Afterthoughts

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 un- derstanding 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 ac- cessible. Likewise, but with some supplementary and sometimes difficult 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 ex- tant 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 with- out 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 re- forms of astronomical canons because the amplitude of their effects is always limited. Keeping in mind these micro-uncertainties, recent research has es- tablished 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 impor- tance 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.

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