (h30) q10 lou guo ming qiu yin chu wang gua sheng Green Frog Croak Earthworms Emerge Royal Gourd Grows

sÈ| (h31) s'i (h32) (h33) q11 kucai xiu micao si xiaoshu zhi Sow-thistle in Seed Delicate Herbs Die Slight Heat Arrives 348 APPENDICES

The Seventy-two Seasonal Indicators qi Initial Indicator Next Indicator Final Indicator chu hou Î ci hou gÎ mo hou ÏÎ

ìß (h34) (h35) DPÎ (h36) q12 tanglang sheng ju shi ming fanshe wu sheng Praying Mantis Born Shrike Begins to Call Mockingbird Silent

h (h37) (h38) ß (h39) lu jiao jie tiao shi ming banxia1 sheng q13 Deer Shed Antlers Cicadas Begin to Sing Midsummer Plant Grows

á± (h40) óê&' (h41) ¿Û.ê (h42) wen feng zhi xishuai ju bi ying nai xue xi q14 Warm Wind Arrives Crickets Settle in the Young Hawks Learn to Walls Fly

ù' Ç (h43) þy (h44) $` (h45) fucao wei ying tu run ru shu da yu shi xing q15 Decaying Grass Be- Ground Humid, Air Heavy Rains Begin to comes Fireﬂies Hot Fall

q± (h46) ç'ª (h47) ¯¨ (h48) q16 liang feng zhi bai lu jiang hanchan ming Cool Wind Arrives White Dew Descends Cold Cicada Chirps

¿Èf (h49) F2¤ (h50) ïÛr (h51) ying ji niao tian di shi su he nai deng q17 Hawks Sacriﬁce Birds Heaven and Earth Grain Presented Begin to be Severe

B¿¼ (h52) Úfh (h53) Nfòæ (h54) hong ying lai xuan niao gui qun niao yang xiu q18 Wild Geese Arrive Black Bird (Swallow) All Birds Store up Return Provisions

1Medicinal Plant corresponding to the Pinellia ternata (or Arum ternatum) according to F. Fèvre and G. Métailié 2005, p. 24. THE 72 SEASONAL INDICATORS 349

The Seventy-two Seasonal Indicators qi Initial Indicator Next Indicator Final Indicator chu hou Î ci hou gÎ mo hou ÏÎ

èÛ[Î (h55) a©ÇV (h56) i (h57) lei nai shou sheng zhi chong pei hu shui shi he q19 Thunder Restrains its Hibernating Creatures Water Begins to Dry Up Sound Close their Burrows

B¼; (h58) ^ái Ò (h59) Äb?º (or )(h60) hong yan lai bin que ru dashui wei ge ju you huang hua q20 Wild Geese Come and Small Birds Enter the Chrysanthemum Bears Stay Great Water and Be- Yellow Blossom come Bivalves

NÛÈ (h61) 'a?a (h62) a©Y» (h63) chai nai ji shou caomu huang luo zhi chong xian fu q21 Wild Dog Offer Prays Plants and Trees Turn Hibernating Insects All Yellow and Drop Burrow in Leaves

i (h64) 2Û (h65) MÖái { (h66) shui shi bing di shi dong yeji rushui wei shen q22 Water Begins to Turn to Ground Begins to Pheasants Enter Great Ice Freeze Waters and Become Large Bivalves

(h67) Fîà2ìª TjAW (h69) hong cang bu jian tianqi shangteng diqi bi se ru cheng dong xiajiang (h68) q 23 Rainbow Hides and is Sky Breath (qi ) All is Closed up: Win- Invisible Rises, Earth Breath ter Sets in Settles

Èf (h70) ø (h71) #Sß (h72) q24 he niao bu ming hu shi jiao liting sheng Yellow Pheasant Silent Tiger Begins to Mate liting2 Emerges

2liting is the name of a plant difﬁcult to identify, respectively called ‘broom-sedge’ and ‘North China iris’ in D. Bodde’s English translation of Fung Yu-lan’s famous His- tory of Chinese Philosophy (Fung Yu-lan 1952–1953, vol. 2, p. 118) and N. Sivin 2009, p. 405. APPENDIX D

OFFICIAL ASTRONOMICAL CANONS

The numerous lists of Chinese astronomical canons published to date cover various temporal intervals and generally present a number of variations induced by chronological micro-uncertainties of limited amplitude.1 The determination of the years of validity of Chinese official astronomical canons is indeed a notoriously difficult problem: the original Chinese sources are not always mutually consistent. For instance, the dates of a large number of astronomical canons listed in the official history of the Yuan dynasty, Yuanshi -ª, are at variance with those of other ancient original sources.2 But is the Yuanshi -ª faulty or on the contrary, are more ancient sources unreliable? In this respect, Wang Yuezhen’s Gujin tuibu zhushu kao *.M, first published much more than one century ago, in 1867, provides a well-argued refu- tation of the Yuanshi dates which should not be forgotten.3 Not all historians of Chinese astronomy follow the famous chronologist, however. Quite recently, N. Sivin has cautiously avoided to choose between the dates of the Yuanshi and those of other sources.4 Therefore, it is certain that the question should be reexamined. Still, it sometimes happens

1See, inter alia, ‘Prolégomènes du P. Hoang à la concordance néoménique’, in Havret and Chambeau 1920, p. 124–128; K. Yabuuchi 1969a/1990*, p. 388–391; the encyclopedia of astronomy COLL. 1980, p. 559–561; Chen Zungui 1984, vol. 3, p. 1399–1407 (this work has been criticized because of various misprints and errors but, even so, it still remains quite useful for it always refers his data to the original Chinese sources. It is thus not too difﬁcult to check what should be corrected or to detect the reasons for uncertainties); Chen Meidong 1995, p. 237–244, N. Sivin 2009, ‘Astronomical Reforms’, p. 38–56; Qu Anjing 2008, p. 629–633. 2See N. Sivin 2009, ibid., p. 43–52. 3See Gujin tuibu zhushu kao, 1936, j. 2, p. 14a, 17b, 21a (notice p. 393 below). 4N. Sivin 2009, ibid., p. 43–52.

© Springer-Verlag Berlin Heidelberg 2016 350 J.-C. Martzloff, Astronomy and Calendars – The Other Chinese Mathematics, DOI 10.1007/978-3-662-49718-0 OFFICIAL ASTRONOMICAL CANONS 351 that recent research provides satisfactory answers to limited aspects of Chinese calendrical chronology. Yet, on the whole, no new large-scale publication has wholly superseded Wang Yuezhen’s influential master- piece, the Lidai changshu jiyao aì5 and research into this very intricate area is rather scarce. Consequently, the various available chronological lists of Chinese astronomical canons, the present one included, remain tentative and none definitely supersedes all others.6 In accordance with the limited purpose of the present work in this respect, the following table is restricted to official astronomical canons issued between ∼104 and AD 1644 and do not indicate the names of the persons credited with authorship.7 Unlike most other lists, however, the important fact that a given astronomical canon has sometimes been officially adopted under several different dynasties has been highlighted in the following one. List of Official Astronomical Canons The following table contains a list of the fifty official Chinese astronomical canons promulgated between 104 BC and AD 1644. It has been elaborated from Xi Zezong .g ’s table (COLL. 1980, p. 559– 561),8 but all tables of the Chinese calendar mentioned above, p. 371 f., have also been consulted.9 An asterisk placed after the name of a given astronomical canon means that its calculations are explained in more or less detail in Wang Yingwei 1998. A little circle indicates that the canon in question belongs to the list of the 42 astronomical canons mentioned in the Yuanshi10 (the importance

5Notice, p. 378 below. 6N. Sivin 2009, ibid., p. 42, rightly remarks that “many dates require further study”. 7They are easily available nevertheless. See, for instance, Th.E. Deane 1989, Ap- pendix D.; N. Sivin 2009, ibid., p. 43–56. 8The specific ordering of these canons slightly differs from one author to the other. The following table lists them according to their first year of official adoption. 9It should be noted that many dates recorded in N. Sivin’s recent list differ from those of other lists. In particular, there is a frequent difference of one year between the initial or final dates of validity of astronomical canons (see Sivin, N. 2009, p. 43–56). In fact, disagreements in this respect often result from the impossibility of obtaining unquestionable dates only from the data contained in Chinese histories. 10Yuanshi, j. 53, ‘li 2’, p. 1178–1188. 352 APPENDICES of this list comes from the fact that it provides the values of a number of fundamental numerical constants, impossible to obtain from other historical sources, but absolutely essential with respect to calendrical calculations).

No. Canons Dynasties Dates 1 Santong◦ ëÙ Han 104 BC–84 2 Sifen◦ °5 Hou Han 85–220 Shu 221–263 Wei 226–236 3 Qianxiang◦ é Wu 223–280 4 Jingchu* ◦ ÿ Wei 237–265 Jin 265–420 Liu Song 420–444 Northern Wei 398–451 5 Sanji jiazi yuan ëSä- Later Qin 384–41711 6 Yuanshi (or Xuanshi) - (or Ú) Northern Liang 412–439 Toba Wei 452–522 7 Yuanjia* ◦ - Liu Song 445–479 Qi 479–50312 Liu Song 502–509 8 Daming* ◦ Liang 510–557 Chen 557–589 9 Zhengguang* ◦ Ñ Toba Wei 523–565 Oriental Wei 534–539 Western Wei 535–556 Northern Zhou 557–565 10 Xinghe* ◦ ·õ Oriental Wei 540–550 Northern Qi 55013 11 Tianbao◦ F1 Northern Qi 551–577 12 Tianhe◦ Fõ Northern Zhou 566–578 13 Daxiang◦ é Northern Zhou 579–581 Sui 581–583

11Many authors write ’517’ here. For a justiﬁcation of the year 417, see Chen Mei- dong 2003a, p. 254. 12The Yuanjia li -K was renamed Jianyuan li -K under the Qi dynasty. 13The Xinghe li has been in force only one year under the Northern Qi dynasty. OFFICIAL ASTRONOMICAL CANONS 353

No. Canons Dynasties Dates 14 Kaihuang ◦ 0 Sui 584–596 15 Daye* ◦ ¼ Sui 597–618 16 Wuyin* ◦ Å× Tang 619–664 17 Linde* ◦ ¡Æ Tang 665–728 18 Dayan* ◦ Tang 729–761 19 Zhide Æ Tang 758–762 20 Wuji* ◦ "S Tang 763–783 21 Zhengyuan* Ñ- Tang 784–806 22 Guanxiang Ìé Tang 807–821 23 Xuanming* ◦ Tang 822–892 24 Chongxuan* ◦ âÚ Tang 893–907 Later Liang 907–923 Later Tang 923–936 Later Jin 936–938 25 Diaoyuan - Later Jin 939–943 Liao 947–994 26 Qintian* ◦ !F Later Zhou 956–960 Song 960–963 27 Yingtian* ◦ TF Northern Song 964–982 28 Qianyuan*◦ - Northern Song 983–1000 29 Daming Liao 995–1125 Jin 1123–1136 30 Yitian ◦ F Northern Song 1001–1023 31 Chongtian* ◦ âF Northern Song 1024–1064 1068–1074 32 Mingtian* ◦ F Northern Song 1065–1067 33 Fengyuan ◦ - Northern Song 1075–1093 34 Guantian* ◦ ÌF Later Zhou 1094–1102 35 Zhantian◦ F Northern Song 1103–1105 36 Jiyuan* ◦ S- Northern Song 1106–1127 Southern Song 1133–113514 37 Tongyuan ◦ Ù- Southern Song 1136–1167 38 Daming ◦ Jin 1137–1181

14According to Chen Zungui 1984, p. 1404, the correct year span is 1106–1166. However, this author leaves unjustiﬁed the year 1166 whereas many chronologists pro- pound the year 1127 instead. 354 APPENDICES

No. Canons Dynasties Dates 39 Qiandao* ◦ ¼ Southern Song 1168–1176 40 Chunxi* ◦ r¦ Southern Song 1177–1190 41 Chongxiu Daming* ◦ ¥Ñ Jin 1181–1234 Yuan 1215–128015 42 Huiyuan* ◦ º- Southern Song 1191–1198 43 Tongtian* ◦ ÙF Southern Song 1199–1207 44 Kaixi* ◦ ¨ Southern Song 1208–125116 45 Chunyou ◦ rÛ Southern Song 1251–125217 46 Huitian ◦ ºF Southern Song 1253–1270 47 Chengtian*◦ WF Southern Song 1271–1276 48 Bentian ÍF Southern Song 1277–1279 49 Shoushi* ◦ 0` Yuan 1281–1384 50 Datong Ù Ming 1384–1644

Metonic Ofﬁcial Astronomical Canons The table on the next page provides the list of all known Chinese Metonic ofﬁcial astronomical canons, their type determined by their two constants α and β (see p. 158 above) and the value of the positive integer α 19k+11 α β k such that β = 7k+4 . (In Chinese sources, and are respectively called zhangsui aÏ and zhangyue a`; sui Ï and yue ` respectively mean ‘solar year’ and ‘lunar month’. Therefore, zhangsui and zhangyue respectively mean ‘number of solar years (or of lunar lonths) contained in a supra-annual zhang period’. The term zhang a, appearing in these two expressions, is the general name of any Chinese Metonic period, generalized or not). The integer k is not recorded in any original Chinese source. Nev- ertheless, we mention it here because a technique of derivation of new fractions from initial ones, consisting in mutual additions of their nu-

15The year 1215 corresponds to the adoption of this astronomical canon by the Mon- gols before their conquest of China, in 1277. See Yabuuchi 1969a/1990*, p. 390. 16These initial and ﬁnal years are uncertain, various authors slightly vary in this respect (micro variations equal to ±1 year in each case). 17From Chen Zungui 1984, p. 1406. However, COLL. 1980 only retains the sole year 1252. OFFICIAL ASTRONOMICAL CANONS 355

N◦ Canons α/β k 1 Santong ëÙ 19/7 – 2 Sifen °5 19/7 – 3 Qianxiang é 19/7 – 4 Jingchu ÿ 19/7 – 5 Sanji ëS ? ? 6 Yuanshi - 600/221 31 7 Yuanjia - 19/7 – 8 Daming 391/144 20 9 Zhengguang Ñ 505/186 26 10 Xinghe ·õ 562/207 29 11 Tianbao F1 676/249 35 12 Tianhe Fõ 391/144 20 13 Daxiang é 448/165 23 14 Kaihuang 0 429/158 22 15 Daye ¼ 410/151 21

Table D.2. List of Metonic constants. merators and denominators, is attested in Chinese sources.18 In the 19k+11 present case, fractions of the form 7k+4 are formally obtained by start- 19 11 ing from the two initial fractions 7 and 4 and by adding a multiple of the numerator (respectively denominator) of the ﬁrst to the numerator (respectively denominator) of the second.19 Moreover, it turns out that the fractions so obtained have values in- 19 11 termediary between the two initial fractions, 7 and 4 :

18See Chen Zungui 1984, ibid., note 3, p. 1447–1448; Chen Jiujin 1984; Liu Dun 1987; Li Jimin 1998. 19This technique evokes the way Farey sequences are obtained (see, for example, E.W. Weisstein 1999, ‘Farey Sequence’, p. 610–611). A list of Chinese Metonic constants decomposed in this way was ﬁrst listed in Chen Zungui 1984, ibid., note 3, p. 1383. 356 APPENDICES

19 19k + 11 11 For k positive integer ≤ ≤ . 7 7k + 4 4 The Metonic constants α and β listed in the preceding table have per- haps been obtained in this way in order to obtain better Metonic approx- 11 imations. However, the fraction 4 is apparently recorded in no extant Chinese source. Yet, similar Metonic fractions also occur in the non- ofﬁcial Kaiyuan taiyi li -HØK (the astronomical canon Taiyi20 from the Kaihuang reign-period (713–741).21

20Literal meaning of this term: ‘The Great One’. See Ho Peng Yoke 2003, p. 36 f. 21Qu Anjing 2005, p. 385, highlights the presence of Metonic fractions of the form 19k+11 235k+136 in this text. APPENDIX E

TIME CONSTANTS

1 The following table contains a partial list of the values of t0 and x0, the time constants introduced in relation with the integer number t(x) of solar years contained in the interval between the two winter solstices of the Epoch and any subsequent year x, in the case of astronomical canons relying on a Superior Epoch:2

t(x) = t0 + (x − x0). In general, Chinese sources indicate the values of these two constants either in inclusive counting (suanjin Õ¼ or suanshang Õî ‘ex- haustive counting’) or in ‘exclusive counting’ (suanwai Õ² ‘external counting’). In the ﬁrst case, the initial year of the interval of years in question is included in the counting, in the second case not.3 Of course, these two modes of reckoning are not limited to the Chinese world.4 In order to avoid irregularities induced by these two possibilities, the values t0 and x0 have always been reduced to the exclusive counting case.

1For a more complete list, see Qu Anjing, Ji Zhigang and Wang Rongbin 1994, p. 154–155. 2See p. 139 above. 3See Gao Pingzi 1987, p. 112. 4See, for example, E.G. Richards 1998, p. 81.

Nº Canon t0 x0 1 Santong 143,1275 −103 2 Sifen 93666 85 3 Jingchu 4045 236 4 Daming 51,939 462 5 Kaihuang 4,129,001 584 6 Daye 1,427,645 608 7 Wuyin 164,341 618 8 Linde 269,881 664 9 Dayan 96,961,741 724 10 Wuji 269,979 762 11 Xuanming 7,070,138 821 12 Chongxuan 53,947,309 892 13 Yingtian 4,825,559 962 14 Qianyuan 30,543,978 981 15 Chongtian 97,556,341 1024 16 Mingtian 711,761 1064 17 Fengyuan 83,185,071 1074 18 Guantian 5,944,809 1092 19 Zhantian 25,501,760 1103 20 Jiyuan 28,613,467 1106 21 Tongyuan 94,251,592 1135 22 Qiandao 91,645,824 1167

5This constant comes from Li Rui AÞ (1765–1814)’s Han Santong shu ëÙ (The Santong Calculation Procedures of the Hanshu), j. 1, in COL-astron, vol. 2, p. 709. 6 We have deduced this constant from the Kaiyuan zhanjing -B, j. 105, in the following way: the number of solar years between the Epoch of the Sifen li and the second year of the Kaiyuan era, 714, is equal to 9995 years (see p. 752 of the edition of the text cited on p. 394 below). Therefore, given that the Sifen li had been ofﬁcially adopted in 85 AD, the number of years from the Epoch of the Sifen li to the year 85 AD is equal to 9995 − (714 − 85) = 9366. TIME CONSTANTS 359

Nº Canon t0 x0 23 Chunxi 52,421,973 1176 24 Huiyuan 25,494,768 1191 25 Kaixi 7,848,183 1206 26 Chunyou 120,267,647 1250

Table E.1. Time constants.

In order to check the values of the above constants, it is useful to calculate the ranks of the sexagenary binomials of several winter solstices and to compare the results with those provided in chronological tables of the Chinese calendar. Let a and b be the numerator and the denominator of the improper fraction expressing the length of the solar year in a given astronomical canon (Appendix F hereafter) and q(x) the winter solstice of the year x. Then:7

q(x) = bin(at,b). For instance, the year 104 BC, or −103, depends on the Santong li (no. 1). The number of years to be taken into account is thus the following:

t(−103) = 143,127 + (x + 103) = 143,127 solar years.

385 562120 Moreover, the solar year of this canon is equal to 365+ 1539 = 1539 days. Consequently, the winter solstice of the year −103 should be calculated as follows:

q(−103) = bin(562,120 × 143,127,1539) = < 0;0 > . Therefore, the sought winter solstice happens on a day #1, or jiazi at midnight and Zhang Peiyu 1990*/1997*’s table conﬁrms this result. In the case of other astronomical canons, the same pattern is used and the following table indicates some other results obtained in the same way:

7See p. 164 above. 360 APPENDICES

Canon Era x t(x) a b day Daming Daming 6 462 51,939 14,423,804 39,491 26 Kaihuang Kaihuang 12 592 4,129,009 37,605,463 102,960 47 Daye Daye 3 607 1,427,644 15,573,963 42,640 7 Wuyin Zhenguan 19 645 164,368 3,456,675 9464 26 Linde Zongzhang 3 670 269,882 489,428 1340 37 Dayan Tianbao 9 750 96,961,767 1,110,343 3040 36 Wuji Jianzhong 1 780 26,999 489,428 1340 #13 Xuanming Dashun 1 890 7,070,207 3,068,055 8400 50 Chongxuan Tiancheng 5 930 53,947,347 4,930,801 13,500 20 Yingtian Kaibao 3 970 4,825,567 3,653,175 10,002 50 Qianyuan Zhidao 1 995 30,543,992 1,073,820 2940 1 Chongtian Huangyou 3 1051 97,556,368 3,867,940 10,590 55 Mingtian Zhiping 3 1066 711,763 14,244,500 39,000 13 Fengyuan Yuanfeng 3 1080 83,185,077 8,656,273 23,700 26 Guantian Yuanfu 2 1099 5,944,816 4,393,880 12,030 6 Zhantian Chongning 3 1104 25,501,761 10,256,040 28,080 32

These results show that the corresponding winter solstices respectively occur on sexagesimal days #27, #48, #8, #27 and so on. Once again, it is easy to check that they are wholly identical with those indicated in all tables of the Chinese calendar. APPENDIX F

SOLAR CONSTANTS

Solar Year and Solar Periods The following table respectively lists the known lengths of the mean solar years and solar periods used in the fifty official astronomical canons promulgated in China from 104 BC to AD 1644. Given that the length of the solar year is a little greater than 365 days, only the fractions expressing its excess over 365 days are retained. Like- wise, the length of the mean solar period being always slightly greater than 15 days, only the fraction expressing its excess over 15 days is indicated. However, since the values of these latter constants are not always explicitly listed in many official astronomical canons, we have reconstituted their values, if need be, by dividing the length of the solar year by 24.

Nº Canon Solar Year – 365 Solar Period –15 385 1010 1 Santong 1539 1539×3 1 7 7 2 Sifen 4 4×8 = 32 145 515 3 Qianxiang 589 589×4 455 402 11 4 Jingchu 1843 1843 + 1843×12 605 535 5 5 Sanji 2451 2451 + 2451×6 1759 1573 7 6 Yuanshi 7200 7200 + 7200×24 150 132 22 7 Yuanjia 608 608 + 608×24

Nº Canon Solar Year – 365 Solar Period –15 9589 8626 5 8 Daming 39,491 39,491 + 39,491×6 1477 1324 1 9 Zhengguang 6060 6060 + 6060×24 4117 3684 1 10 Xinghe 16,860 16,860 + 16,860×24 5787 5170 7 11 Tianbao 23,660 23,660 + 23,660×24 5761 5127 13 12 Tianhe 23,460 23,460 + 23,460×24 3167 2838 5 13 Daxiang 12,992 12,992 + 12,992×8 25,063 22,494 7 14 Kaihuang 102,960 102,960 + 102,960×24 10,363 9315 1 15 Daye 42,640 42,640 + 42,640×8 2315 2068 1 16 Wuyin 9464 9464 + 9464×8 328 292 5 17 Linde 1340 1340 + 1340×6 743 664 7 18 Dayan 3040 3040 + 3040×24

19 Zhide ? ?

328 292 5 20 Wuji 1340 1340 + 1340×6 268 239 7 21 Zhengyuan 1095 1095 + 1095×24

22 Guanxiang ? ?

2055 1835 5 23 Xuanming 8400 8400 + 8400×8 3301 2950 1 24 Chongxuan 13,500 13,500 + 13,500×24

22 Diaoyuan ? ?

1760 40 1 1573 35 26 Qintian 7200 + 7200×100 7200 + 7200×100

1 Unlike previous constants, the value of the solar year is here unusually expressed as a sum of two fractions, the second having a denominator equal to an integer multiple of the denominator of the ﬁrst. See Xin Wudai shi, j. 58, ‘Sitian kao 1’, p. 674; Wang Yingwei 1998, p. 495. SOLAR CONSTANTS 363

Nº Canon Solar Year – 365 Solar Period –15 2445 2185 5 27 Yingtian 10,002 10,002 + 10,002×8 720 642 1 28 Qianyuan 2940 2940 + 2940×2

29 Daming ? ?

2470 2207 3 30 Yitian 10,100 10,100 + 10,100×36 2590 2314 6 31 Chongtian 10,590 10,590 + 10,590×36 9500 8520 5 32 Mingtian 39,000 39,000 + 39,000×6 5773 5178 1 33 Fengyuan 23,700 23,700 + 23,700×24 2930 2628 12 34 Guantian 12,030 12,030 + 12,030×36 6840 6135 35 Zhantian 28,080 28,080 1776 1592 3 36 Jiyuan 7290 7290 + 7290×4 1688 1514 15 37 Tongyuan 6930 6930 + 6930×180 1274 1142 2 38 Daming 5230 5230 + 5230×3 7308 6554 1 39 Qiandao 30,000 30,000 + 30,000×2 1374 1232 25 40 Chunxi 5640 5640 + 5640×100 1274 1142 60 41 Chongxiu Daming 5230 5230 + 5230×90 9432 8455 1 42 Huiyuan 38,700 38,700 + 38,700×2 2910 2621 25 43 Tongtian 12,000 12,000 + 12,000×100 4108 3692 44 Kaixi 16,900 16,900 857 771 1 45 Chunyou 3530 3530 + 3530×8 2366 2127 3 46 Huitian 9740 9740 + 9740×4 1801 1620 7 47 Chengtian 7420 7420 + 7420×8 364 APPENDICES

Nº Canon Solar Year – 365 Solar Period –15 48 Bentian ? ?

49 Shoushi secular variations2 0.21843753

50 Datong 0.2425 0.2184375

2See p. 141 above. 3 The Shoushi astronomical canon says nothing about the consequences of the variations of the tropical year on the lengths of solar periods. APPENDIX G

LUNAR CONSTANTS

The following table provides the values of the lengths of the mean synodic and anomalistic lunar months of Chinese ofﬁcial astronomical canons from 104 BC to AD 1644 (the latter kind of month has never been used in calendrical calculations before 665, that is, before the of- ﬁcial promulgation of the Linde li (665–728). However, it was already known in China much earlier, as soon as the Former Han dynasty, and was used in a number of more involved astronomical calculations). These two sorts of months being respectively a little more than 29 and 27 days long, only their fractional values have been retained.1

N◦. Canon Syn. Month – 29 Anom. Month – 27 43 1 Santong 81 — 499 2 Sifen 940 — 773 3 Qianxiang 1457 — 2419 4 Jingchu 4559 —

3217 5 Sanji 6063 — 47,251 6 Yuanshi 89,052 — 399 7 Yuanjia 752 — 2090 8 Daming 3939 —

1These values have been checked with the help of Chen Meidong 1995, p. 237–244 and Wang Yingwei 1998.

N◦. Canon Syn. Month – 29 Anom. Month – 27 39,769 9 Zhengguang 74,952 — 110,647 10 Xinghe 208,530 — 155,272 11 Tianbao 292,635 — 153,991 12 Tianhe 290,160 — 28,422 13 Daxiang 53,563 — 96,529 14 Kaihuang 181,920 — 607 15 Daye 1144 — 6901 16 Wuyin 13,006 — 711 743 1 17 Linde 1340 1340 + 1340×12 1613 1685 79 18 Dayan 1340 1340 + 3040×80

19 Zhide ? ?

711 743 5 2 20 Wuji 1340 1340 + 1340×37 581 607 132 21 Zhengyuan 1095 1095 + 1095×10,000

22 Guanxiang ? ?

4457 4658 19 3 23 Xuanming 8400 8400 + 8400×100 7163 7486 97 24 Chongxuan 13,500 13,500 + 13,500×100

25 Diaoyuan ? ?

3820 28 4 26 Qintian 7200 + 7200×100 —

2Chen Meidong 1995, p. 240. 3Zhang Peiyu,Wang Guifen et al. 1992, p. 122. 4Wang Yingwei 1998, p. 526. LUNAR CONSTANTS 367

N◦. Canon Syn. Month – 29 Anom. Month – 27 5307 5 5546 6210 6 27 Yingtian 10,002 10,002 + 10,002×10,000 1560 1620 6020 7 28 Qianyuan 2940 2940 + 2940×10,000

29 Daming ? ?

5359 5601 165 30 Yitian 10,100 10,100 + 10,100×10,000 5619 8 5873 594 31 Chongtian 10590 10,590 + 10,590×10,000 20,693 601,471,251 9 32 Mingtian 39,000 39,000×27,807 12,575 33 Fengyuan 23,700 ? 6383 6672 389 34 Guantian 12,030 12,030 + 12,030×10,000 14,899 35 Zhantian 28,080 ? 3868 4043 990 36 Jiyuan 7290 7290 + 7290×10,000 3677 3843 2563 37 Tongyuan 6930 6930 + 6930×10,000

38 Daming ? ?

15,917 76 16,637 7395 39 Qiandao 30,000 + 30,000×100 30,000 + 30,000×10,000 2992 56 3127 9740 40 Chunxi 5640 + 5640×100 5640 + 5640×10,000 2775 2775 6066 41 Chongxiu Daming 5230 5230 + 5230×10,000 20,534 21,461 7310 42 Huiyuan 38,700 38,700 + 38,700×10,000

5This fraction is not irreducible and this is also the case for a few others. 6Wang Yingwei 1998, ibid., p. 526. 7According to Chen Meidong 1995, p. 240, the numerator of the ﬁrst fraction is equal to 1630. Wang Yingwei 1998’s value, 1620, seems more correct (ibid., p. 526). 8Wang Yingwei 1998, ibid., p. 576. 9According to Wang Yingwei 1998, ibid., p. 624, the value of the numerator of this fraction is equal to 601,47 2 ,251 but Chen Meidong 1995, ibid., p. 241, indicates 601,47 1 ,251 instead. 368 APPENDICES

N◦. Canon Syn. Month – 29 Anom. Month – 27 6368 6655 43 Tongtian 12,000 12,000 8967 9372 5396 44 Kaixi 16,900 16,900 + 16,900×10,000 1873 45 Chunyou 3530 ? 5168 46 Huitian 9740 ? 3937 4115 1641 47 Chengtian 7420 7420 + 7420×10,000

48 Bentian ? ?

49 Shoushi 0.530593 0.275546

50 Datong 0.530593 0.275546 Tables of the Chinese Calendar and Bibliography TABLES OF THE CHINESE CALENDAR

All available tables of the Chinese calendar contain at least the following elements:

1. dates of dynastic eras;

2. the numbering of the years, months and days of the Chinese calendar with the sixty binomials (trunks and branches ganzhi Y) of the sexagenary cycle;

3. new moons;

4. the nature of each lunar month (ordinary or intercalary);

5. their number of days, 29 or 30 days (full or hollow months).

Apart from this skeleton service, they differ from one another in various guises, such as their very variable year spans, the astronomical and calendrical elements they take into account (solar and lunar eclipses visible in particular places of China and concordances with Western and non-Western calendars, for example), their approach and their layout. A large number of tables only contain dry lists of dates, without any attempt to explain either how they have been obtained or if they are reliable or differ in some way from those of more ancient tables. Appar- ently, most tables merely reproduce former ones. Nevertheless, a few tables, ancient or modern, attempt to tackle the subject in a critical way and some limit their scope to particular periods in order to take advantage of the most recent archeological ﬁndings. Those available to date are numerous but none is completely satisfactory and that explains why new ones are constantly released. The consultation of several tables at the same time is thus often unavoidable, at least for research purposes. All tables are not equally convenient. As a rule, the most ancient ones are less handy because they suppose an understanding of often implicit

© Springer-Verlag Berlin Heidelberg 2016 371 J.-C. Martzloff, Astronomy and Calendars – The Other Chinese Mathematics, DOI 10.1007/978-3-662-49718-0 372 BIBLIOGRAPHY and non-obvious conventions. On the contrary, the most recent ones often take advantage of the new possibilities of layout and typographical readability made available by sophisticated computer programs. We propose to draw up here not only a mere bibliographical list of available tables but also to provide further details about what can be expected in each case. First, we have tried to indicate not only their first date of publication but also their reprints and to spot possible modifi- cations. Second, we have presented a succinct description of the main characteristics of each table and we have made explicit their year span when this fundamental indication does not already appear in their titles. Lastly, those deemed by us the most important for the scientific study of Chinese chronology have been asterisked. A List of Tables 1. COLL., 2002. Zhonghua wuqian nian changli º"ûOK (Five Thousand Years of Long Chinese Calendrical Chronology), Beijing, Qixiang Chubanshe éÌã. – Takes into account numerous modes of cyclical enumerations for days, months and years, typical of the Chinese calendar such as the nine color palaces, the jianchu and nayin series, etc. provided for all the years of the interval 221 BC–AD 2100. It should be noted, however, that this data should be used with caution because these elements are given regardless of their historical dates of introduction in the calendar. – year span: 2070 BC–AD 2100.

2. *CHEN Yuan, Wn1926/1999*. Ershi shi shuorun biao Þèªi (Chronological Table of New Moons, Ordinary Lunar Months and Intercalary Months in the Twenty Dynastic Histories), Beijing, Zhonghua shuju ºh . – When ﬁrst published, this work was hailed as a major achievement in the ﬁeld. – Unlike most other tables, the exact dates of the beginnings of new dynastic eras are precisely noted (most tables do not go beyond the mere mention of the concerned years in this respect); TABLES OF THE CHINESE CALENDAR 373

– takes the Chinese Muslim calendar into account; – year span: 206 BC–AD 2000.

3. *FANG Shiming ]Z and FANG Xiaofen ] 1987. Zhong- guo shi liri he zhongxi liri duizhaobiao »ªK^õK^E ï (A Table of the Historical Chinese Calendar, With a Concor- dance Between its Dates and those of the Western Calendar), Shang- hai, Shanghai Cishu Chubanshe î[hÌã. – Mentions alternative dates listed in previous chronological tables such as Wang Yuezhen, 1867/1936*/1993*, Lidai changshu jiyao or P. Hoang, 1910/1968* (see p. 374 and 378 below, respectively); – table of names of dynastic eras (p. 881–884) ordered according to the number of strokes of the ﬁrst Chinese character of their names. – year span: 841 BC–AD 2000.

4. *GASSMANN R.H. 2002. Antikchinesisches Kalenderwesen, Die Rekonstruction der chunqiu-zeitlichen Kalender des Fürstentums Lu und der Zhou-Könige, Bern, Peter Lang. – Substantial English overview (p. 431–451); – critical and outstanding reconstruction of antique Chinese calendars (p. 147–347); – year span: 721 BC–467 BC (Lu and Zhou kingdoms).

5. HAZELTON Keith 1984/1985*. A Synchronic Chinese Western Daily Calendar (1341-1661 A.D.), (Ming Studies Research Series, 1), Minneapolis, University of Minnesota (USA), History Depart- ment. – Complete list of all successive days of the Ming dynasty with the indication of their sexagenary binomials and the detailed concordance with Julian or Gregorian Western dates, as the case may be. Very handy.

6. HIRAOKA Takeo ¿)G1990. Tangdai de li êÝK (The Astronomical Canons of the Tang Dynasty), Shanghai, Shanghai Guji Chubanshe î°Ìã. (Chinese transl. from the Japanese of 374 BIBLIOGRAPHY

Todai¯ no koyomi, êGK Kyoto, Kyoto daigaku jimbun kagaku kenkyujo¯ ³K.ßZI.@~X, 1954). – Early reconstitution of the fundamental elements of calendars from the Tang dynasty (618–907) on the basis of ancient calendrical procedures (but their description is omitted; only their results are provided). 7. HOANG Peter [Pierre] 1885/1986*. A Notice of the Chinese Cal- endar and a Concordance with the European Calendar. Zi-Ka-Wei, Printing Ofﬁce of the Catholic Mission. Reedited by Le Cercle Sino- logique de l’Ouest, Rennes, 1986. – Overview of the Chinese calendar (p. 1–34); – year span: 1624–2020. 8. *HOANG Pierre 1910/1968*, Concordance des chronologies néo- méniques chinoise et européenne, 2nd ed., Taichung, Kuangchi Press, (1st ed., Imprimerie de la Mission Catholique, Orphelinat de T’ou-sè- wei, Zikawei, Shanghai). – As the author explains, p. xi, his concordance borrows his dates from the original edition of Wang Yuezhen’s Lidai changshu jiyao released in 1867 (see p. 378 below); – theoretical chronology of the years preceding the beginning of the Christian era according to the calculations of the Zhuanxu li àò K, a computus supposed to have been used some time before 104 BC10 (p. 487–500); – various appendixes (main and partial dynasties; posthumous names of Chinese emperors; names of dynastic eras); – concordance of partial dynasties; – Curiously, Western dates equivalent to Chinese dates follow the proleptic Gregorian calendar from 841 BC to AD 1. Then, from AD 1 to AD 1582, the Julian calendar is more logically used instead. Lastly, as expected, Gregorian dates are provided for later years (see ‘Avertissement’, p. I);

10The Zhuanxu li is one of the ‘six ancient computus’ mentioned on p. 381 below. Zhuanxu is the name of a Chinese mythical emperor. TABLES OF THE CHINESE CALENDAR 375

– succinct but useful historical developments; – year span: 841 BC–AD 2020.

9. *HONG Jinfu ÷° 2004. Liao, Song, Xia, Jin, Yuan wu chao rili õ-"^K (Chronological Tables of the Chinese Calen- dar for the Five Following Dynasties: Liao, Song, Xia, Jin and Yuan), Taipei, Zhongyang yanjiuyuan lishi yuyan yanjiusuo ³@~oa ª+@~X. – Critical reworking of Chen Yuan, 1926/1999* (already mentioned on p. 372 above). In order to ease its consultation and to render im- mediate concordances between Chinese and Western dates, the successive years of the Chinese calendar are granted a full page each and the layout of their months always follows the same pattern, no matter whether they are ordinary or intercalary. The successive days of each lunar month are numbered in several ways: with the sixty binomials of the sexagenary cycle, with their day-number, according to the Chinese and Western numbering systems. The more technical Julian day system is not used. Like Chen Yuan 1926/1999*, provides the exact dates of the beginnings of new dynastic eras; – list of errors detected both in Chen Yuan 1926/1999* and in the first edition of Zhang Peiyu’s chronology of the Chinese calendar (1990, see p. 381 below): without limiting himself to the five successive dynasties of his tables, the author pinpoints twelve such errors in the first publication and nine in the second (introduction, p. iv); – the backmatter of the book provides a number of useful data organized quite conveniently (phonetical transliterations of the names of Mongol emperors, variant appellations of the names of the heavenly stems and the like); – the whole book contains 6195 lunar months composed of 182,941 days and five hundred years.

10. OKADA Yoshiro,¯ )âý , ITO¯ Kazuhiko, þõ OTANI¯ Mitsuo P and FURUKAWA Kiichiro¯ ¡× , 1993. Nihon rekijitsu soran,¯ guchurekijitsu¯ hen, ^ÍK^ÀÌ¥ K^S (A Survey of Japanese Calendars: Annotated Calendars), 20 vol. Tokyo, Honnotomo sha ÍGBã. 376 BIBLIOGRAPHY

– Although it concerns the Japanese calendar, the present work is also highly relevant for the study of the Chinese calendar for both have been established by means of the same astronomical canons for certain year spans.11 More precisely, (a) the Yuanjia li/Genka reki -K,(b) the Linde li/Gihoreki¯ ¡ÆK and (c) the Xuanming li/Senmyoreki¯ K have been respectively adopted in China and Japan during the following year spans:12

(a) 501–509 — 501–691 and 692–697; (b) 692–697 — 697–763; (c) 862–892 — 862–1684, respectively;

– In this admirable publication, unique and unsurpassed to date, the authors have taken into account practically every item that the Jap- anese calendar (and thus the Chinese calendar also) is liable to contain, even its calendrical spirits shen ß.13 They have thus relied not only on the calculation techniques of astronomical canons but also on the modes of insertion of all sorts of elements obtained from hemerological treatises of the concerned periods; – year span: 501 AD–1500 AD.

11. TUNG Tso-pin [Dong Zuobin] l®;, 1960, Zhongguo nianli zong- pu »OKÀH (Chronological Tables of Chinese History), 2 vol., Hong Kong, Hong Kong University Press. – Bilingual introduction, Chinese and English; – concordance between Chinese, Western and Muslim dates; – year span: 2674 BC–AD 2000. (2674 BC corresponds to the beginning of the reign of the mythical emperor Huangdi ?).

12. WANG Huanchun rôÊ et al., 1991. Gong, Nong, Hui, Tai, Yi, Zang, Fo li he Rulüe ri duizhaobiao 2¹/ÑR¡Kõ¯^E

11On Japanese chronological systems, see R. Zöllner 2003. 12See Y. Okada, K. Ito¯ et al. 1993, vol. 1, p. 7; T. Watanabe 1977/1984*, p. 11; M. Sugimoto and David L. Swain 1978, p. 72–73 and p. 254, respectively. 13On calendrical spirits, see A. Arrault 2003, p. 106 f. TABLES OF THE CHINESE CALENDAR 377

ï (622–2050) (Concordance Tables between the Gregorian, Chi- nese, Muslim, Thai, Yi, Tibetan and Buddhist Calendars and Julian Days). Beijing, Kexue Chubanshe I.Ìã. – Brief details about the calendars mentioned in the title (p. 1–10); the Taiping calendar (not mentioned in the title) is also brieﬂy introduced (p. 10); – dates of the 24 solar breaths; – days of the planetary week.

13. WANG Kefu rG and LI Min AÓ, 1996. Zhonghua tongshi dali dian º;ªKÎ (Great Chronology of Chinese History). 3 vol. Chengdu, Sichuan Minzu Chubanshe ° ÓHÌã. – Monumental compilation composed of approximately ﬁve thousand pages and giving all sorts of historical data beyond what is usu- ally included in chronological tables, (various names of Chinese emperors, main historical events, etc.); – year span: 2674 BC–AD 2000.

14. *WANG Yuezhen W_Å, 1866. Lidai changshu a (Long Chronology of the Successive Chinese Dynasties), 50 j., manuscript preserved at the Beijing Library, in COLL., 1983, Beijing tushuguan guji shanben mulu, zi bu ³%h °Íê I (Cata- log of Ancient Texts and Rare Books of the Beijing Library, section devoted to technical works), Beijing, Shumu Wenxian Chubanshe h êZ¤Ìã, p. 1285. See also, in the same catalog, p. 1286, other manuscripts of Wang Yuezhen. – The author, Wang Yuezhen (1812–1881),14 was an Instructor jiaoyu >Ý15 from Kuaiji sub-prefecture ºI, Zhejiang province; he became juren Üß (i.e. ‘licentiate’ or graduate of the provincial examination) in 1836 and he worked on his monumental chronological project for 30 years, from 1836 to 1866. At last, the quality of his

14On Wang Yuezhen, see P. Hoang 1910/1968*, p. xi–xv (notice, p. 374 above); COL-astron, vol. 1, p. 717; Chouren zhuan sanbian ßFë_ 1898/1982*, j. 6, p. 823–826. 15See Ch.O. Hucker 1985, item no. 747. 378 BIBLIOGRAPHY

work, characterized by a rigorous approach,16 has been praised. It turned out, however, that his manuscript was too voluminous and the costs of printing too high. Consequently, his Lidai changshu was never released. Nevertheless, an abridged edition, the Lidai changshu jiyao,17 was issued in 1867. During the second half of the twen- tieth century, archeological ﬁndings have rendered Wang Yuezhen’s work more or less obsolete for everything concerning the more ancient periods, but, even so, it is doubtless that a critical publication of his original manuscript would still be immensely useful for a ﬁne-tuned understanding of the innumerable intricacies of Chinese chronology.

15. *WANG Yuezhen W_Å, 1867/1936*/1993*. Lidai changshu jiyao, fu gujin tuibu zhushu kao aì!*.M (A Concise Handbook of the Long [Chinese] Chronology Calcu- lated According to the Astronomical Canons of the Successive Dy- nasties (shu ), with an Annex Devoted to a Study of the Methods of Predictive Astronomical, Astrological and Hemerological Calcu- lations (tuibu .M) Expounded in [Chinese] Astronomical Canons), Shanghai, Zhonghua shuju ºh , Sibu beiyao collection °I n; 10 j. (chronology) + 2 j. (appendix).

– Among all chronological tables of the Chinese calendar mentioned here, the Lidai changshu jiyao is almost the only one dealing criti- cally with the intricate issues raised by Chinese calendrical chronology. First published in 1867, most less ancient works reproduce its calendrical tables; it has thus not really been surpassed.

– As already noted in the preceding notice, Wang Yuezhen bases his conclusions both on a ﬁrst order knowledge of the mathematical techniques of Chinese astronomical canons and on all sorts of historical sources, main or ancillary. Among the former, he uses, notably, the Twenty-Four Dynastic Histories and Sima Guang ¥ y (1019–1086)’s Zizhi tongjian mulu £»;dê (Chronol-

16See the next notice. 17See P. Hoang 1968, p. xi, (notice, p. 374 above). TABLES OF THE CHINESE CALENDAR 379

ogy of the Comprehensive Mirror to Aid in Government),18 30 j., the most famous extant ancient Chinese chronology.19 His ancillary sources are composed of epitaphs, inscriptions, memorials and all sorts of documents,20 liable to contain authentic dates and therefore to corroborate or invalidate prevailing Chinese dates. – Wang Yuezhen does not list the totality of the successive days Chi- nese lunar years are made of but only the most fundamental ones, that is, as usual, the new moons, the ranks of the intercalary months and the dates of solar breaths. In order to save space, he presents his data year by year, in point form, and he mentions only the following elements: – the sexagenary binomial of the lunar year; – the name of the corresponding dynastic era; – The sexagenary binomial of the ﬁrst new moon of the lunar year and those of a limited number of others; – The sexagenary binomials of intercalary new moons, and one or several solar breaths. – The Lidai changshu jiyao has been successively published three times, (a) 1867 (Tongzhi 6): Liqiang congke princeps edition #

18Concerning this famous work, see Y. Hervouet 1978, p. 169 f. Original text: WYG, vol. 311, p. 321–787. 19Sima Guang’s chronological study covers more than 1300 years, from 403 BC to 959 AD. Its structure is relatively complex and has not been much studied. Roughly speaking, the text is divided into two registers. The ﬁrst one lists the usual elements of the Chinese calendar (new moons, solar breaths, sexagenary binomials, ranks of intercalary months, etc.) together with other indications never recorded in extant Chinese calendars such as, for instance, the entrance of planets in such and such constellation or solar and lunar eclipses. The second one records a list of events of Chinese history taken from the Zizhi tongjian £»;d. After publication, this work of Sima Guang became an unsurpassed reference. During the XVIIIth century evidential scholars such as Li Rui AÞ (1765–1814) and Qian Daxin þ (1728–1804) tackled the subject anew. The ﬁrst began to wrestle with various questions of chronology relating to Chi- nese antiquity; later, he tackled more recent periods. In his turn, Qian Daxin tried to extend the chronology of Sima Guang beyond its endpoint, the year 959 (see his Song, Liao, Jin, Yuan si shi shuorun kao õ-i (Research into the New Moons and Intercalations in the Song, Liao, Jin and Yuan Histories). 20P. Hoang, 1910/1968*, p. xxvii (notice, p. 374 above). 380 BIBLIOGRAPHY

LÑ,(b) 1936: edition made from the ﬁrst one, without mention of corrections, Shanghai, Zhonghua shuju ºh , collection Sibu beiyao °In, and lastly (c), 1993: reproduction of the original manuscript (presumably identical with the copy presently preserved in the Beijing library) (see COL-astron, vol. 1, p. 720–941). – An interesting notice on this important source is DING Fubao Ù É1 and ZHOU Yunqing ø % (ed.), 1957. Sibu zonglu, tianwen bian °IÀFZ_ (Catalog of the Four Bibliographical Departments: Astronomy, Shanghai, Shangwu Yinshuguan (Com- mercial Press), p. 562a–562b. – Another important notice on the same subject is BO Shuren ßW ß in COL-astron, vol. 1, p. 717; – year span: 841 BC–AD 1670.21 The years dealt with in the ten chapters of the book are regrouped in the following way, by clusters of two or three hundreds years, whose limits do not correspond with those of successive dynasties: j. 1: 841 BC–607 BC; j. 2: 606 BC–369 BC; j. 3: 368 BC–141 BC; j. 4: 140 BC–146; j. 5: 147–419; j. 6: 420–617; j. 7: 618–906; j. 8: 907–1126; j. 9: 1127–1367; j.10: 1368–1670.

16. XU Xiqi 6-, 1992. Xinbian Zhongguo sanqian nian liri jiansuo biao ±_»ëûOK^lõ (New Table of Dates for Three Thousand Years of Chinese History), Renmin Jiaoyu Chubanshe ß Ó>Ìã. – Concordance between the dates of the Chinese, Japanese and Mus- lim calendars; – year span: 1500 BC–2050.

17. XUE Zhongsan å ë and OUYANG Yi & , 1940/1957*. Liangqian nian Zhong-Xi li duizhaobiao ËûOKEï (Concordance Between the Chinese and Western Calendars for Two Thousand Years), Beijing, Sanlian shudian ëÐh7.

21i.e. from (Gonghe 1) to (Kangxi 9), the choice of the latter date is justiﬁed by the fact that a perpetual calendar, the Qinding Wannian shu ! 0Oh, was published ca. 1670, under Kangxi’s reign. TABLES OF THE CHINESE CALENDAR 381

– Bilingual introduction (Chinese and English); – year span: 1–2000.

18. *ZHANG Peiyu ùÇ, 1987. Zhongguo xian Qin shi libiao »çªK (Chronological Tables for Pre-Qin China). Jinan, Qi-Lu shushe hã. – Retrospective astronomical calculations (dates of winter solstices, new moons, solar and lunar eclipses); – List of calendrical dates determined from the gu liu li 0K techniques, i.e. those of ‘the six ancient computus’;22 – year span: 1500 BC–105 BC.

19. *ZHANG Peiyu ùÇ, 1990*/1997*. Sanqian wubai nian liri tianxiang ëû"yOK^Fé (Tables and Astronomical Phenom- ena for 3500 years), 1st ed.: Henan Jiaoyu Chubanshe ªP> Ìã; 2nd ed., Zhengzhou, Daxiang Chubanshe éÌã. – composed by a contemporary astronomer and an historian of astronomy from Nanjing Astronomical Observatory;23 – dates of the 24 solar breaths; – Astronomical new moons retrospectively calculated; – The Eight Nodes of the Chinese solar year, ba jie â;; – Ancient and modern eclipses, visible from the main Chinese towns, from 1500 BC to 2052; – year span: 1500 BC–2050 (with a particular treatment for the antique period).

20. ZHENG Hesheng Ð6Î (ed.), 1936/1985*. Jinshi zhong-xi shiri duizhaobiao tª^Eï (A Concordance Between Chi-

22These ‘six ancient computus’ are believed to have been used from the Warring States period to the Former Han. Their calculation techniques have been reconstituted from various sources and are similar to those of the Sifen li °5K but with different Superior Epochs each time. See Chen Meidong 2003a, p. 87–92. Moreover, it should be noted that, here, li K corresponds to ‘computus’ and not to ‘astronomical canon’ because their calculations only concern the luni-solar component of the calendar. 23On the history of this Observatory, see Jiang Xiaoyuan and Wu Yan 2004. 382 BIBLIOGRAPHY

nese, Japanese, Korean and Western Historical Calendrical Dates During the Modern Period), Beijing, Xinhua Shudian ±ºh7. – Interesting historical introduction; – Concordance between the years of the Christian era, Chinese, Jap- anese and Korean dynasties; – Concordance between the Chinese ofﬁcial calendar and the Taiping calendar (1851–1864); – year span: 1516–1941.

Recent Advances (2012–2014) The following three recent publications are particularly innovative: on the one hand, the author has designed his tables of the Chinese calendar with respect to speciﬁc computus (or astronomical canons); on the other hand, each volume contains an in-depth study of the corresponding techniques of calendrical calculations, with references to a large number of primary sources. No less importantly, these tables also contain the results of fundamental calculations. Last but not least, the book precisely relates his tables to the content of authentic calendars (or fragments of calendars) recently discovered at the occasion of archeological surveys.

1. *ZHU Guichang, er}. Zhuanxu rili biao àò^K (Tables of the Zhuanxu li), Beijing, Zhonghua shuju ºh , 2012.

2. –*Taichu rili biao er}. Taichu liri biao H^K (Tables of the Taichu li), Beijing, Zhonghua shuju ºh , 2013.

3. – *Hou Han Sifen rili biao ¡°5^K (Tables of the Sifen li during the Later Han Dynasty), Beijing, Zhonghua shuju º h , 2014.

Computer Programs Computer programs delivering concordances between the Chinese, Ju- lian, Gregorian and other calendars have been devised.24 Among these,

24R. Mercier gives some references and an analysis of the specific difficulties pro- grammers are confronted with in this respect (see R. Mercier, 2002a). TABLES OF THE CHINESE CALENDAR 383 the following utility proposed by the Academia Sinica Department of Information Technology Services is useful for general purposes (see http://sinocal.sinica.edu.tw/, (Chinese language only)). In all cases, such programs are significantly easier to use than printed tables. Still, their degree of reliability is often difficult to check, particularly when their year-span extends over very long periods, inasmuch as they often tend to project mechanically calendrical events into periods where they did not exist. However, the same remark also applies to some printed concordances. Beyond the mere computer reproduction of what is already available, however, programming techniques devoted to particular aspects of Chinese calendrical chronology – such as the possibility of delivering the dates of all events obtainable from a given astronomical canon – would be particularly useful if their underlying procedures and algorithms were made fully explicit, particularly when the corresponding Chinese sources are liable to give rise to multiple interpretations. PRIMARY SOURCES

The following bibliography of primary sources begins with a complete list of Chinese ofﬁcial histories containing chapters devoted to astronomical canons.

Next, references to authentic Chinese calendars handed down to us are also provided because such calendars are essential in order to check our understanding of Chinese calendrical calculations.

Apart from these two fundamental sources, the history of the calendar also requires various other sources, notably works devoted to the reconstruction of the mathematical techniques used in astronomical canons, administrative treatises, biji (pen jottings), manuals of hemerology, inter alia. Consequently, we also present here a number of such sources, listed by subjects and ranked in alphabetical order, by titles.

Astronomical Canons (Dynastic Histories)

In what follows, ofﬁcial astronomical canons are quoted either from the critical edition of the Chinese dynastic histories published in 1975 and 1976 by Zhonghua shuju ºh (Beijing) or from the former Baina- ben yÍ edition, Shangwu yinshuguan ¤h , (Commercial Press, Shanghai), from 1930 to 1936. When the latter is referred to, the name of the edition, ‘Bainaben’, is indicated explicitly, otherwise, the source referred to corresponds to the former. The chapters and page numbers mentioned hereafter refer to the totality of each ofﬁcial astronomical canon and not only to what only concerns the calendar. The dates of birth and death of the historians responsible for the compilation of each dynastic history are indicated when they are known:

Shiji ªB,(Records of the Historian), ca. 100 BC, Sima Qian ¥y Î, (145 BC–86 BC?), j. 26, p. 1255–1287. Hanshu h,(Former Han History), ca. 100, Ban Gu °ü (32–92), j. 21A–21B, p. 955–1026. Hou Hanshu ¡h (Later Han History), 450, Fan Ye oN (398–445) ‘zhi 1–3’, p. 2999–3100. Songshu h (Song History or Book of Song), ca. 500, Shen Yue SV (441–513), j. 11-13, p. 203–325. Weishu ëh (Wei History), 554, Wei Shou ë[ (506–572) et al., j. 107A , p. 2657–2731. Suishu h (Sui History), 636, Wei Zheng ëÇ (580–643) et al., j. 16– 18, p. 385–501. Jinshu ah (Jin History) ca. 646, Fang Xuanling VÚë (579–648) et al., j. 16–18, p. 473–578. Jiu Tangshu êh (Old Tang History), 945, Liu Xu ò, j. 32–34, p. 1151–1292. Xin Tangshu ±êh (New Tang History), 1060, Ouyang Xiu & Ñ (1007–1072) and Song Qi å (998–1061), j. 25–30B, p. 533– 804. Jiu Wudai shi "ª (Old History of the Five Dynasties), 974, Xue Juzheng å&Ñ, j. 40, p. 1861–1880. Xin Wudai shi ±"ª (or Wudai shiji "ªB), (New History of the Five Dynasties), 1072, Ouyang Xiu &Ñ (1007–1072), j. 58–59, p. 705–712. Songshi ª (Song History), 1345, Tuo Tuo ññ [or Toktogha] et al., j. 68–84, p. 1491–2092. Liaoshi õª (Liao History), 1344, Tuo Tuo ññ [or Toktogha] et al., j. 42–44, p. 517–683. Jinshi ª (Jin History), 1343, Tuo Tuo ññ [or Toktogha] et al., j. 21–22, p. 441–532. Yuanshi -ª (Yuan History), 1370, Song Lian c (1310–1381) et al., j. 52–57, p. 1119–1344. Mingshi ª (Ming History), 1739, Zhang Tingyu ùSÛ (1672– 1755) et al., j. 31–36, p. 515–743. PRIMARY SOURCES 387

Extant Calendars The Most Ancient Extant Calendars

The most ancient Chinese calendars handed down to us are mostly composed of strips of bamboo, inscribed with brush and ink and dating back to the Qin (221 BC–207 BC) and Han dynasties (206 BC–220 AD). Authentic calendars on small wood boards are also extant. They have been discovered at various archaeological sites located all over China, notably at Linyi (Shandong), Yinwan (Jiangsu) and Juyan (Inner Mon- golia).25 Insofar as they concern years prior to 104 BC,26 their modes of calculation (if ever they have been so obtained) are not documented and, anyway, their study lies outside the scope of the present work.

Dunhuang Calendars

As already noted,27 fifty much less ancient annotated calendars (juzhu li Ì¥K), essentially from the IXth and Xth centuries, were discovered on the eve of the XXth century at the famous Dunhuang site among thou- sands of various other documents. Most are fragmentary and since their dates are essentially at variance with those of Chinese official calendrical dates, they are mostly non-official. Only three calendars from the same collection agree with official dates, two are from the years 450 and 451 (Taiping zhenjun 11 and 12) and one from 877 (Qianfu 4). The first two, however, are available only from reproductions (see p. 267 above). The original of the third one belongs to the British Library (see p. 296).

Song Calendars

At least six calendars from the Song dynasty (960–1279) are extant. The two most ancient ones are fragmentary. The ﬁrst was discovered at Karakhoto (now Heicheng Ap, Inner Mongolia). It bears no mention of its year but the Chinese historian Deng Wenkuan has shown that it wholly conforms to Chinese ofﬁcial

25Zhang Peiyu 1991; A. Arrault 2002. 26These calendars have been studied, notably, by Zhang Peiyu 1991, ibid., Huang Yi-long, 1999, 2001a , 2001b, 2002. 27See p. 296 above. 388 BIBLIOGRAPHY calendrical dates obtained from the Chunxi li procedures28 for the year 1182 (Chunxi 9). The other fragmentary calendars have the same origin. Their contents agree with the Kaixi li ¨K procedures29 for the year 1211 (Jiading 4). No less importantly, the Kanazawa bunko gZ0, (Yokohama) owns a fragmentary annotated calendar, demonstrably attributable to the year Jiading 11 (1218). Its content has been fully reconstituted from the Kaixi li ¨K procedures – which were in use from 1208 to 1251, and were thus valid in 1218 – by the Taiwanese historian of the calendar Lin Jin-Chyuan.30 Another minute study by Y.Nishizawa is also available.31 Lastly, a complete annotated calendar for the year Baoyou 4 (1256) is also extant.32 Unfortunately, its techniques of calculation rely on the Huitian li ºFK, an almost wholly lost astronomical canon. Nonethe- less, owing to a critical analysis of the values of lunar and solar constants mentioned in the Songshi ª, Yuanshi -ª and Wang Yinglin (1223– 1296) rT¡’s Yuhai Û encyclopedia, the Taiwanese historian Lin Jin-Chyuan has managed to reconstruct its mean elements.33 Y. Nishi- zawa has also published a complete annotated transcription of this calendar.34 Yuan Calendars For the Yuan dynasty, a copy of an ofﬁcial calendar for the year Zhi- zheng 25 (1365) has been discovered at Karakhoto35 by archaeologists. To my knowledge, this calendar is the only extant from the Yuan dynasty. As Zhang Peiyu has shown,36 the calculations for the year 1365

28Deng Wenkuan 2002e. Chunxi (Pure Splendor) is both the name of an astronomical canon and of a reign-period (1174–1189). 29Kaixi (Opening Auspiciousness) is both the name of an astronomical canon and of a reign-period (1205–1208). See Deng Wenkuan 2002f. 30Lin Jin-Chyuan 1998. 31Y. Nishizawa 2005–2006, vol. 3, p. 301–354. 32Chen Zungui 1984 p. 1611 f. and, above all, COL-astron, vol. 1, p. 691–706 (full reproduction). 33Lin Jin-Chyuan 1997, p. 1–27. 34Y. Nishizawa 2005–2006, vol. 3, p. 367–414. 35Zhang Peiyu, 1994 p. 30–58. 36Zhang Peiyu 1994, ibid. PRIMARY SOURCES 389 done with the Shoushi li procedures entirely agree with its content. Its original has been reproduced in the following publication: Li Yiyou A B (ed.). Heicheng chutu wenshu (Hanwen wenshu juan) ApþZ h=ZZhà> (The documents discovered at Karakhoto (Chinese section)), Beijing, Kexue Chubanshe I.Ìã, 1991.37 Ming Calendars For the Ming dynasty (1368–1644), the situation is considerably better. The catalog of rare manuscripts of the National Central Library of the Republic of China, Taipei, lists 53 such calendars, among which a few relating to the same year. In all, they cover the following 46 years: 1417, 1452, 1419, 1482, 1483, 1484, 1506, 1511, 1512, 1519 (2 copies), 1529, 1534, 1535, 1536, 1539, 1540, 1541, 1543, 1545, 1547, 1548, 1549, 1550, 1552 (2 copies), 1554, 1575, 1578 (2 copies), 1581 (2 copies), 1583, 1585, 1586, 1588, 1591, 1592, 1604 (3 copies), 1606 (2 copies), 1608, 1612, 1614, 1617, 1625, 1629, 1632, 1634, 1639, 1643. See Guoli zhongyang tushuguan shanben mulu »ñ³%h Íê (Cata- log of Rare Books of the National Central Library), Taipei, 4 vol., vol. 2, 1967, p. 500–504. These 46 years represent approximately one-ﬁfth of the totality of different calendars issued during the Ming dynasty, from 1417 (Yongle 15) to 1643 (Chongzhen 16). The Beijing National Li- brary also possess a similar number of calendars from the same period: see COLL., 1983, Beijing tushuguan guji shanben mulu, zi bu ³% h °Íê I (Catalog of Ancient Texts and Rare Books from the Beijing Library, zi bu), Beijing, Shumu Wenxian Chubanshe hêZ¤Ìã, p. 1286–1293. (The expression zi bu designates one of the four traditional divisions of Chinese bibliography dealing with technical subjects, notably mathematics and astronomy. For some more details, see note 38, p. 391 below.) Lastly, the following complete reproduction of 105 calendars from the Ming dynasty has recently been released by the Beijing Library, but I did not have access to it: Guojia tushuguan cang Mingdai Datong liri huibian »%h ÙK^_ (Calendars from the Ming dynasty preserved at the National Library), Beijing, Beijing Tushuguan Chubanshe ³%h Ìã, 2007.

37Zhang Peiyu 1994, ibid., p. 58. 390 BIBLIOGRAPHY

Other Primary Sources Apart from authentic calendars, primary sources likely to interest the study of the Chinese calendar, from the viewpoint of its surface or deep structures, are at the same time potentially very important and difﬁ- cult to determine in advance because all sorts of pieces of information are potentially retrievable from a priori unexpected sources, Chinese and non-Chinese, such as those concerning literature, administration or even military art, to quote but a few. Hence the following partial list, recording Chinese, Korean, Japanese, and even forgotten French sources, doubtlessly of interest in this respect.

Collections of Primary Sources 1. COL-astron: REN Jiyu µ (ed.), Zhongguo kexue jishu dianji tonghui, tianwen juan »I.*Î°; FZà (Gen- eral Collection of Chinese Scientiﬁc and Technical Works, Astron- omy), 8 vol., Zhengzhou, Henan Jiaoyu Chubanshe ªP>Ì ã, 1993;

2. COL-math: same reference, same number of volumes as the preceding one, but for mathematical sources;

COL-astron and COL-math are two monumental collections of primary sources, 16 huge volumes in all, each composed of approximately 2000 pages. Both are composed of fac-simile reproductions of various documents (such as inscriptions on bones and shoul- derblades), manuscripts (many from the Dunhuang collection) and major astronomical and mathematical books from all periods. In each case, a general introduction presents these works.

3. WYG: Yinying Wenyuange Siku Quanshu ÅZe°0h (Reproduction of the Siku quanshu Collection Preserved at the Wen- yuange Library), 1500 vol., 1986 edited by Yun Lu .. (1695– 1767) et al., 1500 vol., Taipei, Shangwu yinshuguan ¤h , 1986; PRIMARY SOURCES 391

The Siku quanshu °0h collection (Complete library of the Four Treasuries)38 is a compilation of extant Chinese books initiated by the Qianlong Emperor in 1772.39

Individual Works

4. Chouren zhuan ßF (Biographies of chouren), 46 j., ca. 1810,40 RUAN Yuan ¨- 41 (1764–1849). Taipei, Shijie shuju t&h , 1982.

The Chouren zhuan is a compilation composed of a mosaic of quo- tations, mostly extracted from chapters of Chinese dynastic histories devoted to astronomical canons and related subjects. It has been devised in order to highlight the intellectual proﬁles of ancient Chinese and non-Chinese specialists of computistics, astronomical canons and mathematics, the chouren ß, a term literally meaning ‘specialists whose expertise is transmitted from father to son’.42 This famous work follows the chronology of successive dynasties and is organized like a biographical dictionary. Strictly speaking, however, its biographical component is quite minimal and restricted to a few basic details, such as dates of birth and death, administrative titles and functions. Much more strikingly, the intellectual aspects of the chouren’s activities is on the contrary fully stressed and

38i.e.: (1) canonical books (or classics), (2) histories, (3) ‘masters’ (a section composed, inter alia, of technical texts, notably mathematics and astronomy) and (4) literature. 39See Guy, R. Kent 1987. 40From Wang Ping 1974. 41On Ruan Yuan, see Wang Ping 1974, op. cit., and Betty Peh-T’i Wei 2006. 42This term ﬁrst appears in Sima Qian’s famous Shiji (Records of the Historian), j. 26, p. 1258, and has lastingly designated the members of the Astronomical Bureau. From the end of the Ming dynasty, however, it has gradually been endowed with a less restrictive meaning and has more widely designated specialists of all sorts of mathematical calculations, without the necessary idea of hereditary transmission. According to another explanation, the character chou – the ﬁrst of the compound chouren – is syn- onymous with chou ®, another character having the same pronunciation but meaning ‘to calculate’ or ‘to manipulate divinatory rods’. (See DKW, 7, 21967: 11, p. 8049.) This latter etymology makes sense because the connection between calculation and divination it implies is particularly relevant in the case of China but it is probably too good to be true. 392 BIBLIOGRAPHY

provides precious developments about their technical works and intellectual background. Hence its interest for everything concerning the epistemology of Chinese mathematics, understood as including not only logistics but also mathematical astronomy. No less interestingly, these intellectual biographies are systematically followed by critical appraisals, lun ¡, intended to disclose Ruan Yuan’s opinion on what he regards as the strong and weak points of the Chinese astronomical tradition, with respect to the aspects of the Western tradition he was aware of, from Chinese translations (or rather adaptations) of scientiﬁc Western works due to the initiatives of Jesuit missionaries. In each case, Ruan Yuan highly values the most typical aspects of Chinese ancient astronomy, namely a long- lasting belief in the temporary validity of mathematical techniques, an unceasing search for a greater precision of their forecasts, a constant desire to synthesize all sorts of techniques, even those mutually contradictory, and an overwhelming preference for prediction over explanation, an aspect of Chinese quantitative sciences already dominant in China well before him, even though extremely rare examples to the contrary exist.43 These ideas were lastingly inﬂuen- tial during the entire nineteenth century and several sequels to the Chouren zhuan were successively published, notably LUO Shilin 1ÿ] (1789?–1853)’s Xu Chouren zhuan ßF 1840/1982* (A Sequel to the Chouren zhuan), chapters (juan) numbered from 47 to 52, Taipei, Shijie shuju t&h . (This volume concerns Qing scholars and tends to view mathematics as more and more important than astronomy.)

5. Daxue yanyi bu .L (Complements to the ‘Great Learn- ing’),44 160 j., 1487, QIU Jun w (1420–1495).45 See WYG, vol. 713, p. 72–97.

43See, mostly, Liu’s Hui S ’s proofs reproduced in his celebrated commentary of the Jiuzhang suanshu ÜaÕ (Computational Techniques in Nine Chapters) (263 AD) (English translation in Guo Shuchun, J. Dauben and Xu Yibao 2013). 44The Daxue yanyi .L is a former work by a disciple of Zhu Xi (1130–1200). 45See Wu Chi-hua and Ray Huang 1976. PRIMARY SOURCES 393

The Daxue yanyi bu46 is a manual of administration devoted to various aspects of tasks devoted to members of the public administration such as military affairs, public funds, human resources manage- ment, transportation and even questions pertaining to the technical dimension of astronomical canons, to the extent that it provides the exact values of the shift-constants in the Shoushi li case.47

6. Gujin lüli kao *K (Research into Pitch-Pipes and Astro- nomical Canons, Ancient and Modern), 72 j.,48 1607, XING Yunlu 2 (ca. 1560 – ca. 1620), provincial judge (anchasi ¶D¥) from the Henan province, doctorate in 1680. Reproduced in WYG, vol. 787, p. 413–653.

The Gujin lüli kao revolves around questions of numerology and hemerology in large part but, most interestingly for us, he also provides fully developed examples of calendrical calculations performed with the Shoushi li (1281–1384) and the Datong li (1368– 1644): no less than 24 chapters juan, or one-third of the whole treatise, are devoted to this subject.49 In particular, the Gujin lüli kao provides the details of all the steps of the calculations of a solar eclipse, dated 22/9/1596 (Chinese date 1/VIII*/Wanli 24),50 according to the latter canon51 and of a lunar eclipse, dated 3/4/1605 (Chinese 16/II*/Wanli 33), according to the former canon.52

7. Gujin tuibu zhushu kao *.M (Ancient and Modern Pro- cedural Techniques (shu ) of Predictive (tuibu .M) Astronomical Calculations), 2 j., 1867.

This work is the last part of a very important treatise of calendrical chronology, the Lidai changshu jiyao, succinctly presented on

46WYG, vol. 713. 47WYG, vol. 713, p. 96–97. 48Chen Meidong 2003a, p. 618–620, provides an excellent presentation of the content of this treatise, chapter by chapter. 49Gujin lüli kao *K, j. 36-59, in WYG, vol. 787, p. 413–653. 50See F.R. Stephenson and M.A. Houlden 1986, p. 388. 51Ibid., p. 551–560. 52WYG, vol. 787, p. 561–566. 394 BIBLIOGRAPHY

p. 377 above. It contains a critical list of ofﬁcial and non-ofﬁcial astronomical canons (approximately two hundred in all) and provides precious details such as the values of their fundamental constants, when they are known.

8. Kaiyuan zhanjing -B (Kaiyuan reign-period (713–741) Trea- tise on Astrology), 120 j., ca. 742, QUTAN Xida ~O¾. Quoted from the edition of the text published by Zhongguo Shudian »h 7, Beijing, 1989.

The Kaiyuan zhanjing is most often mentioned in connection with the question of the transmission from India to China, during the early Tang dynasty, of the decimal numeration of position by means of nine written digits and a point representing a zero.53 Moreover, this famous astrological treatise also provides various technical details of interest in Chinese astronomical canons, ofﬁcial or non- ofﬁcial, promulgated before the Tang dynasty and sometimes not known otherwise. Among these, we note lists of calendrical constants recording the number of solar years elapsed since the Supe- rior Epoch and the values of some generalized Metonic constants (see Kaiyuan zhanjing, j. 103, p. 732–741).

9. Lishi yishu Ahöh (Posthumous works of Master Li), LI Rui54 A Þ (1765–1814). Reproduced in COL-astron, vol. 2, p. 701–818.

The following sections of the Lishi yishu analyze the details of calendrical procedures attested in ancient astronomical canons: (a) Han Santong shu ëÙ (The Santong55 Calculation Pro- cedures of the Hanshu), 3 j. See COL-astron, vol. 2, p. 708– 741. (b) Han Sifen shu °5 (The Sifen Calculation Procedures of the Han Dynasty), 3 j. See COL-astron, vol. 2, p. 741–778.

53Yabuuchi 1963a/1988*, p. 6. 54On Li Rui, see note 12, p. 247 above. 55Santong means ‘Triple Concordance Astronomical Canon’, an allusion to its three supra-annual cycles, composed of 19, 76 and 1520 years, respectively. PRIMARY SOURCES 395

(c) Han Qianxiang shu é (The Qianxiang56 Calculations Procedures of the Han Dynasty),57 2 j. See COL-astron, vol. 2, p. 778–797. (d) Buxiu Song Fengyuan shu Ñ- (The Fengyuan58 Calculation Procedures of the Song Dynasty, Revised and Re- constituted), 1 j. See COL-astron, vol. 2, p. 798–802. (e) Buxiu Song Zhantian shu ÑF (The Zhantian59 Cal- culation Procedures of the Song Dynasty, Revised and Recons- tituted), 1 j. See COL-astron, vol. 2, p. 803–806. (f) Rifa shuoyu qiangruo kao ^°iõú3 (Research into the Excess and Default Values of the rifa and shuoyu).60 See COL-astron, vol. 2, p. 807–818. In these various treatises, the author tries to reconstruct the correct values of the fundamental constants of various astronomical canons by taking advantage of fragmentary indications, gleaned here and there, in various sources. Even now, his work remains useful.

10. Linde shu jie Æ (An Explanation of the Linde61 Canon Cal- culation Procedures), 3 j., 1867, LI Shanlan A 62 (1811–1882). From the complete works of Li Shanlan – Zeguxizhai suanxue, J {KÕ. (The Mathematics of the ‘Studio Devoted to the Imitation of the Ancients’), 42 j., 1867, edited by Mo Youzhi B.63

56Qianxiang = Supernatural Manifestation. 57The Qianxiang li was elaborated during the Han dynasty but never ofﬁcially used. See Hou Hanshu, zhi 2 “lüli 2”, p. 3043 (note 1). 58Fengyuan = Oblatory Epoch (N. Sivin 2009, p. 50). 59Zhantian = Augury of Heaven. 60The rifa and shuoyu (literal meanings: the ‘day denominator’ and the ‘lunation remainder’, respectively) are two technical terms referring to the denominators of the fractions used in order to express the length of the solar year and of the lunar month, respectively. 61Linde li = ‘Unicorn Virtue canon’ (an allusion to the rarity and precious character of this mythical animal). 62On Li Shanlan, see Wang Yusheng 1990, Horng Wann-sheng 1991, J.-C. Martzloff 1997*/2006* p. 341–351. 63For an incomplete, but useful, fac-simile reproduction of the Linde shu jie, limited to its two last chapters (j. 2 and 3), see COL-astron, vol. 6, p. 1035–1049. 396 BIBLIOGRAPHY

The Linde shu jie Æ offers a tentative interpretation of the elliptic text of the Xin Tangshu concerning the calculation of true elements according to the Linde li ÆK64 procedures.65

11. Lüli rongtong KÈ; (A Comprehensive Study of Pitch-Pipes (lü) and Astronomical Canons (li)), 4 j., ca. 1590, ZHU Zaiyu (1536– 1611). Reproduced in WYG, vol. 786, p. 556–666.

Like the Shengshou wannian li O.0OK mentioned below, the Lüli rongtong was composed in view of a reform of astronomy.

12. Mengqi bitan 0ç,(Dream Pool Essays), 26 j., 1086, SHEN Gua SÀ (1031–1095). See HU Daojing f¼ Mengqi bitan jiao- zheng 0çlJ (Critical edition of the Mengqi bitan), Shang- hai, Guji Chubanshe °Ìã, 2 vol., 1987.

The pen jotting litterature (biji) B sometimes contain passages of interest in the study of astronomical canons. The famous Mengqi bitan belongs to this category of texts and its seventh juan (chapter), entitled xiangshu éó (Numbers and ‘images’ xiang),66 is particularly interesting in this respect: it is composed of a series of notes about the calendar and astronomy (notably, a proposal of reform of the traditional lunisolar calendar, in favor of a purely solar calendar, based only on the division of the solar year into 24 solar periods). Another very interesting passage of the same juan revolves around the question of the intrinsic limitation of the mathematical predictive techniques of Chinese astronomy.67

13. Shengshou wannian li O.0OK (Perpetual Astronomical Canon Dedicated to the Longevity of our Saint Emperor), 5 j., 1595, ZHU Zaiyu (1536–1611). WYG, vol. 786, p. 451–555.

64Xin Tangshu, j. 26, ‘ li 2 ’, p. 559 f. 65An analysis of its calculation techniques taking into account Li Shanlan’s ideas about the determination of true elements (new moons and the like) is propounded in Liu Jinyi and Zhao Chengqiu 1984. 66This term designates here the moon and the planets. 67For a minute study of this important source, see N. Sivin 1989. PRIMARY SOURCES 397

The Shengshou wannian li is an astronomical canon devoted to the reform of astronomy. It contains a substantial critique of former canons.68

14. Tianwen69 dacheng70 guankui71 jiyao FZWÑì (Compre- hensive Astrological Survey, Humbly Compiled and Limited to the Essentials), 80 j., 1653, HUANG Ding ?ÿ.72

Huang Ding was a Regional Commander zongbing73 Àº of Zhe- jiang province. His treatise is a compilation of Chinese astrological texts from all periods of interest in military affairs. It also includes some precious details, lacking in dynastic histories, concerning the Shoushi li 0`K and even the Huihui li //K.74 Unexpectedly, this rare book has been transmitted to Japan and the Japanese mathematician Seki Takakazu, mentioned on p. 400 below, draws on it abundantly in one of his manuscripts, thus introducing elements of Islamic astronomy into Japan for the ﬁrst time.75

15. Xieji bianfang shu ÜSï]h (A Comprehensive View on the Har- monies Between Cycles and the Distinction Between Allowed and Forbidden Directions), 36 j., ca. 1739, YUN Lu .. et al. See LI Ling Aë (ed.), Zhongguo fangshu gaiguan, xuanze juan »] ÃÌóCà,(An Overview of Chinese Divinatory Techniques,

68On Zhu Zaiyu, see K.G. Robinson and Fang Chaoying 1976; Chouren zhuan (notice, p. 391 above), j. 31, p. 371–378 ; Dai Nianzu 1986; Wang Baojuan 1986. 69In modern Chinese, tianwen FZ means ‘astronomy’ but, in ancient texts, its literal meaning is ‘celestial drawings’ or ‘celestial signs’. It refers to judicial astrology. 70The last term of this title, jiyao ì, means ‘limited to the essential’ and seems incompatible with the idea of comprehensiveness expressed by the term dacheng W. However, the author only intends to stress the fact that his domain of study is vast and that he only claims to have partly mastered it. 71The Chinese term guankui Ñ has an allusive value, impossible to understand without an awareness of its origin. It ﬁrst occurs in Zhuangzi, j. 17, ‘Autumn Flood (Qiu shui Ki)’ and its literal meaning is ‘to gaze at the sky through a bamboo tube’, i.e. ‘to have a limited knowledge’. This is an expression of modesty. 72Biographical notice in Ding Fubao and Zhou Yunqing 1957, p. 52a and 52b. 73Ch.O. Hucker 1985, item 7146. 74Qu Anjing 1995. 75J.-C. Martzloff 1998b. 398 BIBLIOGRAPHY

‘Elections’),76 2 vol., Beijing, Renmin Zhongguo Chubanshe ßÓ »Ìã, 1993.

The text of the Xieji bianfang shu partly occupies the ﬁrst volume (p. 84–464) and the whole of the second one of this publication. It is also included in the section of the famous Siku quanshu, °0 h) collection (WYG, vol. 811, p. 109–1022) devoted to divinatory techniques (shushu lei óv).

Mathematical sources

Chinese mathematical sources have a bearing on the study of astronomical canons for several reasons: quite often, astronomical and calendrical procedures take for granted, without warning, a large number of mathematical techniques such as arithmetical operations, proportionality or fractions, but also particular problems, overtly or covertly concerning astronomy. That is the case, in particular, of many problems of pursuit (between a dog and a rabbit and the like), giving a plausible clue to the logical origin of mathematical procedures concerning the determination of conjunctions between celestial bodies.77 Lastly, certain features of astronomical canons and mathematical treatises are clearly mutually related. In particular, this point is particularly relevant in the case of the history of zero.78 See, for instance:

16. Shushu jiuzhang ó h Ü a (Computational Techniques in Nine Chapters), Yijiatang !GÄ collection, 1842. This famous mathematical treatise is important for the history of mathematics, not only because of its obvious connection with the antique Nine Chapters tradition and its early usage of a complete decimal place-value system of numeration including a written zero, but also because it contains three problems devoted to calendrical and astronomical calculations,

76This technical term designates the choice of auspicious and non-auspicious days. See J. Tester 1989, p. 88 f. 77J.-C. Martzloff 1997*/2006*, p. 140; A. Bréard 2002. 78See p. 125 above. PRIMARY SOURCES 399

notably concerning a famous astronomical canon, the Kaixi li ¨ K (1208–1251), and another one about the motion of Jupiter.79

17. Suanxue qimeng Õ.@ (Introduction to Computational Science) [Zhu Shijie eto (1299), in KODAMA Akihito ÉÛß 1966. Jugo¯ seiki no Chosen kan do-katusi-ji¯ sugaku¯ sho è"tSG > YþCÌó.h (Chinese mathematical books printed in Korea with Movable Type During the Fifteenth Century), Tokyo, privately printed.

Korean and Japanese sources

Korean sources

One of the most important Korean source for Chinese astronomical canons is:

18. Koryoˇ sa/Gaoli shi {ª (1451), j. 50-52, Seoul, Yônse taehakkyo ;¨., Tongbanhak-yôn’guso ].@~X, 1955. Devoted to the ofﬁcial history of the Koryoˇ dynasty (918–1392), this treatise is composed in Chinese and organized exactly like Chinese dynastic histories.80 It expounds the calculations of the Xuanming li/Son- myong ryok81 K (j. 50, p. 81–110) and those of the Shoushi li /Susi ryok82 0`K (j. 51 and 52, p.112–183). The treatise of the Koryoˇ sa devoted to the Xuanming li is particularly precious for the study of Chinese calendrical calculations because what Chinese sources have to offer in this respect is rather limited: the section of the Xin Tangshu concerning this canon83 merely contains lists of numerical constants and astronomical tables and some lapidary ex- planations, but not the details of its calculation procedures. For the same reason, its section devoted to the Shoushi li calculations should

79See Li Yan and Wang Shouyi 1992, p. 82–171; Chen Xinzhuan, Zhang Wenhu and Zhou Guanwen 1992, p. 104–130; Wu Wenjun 2000, p. 390–403. A complete critical English translation of these problems would be highly desirable. 80See K. Pratt and R. Rutt 1999, p. 245. 81Koryoˇ sa, j. 50, p. 81–111. 82Ibid., j. 51 and 52, p. 112–182. 83Xin Tangshu, j. 30A, ‘li 6a’, p. 745–751 (calendrical calculations) and p. 752–770 (positions of the planets and eclipses). 400 BIBLIOGRAPHY

not be neglected.84 Other Korean sources devoted to the Shoushi calculations exist but they have not been much studied.85 Beyond the Shoushi li, the diffusion of islamic astronomy in Korea through the intermediary of the Chinese translation of the Huihui li //K is also quite noteworthy because it provides access to new primary sources, not necessarily previously taken into account by historians of Chinese astronomy.86

Japanese sources

Japanese sources also offer all sorts of resources.87 Among the numerous relevant Japanese sources of interest in the study of Chinese astronomical canons, we note in the ﬁrst place:

19. Juji reki gi kai 0`KÈ (The Shoushi li yi/Juji reki gi Eval- uated),88 6 kan,89 ca. 1720–1730,90 TAKEBE Katahiro I¯Â. Quite legible undated manuscript, Tokyo Univ. (MS T30/95).

20. Juji reki jutsu kai 0`K, 4 kan, ca. 1720–1730, (The Calcu- lation Procedures of the Shoushi li Explained). TAKEBE Katahiro I¯Â, Undated manuscript, Tokyo University (MS T30/99).

21. Juji reki su¯ kai 0`Kó (The Numerical Constants of the91 Shoushi li), 2 kan, ca. 1720, TAKEBE Katahiro I¯Â. Undated manuscript, University of Tokyo (MS T30/102).

84Yuanshi, j. 53 à 55 ‘li 2 to 4’, p. 1153–1264. 85Lee Eun-Hee 1997 and Lee Eun-Hee and Jing Bing 1998. 86See Shi Yunli 2003 (beyond this essential question of sources, this author also raises at the same time the pertinent question of the enormous difﬁculties raised by the conversion of Islamic dates into the Chinese system). 87See K. Yabuuchi and S. Nakayama 2006, p. 2–3. 88The expression Shoushi li yi is the title of a part of the Inception Granting Canon (Shoushi li) devoted to a comparative evaluation of its merits with respect to previous astronomical canons (Yuanshi, ‘li 1’ and ‘li 2’, j. 52 and 53, p. 1120–1189; English translation in N. Sivin 2009, p. 254–388). 89kan à (chapter) corresponds to the juan à of the Chinese. 90Datation from A. Horiuchi 2010, p. 227. 91 The literal meaning of the term su¯ ó is ‘numbers’ and it refers more precisely to ‘numerical constants’ in this context. PRIMARY SOURCES 401

In these three manuscripts, the mathematician TAKEBE Katahiro I¯Â comments in a particularly minute way on the techniques of calculation of the Shoushi li and those of the main Chinese of- ﬁcial astronomical canons by using numerous examples of calculations of calendrical events (winter solstices, new moons, solar and lunar eclipses, notably) whose results are listed in the corresponding Chinese text. In each case, the famous attendant of the shogun¯ Tokugawa Ienobu92 provides the successive steps of the calculations which should be performed without omitting the least detail.

Rare Sources Newly Made Available

A compilation of the following rare Chinese texts from the Ming dynasty (1368–1644) has been issued in 2010:

22. SHI Yunli í ¦ (ed.), Haiwai zhenxi Zhongguo kexue jishu dianji jicheng ² »I.*Î°/W (Rare and Precious Scientiﬁc and Technical Books Preserved Abroad), Hefei, Zhongguo Kexue Jishu Daxue Chubanshe »I.*.Ìã. This book is a reproduction of the following nine rare and precious astronomical texts whose originals are preserved in Korean, Japanese and Taiwanese libraries:

(a) Xuanming li K (The Manifest Enlightenment Astronom- ical Canon), p. 1–32. A different edition of the same text has already been mentioned above (p. 399). (b) Shoushi li licheng 0`KñW (Handy Tables for the Inception Granting Astronomical Canon), p. 33–74. (c) Datong lifa tongggui ÙK°; (A Path to the Methods of the Great Uniﬁcation Astronomical Canon), p. 75–210. (d) Shoushi li genian jiaoshi 0`K&Oø³ (Eclipses Calcu- lated for the Various Years of Validity of the Inception-Granting Astronomical Canon), p. 211–220. (e) Datong li zhu ÙKÛ (Hemerological Annotations for the Great Uniﬁcation Astronomical Canon), p. 221–426.

92See A. Horiuchi 2010, ibid., p. 118. 402 BIBLIOGRAPHY

(f) Da Ming Datong lifa Ù K ° (The Methods of the Great Uniﬁcation Astronomical Canon of the Great Ming Dy- nasty), p. 427–456. (g) Weidu Taiyang tongjing ZH&;5 (General Canon of So- lar Latitudes), p. 457–488. (h) Xuande shinian yue-wu-xing lingfan ÆèO`"ÏÜÙ (The Assaults93 of the Moon and the Five Planets During the Tenth Year of the Xuande Era [1435]), p. 489–506. (i) Huihui lifa //K° (The [Chinese] Muslim Astronomical Canon), p. 507–719.

The importance of these sources, printed during the Ming dynasty, stems not only from their rarity and authenticity but also from the novelty of the interpretations they have recently given rise to or are likely to lead to. For example, the study of (c) has convincingly challenged the accuracy of previous reconstitutions of eclipse calculations based on the usual version of the Datong astronomical canon, incorporated in the Mingshi ª (Ming History).94 Moreover, as the editor of this compilation explains (p. 79) it seems that certain sources have been constantly revised and updated and this mere fact introduces a supplementary degree of complexity in an already extremely entangled history. The editors of the present compilation have aimed at making the content of these ancient texts easily accessible to a modern audience. Consequently, they have discarded the initial layout of the originals in favor of a modernized presentation (i.e. an edition whose layout and typographical conventions follow present standards). For instance, they have displayed the Chinese text in horizontal rather than in vertical lines and they have redesigned astronomical tables. But they have neither modiﬁed the textual content of the originals nor

93This term is a tentative rendering of the Chinese astrological expression lingfan Ü Ù, implying that some celestial body threatens the integrity of another one in various ways, for instance when it conceals it, when it moves from below towards it and so on. See Ho Peng Yoke 1966, p. 36–39. Separate renderings of the technical terms ling Ü and fan Ù are also possible. See the footnotes 40 and 41, p. 16 above. 94See Li Liang, Lu Lingfeng and Shi Yunli 2010. PRIMARY SOURCES 403

the initial complex forms of the Chinese characters. Each time, they have provided a concise description of each text. Even so, a fac- simile reproduction of these important sources would still be highly desirable because the information conveyed is not exactly the same in both cases.

The Jesuit Reform of Chinese Astronomy

The following sources rank among the most important ones on the initial state of the reform of Chinese astronomy:

23. PAN Nai (ed.). Chongzhen lishu, fu Xiyang xinfa lishu zengkan shi zhong âÈKh!õ±°Kh¦èË (Chongzhen reign- period [1628–1644] Treatise on Astronomy with an Addition of Ten Other Works Taken from the Xiyang xinfa lishu). Shanghai, Shanghai Guji Chubanshe îî°Ìã. 2009, 2 vol., 48+2088 p. This long-awaited publication is an extensive collection of early treatises on the European reform of Chinese astronomy95 whose initial collection was called Chongzhen lishu and which was later augmented and renamed Xiyang xinfa lishu (Treatise of Astronomy according to the Western New Methods).

24. XU Guangqi 6 @, Xu Guangqi ji 6 @/ (Collected Works of Xu Guangqi), Shanghai Guji Chubanshe î°Ìã, 1994, 2 vol. Whereas the preceding collection is exclusively devoted to the reform of Chinese astronomy, the present one tackles various other subjects. Its author, Xu Guangqi (1562–1633) – one of the ﬁrst Chi- nese Christian converts, inseparable from the famous Jesuit missionary Matteo Ricci (1552–1610) – was a key ﬁgure of the reform of Chinese astronomy from 1629.96

25. LONGOMONTANUS [C. SEVERIN], Astronomia Danica [. . . ], Amsterdam, 1622. This work, based on Tycho Brahe’s conception of astronomy, is an important set of astronomical tables and techniques which were probably used in competitive eclipse predictions

95Among ancient but still precious studies on this subject, see, notably: Bernard- Maître 1945; R. Malek 1998. 96C. Jami et al. 2001. 404 BIBLIOGRAPHY

established by Chinese authorities, in order to determine the better technique to be retained in view of their reform of Chinese astronomical canons.97

Antoine Gaubil’s History of Chinese Astronomy

The following pioneering work on the history of Chinese astronomy and other related topics – compiled by Étienne Souciet S.J. (1671– 1744) from Chinese documents sent in France to him by the Jesuit missionary Antoine Gaubil (1689–1759)98 – has rightly remained famous:

26. SOUCIET, Étienne (Le P.) Observations Mathématiques, Astronomi- ques, Géographiques, Chronologiques et Physiques, tirées des anciens livres chinois ou faites nouvellement aux Indes et à la Chine, Par les Pères de la Compagnie de Jésus, rédigées et publiées par le P.E. Souciet de la même Compagnie [Mathematical, Astronomical, Geographical and Physical Observations Taken from Ancient Chi- nese Books or Made in India and China by the Fathers of the Com- pany of Jesus, compiled by Father E. Souciet of the Same Company], Paris, Rollin, 3 tomes, 1729–1732. Although Gaubil’s history of Chinese astronomy is difficult to get to grips with because of the somewhat disjointed character of its com- position, the intrinsic interest of the documentation he has gathered for the first time, with a full awareness of the mathematical character of Chinese astronomy, still makes it a work of lasting interest, despite its old age. Gaubil’s work is indeed all the more exceptional given that the perception of the importance of this major aspect of Chinese science has considerably declined after him, save, of course, in China and Japan, as least partly: during the first half of the XXth century, Western historians of Chinese astronomy have often cher- ished endless speculations in the hope of determining the origins of

97K. Hashimoto 1988. 98Long ago, J. Dehergne S.J. gathered precious elements on Gaubil (see J. Dehergne 1944, 1945 and 1973); more recently, R. Simon has edited Gaubil’s correspondence (see Simon 1970) and I. Iannacone has established a list of Gaubil’s manuscripts preserved at the Observatoire de Paris (see I. Iannaccone 2000). PRIMARY SOURCES 405

Chinese astronomy and they have often relied for that purpose on an often incredibly tenuous documentary basis. For example, the in- ﬂuential Léopold de Saussure (1866–1925) – the younger brother of the famous linguist Ferdinand de Saussure99 – does not hesitate to attribute to the ancient Chinese (those who lived two thousand years BC, before the invention of writing in China) the knowledge of ex- traordinary astronomical methods, unheard of among the Egyptians, the Babylonians and the Greeks, on the basis of wild speculations consisting in an overinterpretation of the meaning of a few strings of Chinese characters, taken from the Shujing hB: hence, according to him, the superiority of Chinese antique methods of precision astronomy, long forgotten and superseded in Europe only in recent periods.100 Although, Gaubil had also certainly not been impervious to the charms of unbounded speculation, he lived in a much earlier period and, quite astonishingly, his history of Chinese astronomy is fundamentally based on the voluminous documentation he could ex- tract from Chinese astronomical canons. Quite surprisingly, he was even able to provide exact quantitative details about the initial steps of the calculations of a lunar and a solar eclipse according to the Shoushi li (solar and lunar eclipses of the 22/9/1596 and 3/4/1605, respectively). He also indicates that he has borrowed his technique from Hing-yun-lu [Xing Yunlu] 2, the author of the Gujin lüli kao *K, already mentioned above.101 Insofar as eclipse calculations have played an essential role in the history of Chinese mathematical astronomy, it would certainly still be essential to bring to light the minutiae of the underlying mathematics for a large audience of historians of astronomy and mathematics, given than even now, almost three centuries after Gaubil, only a limited number of

99See L. de Saussure’s biography in P. Pelliot 1925–26. 100L. de Saussure 1930, préface, p. v: (“[les méthodes astronomiques des Chinois de la haute antiquité sont] devenues chez nous depuis deux siècles la base de l’astronomie de précision, mais [. . . ] les Égyptiens, les Chaldéens et les Grecs ne semblent [n’en] avoir tiré aucun parti)” (“[The astronomical methods of the Chinese from high antiquity] have become the basis of precision astronomy in Europe over the two last centuries, but the Egyptians, the Chaldeans and the Greeks have apparently not taken advantage of it”). 101Souciet, ibid., p. 204–207. 406 BIBLIOGRAPHY

historians, such as Li Yong, Zhang Peiyu and Qu Anjing, have tackled the question.102

Philippe de La Hire’s Astronomical Tables

27. La Hire, Philippe de. Tables astronomiques, dressées et mises en lumiere par les ordres et par la magnificence de LOUIS LE GRAND, Dans lesquelles on donne les mouvemens du Soleil, de la Lune et des autres planetes, déduits des seules Observations, et indépendamment d’aucune hypothese [. . . ] par M. DE LA HIRE, Professeur Royal de Mathematiques et de l’Académie Royale des Sciences [Astronomical tables [. . . ], Drawn up [. . . ] by Order of Louis the Great [Louis XIV, the Sun king], Where the Motion of the Sun, the Moon and Other Planets are Deduced from Mere Observations, Independently of Any Hypothesis, by M. de La Hire, Royal Professor of Mathematics and Member of the Royal Academy of Sciences], Third ed. [. . . ], Paris, Montalant, 1735. The mention of this famous work is intended to stress the fact that some of the most significant elements of the Western scientific rev- olution – notably the belief in the possibility of discovering mathematical laws of nature, such as Kepler’s Laws – were still not fully accepted at the highest level, as late as 1735, not only in China, but also in Europe.

102See Li Yong 1996; Zhang Peiyu 1994; Qu Anjing 2008. SECONDARY SOURCES

This bibliography is composed of two parts: (1) collective works and (2) books and articles. Collective works (COLL.) refer to large-scale dictionaries, encyclopedias, catalogs of libraries and collected articles. The other references are composed of books in various languages including, of course, Chinese and Japanese. In particular, Chinese and Japanese periodicals fre- quently have two titles, the ﬁrst in one of these two languages, the other in English or even latin. When such a second title exists, we indicate it explicitly, otherwise we only mention original titles without translations.

COLLECTIVE WORKS – COLL. 1974. Ying-Han tianwenxue cidian zFZ.ÞÎ (Chinese-English Dic- tionary of Astronomy ), Beijing, Kexue Chubanshe I.Ìã. – COLL. 1976. Dictionnaire Français de la langue chinoise, Paris, Institut Ricci, Centre d’Études chinoises. – COLL. 1980. Zhongguo da baikequanshu (tianwenxue) »yIhFZ .(The Great Encyclopedia: Astronomy), Beijing and Shanghai, Zhongguo Da Baikequanshu Chubanshe »yIhÌã. – COLL. 1988. Zhongguo gudai tianxiang jilu zongji »FéBÀ/ (Gen- eral Catalog of Ancient Chinese Astronomical Observations), Jiangsu Kexue Jishu Chubanshe nÁI.*Ìã. – COLL. 1989a. Zhongguo gudai tianwen wenwu lunji »FZZÎ¡/ (Collected Articles on the Chinese Cultural Heritage: the Astronomy of Ancient China), Beijing, Wenwu Chubanshe ZÎÌã, 1989. – COLL. 1989b. Zhongguo tianwen shiliao huibian diyi juan » F Z ª ] _ Ï×à (Materials for the History of Chinese Astronomy, First series), Bei- jing, Kexue Chubanshe I.Ìã. [To my knowledge, the following series have never been released]. – COLL. 1991–1994. Zhongguo xiandai kexuejia zhuanji »¨I.FB (Bi- ographies of Chinese Men of Science from the Modern Period), 6 vol., Beijing, Kexue Chubanshe. I.Ìã. – COLL. 1997. Hanyu da cidian +ÞÎ (Great Chinese Dictionary) 3 vol., Shanghai, Hanyu Da Cidian Chubanshe +ÞÎÌã. – COLL. 1999. Hanyu fangyan da cidian +]ÞÎ (Great Dictionary of Chinese Dialects), Collective work published by the Fudan University (Fudan daxue

ÆË.) and the University of Foreign Languages (Kyoto) (Kyoto Gaikokugo Daigaku ³K²»+.), Beijing, Zhonghua Shuju ºh . – COLL. 2001a. Zhangjiashan Han mu zhujian : 247 hao mu, ù )z Þ°Úr) (The Bamboo Slips Discovered in a Tomb at Zhangjiashan; The Tomb no. 247), Beijing, Wenwu Chubanshe ZÎÌã. – COLL. 2001b. Grand dictionnaire Ricci de la langue chinoise. Six volumes (more than 1000 pages each), with one supplementary volume devoted to special topics (Chinese Administration, Astronomy and the Calendar, Medicine, etc.) and indices, Institut Ricci, Paris-Taipei, Desclée de Brouwer. – COLL. 2005. Le manuel des éclipses (Institut de Mécanique céleste et de Calcul des Éphémérides, Obseervatoire de Paris), EDP Sciences.

BOOKS AND ARTICLES AKHMEDOV, A. A. (ed.), 1994 [Russian language]. Ulughbeg Mukhammad Tara- gaï (1394–1449) “Zidzh, Novye Guraganovy Astronomicheskie Tablitsy”(Ulug-Beg Muhammad Taragaï (1394–1449): Z¯ıj, New Astronomical Tables), Tashkent. ANG Tian Se, 1976. ‘The Use of Interpolation Techniques in the Chinese Calendar’, Oriens Extremus, Wiesbaden, vol. 3, no. 2, p. 135–151. – 1979. I-Hsing [Yixing × ] (683–727 A.D.): His Life and Scientiﬁc Work, Kuala Lumpur, thesis submitted for the Degree of Doctor of Philosophy to the Faculty of Arts and Social Sciences, University of Malaya. [Unpublished PhD Dissertation, accessible at the Needham Research Institute, Cambridge]. ARRAULT Alain, 2002. ‘Les premiers calendriers chinois du IIe siècle avant notre ère au Xe siècle’, in J. LE GOFF, J. LEFORT and P. MANE (dir.), Les calendriers, leurs enjeux dans l’espace et dans le temps, Colloque de Cerisy, du 1er au 8 juillet 2000, Paris, Somogy Éditions d’Art, 2002, p. 169–191. – 2003. ‘Les calendriers de Dunhuang’, in M. KALINOWSKI (dir.), Divination et société dans la Chine médiévale, Paris, Bibliothèque nationale de France, 2003, p. 85–123. – 2004. ‘Jianlun Zhongguo gudai liri zhong de nian ba xiu zhu li’, ¡» K^ÝQâÚ¥K (On the Twenty-Eight Mansions in the Ancient Chinese Calendar), Dunhuang Tulufan yanjiu õó#j@~, Beijing, Zhonghua Shuju ºh , vol. 7, p. 410–421. – 2014. ‘Les calendriers’, in Jean-Pierre DRÈGE (dir.) avec la collaboration de Costantino MORETTI, La fabrique du lisible, La mise en page des manuscrits de la Chine ancienne et médiévale, Paris, Collège de France, Institut des Hautes Études Chinoises, p. 99–111. ARRAULT Alain et MARTZLOFF Jean-Claude, 2003. ‘Notices [sur les calendriers de Dunhuang]’, in M. KALINOWSKI (dir.), Divination et Société dans la Chine médiévale, étude des manuscrits de Dunhuang de la Bibliothèque nationale de France et de la British Library, Paris, Bibliothèque nationale de France, 2003, p. 141–211. ASCHER Marcia, 1991. Ethnomathematics, A Multicultural View of Mathematics, Paciﬁc Grove (Californie), Brooks-Cole Publishing Cie. SECONDARY SOURCES 409

– 2002. Mathematics Elsewhere, An Exploration of Ideas Across Cultures, Princeton, Princeton University Press. AUBIN Françoise, 2005. ‘Short Note About Islamic Mathematics and Astronomy in China’, Études Orientales, Paris, no. 23–24, p. 39–56. BÄCKER Jörg, 2007. ‘Sur l’origine des signes cycliques chinois, quelques implications cosmologiques et mythologiques’, in Ch. LE BLANC et R. MATHIEU (dir.), Ap- proches critiques de la mythologie chinoise, Montréal, Les Presses de l’Université de Montréal, 2007, p. 51–85. BALACZ E. (ed.), 1961. ‘L’histoire comme guide de la pratique bureaucratique (les monographies, les encyclopédies, les recueils de statuts)’, in W. G. BEASLEY and E. G. PULLEYBLANK, Historians of China and Japan, London, School of Orien- tal and African Studies, 1961, p. 78–94. BAZIN Louis (ed.), 1991. Les systèmes chronologiques dans le monde turc ancien, Budapest, Paris, Akadémiai Kiadó et Éditions du CNRS. BECK M. (ed.), 1990. The Treatises of Later Han: their author, sources, contents and place in Chinese historiography, Leyden, E. J. Brill. BEER A., HO Ping-yu, LU Gwei-djen, NEEDHAM Joseph et al., 1961. ‘An 8th- Century Meridian Line: I-Hsing’s Chain of Gnomons and the Prehistory of the Metric System’, Vistas in Astronomy, Elsevier Science, vol. 4, p. 3–28. BÉHAR Pierre, 1996. Les langues occultes de la Renaissance, Paris, Desjonquières. BERNARD-MAÎTRE Henri, 1945. ‘Les adaptations chinoises d’ouvrages européens : bibliographie chronologique. I. Depuis la venue des Portugais à Canton jusqu’à la Mission française de Pékin (1514–1688)’, Monumenta Serica, Los Angeles, vol. 10, p. 1–57 and 309–388. BIÉMONT Émile, 2000. Rythmes du temps, Astronomie et calendriers, Paris, Bruxelles, De Boeck. BLAY M., HALLEUX R. et al. (ed.), 1998. La science classique (XVI e–XVIII e siè- cles), dictionnaire critique, Paris, Flammarion. BO Shuren ßWß, 1983. ‘Shitan Santong li he Taichu li de butong dian’ "ëÙ KõHKÝ!F (The Differences Between the Santong li and the Taichu li), Ziran kexue shi yanjiu QI.ª@~ [Studies in the History of Natural Sciences], Beijing, vol. 2, no. 2, p. 133–138. – 1996. Zhongguo tianwenxue shi »FZ.ª (A History of Chinese Astronomy), Taipei, Wenjin Chubanshe ZùÌã. – 2003. Bo Shuren wenji ßWßZ/ (Collected Works of Bo Shuren), Hefei, Zhong- guo Kexue Jishu Daxue Chubanshe »I.*.Ìã. BODDE Derk, 1975. Festivals in Classical China, New Year and Other Annual Obser- vances during the Han Dynasty, 206 BC – AD 220, Princeton, Princeton University Press. BOUCHET Ulysse, 1868. Hémérologie ou traité pratique complet des calendriers julien, grégorien, israélite et musulman. Ouvrage approuvé par l’Académie des Sciences de l’Institut Impérial de France, Paris, E. Dentu. BRÉARD, A. ‘Problems of Pursuit: Recreational Mathematics or Astronomy?’, in Y. DOLD-SAMPLONIUS, J. W. DAUBEN , M. FOLKERTS and B. van DALEN, 410 BIBLIOGRAPHY

2002, Stuttgart, Franz Steiner Verlag, p. 57–86 [This article contains a large number of speciﬁc examples of problems of pursuit]. BREDON Juliet and MITROPHANOW Igor, 1927. The Moon Year, A Record of Chi- nese Customs and Festivals, Shanghai, Kelly and Walsh. BRIND’AMOUR Pierre, 1983. Le calendrier romain, recherches chronologiques, Ottawa, Éditions de l’Université d’Ottawa, Collection d’Études Anciennes de l’Uni- versité d’Ottawa, no. 2. BRITTON John P., 1999. ‘Lunar Anomaly in Babylonian Astronomy’ in N. M. SWER- DLOW (ed.), Ancient Astronomy and Celestial Divination, Cambridge (Mass.), The MIT Press, 1999, p. 187–254. BRUIN Frans and Margaret, 1977. ‘The limit of Accuracy of Aperture Gnomons’, in Y. MAEYAMA and W. G. SALZER (ed.), Prismata, Naturwissenschafts-geschicht- liche Studien für Willy Hartner, Franz-Steiner, 1977, p. 21–42. CALLATAŸ Godefroid de, 1996a. Annus Platonicus, A Study of World Cycles in Greek, Latin and Arabic Sources, Louvain, Université Catholique de Louvain, Institut Orientaliste, Louvain-La-Neuve. – 1996b. Ikhwân al-S. afâ, Les révolutions et les cycles (Épitres des Frères de la Pureté, xxxvi), Traduction de l’arabe, introduction, note et lexique. Beyrouth, Al-Bouraq Éditions, Louvain-la-Neuve, Academia-Bruylant. CAUTY André, 2012. Multiversalité du temps, du calendrier et du zéro Maya. Villejuif, Amerindia, Chantiers Amerindia, A.E.A.– CELIA. CHAVANNES Édouard et PELLIOT Paul, 1913. ‘Un Traité Manichéen retrouvé en Chine’, Journal Asiatique, Paris, 11e sér., tome 1, p. 99–199. CHE Yixiong ×., 1984. ‘Jingchu li chu yi’, ÿK È (Some Remarks on the Jingchu li) in ZHANG Yuzhe ùòï (ed.), Tian wen: Zhongguo tianwen shi yanjiu, diyi ji F® »FZª@~ Ï×ì (Questions of Astronomy: Research into the History of Chinese Astronomy, First serie) Jiangsu Kexue Jishu Chubanshe n ÁI.*Ìã, p. 92–110. CHEMLA K. et GUO Shuchun (tr.), 2005. Les Neuf Chapitres, le classique mathéma- tique de la Chine ancienne et ses commentaires ; édition critique traduite, présen- tée et annotée ; glosssaire des termes mathématiques chinois anciens ; préface de Geoffrey LLOYD. Paris, Dunod. CHEN Hao W, 2007. ‘Tulufan Yanghai yihao mu chutu wenshu niandai kaoshi’ # jõ×r)þZhOÕ (The datation of the documents discovered in the tomb no. 1, at Yanghai, Xinjiang province) Dunhuang Tulufan yanjiu õó# j@~ vol. 10, p. 11–20. CHEN Jiujin Wò,1983. ‘Zhongguo gudai shizhi yanjiu jiqi huansuan’, » `×@~CÍðÕ (Research into Ancient Chinese Systems for Measuring Time and Their Equivalences), Ziran kexue shi yanjiu QI.ª@~ [Studies in the History of Natural Sciences], Beijing, vol. 2, no. 2, p. 118–132. – 1984. ‘Diaorifa yanjiu’ ^°@~ (An Examination of the diaorifa Method [An arithmetical technique based on the repeated addition of the numerators and denominators of given fractions]), Ziran kexue shi yanjiu QI.ª@~ [Studies in the History of Natural Sciences], Beijing, vol. 3, no. 3, p. 245–250. SECONDARY SOURCES 411

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Chinese or Japanese Names

Bo Shuren ßWß Huang Yi-long ?×¹ Che Yixiong ×. Ikeda On oâá Chen Hao W Ito,¯ Kazuhiko þõ Chen Jing W Ji Zhigang S â Chen Jiujin Wò Jiang Xiaoyuan nLæ Chen Kaige Wu Jing Bing ÿ Chen Meidong WY Kawahara, Hideki æ|p Chen Xiaozhong WL Kodama, Akihito ÉÛß Chen Yongzheng WÕÑ Lai Swee Fo äõ Chen Yuan Wn Lee Eun-Hee AX Chen Zhanyun W"2 Li Caiping Aýµ Chen Zungui Wñ Li Chongzhi Aâ Dai Nianzu ZFÞ Li Heng A Deng Wenkuan ÑZ´ Li Rui AÞ Dong Yuyu lðE Li Yan A? Du Yu Eï Li Yong AK Fujieda, Akira ;c Li Yongkuang AÕ Fukunaga, Mitsushi ÉÕ ¥ Lin Jin-Chyuan ô Gao Pingzi {¿ Liu Dun Gongyang Gao 2~{ Liu Hongtao ÷} Guo Moruo J¯u Liu Jinyi i Guo Shoujing JF¯ Liu Xin _ Guo Shuchun JhÊ Lu Yang ³ Han Qiheng 8Í Luo Zhenyu 1MÛ Hashimoto, Keizo [Í¯C Ma Mingda y¾ Hirose, Hideo Â|. Nakayama, Shigeru v Hong Jinfu ÷° Nishizawa, Yus¯ o¯ gá Horng Wannsheng ÷0ß Ohashi,¯ Yukio [ãSG

Okada, Yoshiro¯ )âý Xin Deyong ÆK Onozawa, Seiichi MgÞ× Xu Guangqi 6 @ Pan Nai Xu Xiqi 6- Qian Baocong þ/ Xu Yibao 6L1 Qin Jiushao çÜp Xu Zhentao 6Mt Qu Anjing `H³ Yabuuchi, Kiyoshi 9/ Ruan Yuan ¨- Yamada, Keiji âÈÉ Shen Chaying S-z Yamanoi Yu¯ . Sima Qian ¥yÎ Yan Dunjie õo Sun Xiaochun r Yano, Michio ´M¼. Takebe, Katahiro I¯Â Yao Dali æ Uchida, Masao /âÑv Ying Zong z Wang Baojuan r Yixing × Wang Guifen rr Zeng Xiongsheng .ß Wang Lixing rñ· Zhang Peiyu ùÇ Wang Mang r Zhang Shuli ù~ Wang Rongbin rvü Zhang Zixin ù* Wang Shouyi rFL Zhao Chengqiu ?K Wang Xi rw Zhou Yiping ø×¿ Wang Xiaohu W Zhu Wenxin eZÃ Wang Yingwei rT Zhu Xi ew Wang Yuezhen W_Å Zhu Zaiyu eµ Watanabe, Toshio ,\AG Zhuang Shen å Wu Jiabi w Zhuang Weifeng Wu Zetian JF Zhuangzi Xi Zezong .g Zu Chongzhi Þ]

Various Items

Amoy (Xiamen) Chouren zhuan ßF bajie â; chunjie Ê; ce da Chongzhen lishu âÈKh Daming huidian ºÎ cha - Daming li K chunjie Ê; dan í Chunqiu ÊK Datong li ÙK GLOSSARY 443

Dayan li K Jihe yuanben ¿¢æÍ di zhong 2 Jingchu li ÿK dizhi 2Y Jinshu ah dongzhi jiu gong Ü du Jiu Tangshu êh Duanwu Ð= Jiuzhang suanshu ÜaÕ Duanyang Ð& Jiuzhi li ÜÆK du-liang-heng É Jiyuan li S-K Dunhuang õó juzhu li Ì¥K fangcheng zhengfu ]Ñ Kaixi li ¨K Fantian huoluo jiuyao YFj1 Kaiyuan zhanjing -B Ü¸ kaogu Futian li ÐFK kaozhengxue J. Gongyang zhuan 2~F Koryoˇ sa/Gaoli shi {ª gu liu li 0K kong è gua ß layue 6` Guantian li ÌFK li (astronomical canon) K Gujin tuibu zhushu kao *.M li (principle of organization) § Liji B Guoyu »+ lipu KH Han Sifen shu °5 liri K^ Hanshu h Lidai changshu jiyao aì Hanyu fangyan da cidian +] ÞÎ lifa K° hao m Lifa xizhuan K°F Hou Hanshu ¡h Linde li ¡ÆK hou jiuyue ¡Ü` ling ë Huainan zi P lingfan li ÜÙK Huangji li 0ÁK Lixue dazhi K.¼ Huihui li //K li zhou Kø Huitian li ºFK Lüli rongtong KÈ; jiaguwen äzZ mi Jiaguwen heji äzZ)/ Mie Ü jianchu t min li ÓK jiaoying øT Mingshi ª jie ; Mingtian li FK 444 GLOSSARY miri ^ Shiji ªB Mishu geju éhe Shoushi li 0`K Mishu jianzhi éh½ shu (numbers, quantities) ó Mo ^ shu (procedures) mulu ê Shujing hB nayin û¯ shuo i nian O Shuowen jiezi 1ZC nianshen Oß Shushu jiuzhang óhÜa qi si li °ñ Qingming z Sifen li °5K Qintian jian !F½ Sitian jian ¥F½ qiying T Songshi ª qizheng chandu li ÚÅÜK Suanjing shishu ÕBèh quan º suanshu Õ qubian ` Suanxue qimeng Õ.@ rili ^K sui Ï rishen ^ß Suishu h ruli áK Sunzi suanjing ÕB runxian § Taishi Hª runying T Taishi jian Hª½ runyu õ Taishi li K runyue ` Taishi yuan Hªo ruqi á Taisho¯ shinshu¯ daizoky¯ o¯ Ñ±ò ruzhuan á» B san zheng ëÑ Tianbao li F1K san fu ë tiangan F Santong li ëÙK tian guan F shangli îK Tianhe li FõK shangxian î< tianwen FZ shang yuan î- tianwen yue FZ` she ã tianxue F. Shengshou wannian li O.0O tianyuan yi F-× K tong Ù shi ` tongshu ;h shi ci ´g Tongtian li ÙFK Shi geng tuibu .M GLOSSARY 445

Tumubao þa¡ Yingri 2^ tuwang þr yin-yang Y& wang T Yitian li FK wangli rK yongjiu Õò wangwang ?÷ Yuandan -Ë wannian 0O Yuanjia li -K ëh Weishu yuannian -O n Wu Beizhi Yuanshi -ª Å×K Wuyin li Yuanxiao - H Xi’an yue ` ì< xia xian yuejian ganzhi `Y Xiamen (Amoy) yueshen `ß xiao Yufodan ¡ xiao xue . Yulanpen Þ 3 Xieji bianfang shu ÜSï]h Yusi jing úB xin ± Zhantian li FK Xu Guangqi ji 6 @/ Ñ Xin Tangshu ±êh zheng Ñ` Xinghe li ·õK zhengyue ÆK Xiyou ji »B Zhide li ¥& Xuanming li K Zhongyang øT Xuanye zhouying Xunzi, ‘zhenglun’ /, Ñ¡ zhuanying »T Xuri Ì^ Zhu Xi quanshu ewh Yao dian Î ziran Q yaosenwen ]Z Zhoubi suanjing øgÕB Yijing |B Zhouli ø ying T zhu [ INDEX OF NAMES

Akhmedov, A.A., 22 Chavannes, É. (1865–1918), al-Sanjuf¯ın¯ı, 21 91, 92 Ang Tian Se, 55, 119, 132, 334 Che Yixiong, 274 Arrault, A., 34, 67, 75, 79, 82, Chen Hao, 268 88, 91, 93, 98, 99, 219, Chen Jing, 109 271, 274, 296, 329 Chen Jiujin, 5, 21, 30, 62, 109 Ascher, M., 333 Chen Kaige, 132 Aubin, F., 21 Chen Meidong (1942–2008), 5, 8, 28, 39, 55, 81, 93, 132, Béhar, P., 333 133, 169, 201, 202, 276, Bäcker, J., 82, 83 279 Bazin, L., 82, 94 Chen Xiaozhong, 5, 54 Biémont, É., 100 Chen Yongzheng, 99 Billard, R. (1922–2000), 335 Chen Yuan (1880–1971), 78 Blay, M., 45 Chen Zhanyun, 153 Bo Shuren (1934–1997), 5, 6, Chen Zungui (1901–1991), 56, 137, 194 62, 66, 70, 80, 86, 241, 261 Bodde, D. (1909–2003), 39, 63, Chionades, G., 129 97, 99, 101 Chrisomalis, S., 333 Bouchet, U., 29, 108, 149 Chu Pingyi, 41 Bredon, J., 97 Clavius, C. (1538–1612), 45– Briggs, H. (1561–1639), 131 46 Brind’Amour, P. (1941–1995), Cook, A., 36 99 Copernicus, N. (1473–1543), Britton, J.P., 157 24, 45 Couvreur, S. (1835–1919), 58, Callataÿ, G. de, 333 59, 67 Callipus (ﬂ. 330 BC), 243 Coyne, G.V., 29, 149 Cauty, A., 86, 120 Cullen, C., 65, 66, 92 Chambeau, G., 31, 61 Chang Chih-ch’eng, 41 Dai Nianzu, 303

Dalen, Benno van, 21, 109 Forte, A., 91 Danton, G.-J. (1759–1794), 49 Fujieda, A., 297 De Ursis, Sabatino S.J. (1575– Fukunaga, M., 63 1620), 93 Fung Yu-lan (1895–1990), 63 Deane, T.E., 54, 59 Dehergne, J. (S.J.) (1903– Galileo, G. (1564–1642), 45 1990), 4 Gao Pingzi (1888–1970), 132, Deng Wenkuan, 14, 24, 25, 34, 242 75, 86, 95, 267, 268, 272, Gassmann, R.H., 78 276, 279, 290, 296–298, Gaubil, A. (1689–1759), 4, 110, 300, 315, 316 198 Denys the Little, see Dionysius Gernet, J., 49, 97, 259, 337 Exiguus Ginzel, F.K., 333 Dershowitz, N., 79, 108 Goddu, A., 45 Dicks, D.R., 243 Golvers, N., 16, 54 Dionysius Exiguus (ca. 475– Gongyang Gao (Warring States 550), 119 period (403–222 BC)), 267 Doggett, L.E., 29, 136 Grafton, A., 136 Dong Yuyu, 5 Graham, R.L., 9, 167 Dorotheus of Sidon (end of 1st Granet, M. (1884–1940)), 35 cent. AD), 333 Guo Moruo (1892–1978), 80, Du Shiran, 56 84 Du Yu (222–284), 46 Guo Shoujing (1231–1316)), Dumoulin, C., 136 55, 110 Dux, G., 47 Guo Shuchun, 138

Eade, J.-C., 333 Halleux, R., 45 Eberhard, W., 97 Han Qiheng, 227 Elia, P. d’ S.J. (1890–1963), 94 Hannah, R., 157, 243 Elman, B.A., 55, 197, 247 Harris, J., 78 Engelfriet, P., 45 Hashimoto, K., 24, 42 Escher, M.C. (1898–1972), 25 Havret, H., 31, 61 Herschel, J. (1792–1871), 136 Fairbank, J.K. (1907–1991), 14 Hirose, H., 6, 205, 210, 212 Febvre, Lucien (1878–1956), Ho Peng Yoke, 15, 16, 51, 54, 120 55 Forke, A., 271 Hoang, P., 73, 76, 77, 100 INDEX OF NAMES 449

Hong Jinfu, 336 La Hire, Philippe de (1640– Hopkirk, P., 296 1718), 39 Horiuchi, A., 198 Lænsberg, M., 29 Horng Wann-sheng, 247 Lai Swee Fo, 28, 54, 213 Hoskin, M.A., 29, 149 Lam Lay Yong, 119 Hu, W.C., 97 Laurent, D., 65 Huang Chun-chieh, 32 Leduc, J., 141 Huang Yi-long, 15, 27, 33, 41, Lee Eun-Hee, 198, 277 296, 336, 337 Lefort, J., 30 Hummel, A.W., 247 Lehoux, D., 50 Le Blanc, Ch., 17, 66 Ikeda On, 267 Li Caiping, 227 Isahaya, Y., 22 Li Chongzhi, 56, 78 Ito,¯ K., 279, 329 Li Feng, 80 Li Rui (1768–1817), 132, 247 Jardine, N., 39 Li Yan (1917–1988), 125, 334 Ji Zhigang, 7, 30, 109, 171, Li Yong, 7, 199, 205 181, 195 Li Yongkuang, 97, 102, 273 Jiang Xiaoyuan, 17 Libbrecht, U., 57, 118, 125, Jing Bing, 198 128, 130 Jones, A., 128 Lin Jin-Chyuan, 7, 192, 213, 261, 267, 268, 317, 327 Kalinowski, M., 81, 82, 92, 93, Liu Hongtao, 133 96, 271 Liu Jinyi, 192 Kallipos, see Callipus Liu Xin (ca. ?–AD 23), 65 Kawahara, H., 65 Loewe, M., 25, 81, 94 Kennedy, E.S. (1912–2009), 23 Longomontanus, see Severin, Kepler, J. (1571–1630), 24, 40, Christian 41, 53 Lu Yang 2008, 5 King, D.A., 22 Luo Zhenyu (1866–1940), 14 Kistemaker, J., 93 Knobloch, J., 57 Ma Mingda, 21, 109 Knuth, D.E., 9, 167 Macrobius (Vth century, philos- Kodama, A., 125, 127 opher and philologist), 100 Krupp, E.C., 65 Maeyama, Y., 335 Kugler, F.X., 129 Major, J.S., 17, 66 Kurath, H., 65 Mak, Bill M., 91, 333 450 INDEX OF NAMES

Martzloff, J.-C., 24, 34, 45, 46, Ovid (43BC–AD17 or 18), 67 57, 75, 91, 109, 119, 141, 296, 297, 300, 329 Pan Nai, 24, 93, 109 Maspero, H. (1883–1945), 30 Panaino, A., 64 Mathieu, R., 17, 66 Parisot, J.-P., 136 Meeus, J., 314 Patashnik, O., 9, 167 Mercier, R., 39, 129, 335 Pedersen, O., 29, 149, 211 Merzbach, U.C., 108 Pelliot, P. (1878–1945), 74, 91, Meton of Athens (ca. 430 BC), 92 157 Peterson, J., 303 Mithra (solar divinity), 91 Peurbach, G. (1423–1461), 21 Mitrophanow, I., 97 Plethon, G.G. (ca. 1360–1452), Monier, R., 47 129 Mosshammer, A.A., 119, 149 Plofker, 333 Prager Branner, D., 80 Nakayama, S. (1928–2014), 5, Ptolemy, C. (ca. 85–165), 21, 39, 43, 50, 109, 140, 153, 24, 50, 211, 212 195, 215 Pulleyblank, E.G., 80, 131 Nan Wang, 15 Napier, J. (1550–1617), 131 Qian Baocong, 334 Nas.¯ır al-Din T. us¯ ¯ı (1201–1274), Qin Jiushao (ca. 1202–1261), 22 125, 128 Needham, J. (1900–1995), 14, Qu Anjing, 6, 7, 30, 109, 171, 53, 66, 91, 92, 129, 130, 181, 191, 195, 199, 212, 244 227, 334 Neugebauer, O. (1889–1990), Qubilai (emperor, founder of 157, 234 the Yuan dyn. in 1277), 19, Nishizawa, Y., 263, 269, 279, 55 296, 298 Nivison, D.S., 93 Ramsey, S.R., 131 North, J.D., 13 Ramus (Pierre de la Ramée) (1515–1572), 39 Ohashi,¯ Y., 234 Reingold, E.M., 79, 108 Oirats (Mongolian related peo- Ricci, Matteo S.J. (1552–1610), ple), 41 46, 112 Okada, Y., 279, 299, 329 Richards, E.G., 29 Onozawa, S., 63 Rocca-Serra, G., 243 INDEX OF NAMES 451

Romano, A., 45, 54 Teiser, S.F., 98 Rotman, B., 123 Tester, J., 58 Ruan Yuan (1764–1849), 197 Tihon, A., 129 Tokugawa Ienobu (shogun)¯ Saint Ronan, 65 (government: 1709–1712), Samsó, J., 22 198 Scaliger, J.-J. (1540–1609), 136 Toomer, G.J., 50 Schafer, E.H., 28, 52, 54 Troesch, A., 30 Schuh, D., 333 Trombert, E., 97 Segonds, A., 39 Tropfke, J., 108, 125 Severin, Christian, 18 Twitchett, D.C., 14 Shakespeare, W., 64 Tycho Brahe (1546–1601), 18, Shaughnessy, E.L., 80 24 Shen Shaying, 97 Sigler, L.E., 108 Uchida, M., 6, 10, 189, 190, Sima Qian (135–86 BC), 52 277, 279, 280, 285 Simon, G., 53 Ulugh-Beg (1394–1449), 22 Sivin, N., 5, 8, 17, 21, 23, 43, Ursis, see De Ursis 47, 48, 50, 55, 59, 132, Vandermeersch, L., 53, 62, 63, 138, 201, 215, 244, 317 271 Smith, R.J., 16, 27, 33, 94, 319 Vernus, M., 29 Souciet, É. (1671–1744), 4, 110 Vogel, H.U., 36, 47, 48 Steele, J.M., 274, 277 Stein, Sir Marc Aurel (1862– Wang Baojuan, 54 1943), 297 Wang Guifen, 279, 336 Stern, S., 157 Wang Lixing, 30 Sugiki, T., 333 Wang Mang (emperor (reign: Sugimoto, M., 198, 277 9–25 AD)), 56, 100 Sun Xiaochun, 5, 93 Wang Rongbin, 7, 30, 109, 171, Sung, Z.D., 57 181, 195, 221, 224, 334 Swain, D.L., 198, 277 Wang Shouyi, 125 Wang Xi, 97, 102, 273 Takebe, Katahiro (1664–1739), Wang Xiaohu, 5, 16, 42 198 Wang Yingwei (1877–1964), Tasaka, K., 19, 213 56, 133, 171, 195, 274 Taub, L.C., 50 Wang Yuezhen (1812–1881), Teboul, M., 132 100 452 INDEX OF NAMES

Watanabe, T., 91 Yao Dali, 21 Welch Bjaaland, P., 97 Ying Zong (Chinese emperor, Weschler, H.J., 55 reign: 1435–1449), 41 Whitﬁeld, S., 91, 296, 297 Yixing (683–727), 55 Wilkinson, E.P., 78, 97 Wu Yiyi, 51 Zeng Xiongsheng, 5 Wu Zetian (empress (reign: Zhang Peiyu, 5, 7, 8, 11, 100, 684–704)), 101 171, 199, 202, 205, 253, Wylie, A. (1815–1887), 91, 92, 276, 279, 304, 336 129, 130 Zhang Shuli, 54 Zhang Zixin, 180 Xin Deyong, 56 Zhao Chengqiu, 192 Xu Guangqi (1562–1633), 18, Zhou Yiping, 97 46 Zhu Wenxin (1883–1939), 132, Xu Zhentao, 5 241 Yabuuchi, K. (1906–2000), 5, Zhu Xi, 50, 51 109, 213, 215 Zhu Zaiyu (1536–1611), 28, 37 Yamada Keiji, 55 Zhuang Shen, 91 Yamanoi, Y., 63 Zhuang Weifeng, 5, 274 Yan Dunjie (1917–1988), 56, Zhuangzi (late 4th century BC), 121, 241, 279, 334 267 Yang Lien-sheng, 92 Zu Chongzhi, 56 Yano, M., 21, 109 Zürcher, E. (1928–2008), 32 INDEX OF SUBJECTS

60/a/b, 144 arithmetical operations, 36, 53, < a;b >, 142 111, 114, 119, 121, 123, a;b,c, 143 124, 189 < a mod 60;b >, 142 armillary sphere, 53 bin(x,y), 142 Artha-sastra, 234 days/a/b, 144 Ashmolean Museum (Oxford), ⌊ ⌋ x , 10 128 x mod y, 10 astrology, 13, 16, 28, 33, 58, α, β, γ (Metonic constants), 334; also called tianxue 158 (the science of the heav- administration, 33 ens), 28; and military af- age of the moon, 9, 29, 149, fairs, 28, 55; elections 164, 201–203, 262 (choices), 58; Greek, 129; Almagest, 24, 50 horoscopic, 129, 130; Almanach de Liège, 29 Greek papyri, 128; incep- almanacs, 29 tions, 58; Indian inﬂuence, Amoy (Xiamen) (place name), 81; judicial, 4, 54 92 Astronomica Danica (C. Sev- animals; bones of, 62; cyclical, erin), 18 81, 297; lost, 92; self mov- astronomical canon(s), 5, 6, 13, ing things, 46 17, 19, 21–25, 28, 31, 32, archaeoastronomy, 65 34–37, 42, 52, 53, 55, 63, arithmetic, see also logistics, 65, 86, 107, 109–112, 114– 35, 213, 242, 262; mean- 118, 120–122, 130–132, ing of shu (numbers or 220, 221, 224, 235, 241, calculations), 36; com- 242, 259, 261, 267, 274, mercial, 108; Dunhuang 277, 286, 295, 300, 317, manuscript, 119; rules of, 325, 328–330; atypical, 53; written, 120 109; centesimal number arithmetical mean, 25 system, 109; key ideas, 37–

41; later, 197–217, 303, tional, 11, 55, 108; predic- 308, 323, 329, 330; mean tive techniques, 49, 111; elements, 157–167, 220, table(s), 8, 109–111, 117, 278; mentioned in the Chi- 122, 123, 192–194; Tri- nese Tripitaka, 122; no ar- bunal, 4; Chinese Muslim; tificial days, 234; predic- Bureau, 19; Greek, 129; tive mathematics, 111; re- sexagesimal number sys- form(s), 42–52, 259; their tem, 112; Indian; sexages- name(s), 55–59; true ele- imal number system, 112; ments, 169–195, 312 non-Chinese; Chinese per- astronomical system, 17 ception of, 19; positional, 334; tables, 127; technical astronomical tables, 8, 109, and contemporary, 3; with- 110, 117, 120–122, 127, out hypotheses, 38 128, 131, 146, 199, 204; day and night durations, Bahnasa (place name (mod- 308; European, 18, 109; lu- ern Egypt), Oxyrhynchus nar, 8, 123, 185, 191; mod- in antiquity), 128 ern, 202; Philippe de la bajie (the Eight Nodes (solar Hire, 40; political factors, breaths),64, 75 41; Rudolphine, 40; shad- Beatty sequences, 30 ows, 111; solar, 8, 41, 109, Beijing, 74, 93, 131; Library, 123, 180–185; Venus, 123, 16; capital of China, 41; 124; zero, 122 latitude, 41 astronomy, 13, 108; non- Bibliothèque nationale de Chinese; and theology, 53; France, 74 Arabo-Persian; sexagesi- binary indicators, 184, 185 mal number system, 112; bindu (a dot (Skr.)), 120 Chinese; Bureau of, 4, 52– binomials, 9 55, 128, 153; history, 5, 49, branche(s), 73, 76, 80–84, 86, 73, 110; ignorance of tradi- 95, 269, 272, 274 tional Chinese astronomy Breton ethnology, 65 in 18th China, 4; Islamic British Isles, 64 (transl.), 109; its limitation, Buddhist sutras, 122 41; its mathematical char- Bureau of Astronomy, 52 acter, 4, 5; Jesuit reform of, 4, 18, 23, 63, 197; posi- calculating instruments, 121 INDEX OF SUBJECTS 455 calendar(s) Chinese; non- 208, 219, 221, 223, 226, official, 74; contemporary; 232, 233, 289; uniform and Hong Kong, 25; Taiwan, linear, 32; units of time, 31; 25; counterfeit, 319; Dun- double history, 32; histori- huang, 25, 34, 82, 219, cal sources, 29; public, 29; 267, 296, 301; at vari- secret, 29; surface, 9, 29, ance with official calen- 49, 61–63, 66, 70, 71, 76, dars, 79; P3247 vº, 74; Pel- 135, 148, 151, 152, 160, liot collection, 74; plan- 163, 219, 220, 222–224, etary week, 92; errors of 232, 233, 251, 254, 258, dates corrected by Huang 264, 289, 308; discrete, 30; Yi-long, 336; lunisolar, 9; history, 33; lunar month, manuscripts, 34; Ming dy- 151, 233; monthly, 73 nasty, 5, 34; paradoxical Callipic period (or cycle), 243, object(s), 24–27; popular, 244, 255–258, 260 15; printed, 34; private, canon (= κανων´ (Greek)), 23 319; Qing dynasty, 33; Canon of Yao (Yao dian), see Shang-Yin dynasty; sexa- Shujing genary, 62; Song dynasty, Carmen Astrologicum, 333 5; sources for its history, ce (divinatory rods), 53 33–35 celestial anomalies, 51 calendar(s) non-Chinese; Bre- celestial motions; multifactorial ton, 65; Celtic, 64; Grego- character, 51 rian, 63, 73, 97, 149, 258; celestial officers (tian guan), 52 simplified version, 77; He- century (in history), 141 brew; division of the day, cha (difference (shift con- 108; Julian, 73, 91, 101, stants)), 201 136, 149, 216, 217, 253, Chinese Thought (M. Granet), 258, 265, 295, 296, 313; 35 Roman pre-Julian; years of Chongzhen lishu (Chongzhen confusion, 99; Zoroastrian reign-period (1628–1644) gah¯ anb¯ ar¯ , 64 Treatise on Astronomy), 23 calendrical norms, 76 Chouren zhuan (Biographies calendrical structure, 29–35; of Hereditary Specialists), deep, 9, 29, 62, 63, 70, 73, 111, 131, 198, 247 135, 136, 147, 151, 157, Christians; Nestorian stele, 91 456 INDEX OF SUBJECTS

Chunjie (Spring Festival), 97 jianchu, 94–95, 298, 300, Chunqiu (Spring and Autumn 321; Metonic, 29, 136, Annals), 267 149, 157, 243, 244, 254, clepsydra, 30 255, 258, 262; simple or COL-astron (abbrev.), 16, 20, generalized, 9; nayin, 96, 113, 122, 124, 247, 267, 297, 313, 321; number of 298 elements, 79; of the sea- COLL. (abbrev.), 25, 112 sons, 39; pseudo, 94, 273, color palaces (the nine) (jiu 298, 300; quasi-Buddhist, gong), 87–90, 279, 297, 50; sexagenary, 24, 26, 317, 319, 321 30, 32, 83, 86, 88, 134– computistics, 12–14, 23, 29 137, 144, 174, 200, 201, computists, 29 204, 214, 255, 272; divina- computus, 17, 23 tory inscriptions, 84; enumeration of years, 89; ori- concrete mathematics, 9 gin, 83; simple, 79; si- correlative thinking, 36 multaneous, 79; solar, 94, cosmology, 24 136; supra-annual, 79, 134, counting-rods, 53, 119–121, 136; 5 years, 86; 12 years 126 (animals), 82; 60 years, criterion, see also quasi- 89; 76 years, 86; 180 criterion, 49; full and hol- years, 89; 1520 years, 244; low months, 159, 162, 253, 1539 years, 58; 4560 years, 264; intercalary month, 244; Triple concordance 151, 248, 255, 264, 308; (san tong), 58; with redu- of superiority, 197; shifts plications, 79, 96, 273 of moon phases, 247, 248, 308 da (full lunar month), 70 cult of soil god (she), 269, 270, Daming huidian (Collected 273, 297, 318 Statutes of the Ming Dy- cultural manifolds, 5 nasty), 28 cycle(s), 69, 79, 119, 297, Daming li (astron. canon), 42, 298; 28 mansions, 93; as- 43, 56, 160 tronomical, 53; Callipic, dan (one bushel), 48 243, 244, 255–258, 260; date of birth, 54 denary, 80, 296; duodec- Datong li (astron. canon), 18, imal, 73, 81, 82, 272; 58, 63, 109, 117, 118, 145, INDEX OF SUBJECTS 457

169, 197, 199, 208, 210– year, 138; technical terms, 216, 219, 222, 303, 304, 35; zhuanying, 202 308, 316–318, 321, 323, di zhong (the middle of the 329, 330 Earth), 41 Dayan li (astron. canon), 32, divination, 33, 52, 57, 58, 81, 37, 42, 43, 46, 57, 111, 146 112, 115, 121, 132, 143, divinatory rods, 53, 111 146, 171, 182–184, 190, DKW (abbrev.), 46, 64, 74, 80 224 Dog star, see Sirius Daye li (astron. canon), 55, 115, dominical letter, 29 116, 166, 221 dongzhi, see winter solstice double-hour(s), 30, 79, 81, 213, De Revolutionibus (Coperni- 214, 320 cus), 24 du (degrees), 123 deep waters or rivers, 219 du-liang-heng (units of length, definition(s), 131; astronomi- capacity and weight), 36 cal systems, 17; Contem- Dunhuang manuscripts, 74, porary Epoch, 135; cy- 296; almanacs S-P6 rº and cle(s) with reduplications, S612 rº, 82; arithmetic, 79; day, 62; emerging 119; calendars, 25, 34; year, 138; epact, 149, 177; deep waters, 219; deviant Epoch of the Shoushi li, dates, 335; for the year 200; in a military ency- 877 (S-P6 rº), 296; for the clopedia, 55; intercalary years 450 and 451, 267; month(s), 76, 99; lunar planetary week, 92; festi- month(s), 148, 151; Mie vals; socio-economic as- points (first type), 221; Mie pects, 97; non-valid names points (second type), 221; of dynastic eras, 79 Mo points, 221; monthly dynastic era(s), 31, 55, 135, epact, 150; primary con- 139, 271; concordance ta- stants, 9, 134; ruli, 179; bles, 78; naming system ruqi, 172; ruzhuan, 210; first attested in 140 BC, 78; secondary constants, 9, Taisho¯ (1912–1926), 122 134; simple cycles, 79; si- multaneous cycles, 79; so- East Asian Science, Technology lar month(s), 147, 148; Su- and Medicine, 5 perior Epoch, 136; support East-Syrian Church, 91 458 INDEX OF SUBJECTS

Easter; date, 13; moon, 149; ta- Datong li, 216, 329; ble of Dionysius Exiguus, Shoushi li, 140, 200–202, 119 204, 303; and astronomi- ecclesiastical moon, 13 cal observations, 135; pro- eclipses, 109; A. Gaubil’s cedural complexity, 135; French transl. of Chin. deep structure, 135; Julian proc., 198; and Chin. cal. day, 136; multiple, 31; Su- calculations, 180; cycle(s), perior (shang yuan), 11, 274; forecasts, 3, 11, 32, 135–141, 144, 150, 158, 274, 334; limits, 275; lu- 161–164, 166, 167, 200, nar; mentioned in the cal. 329, 330, 333; Jingchu li, of the year 451, 16; on May 261, 262; Sifen li, 241, 245, 15, 1631, 18; on May 4, 254, 256; Tongtian li, 140; 1632, 18; partial, 268; un- Xuanming li, 277; defini- usual mention of, 268; ob- tion, 136; does not belong servations, 42; solar and lu- to the surface structure, nar; not recorded in Chi- 135; its fictitious character, nese calendars, 16 135; located outside histor- elections, 318 ical time, 139; procedural elementary knowledge (xiao simplicity, 136 xue), 111 Euclidis Elementorum Libri XV emerging year, 138 (Clavius), 45 ‘ev-šambat (‘Sunday’ (Pehlvi)), empirical verifications, 38 91 English (modern and middle), ezich (Lat., from z¯ıj (Ar.)), 23 64 epact, 9, 29, 147, 149, 150, 158, faithful simplifications, 10 163, 164, 166, 175, 177, fangcheng zhengfu (arrays with 209, 210, 262, 263, 283; positive and negative quan- approximate, 149; defini- tities), 126 tion, 149; initial, 201; Fantian huoluo jiuyao (Tantric mean, 203, 279, 280, 304; Buddhist text), 130 monthly, 150; true, 149, festivals (Chinese); Duan- 304 yang, 98; Duanwu (Dou- ephemeris, 16 ble Fifth), 98; Qing- epoch, 31, 51, 66, 134–136, ming (Tomb sweeping), 140; Contemporary, 135; 98; Winter Sacrifice, 101; INDEX OF SUBJECTS 459

Yulanpen (Ghost), 98; geometry, 24, 45 Yuanxiao (Lantern), 98; gnomon; as an instrument, 30, Yuandan (New Year or 51, 53; shadows, 39, 108, Spring Festival), 97; Yu- 111, 113 fodan (Bathing Buddha), golden number, 29 98; Zhongyang (Double Gongyang zhuan (a commen- Ninth), 98 tary), 267 festivals non-Chinese; Celtic graduations (waterclocks), 213 New Year, 65; Halloween, graha (upholders (astrological 65 ‘planets’)), 109 fictitious celestial entities, 15 Grand dictionnaire Ricci de la five phases, 112 langue chinoise, 112 fossil constants, 166 Greece, 47 fraction(s), 10, 162, 164, 185, Gregorian calendar (or dates), 190, 192–194, 216, 221– 63, 73, 77, 149, 258 228, 230, 232, 233, 243, gua (hexagrams), 53 246–248, 254, 256, 263, Guantian li (astron. canon), 195 264, 326, 328; centesimal, Gujin tuibu zhushu kao 113; compound, 171, 172; (Research into astron. decomposition of, 116; ex- canons (Wang Yuzhen), act, 171; improper, 158, 202 163; irreducible, 222; lu- Guoyu (Early Chin. text, 5th nar month, 115; monetary century BC), 267 units, 108; non-decimal, 37; reduced to a com- h. alakim (Heb., time unit), 108 mon numerator, 117; rep- Han Sifen shu (The ‘Quarter resented with a single Chi- Remainder’ Procedures (Li nese character, 116; sex- Rui)), 247 agesimal, 128; solar year, Hanshu (Han History), 14, 35– 116; solar and lunar peri- 38, 65, 66, 336 ods, 117; sum of two, 278 Hanyu fangyan da cidian Fudan University, 131 (Great Dictionary of Chi- Futian li (astron. canon), 22, nese Dialects), 131 109, 122, 195 hao (hair), 47 heavenly stems (tiangan), 80, geocentrism, 41 82 geomancy, 85 Hellenistic Egypt, 47 460 INDEX OF SUBJECTS hemerology, 33 irregularities within general hexagram(s), 53, 57, 112 regularities, 259 history; short-term, 33 History of Chinese Philosophy Jesuit(s), 4, 16, 23, 45, 46, 49, (Fung Yu-lan), 63 63, 81, 93, 109, 153, 197 Horner’s schema, 208 jiaguwen (oracle bones), 62 jianchu horoscope, 54 (pseudo-cycle), 94–95, 269–273, 313, 318 Hou Hanshu (Later Han His- jiaoying (draconitic shift con- tory), 49, 76, 86, 166, 234, stant), 203 241, 244, 248 jie (node(s)), 75 hou jiuyue (intercalary month Jihe yuanben (Chin. transl. of after the ninth month), 70 Euclid’s Elements), 46 Huainan zi (The Masters of Jingchu li (astron. canon), 56, Huainan), 16, 66 114, 115, 144, 261–274 Huangji li (astron. canon), 180, Jinshu (Jin History), 46, 47, 55, 181 75, 115, 268 Huihui li (astron. canon), 18, Jiu Tangshu (Old Tang His- 21, 109, 303 tory), 171, 180 Huitian li (astron. canon), 49 Jiuzhang suanshu (Computa- tional Techniques in Nine infinite space, 53 Chapters), 138 intercalary; delay (runying), Jiuzhi li (astron. canon), 109, 201; limit (runxian), 159, 120, 130 163, 164, 177, 278, 280, Jiyuan li (astron. canon), 42, 304; month(s) (runyue), 9, 43, 137, 167, 174, 191, 69, 70, 75, 76, 86, 87, 194, 274 90, 99–103, 138, 148, 150– Julian calendar (or dates), 73, 153, 157–161, 163, 165, 149, 258 176, 247, 248, 254–256, Jupiter; entrance of the sun 263, 264, 270, 273, 278, into its stations, 317, 321; 280, 289, 308; remainder planet, 15; sidereal revolu- (runyu), 150; year, 69, 70, tion, 63, 86, 271; signs of 139, 159, 160, 248, 253, the zodiac, 317; the Year 255–258, 260, 263, 304, Star, 51 308 juzhu li (annotated calendar(s)), inverted tree (arbor inversa), 83 15 INDEX OF SUBJECTS 461

Kaixi li (astron. canon), 49, Mie, 226, 328, 330; Mo, 167, 176, 178, 182, 192, 225, 325, 326; of Heat, 325 270, 273 Kaiyuan zhanjing (astrological limits; eclipses, 275 book), 81, 109, 241 Linde li (astron. canon), 180, kaogu (investigation of past 185, 190–193, 278, 336 events), 42 Linear congruences, 57 kaozhengxue (evidential stud- Lingfan li (‘calendars’ of ‘co- ies), 247 ercitions’ (a sort of astro- Koryoˇ sa/Gaoli shi (Korean logical ephemeris)), 16 History), 174, 180, 277 lipu (calendrical calculations), ks. aya-tithi (Skr.), 234 14 Kyoto University, 131 liri (calendar), 14 Lixue dazhi (The Almagest), 24 La pensée chinoise (M. Granet), lizhou (anomalistic month), 278 35 logistics, 36, 111 laws of Kepler, 40, 41 Lüli rongtong (astron. canon), layue (month in which the Win- 37 ter Sacriﬁce was held), 101 lunisolar coupling, 9, 73–76, li (principle of organization), 50 99, 152, 161, 249, 273 li (astronomical canon), 13, 19, 24 Manifest Enlightenment canon, Li Heng (emperor (reign: 756– see Xuanming li 762)), 101 manicheism, 91 Liber Abaci, 108 Maple (from Maplesoft), 10 Lidai changshu jiyao (cal. ta- Mars (planet), 15 ble), 202, 336 martyrolog, 13 lifa (predictive methods of as- mathematical astronomy, 13 tronomical canons), 17, 50 Mathematical Syntaxis, see Al- LIFA (abbrev.), 5, 66, 241, 276, magest 321 mathematics; as the best ap- Lifa xizhuan (History of West- proximation to theology, ern Astronomy), 24 45; their limitation, 107 Liji (The Ritual), 67 mean elements, 144–146, 153, limit; ﬁnal, 184; initial, 184; 157–167, 169–171, 176, intercalary, 159, 163, 164, 182, 201, 262, 277–280, 176, 177, 278, 280, 304; 304, 305, 309; anomalis- 462 INDEX OF SUBJECTS

tic month, 169, 179, 185; mi (Sunday), 91, 92, 290, 295, apparent lunar daily side- 299 real motion, 182; epact, micro-uncertainties (dates), 336 203, 279, 280, 304; lu- Midsummer Night’s Dream (A) nar; diurnal motion, 208, (Shakespeare), 64 211; longitude, 212; lu- Mie (points and days), 219– nar month(s), 71, 175, 236, 325–332 330; new moon(s), 157, Mie limit (Miexian), 226, 328, 159, 175, 186, 194, 200, 330 203, 207, 212, 279, 280, Mingshi (Ming History), 18, 52, 283, 285–287, 289, 290, 118, 198, 202, 210–216, 305, 307, 308, 328, 330; 219, 220, 308, 309, 317, phase(s) of the moon, 115, 322 169, 172, 176, 177, 179, Mingtian li (astron. canon), 195 186, 203, 207, 209, 246; Ministry of Rites, 54 seasonal indicators, 203; minli (popular calendar), 15 solar; period(s), 157, 190; miri (Sunday), see mi breath(s), 159, 170, 173– Mishu geju (Board of the 175, 177, 178, 203, 280, Gallery of Secret Writ- 281, 289, 305, 326; longi- ings), 52, 54 tude, 195, 212; period(s), Mishu jianzhi (Records of Se- 171; year(s), 158, 171, cret Writings), 19 210; synodic month, 169; terminology, 147; tropical Mo (points and days), 219–236, year, 203; value(s), 305 325–332 Mo limit (Moxian), 225, 325, Mercury (planet), 15 326 Messager Boiteux, 29 Mogao caves (near Dunhuang), metamorphosis, 67 296 method of deviations (R. Bil- month(s); anomalistic, 169, lard), 335 170, 179, 181, 185, 192, Metonic; calculations, (158, 193, 202, 205, 208, 210, )163; canon(s), 157, 242, 278, 286, 304; draconitic, 254, 259; constants, (157, 32; intercalary, 9, 14, 70, )158, 242, 261, 278; cy- 99–101, 138, 148, 150– cle(s), 9, 29, 136, 149, 243, 152, 157–161, 163, 165, 244, 254, 255, 258, 262 176, 247, 248, 254–256, INDEX OF SUBJECTS 463

263, 264, 273, 278, 280, of, 26, 257; smaller than a 289, 304, 308, 335; 11* solar month, 152; synodic, or 12* are rare, 161; def- 110, 134, 169, 176, 205, inition, 151; rank, 159; 278, 304; true, 153; Win- skipped, 86; irregular, 101; ter Sacrifice, 101; solar, 61, lunar, see also synodic, 30, 147, 148, 150–152; length, 47, 61, 64, 69, 70, 73, 154; true, 153 74, 79, 83, 87, 88, 90, 91, moon, see also age, Easter, ec- 96, 99–102, 134, 137, 138, clesiastical, epact, mean el- 147, 148, 150–152, 157– ements, phases, true ele- 159, 161–164, 175, 177, ments 242, 246, 248, 249, 254, mulu (planetary ephemeris), 15 256–258, 260–262, 264, mythology, 33 265, 269–271, 273, 290, 297–299, 301, 312, 313, Nanjing; capital of China, 41; 316–319, 321, 327; astro- latitude, 41; Observatory, nomical, 73, 138; calen- 336 drical spirits, 81; eleventh, Nanjing University, 7 87, 90, 100, 207, 243, 271; Nanking, see Nanjing enumeration, 86, 87, 90; National Cheng-Kung Univer- first, 69, 75, 90, 96; full or sity, 7 hollow, 9, 26, 70, 101, 159, natural regularities, 13 251, 253, 254, 256–258, navagraha (the Nine Demons 264, 273, 279, 321; length, (astrological entities)), 26, 115, 116, 118, 146, 109, 130 154, 163, 328; mean, 163, nayin (cycle), 96, 318 330; missing, 101; names, negative numbers (or quanti- 31, 69, 99, 101, 273, 299; ties), see positive or nega- non-intercalary, 69; num- tive (numbers or quantities) ber of, 69, 99–101, 158, Nestorians, 90 159, 161, 163, 243; num- nian (year), 62 bered backwards, 89; num- nianshen (yearly spirits), 271 bering, 290; ordinary, 69, Nine Chapters, 111, 118, 138 70, 86, 90, 157; percentage nine roads of the moon, 275 of full and hollow months, Nishizawa, Y., 219 71; rank, 31, 77; sequences non-Metonic canons, 163 Northwest University (Xi’an), 7 464 INDEX OF SUBJECTS novelty, 49 paper, 33, 34, 267, 268, 296, number systems; centesimal, 297; rare commodity, 33; 108–110, 113, 117; Fu- scientific analysis, 268 tian li, 122; Shoushi li, Pehlvi language, 91 141, 200; of Indian ori- Peking, see Beijing gin, 109; decimal, 30, 37, People’s Republic of China, 77 108, 110–113, 118, 119, Persian language, 91 128, 169, 186, 188; digits, phase(s) of the moon, 107 110; approximations, 263; phonograms, 80 decimal submultiples, 113; piecewise functions, 9, 205, positional, 119; degrees; 210, 211 non-decimal, 123; non- pitch-pipes, 36, 37 sexagesimal, 123; mixed, planetary ephemeris, 15, 16; for 111; non-decimal, 37; sex- the year 1531, 16 agesimal, 108, 109; their planetary week, see week, 92 variability, 107 pocket computer Sharp PC numerical constant(s), see also 1475, 10 Metonic constants, 6, 50, politics, 41–42, 50, 54–56, 96 158, 192, 211, 278, 287, 330; astronomical, 12, 27, Portuguese language, 93 51, 182, 216; fossil, 9, positive or negative; integer, 166; interval, 201; lunar, 89; lunar inequality, 192; 157, 164, 224, 261, 277, quantities, 126, 183, 338; 278, 303; primary, 9, 134, small corrections, 145 185, 200, 205, 207, 241, precision; Chin. astron. param- 275; secondary, 9, 134, eters and constants, 3, 12, 135, 241, 325; shift, 51, 202; fictitious, 31; in- 201–203, 216, 217; solar, creased, 31, 197; lack of, 157, 164, 224, 261, 277, 118; limited to one day, 278, 303 43; of our historical knowl- numerology, 36, 37, 111, 117 edge, 33; predictive power, nychtemeron, 62 13, 27, 42, 49; quantifiable (Epochs), 135; surface ver- occult sciences, 53 sus deep structures, 62 overlined circle, 130 prediction, 42, 44; and retrodic- Oxyrhynchus (place name (an- tion, 42; eclipses, 180, 274, cient Egypt)), 128 334; fate of individuals, 54; INDEX OF SUBJECTS 465

not explanation, 17; preci- Roman abacus, 121 sion, 49, 197; wrong, 46 Roman Empire, 128 printing, 33 Rudolphine tables (Kepler), 40 programming techniques, 10 ruli (parameter), 170, 179, 194, proleptic calculations, 143, 202 280, 286, 287 pronunciation of kong (zero), runying (intercalary delay), 201 131 ruqi (parameter), 170, 172–179, 186, 188, 190, 209, 280, qi (breath or energy), 50, 63 281, 284, 287 Qintian jian (Directorate of the ruzhuan (parameter), 170 Celestial Veneration), 52, 54 sacrifice(s); of dogs, 99; Winter qiying (solar breaths delay); de- (la), 273, 274 lay, 201 Saint Ronan (tomb of), 65 qizheng chandu li (planetary san fu (canicular days), 99, 290, ephemeris), 15 296, 297 quasi-criterion, 9, 159–161, san zheng (three norms (be- 163, 263, 304; intercalary ginning of Chinese lunar limit, 159 years)), 76 quasi-table(s), 9, 192–194, 286, Sanjuf¯ın¯ı z¯ıj (astron. handbook), 288 21 qubian (clever modification), Sanskrit language, 81, 98, 109, 44 122, 130, 234 Santong li (astron. canon), 58, reductio ad absurdum, 44 132, 198 reforms of Chinese astronomi- Sasanian world, 91 cal canons, 25 Saturn (planet), 15 regaim (Heb., time unit), 108 Saturnalia (by Macrobius), 100 religion, 33 saura-dina (Skr.), 234 Republic of China, 78 saura-divasa (Skr.), 234 Resplendent Heaven canon, see Science and Civilisation in Daming li China, see also Needham, retrodictive calculations, 42 J., 14 ri (day), 62 seasonal indicators, 107, 112 rili (calendar), 14 secular variations of the tropi- rishen (daily spirits), 271 cal year, 51, 140, 141, 153, rod-numerals, 125, 126 198 466 INDEX OF SUBJECTS seven days week, see week Sifen li (astron. canon), 56, Shang norm, 76 241–261 shangli (calendar for princes), signs (+ and −), 184, 185 15 Silk Road, 296, 298 shangxian (first quarter of the Sirius (heliacal rising of), 99 moon), 70 Sitian jian (Directorate of the Shengshou wannian li (astron. Celestial Administration), canon), 28 52 shi (seasons or propitious mo- six ancient computus (gu liu li), ments), 58 23 shi ci (erratic planetary behav- SIXIANG (abbrev.), 31, 38, 48, ior), 51 50 Shi geng (Beginning of plough- slaves, 47 ing ceremony), 273, 274 solar breath(s) (qi), 9, 11, 64– shift constants, 201, 202, 203, 67, 73–75, 137, 145, 147, 216, 217 148, 151, 152, 161, 170, Shiji (Records of the Historian), 172–177, 179, 182, 186, 52, 143, 336 209, 243, 246, 249, 253, Shoushi li (astron. canon), 5, 255, 263, 265, 273, 281, 42–46, 49, 50, 57, 58, 109, 285, 290, 295, 297, 300, 110, 117, 132, 135, 140, 305, 306, 308, 316, 318, 154, 169, 197–204, 211, 326, 327, 329; and inter- 212, 215, 216, 222, 303, calary months, 76; consec- 317 utive, 66, 157, 181, 327; shu (numbers, quantities), 36 delay, 201; even, 75, 76, Shuihudi (place name (Hubei 152, 269–271, 273, 313, province)), 94 319; internal, 74; mean, Shujing (Canon of History), 15, 147, 165, 170, 173–175, 57, 58 177, 178, 203, 280, 289, shuo (first day of the lunar 305, 326; number system, month), 69 243; numbering, 176; odd, Shuowen jiezi (dictionary), 74 74–77, 150, 151, 248, 264, Shushu jiuzhang (math. book), 273, 289; odd and consec- 118, 125, 127, 128 utive, 148, 153, 262; tem- si li (Enthronements of the four porary list, 160; true, 170, seasons), 64, 68, 98 172, 173, 281, 283, 285 INDEX OF SUBJECTS 467 solar period(s), 26, 36, 63, 68, Taiping zhen jun shiyi [shi’er] 112, 117, 170, 171, 181, nian li [ri] (calendars 182, 186, 189, 190, 241, (years 450 and 451)), 267 246, 248, 261, 262, 277, Taishi jian (Office of the Grand 281, 329; mean, 144, 190; Historiographer), 52 true, 190, 284 Taishi li (astron. canon)), 56 solar terms, see solar breath(s) Taishi yuan, see Taishi jian Songshi (Song History), 37, 47, Taisho¯ shinshu¯ daizoky¯ o¯ (Re- 49, 53, 111, 113, 123, 167, vised Tripitaka)), 122 174, 176, 179, 191, 194, technical terminology, 8 195 ten days week, see week Songshu (Book of Song), 137, terrestrial branches (dizhi), 80– 268, 274, 276 82 Sovereign-Pole canon, see terrestrial branches, (dizhi), 73 Huangji li tianguan (celestial officials), 52 spontaneous generation, 67 Tianbao li (astron. canon, 116 structure, see calendrical struc- Tianhe li (astron. canon), 116 ture tianwen (judicial astrology), 54 Suanjing shishu (math. book), tianwen yue (astronomical 119 months), 73 suanshu (logistics), 36 tianyuan yi (celestial unit), 125 Suanxue qimeng (math. book), Tibet (Mongol viceroy of), 21 125, 126, 130 time; discrete, 30; instruments sui (solar year), 62 of measure, 32; linear, 32; Suishu (Sui History), 54, 89, local, 30; origin, 31; schol- 116, 166 arly, 32; units of, 32 Sunday, 29 tithi (artificial lunar day (Skr.)), sunya (void, (Skr.)), 122 234, 236 Sunzi suanjing (math. book), tong (cycle), 58 119 tongshu (almanacs), 15 support days, 140 Tongtian li (astron. canon), 42, support year, 138, 223, 242 43, 50, 57, 140, 153, 201, 317 Tables astronomiques (Philippe translation(s), 28, 47, 57, de La Hire), 40 58; Jinshu (Eng.), 55; tables of logarithms, 131 Sunzi suanjing (Eng.), 119; 468 INDEX OF SUBJECTS

Shoushi li (Eng., Jap.), 5; the moon, 179, 207, 308; Arabo-Persian z¯ıj (Chin.), solar; breath(s), 170, 172– 21; cal. of the year 1417 174, 281, 285; month(s), (1st month), 318–322; 153; period(s), 284; termi- Chinese primary sources nology, 147 (Eng.), 7; Clavius’s com- trunk(s), 80, 81, 83–85, 99, 269, mentary to Euclid’s Ele- 274, 299 ments (Chin.), 45; from tuibu (prediction and retrodic- the Chinese, 7, 56, 58; tion of celestial phenom- absence of consensus, 8; ena), 42 from the Sanskrit, 81; in Tumubao (place name), 41 medieval China, 109; in Turkish world, 82 the present work, 61; in- tuwang (period of governance dependent of philology, of the Earth), 68, 290–295, 63; lists of numerical con- 298 stants, 132; ms. of the twenty-eight mansions (the), year 450, 268–271; names 46, 92–94, 314, 317, 318 of Chinese astronomical Tycho Brahe, 24 canons, 55–59; of Buddhist texts (Chin.), 141; of the ullambana (hanging down terms Mo and Mie, 219; (Skr.)), 98 passage from Scaliger’s ullampana (salvation (Skr.)), 98 works, 136; problem from unpredictability, 44, 49, 73, the Shushu jiuzhang, 125 258, 259, 312 Tribunal of mathematics, 4 Uyghur; language, 94; roll troglodyte Buddhist cave com- manuscript, 94 plex, see Mogao caves true elements, 63, 144–146, Venus (planet), 15 149, 152, 153, 157, 169– 195, 201, 209, 280, 304, wall paintings; Dunhuang; zo- 309, 312; angular veloci- diac, 81 ties, 212; daily lunar mo- wang (full moon), 70 tion, 185; epact, 149, 304; wangli (royal calendar), 15 new moon(s), 199, 202, wangwang (disparition), 98 204, 211, 289, 290, 307, wannian (perpetual), 49 308, 321, 327; phase(s) of waterclocks, 213 INDEX OF SUBJECTS 469 week; of ten days, 92; plan- Xiyou ji (Journey to the West), etary, 32, 33, 79, 90–93, 15 136, 279, 295, 318, 321 Xu Guangqi ji (Collected Works Weishu (Wei History), 75 of Xu Guangqi), 18 Western Jin dynasty, 46 Xuanming li (astron. canon), 42, winter solstice(s), 39, 41, 43, 43, 57, 165, 174, 180, 186, 63–65, 73, 75, 100, 113, 187, 192, 277–300, 328, 136–141, 143, 144, 147, 329, 336 150, 158, 161, 162, 164, Xuanye (inﬁnite space), 53 166, 167, 175, 177, 182, Xunzi, ‘zhenglun’, 57 188, 200–204, 209, 216, Xuri (other appellation of Mie 241, 243, 245, 248, 254, days), 219, 312, 314, 318, 255, 261, 262, 271, 274, 319, 330, 331 277, 279, 303, 305, 321 xylographic printing, 126 written calculation, 120 Wu Beizhi (Treatise on Arma- yakšambah (Sunday (Persian)), ment Technology), 55 91 yaosenwen Wuyin li (astron. canon), 65, (Sunday (Nestorian 145 inscription)), 91 year(s); bissextile (Julian or WYG (abbrev.), 28, 37 Gregorian calendars), 73; emerging, 138; intercalary, Xi’an (place name), 6, 91 69, 159, 160, 253, 255– Xia norm, 76, 87 258; lunar, 30, 31, 61, 69– xia xian (last quarter of the 73, 78, 83, 87, 90, 137, moon), 70 256, 271; beginning, 76– xiao (hollow lunar month), 70 78; enumeration, 88, 94; xiao xue (elementary knowl- ﬁrst month, 69; number of edge), 111 days, 69, 71; number of Xieji bianfang shu (Treatise on full and hollow months, 71; Hemerology), 91, 94, 95, ordinary, 69, 70, 159, 257, 271, 272, 299, 319 258; of birth, 297; of con- xin (novelty), 48 fusion, 99; sidereal, 110, Xin Tangshu (New Tang His- 182; solar, 9, 36, 39, 56, tory), 46, 49, 53, 111, 120, 61–64, 66–69, 73, 86, 111– 121, 124, 277 113, 134, 136, 137, 157, Xinghe li (astron. canon), 116 158, 161, 163, 166, 170, 470 INDEX OF SUBJECTS

177, 181, 182, 203, 209, (Skr.)), 120; circle, 118, 210, 215, 216, 241–245, 125, 127, 128, 130, 131; 248, 254, 255, 261, 277, overlined, 128, 129; dot, 280, 303, 321, 325; be- 120; Greek, 129; history ginning, 63; length, 116, of, 128; Indian, 120, 122, 134, 162, 171; mean, 210; 124, 128; kong, 121, 122, number of, 161; quadripar- 338; ling, 130, 338; lunar tition, 64, 65; sexagenary inequality, 192, 193; miss- numbering, 244; support, ing units, 143; nulla (Lat.), 9, 138, 158, 177, 245, 249, 119; numbering of binomi- 255, 262, 279, 283, 304; als, 142; ordinal, 120, 309; tropical, 51, 134, 198, 203 quan, 131; separation sym- Yijing (Book of Changes), 57, bol, 129; sunya, 122; uni- 111, 112, 146 versal, 118; void space(s), yin-yang, 39 119 Yingri (other appellation of Mo Zhantian li (astron. canon), 57 days), 219, 318, 329, 330 zheng (norm), 69 Yitian li (astron. canon), 195 zhengyue (ﬁrst month of the lu- yongjiu (perpetual), 50 nar year), 69 Yuanjia li (astron. canon), 55, Zhide li (Perfect Virtue canon), 137, 222 57 yuannian (initial year of a dy- Zhou norm, 76 nastic era), 78 Zhoubi suanjing (Zhou Dynasty Yuanshi (Yuan History), 48, 50, Canon of the Gnomon), 66 110, 118, 141, 146, 201, Zhouli (The Rites of Zhou), 46 208, 213, 215, 317 zhouying (sidereal shift con- yue (lunar month), 69 stant), 203 yuejian ganzhi (sexagenary zhu (unit of weight), 36, 48 enumeration of lunar Zhu Xi quanshu (Complete months), 86 Works of Zhu Xi), 50 yueshen (monthly spirits), 271 Zhuangzi Yusi jing, 333 (book), 267 zhuanying (anomalistic age of zero, 118–131, 182, 188, 221, the moon), 202 222, 224, 227, 338; astro- zich (Lat., from z¯ıj (Ar.)), 23 nomical, 125–128; Baby- z¯ıj (astronomical handbook lonian, 129; bindu (a dot (Ar.)), 21–23 INDEX OF SUBJECTS 471

Z¯ıj al-Sanjar¯ı (astronomical ziran (that-which-is-of-itself- handbook), 129 what-it-is), 47 zodiac, 81; introduction in Zijinshan Observatory (Nan- China, 81; signs of the, jing), 5 130, 317