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Jean-Claude Martzloff Astronomy and Calendars – The Other Chinese Mathematics 104 BC – AD 1644 Astronomy and Calendars – The Other Chinese Mathematics Jean-Claude Martzloff Astronomy and Calendars – The Other Chinese Mathematics 104 BC–AD 1644 123 Jean-Claude Martzloff East Asian Civilisations Research Centre (CRCAO) UMR 8155 The National Center for Scientiﬁc Research (CNRS) Paris France The author is an honorary Director of Research. After the publication of the French version of the present book (2009), he has been awarded in 2010 the Ikuo Hirayama prize by the Académie des Inscriptions et Belles-Lettres for the totality of his work on Chinese mathematics. ISBN 978-3-662-49717-3 DOI 10.1007/978-3-662-49718-0 ISBN 978-3-662-49718-0 (eBook) Library of Congress Control Number: 2016939371 Mathematics Subject Classiﬁcation (2010): 01A-xx, 97M50 © Springer-Verlag Berlin Heidelberg 2016 The work was ﬁrst published in 2009 by Honoré Champion with the following title: Le calendrier chinois: structure et calculs (104 av. J.C. - 1644). This work is subject to copyright. All rights are reserved by the Publisher, whether the whole or part of the material is concerned, speciﬁcally the rights of translation, reprinting, reuse of illustrations, recitation, broadcasting, reproduction on microﬁlms or in any other physical way, and transmission or information storage and retrieval, electronic adaptation, computer software, or by similar or dissimilar methodology now known or hereafter developed. The use of general descriptive names, registered names, trademarks, service marks, etc. in this publication does not imply, even in the absence of a speciﬁc statement, that such names are exempt from the relevant protective laws and regulations and therefore free for general use. The publisher, the authors and the editors are safe to assume that the advice and information in this book are believed to be true and accurate at the date of publication. Neither the publisher nor the authors or the editors give a warranty, express or implied, with respect to the material contained herein or for any errors or omissions that may have been made. Cover illustration: After an illustration from Sun Jianai 孫家鼐 et al. Qingding Shujing Tushuo 欽定書經 圖 說 (Imperially Commissioned Illustrated Edition of the Classic of History), ﬁrst chapter, 1905. This late picture represents the measurement of the Sun s shadow at the summer solstice with a gnomon and its shadow template, in legendary Chinese antiquity. It results from an interpretation of a short passage of the Shujing 書經. Printed on acid-free paper This Springer imprint is published by Springer Nature The registered company is Springer-Verlag GmbH Berlin Heidelberg Omnem movere lapidem (To leave no stone unturned) D. Erasmus, Adagiorum collectanea, Paris, 1506-1507, I-4-30 To France CONTENTS List of Illustrations . . . . . . . . . . . . . . . . . . . . xvii Acknowledgements . . . . . . . . . . . . . . . . . . . . . . . xix Abbreviations and Conventions . . . . . . . . . . . . . . . . . xxiii Foreword xxvii Initial Foreword xxxi I Chinese Astronomical Canons and Calendars 1 Preliminary Observations The State of the Art . . . . . . . . . . . . . . . . . . . . . Methodological Orientations . . . . . . . . . . . . . . . . Computistics and Predictive Astronomy . . . . . . . The Paradox of the Chinese Calendar . . . . . . . . . . . . The Calendar and its Calculations . . . . . . . . . . . . . The Difficulty of Access to Astronomical Knowledge The Surface and Deep Structures . . . . . . . . . . . . . . Two Notions of Time . . . . . . . . . . . . . . . . . The Double History of the Calendar . . . . . . . . . Historical Sources (Surface Structure) . . . . . . . . Historical sources (Deep Structure) . . . . . . . . . . Numbers . . . . . . . . . . . . . . . . . . . . . . . . . . . The Key Ideas of Astronomical Canons . . . . . . . . . . Political and Cultural Factors: An Example . . . . . The Reforms of Astronomical Canons . . . . . . . . . . . The Bureau of Astronomy . . . . . . . . . . . . . . . . . The Names of Astronomical Canons . . . . . . . . . . . . 1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3 3 7 12 24 27 28 29 30 32 33 34 35 37 41 42 52 55 x CONTENTS 2 Description of the Chinese calendar Limitation and Scope . . . . . . . . . . . . . . . . . . . . . Fundamental Components . . . . . . . . . . . . . . . . . . The Day . . . . . . . . . . . . . . . . . . . . . . . . . The Solar Year . . . . . . . . . . . . . . . . . . . . . The Twenty-Four Solar Breaths . . . . . . . . . . . . . The Seventy-Two Seasonal Indicators . . . . . . . . . The Five Phases . . . . . . . . . . . . . . . . . . . . . The Lunar Year . . . . . . . . . . . . . . . . . . . . . Lunar Months, Ordinary and Intercalary . . . . . . . . The Structure of the Lunar Year . . . . . . . . . . . . The Percentage of Full and Hollow Months . . . . . . Local Patterns of Full and Hollow Months . . . . . . . The Astronomical Months and the Lunisolar Coupling The Beginning of the Lunar Year . . . . . . . . . . . Dynastic Eras and Concordance Tables . . . . . . . . . Cycles and Pseudo-Cycles . . . . . . . . . . . . . . . . . . Definitions . . . . . . . . . . . . . . . . . . . . . . . . The Denary Cycle . . . . . . . . . . . . . . . . . . . . The Duodecimal Cycle . . . . . . . . . . . . . . . . . The Inverted Tree . . . . . . . . . . . . . . . . . . . . The Sexagenary Cycle . . . . . . . . . . . . . . . . . Various Uses of the Sexagenary Cycle . . . . . . . . . The Nine Color Palaces . . . . . . . . . . . . . . . . . The Planetary Week . . . . . . . . . . . . . . . . . . The Twenty-Eight Mansions . . . . . . . . . . . . . . The Jianchu Pseudo-Cycle with Reduplications . . . . The Nayin Cycle with Reduplications . . . . . . . . . Other Aspects . . . . . . . . . . . . . . . . . . . . . . . . . Festivals and Annual Observances . . . . . . . . . . . Irregular Years . . . . . . . . . . . . . . . . . . . . . II Calculations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 61 61 62 62 62 63 66 68 69 69 70 71 72 73 76 78 79 79 80 81 82 83 86 87 90 92 94 96 96 96 99 105 3 Numbers and Calculations 107 Modes of Representation of Numbers . . . . . . . . . . . . . 107 CONTENTS xi Various Zeroes . . . . . . . . . . . . . . . . . . . The Zero-Circle . . . . . . . . . . . . . . . . . . . The History Zero Revisited . . . . . . . . . . . . . Numerical Constants . . . . . . . . . . . . . . . . The Epoch . . . . . . . . . . . . . . . . . . . . . . The Superior Epoch . . . . . . . . . . . . . . . . . The Support Year . . . . . . . . . . . . . . . . . . The Emerging Year . . . . . . . . . . . . . . . . . Numbers of Years from the Epoch . . . . . . . . . Changes of Origin . . . . . . . . . . . . . . . . . . Support Days . . . . . . . . . . . . . . . . . . . . Binomial Representations . . . . . . . . . . . . . . Fractional Representations . . . . . . . . . . . . . Mean and True Elements . . . . . . . . . . . . . . Definitions . . . . . . . . . . . . . . . . . . . Historical Aspects . . . . . . . . . . . . . . . Notation and Terminology . . . . . . . . . . Fundamental Elements . . . . . . . . . . . . . . . The Last Solar Breath of a Lunar Year . . . . . . . The Numbering of New Moons . . . . . . . . . . . The Lunisolar Shift . . . . . . . . . . . . . . . . . Introduction . . . . . . . . . . . . . . . . . . The Epact . . . . . . . . . . . . . . . . . . . The intercalary remainder (Runyu) . . . . . . The Monthly Epact and the Intercalary Month Consequences . . . . . . . . . . . . . . . . . Pathological Calendars . . . . . . . . . . . . 4 Mean Elements Mean Elements in Practice Metonic constants . . . . . Metonic Calculations Justifications . . . . Non-Metonic Canons . . . Calculation Variants . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 118 125 128 131 135 136 137 138 138 139 140 141 143 144 144 145 146 147 148 148 149 149 149 150 150 152 153 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 157 157 157 158 161 163 166 xii CONTENTS 5 True Elements (618–1280) Introduction . . . . . . . . . . . . . . . . . . . . . . . . . True Solar Breaths . . . . . . . . . . . . . . . . . . . . . Some Peculiarities Leading to Simplifications . . . . A Technical Term: The ruqi . . . . . . . . . . . . . . . . A General Mode of Calculation of the ruqi . . . . . Another Mode of Calculation . . . . . . . . . . . . . The Calculation of the ruqi from Mean Solar Breaths Another Technical Term: The ruli . . . . . . . . . . . . . The ruli . . . . . . . . . . . . . . . . . . . . . . . . Tables and Interpolation Techniques . . . . . . . . . . . . Solar Tables . . . . . . . . . . . . . . . . . . . . . . Lunar Tables . . . . . . . . . . . . . . . . . . . . . The Solar Correction . . . . . . . . . . . . . . . . . . . . Further Remarks On the Solar Correction . . . . . . The Lunar Correction . . . . . . . . . . . . . . . . . . . . Calculations Without Tables . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 169 169 170 170 172 173 174 175 179 179 179 180 185 186 190 191 195 6 Later Astronomical Canons The Supremacy of the Inception Granting Canon The Two Last Astronomical Canons . . . . . . . Units of Time . . . . . . . . . . . . . . . . . . . The Epoch . . . . . . . . . . . . . . . . . . . . . Concordances with Julian Dates . . . . . . . . . The Reform of the Shift Constants . . . . . . . . Mean Elements . . . . . . . . . . . . . . . . . . Justifications . . . . . . . . . . . . . . . . True Lunar Phases . . . . . . . . . . . . . . . . . True New Moons . . . . . . . . . . . . . . True Lunar Phases . . . . . . . . . . . . . Notes . . . . . . . . . . . . . . . . . . . . Horary System . . . . . . . . . . . . . . . . . . The Durations of Day and Night . . . . . . . . . The Epoch . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 197 197 198 200 200 201 202 203 204 204 204 207 207 213 215 216 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . CONTENTS 7 III xiii Mo and Mie days Introduction . . . . . . . . . . . . . . . . . Definitions . . . . . . . . . . . . . . . . . . Immediate Consequences of the Definitions Calculations Techniques . . . . . . . . . . Justifications . . . . . . . . . . . . . Supplementary Results . . . . . . . . . . . Justifications . . . . . . . . . . . . . The Indian Origin of Mo and Mie Days . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 219 219 221 222 223 226 232 232 234 Examples of Calculations 239 8 The Quarter-Remainder Canon Importance . . . . . . . . . . . . . . Fundamental Parameters . . . . . . . The Year 119 . . . . . . . . . . . . . Initial Calculations . . . . . . . Another Procedure . . . . . . . Other Solar and Lunar Elements General Structure . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 241 241 241 242 242 243 246 254 9 The Luminous Inception Canon Importance . . . . . . . . . . . . . . . . . . Fundamental Parameters . . . . . . . . . . . The Years 450 and 451 . . . . . . . . . . . . The Years 450 and 451 . . . . . . . . . . . . The Calendar of the Year 450 . . . . . . Guidelines . . . . . . . . . . . . . . . . Translation . . . . . . . . . . . . . . . Notes . . . . . . . . . . . . . . . . . . The Two Lunar Eclipses of the Year 451 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 261 261 261 262 267 268 268 269 271 274 . . . . . . . . . . . . . . . . . . . . . 10 The Manifest Enlightenment Canon 277 Importance . . . . . . . . . . . . . . . . . . . . . . . . . . . 277 Fundamental Parameters . . . . . . . . . . . . . . . . . . . . 277 The Year 877 . . . . . . . . . . . . . . . . . . . . . . . . . . 279 xiv CONTENTS Former Studies . . . . . . . . . . . The Mean Elements of the Year 877 The True Elements of the Year 877 . True New Moons . . . . . . . . . . The Calendar of the Year 877 . . . . . . . A Printed Almanac of the Year 877 . . . . General Presentation . . . . . . . . Some More Details . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11 The Great Unification Canon Its importance . . . . . . . . . . . . . . . . . . . Fundamental Parameters . . . . . . . . . . . . . The Year 1417 . . . . . . . . . . . . . . . . . . . The Intercalary Character of the Year 1417 . The Mean Elements of the Year 1417 . . . The True Moons Phases of the Year 1417 . The Determination of the Intercalary Month Other Moon Phases . . . . . . . . . . . . . Cycles and Pseudo-Cycles . . . . . . . . . Justifications . . . . . . . . . . . . . . . . A Calendar for the Year 1417 . . . . . . . . . . . Presentation . . . . . . . . . . . . . . . . . Monthly Structure . . . . . . . . . . . . . . Translations . . . . . . . . . . . . . . . . . 12 Mo and Mie Days Preliminary Remarks . . . . . . . . . . . The Mo days of the year Jiading 11 (1218) The Mie Days of 877 . . . . . . . . . . . The Mo Days of 1417 . . . . . . . . . . The Mie Days of 1417 . . . . . . . . . . . Afterthoughts . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 279 279 280 289 290 296 296 298 . . . . . . . . . . . . . . 303 303 303 304 304 305 305 308 308 313 313 315 315 316 318 . . . . . 325 325 325 328 329 330 333 Appendices 339 Appendix A The sexagenary cycle 341 CONTENTS xv Appendix B The Twenty-Four Solar Breaths 343 The Twenty-Four Solar Breaths (104 BC – AD 1644) . . . . . 344 The Lunisolar Coupling . . . . . . . . . . . . . . . . . . . . . 345 Appendix C The Seventy-Two Seasonal Indicators 346 Appendix D Official Astronomical Canons 350 List of Official Astronomical Canons . . . . . . . . . . . . . . 351 Metonic Official Astronomical Canons . . . . . . . . . . . . 354 Appendix E Time Constants 357 Appendix F Solar Constants 361 Appendix G Lunar Constants 365 Bibliography 371 Tables of the Chinese Calendar 371 A List of Tables . . . . . . . . . . . . . . . . . . . . . . . . . 372 Recent Advances (2012–2014) . . . . . . . . . . . . . . . . . 382 Computer Programs . . . . . . . . . . . . . . . . . . . . . . . 382 Primary Sources The Astronomical Canons in the Dynastic Histories Extant Calendars . . . . . . . . . . . . . . . . . . The Most Ancient Extant Calendars . . . . . Dunhuang Calendars . . . . . . . . . . . . . Song Calendars . . . . . . . . . . . . . . . . Yuan Calendars . . . . . . . . . . . . . . . . Ming Calendars . . . . . . . . . . . . . . . . Other Primary Sources . . . . . . . . . . . . . . . Collections of Primary Sources . . . . . . . . Individual Works . . . . . . . . . . . . . . . Mathematical sources . . . . . . . . . . . . . Korean and Japanese sources . . . . . . . . . Japanese sources . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 385 385 387 387 387 387 388 389 390 390 391 398 399 400 xvi CONTENTS Rare Sources . . . . . . . . . . . . . . . The Jesuit Reform of Chinese Astronomy Antoine Gaubil . . . . . . . . . . . . . . Philippe de La Hire . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 401 403 404 406 Secondary sources 407 COLLECTIVE WORKS . . . . . . . . . . . . . . . . . . . 407 BOOKS AND ARTICLES . . . . . . . . . . . . . . . . . . . 408 Glossary 441 Index of Names 447 Index of Subjects 453 LIST OF ILLUSTRATIONS 1.1 1.2 1.3 The lunar eclipse on May 4, 1632 . . . . . . . . . . . . 18 A Chinese planetary ephemeris for the year Jiaqing 10 (1531) . . . . . . . . . . . . . . . . . . . . . . . . . . . 20 Philippe de la Hire’s astronomy without hypotheses . . . 40 2.1 The earliest known representation of the sexagenary cycle 84 3.1 A. Gaubil’s full awareness of the centesimal number system . . . . . . . . . . . . . . . . . . . . . . . . . . Yearly table of gnomon lengths . . . . . . . . . . . . . Zeroes in a table of the motion of Venus . . . . . . . . Types of zeroes in two important mathematical texts from the Song and Yuan dynasties . . . . . . . . . . . the cuneiform zero . . . . . . . . . . . . . . . . . . . 3.2 3.3 3.4 3.5 . 110 . 113 . 124 . 127 . 129 10.1 A part of the ninth month of the printed almanac of the year 877 . . . . . . . . . . . . . . . . . . . . . . . . . . 301 11.1 The division of the months of the calendar for the year 1417 into nine zones . . . . . . . . . . . . . . . . . . . 317 11.2 The thirteen first days of the first lunar month of the year 1417 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 323 ACKNOWLEDGEMENTS The idea at the origin of this study of the Chinese calendar and its calculations was sparked by the observation that surprisingly little research has been conducted into this domain until now, but it is also the consequence of a much more ancient interest in the history of Chinese mathematics and the mathematical aspects of Chinese astronomy. It would not have been completed without the support of the National Center for Scientific Research (CNRS) and other research centers into sinology and the history of science, and without repeated contacts with professionals of these domains all over the world. The professors Jacques Gernet (honorary professor at the Collège de France) and Jean Dhombres (emeritus director of research at the CNRS and head of study at the École Pratique des Hautes Études en Sciences Sociales (EHESS)) have both actively encouraged its realization and it is a pleasure for me to express my gratitude to them in the first place. In France, the completion of the Grand dictionnaire Ricci de la langue chinoise (Great Ricci Dictionary of the Chinese Language) (Paris, Desclée de Brouwer, 2001), in which I have been involved for many years as regards Chinese mathematics, astronomy and the calendar, has been an occasion for me of frequent contacts with members and researchers of the Ricci Institute, notably the late Father Claude Larre S.J. and Élisabeth Rochat de la Vallée. These philological works have then stimulated my interest in all sorts of related issues. Above all, the CNRS team devoted to research into Chinese civilization, where I have worked with other sinologists, has been essential. More particularly, I have been involved in a collaborative research project on divination and society in Medieval China, notably with my colleagues Marc Kalinowski (Head of Study at the École Pratique des Hautes Études (EPHE)) and Alain Arrault (member of the École Française d’Extrême Orient (EFEO)) from 1999 to 2003. On this occasion, we have been lastingly interested in the description, classification and dat- xx ACKNOWLEDGEMENTS ing of about fifty non-official Chinese manuscript calendars from the Dunhuang collections (781–993) held by the Bibliothèque nationale de France and the British Library, mainly. During these years, I had direct exchanges with many Chinese researchers, invited by our research team or met on the occasion of international congresses, notably the Taiwanese professors Ping-yi Chu (Academia Sinica, history and philology), Daiwie Fu (Tsing-Hua University, history department), Yi-long Huang (Tsing-Hua University, history department and later Academician from the Academia Sinica), Wann-Sheng Horng (Taiwan National University mathematics department) and Deng Wenkuan (Research Institute into the Chinese Cultural Heritage, Beijing). All of them have then kindly answered my queries and have kept me informed of ongoing research into the Chinese calendar or related subjects, connected in one way or another with the history of Chinese mathematics. During the first week of July 2000, I participated in a conference on calendars in general, organized by the historians of the Middle Ages Jacques Le Goff and Perrine Mane and the mathematician Jean Lefort. Then, the multi-faceted aspects of calendrical time in various cultures have usefully highlighted the similarities and the peculiarities of the Chinese case comparatively with so many other possibilities of calendars. Some time before, regular contacts with Tony Lévy (Hebrew mathematics, CNRS), Pierre-Sylvain Filliozat (Sanskritist, Académie des Inscriptions et Belles-Lettres, CNRS, EPHE), André Cauty (Amerindian linguistics, Bordeaux University) and Jim Ritter (Babylonian mathematics, Paris VIII University) have brought to my attention the problems raised by written numerations and zero in general. In England, I have visited two times the Needham Research Institute in Cambridge, in December 1997 and December 2005. Christopher Cullen (the present Director of this Institute, successor of Professor G.E.R. Lloyd) and the librarian John Moffett, have facilitated my access to its precious documentation and allowed me to work in particularly favorable conditions. I have also met there the historian of astronomy Raymond Mercier (Cambridge University) in Cambridge and elsewhere. In China, at Beijing, multiples invitations to the Institute of Mathematics and Systems Science (Academia Sinica) and the continued support of Professor Li Wenlin – whose role in the development of the his- ACKNOWLEDGEMENTS xxi tory of mathematics in contemporary China has been essential – have allowed me to meet, on numerous occasions, Chinese researchers specializing in the history of Chinese mathematics and mathematical astronomy, notably the late Chen Meidong, Liu Dun, Guo Shuchun and Sun Xiaochun. Wang Yusheng too has always been ready to meet efficiently all my requests with an unequaled joviality and receptiveness. In September 1993, I could meet the professors Zhang Peiyu and Li Yong, specialists of Chinese astronomy, at the Nanjing Zijinshan Observatory. At the same time, Professor Xuan Huancan (Nanjing University, Astronomy Department) has kindly forwarded to me recent publications on the subject over many years. At Xi’an and elsewhere, I also have had extensive exchanges with Qu Anjing (Department of Mathematics, Northwest University, Xi’an) and I would like to warmly thank all these researchers and more generally all those who have made my work in China easier. Lastly, I would like to express my gratitude to the librarians of the Institut des hautes études chinoises, those of the Sorbonne University (Paris) and all the members of the research team devoted to the study of Chinese civilization to which I belong and where I have worked with pleasure and enthusiasm for so many years. ABBREVIATIONS AND CONVENTIONS General Abbreviations − indicates a negative year; . As usual, a dot is used to separate the integer and fractional part of a decimal number. Not to be confused with other similar notations, introduced in Chapter 3 of this work and only concerning non-decimal numbers used in Chinese calendrical calculations; j. juan, book chapter (sometimes ‘book’), literal meaning :‘roll’, an allusion to antique Chinese books, similar to the volumina of Greek and Roman Antiquity. Bibliographical Notations * When a book has been edited several times, an asterisk marks the years of publication of the consulted versions; When the publication of some work extends over several years, its initial and final years are separated by a dash; / Slash between the various years of edition of a work published several times; Other Abbreviations abbrev. abbreviation Ar. Arabic astron. ‘astronomy’ or ‘astronomical’ ca. Latin: circa. Means ‘approximately’ cal. calendar(s) or calendrical Chin. Chinese xxiv denom Eng. f. ABBREVIATIONS AND CONVENTIONS denominator English after a page number means ‘and the following pages’ (also noted ‘ff.’ in other works) fl. floruit. Indicates when somebody was active (Latin: flourished). Heb. Hebrew Jap. Japanese Lat. Latin ms. manuscript numer numerator proc. procedure(s) Skr. Sanskrit transl. translation Lunar Phases and Special Months FM full moon; FQ first quarter; LQ last quarter; NM new moon (also very often noted ni for some i); man anomalistic month msyn synodic month Reference Works COLL. collective work; COL-astron Zhongguo kexue jishu dianji tonghui, tianwen juan dddddddddd,ddd (General Collection of Chinese Scientific and Technical Works, Astronomy), 1993 (full reference p. 390); COL-math same reference but for mathematics (see also p. 390); ABBREVIATIONS AND CONVENTIONS DENG-2002 DENG-2010 DKW LIFA SIXIANG WYG xxv DENG Wenkuan, Dunhuang Tulufan tianwen lifa yanjiu d d d d d d d d d d d (Research into Calendars and Astronomy from Dunhuang and Turfan), Lanzhou, Gansu Jiaoyu Chubanshe dddd ddd, [collection of articles] DENG Wenkuan, d d d, Dunhuang tianwen lifa kaosuo d d d d d d d d (Astronomical Research and Calendar Manuscripts from Dunhuang) Shanghai, Shanghai Guji Chubanshe dddddd d, [collection of articles]. Morohashi Tetsuji d d d d. Dai kanwa jiten d d d d d (Great Chinese-Japanese Dictionary), Tokyo, 1960. References to this dictionary are given as follows: volume-page-item number; Zhang Peiyu ddd, Chen Meidong ddd et al., 2008. Zhongguo gudai lifa d d d d d d (Ancient Chinese Astronomical Canons), Zhongguo tianwenxue shi daxi dddddddd (Great Encyclopedia of Chinese Astronomy), Beijing, Zhongguo Kexue Jishu Chubanshe, ddddddddd; Chen Meidong ddd, 2008. Zhongguo gudai tianwenxue sixiang ddddddddd (Ancient Chinese Astronomical Thinking), Zhongguo tianwenxue shi daxi dddddddd (Great Encyclopedia of Chinese Astronomy), Beijing, Zhongguo kexue Jishu Chubanshe, ddddddddd; Yingyin Wenyuange siku quanshu dddddddd d (Reproduction of the Siku quanshu Collection Preserved at the Wenyuange Library), 1500 vol., 1986 (full reference p. 390); is pinyin The Transliteration of Chinese The transliteration of Chinese used here is the pinyin phoneticalsystem , a system adopted in 1958 by the People’ s Republic of China and now widely accepted all over the world in specialized and non-specialized xxvi ABBREVIATIONS AND CONVENTIONS publications alike, even though the older Wade-Giles system is still rather widespread in the English speaking world. At the same time, it has been impossible to avoid all sorts of other transcriptions of Chinese names and notions established by custom. But their pinyin equivalents or even their original Chinese written forms have also been indicated when necessary. Chinese Dates Chinese dates traditionally rely on various and more or less complex formulations. More simply, we have uniformly adopted here the simple format ‘day/lunar month/ lunar year’, where lunar months are denoted by Roman numerals in order to avoid any confusion with Julian or Gregorian dates. FOREWORD The study of the ancient Chinese mathematics used for astronomy and the calendar proves that it differs significantly from that of the wellknown ‘Nine Chapters’ tradition. It consist in unceasingly reworked procedures devoted to the prediction of celestial phenomena and the calendar. This study reveals unexpected results, notably, inter alia, nondecimal number systems and a form of written zero not attested elsewhere, the weight of numerology, the strong link between predictive mathematics and divination and the predominance of empirical observation over theory. These results are of interest not only in the history of Chinese mathematics but also, more generally, in the history of science, including astronomy. Furthermore, a comparison between Chinese and non-Chinese ancient approaches reveals both numerous points of contacts and striking dissimilarities, notably a lasting Chinese belief in the impossibility of long-term mathematical predictions. Lastly, numerous examples of calculations support the general description of mathematical patterns underlying calendrical calculations. The present study is a logical extension of my former A History of Chinese Mathematics (Springer, 1996 and 2006) and, as such, is primarily intended for readers interested in the cultural history of Chinese mathematics, with or without any sinological background. Given the multifactorial nature of the history of mathematics, various subjects, which are an integral part of the history of Chinese mathematics, have also been introduced in the first part of the book, notably the influence of politics on mathematics, the Bureau of astronomy, the secret character of astronomical canons, their names, the importance of numerology and divination. However, these various subjects have always been subordinated to the mathematical aspect of the present study, even though each of them could easily have led to the redaction of a sizeable monograph. By committing to this perspective, I have tried to highlight a number of elements generally not taken into account in available his- xxviii FOREWORD tories of Chinese mathematics, notably a piecewise conception of variable phenomena in terms of phases depending on yin-yang conceptions, tests (or quasi-criterions) delivering only probable results, unusual astronomical tables containing not only lists of predetermined coefficients but also terse procedural instructions (‘quasi-tables’), a negative mode of definition – the sole in all extant ancient Chinese mathematics, apparently –, concerning intercalary months. Moreover, beyond punctual peculiarities of this mathematics and, more generally, of Chinese calendrical calculations constituting the core of this work, a broader issue, of interest in the comparative history of science, has also been addressed in order to catch a glimpse of fundamental Chinese conceptions concerning the nature, function and strength of mathematics and the possibility of “laws’ of nature. Starting from a corrected and updated version of its former French version, this book has been modified and organized as follows: • more importance has been granted to methodological problems (Chapter 1); • the analysis of the fundamental but difficult notion of li d, respectively meaning ‘calendar’ in general and ‘mathematical astronomy’, ‘astronomical systems’ or ‘astronomical canons’ in a technical sense, has been significantly developed from a comparative perspective, mainly implying mathematical astronomy from the Islamic and Greek worlds, in the first chapter of this book; • the analysis of the Chinese belief in the artificial nature of mathematics, and in the impossibility of obtaining immutable predictive techniques based on mathematics, has been reexamined in order to show that, on the contrary, unbounded mathematical predictions were also regarded temporarily as an abstract possibility in the Chinese late medieval and pre-modern context. In the long run, however, the very possibility of obtaining a mathematical formulation of such techniques has continued to be strongly called into question in China; • the presentation of Chinese mathematical techniques, including those of interpolation and the analysis of solar and lunar tables FOREWORD xxix has been much more developed. Moreover, the notions of quasicriterions and quasi-tables have been introduced in order to highlight essential aspects of these other Chinese mathematics; • a fully worked out example has been added which details the calculation of the dates on which two lunar eclipses occur, recorded in a manuscript calendar for the year 451; • the bibliographies of primary and secondary sources have been updated in order to take the latest developments (2014) into account; • last but not least, a large number of further details and new figures have been inserted in various places of the main text. The latter concern, for instance, a Chinese planetary ephemeris, the question of an astronomy without hypotheses, the centesimal system, a schema of an eclipse prediction, the oldest known inscription showing the whole Chinese sexagesimal cycle and, notably, zero. • a new final section, ‘afterthoughts’, indicates possible directions of interest for future research into the field, concerning, notably, the extension of the present investigation to the mathematical aspects of astronomical canons beyond calendrical calculations, chronological problems and topics concerning more particularly the history of mathematics. Moreover, like its former version, this book can be used in various ways: A number of sections can be consulted independently. That is the case for the appendixes and bibliographies, of course. In particular, detailed and updated presentations of almost all available tables of the Chinese calendar, concordance tables and various primary sources have been propounded. As for the body of the book itself, chapters 1 to 4 are certainly prerequisites but, whereas the first chapter does not involve technical developments and can be read independently, on the contrary, all the notations, definitions and notions introduced in chapters 3 and 4 are constantly used everywhere in the sequel. xxx FOREWORD More particularly, the fourth chapter grants access to all subsequent developments relying on mean elements which are used in one way or another, exclusively or partly, in all following chapters. In its turn, the fifth chapter introduces true elements (as opposed to mean elements) and related notions intervening in Chinese calendrical calculations. As such it is thus a prerequisite for the examples of calculations developed in the tenth chapter. The sixth chapter concerns the two latest systems of calendrical calculations (from 1281 to 1644) and the related example of calculations developed at length in the eleventh chapter cannot be read independently. In the same order of ideas, almost everything contained in the seventh chapter is self-contained, but the examples of related calculations presented in the twelfth chapter also rely on techniques of calculation of mean elements introduced in the fourth chapter. Lastly, I add that I have entirely composed the French and English versions of the present book from the latest versions of MiKTeX1 and Texmaker2 in order to produce a pdfLaTeX output. Moreover, in order to avoid reencoding all previous files of the French original I had to enter Chinese characters from their Big5 encoding. Consequently, a rare Chinese character, whose pinyin transcription is chong and which is used in the name ‘Zu Chongzhi’ was not available. However, I could replace it by an homophone, d, having a nearly identical graph: it has merely one dot in excess on its left part which should be removed in order to obtain the missing character.3 Moreover, for the same reason, some Japanese words have been reproduced from their ancient forms. However, the correspondence with those now in use can be easily retrieved from current dictionaries.4 1 http://miktex.org/. 2 http://www.xm1math.net/texmaker/. 3 The three Chinese characters used in this book for ‘Zu Chongzhi’ are the following: ddd. 4 For instance: Nelson, A.N., The Modern Reader’s Japanese-English Character Dictionary, 2nd Revised Edition, Charles E. Tuttle Company, 12th printing, 1981. INITIAL FOREWORD This purpose of this book is to highlight some of the most fundamental mathematical structures underlying the calculation techniques used for the construction of the Chinese historical official calendar, in a way that makes it possible to efficiently determine its main elements over as large a number of years as possible, from a preliminary description of its invariant structure. Apart from technical matters, however, great importance has also been granted to the wider context of this mathematics, particularly its epistemological aspect, which is so important in order to understand its nature, purpose and function. Unofficial Chinese calendars and non-Chinese calendars currently also used in China, such as the sinicized Muslim calendar, are not included in the present study for the following reasons: we wholly ignore the calculation techniques of the former whereas the latter rely on mathematical techniques utterly different from those of the Chinese calendar. The historical period retained spans the years of the interval 104 BC– AD 1644, a choice dictated both by the state of manuscript and printed sources handed down to us and by the overall unity of Chinese calendrical calculations developed between these two limits, extending over more than seventeen centuries. In the case of more ancient years, we do not possess any historical document explaining calendrical calculations while, on the contrary, numerous and detailed treatises are available for most years posterior to 104 BC. From 104 BC to AD 1644, Chinese calendrical calculation techniques have never ceased to belong to the same family, for they have always been designed in the form of lists of procedures, always formulated and organized in the same way. Overall, calendrical events are mostly dealt with like those of astronomy by only seeking to obtain the best possible precision, irrespective of their unceasingly variable underlying techniques. No less characteristically, the modes of representation xxxii INITIAL FOREWORD of numbers and the technical terminology they rely on are eminently unstable; they also assign arbitrary patterns to numbers by taking avail of numerological correlations. By contrast, the period following the year 1644 marks a break with previous traditions: the reform of Chinese astronomy then undertaken successfully by Jesuit astronomers has resulted in a dependency of Chinese calendrical calculations on techniques previously unheard of in China – trigonometry, geometry, logarithms, . . . – used in Renaissance Europe and directly in line with Greek mathematics and astronomy. Hence a discontinuity between former and newer Chinese practices which would doubtlessly deserve a study in its own right. Yet, when certain features of Chinese calendrical calculations from this later period can clarify older practices for one reason or another, we have not ruled out comparisons. Although, as already noted, official Chinese calendrical and astronomical techniques developed between 104 BC and AD 1644 belong to the same family, its members are extremely numerous and seemingly far apart from each other: in the intervening years, they have been unceasingly reformed no fewer than fifty times. Nothing of the sort exists anywhere else than in the Chinese world. In order to highlight the main structures organizing this wealthy repository of mathematical techniques, the following approach has been retained: First, noting that the backbone of the Chinese official calendar has remained identical to itself over time, we have attempted to highlight its invariant structuring principles and ideas. Second, we have deliberately chosen to formulate in as general a way as possible the abstract techniques underlying Chinese calendrical calculations and the numerical results they lead to. By contrast, their profuse philological and syntactic peculiarities have not been our priority, even though this aspect of the question is certainly also of primary importance for other purposes. Given the inchoative state of this field of study, however, practically everything remains to be done, even when taking into account the always increasing number of specialized publications. Therefore, the need to clarify the general structure of calendrical calculations has seemed more pressing than a study focusing on the INITIAL FOREWORD xxxiii exploration of the quasi-tropical jungle of its specific linguistic manifestations. Nevertheless, such questions have been highlighted when it was obviously desirable to take them into account. After numerous tentative steps and false trails, we have elaborated a first version of a technique for describing Chinese calendrical calculations from a limited set of ad hoc notions and notations. Thanks to this tool, it turned out that only a limited number of such techniques exists, modulo a residual number of recalcitrant processes, either seemingly difficult to gain access to because of the incomplete character of Chinese sources or their apparent obscurity, mainly. More precisely, Chinese calendrical techniques fall under two fundamental types: those relying on mean elements and those admitting both mean and true elements, these two notions having their usual astronomical meaning. On this basis, we have fully described the structural core of all the techniques of the first type and sketched an outline of the second ones, so that it became easier to figure out the overall rationale of the innumerable techniques used to calculate Chinese official calendars from 104 BC–AD 1644. Hence, also, the conquest of a sort of autonomy enabling us to perform Chinese calendrical calculations in various but equivalent ways, recorded or not in original sources. It follows therefrom that it is not always necessary to follow original procedures to the letter in order to obtain exactly the same results, to grasp their scope, to deduce some of their consequences and to provide answers to, notably, the following questions: • Is a calculation technique so obtained sufficient in order to restore exactly the content of authentic Chinese official calendars? • Is it generalizable? Although the small number of extant authentic calendars issued between 104 BC and AD 1644 precludes the possibility of giving a final and general answer to these questions, it remains that once a calculation method has been formulated, its becomes possible to determine the putative content of Chinese official calendars, even in the case of years for xxxiv INITIAL FOREWORD which authentic calendars are not extant. Hence the possibility to retrieve the theoretical dates of a large number of calendrical events, well beyond the most fundamental ones listed in available tables of the Chinese calendar and concordance tables. But, of course, this is not always possible in all cases if only because not all Chinese calendars have been obtained only from calculations but also from political decisions. Beyond reconstructed calendrical dates, it is sometimes also possible to deduce completely and globally the general structure of calendars obtained from certain types of procedures, notably those based only on mean elements. Furthermore, the practice of these calculations shows that the modes of representation of numbers used in calendrical calculation – and more generally in all Chinese astronomical canons too – are unexpectedly not based on decimal representations. Moreover, it also happens that number representations also rely on a particular form of written zero, practically never mentioned by historians of mathematics. It is of course hardly necessary to stress the importance of these two results which are of interest not only for the history of the Chinese calendar and astronomy but also, more broadly, for the history of mathematics since they drastically challenge the usual idea that, over its very long history, China would have been only aware of nothing else than a decimal and positional system of numeration, any other possibility being ruled out. In order to present these results, the present work has been divided into three parts: The first part expounds the principles on which this study is based. Then a presentation of the history of the Chinese calendar follows, both from the perspectives of its specific content and calculations, together with related questions of interest, such as a list of all irregular years of the Chinese calendar. The second part focuses on the fundamental technical, mathematical and astronomical aspects essential to any description of Chinese calendrical calculations: the representation of numbers and numerical constants, astronomical and non-astronomical, the question of the determination of the origin of time and other technical notions peculiar to lunisolar calendars, such as the lunisolar shift, or epact, and the rule for determining intercalary months. On this basis, the details of par- INITIAL FOREWORD xxxv ticular techniques of interest in the calculation of the Chinese calendar are tackled by distinguishing those using either mean elements, true elements or both sorts of elements. Then, two little known elements of the Chinese calendar, namely the Mo d and Mie d days, are introduced with a wealth of details because they tend to prove the influence of Indian culture on the Chinese calendar. The third and last part contains some examples of fully developed calculations for calendars of given years, their results being compared, when possible, with the content of authentic calendars and when not, with the calendrical data listed in chronological tables of the Chinese calendar. In addition, a large number of examples of specific calculations are provided everywhere in the main text. Thereafter, a series of appendixes also provides systematic tables giving lists of numerical constants indispensable when performing calendrical calculations, together with a chronological list of Chinese official astronomical canons. Lastly, the bibliography of primary sources presents the most important references concerning both the study of the Chinese calendar, its calculations and Chinese chronology. In its turn, the bibliography of secondary sources contains an extensive list of publications in Chinese, Japanese and Western languages. Chinese calendrical calculations in general constitute a vast domain. However, they represent only a minute fraction of the wider domain of Chinese mathematical astronomy. So far, however, this astronomy has often been regarded as reducible to a purely qualitative science, based on an accumulation of precise observations and no mathematics, contrary to the other great astronomies from Antiquity and the Middle Ages. If, by stressing the role of mathematics with respect to the limited domain of Chinese calendrical calculations, the present work can encourage historians of science and sinologists to take a greater account of the eminently mathematical character of traditional Chinese astronomy, our objective would have been achieved. Part I Chinese Astronomical Canons and Calendars CHAPTER 1 PRELIMINARY OBSERVATIONS The State of the Art So far, research into the Chinese calendar has mostly been undertaken by two independent families of historians: social sciences historians and historically-minded astronomers. The former have essentially focused their efforts on non-technical primary sources, often retrieved from archaeological excavations, and by using approaches typical of historical research: textual criticism, philology, chronology, social and intellectual history, notably. They have thus described the material aspect and the content of extant calendars, analyzed their cultural background, including their religious aspects, and tracked their modifications over long historical periods, taken advantage of their findings in order to solve difficult chronological puzzles, such as the dating of fragmentary manuscript calendars bearing no explicit indication of year. Despite the wide-ranging scope of their research, their attention to the smallest details and their interest in all sorts of issues raised by the history of the calendar, however, these historians have not often integrated calendrical calculations into their investigations. The latter, on the contrary, have quite often tackled the subject from the perspective of contemporary technical astronomy, in order to evaluate the degree of precision of ancient Chinese astronomical parameters and eclipse forecasts and to compare the achievements of Chinese and Western astronomy. They have thus often interpreted ancient procedures in the light of modern knowledge in order to determine objective or purported Chinese contributions to the progress of astronomy. Whereas social historians have been impeded by the paucity of extant authentic calendars, astronomers have not been as limited as social © Springer-Verlag Berlin Heidelberg 2016 J.-C. Martzloff, Astronomy and Calendars – The Other Chinese Mathematics, DOI 10.1007/978-3-662-49718-0_1 3 4 PRELIMINARY OBSERVATIONS historians by the availability of ancient technical sources: as will be noted later, extant primary sources are voluminous and generally easily accessible, even though they only represent a small fraction of what has been preserved in this domain.1 However, they have been significantly hampered by their unbelievable technical difficulty, a consequence of severe and lasting process of acculturation, prompted both by the secret character of traditional Chinese astronomy and by its successful reform, initiated during the first half of the seventeenth century by Jesuit astronomers and based on principles radically different from those previously in effect. As noted long ago by Antoine Gaubil S.J.2 (1689–1759), the famous pioneer-historian of Chinese astronomy, even the members of the Chinese Bureau of Astronomy – called by him the ‘Tribunal of mathematics’ – were almost wholly ignorant of ancient Chinese astronomical techniques in use before this reform: “[. . . ] les Chinois qui compôsent ce Tribunal ne savent prèsque rien de l’Astronomie qui étoit en usage avant la venuë des Jésuites.”3 (“[. . . ] the Chinese members of this Tribunal are almost completely unaware of the astronomy in use before the arrival of the Jesuits.” Consequently, despite the remarkable development of research into this forgotten subject from the middle of the seventeenth century, in Japan, China and, to a lesser extent, but somewhat later, in Europe,4 our knowledge of Chinese calendrical calculations has lastingly encountered considerable difficulties and research into this field has often remained much less developed than the non-mathematical aspect of astronomy to the extent that Chinese astronomy is sometimes presented as a purely 1 See p. 34 below. excellent analysis of A. Gaubil’s works was published long ago by J. Dehergne (S.J.) See J. Dehergne 1944 and 1945. 3 See É. Souciet 1732, tome 3, p. 238 (notice p. 404 below). The choice of the term ‘Tribunal’ consistently used by the Jesuits in order to designate this Chinese official institution probably comes from their perception of the powerful astrological component of Chinese astronomy which rightly appeared to them as heavily dependent on judicial astrology. Likewise, their mention of mathematics rather than astronomy rightly highlights the mathematical character of this astronomy. 4 On this point see the bibliographical notices p. 390 f. below. 2 An THE STATE OF THE ART 5 qualitative discipline (see, for instance, J. Needham 1959’s influential work). Significantly, the Japanese translation of a major corpus of mathematical astronomy from the Yuan dynasty (1277–1367), the Shoushi li,5 was undertaken by the then most influential Japanese scholars in the field, K. Yabuuchi (1906–2000)6 and S. Nakayama (1928–2014), more than fifty years ago, in 1960, but it took no less than fifty years before a tiny, but everywhere dense, volume (176 pages) could be eventually released, in 2006, six years after the death of the first author and after reiterated delays resulting from the difficulty of figuring out its planetary theory.7 No less importantly, since before it began, this interminable Japanese project was closely related to a long-awaited English translation of the same source by the influential historian of Chinese science N. Sivin (Pennsylvania University), published three years later, in 2009.8 Likewise, but later, a still more ambitious Chinese encyclopedic project of a complete survey of ancient Chinese astronomy, initiated by leading Chinese historians of astronomy and Chinese astronomers from the major Chinese astronomical observatories, first and foremost the late Chen Meidong ddd (1942–2008), Bo Shuren ddd (1934–1997) and Zhang Peiyu ddd (Purple Mountain Observatory, Nanjing), was launched a little before 1980, but brought to completion only a little less than thirty years later, from 2007 to 2009.9 This time, these efforts have 5 For a critical analysis of the meaning of this term and similar ones, see p. 55 below. 6 For an adequate presentation of K. Yabuuchi’s life and works see East Asian Science, Technology and Medicine (2001, no. 18). 7 K. Yabuuchi, and S. Nakayama 2006. 8 See N. Sivin 2009, p. 616: “Nakayama and I had discussed the project since before it began.” Moreover, N. Sivin has been, inter alia, at the origin of the enlightening notion of ‘cultural manifolds’ introduced in order to advocate the taking into account of all the dimensions and mutual interrelations of the historical phenomenon under consideration. Since then, several outstanding studies of the Chinese calendar under the Song and Ming dynasty developed in such a direction have been issued since 2000, approximately. See, notably, Dong Yuyu 2004 (Song calendars); Sun Xiaochun and Zeng Xiongsheng 2007 (Song calendars); Wang Xiaohu 2011 (Ming calendars). 9 LIFA, p. 721–724. We have also retained here the following volumes of this project: Chen Jiujin 2007a and 2007b, Chen Meidong 2008, Chen Xiaozhong and Zhang Shuli 2008, Lu Yang 2008, Xu Zhentao 2007, Zhuang Weifeng 2009. 6 PRELIMINARY OBSERVATIONS resulted in an impressive collection, composed of nine monumental volumes, representative of Chinese research into this area (mathematical astronomy, biographies of ancient astronomers, the Bureau of astronomy and ancient astronomical education, ancient astronomical thinking, the transition from ancient astronomy to modern astronomy, ancient astrology, records of celestial phenomena, the astronomies of Chinese minorities, dictionary of Chinese astronomy). Overall, however, what concerns more particularly the Chinese calendar has often been either left behind the scenes or undifferentiated from astronomy. Whereas this long-term accomplishment is representative of the state of the art, a large number of articles and many books have also been published by individual authors and our bibliography eloquently witnesses the vitality of contemporary Chinese research into Chinese astronomy. Most notably, among recent studies, one notes those of Qu Anjing (mathematician, Northwest University, Xi’an) d d d.10 In spite of the importance of these numerous works, however, publications more particularly devoted to a comprehensive and operational description of Chinese astronomical and calendrical calculations are rare. Among these, I have had access to the following books and articles: 1. research into the Shoushi li calculations and other types of calendrical calculations from the Tang dynasty (Uchida Masao d ddd and Hirose Hideo dddd (Japanese historian of the calendar and astronomer, respectively));11 2. Bo Shuren d d d’s article12 devoted to a complete and convincing explanation of the Jiyuan li ddd (Era Epoch canon)13 mathematical procedures; 10 Qu Anjing 2005 and 2008. Further references, important for the identification of the meaning of numerical constants used in calendrical and astronomical calculations, are also indicated below, p. 131 f.). 11 M. Uchida 1975 and H. Hirose 1979. 12 Bo Shuren 2003, p. 369–447. 13 1106–1127. METHODOLOGICAL ORIENTATIONS 7 3. a reconstitution of the calculations of a calendar for the year 1365, according to the Shoushi li ddd techniques (Zhang Peiyu d dd);14 4. The calculation of the official calendar for the year 664 according to the Linde li ddd techniques (Qu Anjing ddd, Ji Zhigang ddd and Wang Rongbin ddd, Northwest University, Xi’an));15 5. a Ph.D. dissertation entirely devoted to the Shoushi li dd calculations (Li Yong dd, Nanjing University);16 6. most publications of Lin Jin-Chyuan ddd (Professor, Department of Chinese Literature, National Cheng-Kung University d ddddd).17 When necessary, other works of interest in the same area are also duly cited in the sequel. Methodological Orientations Ideally, complete English translations of the most essential primary sources concerning the mathematics of the Chinese calendar would be desirable since they would ensure an increased accessibility to this difficult subject, presently limited to a very restricted number of specialists. Given the present inchoative state of our knowledge of the subject, however, the magnitude of the task is daunting: the volume of the relevant sources is so monumental that a realistic project of translation would probably involve teams of highly specialized translators working continuously, for many decades, if not longer. Moreover, given than the methodological, historical, philological and epistemological problems are colossal, it is not certain that the result would be ipso facto satisfactory because, despite the remarkable efforts of sinologists and historians of Chinese sciences, the texts of all available primary sources have still not always been established with a sufficient degree of reliability. 14 Zhang Peiyu 1994. Anjing, Ji Zhigang and Wang Rongbin 1994. 16 Li Yong 1996. 17 Lin Jin-Chyuan 1997 to 2008. 15 Qu 8 PRELIMINARY OBSERVATIONS Therefore, not everything they contain can be translated safely without a prior deep understanding of their astronomical and mathematical tenets. For instance, numerous Chinese astronomical tables have reached us in a poor state and still need heavy conjectural emendations.18 No less fundamentally, the procedures underlying Chinese calendrical and astronomical calculations are characterized by such a profuse, variable, unusual and unstable technical terminology that in many cases, extensive translations would necessarily reflect an authentic but almost unmanageable complexity, all the more so that, more generally, there is little consensus among historians of Chinese science on issues concerning the translation of Chinese technical terms. Anyway, translations cannot but rely on a profusion of philological details and the creation of a deluge of neologisms in order to avoid an untimely overflow of untranslated technical terms. For example, the English translation mentioned above introduces a little more than two hundred and fifty English translations of technical terms, or expressions, and as many explanations each time.19 However, despite its great usefulness, this particular case reveals only a fraction of a long story, spanning numerous centuries and dynasties. It is thus not surprising that many difficult philological problems are still either unsolved or solved in an unsatisfactory manner. Therefore, it is certainly important to continue to explore them. In this respect, we have, in particular, tackled here this sort of problem in a reverse manner, by taking avail of the way certain medieval Chinese tried to translate foreign terms when they were confronted with non-Chinese astronomical issues.20 Still, not all problems of translation can always receive satisfactory solutions in all cases and we have noted present limitations in this respect when necessary. Fortunately, however, translations other than those concerning technical terms generally raise much less difficult issues. Therefore, we have often proposed here a number of translations of various passages of interest, including partial translations of authentic calendars. 18 See Chen Meidong 1984 (solar tables); Chen Meidong and Zhang Peiyu 1987 (lunar tables). Naturally, the same remark also apply to a very large number of other technical questions. 19 Sivin, N. 2009, p. 597–615. 20 See, for instance, “The Technical Meaning of the Term Li d”, p. 16 below. METHODOLOGICAL ORIENTATIONS 9 Yet, the mere linguistic dimension of the subject is only an aspect of the question. Translations cannot as such reflect the mathematical structures at work in these difficult Chinese technical sources any more than translations of sentences from a foreign language into English can spontaneously reveal its grammar. That is precisely why the present work is more particularly engaged in a preliminary and limited investigation concerning both the structure of the mathematics underlying the most important available Chinese sources concerning calendrical calculations and their main epistemological features, in order to disclose structural characteristics typical of such calculations and their kind of mathematics, in general. To this end, insofar as the Chinese calendar is technically a lunisolar calendar, we have freely used current notions typical of this widespread kind of calendar, notably the solar year, full or hollow lunar months, intercalary months, the epact (or age of the moon), simple or generalized Metonic cycles, fixed and mobile dates and the like. Yet, a number of peculiarities of Chinese calendars and calendrical calculations cannot always be analyzed in a sufficiently precise manner with such tools. We have thus also coined a number of ad hoc notions such as ‘surface structure’, ‘deep structure’, ‘luni-solar coupling’, ‘primary constants’, ‘secondary constants, ‘fossil constants’ ‘support year’, ‘quasi-criterion’, ‘quasi-table’, or reintroduced ancient but enlightening translations such as ‘solar breaths’, instead of ‘solar terms’. Moreover, a long preliminary practice having proven that Chinese calendrical calculations mostly depend on clear-cut procedural instructions, leading to the same results independently of the way arithmetical operations are performed, we have borrowed some of our notations not only from current mathematics but also, and in large part, from concrete mathematics, that is “the controlled manipulation of mathematical formulas, using a collection of techniques for solving problems”.21 We have thus, notably, used mathematical variables, ordered pairs of numbers (called here ‘binomials’) and piecewise functions, the latter being particularly important with respect to the fact that, in ancient and medieval China, astronomical phenomena are fundamentally described in terms of successive phases, where some quantity is greater, smaller or 21 R.L. Graham, D.E. Knuth and O. Patashnik 1990, p. VI. 10 PRELIMINARY OBSERVATIONS equal to its mean value. Moreover, the notations ‘⌊x⌋’ and ‘x mod y’ have been particularly useful. The first represents the integer part of a number x and the second is defined as follows: ⌊ ⌋ x x mod y = x − y × y def (1.1) where x and y are not necessarily integers but also rational numbers, corresponding either to the quotient of two integers or to a fraction. Owing to this modus operandi, we have been able to obtain exact numerical results, wholly conforming to those which would have been delivered by the Chinese procedures, in a large number of cases, because Chinese calendrical calculations often depend on exact arithmetical calculations with fractions. In some cases, however, we have been compelled to use decimal approximations, for instance when the original procedures rely on numerical approximations and do not describe the different steps of their calculations in a sufficiently explicit way. Moreover, we have not attempted to reconstruct the way arithmetical operations might have been performed in each case because any attempt in such a direction would necessarily have relied on numerous more or less valid presuppositions, nothing of the sort ever having been fully explained in Chinese sources. Sometimes too, we have had recourse to some simplifications of our mathematical procedures but in such a case, we have only admitted ‘faithful simplifications’, that is simplifications fundamentally following the same successive steps and leading to the same results as those which would have been obtained from original procedures. For example, we have partly tackled interpolation techniques in this way by making explicit and generalizing a fruitful idea initially developed by Uchida Masao.22 Moreover, we have also checked systematically our calculations by using various programming techniques. At the very beginning, many years ago, we relied on the pocket computer Sharp PC 1475 because it was both programmable and able to deliver up to twenty decimal digits. Later, we also have taken advantage of initial versions of Maple, 22 See p. 189 below. METHODOLOGICAL ORIENTATIONS 11 then distributed by Springer, because its ability to handle symbolic expressions renders the programming of Chinese calendrical calculations straightforward. In this way, we have been able to compare our understanding of Chinese calendrical calculations with the content of authentic Chinese calendars. In particular, the examples of calculations propounded in this book, aiming at recalculating the content of extant Chinese calendars, have been checked using this technique. But, when such calendars do not exist, we have compared our calculations with the content of available calendrical tables, notably by having recourse to those of Zhang Peiyu’s tables,23 because they present the advantage of indicating the dates of solar breaths,24 and not only those of new moons. Of course, we have been constantly confronted with the justifications of numerous Chinese procedures because nothing of the sort exists in the purely procedural Chinese texts. Sometimes, they are self-explanatory, sometimes not. We have thus attempted to provide justifications of our own in many cases. We have not addressed all calendrical calculations, however, for it is a fact that they are so voluminous and so unexplored that we have been compelled to limit the scope of our work in various respects. On the whole, the present study has been restricted to temporal phenomena even though questions implying developments of positional astronomy (inter alia, forecasts of eclipses and conversions of celestial coordinates) would have been required in a limited number of cases.25 Likewise, the present approach is strictly limited to calendrical calculations and aims in no way at the wider domain of astronomical calculations. For instance, our definition of the Superior Epoch (the origin of time) has been deliberately restricted to calendrical calculations and any generalization to the wider field of astronomy has been avoided because it would have implied a significantly different approach.26 Moreover, all calendrical calculations dealt with here have been granted the same importance, regardless of their astronomical or non-astronomical nature for, if the calendar is an object of historical interest, it must certainly be 23 Full reference p. 381 below. solar breaths, see p. 63 below; on their dates in general, see also p. 344 below. 25 Nevertheless, the calculation of the days of occurrence of two lunar eclipses, in a calendar from the year 451, is exceptionally given below, p. 274. 26 See note 79, p. 31 below. 24 On 12 PRELIMINARY OBSERVATIONS regarded as a whole, without granting more importance to certain topics on the grounds that they are more important for us. Nevertheless, it turns out that, despite their limited scope, Chinese calendrical calculations are essentially conceived and organized in the same way as more general astronomical calculations. Consequently, epistemological problems concerning the nature of mathematics and the very important question of the Chinese perception of the relation of mathematics with the physical world and calendrical time also concern calendrical calculations. Hence our taking into consideration of such epistemological concerns, including some of their comparative aspects, so essential for the history of science in general. Apart from the study of the mathematical predictive procedures of the Chinese calendar and their epistemology, however, we have not been involved in an astronomical analysis which is often regarded as the kernel of the subject in previous works and which would have involved, inter alia, the retrospective evaluation of ancient Chinese calendrical and astronomical calculations with respect to modern astronomical knowledge or the determination of the precision of their astronomical constants. Rather, insofar as these calculations and their underlying mathematics are still not fully understood, we believe that the most fundamental task is now to start from the beginning, that is to try to understand and to take at face value available Chinese sources concerning calendrical calculations as they were intended to be used originally: collections of mathematical procedures aiming at establishing the calendar of any year, given in advance. The focus of the present work has thus been limited to an as precise as possible understanding of the structure of the mathematics of Chinese calendrical calculations in order to propose an as faithful as possible operational description of the procedures they rely on. Nevertheless, when necessary, we have also indicated still unsolved problems of interest concerning calendrical calculations with the hope of stimulating future investigations into this area. Computistics and Predictive Astronomy In a Western context, the science of calendrical calculations, computistics, is a highly specialized and somewhat esoteric area of knowledge, dealt with apart in particular sources and never confused with astron- METHODOLOGICAL ORIENTATIONS 13 omy, even though a more or less important amount of astronomical knowledge is required in one way or another. In many cases, religious topics, such as the determination of the date of Easter, the dates of fixed and movable feasts and the martyrolog, a catalog of martyrs (or saints) ranked in calendrical order, constitute the core of the subject.27 On the contrary, Chinese calendrical calculations are not developed in independent sources but are included in the more general domain of predictive astronomy and astrology. Any study of Chinese calendrical calculations is thus unavoidably confronted with Chinese sources having a much wider scope than treatises of computistics. Therefore, they should be studied globally in order to avoid dissociating artificially what is not dissociated in the Chinese context. Still, the present study is limited to calendrical calculations, that is a modest fraction of the whole, but this limitation is only a consequence of the hitherto incredible difficulty and unmanageable volume of original sources given the present advancement of research into such a domain. Remarkably, these sources are far from limited to technical matters. On the contrary, they are also characterized by a fairly well developed epistemological aspect concerning the nature of mathematics, their adequacy or non-adequacy to the description of the physical world, their predictive power and precision, the existence or non-existence of natural regularities, possibly leading or not to the notion of “laws” of nature.28 Well beyond putative Chinese equivalents of the problem of the date of Easter or the arcanes underlying the bizarre notion of ecclesiastical moon, we are thus facing here a variety of issues not limited to the problem of the calendar but also of interest to the much wider circle of historians of mathematics, astronomy and science. This unexpected aspect of Chinese calendrical calculations can be explained in various ways but, given the present state of our knowledge, it is best apprehended from a renewed analysis of a key term, namely the polysemous term li d which is generally believed to mean nothing else than ‘calendar’ even though its scope is in no way so restricted. 27 On 28 See general characteristics of the Western calendar, see J.D. North 1983. below, ‘The Key Idea of Astronomical Canons’, p. 37 f. 14 PRELIMINARY OBSERVATIONS The Non-Technical Meaning of the Term Li d Chinese-English dictionaries and reference works generally explain that, when understood in a non-technical sense, li d only means ‘calendar’ or even ‘the calendar’. Likewise, many influential contemporary sinological works also take this equivalence for granted.29 In fact, li d refers not only to ‘the’ calendar but also to all sorts of calendars, sometimes very different from each other and also, more widely, to artefacts different from calendars and obtained likewise from predetermined calculations, save in the case of the most ancient periods. For the earliest periods (Qin and Han dynasties, essentially), the famous Chinese historian Luo Zhenyu ddd (1866–1940) has introduced the idea of regarding the term lipu dd as sorts of calendars (apparent literal meaning: ‘calendrical tables’) and he has been followed by sinologists, although this expression rather means ‘calendrical calculations’, as the bibliographical chapters of the Hanshu (Han History) indicate.30 Still, the expression liri dd (successions of days) is also documented in original sources and, in all cases, whatever their names, the content of extant calendars from antiquity is limited to hardly more than a lunisolar and sexagenary enumerating kernel or even some fraction of it. Later, the term liri became generic and was used even for less ancient calendars.31 Sometimes too, the term rili dd, (having the same meaning) was substituted with liri in a somewhat confusing way since 29 The fifteen volumes of the authoritative Cambridge History of China all presuppose an equivalence between the li d and the calendar regardless of its general or technical context of utilization (see D.C. Twitchett, J.K. Fairbank et al., 1978–2015) and the same remark apply to a considerable number of important sinological publications, even those devoted to the history of Chinese sciences and techniques. In particular, in the part of his monumental Science and Civilisation in China devoted to the history of Chinese astronomy, the eminent British biochemist and sinologist J. Needham (1900–1995) asserts that the scientific interest of the li d is minor because the study of the calendar and of its theory (or computistics) concern the history of sciences only marginally: “The whole history of calendar-making, [. . . ], is that of successive attempts to reconcile the irreconcilable, and the numberless systems of intercalated months [. . . ], and the like, are thus of minor scientific interest” (J. Needham 1959, p. 390). 30 For a minute analysis of this term and numerous references, see Lin Jin-Chyuan 1998, p. 39–40. 31 For more philological details, see Deng Wenkuan 2010a and A. Arrault 2014, p. 109–110. METHODOLOGICAL ORIENTATIONS 15 these two permuted Chinese characters designate not only the calendar but also administrative daily reports.32 From the Tang dynasty onwards, the term juzhu li ddd (annotated calendars) became widespread and refers to significantly more complex calendars, taking into account not only usual lunisolar phenomena but also all sorts of hemerological and divinatory prescriptions, wholly independent of astronomy but documented, much more anciently, in a noncalendrical context. Still later, the terms minli dd (popular calendar), wangli dd (royal calendar, i.e. a calendar intended for the emperor and imperial princes), dd shangli (calendar intended for the latter) or even tongshu dd (almanacs)33 were commonly used. This short and incomplete list, regrouping various sorts of Chinese calendars from widely different periods of Chinese history, does not exhaust what the term li d possibly refers to because it also designates various kinds of astronomical and astrological ephemeris, having nothing to do with calendars as such, despite the fact that they are likewise organized according to the same successions of days and lunar months. The most important of these are the planetary ephemeris which are sometimes called qizheng chandu li ddddd (li for the degrees of the ‘Seven Governors’ along their paths).34 They concern the sun, the moon and the five classical planets – Jupiter, Mars, Saturn, Venus and Mercury (Mu d, Huo d, Tu d, Jin d, Shui d), respectively. Other sorts of more or less similar ephemeris are also documented, notably those taking into account four fictitious celestial entities of Indian origin, namely those associated with the nodes of the moon’s orbit and other invisible celestial entities.35 In both cases, these various ephemeris are often loosely referred to as li, ‘calendars’ in numerous Chinese contexts. Sometimes, however, they are also more precisely called ‘catalogues’ or ‘lists’ [of astronomical events] (mulu dd). For example, the full title of one of them is 32 Huang Yi-long 1998, p. 431 and note 6, p. 457. term Tongshu literally means ‘general book’; it occurs in chapter 23 of the famous novel Xiyou ji ddd (Journey to the West), first printed around 1570. See also Nan Wang 1992; Huang Yi-long 1996. 34 See Ho Peng Yoke 1986, vol. 1, p. X . The classical term ‘Seven Governors’ first appears in the Shujing, chapter ‘Yao dian’ dd (The Canon of Yao (a mythical emperor)). 35 Huang Yi-long 1993. 33 The 16 PRELIMINARY OBSERVATIONS Da Ming Jiaqing shinian suici xinmao wuxing fujian mulu ddddd ddddddddddd (Catalogue [or list] of periods of visibility or invisibility of the Five Planets for the tenth year of the Jiaqing era (1531), a xinmao36 year)37 (Fig. 1.2, p. 20). A few other examples of such ephemeris can be cited.38 When Jesuit astronomers first noticed them, they rightly called them ‘ephemeris’, not ‘calendars’.39 Other documents, such as the Lingfan li ddd (‘Calendars’of ‘coercitions’40 and ‘encroachments’41 ) also exist. Like the former, they constitute another variety of ephemeris, impossible to mistake for ordinary calendars. Lastly, apart from calendars and ephemeris, the li d mathematical techniques also lead to specific and very important reports concerning the predicted circumstances of eclipses (Fig. 1.1, p. 18), never recorded either in calendars or ephemeris, as far as we know.42 In all cases, and independently of their obvious astrological purposes, however, it should be noted that, contrary to calendars, ephemeris and eclipse reports only provide mere ‘objective’ lists or records of predicted astronomical events, without divinatory interpretations. The Technical Meaning of the Term Li d According to the Huainan zi ddd (The Masters of Huainan) – a famous syncretic treatise from the second century BC –, the term li d des36 xinmao is the twenty-eighth sexagenary binomial. fac-simile reproduction of this rare ephemeris appears in COL-astron, vol. 1, p. 709–715 (no indication of origin is provided but, from the seal of the owner reproduced on its first page, it is (or has been) in possession of the Beijing Library. An inspection of its content shows that it is not limited to what its title indicates: it also contains, inter alia, the positions of the sun, the moon and the four fictitious celestial entities, already alluded to on page 15 above. 38 Wang Xiaohu 2011 (Ming dynasty), p. 106; R.J. Smith 1991, p. 76–77 (Qing dynasty). 39 N. Golvers 1993, p. 73 f.; N. Golvers 2003, p. 467 f.: “Ephemerides Sinicae, sive motus septem/Planetarum anni Christi 1679 [. . . ]”. 40 ling d is an astrological term used when a celestial body moves upwards from below towards another. See Ho Peng Yoke 1966, p. 38–39. 41 fan d means that a celestial body passes the side of another and extends its rays towards the latter. See Ho Peng Yoke, ibid., p. 36–37. 42 A manuscript calendar dated 451 mentions two lunar eclipses (see p. 267 f. below). But not a single other such example has been detected. 37 A METHODOLOGICAL ORIENTATIONS 17 ignates predictive mathematical techniques applied to the determination of the positions of the sun, the moon and the planets.43 The Chinese also have had at their disposal other terms denoting what is intended less vaguely: they also call these techniques lifa d d (the li methods) or even shu d (procedures, techniques or recipes). Of course, the scope of this latter term is extremely wide but at least, it has the merit of highlighting the procedural aspect of the li d calculations. Available Translations of the Term Li d Different from ‘Calendar’ In order to avoid narrowing the scope of the li d to calendrical calculations as though this term was the Chinese equivalent of ‘computus’, a small number of historians of Chinese astronomy, notably N. Sivin,44 have propounded a much more adequate rendering: li d = ‘astronomical system’. In general, the notion of ‘system’ calls to mind fixed plans, or a set of rules, organized in such a way that all its parts work or fit together. With respect to the history of astronomy, it also evokes world systems, such as those of Ptolemy or Copernicus. In the Chinese context, however, nothing of the sort exists because Chinese mathematical astronomy focuses on prediction and never on explanation. Nevertheless, N. Sivin defines ‘astronomical systems’ much more restrictively and differently as follows: “[a li d is] a step-by-step sequence of computations that generates [. . . ] forecasts and assembles them to make a complete ephemeris. That set of procedures I call an “astronomical system”.45 Therefore, thus defined, astronomical systems are analogous to astronomical canons. Another rendering, ‘mathematical astronomy’, is more widespread. However, its scope is in no way limited to mathematical procedures but also encompasses, more generally, anything quantitatively linked with the heavens, in one way or another, from counting the number of visible stars to relativity theory. Therefore, its usage tends to lend weight to 43 See, Ch. Le Blanc and R. Mathieu 2003, VIII, p. 337 (French translation); J.S. Major et al. 2010. For other references on astrology, see also Jiang Xiaoyuan 1991 and 1992. 44 N. Sivin 2009, p. 39. 45 idem. 18 PRELIMINARY OBSERVATIONS Figure 1.1. The predicted circumstances of the partial lunar eclipse on May 4, 1632. The attached Chinese main text provides precise quantitative data (instants of first and last contact, maximum eclipse, instant of the first contact for 12 Chinese provinces). In other cases from the same period, three systems of astronomical predictive calculation were used, namely the Chinese Datong li ddd (Great Unification canon), the Huihui li ddd (Muslim Canon) and a non-identified European set of astronomical tables (perhaps those of Christian Severin’s Astronomica Danica, a disciple of Tycho Brahe also called Longomontanus). From Xu Guangqi ji dddd (Collected Works of Xu Guangqi, Shanghai, 1984, vol. 2, p. 396) (On this work, see the notice on p. 403 below. Note also that the English translations contained in the above figure have been added). Concerning another lunar eclipse (on 15 May, 1631), the Ming History also states that European predictive calculations were much more precise than the two others (Mingshi, j. 31, ‘li 1’, p. 531). Numerous other such examples explain the success of the reform of Chinese astronomy according to European principles. Concerning the various Chinese sources mentioned here, see p. 403 below. METHODOLOGICAL ORIENTATIONS 19 the non-historical idea of the existence of an autonomous astronomical science having the same purpose from the most ancient times to the present. Regardless, we are confronted in both cases with external characterizations of what a li d is. Quite differently but in a complementary way, an internal approach, that is, an approach starting from what ancient Chinese texts have to say on this issue, can also throw some light on what the technical dimension of the li was believed to be in different periods of Chinese history. The Chinese Perception of Foreign Astronomies The Chinese perception of non-Chinese astronomies during different historical periods can also help us to grasp the scope of the term li d and, in this respect, the periods of contact between China and its neighbors are particularly important. In particular, Chinese and Islamic astronomies confronted one another during the Yuan period (1277–1367) and the astronomical exchanges which took place at that time have been sufficiently important to leave a trace, however small, concerning the problem of the meaning of the term li d: a forgotten gloss, buried in the Mishu jianzhi dd dd (Records of Secret Writings), printed ca. 1350, provides an equivalence between the term li d and a non-Chinese term, jichi dd, in reference to the following lapidary elucidation of the nature of a collection of foreign books, preserved at the Northern Observatory:46 “[The library holds] a number of jichi from various schools in 48 volumes, i.e. li [handbooks]” dd ddddddd47 In a purely Chinese context, the term jichi does not exist but, with respect to the history of astronomy, its identification with a phonetic transliteration of the Arabic term zı̄j (astronomical handbook) is immediate, even if the pronunciations of Chinese, Arabic or other nonChinese languages such as Persian, most probably liable to have been 46 i.e., the Chinese Muslim Observatory founded by Qubilai in 1271, in order to supplement the native Chinese Bureau of Astronomy, located at Dadu dd, the Yuan capital (now Beijing). 47 Quoted from K. Tasaka 1957, p. 101, item no. 10. 20 PRELIMINARY OBSERVATIONS Figure 1.2. A Chinese planetary ephemeris for the year 1531 and the first fourteen days of its first month, with: (a) the days and instants of occurrence of its moon phases and spring equinox, (b) the daily positions of the sun and moon with respect to the 28 Chinese mansions, (c) the entry of the sun into the Jupiter stations, (d) the daily positions, in the same mansions, of the five classical planets, with further indications concerning their stations, retrogradations and variations of angular velocity (slackening or hastening). For a reproduction of the original, see COLastron, vol. 1, p. 709 (Original Chinese characters have been replaced by modern types and little squares indicate illegible characters. Moreover, the division of the page into rectangular zones has been made visible (original straight lines are not wholly apparent due to the limited quality of the woodblock printing)). METHODOLOGICAL ORIENTATIONS 21 involved in the process of transmission of foreign astronomical texts in China, during the Yuan period, were certainly slightly different from modern usage.48 Therefore, the Arabic term zı̄j was doubtlessly perceived by the Chinese of the Yuan period as similar to their li d category. Should we then interpret this similarity as a sort of more or less rough approximation, induced by the widely different Chinese and Islamic cultural contexts or, on the contrary, are zı̄j and li d close epistemological categories? To address this issue, we first note that, at least in one important case, the identity is complete because a Chinese translation of an AraboPersian zı̄j was referred to as a li d – the translation in question was entitled ‘Huihui49 li’ ddd (Muslim Astronomical canon). It was prepared in 1383, shortly after the replacement of the Yuan dynasty by the Ming (1368–1644). Although this Chinese zı̄j is apparently no longer extant, it was certainly deemed very important for it was reworked one century later, in 1477. An historian of astronomy, Benno van Dalen, has shown that it is closely related to an extant Arabic zı̄j, the Sanjufı̄nı̄ zı̄j, an Arabic astronomical handbook by a certain al-Sanjufı̄nı̄, written in 1366 for the Mongol viceroy of Tibet and presently kept in the Bibliothèque nationale de France (manuscrit arabe 6040).50 More generally, the formal similarities between li d and zı̄j treatises are striking.51 Still, beyond these remarks, a comparison between the nature of the zı̄j and li d handbooks52 is in order inasmuch as their scopes are a priori widely different because the zı̄j handbooks are apparently based 48 In early modern Persian, the same or similar notions are associated with the same term zı̄j or with close others ones, spelt slightly differently. See Mercier, R. 2004, p. 454. 49 The Huihui are the Chinese Muslims. On this people and the history of the Chinese conceptions about their origin, see Yao Dali 2004. 50 Benno van Dalen 2002a, p. 336–338. See also: same author, 1999, 2000, 2002b; Chen Jiujin 1996; M. Yano 1999; Ma Mingda 1996 (reproduction of all primary Chinese sources about the Huihui li known in 1996), F. Aubin 2005. 51 These similarities are rightly noted in N. Sivin 2009, p. 38, in the following form: “[there is] a basic similarity between the Chinese technical literature, Islamic tractates – most historians of Muslim astronomy translate zı̄j as "table," although they are actually handbooks – and Western treatises from Claudius Ptolemy (ca. 100–ca. 165) to Georg von Peurbach”. 52 For further details about these handbooks and the related Chinese ancient sources, see p. 34 below. 22 PRELIMINARY OBSERVATIONS on Ptolemaic geometrical models whereas, on the contrary, Chinese li d exploit purely numerical models, independent of any predetermined cosmological idea and geometrical patterns. However, as historians of Islamic astronomy have clearly shown, the voluminous zı̄j literature is no more characterized by the nature of its astronomical methods than by a fixed content. Rather, it consists of handbooks open to so significant variations that some of them are independent of the Ptolemaic tradition, just like the Chinese li d handbooks: “The earliest Islamic zı̄djs – [zı̄j] from the 2nd/8th century were based on Indian and Persian models but in the 3rd/9th century the Ptolemaic tradition was introduced and predominated, if not universally. After the 4th/10th century, regional schools of astronomy developed in the Islamic world, with different authorities and different interests and specialities [. . . ]. The first Islamic Zı̄djs – [zı̄j were part of an Indo-Persian tradition which has a pre-Ptolemaic Greek origin.”53 What is more, the circulation of knowledge between these various sorts of astronomical handbooks was a reality, even when they belonged to different epistemological categories. For instance, the Chinese li d and the non-Chinese zı̄j handbooks were not necessarily independent of each other in the sense that, sometimes, purely Chinese notions were incorporated in the latter.54 Quite strikingly, the famous zı̄j of the great Uzbek astronomer Ulugh-Beg (1394–1449), contains data concerning the Chinese calendar such as the cycle of the twelve animals, the sexagenary cycle and the division of the Chinese solar year into 24 solar periods (jieqi dd).55 Similarly, the no less famous Zı̄j-i-Īlkhānī compiled by Nas.ı̄r al-Din T.ūsı̄ (1201–1274) contains more complex elements of Chinese origin56 such as a Chinese value of the anomalistic month and a table for the solar equation built from a parabolic interpolation scheme.57 53 D.A. King and J. Samsó 2007. reverse problem of the influence of non-Chinese astronomies on Chinese astronomy is open. 55 See A.A. Akhmedov 1994, p. 33–35. 56 Until recently, this famous source was wrongly associated with the Chinese-Uighur ‘calendar’. See Y. Isahaya 2009. 57 This table comes from the Futian li ddd, (Tallying with Heaven canon), compiled in China between 780 and 783. See Y. Isahaya 2009, p. 32–33. 54 The METHODOLOGICAL ORIENTATIONS 23 Similarly, the more ancient astronomical canons of the ancient Greek and Latin Medieval worlds were not necessarily independent of one another and were sometimes called zich, ezich and καν ών , respectively.58 As E.S. Kennedy explains: “The Greek word καν ών , in meaning very close to zı̄j, has likewise been Arabicized, as qānūn, and the two words are sometimes used interchangeably [. . . ], from Arabic or Persian, the word zı̄j entered Byzantine Greek as ζ η̃ζ ι .”59 Therefore, notwithstanding deep cultural and historical variations, astronomical canons and Chinese li treatises refer rather well to a similar corpus of technical texts, well beyond the particular case of the Yuan dynasty. Consequently, the li d handbooks will be regarded as the Chinese equivalents of ‘astronomical canons’ or, more simply, ‘canons’.60 Nevertheless, in the same way as terms such as zı̄j are freely used in works about Arabic astronomy, the term li d will sometimes also be left untranslated. Moreover, despite this very general characterization, the most ancient li d techniques were apparently limited to the luni-solar component of the calendar, as the case of the so-called six ancient li dd d from the Qin and Former Han periods61 tends to suggest. Therefore, in their case, the equivalence li d = computus seems more appropriate. In a completely opposite direction, the Chinese situation began to change significantly with the arrival of Jesuit missionaries in China from the end of the sixteenth century: the monumental Jesuit encyclopedia which was at the basis of the reform of Chinese astronomy undertaken from 1628, entitled Chongzhen lishu dddd (Chongzhen reign-period (1628–1644) Treatise on the Li d), is neither a treatise of computistics nor an astronomical handbook but, on the contrary, a monumental 58 See E.S. Kennedy 1956, p. 3. the Greek letters zêta, êta and zêta-iota respectively correspond to the pronunciation of the Arabic z, ı̄ and j. Therefore a Greek phonetical transliteration of the word zı̄j made its way back into the Greek world despite the existence of the much more ancient Greek term καν ών endowed with the same meaning. See E.S. Kennedy 1956, ibid., p. 3. 60 when referring to the technical part of the Shoushi li ddd, N. Sivin also uses the term ‘canon’. See N. Sivin 2009, p. 389. 61 See p. 381 below. 59 Here 24 PRELIMINARY OBSERVATIONS compilation of Chinese adaptations of outstanding Western astronomical works such as those of C. Ptolemy, Tycho Brahe, J. Kepler and even N. Copernicus.62 In other words, the totality of Western astronomy was then identified with the li d realm despite obvious differences between both, if only because former Chinese li d treatises are merely procedural, non-geometrical and do not involve in the least questions of cosmology, which were and remained remarkably and lastingly absent from the Chinese context of calendrical and astronomical calculations. In such a case, li d was thus taken as equivalent to ‘astronomy’ in general and likewise, C. Ptolemy’s famous Almagest (or Mathematical Syntaxis), abundantly quoted in this encyclopedia, was regarded as a sort of li d and referred to as the Lixue dazhi dddd (Comprehensive Survey of Knowledge Concerning the li).63 Likewise, all other Western works of astronomy, even N. Copernicus’s De Revolutionibus or later works, such as those of Tycho Brahe and J. Kepler, were similarly regarded as examples of treatises about the li.64 The Chinese Calendar, A Paradoxical Object In spite of numerous differences between the most ancient varieties of Chinese calendars and those from later periods, the Chinese luni-solar calendar – viewed as a unique abstract artefact modulo some variations of its hemerological component determined by its various groups of users – is as much a powerful repository of archaic traditions as other calendars. It thus gives the impression of traveling down the ages without being much damaged by the ravages of time. While not easily admitting the rejection of ancient components, it does not confirm novelties without long delays. When, against all expectations, certain of these were eventually retained, their permanency was firmly ensured. For instance, the continuous sexagenary numbering of day cycles, initiated very approximately towards the end of the second millennium BC, is still prominent in contemporary Chinese calendars. Similarly, as the Chinese historian of the calendar Deng Wenkuan has noted, numer62 J.-.C. Martzloff 1998a, notably, and many other previous studies. See Lifa xizhuan dddd (A History of the Western li techniques – Western astronomy in fact –), vol. 2, p. 1991 of Pan Nai’s edition of the Chongzhen lishu (see p. 403 below). 64 See K. Hashimoto 1988; Pan Nai 1993 and 1994; J.-C. Martzloff 1998a, ibid. 63 This wording is of course an equivalent of Almagest. THE PARADOX OF THE CHINESE CALENDAR 25 ous hemerological and divinatory components of the Chinese calendar, first attested in Chinese calendar manuscripts from Dunhuang (IXth and Xth centuries), are still present in popular calendars distributed in Hong Kong.65 Obviously, such examples could be multiplied.66 At the same time, Chinese astronomical canons have been reformed at a rapid pace and calendars have been modified accordingly: from 104 BC to AD 1644, the Chinese imperial authorities elaborated no less than ninety such projects.67 Still, although the number of those officially adopted is significantly lower, it already amounts to approximately fifty:68 in a little more than seventeen centuries, astronomical canons have thus been reformed once every thirty-five years on average, that is incredibly often, even though this number cannot be easily reduced to a misleading arithmetical mean: the most successful reforms have lasted up to two or three centuries while the shortest ones have fallen into obsolescence after a few years only. Such a frenzy of change, such a sustained commitment to calendar reform exists nowhere else than in China. Nothing of the sort has happened in any other civilization, not even remotely. China constitutes a unique case in this respect. Seen from the perspective of its reforms, far from being a depository of immutable traditions, the Chinese calendar appears on the contrary as the paragon of change, the opposite of its manifest conservative nature. Insofar as it appears mobile and immobile at the same time, the calendar has thus the appearance of a truly paradoxical object, similar in its own way to the impossible, but seemingly plausible pictorial constructions, vividly representing simultaneously ascending and descending staircases, imagined by the talented Dutch artist, M.C. Escher. Now, if we stop viewing the calendar from the two different viewpoints of its factual content and reforms, without attempting to establish any connection between both, it becomes possible to overcome this 65 Deng Wenkuan 2010a, p. 60–78, and same author, 2006. confirmed by the British sinologist M. Loewe in the case of an earlier period, “Some of the esoteric signs or expressions of the Ch’in and Han almanacs persist on copies of the calendars drawn up by officials of imperial government; they may be seen today in the calendars that adorn the walls of a bank in Taiwan or Hong Kong, or in some of the manuals for guidance printed in Japan” (M. Loewe 1994, p. 18–19). 67 See COLL. 1980, p. 559–561. 68 See Appendix D, p. 350. 66 As 26 PRELIMINARY OBSERVATIONS paradox. In fact, we are faced with two related aspects of a complex reality: on the one hand, the calendar is a familiar object pertaining to daily life and, on the other hand, a set of more or less complex mathematical rules used to construct specific calendars and modified more or less deeply at the occasion of a reform. Therefore, we can regard the calendar relating to a determined set E of years as a bipartite structure (A, B) where A represents the calendar from the perspective of its calculation techniques and B the manifest structure of the corresponding annual calendars. Insofar as the Chinese calendar is the result of well-defined calculations, when a set A of mathematical techniques is given, B is uniquely determined. The reverse is not true however: the knowledge of the specific calendar of a given year does not give access to its underlying mathematical techniques if they are not previously known. In other words, the relation between A and B is not reversible. In order to really understand the mutability or immutability of the calendar over time, A and B must be distinguished from one another and studied apart in order to evaluate the consequences of the changes that affect A over B, if need be, given that, in its turn, B can always be decomposed into two subsets, the first immutable and the second variable. The fixed part of B is easy to describe and is a consequence of its enumerative structure and of its invariant lunisolar framework: the successive days of the calendar have always been enumerated, cyclically and without any discontinuity, from one to sixty by means of the sexagenary cycle, not only from 104 BC to AD 1644, but also from a much earlier indeterminate origin. Moreover, always during the same time-interval and even beyond, the lunar months of the Chinese calendar have always been composed of 29 or 30 days, any other possibility being definitely excluded. Likewise, the number of solar periods of the Chinese calendar has never been different from 24 and this enumeration could be easily extended to its other components. In its turn, the variable part of B is less immediately obvious but it clearly manifests itself through its types of successions of full (30 days) or hollow (29 days) lunar months. As will be explained in detail below, certain years are merely composed of alternated sequences of such months while others follows various other THE CALENDAR AND ITS CALCULATIONS 27 patterns. Some very particular years contain consecutive sequences of 3 or even 4 lunar months of 30 days each and, more generally, the attested types of sequences of lunar months of different lengths are surprisingly varied. But these variations are in no way the consequence of independent modifications affecting B but only the result of calculations determined by A. Hence the interest of a separate study of both structures. Lastly, another important but less immediately obvious aspect of the question hinges on the fact that the variability of B can be attributed to the degree of conformity of its lunisolar dates with those of the corresponding astronomical phenomena, such as solstices, equinoxes or lunisolar conjunctions. Hence a supplementary degree of complexity determined by changes affecting astronomical constants used in calendrical calculations. This question will not be tackled here, however, because our purpose is limited to an analysis of the structure of calendrical calculations and aims in no way at solving the wider question of an evaluation of their astronomical relevance or precision. The Calendar and its Calculations As a visible symbol of the imperial origin of daily schedules for millions of people, the tangible calendar practically reaches all the members of the Chinese society it is intended for, from the most eminent personalities to the most anonymous commoners. The calendar determined by this political origin is nevertheless not an object whose content is uniquely determined once and for all because its hemerological content was adapted to various target audiences.69 However, since we limit ourselves to the overall structure of the official calendar, the lunisolar and enumerative skeletons of its various versions are certainly identical. 69 See R.J. Smith 1991, p. 76–77 “[...] we find considerable variation in size and type, even for calendars bearing the same date. In part, these variations can be explained by different target audiences [...] Certain variations in calendars, then, reflected distinctions in the ethnicity, status, administrative responsibilities and personal concerns of the respective recipients within the Qing social and political hierarchy”. Although these remarks apply to the Qing period (1644–1911), they obviously also concern earlier periods since the Chinese calendar contains a very large number of prescriptions concerning daily activities which can certainly not have been identical in different social groups. In particular, this point is wholly confirmed in the case of the Ming dynasty (see Huang Yi-long 1998). 28 PRELIMINARY OBSERVATIONS As a set of mathematical techniques, Chinese calendrical calculations form a covert body of knowledge, kept secret as long as the astronomical canon used for its calculations remained in force. Consequently, the manufacturing process of the State calendar can be compared to the techniques of production of banknotes: both domains are covered by a State monopoly and are veiled in secrecy.70 The Difficulty of Access to Astronomical Knowledge For this very reason, when the Prince Zhu Zaiyu ddd (1536–1611) decided to reform the Chinese calendar on its own initiative, at the end of the sixteenth century, for instance, he had to face innumerable difficulties because of his lack of access to the techniques of calculation of the calendar then in force. Under the Ming dynasty, not a single treatise explaining this intricate subject had been overtly released and specialists who could have helped him in this respect were extremely rare. After having sought advice from former experts, gleaned tidbits of information in administrative manuals and carried out systematic crosschecks between relevant data and outdated but apparently similar techniques, freely available in Chinese dynastic histories, he finally managed to get a rough idea of the intricacies of the subject.71 70 In this respect, and in the particular case of the Tang dynasty (but this observation can be generalized) Lai Swee Fo 2003, p. 342, aptly writes that “The Imperial Observatory was probably the most ‘secret’ organization in Tang government. Officials and minor clerks working there were barred from communicating with other court officials and civilians, to prevent the leak of sensitive information”. For the same period, see also the pertinent remarks of E.H. Schafer 1977, p. 12–13. For the beginning of the Ming dynasty, Chen Meidong 2003a, p. 555, mentions a decree from the Daming huidian dd dd (j. 223) stating that “the staff members of the Bureau of Astronomy should remain under house arrest during their lifetime; the duty of their children and grandchildren is the study of astrology (tianxue dd, literally ‘the science of the heavens’) and the mathematics of astronomical canons to the exclusion of any other domain. Those who will be judged to be incompetent in this respect will be sent to our southern maritime borders in order to supplement our troops.” ddddddddddddddddd ddddddddddddddddddddddd. On the Bureau of Astronomy, see p. 52 below. 71 These details are given in Zhu Zaiyu’s lengthy preface to his Shengshou wannian li ddddd (WYG, vol. 786, p. 451–459), an essential piece of information with respect to the question of the reform of astronomical calculations towards 1600 which would certainly deserve a full English translation. THE SURFACE AND DEEP STRUCTURES 29 Obviously, this situation is radically different from the prevailing state of affairs in Europe from the XVIIth to the XIXth centuries, at least, where concepts and technical notions pertaining to computistics were not deemed secret. In particular, specialized concepts referring to the ecclesiastical calendar, such as the dominical letter,72 the golden number73 or the epact,74 were often made available and explained in the body of the civil calendar itself, in such a way that, as the content of many extant almanacs shows, their users could deduce various elements of interest for them if they wished, not only in the case of the current calendar, but also for those of past or future years.75 The Surface and Deep Calendrical Structures In order to further clarify the opposition between the tangible Chinese official calendar and its abstract calculation techniques, we now assert that everything concerning public calendars depends on a ‘surface structure’ while, on the contrary, everything connected with secret and hidden calendrical calculations relies on a ‘deep structure’. Borrowed from linguistics, this opposition offers the advantage of dissociating the two complementary aspects, public versus secret, of the calendar and to avoid confusing what should be all the more distinguished, that the two sides of this same coin are widely different for various other reasons: their historical sources are not the same in both cases and the concepts of time revealed by the analysis of their two structures are strikingly different. 72 The dominical letter of a particular year is one of the seven letters A, B, C, . . . . attributed to its first Sunday in January. It moves one letter backwards from one year to the next. 73 The number of a year inside the Metonic cycle of 19 years is called its ‘golden number’. 74 The epact of a given year is an integer denoting the ecclesiastical age of the moon on a certain date. On the various technical notions defined by computists, see, inter alia, U. Bouchet 1868; G.V. Coyne, S.J., M.A. Hoskin and O. Pedersen, 1983; E.G. Richards 1998; L.E. Doggett, 1992. 75 M. Lænsberg’s Almanach de Liège or the famous Messager Boiteux (see M. Vernus 2003, p. 29) are examples of such calendars, among many others. 30 PRELIMINARY OBSERVATIONS Two Notions of Time Fundamentally, the calendar can be described as a sequence of astronomical and non-astronomical events, listed in a determined temporal order. However, the resulting sequences differ, depending on whether its surface or deep structure is at stake. In surface calendars, the fundamental unit of time is of course the day. The resulting time is thus discrete, even though it sometimes happens that the calendar localizes certain events more precisely than according to their sole day of occurrence, owing to various subdivisions of the day into, for example, 12 double-hours and other finer units of time.76 Consequently, these days are strung like pearls, one after the other, but they are also regrouped by months and years, in much the same way arithmetical quantities are counted by tens, hundreds and thousands in the decimal system, for example, except that the resulting structure is of course significantly less regular, the number of days contained in a lunar month and a lunar year being variable.77 This discrete time is quite particular, however, because the temporal horizon unveiled by its various cyclical enumerations imposes various sort of limits, from a few days (in the case of the ten-day week) to the sixty days of the sexagenary cycle or longer sequences. Moreover, similar patterns are also frequently transferred to months and years which are thus enumerated as though they were days. For instance, months are enumerated cyclically by groups of sixty covering five years each time, and years form in their turn various supra-annual clusters. The notion of time that rules over the destiny of the surface calendar is thus also a highly cyclical, fragmented and local sort of time. In spite of these 76 On the horary subdivisions of the day and the corresponding instruments for measuring time such as the clepsydra and the gnomon, H. Maspero 1939 still remains quite useful, even though it was published a long time ago. Of course, other more recent articles are also of interest inasmuch as they develop more precisely the mathematical aspect of the question. See Chen Jiujin 1983; Wang Lixing 1986, (lengthy article giving precious references to ancient Chinese sources), Qu Anjing, Ji Zhigang and Wang Rongbin 1994, p. 236–247. 77 The abstract study of calendars in general has given rise to interesting mathematical developments, based on the notion of discrete lines and generalized Beatty sequences (see A. Troesch 1998). I would like to thank J. Lefort for handing down to me a copy of this article. THE SURFACE AND DEEP STRUCTURES 31 various temporal limitations, however, more extended periods of time have been taken into account, by means of the system of dynastic eras and calendrical dates, equivalent in one way or another to triplets of numbers indicating the rank of a lunar year, the rank of a lunar month and a day number. The way these dates are really expressed is however generally significantly more involved than what this schematic presentation implies if only because Chinese lunar months are often referred to in very different manners by using a bewildering number of literary variants.78 By contrast, the time of the calendrical deep structure is much more regular because, in its case, time is reckoned from a fixed origin, the epoch. The position of all subsequent events79 is thus uniquely determined by a unique temporal parameter t, possibly only positive or also admitting negative values, as though instants of time were represented geometrically by points, ordered on a line extending indefinitely, either towards the future only or in both directions. All events liable to be determined by a given set of calendrical procedures thus become fundamentally comparable from the values of their respective time parameters. Together with its remarkable regularity, the time of the calendrical deep structure also possesses another characteristic induced by an unceasing Chinese quest for an always increased precision: time determinations are required to be as precise as possible, even though the degree of precision so obtained is often purely imaginary as being often the consequence of fictitious calculations, not based on controlled measurements. In practice, the units of time peculiar to the calendrical deep structure rely on very small units determined by reiterated artificial subdivisions of the day, so tiny that no measuring instrument then available in China would have had a sufficient power of resolution in order to appreciate the situation in any realistic way. 78 Havret and Chambeau 1920, p. 17, provides an impressive list of such variants: for instance, the first month of the lunar year possesses no fewer than twenty-one synonyms. 79 In the more complex case of astronomical calculations, it sometimes happens that a same canon uses multiple epochs, for example one for the planets and another one for other phenomena. See SIXIANG, p. 350–359 . 32 PRELIMINARY OBSERVATIONS In the Dayan li (Great Expansion canon), for instance, the length 645 1322 of the draconitic month80 is equal to (27 + 3040 + 10000×3040 ) days and thus rely on a wholly fictitious division of the day into no less than 30, 400, 000 parts. With such tiny units of time, the assignment possibilities of mathematically calculated calendrical events are so numerous that as soon as two time-values t(e1 ) and t(e2 ) are associated with any two events e1 and e2 in the deep structure of the calendar, it would seem possible to determine a third event e3 , such that t(e1 ) < t(e3 ) < t(e2 ), no matter how close to each other e1 and e2 are, as though the time variable were continuous but, of course, this impression is illusory because the calendrical deep structure relates all events to some smallest unit of time so that, in fact, an infinity of putative events are indistinguishable from one another. To sum up, the time of the calendrical deep structure is linear and mathematical. Socially, it can also be regarded as a scholarly time because the conception of astronomical canons essential for calendrical and astronomical calculations supposes a creative mastery of often complex mathematical techniques.81 The Double History of the Chinese Calendar If the two surface and deep structures of the calendar stand out due to their different conceptions of time and their peculiar modes of localization of calendrical events, they are also deeply different from each other from an historical perspective. Overall, the rate of change typical of the surface calendar is extremely slow: the appropriate unit of time revealing its changes oscillate between quite a number of centuries and one millennium or more. When was the continuous numbering of days by means of the sexagenary cycle deliberately and constantly used in Chinese calendars without disruption? When was the planetary week first incorporated into Chinese calendars? In the first case our ignorance is enormous, in the second, we 80 A type of lunar month used in eclipse calculations. studies of Chinese notions of time practically always concentrate their analysis on their discrete aspects and never on the scholarly time typical of calendrical and more general astronomical calculations. The study of time measurement and of instruments of measure of time has nevertheless often been attempted (see Huang Chunchieh and E. Zürcher 1995). 81 Available THE SURFACE AND DEEP STRUCTURES 33 have to resign ourselves to a precision of the order of a century. Since we are unable to answer precisely, we are bound to admit that a long-term history is appropriate in its case. By contrast, despite some limited chronological uncertainties, the extremely numerous reforms of Chinese astronomy are reasonably well documented and dated.82 Consequently, the history of the Chinese calendrical deep structure requires a short-term history. Sources for the History of the Calendar (Surface Structure) The history of the Chinese surface calendar can be tackled from various calendar manuscripts and printed sources as well as all sorts of ancillary sources, concerning, notably, hemerology, administration, astrology, divination, religion and mythology. Of course, all documents containing calendrical dates are also extremely precious, all the more so that they were generally designed for all sorts of purposes having nothing to do with the calendar as such. More widely, various non-Chinese sources are also of great interest because numerous Chinese calendars contain, overtly or covertly, various elements foreign to Chinese culture such as, for instance, the planetary week.83 Among these various sources, Chinese authentic calendars have of course a vital interest but only an exceedingly small number of them have been handed down to us, even though millions of copies of the Chinese official calendar have been widely distributed all over the Chinese Empire each year84 after the invention of xylographic printing on paper from the middle of the Tang dynasty (618–907). In fact, all sorts of reasons explain the phenomenon: the Chinese calendar was an everyday object and the thin Chinese paper not very robust. The preservation of the calendar beyond its expiry date had no interest for its users and moreover, paper was a rare and expensive commodity, frequently reused for various other purposes. Moreover, the disorders inherent in dynastic 82 See Appendix D, p. 350. p. 90 below. See also (but for a later period) R.J. Smith 1992, p. 33 f.: ‘The Introduction of New Elements’. 84 See Huang Yi-long 1998, (p. 432 and note 12, p. 458). The number of copies distributed in the Song Empire exceeded three million in 1328. Concerning the later Qing (1644–1911), R.J. Smith 1991, (p. 75) mentions a similar number of copies: “In all, about 2,340,000 [versions of the Qing calendar] were officially printed each year”. 83 See 34 PRELIMINARY OBSERVATIONS changes and numerous other periods of turbulence have also certainly not played a less important role in China than elsewhere in this respect, inasmuch as the Chinese official calendar is not a neutral object but, on the contrary, one of the fundamental symbols of the control over social time by imperial authorities. A search across multiple catalogues concerning sinological libraries all over the word and an attention to articles devoted to specific calendars shows that the number of official or non-official extant Chinese calendars, relating to the years of the interval 104 BC–AD 1644,85 is of the order of several hundred as long as incomplete and even fragmentary calendars, sometimes limited to a scratch of paper, are taken into account. As could be surmised, however, the most numerous extant authentic calendars are by far those from the Ming dynasty.86 Sources for the History of the Calendar (Deep Structure) Very few original treatises offering a direct access to the history of the traditional Chinese calendrical deep structure have been handed down to us.87 Nevertheless, the situation is not as bad as one might suspect because slightly later treatises, devoted to the historical presentation of former ones, are extant. The treatises in question are the astronomical canons included in Chinese official histories.88 They were compiled after the fall of a dynasty by teams of highly specialized historians and, quite remarkably, they provide both the detail of their mathematical techniques and 85 For an overview of early calendars from the Han dynasty and a little more ancient periods (up to the third century BC), see A. Arrault, 2002 and Deng Wenkuan, 2006, p. 3. In particular, Deng Wenkuan states that more than sixty calendars from the Han dynasty have reached us. 86 See p. 389 below. Apart from these calendars, the most extensive collection of ancient Chinese calendars is the one discovered at Dunhuang at the beginning of the twentieth century. Most are fragmentary manuscripts but some are printed copies. See A. Arrault and J.-C. Martzloff 2003, p. 203 and 204 ; A. Arrault 2014. 87 These treatises are late and concern the Ming dynasty. See p. 401 below. 88 The other technical treatises included in these histories deal with rituals, extraordinary phenomena, civil and military administration, geography, fiscal economy and laws. For a general presentation, see, for instance, E. Balacz 1961; M. Beck 1990 and D. Twitchett 1992. NUMBERS 35 sizeable developments about their history, together with epistemological considerations and critical evaluations of former astronomical canons. From one treatise to another, the space devoted to a given astronomical canons is variable and not all dynastic histories have one: astronomical canons are included in sixteen Chinese dynastic histories out of twenty-four.89 Overall, their spirit and mode of composition is generally more or less modeled after the most ancient of them, namely the astronomical canon of the Hanshu (History of the Former Han),90 even though noteworthy variations occur. As a rule, their goal is to advise their learned readers of the weak and strong points of past predictive astronomical techniques and to show why certain ancient astronomical canons were more successful than others. Quite interestingly too, the technical aspect of calculations is generally developed at length and treated in a quite general and procedural way, without specific examples of calculations nor rational justifications and almost never definitions of technical terms. Their study can thus apparently easily lead to multiple interpretations. Still, when taking into account the Chinese epistemological and historical context as well as the invariant structures concealed under the multifarious formulations of what often refers, in reality, to a same and unique procedure, repeated again and again in different canons in various ways, many possibilities of variant interpretations vanish. In a different order of ideas, episodic developments scattered in some astronomical canons also allow us to understand that numbers have not only an arithmetical function but also a vivid symbolic function. Numbers Practically all facets of symbolic aspects of numbers in Antique China have been masterfully analyzed in M. Granet’s celebrated masterpiece, La pensée chinoise (Chinese Thought).91 Although this famous French sinologist exclusively relied on sources significantly more ancient than dynastic histories, the Chinese have often continued to regard numbers in the same way, even between 104 BC and AD 1644. They have thus lastingly approached numerical phenomena both under an arithmetical 89 See p. 385 f. below. j. 21A, ‘lüli zhi’. 91 M. Granet 1934/1968*. 90 Hanshu, 36 PRELIMINARY OBSERVATIONS angle and in numerological terms, numbers being associated at the same time with calculation techniques and with arbitrary correlations between all sort of unrelated domains, notably units of measurement, calendrical elements and numbers. In the sections of astronomical canons devoted to numerological considerations, numbers, shu d,92 are thus presented as operating devices from a double point of view. First, they are the cornerstone of logistics, suanshu dd (literally ‘calculation procedures’), a technique of manipulation of numbers from arithmetical operations, in order to provide quantitative answers to all sorts of problems (commercial transactions, partnership, areas and volumes, measurement of the distances of inaccessible objects and the like). Moreover, and most importantly, such numbers were above all quantities and they were thus essentially used in order to assign a quantitative value to objects having a length du d, a capacity liang d, a weight heng93 d or a duration. Second, they were regarded as essential ingredients of multifarious numerological correlations. Hence, in the case of the calendar and astronomy, arbitrary connections between temporal and non-temporal units of measurement, independently of any instrument, even though some were available at least from the Han period.94 For instance, the link between units of weight and time was ensured through the equivalence between the two units of weight called liang and zhu (1 liang d = 24 zhu d) and the solar year on the grounds that the latter is likewise divided into 24 solar periods. In the same way, since one jun d contains 30 jin d (the jun and the jin are also units of weight) the correlation with one of the two possible numbers of days of the Chinese lunar month was also similarly ensured.95 Less directly, but always in the same spirit, the capacities of pitch-pipes were evaluated, in their turn, from the number of grains of millet they contained and the number in question was used the Warring State period shu d also meant ‘the calculation of calendars’ rather than ‘arithmetic’. See C.A. Cook 2011, p. 305. 93 Hanshu, ibid., p. 966 f. 94 See H. Maspero 1939. 95 See H.U. Vogel 1994, p. 139. 92 During THE KEY IDEAS OF ASTRONOMICAL CANONS 37 in order to establish correlations with the sound they produced and the lengths of calendrical time-intervals.96 Such examples could be multiplied but a careful examination of Chinese astronomical canons also shows that from 104 BC to AD 644, the Chinese vision of numbers is more complex that what is explicitly stated and overtly stated in Chinese sources. For instance, whereas the Hanshu explicitly asserts the fundamental character of the decimal numeration system,97 numerous non-decimal modes of representation of numbers, based in various ways on irregular sets of non-decimal fractions, are introduced without warning everywhere and in relation with numerological concerns. Nevertheless, the numerological aspect of numbers, which is still extremely prominent during the Tang dynasty, especially in the case of the celebrated Dayan li,98 has tended to decrease over time but not in a linear way: from the Song, Chinese astronomical canons do not develop the subject any more, even though they are still entitled lüli zhi dd d (Treatises on Pitch-Pipes and Astronomical Canons), thus following the ancient naming pattern first established in the Hanshu.99 At the end of the sixteenth century, however, an important project of reform of astronomical canons tried to reinvigorate this antique tradition.100 But the new reform was rejected and no further attempt to establish a link between numerology and calendrical calculations was ever attempted again. The Key Ideas of Astronomical Canons In spite of the almost inextricable maze of historical events always presented with a wealth of details, the historical parts of Chinese astronomical canons follow a remarkably invariant direction: their key issue is always the reform of astronomical canons and the small number of 96 H.U. Vogel, 1994, ibid., p. 137. j. 21A, ‘lüli zhi 1’, p. 656. 98 See p. 111 below. 99 Chapters 68 to 84 of the Songshi are all entitled lüli, probably by analogy with the titles of the astronomical canons of preceding dynasties, but this usage is somewhat fossilized for it does not reflect the actual content of the corresponding texts. 100 This project of reform is still extant. See Zhu Zaiyu’s’s magnus opus Lüli rongtong dddd (A Comprehensive Study of Pitch-Pipes lü d and Astronomical Canons li d). See WYG, vol. 786, p. 556 f. 97 Hanshu, 38 PRELIMINARY OBSERVATIONS guiding principles put forward in this respect are always the same. It is therefore possible to get a global view of the conceptions determining the global trend of ideas behind these reforms independently of their countless factual peculiarities. The fundamental idea, unceasingly rehearsed in ancient and less ancient sources,101 is that predictive calculations must agree with heavenly appearances. As the Hanshu already states: “The verification of ancient astronomical canons is to be sought in [their conformity with] the heavens.” “It is necessary to conform [oneself] with heavenly [appearances] in order to obtain agreements [between predictive calculations and astronomical observations] and not the other way round.” ddddddddd102 dddddd dddddddddd103 The first lapidary sentence means that predictive techniques must depend on empirical observations and adjusted accordingly. The second one reiterates the same idea but renders its injunction more explicit: the subordination of mathematical predictive techniques to empirical observations is much more essential than the reliance on ingenious mathematical principles owing nothing to preliminary empirical verifications and a priori deemed superior to others for whatever reason. To put it another way, no method, no presupposition, no mathematical idea can dictate its terms to the heavens, define how it should act, attribute it compelling prerequisites, celestial appearances being then described in terms of preestablished principles. Consequently, the Chinese have focused their study of celestial appearances on direct or indirect observations by taking avail of astronomical and time-keeping instruments, independently of intangible axioms. From the enormous weight granted to astronomical observations and the correlative rejection of rigid principles, it would seem that we are faced here with the Chinese equivalent of an astronomy without hypotheses, an astronomy which first appeared in Europe during the second half of the sixteenth century but remained subsequently marginal: 101 For more details, see SIXIANG, p. 330–358. j. 21A, ‘lüli zhi 1’, p. 978. 103 Jinshu, j. 18, ‘lüli 3’, p. 564. 102 Hanshu, THE KEY IDEAS OF ASTRONOMICAL CANONS 39 first advocated by Pierre de la Ramée (1515–1572) (Ramus) in a rather ambiguous form,104 it was taken to the letter more than one century later, by the French mathematician and astronomer Philippe de La Hire (1640–1718) (Fig. 1.3, p. 40). However, while maintaining, in their written declarations, the prominence of observations with respect to theories intended to organize an inherently meaningless observational chaos, the Chinese have never completely taken advantage of empirical measurements independently of any underlying principles. For example, in order to determine the length of the solar year, defined as the mean time between two consecutive winter solstices, they carried out series of measurements of meridian lengths of gnomon shadows before and after the day preceding and following its empirical date of occurrence and by assuming that these variations are symmetrical against one another, as a consequence of their belief in a kind of mirror correlation between meridian shadows determined by the symmetrical variations of the yin and yang factors supposedly attached to the cycle of the seasons.105 By the same token, they also used the weighting of earth and charcoal against one another in order to determine the instant of the winter solstice as a result of occult yin-yang influences behind the scenes. To this end, earth and charcoal were suspended from the two ends of a scale and when the charcoal became heavier and moved downwards, the phenomenon was taken as an irrefutable proof of the occurrence of the winter solstice. This time, the rationale at the origin of this experimentation consists in a belief in the occult influence of the yin factor which is supposed to constantly increase at the expense of its yang counterpart, until the point of rupture marked by the winter solstice.106 104 See N. Jardine and A. Segonds 2001. slightly erroneous for no such a symmetry exists. See S. Nakayama 1969, p. 247–256; Chen Meidong, 1995, p. 50–64; R. Mercier 2003. 106 The American sinologist D. Bodde 1975, (p. 175) explains this method in the following way: “we may speculate that the probable reason why the charcoal should allegedly become heavier at the arrival of the yang is that charcoal burns and therefore pertains to fire which is yang. By the same token, earth would be said to become heavier with the arrival of the yin because the element earth [...] pertains to the yin”. 105 This method is 40 PRELIMINARY OBSERVATIONS Figure 1.3. In the preface to his astronomical tables, first published in 1702, Philippe de La Hire asserts that Kepler’s Rudolphine tables were significantly not in accordance with celestial appearances and he attributes this defect to the inaccuracy of the famous laws discovered by the celebrated astronomer. Hence his proposal of elaborating tables from mere observations, without taking into account either circles or ellipses, or any other system. Partial translation: “I knew that the Rudolphine Tables [i.e. Kepler’s tables] were significantly at variance with celestial appearances [. . . ] and that [this defect] was due to Kepler’s hypothesis. Therefore, I have undertaken to draw up my tables on the sole basis of observations without relying on any system [. . . ]”. From the preface to the third edition of his Tables astronomiques, Paris, 1735, p. vii (copy preserved at the Bibliothèque nationale de France, V-8417). Notice p. 406 below. From a scientific standpoint, while the first method makes sense despite its approximation, this second does not, of course. Yet, they do not substantially differ that much from one another since they both depend on the belief in the existence of two fixed yin and yang principles, increasing and decreasing in a symmetrical way, not warranted by any THE KEY IDEAS OF ASTRONOMICAL CANONS 41 observational basis. The same remark also applies in numerous other cases such as, for example, the variant of the preceding method based on the insertion of ashes in a pipe, the instant of the winter solstice being determined by their dispersion, always as a consequence of an interplay between yin and yang elements.107 Likewise, and most importantly, the fact that the winter solstice was respectively attached to the maximum and to the minimum of yin and yang factors was also reflected in the structure of all Chinese tables of the solar inequality, which are always designed in such a way that the maximum solar velocity occurs at the winter solstice and not at the passage of the sun to the perigee, as one of Kepler’s laws would imply. More generally too, practically all other Chinese astronomical tables display similar features, induced by yin-yang presuppositions. Hence an important intrinsic limitation of Chinese mathematical astronomy, comparable in its own way with the lasting limitation of Western astronomy induced by geocentrism. Political and Cultural Factors: An Example Sometimes, political factors have also influenced astronomical tables. For example, in 1449 and 1450, a little more than ten years after the transfer of the Chinese capital from Nanjing to Beijing, the official astronomical tables indicating the lengths of the day and night were adapted to the latitude of Beijing. However, this objective astronomical modification was not well accepted because it provoked substantial modifications of the former hemerological structure of the calendar with respect to the timing of its auspicious and inauspicious daily activities. Moreover, the simultaneous capture of the then reigning emperor, Ying Zong, by the Oirats – who was then held hostage by them at Tumubao (in Hebei province) was interpreted as a proof of the noxious character of this modification. Consequently, his successor ordered to restore the former Nanjing latitude for calendrical calculations. Subsequently, this move backwards was all the more welcomed that the latitude of Nanjing was deemed the same as that of ‘the middle of the Earth’, di zhong d d, the symbolic geographical center of the Middle kingdom, a place 107 Chu Pingyi 1997, p. 11. This method was still in use in China in 1664. See Huang Yi-long and Chang Chih-ch’eng 1996. 42 PRELIMINARY OBSERVATIONS more in agreement with the mythical requirements of the calendar with respect to political power than any other.108 The Reforms of Astronomical Canons At the occasion of a reform of predictive techniques, various methods based on direct or indirect empirical observations were used in order to evaluate rival astronomical canons and to rank them according to the degree of precision of their predictions or retrodictions of future or ancient celestial phenomena. The Chinese sources refer to these predictions by using the terms tuibu dd and kaogu dd. The first literally means ‘to push’ (tui), or ‘to infer’, the paces (bu) [of celestial bodies], and the second ‘to investigate past events’ as if it were a kind of archaeology.109 These two sorts of evaluations are complementary but, of course, not equivalent in practice: while the observation of certain celestial phenomena such as lunar or solar eclipses generally requires a certain waiting-time, retrospective verifications need no other delay than the amount of time needed to perform the relevant retrodictive calculations. Chinese astronomical canons contain numerous examples of such predictive and retrodictive tests. Sometimes, a degree of accuracy, limited to a one day range, was deemed sufficient, sometimes not, but higher performances were increasingly sought after.110 For instance, the Shoushi li ddd (Inception-Granting canon) (1281–1384), provides a very interesting example of a test devised in order to determine the best ancient or modern astronomical canons among the following, deemed the best-performing, namely the Dayan li ddd (Great Expansion canon) (729–761), the Xuanming li dd (Manifest Enlightenment canon) (822–892), both from the Tang dynasty, the Jiyuan li dd d (Era-Epoch canon) (1106–1166), the Tongtian li ddd (Concordwith-Heaven canon) (1199–1207), the Daming li ddd (Great Enlightenment canon)111 (1137–1181), from the Song dynasty and lastly, the Shoushi li ddd. After having established the list of the advances and delays of their respective retrodictive calculations, concerning 49 108 Wang Xiaohu 2012. bisyllabic term kaogu means ‘archaeology’ in modern Chinese. 110 K. Hashimoto 1979. 111 Several different astronomical canons bear this name. See p. 55 f. below. 109 The THE REFORMS OF ASTRONOMICAL CANONS 43 control-dates of past winter solstices,112 recorded in ancient sources believed to be reliable,113 extending over a time interval of more than two thousand years and evaluated with an accuracy limited to a single day,114 the following statistic was established: “[The preceding instances], on the right,115 span an interval of time of more than 2160 years since the time of Duke Xian from the Spring and Autumn period116 and concerns the computation of 49 events117 by using the six following astronomical canons: the Dayan li, the Xuanming li, the Jiyuan li, the Tongtian li, the Daming li and the Shoushi li. The Dayan li agrees 32 times with the observational records and fails 17 times, the Xuanming li agrees 26 times and fails 23 times; the Jiyuan li agrees 35 times and fails 14 times; the Tongtian li agrees 38 times and fails 11 times; the Daming li agrees 34 times and fails 15 times. [Lastly], the Shoushi li agrees 39 times and fails 10 times.” ddddddddd dddddddddd dddddddd dddddddddddddddddddddddddddd dddd dddddd ddddddddd dddddddd dddddddd ddddddddddddddd dddd ddddddddddd ddddddddddddddd d ddddd118 The Shoushi li obtains the highest score (39 successes/10 failures). It should therefore be declared the winner of the competition. However, instead of feeling satisfied with this result, the historical records of the Yuanshi question the validity of this statistical evaluation in a particu112 Yuanshi, j. 52, ‘li 1’, p. 1132–1138. 113 We have not attempted here to check the astronomical validity of the corresponding Chinese data since we are only interested in the Chinese mode of argumentation. For an adequate analysis of the question, however, see S. Nakayama 1969, p. 247–249 and N. Sivin 2009, p. 283 and p. 286–287. 114 Only the sexagenary days of the winter solstices in question are recorded. Therefore, a better precision would be meaningless. 115 Chinese writing progresses from the right to the left. Consequently, the author refers to what has already been written as being ‘on the right’. 116 No critical analysis of the underlying chronology is attempted here, we only note that the year allusively referred to in this text corresponds to 884 BC. See, for instance, S. Nakayama 1969, ibid., p. 248 and N. Sivin 2009, ibid., p. 283. 117 The context shows that the events in question are winter solstices. Moreover, the text of the Yuanshi handed down to us is inaccurate in this respect because it only mentions forty-eight winter solstices. 118 Yuanshi, j. 52, ‘li 1’, p. 1138. See also N. Sivin 2009, p. 283. 44 PRELIMINARY OBSERVATIONS larly original way, revealing a very unusual assessment of the notion of celestial regularities. In order to show that the Shoushi li is really superior to its competitors, they explain that discrepancies between retrodictive calculations and control-dates are possibly the result of unpredictable astronomical irregularities and not of mathematical inaccuracies. Therefore, no matter how well the concerned predictive or retrodictive mathematics were designed, potential celestial abnormalities beyond their control can never be excluded. In order to obtain such a conclusion, the historians of the Shoushi li rely, in fact, on a sort of reductio ad absurdum – presented here in a very summarized form – by saying that if certain calculations are cleverly modified (qubian dd), it becomes possible to transform certain erroneous retrodictions into correct ones but at the expense of rendering false certain other calculations initially correct. Therefore, any modification of initial calculations is deemed less desirable than the statu quo. Lastly, they conclude from this reasoning that the correctness of the Shoushi li calculations cannot be challenged. In other words, they are fundamentally correct even though certain of their results are false. Then, given the strangeness of this conclusion, they also suggest that the Shoushi li errors arise as a result of a temporarily erratic solar behavior and are not the consequence of faulty retrodictive mathematics. What is at stake here is thus nothing less than the temporary possibility of an erratic solar motion: “The degrees of the solar motion have become erratic” ridu shixing dddd.119 In the case of the above retrodictions, the usual solar regularities have been so exceptional that no predictive mathematics could have saved the phenomena, any attempt to modify initial calculations leading to a worsening of the situation. In other words, if the sun had not been subjected to a series of temporary but unpredictable violations of its regular cinematic behavior, the Shoushi li calculations would have been wholly correct.120 Therefore, perfectly flawless mathematics can lead to false predictions and conversely, when mathematical predictions are not corrobo119 Yuanshi, 120 Yuanshi, j. 52, ‘li 1’, p. 1139. Ibid. j. 52, p. 1139 and 1140. THE REFORMS OF ASTRONOMICAL CANONS 45 rated with reliable observational records, the falsity of the underlying mathematics cannot be asserted. Clearly, this type of analysis, assigning to the sun a kind of cinematic freedom inaccessible to rational analysis, excludes all possibilities of developing any belief in the existence of immutable mathematical predictive techniques, at least in the case of the sun. Almighty mathematics, that is divine mathematics, endowed with an unlimited predictive power is thus excluded per advance and the ‘great book of nature’ cannot have been written in the language of mathematics as Clavius, the famous Jesuit mathematician, imagined121 a few years before Galileo.122 Clearly the Chinese conclusion is also not alien to the idea of the pre-eminence 121 Clavius is a central character in the development of mathematics from the end of the sixteenth century, not only in Europe but also in China. A great number of his works have been translated or adapted into Chinese during the first half of the seventeenth century. See H. Bernard-Maître 1945; J.-C. Martzloff 1997*/2006*, p. 21–22, 375, 383–385; P. Engelfriet 1998 (study of the Chinese translation of Clavius’s commentary to Euclid’s Elements). 122 In his Prolegomena to his Euclidis Elementorum Libri XV [. . . ], first published in 1574, Clavius explains in substance that Euclid’s geometry should be taught before any other subject, even theology, for it prepares for access to all sorts of hidden truths owing to the certainty of its demonstrations. For him, mathematics is so fundamental that he assimilates geometrical figures to the letters of an alphabet whose combinations open the way to the mathematical intelligibility of the world, in the same way as the letters of an alphabet form meaningful words by their combinations, and he concludes that “To submit the world – Nature and God’s immense work – in its totality to the sight of our mind and offer it to our contemplation is the duty and the beneficial effect of geometry.” (A. Romano 1999, p. 141, J.-C. Martzloff 2013.). By expressing himself in this manner, Clavius offer a synthesis of Aristotelian, Platonician and Patristic traditions (theory of demonstration, ideality of geometrical constructions; occult, yet intelligible, character of number and belief in the inherently mathematical structure of the world), explicitly mentioned by him in his Prolegomena and already prominent before him (see, for instance, A. Goddu 2010, p. 216, citing N. Copernicus: “Mathematics serves [. . . ] as the best approximation to theology because through exercise it can lead us to the celestial divinities where by means of its arithmetical and geometrical proofs we find nothing obscure and nothing disordered.”). As A. Romano 1999, idem., p. 141, also remarks, Galileo says nothing else when he states that “Philosophy is written in that Great book, the universe, which stands continually open before our eyes, but it cannot be understood without first learning the symbols in which it is written, namely triangles, circles and geometrical figures without whose help it is impossible to grasp a single word of it.” (on Galileo, see also M. Blay’s article ‘Mathématisation’ in M. Blay, R. Halleux et al., 1998, p. 604). 46 PRELIMINARY OBSERVATIONS of unceasing experimental evaluations leading to the idea that predictive mathematics is intrinsically limited.123 Although the above quotation only concerns a single astronomical canon, the Shoushi li, this conclusion was not limited to the sole case of the Mongol period. Several centuries earlier, another important astronomical canon, the Dayan li,124 (New Tang History) focused on the same issue. Still more anciently, other ideas about the limitation of predictive mathematics are also attributed to Du Yu dd125 (222–284). In a passage from the astronomical canon of the Jinshu126 (Jin History), ca. 646, infinitesimal discrepancies, only detectable from long-term observational records, are said to slowly accumulate their effects over very long periods of time in such a way that wrong mathematical predictions can never be avoided sufficiently in advance: “Celestial motions are endless. While traveling across their respective mansions, the sun, the moon and the planets are ‘moving things’.127 123 Historians of science rightly oppose the Chinese and Greek astronomical traditions on the grounds that the first is ‘algebraic’ and the second ‘geometrical’. In addition, the Chinese conviction of the artificial character of mathematics, taken by them as an artifact among others and in no way an exact representation of phenomena providing access to immutable truths, is also absolutely essential. The lasting Chinese conviction of the deeply historical character of mathematics, never removable from their historical time and the particular conditions of their elaboration is essential. Therefore, the very idea of an axiomatico-deductive mathematical system such as Euclid’s Elements (Jihe yuanben dddd) – which was first partially translated into Chinese in 1607 by the Jesuit missionary Matteo Ricci (1552–1610) from Clavius’s commentary of this famous work, with the help of his Chinese collaborator and Christian convert Xu Guangqi (1562– 1633) ddd – could not be accepted by them as the ultimate paragon of immutable truths. Hence the later reduction of this geometrical mathematics to mere calculation techniques by the most influential Chinese mathematicians from the seventeenth and eighteenth centuries. See J.-C. Martzloff 1993–1994. 124 Xin Tangshu, j. 27B, ‘li 3b’, p. 625–626. 125 Renowned scholar, chronologist and military general of the emperor Wu Di (265– 290), under the Western Jin dynasty. 126 On the Jinshu, see Ho Peng Yoke 1966. 127 dong d = moving, wu d = thing. In general, the term dongwu dd designates living beings, animals, with respect to the fact that they are able to move freely in space and such a meaning very anciently is already attested in the Zhouli (The Rites of Zhou), one of the canonical texts from Chinese antiquity. See DKW, 2-394-2390:107. THE REFORMS OF ASTRONOMICAL CANONS 47 When a thing128 d moves, its motion is not [wholly] regular even though it is always possible to confine the degrees of its motion within [certain] limits. When days accumulate into months and months into years, new and old [elements] are mutually interdependent and deviations [from the original state of their motions] in the order of the thickness of the tip of a hair129 cannot but become manifest.130 This is a natural131 principle. [. . . ] These errors being initially not greater than the tip of a hair, they cannot be detected but they continue to accumulate. When they show up, they provoke [predictive] errors of moon phases. Then, it becomes impossible to maintain the statu quo and to avoid reforming astronomical canons in order to take these discrepancies into account.” dddd ddddddddd ddddddddddd dddd ddddddd ddddd ddddd dddddd ddddddd dddddddddddddddddd dddddd ddddd d dddddd ddddddddddd132 The idea of the intrinsic long-term irregularity of celestial motions has been constantly rehearsed subsequently either by repeating general statements133 or from specific metaphors, intended to illustrate the fact Curiously, a similar conception exists in Roman Law from Antiquity, where slaves and animals are called res per se moventes (self moving things). See R. Monier 1942, p. 253. 128 Viewed in isolation, the term wu is quite vague and only means ‘thing’, or ‘things’, but in the present context it must be understood as a synonym of the bisyllabic term dongwu, animal-like ‘moving thing’, already present in the preceding sentence. See also N. Sivin 1989, p. 177: “in Greece and Hellenistic Egypt, a source that determined its own motion would be divine; in China, it was an animal-like ‘moving-thing”’. 129 mo means ‘end’ and hao ‘hair’, but it is also the one-thousandth part of any principal unit. 130 These deviations are to be understood as [minute] discrepancies between observable positions of celestial bodies and predictive calculations. 131 This translation supposes that ziran dd = natural, but its literal meaning is exactly ‘that-which-is-of-itself-what-it-is’ (H.U. Vogel and G. Dux 2010, p. 7). 132 Jinshu, j. 18, ‘lüli 3’, p. 563–564. 133 For instance, the Xin Tangshu maintains that deviations between predictive calculations and observations cannot but appear in the long run: qi jiu er bu neng wu chate zhe ddddddddd (Xin Tangshu, j. 25, ‘li 1’, p. 533). Similarly, the Songshi uses a quasi-identical formulation in order to state the intrinsic imperfection of astronomical canons: bu neng wu te dddd and youjiu bu neng wu cha dddddd (Songshi, j. 68, ‘lüli 1’, p. 1492 and ‘lüli 15’, p. 1945 resp.). More generally, all sorts of Chinese sources, astronomical and non-astronomical alike, constantly develop the same idea. 48 PRELIMINARY OBSERVATIONS that initially undetectable errors are cumulative and thus bound to show up. For instance: “When a thing is weighed with a series of weights of one ounce zhu,134 a noticeable difference [between its exact and measured weights] necessarily occurs when the measured weight is of the order of one bushel dan.135 What about numbers attached to what is formless, then?” dddddd ddddd ddddddddd136 In other words, when we try to weigh a heavy object by using an immense number of small weights, an error will necessarily become measurable. Therefore: “The heavens are subjected to irregular motions while astronomical canons rely on fixed methods. Consequently, errors necessarily arise in the long term and therefore, reforms are unavoidable once they have become detectable.” dddddddd dddddddd ddddddddd ddd dddddd137 This is exactly why the research of new mathematical predictive techniques always continued without respite, even after the enactment of an official reform, the newly adopted canon being always subjected to experimental controls in spite of its recognized superiority over its former rivals. Significantly, the most common expression found in Chinese astronomical canons in this respect is ‘new astronomical canons’ (xin li d d and xin shu dd). Moreover, everything mentioned in their historical part is practically reduced to the exposition of unceasing controversies bearing on the empirical appropriateness of predictive techniques. Hence a continual search for novelty, everything elaborated more or less See N. Sivin 1989 (comparative study of the question of the limits of empirical knowledge in the Chinese and Western classical worlds), H.U. Vogel 1996 (new remarks on the same subject), SIXIANG, p. 379 (remarks on the limitation induced by naked-eye astronomical observations). 134 The zhu d is a minute unit of weight, equivalent to one hundred grains of millet, or approximately 0.64 grammes. 135 The dan d is a unit of weight equivalent to approximately 120 pounds or 70 kg. 136 Songshi, j. 71, ‘lüli 4’, p. 1618. 137 Yuanshi, j. 52, ‘li 1’, p. 1119. THE REFORMS OF ASTRONOMICAL CANONS 49 successfully in past times and given a final fixed form being ipso facto subjected to a process of degeneration. The notion of novelty, xin d, is thus the cardinal notion of Chinese astronomical canons. It first appears in the Hou Hanshu (Later Han History) and in practically all subsequent Chinese astronomical canons.138 As a famous French revolutionary139 would have proclaimed in the same situation: “We need novelty, still more novelty, novelty forever!”140 In a world of unceasing change, an astronomical canon was all the more rated highly than it turned out to be precise. No other decision criterion was put forward, even though the interpretation of what was deemed precise, and what was not, required the taking into account of the purported possibility of an unpredictable but temporarily aberrant behavior of celestial motion. In fact, as the history of Chinese astronomy witnesses, the successive official astronomical canons have never ceased to be deemed imperfect and always as ‘biodegradable’ as before, while offering more and more punctually precise predictions. This trend was challenged, however, when some titles of surface calendars from the end of the Southern Song period began to claim the ‘perpetual’ character of the astronomical canons they had been obtained from:141 they were said to be ‘perpetual’ (wannian dd, literally ‘[valid for] ten thousand years’ i.e. ‘always’). Likewise, despite their maintained adherence to the principle of limitation of the predictive power of astronomical canons,142 the authors of the Shoushi li claimed still more Hanshu , zhi 1, ‘lüli 1’, p. 3028; Xin Tangshu, j. 25, ‘li 1’, p. 534 (xin li d d); ibid., p. 536 (xin shu dd); Songshi , j. 68, ‘lüli 1’, p. 1492 (xin li dd). 139 G.-J. Danton (1759–1794). 140 On the temporary character of Chinese mathematics, J. Gernet 2005, p. 54–55, underlines the opposition between discursive reasonings of Greek origin, based on the idea of the existence of stable and eternal truths, and Chinese combinatorial thinking, ruled by relative and temporary truths, only related to specific times and places [. . . ]. He also notes (ibid.) that the European and Christian world-view introduced into China by the Jesuits during the XVIth and XVIIIth were difficult to reconcile with Chinese conceptions: the former relied on medieval scholastic reasoning and assumed the existence of eternal laws, granted to nature by a Creator and almighty God, whereas the latter supposed the relativity of times and spaces. 141 The Kaixi li (1208–1251) and the Huitian li (1253–1270). 142 Yuanshi, j. 52, p. 1119. 138 Hou 50 PRELIMINARY OBSERVATIONS clearly its eternal character (yongjiu dd) .143 However, given the close relationship between astronomy and politics in the Chinese world, these facts can possibly be understood as referring to an affirmation of the unbounded character of imperial power. Or perhaps, the Chinese from the Mongol period have on the contrary directly or indirectly borrowed the notion of eternity from C. Ptolemy through the intermediary of Muslim astronomers which were then present in China and were of course fully aware of the Greek astronomical tradition (see the beginning of the Almagest: “It is this love of the contemplation of the eternal and unchanging which we constantly strive to increase [...]”)144 If this occurred, an entirely new conception, possibly also dependent on the rejection of the quasi-Buddhist cycles of creation, downgrading and rejection of official astronomical canons, has temporarily gained momentum towards 1280. However, this new conception can also be related to the apparition, during the eleventh century, of a new current of thought, going against China’s long-standing tradition, and asserting that the quasi-infinitesimal deviations between astronomical observations and systematic calculations are potentially analyzable in terms of fixed mathematical principles. As noted in the collected works of the influential Zhu Xi d d (1130–1200): “It is not true that celestial motions are undetermined. Differences between [calculated and apparent] motions are also constant quantities.” dddddd [. . . ] ddddddddddd145 Moreover, these quasi-infinitesimal variations were not only supposed to be regular but they were also given an explicit mathematical formulation in the Tongtian canon (1199–1207) and in the Shoushi canon (1281–1384),146 for the first time. Still, Zhu Xi was not even 143 Yuanshi, j. 52, ‘li 1’, p. 1120. from G.J. Toomer 1984, p. 37. See also L.C. Taub 1993. 145 SIXIANG, p. 379 (the quotation is taken from the chapter of Zhu Xi’s complete works (Zhu Xi quanshu dddd) devoted to astronomical canons (lifa dd, second part), included in his reflections on the li d (the principle of organization) and qi d (breath or energy)). See also N. Sivin 1989, p. 176 f. Here, we also note in passing that the notion of natural regularities and even of “laws” of nature in a general sense, including the case of astronomy, were already present in Roman antiquity. See D. Lehoux 2012, p. 47–76. 146 S. Nakayama 1982 explains the Chinese situation by means of the two following formulae, T = 365.2425 − 0.0000021166t and T = 365.2425 − 0.000002t, where the 144 Quoted THE REFORMS OF ASTRONOMICAL CANONS 51 mentioned in the technical parts of Chinese Histories devoted to astronomical canons. Was it because he never explained how his idea should be implemented in practice? Or was it the consequence of something else? Anyway, his radically new idea was not lastingly successful and, although deemed ‘perpetual’ or ‘eternal’, the Shoushi li was already subjected to some modifications less than ten years after its adoption, when some of its fundamental astronomical constants were corrected (adjustments of the values of its shift constants).147 Sic transit gloria mathematicarum. Nevertheless, the revolutionary idea of Zhu Xi and later thinkers was not entirely rejected but the practical possibility of creating sufficiently powerful predictive mathematics was eventually ruled out when the authors of the Mingshi (Ming History) (1755) observed that: Later [predictive] methods supersede former ones and have become increasingly precise after reiterated reforms; this point is most clearly established in astronomical canons. The Tang canon states that the sky is a moving thing and that deviations [of its motion] show up in the long run. Repeated reforms of [predictive] methods are thus unavoidable in order to get better agreements between [calculations and observations]. Still, although this theory seems correct, it is not in fact. [. . . ] This is because celestial motions are extremely robust and positively constant, to the extant that there is no difference between the past and the present. [. . . ] Taking into account what is stated in dynastic histories, the Year Star, Jupiter, has slipped its stations (shi ci dd)148 and the sun has ‘lost’ the degrees of its motion.149 Yet, one also remarks that nothing of the sort has ever been observed in modern or contemporary periods. In fact, the degrees attached to celestial motions are multifactorial and human intelligence limited. How could we detect everything clearly without the slightest error from a gnomon a few inches long and by looking variations of the length of the tropical year are expressed as a function of the number of years elapsed from the epoch, t (both are easy consequences of mathematical procedures explicitly stated in the Chinese sources but not their direct algebraic transcriptions: in these two formulae, the variable t is equal to an integer number of years but the two mentioned Chinese canons are interested in the variations of the mean value of the tropical year by steps of one century (see p. 140 f. below)). 147 See p. 201 below. 148 According to Ho Peng Yoke 1966, p. 36, shi ci dd (slipping stations) is used in connection with irregularities of retrograde motion. 149 On the more general problem of celestial anomalies, see Wu Yiyi 1990. 52 PRELIMINARY OBSERVATIONS upwards in order to measure the celestial vault? It is only possible to synthesize the ideas of Ancient and Modern authors, to extend their efforts and to correct [their methods] more broadly in order to hope to reach an agreement [between predictive calculations and observations]. It is thus impossible to devise immutable [predictive] methods.150 dddddd ddddddd ddddddddddddd dddd d dddddddddddddddd ddddddd. . . dddd dd ddddd ddddddddd. . . dddddd ddddd dd ddddddddddddddddddddddddddddd ddd ddddddd dddddddddddddddddddd dddddddddddddddddddddddd The Bureau of Astronomy The sustained interest in astronomical canons over long successions of centuries was made possible only through the political and permanent financial support of a State structure, generally named ‘Bureau of Astronomy’ by historians of China, but referred to quite differently in Chinese primary sources. During various periods and dynasties, this ‘Bureau’ was named, inter alia: Taishi jian ddd (Office of the Grand Historiographer/Astrologer), Taishi yuan ddd (same meaning), Sitian jian ddd (Directorate of the Celestial Administration), Qintian jian d dd (Directorate of the Celestial Veneration) and Mishu geju ddd d (Board of the Gallery of Secret Writing).151 As the two first appellations indicate, a lasting link unite history and divination and the most prominent figure associated with this institution is precisely the famous historian Sima Qian, the author of the celebrated Shiji (Records of the Historian). Indeed, Historians were not only responsible for keeping track of political events but also for recording, updating and interpreting celestial archives in order to develop new astronomical canons, always regarded as quantitative methods of divination. Significantly, some aspects of these questions are dealt with for the first time in two particular chapters of the Shiji.152 No less importantly, 150 Mingshi, j. 31, ‘li 1’, p. 515. Schafer 1977, p. 13. 152 Shiji, j. 26 and 27 (calendar, on the one hand and ‘celestial officials’, tian guan d d, i.e. astrological enumeration of stars, planets and asterisms), on the other hand. 151 E.H. THE BUREAU OF ASTRONOMY 53 precise dating of past and present regular or irregular heavenly phenomena, over very long periods of time, were also a part of their routine work because celestial phenomena were not only interpreted as portents but also as reference events, used to check the validity of new mathematical techniques with respect to their retrodictive power. Hence the elaboration of rational divinatory techniques,153 if we may say so, by means of preestablished mathematical calculations, endowed with a computational structure perfectly following the universal rules of arithmetic. No less significantly, in Chinese astronomical canons from various periods, but in the first place in those from the Tang period, it often happens that numbers associated with astronomical cycles are metaphorically called ‘hexagrams’ (gua d) or, more often, ‘divinatory rods’), (ce d154 exactly as if astronomers were handling such rods, similar to those used by Chinese seers. More generally, all knowledge now incorporated into the history of astronomy was explicitly said to have its origins in occult sciences. As clearly stated in the Xin Tangshu: “The making of instruments in order to observe to sky and the earth, the armillary sphere, the gnomon of the Zhou, the Xuanye teachings [d d]155 and also the stars manuals, the astronomical canons, are products of occult sciences.” ddddddddddddddddddddddddddddd dddddd156 153 The association of these two notions may seem definitely contradictory but as L. Vandermeersch 1980, p. 285–315, has suggested, albeit in the limited case of a quite different period of Chinese history, this is not always necessarily so. In fact, scientific ideas from all times and periods sometimes have had their origin in all sorts of preconceptions. The paramount importance of Kepler’s ideas establishing a link between astronomy and theology and his vision of celestial bodies representing an image of the Trinity, while revolving around the Sun, is only one famous example of this assertion among many others (see G. Simon 1979). 154 The term ce refers to the fact the Chinese generally used small rods in order to perform arithmetical operations. See Xin Tangshu, j. 28A, ‘li 4a’, p. 637 f.; j. 29, ‘li 5’, p. 697 f.; Songshi, j. 79, ‘ lüli 12 ’, p. 1848 f. 155 Literally ‘the darkness of the night’, a later Han vision of infinite space. See J. Needham 1959, p. 219–220. 156 Xin Tangshu, j. 31, ‘tianwen 1’, p. 805. 54 PRELIMINARY OBSERVATIONS Likewise, the other appellations of the Bureau of Astronomy are intended to emphasize the political role of celestial divination because they clearly refer to some sort of political organization or agency: each time, they have to do with divination from celestial phenomena, that is, more exactly, judicial astrology, the art of judging the reputed occult influence of stars and planets, real or imaginary, upon the fate of empires and human affairs in general and not the prediction of the fate of individuals from their date of birth and their horoscope. Most interestingly, this fundamental point is clearly underlined in the Suishu dd (Sui History) where the term tianwen (astrology) dd is defined as follows: “Tianwen consists in the observation of the changes the stars are submitted to in order to examine political matters” ( ddddddddddddd ddddd).157 In addition, the second expression, Qintian jian ddd, reveals the existence of a sort of sacredness which is more precisely made obvious when noting that the Bureau of Astronomy was connected with the Ministry of Rites, one of the six Ministries of Imperial China.158 Lastly, the third expression Mishu geju dddd stresses not only the secret character of judicial astrology, a politically sensible science, but also its connection with writings, that is, in fact, archives, containing, notably, records of past observations. This preliminary survey could be easily supplemented by long lists of similar names but, more importantly, previous studies have also clearly shown that a more or less numerous technical staff, of up to one hundred members, if not more, was attached to the various services of the Bureau of Astronomy from one dynasty to the next.159 Among their most fundamental tasks, those concerning the systematic observation of the heavens understood in a broad sense, that is by including atmospheric phenomena, were regarded as essential. Hence the importance of instruments of observation, of time measuring instruments, the organization of long-term astronomical expeditions in or157 Suishu, j. 34, ‘jingji zhi 3’, p. 1021. Chen Xiaozhong and Zhang Shuli 2008 ; for a particularly enlightening article on this subject (but for a slightly later period) see also A. Romano 2004. 159 See T.E. Deane 1989; E.H. Schafer 1977, p. 8–20 (Tang); Lai Swee Fo 2003 (Tang); Ho Peng Yoke 1969 (Ming); Wang Baojuan 1994a (Song) and 1994b (Liao, Jin and Yuan); and, for a later period, N. Golvers 1993, p. 81–87, notably. 158 See THE NAMES OF ASTRONOMICAL CANONS 55 der to perform all kinds of observations between places situated several thousands kilometers from one another,160 and the recording of ancient and modern observations. These records were then interpreted in order to detect warning signs of a hazardous situation such as war.161 More generally, divination was omnipresent and hemerological techniques were used for various purposes such as the selection of auspicious and inauspicious days for events such as the marriage or funeral of members of the royal family, the publication of the calendar for the coming year,162 the conception of predictive calculation techniques for the calendar, positional and astrological astronomy, the specific calculation of various calendars and ephemerides from year to year and lastly, the training of students in these various fields163 in order to ensure continuity, without forgetting the safeguarding of secrecy. The Names of Astronomical Canons As political symbols of their imperial origin, it comes as no surprise that Chinese official astronomical canons often bear names identical with those of dynastic eras. For instance, Yuanjia dd (Epochal Excellence) and Daye dd (Great Patrimony) are both the names of official astronomical canons and dynastic eras. As can be readily checked, however, they do not concern exactly the same intervals of time: the Yuanjia li most famous one was due to the monk Yixing dd (683–727). It took place in 724 and led to observations made in thirteen stations, ranging from 52.3◦ to 17.4◦ in latitude. See A. Beer et al. 1961; Ang Tian Se 1976; Chen Meidong 2003a, p. 366 f.; Chen Jiujin 2007a, p. 226–235. Another famous expedition was undertaken by Guo Shoujing ddd (1231–1316) and others by order of the Emperor Qubilai, in 1279. See Yamada Keiji 1980, p. 136 f., Chen Meidong 2003a, ibid., p. 536 f.; Chen Meidong 2003b, p. 201–206; N. Sivin 2009, p. 577–579. 161 The importance of astronomy in Chinese military art was so conspicuous that, in his translation of the astronomical chapters of the Jinshu (Jin History), the historian Ho Peng Yoke bases his study of Chinese astronomical/astrological terminology almost wholly on definitions taken from a military encyclopedia dating from 1628, the Wu Beizhi ddd (Treatise on Armament Technology). See Ho Peng Yoke 1966, p. 34–41. 162 H.J. Weschler 1985. 163 Beyond the restricted circle of the members of the Astronomical Bureau, some epistemological aspects of astronomy and of astronomical canons have sometimes played a role in the examinations intended for recruiting functionaries during the Ming dynasty. See B.A. Elman 2000, p. 468 f. 160 The 56 PRELIMINARY OBSERVATIONS was adopted under three different dynasties, from 445 to 509, while the Yuanjia era lasted only from 424 to 453. Slightly differently, the Daye canon was in force from 597 to 618 while the Daye era extends from 605 to 618. Sometimes too, a unique astronomical canon is referred to in Chinese sources under different appellations. For example, the Jingchu li dd d (Luminous Inception canon) of the Wei dynasty (237–265) was also called Taishi li ddd (Tranquil Beginning canon),164 from 265 to 420, under the Jin dynasty.165 This versatility, typical of political symbols and slogans, also claims its conservatism in a direct manner. Consequently, the same appellation has sometimes been attributed to several canons from various periods of time, technically distinct or not from one another. This is the case, notably, of the Daming li ddd (Great Enlightenment canon).166 The first was designed by the unrivaled calculator Zu Chongzhi dd d (429–500);167 the second was apparently identical with it168 but the third was distinctively different.169 Obviously, the name of a given astronomical canon does not give a clue to its dates any more than to the nature of its techniques save in the very atypical case of the Sifen li ddd, an expression meaning ‘Quarter-remainder canon’, an obvious allusion to the length of its solar year, 365 14 days. In the list given on p. 351 below, the names of astronomical canons have not been translated into English not only because translations would have raised more or less difficult problems but, above all, because doing so would have implied a very consequent critical apparatus if not a particular monograph each time.170 Nevertheless, some translations are 164 Taishi is the name of several dynastic eras. See Li Chongzhi 1981/2006*, p. 20, 58 and 59. 165 Chen Zungui 1984, p. 1400. 166 For their dates, see the items 8, 29 and 38 in the list of astronomical canons provided in Appendix D, p. 351 below. 167 Du Shiran 1992 (biography of Zu Chongzhi) Wang Yingwei 1998, p. 338–350, Yan Dunjie 2000 (technical presentation of the Daming li procedures). 168 Chen Meidong 2003a, p. 491. 169 Wang Yingwei 1998, ibid., p. 707–739. 170 A recent book addresses this issue but only in the particular case of Wang Mang’s Xin dynasty (AD 9–25) See Xin Deyong 2013. THE NAMES OF ASTRONOMICAL CANONS 57 unquestionably straightforward and do not need complex justifications. For instance, Zhantian li ddd means nothing else than ‘Augury of Heaven canon’ (d = augury and tian d = heaven). In the same order of ideas, Dayan li (Great Expansion canon) designates metaphorically a famous astronomical canon from the Tang dynasty by giving him the name of a famous divination method from the Yijing, the famous Book of Changes.171 Likewise, and not surprisingly, some names of astronomical canons also come from less well known parts of the Yijing. For instance, the names Zhide li ddd (Perfect Virtue canon) and Tongtian li dd d (Concord-with-Heaven canon) are both taken from this famous classic.172 In its turn, the name Xuanming li ddd (Manifest Enlightenment canon) comes from a chapter of Xunzi dd (fl. ca. 312 BC)’s extant writings, stating the absolute power of the monarch.173 Still, some other appellations are more intricate. For example, the astronomical canon Shoushi li174 ddd is often called ‘Season Granting system’ by Anglo-Saxon historians of Chinese astronomy and, insofar as shou d shi d and li d possibly means (1) ‘to give to, to confer, to transmit, to impart, to communicate’, (2) ‘season’ and (3)‘astronomical canon’ or ‘astronomical system’, respectively, this translation is formally exact. However, the character shi d also has an important connotation that the following key sentence of the Shujing dd (Canon of History) (one of the Confucian classics) tends to illustrate: [The Emperor Yao] commanded Xi and He to revere the wide heavens and to observe the sun, the moon and the stars in order to communi171 The Dayan method is a divinatory technique based on a division of 49 divinatory stalks into two arbitrary groups and on various further manipulations. See Yijing, ‘Xici shang’ (‘The Great Appendix, first section’, ix, p. 291 in Z.D. Sung 1976). Later, the same appellation was attributed to a technique of resolution corresponding to linear congruences in one unknown. See U. Libbrecht 1973, p. 296 f.; J.-C. Martzloff 1997*/2006*, p. 316, notably. 172 Yijing, hexagram Qian and ‘Xici’, shang, ch. 6, p. 282 and p. 3, respectively in Z.D. Sung 1976. 173 Xunzi, ‘zhenglun’ dd (Rectifying Theses). See J. Knobloch 1994, p. 32. 174 For an explanation of the meanings of the names of Chinese astronomical canons, see p. 55 below. 58 PRELIMINARY OBSERVATIONS cate respectfully, to the people, auspicious times for initiating [their] activities (d) .175 ddddddddddddddddd176 Here, this alternative translation is possible because the character shi d of this key sentence is liable to mean not only ‘seasons’ but also ‘moments propitious for performing determined daily activities’: one of the duties of the emperor precisely consists in determining such moments from the observation of seasonal and heavenly phenomena, i.e. by astrological means. Therefore, what is at stake here is not only the mere astronomical and objective dimension of the calendar but also the role of the supreme Chinese political authority in the harmonious calendrical organization of the social life of its subjects, according to a set of beliefs heavily immersed in a social mode of organization of calendrical time governed by astrology and divination. Another possible translation deliberately stressing this essential aspect of the intended meaning of the expression Shoushi li would thus be ‘astronomical canon intended to deliver the [beginnings of the] propitious moments for initiating various daily activities’ or, more simply, something like ‘Inception Granting canon’, because ‘inception’ is a key technical astrological term having such a meaning.177 Some translations can also possibly raise various other difficulties. For instance, despite the fact that the character d used the name Datong li ddd, – a famous astronomical canon from the Ming dynasty – possibly refers to supra-annual cycles,178 it cannot be interpreted in this way because the Datong li has no supra-annual cycles. In fact, this appellation also comes from the Shujing and means something like 175 The present translation of this passage of the Shujing is my own and the rendering of shi d has been purposely emphasized. 176 S. Couvreur 1950b, p. 3. 177 The astrological technical term ‘election’ (meaning ‘choice’ of auspicious moments) would also have been adequate. See J. Tester 1989, p. 88. 178 The idea of supra-annual cycles is typical of one of the most ancient Chinese astronomical canons, the Santong li ddd. (no. 1), where tong d really refers to three supra-annual cycles composed of 1539 years each. THE NAMES OF ASTRONOMICAL CANONS 59 ‘Great Unification canon’ (of the Chinese empire).179 Hence the following proposal of rendering: ‘Great Unification canon’. For the time being, however, the available translations of the names of astronomical canons constitute as such a useful basis for future research into their direct, indirect or allusive meanings but there is no need to duplicate them all here since English publications mentioning them are easily available.180 179 S. Couvreur 1950b, ibid., p. 193 (the text alludes to a period when the unification of the Chinese empire was not yet achieved and, in the eyes of Ming rulers, this unity has been precisely restored by them). 180 See T.E. Deane 1989, p. 490–499; N. Sivin 2009, p. 43–53. CHAPTER 2 GENERAL DESCRIPTION OF THE CHINESE CALENDAR Limitation and Scope This chapter addresses the question of the Chinese official calendar from the perspective of its surface structure. It will thus be temporarily regarded as a discrete architecture, made up of particular sequences of days, essentially grouped into solar or lunar months and solar or lunar years. For the sake of simplicity, it will also be described in a most systematic and simplified way as possible, in order to highlight its most prominent and invariant aspects from 104 BC to AD 1644. Nonetheless, examples of anomalous years and other striking but local peculiarities, not liable to modify significantly its overall picture, will also be taken into account. In the same spirit, the presentation of the prolific terminology of the Chinese surface calendar1 will be limited to the essentials because an immersion into endless philological details would perhaps have been detrimental to the perception of its overall pattern: the purpose of this study is not the writing of a dictionary but only the description of a not so obvious structure. Yet, some salient points of interest in this respect will be taken into account and new translations of key terms will be propounded when appropriate. 1 Overall, this point has been adequately dealt with by Havret and Chambeau 1920. © Springer-Verlag Berlin Heidelberg 2016 J.-C. Martzloff, Astronomy and Calendars – The Other Chinese Mathematics, DOI 10.1007/978-3-662-49718-0_2 61 62 DESCRIPTION OF THE CHINESE CALENDAR The Fundamental Components of the Chinese Calendar The Day The most fundamental unit of time of the Chinese calendar is the nychtemeron, or a day plus a night. More simply, it will be referred to as ‘the day’ in the following. In Chinese sources, its name is ri d (literally ‘the sun’) even if its apparent meaning excludes the idea of night. This appellation is naturally as vague as those attested in Western languages but what is intended in such and such a case is generally sufficiently explicit from the context. Calendars from various periods define the beginning and end of the day in various ways. During our period of study, the Chinese day extends from any instant of midnight to the next.2 For obvious reasons, the day is generally the smallest unit of time of Chinese surface calendars but smaller units also appear occasionally. Some calendars, from the Tang and later periods, record the calculated occurrence of certain phenomenon such as the instants of sunrise or sunset, the lengths of day and night and the like. In such a case, however, as already noted above, p. 30, only a system of time units peculiar to the surface calendar is used.3 These calculated instants are thus expressed with systems of units completely different from those used in the deep structure and, in particular the precision they allow is not identical in both cases. Nevertheless, the two structures are not independent from each other and the existence of different systems of units of time in each case imply the existence of techniques of conversion. The Solar Year The solar year of the Chinese calendar is often called nian d or sui d and these two terms already occur in oracular inscriptions on tortoise shells and bones of animals (jiaguwen ddd).4 The first term 2 For earlier periods, the calendrical day has not always been defined in such a way. For instance, as explained by L. Vandermeersch 1980, p. 319, in the Shang-Yin sexagenary calendar the day extends from one sunrise to the next. 3 For a specific example (Ming dynasty), see p. 215 below. More generally, see also Chen Jiujin 1983; Chen Zungui 1984, p. 1343–1348. 4 L. Vandermeersch 1980, ibid., p. 326. FUNDAMENTAL COMPONENTS 63 means at the same time ‘crop’, ‘harvest’ or ‘crop year’, in relation to the two successive crops of a year, the first concerning millet and the second wheat. The second term refers to the twelfth part of the sidereal revolution of Jupiter – ‘the Year Star’ (suixing dd) –, a duration approximately equal to one solar year (sui d). In the surface calendar, the solar year is defined as equal to the integer number of days between two calculated consecutive winter solstices. Hence its length, 365 or 366 days. Moreover, from 104 BC to AD 1644, the beginning of the Chinese solar year falls between December 25 and December 21 in the Julian or Gregorian calendar, as the case may be. The Twenty-Four Solar Breaths The division of the length of the solar year by 24 gives rise to as many intervals of equal lengths in the calendrical deep structure and unequal lengths in the surface structure (15 or 16 days).5 The 24 particular days of the surface calendar determined by the beginnings of these intervals are denoted here by q1 , q2 , . . . , q24 .6 They represent the ‘breaths’7 of the solar year, qi d, with reference to the dynamic and vital energy principle, manifesting itself through the fluctuations of the yin and yang.8 Moreover, in accordance with a well-established practice, we still call a ‘solar period’ any interval extending between two consecutive solar 5 These numbers of days are only valid from 104 BC to AD 1644. After 1644, the Jesuit astronomers responsible for the reform of Chinese astronomy have substantially modified Chinese calendrical calculations by linking the Chinese solar year to the true motion of the sun instead of merely dividing it into 24 equal time-intervals. Hence variable intervals, possibly composed of 14, 15 or 16 days each, from 1645 onwards (the year 1644 still uses the Ming astronomical canon, the Datong li ddd ). 6 The same notation is also used hereafter with respect to the corresponding deep structure, in which case the qi are determined instants of time. 7 This rendering appears first in D. Bodde’s English translation of Feng Yu-lan’s History of Chinese Philosophy (Fung Yu-lan 1952–1953, vol. 2, p. 114). Some other translations, sometimes independent of philology and of the Chinese cultural context, such as ‘solar terms’, are also widely used. 8 L. Vandermeersch 1980, p. 329, remarks that the Chinese calendarists identified the year with a vast respiration, as soon as they became aware of its existence and that they gave it the dynamic structure of a breath, organized in inspirations and exhalations. On the notion of qi d and its multiple renderings (air, vapor, stream, vital force, ether, material force, energy, etc.), see S. Onozawa, M. Fukunaga and Y. Yamanoi 1978/1984*’s illuminating study. 64 DESCRIPTION OF THE CHINESE CALENDAR breaths. Four breaths mark the two solstices and the two equinoxes, i.e. the winter solstice, dongzhi dd (literally. The culmination of winter), the Summer Solstice, xiazhi dd (the culmination of summer), the Spring Equinox, chunfen dd and the Autumn Equinox, qiufen d d. Four other breaths, distinct from solstices and equinoxes, indicate the beginnings of the four seasons and are called ‘Enthronement of Winter’, lidong dd, ‘Enthronement of Spring’, lichun dd, ‘Enthronement of Summer’, lixia dd and ‘Enthronement of Autumn’, liqiu d d. They are collectively referred to as the ‘Four Enthronements’, si li d d. Taken together, the two Solstices, the two Equinoxes and the ‘Four Enthronements’ are called ‘the Eight Nodes’, bajie dd.9 Moreover, the quadripartition of the solar year into four intervals, in two different ways by means of its solstices and equinoxes on the one hand and its Four Enthronements on the other hand, determines two sorts of seasons: • the four usual astronomical seasons, determined by the calendrical solstices and equinoxes; • the four civil seasons determined by the ‘Four Enthronements’. Consequently, the beginning of the winter season, with respect to the second quadripartition of the solar year, does not start from the winter solstice but from an instant located one month and a half earlier, between November 9 and 6, always between 104 BC and AD 1644. Similarly, Spring ‘starts in winter’ if we may say so, always one month and a half earlier than the Spring Equinox and the same holds true for the two other seasons. The Chinese calendar shares this peculiar organization of its seasons with other calendars, such as the Zoroastrian calendar gāhānbār10 and with the Celtic calendar, from the British Isles and elsewhere in Europe. That is why the terms midsummer and mid-somer both refer, in modern and middle English, to the night closest to the summer solstice (mid-somer night). In this way, summer begins one month and a half earlier than the astronomical summer, starting from the summer solstice. This is precisely what the playwright Shakespeare refers to in his tragedy A Midsummer Night’s Dream, a title generally reduced to 9 DKW, 10 A. 2-1095, 1450:310. Panaino 1990. FUNDAMENTAL COMPONENTS 65 ‘Le songe d’une nuit d’été’ (A Summer Night’s Dream) in French translation. Likewise, the term midwinter is also an appellation of the winter solstice whereas mid-autumn and mid-spring have similar meanings.11 Apart from the eight breaths determining the beginnings of the two sorts of seasons of the Chinese calendar, astronomical and civil, five other breaths mark the periods of heat and cold and seven other announce various aspects of atmospheric precipitations or humidity. Lastly, one of the four remaining breaths, qingming dd (Pure Brightness), occurs towards the beginning of April and evokes the clarity of the atmosphere while the three others evoke various natural transformations linked with agricultural activities: jingzhe dd (Waking of Insects), between March 7 and 11, xiaoman dd (Grain Full), between May 22 and 27, and mangzhong dd (Bearded Grain), between June 6 and 10.12 With respect to our period of study, the complete list of the 24 solar breaths first appears in the astronomical canon of the Hanshu13 due to Liu Xin dd (32 BC ? – 23).14 Their order of enunciation, however, is not wholly identical with the one generally followed during the greater part of Chinese history and still now. The breaths q5 –q6 , on the one hand and q8 –q9 , on the other hand, are interchanged.15 Likewise, the Wuyin li ddd (Fifteenth-Year Epoch canon), from the beginning of 11 See H. Kurath et al. 1975 and the interesting remarks of the archaeoastronomer E.C. Krupp 1994, p. 195: “The Celtic New Year took place in early November [. . . ]. This falls about midway between the autumnal equinox and the winter solstice and was traditionally the start of winter in the British Isles”. The same author also notes that this kind of winter beginning coincides with Halloween. Lastly, always along the same lines, the ethnologist D. Laurent has also proven – from a very particular question of Breton ethnology, viz a ritual of circumambulation (penitential long march following a path in the form of a quadrilateral twelve kilometers long through the countryside, starting from and arriving at the tomb of Saint Ronan, Irish bishop from the High Middle Ages) – that the Breton calendar also follows the same quadripartition of the solar year. See D. Laurent 1990. 12 See Appendix B, p. 343. 13 Hanshu, j. 21B, ‘lüli zhi 2’, p. 1005-1006. 14 On Liu Xin, see H. Kawahara 1989; C. Cullen 1996 and 2004, p. 31 and p. 27, respectively. 15 C. Cullen 1996, ibid., p. 108. 66 DESCRIPTION OF THE CHINESE CALENDAR the Tang dynasty,16 rely on a list where q5 and q6 are inverted.17 These exceptions were temporary, however, and their most frequent order of enunciation18 is the same as the one already adopted in the Huainan zi ddd19 (second century BC), a treatise more ancient than the Hanshu and the Zhoubi suanjing20 dddd. Irrespective of their order of enunciation, the solar breaths can in no way apply stricto sensu to the whole of China as climatic benchmarks, given the extent of its territory. Only the basin of the Yellow river or slightly more northern regions would be adequate in this respect.21 Certain Southern regions do not get snowfalls while other lands are used for rice-growing instead of cereals. In fact, the Chinese official calendar remained identical everywhere until 1644. Initially restricted to a meteorological and agricultural particular situation, the solar breaths of the Chinese calendar were thus used everywhere alike in the Chinese world, even in regions having quite different characteristics. The Seventy-Two Seasonal Indicators A refinement of the notion of solar breaths leads to a finer subdivision of the solar year into 72 seasonal indicators qishi’er hou dddd.22 Like the solar breaths, the seasonal indicators are associated with 72 particular days of the surface calendar and the interval of time between two consecutive indicators contains most often 5 days, and sometimes one more day, 5 or 6 times within a same solar year. 16 Wuyin dd is the fifteenth sexagenary binomial. Concerning here the enumeration of lunar years, it represents the year 618, the epoch of the Wuyin li ddd, the first year computed with its procedures being the year 619. See LIFA, p. 458. 17 Xin Tangshu, j. 25, ‘li 1’, p. 539. 18 See Appendix B, p. 343 below. 19 See Ch. Le Blanc and R. Mathieu 2003, p. 115 f. On the history of the 24 solar breaths before the Han dynasty, see also J.S. Major 1993, p. 90 f., Chen Zungui 1984, p. 1376–1380. 20 C. Cullen 1996, p. 108. 21 See J. Needham 1959, p. 405: “The names of the periods [i.e. the 24 solar periods delimited by consecutive solar breaths] suggest that the list was first established in, or north of, the Yellow River valley”. 22 Hou d means ‘state’, ‘symptomatic moment’ (of a disease), ‘time when something happens’, notably. FUNDAMENTAL COMPONENTS 67 Although the 72 seasonal indicators are already mentioned very early in a chapter of the Liji dd 23 (The Ritual), one of the Confucian classics, they are believed to have been first incorporated into the calendar many centuries later, at the earliest, during the Northern Wei dynasty (386–534),24 but no authentic calendar from this period mentioning them has reached us. Moreover, their appellations have known a number of variants, which have been fixed once and for all after the carving of the Liji dd and other Confucian classics on steles in 837.25 Rather differently from the 24 solar breaths, each seasonal indicator bears a name relating to a series of real of fictitious natural phenomena, reflecting an archaic Chinese conception, inclined to put on an equal footing meteorological, agricultural or zoological natural phenomena, together with fantastic interpretations of climatic changes occurring during a year. For instance, certain seasonal indicators refer to the melting of the ice, the rumble of thunder, the growth of buds, the thawing of source water, the flowering of peach trees, the arrival of the martins, the fall of dear antlers and other common natural events. By contrast, other indicators signal the quasi-Ovidian metamorphosis of eagles into turtle doves, of moles into quails, of sparrows into shells or of birds into oysters, after their dives into the sea. Lastly, the spontaneous generation is also present through the generation of fireflies from rotten grass.26 The 72 days of the calendar associated with a seasonal indicator are not determined independently of the 24 solar breaths. On the contrary, they form a finer subdivision of the solar year, coinciding partly with them, since 24 different seasonal indicators fall exactly on the same day as a solar breath. When a coincidence between a solar breath and a seasonal indicator exists, the latter is called an ‘initial indicator’, chu hou dd, while the two next ones are respectively called the ‘following indicator’, ci hou d d, and ‘final indicator’, mo hou dd.27 Naturally, this final indicator is also the Initial Indicator of the next solar breath. 23 See Liji, ‘Yueling’, in S. Couvreur 1950a, t. 1, 1st part., p. 390–410. Weishu, j. 107A and 107B, ‘lüli zhi 3a–3b’, p. 2679–2681 and p. 2716–2718. 25 A. Arrault 2003, p. 102. 26 See Appendix C. 27 Ibid. 24 See 68 DESCRIPTION OF THE CHINESE CALENDAR The tripartition of each solar period resulting from this arrangement produces 3 smaller periods having a variable number of days. Depending on whether the solar period in question is composed of 14, 15 or 16 days, their respective numbers of days are equal to 5, 4, and 5 days, 5, 5 and 5 days or 6, 5 and 5 days, not necessarily in this order. The Five Phases The five phases, wu xing dd (jin d Metal, mu d Wood, huo d Fire, tu d Earth and shui d Water), are key cosmological categories supposed to rule over (wang d) portions of the solar year each in their turn, on the basis of its partition into five connected or disconnected sets of days, composed of approximately 73 days (73 × 5 = 365)28 each. In a highly original way, four of these shortened seasons, or pseudoseasons, form a single block whereas the fifth one, corresponding to the period of governance of the Earth, tuwang dd, consists of four disjoint intervals, having altogether the same length as each of the four other pseudo-seasons. The first day of the four pseudo-seasons is always one of the ‘Four Enthronements’ of ordinary civil seasons (si li dd , i.e. Enthronement of Spring, Summer, Autumn, Winter) recording 73 days each, approximately. By contrast, the Earth ‘season’ is made up of four disjoint intervals, generally having 18 days each, and extending from the day following the last day of one of the four pseudo-seasons to the next Enthronement. With this pattern, the Earth phase is seen as ensuring a balanced transition between the four seasons, whereas Water, Fire, Wood and Metal respectively govern winter, summer, spring and autumn. In practice, calendars signal this quinary division of the solar year by merely inserting the two characters tuwang dd above each column of text marking their initial days, other indications being superfluous since the Enthronements of the pseudo-seasons always coincide with already known elements of the calendar. 28 Weishu, j. 107A, ‘lüli zhi 3a’, p. 2677 (most ancient reference to this topic). FUNDAMENTAL COMPONENTS 69 The Lunar Year Depending on its ordinary or intercalary character, the Chinese lunar year is composed of 12 or 13 lunar months. The time span composed of 12 lunar months is conventionally called ‘a lunar year’ but this appellation does not refer to any astronomical lunar cycle, the term year being used analogically, with reference to the solar year, 12 lunar months being approximately 11 days less than a solar year of 365 days while 13 lunar months have 19 more days. More exactly, apart from exceptional cases listed at the end of the present chapter, an ordinary Chinese lunar year is composed of 353, 354 or 355 days, or from 10 to 13 fewer days than a solar year whereas an intercalary lunar year has either 383, 384 or even 385 days, as the case may be. Lunar Months, Ordinary and Intercalary The lunar months of the Chinese calendar are obtained from calculated new moons and are represented by the character yue, d, derived from a pictogram of the moon. They begin on the day where the calculated new moon, shuo d, falls and extend to the day preceding the following new moon, marking the beginning of the next lunar month. The first month of the lunar year is called zhengyue dd, an appellation intended to draw the attention to the fact that its position with respect to the solar component of the calendar is not fixed once and for all but changes with respect to some official lunisolar norm zheng (d) (see p. 76 below). In other words, the first month of the Chinese calendar is not merely a first month but also a distinguished month, having a particular position marking the kind of lunisolar norm chosen in order to establish a connection between the solar year and the beginning of the lunar year. By contrast, the names of the other ordinary months are regularly numbered from two to twelve. From the second to the twelfth, the successive lunar months are thus respectively named eryue dd; (second month), sanyue dd; (third month) and so on, up to shi’eryue dd d (twelfth month). When a lunar year is intercalary, the number of lunar months becomes equal to 13 but non-intercalary lunar months are still numbered 70 DESCRIPTION OF THE CHINESE CALENDAR as if the lunar year were ordinary and its intercalary month is inserted between two ordinary lunar months. The first of these is then called i-yue and the second (i + 1)-yue but never (i + 2)-yue. In its turn, the intercalary month, is called run i yue d i d (run = intercalary). Of course, its true arithmetical rank is i + 1. Consequently, we will use the notation i, i∗, i + 1, . . . where i∗ is the intercalary month. However, the temporary notation i, i + 1, i + 2 will be maintained as long as its position will not have been established, even when the lunar year is known to have 13 months, because the rank of an intercalary month cannot be determined from the surface calendar alone but only from more or less complex calculations depending on its deep structure. The position of an intercalary month is not fixed once for all; it can occur absolutely anywhere, at the beginning, middle or end of the lunar year. There is an exception, however, but it does not concern calendars issued between 104 BC and AD 1644. In high antiquity, it was probably systematically placed after the twelfth month, at the very end of the lunar year. During the Qin dynasty, from 221 to 206 BC, it was placed after the ninth month. Hence its name hou jiuyue ddd (posterior ninth month).29 Ordinary and intercalary months can be full or hollow and the number of days of an intercalary month, i∗, is not connected in any way with the hollow or full characters of the months i and i + 1. Full and hollow months are respectively called da d ‘long’ and xiao d ‘short’. Within a lunar month, three particular days are associated with moon phases other than a new moon: the first quarter shangxian dd, the full moon wang d, and the last quarter xia xian dd. The number of days between these different phases of the moon is variable and equal to 6, 7 or even 8 days. The Structure of the Lunar Year The possible numbers of days of an ordinary or intercalary lunar year, already indicated above, p. 69, can be obtained from various positive integers x and y of full (30 days) and hollow months (29 days), both smaller than 12 or 13 and such that: 29 Chen Zungui 1984, p. 1383 and p. 1422–1423. FUNDAMENTAL COMPONENTS 30x + 29y = 353, 354, 355, 383, 384 or 385 days. 71 (2.1) As can be readily checked, each of these six equations has a unique solution (x, y). For example, lunar years of 353 days can only be obtained from the pair of full and hollow months (5, 7) and in no other way. Likewise, the other sorts of lunar years, composed of 354, 355, 383, 384 and 385 days, correspond to the solutions (6, 6), (7, 5), (6, 7), (7, 6) or (8, 5), respectively. The relative proportion of these various sorts of years is quite variable however. Without going into the details, years of 353 or 385 days are extremely rare. The years Zaichu 2 (690) and Chongzhen 15 (1642) are examples of each. The Percentage of Full and Hollow Months Given that the mean value of the lunar month is approximately equal to 29.53 days, two mean lunar months are 29.53 × 2 = 59.06 days long while the number of days of a full and a hollow month is only equal to only 29 + 30 = 59 days. If the numbers of hollow and full months were equal in all lunar years, the deficit of 0.06 days would thus inevitably lead to an indefinitely increasing drift of the calendar. Therefore, the numbers of full and hollow month cannot always remain identical. More precisely, an alternating sequence of 2x lunar months, x full and x hollow, produces a shift of d = 2 × 29.53x − (30x + 29x) days or 0.06x days (or, of course, a slightly different value according to the chosen mean value of the lunar month). With x = 1000, for instance, the shift already reaches sixty days or two full months. If the lunar months of the surface calendar are to remain in phase with the mean lunar month, these sixty days must be added in one way or another to as many hollow months because no full month can have more than thirty days while, on the contrary, a hollow month can always be transformed into a full month with the addition of a single day. Therefore, such an alternating series of months leads to a calendar composed of respectively 1000 − 60 = 940 hollow months and 1000 + 60 full months. Hence a proportion of 940/2000 = 47% and 1060/2000 = 53% full and hollow months, respectively. Naturally, the same result could also be easily obtained by counting the total numbers of full and hollow months contained in a great number of years listed in any table of the Chinese calendar. But 72 DESCRIPTION OF THE CHINESE CALENDAR should a given lunar year necessarily have more full months than hollow months? Local Patterns of Full and Hollow Months Not at all: in fact, the local distribution of full and hollow months, with respect to particular years, reveals quite heterogeneous patterns. From tables of the Chinese calendar, it is easy to note that fully regular years, i.e. years composed of a simple alternation of full and hollow months (either 29, 30, 29, 30, . . . days or 30, 29, 30, 29, . . . days) are rather common. For example, the years 182, 186, 191, 347, 351 and 392, are made of a regular succession of full (F) and hollow(H) months, beginning with either a full month (like the first four years) or a hollow month (like the last two years). Other years, such as the year 183, obey a pattern of type F H F F H F, . . . displaying two consecutive full months in third and fourth position. This kind of succession of lunar months is also rather frequent, particularly in calendars prior to the Tang dynasty, but the ranks of the two successive full months are not necessarily always identical. In fact, they are not restricted in any way. In the case of the year Huangchu 4 (223), for example, the two full months are the ninth and the tenth. Still other years contain not only two consecutive full months but also two consecutive hollow months at the same time. Such is the pattern of the year Zhenyuan 13 (797): H F H F H H F H F H F F , for instance. Sometimes too, three or even up to four full months follow themselves uninterruptedly. For example, the year Dali 12 (777) follows the pattern H F H H F H F H F F F H and is doubly irregular, owing to its simultaneous successions of two hollow and three full consecutive months. Similarly, the year 769 (F H F H H H F F H F F F ) displays at the same time three consecutive hollow months and also two groups of two and three consecutive full months, respectively. Lastly, the year Shengong 2 (697) (F H F H H F H F H F F F F ) has four consecutive full months. This last case is nonetheless the only one of its kind. The arrangements of full and hollow months actually attested in calendars thus reflect various patterns and an examination of available ta- FUNDAMENTAL COMPONENTS 73 bles of the Chinese calendar clearly reveals that the peculiarities of any given year in this respect cannot be deduced from the monthly surface structure of previous lunar years, independently of the complex calculations of its deep structure. For calendar users, the actual successions of full and hollow months are unpredictable from year to year even though they have been obtained from wholly deterministic processes. Obviously, these characteristics are very different from those of the Julian and Gregorian calendars where January always has 31 days, February 28 or 29 days, according to the regular pattern of bissextile years, March 31 days and so on. It is probably for this reason that the full list of lunar months, with the indication of their full or hollow character, is often explicitly mentioned in the preliminary part of Chinese calendars, even though this kind of data is redundant since the body of the calendar itself necessarily also indicates the number of days of each lunar month. The Astronomical Months and the Lunisolar Coupling Towards the end of the nineteenth century, historians of Chinese astronomy have enumerated the ordinary lunar months of the Chinese calendar by means of the twelve terrestrial branches dizhi dd of the duodecimal cycle,30 and have called them ‘astronomical months’ tianwen yue ddd, because the structure of the Chinese calendar is such that its lunar months are connected to the solar year, that is to an astronomical type of year,31 in a fixed manner: by construction, the first astronomical lunar month contains the winter solstice – the first solar breath, q1 . Still, this q1 is not necessarily identical with its astronomical counterpart, but it belongs to any of the 29 or 30 days of the lunar month in question. Then, solar breaths are enumerated from q1 and each lunar month is calculated in such a way that it always contains a unique odd solar breath, located in any of its 29 or 30 days and never in another lunar month. The fixed – and one-to-one – correspondence between the twelve ‘astronomical’ lunar months, mi , and the twelve odd solar breaths, qi , is thus the following: 30 See p. 81 below. for instance, P. Hoang 1910/1968*, p. III. 31 See, 74 Months Breaths DESCRIPTION OF THE CHINESE CALENDAR m1 q1 m2 q3 m3 q5 m4 q7 m5 q9 m6 q11 m7 q13 m8 q15 m9 q17 m10 q19 m11 q21 m12 q23 This correspondence, defining what will be called ‘the lunisolar coupling’ in what follows, is of course essential but it should be noted that, in practice, lunar months are not enumerated only as shown in this table because the way the beginning of the lunar year is determined is generally not such that the first month of the lunar year m1 is coupled with the winter solstice q1 (see The Beginning of the Lunar Year, p. 76 below), the first odd solar breath. In Chinese sources, odd solar breaths are called zhongqi dd, a term often left untranslated or interpreted as meaning ‘median qi’, on the basis of a classical gloss stating that zhongqi appear in the middle of lunar months.32 Yet, solar breaths most often occur elsewhere. Consequently, the rendering ‘zhong d = middle’ seems dubious, even though ‘middle’ is one of the possible meanings of this d. In fact, the Shuowen jiezi dddd (ca. 100 BC) – a famous etymological dictionary of single Chinese characters from the Han period – gives weight to another equivalence, namely ‘zhongqi = internal breath’ for it defines the character zhong d in the following way: zhong nei ye ddd, i.e. “zhong means inside.”33 Any individual odd solar breath is thus attributed to some day of the lunar month coupled with it, no matter whether the day in question belongs to its beginning, middle or end. What is important is its occurrence inside the same month: only the fact that a fixed odd solar breath always ‘resides’ inside a given month, determined once and for all by the lunisolar coupling, is important. The ‘refusal of entry’ of a solar breath into other lunar months than the one determined by the lunisolar coupling is a fundamental principle of all official Chinese calendars belonging to our period of study (unofficial Chinese calendars do not necessarily respect this principle).34 32 DKW , 1-291-77:117. jiezi, j. 1a, p. 14 (from the edition of the text published in Beijing in 1965 by Zhonghua shuju). 34 For instance, the non-official calendar P3247 vº, from the Pelliot collection of Dunhuang manuscripts preserved at the Bibliothèque nationale de France, Paris, designed 33 Shuowen FUNDAMENTAL COMPONENTS 75 Solar breaths which are not internal – those of even order – are called jie d, that is ‘bamboo nodes’, ‘joints’ or still ‘articulations’. However, this term is not very precise since it more generally qualifies any solar breath, whether of even or odd order. For example, the expression dbajie dd ‘the Eight Nodes’ (see p. 64 above), designates a group of eight solar breaths, some being odd and others even, such as q4 and q1 . Even solar breaths, however, are not subjected to the lunisolar coupling. Consequently, they are liable to belong to two different lunar months. The number of days between two consecutive solar breaths being equal to approximately 15 days, when an odd solar breath happens before the middle of its month, the preceding even solar breath necessarily belongs to the preceding month. By contrast, when an odd solar breath happens after the middle of its month, the preceding even solar breath belongs to the same lunar month. Therefore, it is impossible to determine once and for all whether or not a given lunar month contains a given even solar breath. However, any ordinary lunar month certainly contains two solar breaths, one of odd order, known in advance as a consequence of the lunisolar coupling, and the second one of even order: if q2i+1 belongs to the lunar month m then it also contains either q2i or q2i+2 but not both. In other words, the pair of solar breaths associated with m is either (q2i , q2i+1 ) or (q2i+1 , q2i+2 ). In spite of this double possibility, the Chinese regroup odd and even solar breaths into inseparable pairs, linked to the same lunar month each time, as follows: (q24 , q1 ), (q2 , q3 ), . . . , (q22 , q23 ). For example, q24 and q1 are respectively called shiyiyue jie dddd and shiyiyue zhong dddd, that is ‘even solar breath of the eleventh month’ and ‘odd solar breath of the eleventh month’.35 With such a nomenclature, each solar breath of even order is artificially linked with the same month as the following solar breath, as if it were depending on the lunisolar coupling whereas nothing of the for the year Tongguang 4 (926), violate the lunisolar coupling since the second day of its intercalary month 1* contains the odd solar breath q5 (Rain Water) which normally only belongs to the first lunar month. See Deng Wenkuan 1996, p. 390; A. Arrault and J.-C. Martzloff 2003, p. 156–158. 35 The complete list of these appellations appears, in particular, in the following sources: Weishu, j. 107B, ‘lüli zhi 3b’, p. 2703–2704; Jinshu, j. 18, ‘lüli 3’, p. 541– 543. 76 DESCRIPTION OF THE CHINESE CALENDAR sort exists. As might be expected, the case of intercalary months is quite different since they are defined in the following way: Definition 2.1 (Intercalary month) With respect to the surface structure of the calendar, a month devoid of any odd solar breath is an intercalary month.36 From this definition, it follows that intercalary months are excluded from the lunisolar coupling and that, when a month is intercalary, there is a couple of odd solar breaths respectively located just before its first day and after it last day, respectively. In other words, this particular kind of month is included in the solar month determined by the two odd solar breaths in question. Consequently, it necessarily contains an even solar breath falling in its middle. The Beginning of the Lunar Year In general, the beginning of the Chinese lunar year is mostly fixed in three ways, traditionally believed to have been determined by the Xia d, Shang d (or Shang-Yin dd) and Zhou d calendrical norms, the san zheng dd, respectively, in reference to the supposed modes of determination of the beginnings of lunar years during these mythical or historical dynasties. In addition, some Chinese concordance tables also indicate an unnamed fourth way of determining the beginning of the lunar year, supposed to have existed from 324 to 256 BC, 255 to 207 BC and 206 to 103 BC.37 More precisely, the way lunar months are enumerated according to these four possibilities needs to take into account the following enumerative elements of the Chinese calendar: 1. the terrestrial branches, composed of twelve elements,38 enumerated as follows: hai d, (the twelfth branch) to zi d, chou d, . . . i.e. in the unusual order 12, 1, 2, . . . , 12; 36 See Hou Hanshu, zhi 3, ‘lüli 3’, p. 3058. p. IV. This putative fourth possibility is not often mentioned in contemporary publications and should be reexamined to the light of contemporary archeological findings. 38 See p. 81 below. 37 P. Hoang 1910/1968*, FUNDAMENTAL COMPONENTS 77 2. the 12 odd solar breaths enumerated in the following order: q23 , q1 . . . , q21 ; 3. the ranks attributed to the lunar months according to the four modes of determination of the beginning of the year (four last lines). Then, the following table indicates the ways these various elements are associated with each other: 12 1 2 3 4 5 6 7 Terrestrial branches hai zi chou yin mao chen si wu d d d d d d d d Odd Breaths q23 q1 q3 q5 q7 q9 q11 q13 Lunar Months Xia d 10 11 12 1 2 3 4 5 Shang d 11 12 1 2 3 4 5 6 Zhou d 12 1 2 3 4 5 6 7 Fourth Possibility 1 2 3 4 5 6 7 8 8 9 10 11 wei shen you xu d d d d q15 q17 q19 q21 6 7 8 9 7 8 9 10 8 9 10 11 9 10 11 12 Table 2.1. The four possible modes of enumeration of the Chinese lunar months. We now have everything we need in order to determine the beginning of any lunar year of the Chinese calendar and, of course, to enumerate all its other months. For instance, this table indicates that, according to the Xia norm, the first lunar month is coupled with the solar breath q5 and associated with the third terrestrial branch, yin d. This example might seem quite particular, perhaps, but, in fact it is extremely important because, in practice, the years of the interval 104 BC–AD 1644, have known no other norm than the Xia d norm39 save the two following temporary exceptions: the Zhou d norm has been restored under the Tang dynasty (618–907), between 690 and 700 and in 762. More generally, it has not been modified later and is still observed now, in non-official traditional Chinese calendars, always popular in China.40 39 See P. Hoang 1910/1968*, p. III and IV. official People’s Republic of China calendar is a simplified version of the Gregorian calendar, resulting from a calendar reform initiated in 1912 but accepted after many vagaries, many years after the establishment of the Chinese Republic in 1911. 40 The 78 DESCRIPTION OF THE CHINESE CALENDAR Dynastic Eras and Concordance Tables The naming of Chinese lunar years by means of dynastic eras is first attested in 140 BC41 and has been followed until the overthrow of the last Chinese dynasty, in 1911. According to this system, each dynasty is divided into one or several eras having a particular name and their successive lunar years are enumerated in the following way: the first is called yuannian dd (initial year), the second ernian dd, the third sannian dd and so on. The same technique is of course quite widespread outside China too, successive dynastic eras follow one another without regularity: sometimes, certain eras last several decades while others are reduced to one or two years. No less strikingly, identical names of dynastic eras often refer to different dynasties and periods of time. But beyond this already significant complexity, the overall system is still more intricate because the Chinese Empire has often been divided. Hence the existence of parallel dynasties, making the Chinese chronological system almost hopelessly irregular. It therefore seems uneasy to get one’s bearings in this chronological chaos. However, available concordance tables42 between the Chinese and Western calendars have solved the question at best, notwithstanding a number of difficulties not often taken into account. For instance, the precise dates of the beginnings of such and such a new dynastic era do not necessarily correspond with the beginnings of lunar years, but this point is seldom clarified.43 Moreover, unexpected delays between the official adoption of a new dynastic era and the impact of this decision over the Chinese territory sometimes exist: in certain peripheral regions, In fact, the old Chinese calendar was abolished by the Nationalist government only in 1928. Moreover, in the system adopted in the Republic of China, years are counted inclusively from 1912, the first year of the Republican era. See L.J. Harris 2008; E.P. Wilkinson 2012, p. 507. 41 Li Chongzhi 1981/2006*, p. 1. Other naming peculiarities concerning more ancient times have been convincingly highlighted by R.H. Gassmann 2002 (see also p. 373 below). 42 For an overview of these tables, see p. 371 below. 43 Chen Yuan 1926/1999* is an exception in this respect. CYCLES AND PSEUDO-CYCLES 79 dynastic eras have been episodically left unmodified up to several years after their disappearance.44 Cycles and Pseudo-Cycles Definitions The Chinese calendar contains numerous cycles applied to the enumeration of its discrete units: days, lunar months, years and even doublehours dividing the day into twelve equal parts. The most fundamental cycles are composed of a small number of elements, from the seven days of the planetary week to the sixty elements of the sexagenary cycle but, when combined with each other, they also give rise to a number of supra-annual cycles. Cycles can be simple, simultaneous or with reduplications and the latter can also be referred to as ‘pseudo-cycles’, the resulting sequences being not necessarily cyclical.45 Hence the three following informal definitions: Definition 2.2 (Simple Cycles) Simple cycles are those composed of any discrete sequence of elements enumerated cyclically. Definition 2.3 (Simultaneous Cycles) Simultaneous cycles are those obtained from the simultaneous enumeration of several simple cycles.46 Definition 2.4 (Reduplications) When numbering instructions lead to repetitions of elements according to more or less complex rules, the resulting sequences are said to have ‘reduplications’. They can be cyclical or not. The chronology of the introduction of these various kinds of cycles and pseudo-cycles into the official Chinese calendar is not well known, even to within several centuries. 44 This point has been clearly established for the region of Dunhuang . See A. Arrault 2003, p. 93. 45 See p. 94 below. 46 N. Dershowitz and E.M. Reingold, 1997, p. 19 f. offer a useful mathematical presentation of this notion. 80 DESCRIPTION OF THE CHINESE CALENDAR The Denary Cycle The denary cycle, most commonly referred to as the ‘heavenly stems’, tiangan dd, or trunks, is a simple cycle composed of the following ten elements: jia yi bing ding wu ji geng xin ren gui d d d d d d d d d d These trunks are probably more ancient than the origin of writing in China. From recent investigations, it appears that they were crucially related to the calendar in the same way as the terrestrial branches (see p. 81 below), in a period were writing was still not used for other purposes.47 During these remote periods, they were used for counting days by decades48 (xun d) and, in particular, most Chinese classics mention this notion.49 Their etymology is obscure. All sorts of hypothesis have been formulated in this respect, but none has definitely gained the favor of sinologists. The historian of astronomy Chen Zungui believes that they derive from pictograms representing the head, the neck, the shoulders and other parts of the human body.50 The famous man of letters and historian of ancient China, Guo Moruo ddd (1892–1978), associates them, more generally, with representations of the body of a fish (head, viscera, tail, etc.) and daily life objects (knifes, spears, halberds).51 Starting from a quite different kind of hypothesis which would be revolutionary if it were confirmed, the Anglo-Canadian sinologist and linguist E.G. Pulleyblank, has supposed that the ten trunks have been used as phonograms during the second millennium before our era, that is as purely phonetical symbols used in order to indicate the pronunciation of Chinese words.52 But this hypothesis has been rejected later by his author and not a single specific example of such a usage of trunks has been established. 47 Li Feng and David Prager Branner 2011, p. 28–29. Shaughnessy 1999, p. 20. 49 DKW, 5-748:13746. 50 Chen Zungui 1984, note 3, p. 1352. 51 Ibid., note 2, p. 1353. 52 E.G. Pulleyblank 1991a. 48 E.L. CYCLES AND PSEUDO-CYCLES 81 Anyway, during their long history, the trunks have become abstract symbols, used for naming all sorts of discrete series, in no way limited to calendrical matters. For instance, they have been used in order to designate unnamed things in mathematical problems from the Han period, anonymous characters in a play, and the equivalent of letters in geometrical figures from the beginning of the seventeenth century, when mathematical works of European origin were first translated into Chinese by Jesuit missionaries. The Duodecimal Cycle The duodecimal cycle, also referred to as the ‘terrestrial branches’ dizhi, dd is a simple cycle composed of the following elements: zi chou yin mao chen si wu wei shen you xu hai d d d d d d d d d d d d In calendars, they appear either as second elements of sexagenary binomials or independently. In the latter case, they serve to record the twelve Chinese double-hours or, notably, the cycle of the twelve animals (Table 2.2 below). In particular, this famous zoomorphic cycle is extremely ancient and already appears in manuscripts unearthed at Shuihudi (Hubei, third century BC).53 Its history is complex. Funerary statuettes representing either animals or hybrid beings, half-animal halfhuman, have been discovered. Moreover, it has been established that the twelve animals were not associated only with lunar years but also with lunar months and calendrical spirits and that they were divided into two modes of divination, depending on either the year of birth54 or the date 53 M. Loewe 1994, p. 214 f. and M. Kalinowski 2003, p. 228–229, mentions numerous references in this respect. 54 Historically, the twelve animals have nothing to do with the zodiac since they do not refer to zones of the celestial sphere. Nonetheless, the zodiac has been transmitted in China under the Sui dynasty (589–618) at the latest through the diffusion of Buddhism (Chinese translations of Sanskrit works). Moreover the zodiac is mentioned in a famous treatise of astrology also influenced by Indian culture, the Kaiyuan zhanjing d ddd (Kaiyuan reign-period (713–742) Treatise on Astrology). Lastly, various wall paintings from the Xth –XIIth centuries, from Dunhuang and elsewhere, also witness the introduction of the zodiac in China (Xia Nai 1989, p. 306 f., Chen Meidong 2003a, 82 DESCRIPTION OF THE CHINESE CALENDAR of birthday.55 Drawings of the twelve cyclical animals sometimes exist in ancient almanacs. In the two almanacs S-P6 rº and S612 rº from Dunhuang, dated 877 and 978, for example, they are represented either directly or in the form of designs appearing on the hats of functionaries.56 But, as a rule, one cannot expect to see them in calendars, save indirectly, from their tacit correspondence with each term of the duodecimal cycle. zi chou yin mao chen si wu wei shen you xu hai d shu d Rat d niu d Ox d hu d Tiger d tu d Rabbit d long d Dragon d she d Snake d ma d Horse d yang d Goat d hou d Monkey d ji d Rooster d gou d Dog d zhu d Pig Table 2.2. The twelve animals and their correspondence with the twelve terrestrial branches. The Inverted Tree In writings about the Chinese calendar, the two enigmatic appellations ‘heavenly stems’ and ‘terrestrial branches’ are generally left unexplained. Fortunately, however, a researcher from Bonn University, Jörg Bäcker, has recently established that they are not earlier than the p. 394–396). Nevertheless, authentic Chinese calendars which have been handed down to us never refer to it. 55 M. Kalinowski 2003, ibid., p. 228–229. On the less ancient cycle of the twelve animals in the Turkish world, see L. Bazin 1991, p. 123 f. 56 A. Arrault 2003, p. 201 and p. 183. CYCLES AND PSEUDO-CYCLES 83 Han dynasty and that they are related to Indian cosmological ideas and a number of various other ancient traditions. More precisely, they evoke the image of an inverted tree whose stem (or trunk, including its roots) ‘sinks’ into the sky while its branches ‘rise’ to the earth. We are thus here in presence of the most archaic form of the cosmic tree, a tree which is omnipresent in Indian philosophy from the Vedic period.57 More generally, this arbor inversa is also documented in the Arabic, Hebraic, Icelandic, Finnish and Siberian traditions.58 A new insight into an apparently unsolvable problem has thus been obtained from a wide understanding of non-Chinese questions. The Sexagenary Cycle Among the numerous enumerating techniques for days, lunar months and years, the sexagenary cycle is the backbone of Chinese calendars, from the oldest to the latest. Formally, it can be described in terms of ordered pairs, or binomials (ai , b j ) 1 ≤ i ≤ 10 and 1 ≤ j ≤ 12, where the ai and b j are respectively a trunk and a branch, enumerated simultaneously and cyclically. The first ten binomials are thus (a1 , b1 ), (a2 , b2 ), . . . , (a10 , b10 ). Then, the ten trunks being exhausted, they are then reused from a1 so that the eleventh binomial is (a1 , b11 ) and the twelfth (a2 , b12 ). Similarly, the enumeration starts anew from b1 . Hence the new pairs (a3 , b1 ), (a4 , b2 ), (a5 , b3 ) . . . and so on, up to (a10 , b12 ), the sixtieth. The binomials so listed are of course all different and, beyond the last, the same enumeration technique reproduces endlessly the same ones which are thus more precisely called ‘sexagenary binomials’. The earliest full representation of the whole cycle dates back to the Shang-Yin dynasty, where these binomials are listed into six successive 57 “Un tel arbre, avec ses racines dans le ciel et ses branches pendant vers le bas est omniprésent dans toute la philosophie indienne depuis l’époque védique; see RigVeda, I, 27, 7: “C’est vers le bas que se dirigent les branches, c’est en haut que se trouve sa racine, que ses rayons descendent sur nous!” ” (“Such a tree, with his roots in the sky and his branches hanging down, is omnipresent in Indian philosophy since the Vedic period; see Rig-Veda, I, 27, 7: “Its branches head downwards, its stem is situated upwards, its rays stream downwards upon us!” ”) (J. Bäcker 2007, p. 64). 58 J. Bäcker, ibid., p. 64–65, provides numerous references in this respect. 84 DESCRIPTION OF THE CHINESE CALENDAR Figure 2.1. The earliest known representation of the sexagenary cycle is recorded in divinatory inscriptions on bones and turtle shells from the Shang period. The diagram on the right shows the written part of a shoulderblade, the only extant piece from this period containing a complete list of the sexagenary binomials. From this reproduction and from the correspondence between the ancient and modern forms of the trunks and branches given on the left, it appears that this list is composed of six columns, enumerated in canonical order from (1, 1), jiazi dd, to (10, 12), guihai dd, in groups of ten. See Guo Moruo 1978–1982, item no. 37986. CYCLES AND PSEUDO-CYCLES 85 trunks columns, each composed of ten elements (Fig. 2.1), p. 84). Later, the same enumerative pattern, highlighting likewise the six decades of the sexagenary cycle, has been often used. But when correlations between the sixty binomials and other cycles had to be displayed, circular patterns, such as those used in geomancy, are also extremely common. A simplified notation, more appropriate for mathematical purposes, is obtained by replacing the ai and by bi , by their ranks in their respective series. Hence binomials such as (1, 1) or (7, 11) instead of (a1 , b1 ) and (a7 , b11 ) or (jia, zi) dd and gengxu dd), respectively. Still more simply, any binomial will also be designated by its rank, denoted #1, #2 . . . #60 instead of 1, 2, . . . 60 in order to avoid any confusion with ordinary integers. For instance, #54 will refer to (4, 6) or dingsi dd. 1 2 3 4 5 6 7 8 9 10 1 1 2 3 51 2 13 4 52 3 14 25 4 15 26 37 16 27 38 49 28 39 50 40 branches 6 7 31 42 53 43 54 5 55 6 17 7 18 29 19 30 5 41 8 9 21 32 10 11 11 22 33 44 12 23 34 45 56 24 35 46 57 8 36 47 58 9 20 12 48 59 10 60 Table 2.3. the sixty sexagenary binomials. In order to determine the rank of a given binomial (a, b), an easy rule of thumb is available59 but the most straightforward method still consists in using a double-entry table (Table. 2.3), the reverse correspondence being also easily available at the same time. However, no such table is attested in Chinese sources. Rather, binomials were often listed as in Fig. 2.1, p. 84 above, and given that with such a pattern the sixty binomials are regularly listed in six successive columns, composed 59 See Appendix A. 86 DESCRIPTION OF THE CHINESE CALENDAR of ten binomials each, the top one always beginning with jia d, some more direct mnemotechnical rule, for determining their ranks and for the reverse operation, have probably been used instead.60 Various Uses of the Sexagenary Cycle Enumeration of Days The sexagenary cycle was first used for enumerating sequences of days at a very early date, impossible to determine precisely but probably going back to the Shang-Yin dynasty (1765–1122 BC). Historians of China also believe that the same technique has been used without any interruption, from an unknown early period, until now. However, the days so enumerated have not necessarily always been defined in the same way. Enumeration of Years From the Spring and Autumn period (722–481 BC), the twelve branches have been used in order to number years by analogy with the sidereal period of revolution of Jupiter, a period approximately equal to twelve years.61 Much later, during the Later Han at the earliest, the sexagenary binomials served the same purpose.62 This innovation made its way into calendars much later, however, viz. in those of the Tang dynasty (618– 907), many centuries later. Enumeration of Months During the Tang dynasty, the sexagenary cycle was also used for enumerating ordinary lunar months, according to a supra-annual cycle composed of 12 × 5 = 60 months or 5 years.63 In this manner, the sixty binomials have been associated one by one with successive ordinary lunar months, intercalary months being skipped. 60 Simultaneous cycles similar to the sexagenary cycle are of course also attested elsewhere than in China. For the most recent presentation reflecting the state of the art in the Mesoamerican domain, see A. Cauty 2012. 61 Chen Zungui 1984, p. 1358–1363. 62 The astronomical canon of the Hou Hanshu (zhi 3, ‘lüli 3’, p. 3061–3062) contains a table indicating the sexagenary numbers of the first lunar years of a series of supraannual periods (or cycles) composed of 76 solar years each. 63 Deng Wenkuan 1998a, p. 613, ‘yuejian ganzhi’ dddd (sexagenary enumeration of lunar months). CYCLES AND PSEUDO-CYCLES 87 Less obviously, but in accordance with the Xia norm, used during the majority of years between 104 BC and AD 1644,64 this lunar enumeration always begins with an eleventh lunar month. Consequently, the months corresponding to #1 and #2 are respectively an eleventh and twelfth month of the lunar year preceding the first lunar year of this enumeration. In its turn, the month #3 is the first lunar month of the first year so enumerated. From this peculiarity, it follows that the whole cycle is exhausted after 5 years composed of 12 enumerated months each (and some non-enumerated intercalary months). Hence the following table, showing the correspondence between sexagenary binomials and lunar months, over a period of five consecutive years: 1 2 3 4 5 Months 6 7 8 9 10 11 12 Year 1 #3 #4 #5 #6 #7 #8 #9 #10 #11 #12 #13 #14 Year 2 #15 #16 #17 #18 #19 #20 #21 #22 #23 #24 #25 #26 Year 3 #27 #28 #29 #30 #31 #32 #33 #34 #35 #36 #37 #38 Year 4 #39 #40 #41 #42 #43 #44 #45 #46 #47 #48 #49 #50 Year 5 #51 #52 #53 #54 #55 #56 #57 #58 #59 #60 #1 #2 Table 2.4. The sexagenary enumeration of lunar months. With this quinary pattern, the binomials associated with any month of any given year are easily obtained. For example, since the binomial of the 11th month of the year 803 (Zhenyuan 19) is (1, 1) or jiazi dd,65 all 11th months of years of the form 803 + 5k, k = . . . − 2, −1, 0, 1, 2, . . . are associated with the same binomial. However, the starting date of this enumerating system is unknown. The Nine Color Palaces The nine color palaces, jiu gong dd, are small squares divided into nine smaller squares, or ‘palaces’, gong d, containing the following seven names of colors: white bai d, black hei d, azure bi d, green lü d, yellow huang d, red chi d, and purple zi d, arranged in such a way that these squares are associated in a fixed way with numbers from one to nine (Table 2.5). 64 See p. 76 above. fact readily obtained from any extensive table of the Chinese calendar. 65 This 88 DESCRIPTION OF THE CHINESE CALENDAR A replacement of the colors by their associated numbers also shows that the nine corresponding squares are all different and follow a quite regular pattern: when they are listed in their order of succession attested in calendars and replaced by the number of their central square, they succeed one another in reverse order (Table 2.6). white black azure green yellow red purple bai hei bi lü huang chi zi 1, 6, 8 2 3 4 5 7 9 Table 2.5. The correspondence between colors and the central numbers of color palaces. 9 8 4 4 3 8 5 1 6 1 9 5 1 5 7 3 2 2 7 6 → 8 7 3 → 3 2 7 4 9 5 9 8 4 9 4 6 2 1 1 6 5 → 7 6 2 → 2 1 6 3 8 4 8 7 3 8 3 5 1 9 9 5 4 → 6 5 1 → 1 9 5 2 7 3 7 6 2 7 2 4 9 8 8 4 3 → 5 4 9 → 9 8 4 1 6 2 6 5 1 6 1 3 8 7 7 3 2 Table 2.6. The nine color palaces and their representative numbers (central squares). In addition, a further examination of these squares also reveals that each of them is deduced from its predecessor by first replacing their central one by nine and by subtracting one unit from all other digits and then by a series of similar subtractions. Like the sexagenary cycle, the nine color palaces are used in order to number years, months and even days. But the latter mode of enumeration has apparently been used very scarcely.66 66 To my knowledge, only four examples of days numbered by means of color palaces are documented in extant Chinese calendars. See A. Arrault 2003, p. 109. CYCLES AND PSEUDO-CYCLES 89 Enumeration of Years The beginning of the enumeration of years with color palaces is traditionally attributed to the year 604 (Renshou 4, Sui dynasty).67 The years 604, 605, 606 . . . are thus successively numbered 1, 9, 8 and so on, in reverse order. Incidentally, it also happens that the sexagenary binomial of the initial year, 604, is jiazi dd or (1, 1). The color palaces repeating themselves every nine years, the years of the Chinese calendar fall accordingly into nine categories, as the following table indicates: Palaces Years 1 9k+1 2 9k 3 9k–1 4 9k–2 5 9k–3 6 9k–4 7 9k+4 8 9k+3 9 9k+2 Table 2.7. The enumeration of years with color palaces (k is a positive or negative integer). Moreover, given that 604 mod 9 = 1, the years associated with the first color palace are such that x mod 9 = 1. Likewise, those associated with the ninth color palace are of the form x = 9k + 2. More generally, years mutually differ according to the value of x mod 9. Hence the above table. Color palaces are also linked to the sexagenary enumeration of years, thus producing 3 supra-annual periods of 60 years each: the least common multiple of 9 (number of color palaces) and 60 (number of sexagenary binomials) is indeed equal to 180 and 3 cycles of 60 years exhaust all possibilities in this respect. The first such cycle of 60 years is called shang yuan , dd (initial cycle), the second zhong yuan dd (median cycle) and the third xia yuan dd (final cycle).68 Enumeration of Lunar Months An analysis of calendars having their months numbered backwards with color palaces shows that the first months of jiazi years have an eight in the central square of their color palace. 67 This date is first mentioned in the Suishu (j. 69, ‘liezhuan 34’, p. 1611) See A. Arrault 2003, ibid., note 100, p. 109. 68 Suishu, ibid. 90 DESCRIPTION OF THE CHINESE CALENDAR From this fact and from a backwards enumeration of the following months it follows that the first month of the second and third years are respectively equal to 5 and 2, intercalary months being skipped, as usual. Consequently, as the starting year is a jiazi dd year, the first months of three successive years are always numbered 8, 5 and 2. More generally, the following table gives the enumeration of all other months: Month no. 1st Palace Year 2nd Years 3rd Year 1 2 3 4 5 6 7 8 9 10 11 12 8 7 6 5 4 3 2 1 9 5 4 3 2 1 9 8 7 6 2 1 9 8 7 6 5 4 3 8 5 2 7 4 1 6 3 9 Table 2.8. The enumeration of lunar months with color palaces represented by the number of their central square. More mathematically, this regularity comes from the fact that the least common multiple of the number of color palaces, 9, and of the number of ordinary months in a year, 12, is equal to 36 or three lunar years. Once again, the starting year of this enumeration is unknown but, for example, when starting enumerating backwards lunar months from the first color palace and from the eleventh month of the year 603, in order to obtain an enumeration starting from the same year as the one first historically used for enumerating years with them, the first month of the year 604 is numbered ‘8’. Then, the numbers of the successive years reproduce themselves indefinitely in the order 8, 5, 2 and since the year 604 is of the form 3k + 1, the first months of all years of the same form are also numbered ‘8’. Likewise, the first month of years 3k + 2 and 3k are respectively associated with the color palaces 5 and 2. The Planetary Week week69 The planetary was first introduced in China by the so-called Nestorians or, more exactly, the members of the East-Syrian Christian 69 The planetary week and its seven days is extremely ancient and is so called with reference to a conception according to which the sun and the moon, respectively associated with Sunday and Monday, are included among the planets, the antique notion CYCLES AND PSEUDO-CYCLES 91 Church (Christian community of the Sasanian world),70 in 781, and a Chinese neologism meaning Sunday was then coined for the first time: as the famous French sinologist Paul Pelliot has shown, the word yaosenwen ddd, inscribed at the end of the famous Nestorian stele discovered at Xi’an and dated to the 7th day of the 1st [lunar] month of the Jianzhong era of the Tang dynasty (Julian date: Sunday71 4/2/781) – corresponds to a Chinese phonetic transliteration from the Pehlvi evšambat72 meaning ‘Sunday’.73 As far as we know, this term is a hapax and has not been recorded in Chinese calendars handed down to us. From the tenth century of our era, approximately, Sundays of the planetary week have made their way into Chinese calendars and have been denoted by the character mi d or its homophone mi d. No matter which mi was used, the introduction of the planetary week in Chinese calendars was probably triggered by the diffusion of manicheism from Persia, mi being probably a transcription of mir, the name of the solar divinity (Mithra).74 If the mi Sunday is conspicuous, other weekdays also appear in calendars and, from comparative chronology, we know that their Julian day numbers mostly coincide with those of the corresponding weekdays used everywhere in non-Chinese regions of the world.75 of planet meaning ‘wandering star’ as opposed to ‘fixed stars’ and not ‘celestial body orbiting the sun or another star’. On the history of the planetary week outside China, see F.H. Colson 1926 and, above all, E. Zerubavel 1985. 70 See Bill M. Mak 2014, p. 105. 71 Most importantly, this date is also a Sunday in both Persian and Indian calendars. See Bill M. Mak 2014, op. cit., note 70, p. 119. 72 This term is equivalent to the modern Persian yakšambah. 73 P. Pelliot 1996 (edited by A. Forte), p. 309. 74 Numerous studies on this subject have been published. The oldest, but still valuable ones, are A. Wylie 1897/1966* and É. Chavannes and P. Pelliot 1913, p. 171 f. More recent publications such as J. Needham 1959, p. 204; Zhuang Shen 1960; S. Whitfield 1998, p. 6 and A. Arrault 2003, p. 100 are also useful. Lastly, various Chinese sources also mention this question, notably the very important Xieji bianfang shu, j. 1, p. 98–99 (notice p. 397 below). 75 A. Arrault and J.-C. Martzloff 2003, p. 100. We also note in passing that Japanese calendars from much later years (1606 and 1648) also have weekdays but the historian of the calendar T. Watanabe 1977/1984*, p. 89, has shown that their dates, deduced from their Julian day numbers, are not identical with the corresponding non-Japanese weekdays. 92 DESCRIPTION OF THE CHINESE CALENDAR In the Chinese context, however, Sundays and other weekdays are disconnected from our familiar alternation between work and rest since the Chinese week was based on ten day sequences.76 On the contrary, the association between weekdays and planets was used in order to determine the auspicious or inauspicious character of all sorts of daily activities. At Dunhuang, during the tenth century, for example, Sunday was deemed auspicious for traveling or searching for lost animals and personal belongings.77 From the testimony of the famous British Protestant missionary and sinologist A. Wylie (1815–1887), valid for the end of nineteenth century, as well as those of the French sinologists É. Chavannes and P. Pelliot (1878–1945), Sundays and other days of the planetary week were still in use in Fujian province at the beginning of the twentieth century.78 The Twenty-Eight Mansions While the planetary week has lastingly survived in South China, it has also surreptitiously interfered with the ancient Chinese system of the twenty-eight mansions79 ershiba xiu dddd, a few centuries after its 76 See Yang Lien-sheng, 1969b. Arrault 2003, p. 100. M. Kalinowski 2003, p. 237–238, also give other such details for days other than Sundays. 78 “When at Amoy [Xiamen, Fujian province] I procured a copy of the Almanac [...] the mih jih [miri dd] was certainly recorded throughout under every Sunday [...]”. See A. Wylie 1897/1966*, p. 87. The present mih jih corresponds to the pinyin transliteration miri, the mi being the same as above while ri means ‘day’. Consequently miri literally means ‘Sun day’. See also É Chavannes and P. Pelliot 1913, p. 173: “[. . . ] c’est au Fou-kien [Fujian] que, jusqu’à nos jours, le souvenir du dimanche, jour du soleil, a survécu, et sous une appellation sogdienne, or c’est au Fou-Kien même que, du XIe siècle au XIIIe siècle, nos textes historiques attestent la présence et l’importance de communautés manichéennes.” (It’s in Fujian that, until now, the memory of Sunday, understood as a Sun day, has perpetuated itself under a Sogdian appellation. But it’s precisely in this province that, from the XIth to the XIIIth centuries, our historical texts witness the presence and the importance of Manichean communities.) 79 The term ‘mansion’ comes from the Latin mansio, a term meaning ‘dwelling place’, ‘lodge’, ‘station’. In Chinese traditional astrology and astronomy, it refers to the places where certain moving celestial bodies such as the sun, the moon, planets or comets give the impression of resting temporarily. These mansions are often oddly called ‘lunar mansions’, without much basis (C. Cullen 2011); their possible connection with other stellar systems attested in India and in the Islamic world has given rise to all sorts of insoluble speculations. For a history of the subject, see J. Needham 1959, p. 242 f.; 77 A. CYCLES AND PSEUDO-CYCLES 93 first apparition in a Chinese context.80 Most interestingly, as indicated in the following table, each weekday has then been associated with four different mansions. For example, Sunday successively corresponds to Fang, Xu, Mao and Xing:81 Sunday 4 Monday 5 Tuesday 6 Wednesday 7 Thursday 8 Friday 9 Saturday 10 Fang Xin Wei Ji Dou Niu Nü d d d d d d d 11 12 13 14 15 16 17 Xu Wei Shi Bi Kui Lou Wei d d d d d d d 18 19 20 21 22 23 24 Mao Bi Zi Shen Jing Gui Liu d d d d d d d 25 26 27 28 1 2 3 Xing Zhang Yi Zhen Jiao Kang Di d d d d d d d Table 2.9. Correspondence between the days of the planetary week and the twenty-eight mansions, traditionally enumerated from Jiao d (1) to Zhen d (28) (right column). The transition from the weekday to the 28 mansions cycle does not seem to have been abrupt, however, and the two systems probably coexisted for several centuries.82 Moreover, even after the disappearance of the explicit mention of weekdays in Chinese official calendars, their underlying existence was still recognizable. Most remarkably, in the first known report on the Chinese calendar ever written in a European language, Portuguese, (August 1612, Beijing), the Italian Jesuit missionary Sabatino De Ursis (1575–1620) judiciously notes that: see also D.S. Nivison 1989 (proof of existence of two different antique versions of the twenty-eight mansions); Pan Nai 1989*/2009* (very detailed examination of the question based on all known Chinese sources); Sun Xiaochun and J. Kistemaker 1997, p. 26– 28 (ancient Chinese star catalogs); Chen Meidong 2003a, p. 67–72 (recent overview of the subject). 80 M. Kalinowski 1996 provides a careful study of the subject. 81 Tentative renderings: Fang: Chamber, Xu: Tumulus ; Mao: Pleiads ; Xing: Stars; Xin: Heart; Wei: Rooftop; Bi: Net; Zhang: Strung Bow; Wei: Horn; Shi: Hall; Zi: Beak; Yi: Wings; Ji: Winnowing-basket; Bi: Wall; Shen: Triad; Zhen: Chariot Baseboard; Dou: Dipper; Kui: Crotch; Jing: Well; Jiao: Horn; Niu: Ox; Lou: Pasture; Gui: Devils; Kang: Neck; Nü: Serving-Maid; Wei: Stomach; Liu: Willow; Di: Root. For recent critical observations on this subject see Sivin, N. 2009, p. 90–94. 82 A. Arrault 2003, p. 101 and, same author, 2004. 94 DESCRIPTION OF THE CHINESE CALENDAR “The Chinese have 28 constellations [. . . ] This is really equivalent to the number 28 of our solar cycle. [. . . ] The four constellations corresponding to the sun always fall on Sunday, those corresponding to the moon always fall on Monday, those corresponding to Mars on Tuesday, and so forth. [. . . ] Christian thus, looking at the calendar, know that the day under which there is one of these four characters [associated with the sun] is a Sunday.”83 The Jianchu Pseudo-Cycle with Reduplications The pseudo-cycle jianchu dd is so-called from its two first terms, jian d and chu d. Like many other components of the Chinese calendar, its terms are associated with the auspicious or inauspicious characters of the days they are associated with.84 Its twelve terms are: jian d (instauration), chu d (removal), man d (plenitude), ping d (balance), ding d (settlement), zhi d (stability), po d (destruction), wei d (danger), cheng d (maturity), shou d (reception), kai d (openness), bi d (closure). In his outstanding study of Turkish calendars, Louis Bazin highlights the divinatory character of this series and calls it ‘la série des douze présages’ (‘the twelve oracle series’); he also analyses the Uyghur roll manuscript ML 21, no. 11 (1202 AD) with a great wealth of details and concludes that the twelve Chinese terms have been transliterated phonetically into the Uyghur language, under the manifest forms kin, čuu, man . . . 85 The jianchu pseudo-cycle is also documented much more anciently in bamboo strips, discovered in 1975 at Shuihudi, Yunmeng, Hubei province, in the remains of a tomb sealed in 217 BC. We are dealing here with a series also beginning with the two first jianchu terms but its rules and purpose are still not well understood save that, unlike its later version, it was then also used in the enumeration of years.86 At least from the Tang dynasty, its rules of association with calendrical days have been stabilized as follows: 83 P. d’Elia 1960, p. 73. a later period (Qing dynasty (1644–1911) R.J. Smith 1992, p. 81, remarks that the auspicious or inauspicious character of these terms is not fixed once for all. See also Xieji bianfang shu, j. 4, p. 169 f. (Notice p. 397 below). 85 L. Bazin 1991, p. 286–292. 86 M. Loewe 1994, p. 215 and 220 f. (‘the system of oracles’). 84 For CYCLES AND PSEUDO-CYCLES 95 A (Starting Point) Let x be a given year. Then, the first term of the jianchu pseudo-cycle, jian d, is attributed to the first day posterior to the even solar breath q4 , lichun (Enthronement of Spring), having a sexagenary binomial equal to either #3, #15, #27, #39 or #51 (binomials of the form 3 + 12k, k = 0, 1, 2, 3, 4); B (Reduplications) Let us suppose that an even solar breath falls on a certain day n. Then, the days n and n − 1 are assigned the same jianchu term;87 C (Temporary Cyclical Character) As long as condition B is not fulfilled, the jianchu terms are enumerated cyclically. From these rules it follows that: 1. days cannot be enumerated with the jianchu terms without a prior determination of the solar breaths of the year x. Hence a fundamental difference between the jianchu pseudo-cycle and many other cycles of the Chinese calendar, such as the sexagenary cycle, whose enumeration is never affected in any way by calendrical calculations; 2. the rule A can be formulated differently by mentioning explicitly the concerned binomials – (1,3), (3,3), (5,3), (7,3) and (9,3) – instead of their ranks. In this way, the fact that all their second terms are all equal to 3 not only becomes obvious but also explains why the original Chinese rule states that the first day posterior to lichun and whose branch is the third one, yin d, is associated with jian d;88 3. the rule B refers to even solar breaths. The terms of the jianchu pseudo-cycle are thus reduplicated twelve times during an interval of time equivalent to a calendrical solar year. 87 Deng 88 Xieji Wenkuan 1998b. bianfang shu, j. 4, p. 169–170 (see p. 397 below). 96 DESCRIPTION OF THE CHINESE CALENDAR The Nayin Cycle with Reduplications The nayin dd cycle with reduplications concerns the cyclical enumeration of days and years by groups of sixty. Its name literally means ‘induced sounds’ and alludes to the following correspondence between the five notes of the Chinese scale and the five phases:89 1. Metal Jin d, note Shang d ; 2. Fire Huo d, note Zhi d ; 3. Wood Mu d, note Jue d ; 4. Earth Tu d, note Gong d ; 5. Water Shui d, note Yu d. However, this correspondence remains tacit and, in practice, the sixty elements of this new cycle are obtained by enunciating twice each phase and by establishing a correspondence with the sixty sexagenary binomials, represented by their rank, from #1 to #10, #11 to #20 and so on. Now, let us write 1, 2, 3, 4, 5 instead of Jin, Huo, Mu, Tu and Shui.90 Then #1, #2, #3, #4, . . . are respectively associated with 1, 1, 2, 2,. . . or Jin, Jin, Huo, Huo, . . . (Table 2.10 below). For instance, the first day of the first month of the year Yongle 15 (1417) is equal to (5, 1), or wuzi (#25) (see p. 314 below) and, from this table, #25 corresponds to 2, ‘Fire’. Consequently, the following days of the same month respectively correspond to 2, 3, 3, 5, 5, 1, 1, . . ., or, more explicitly Huo, Mu, Mu, Shui, Shui, Jin, Jin, and so on. Other Aspects of the Chinese Calendar Festivals and Annual Observances The Chinese calendar has a large number of festivals and annual observances that bear witness to a rich and complex history, extending over several millennia and still awaiting its historian of religion, mythology, ritual, social phenomena and politics for a global and multidimensional 89 M. 90 M. Kalinowski, 2003, p. 220–222. Kalinowski, 2003, ibid., p. 221. OTHER ASPECTS 97 presentation of the subject. In the following, we limit ourselves to the determination of the dates of the main Chinese traditional festivals.91 As usual, it is necessary to distinguish between fixed and movable festivals. In the sequel, the term ‘fixed’ designates any festival having a fixed date relatively to its lunar or solar component. Contrary to what might be expected, the following festivals and special days are not necessarily explicitly indicated in traditional calendars from our period of study. The most famous is, of course, the New Year Festival, held on the first day of the first lunar month (1/I). Documented from the Southern and Northern dynasties (420–589),92 it was called Yuandan dd for a very long historical period. In a modern context, however, this term designates the non-Chinese Gregorian 1st January while the original Yuandan corresponds to the Spring Festival, Chunjie dd, because of the proximity between the first day of the Chinese lunar year and the Enthronement of Spring, lichun, q4 . 91 For more details, outstanding works concerning specific festivals, or certain of their aspects, are already available. See, in particular, J. Bredon and I. Mitrophanow 1927, W. Eberhard 1952; Li Yongkuang and Wang Xi 1995 (general works) ; J. Gernet 1959 (last years of the Southern Song dynasty (1250–1276)); D. Bodde 1975 (Han dynasty); Zhou Yiping and Shen Shaying 1991 (dictionary); Tun Li-ch’en 1965 (customs and festivals in Peking); W.C. Hu 1991 (the New Year and its folklore); P. Welch Bjaaland 1997 (the New Year, modern times); E. Trombert 1996 (Dunhuang, socio-economical aspects); E.P. Wilkinson 2012 (all periods and all aspects of the subject, p. 524–526). 92 Li Yongkuang and Wang Xi 1995, p. 169. Binomials No. #1–#10 #11–#20 #21–#30 #31–#40 #41–#50 #51–#60 Phases No. 1 2 5 1 2 5 1 2 5 1 2 5 2 5 4 2 5 4 2 5 4 2 5 4 3 4 2 3 4 2 3 4 2 3 4 2 4 1 3 4 1 3 4 1 3 4 1 3 1 3 5 1 3 5 1 3 5 1 3 5 Table 2.10. The nayin cycle with reduplications and the correspondence between the sexagenary binomials and the five phases (denoted 1, 2 . . . 5). 98 DESCRIPTION OF THE CHINESE CALENDAR Among the most noticeable other fixed festivals with respect to lunar dates, we note, in particular, the Lantern Festival, Yuanxiao dd, (15/I); the Bathing Buddha Festival (anniversary of his birth) Yufodan dd d, (8/IV); the Double Fifth festival, Duanwu dd, so-called because it falls on 5/V;93 the Ghost festival, Yulanpen ddd, a name probably originating from the Sanskrit ullambana (hanging down) or ullampana (salvation) and having given rise to all sorts of conjectural interpretations.94 During this religious festival, held on 15/VII, people from all walks of life present offerings to Buddhist monks to gain salvation for their ancestors. The ‘Double Ninth’, Zhongyang dd, (9/IX), should also be mentioned. An exhaustive historical list would be considerably longer. Other festivals, or annual observances, occur on fixed or movable dates with respect to the solar component of the Chinese calendar. They fall into the two following categories: (a) event coinciding with a solar breath; (b) event falling either a fixed number of days after a solar breath or determined by a more complex rule, generally linked to peculiarities of the sexagenary numbering of days. For example, the ‘Tomb sweeping Festival’, Qingming dd, (‘Pure Brightness’) belongs to the first category since it coincides with the solar breath q8 , qingming dd. By contrast, the wangwang dd days (‘disparition’) belong to the second category since they occur a fixed number of days after the ‘Enthronements of the four seasons’, si li dd, that is after the four following solar breaths: q4 , q10 , q16 and q22 . More precisely, the three spring wangwang dd fall 7 days, 2 × 7 days and 3 × 7 days after the Enthronement of Spring, lichun, q4 ; the three summer wangwang fall 8 days 2 × 8 days and 3 × 8 days after the Enthronement of Summer, lixia, q10 ; the Autumn wangwang 9 days, 2 × 9 days 3 × 9 days, after the Enthronement of Autumn liqiu q16 . Lastly, the three winter wangwang falls 10 days, 2 × 10 days 3 × 10 days, respectively, after the ’Enthronement of Winter’, lidong, q22 .95 93 Other names: Duanyang d d (literally: beginning of the sunny season) and ‘Dragon Boat festival’ (boat races and competitions were held on that occasion). 94 See S.F. Teiser 1988, p. 22. 95 A. Arrault 2003, p. 106. OTHER ASPECTS 99 Similarly, the san fu dd (the three days of concealment associated with days of scorching heat or canicular days, dog days) mark the three first days initiating three decades of scorching heat. They are respectively called chufu dd, zhong fu dd and houfu dd or mofu dd, i.e., initial, median and final fu (the meaning of the term fu is explained in the next paragraph). They fall either after the Summer Solstice or after the Enthronement of Autumn.96 In both cases, the corresponding days have binomials whose first term is the trunk. geng d97 Under the Tang dynasty (618–907), for example, they coincide with three days posterior to the Summer Solstice, q13 , whose trunk is geng. More mathematically, the first term of their binomials is ‘7’. In antiquity, the san fu were associated with sacrifices of dogs as a means of prevention against epidemics, insects and various other evils triggered by the hot and humid weather of the sixth month.98 Irregular Years A systematic perusal of almost any table of the Chinese calendar reveals the existence of various sorts of irregular lunar years: the appellations of their lunar months is atypical if not aberrant; the number of their lunar months is neither equal to twelve nor thirteen; they violate the definition of intercalary months so that the coupling between odd solar breaths and lunar months is broken. By analogy with the disorders of the Roman calendar – well-known for containing several years having a number of days in severe disagreement with the length of the solar year – such years could be called ‘years of confusion’.99 Still, the Chinese 96 Chen Yongzheng 1991, p. 273. Arrault 2003, ibid. p. 104. 98 D. Bodde 1975, p. 320: “In Ch’in the warding off was done by dismembering a dog at each of the four gates of the capital”. See also the other remarks of the same author about the persistence of these sacrifices under the Han dynasty (p. 320) and the similarity between the Chinese and Western worlds of Roman times “In duration, season and association with heat, there are similarities between the three Chinese Fu [. . . ] and the ‘Dog days’ of the Western world. The latter’s dating, as far back as Roman times, was determined by the heliacal rising of the Dog Star Sirius” (p. 321). Hence a possible analogy with the Chinese case inasmuch as the Chinese character d is composed of two disjoint parts, the first one, (right part) designating a dog d and the other one, a man (left part). 99 The Roman pre-Julian calendar also has numerous other irregular years (see P. Brind’Amour 1983, ch. 2, p. 27–123). For instance, the Roman year 708 AUC (ab 97 A. 100 DESCRIPTION OF THE CHINESE CALENDAR irregularities are quite different: the Roman aberrations were the consequence of a careless management of the calendar whereas the Chinese oddities were essentially provoked by political decisions, consisting in extravagant modifications of the beginnings and ends of certain years. Without going into too much detail we now provide their complete list and an overall description of their major characteristics:100 104 BC (Former Han, Taichu 1) This year contains 15 lunar months numbered 10, 11, 12, 1, 2, . . . , 9, 10, 11, 12. Beginning with a tenth month and finishing with a twelfth, it has two tenth, two eleventh and two twelfth months, respectively located at its beginning and end. Among its fifteen months, seven are full and eight hollow. Hence a total of 442 days (7×30+8×29 = 442). Despite this out of the ordinary number of days, none is intercalary even though its first tenth month would be an ideal candidate for such a qualification since it apparently contains no odd solar breath.101 8 (Former Han, Chushi 1) This year plainly contains twelve months (six full and six hollow) and 354 days. Contrary to all logic, however, it has an intercalary month, 1*, but no twelfth month, as a consequence of Wang Mang’s decision to start the beginning of the year 9 with its twelfth month.102 23 AD ( Xin (Wang Mang), Dihuang 4) With its 13 months and 384 days, this year does not seem particularly noteworthy but its two 12th months are not determined by any intercalation process and, contrary to what could be expected, its winter solstice does not belong to its 11th month but to its first twelfth month. urbe condita), corresponding to 46 BC, was first dubbed annus confusionis ultimus (the final year of confusion) by the Vth century philosopher and philologist Macrobius in his Saturnalia (P. Brind’Amour, ibid., p. 27). According to É. Biémont 2000, p. 225, this year contains 432 days and the preceding, 707 AUC, 378 days. 100 The present list is slightly more complete than the one established by P. Hoang 1910/1968*, p. VII f., long ago. Further historical details of great interest are also given in E.P. Wilkinson 2012, p. 503–504. 101 See Zhang Peiyu 1990*/1997*’s table. 102 Wang Mang is the founder of the Xin dynasty, in 9 AD. See Wang Yuezhen 1867/1936*/1993*, j. 4, p. 12b–13a, 1936 edition (notice p. 378 below). OTHER ASPECTS 101 237 AD (Wei, Qinglong 5) No third month this time. Hence an unusually short year, containing eleven months (five full and six hollow) and only 324 days. However, its third month is missing and consequently, its last month is still the twelfth. Next, the four years 689, 700, 761 and 762 AD have a number of days significantly lower or greater than a plain lunar year, oscillating between 295 and 444 days. The years 689 and 761 are wholly contained in the same Julian year while, on the contrary, the years 700 and 762 span three such years. The first two and the last ones respectively belong to the reign of the empress Wu Zetian (reign 684–704) and of the emperor Li Heng (reign 756–762). In these four cases, the calendar does not always respect the usual appellation of lunar months and violate the rule of insertion of intercalary months: 689 AD (Yongchang 1) This year is entirely contained in the year 689 (beginning: 27/1/689, end: 17/12/689). Its number of months is the same as that of the year 237 AD already mentioned above, eleven, but with six full months and five hollow months instead. Hence its 6 × 30 + 5 × 29 = 325 days. Contrary to all logic, its intercalary month double its ninth month even though its number of lunar months is smaller than thirteen. 700 AD (Jiushi 1) Contrary to the preceding year, this one is composed of fifteen months, nine full and six hollow. Hence its 444 days (9 × 30 + 6 × 29 = 444). With such an exceptional length, this endless year begins on 27/11/699 and finishes on 12/2/701, thus extending over the three Julian years 699, 700 and 701. For once, its intercalary month, 7*, is not exceptional but the naming of its successive months is profoundly perturbed since each of its first three months seemingly correspond to a first month of the year: its first month, regularly called zhengyue dd, follows the usual practice but, on the contrary, its second month, layue d d, refers to the month in which the Winter Sacrifice was held, the corresponding day being considered in Han times as another beginning of the lunar year.103 Lastly, its third month is called 103 D. Bodde 1975, ch. 3, p. 49: “Of the five annual beginnings listed in the preceding chapter, unquestionably that known as the la was, above all others, regarded by Han 102 DESCRIPTION OF THE CHINESE CALENDAR yiyue dd ‘first month’ since it contains the Beginning of Spring, another possibility referring to a plausible beginning of the solar year this time. Designed in this way, the year 700 thus has three successive first months. Consequently, its other lunar months are enumerated from its third month so that its last month is still the twelfth although it should be the fifteenth. (Table 2.11). Months n◦ Names Meanings 1 zhengyue dd First Month 2 layue dd Winter Sacrifice Month 3 yiyue dd First Month 4 eryue dd Second Month ..................................................... 9 qiyue dd Seventh Month 10 run qiyue ddd seventh month 11 bayue dd Eight Month ..................................................... 14 qiyue ddd Eleventh Month 15 qiyue ddd Twelfth Month Table 2.11. Anomalous month names (first year of the Jiushi era (700 AD)). 761 (Shangyuan 2) At the extreme opposite, this year has only 295 days (5 × 30 + 5 × 29 = 295). It begins on 10/2/761 and ends on 1/12/761 in the same Julian year. Less baroque than the two preceding years, save of course its unlikely length, this dwarf year has no intercalary month and its months are numbered regularly so that its last month is a tenth month and not a twelfth. 762 AD (Baoying 1) This year has 14 months and 413 days obtained from the combination of 7 full months and seven hollow months (7 × 30 + 7 × 29 = 413). The Julian dates of its first and last days, respectively correspond to the following dates: 2/12/761 Chinese as being the real New Year”. To be sure, this Han feast has still been observed later. See, for instance, Li Yongkuang and Wang Xi 1995, p. 199 f. OTHER ASPECTS 103 and 18/1/763. Quite regularly too, this year has no intercalary month but its fourth and fifth months are doubled in a very original way: up to the fifth month, called ‘fifth month’, wuyue, the names of its months are regular but its two following months are respectively called siyue, fourth month and wuyue, fifth month, the succession of months in the corresponding part of the calendar being thus 4, 5, 4, 5, and giving the illusion of going back in time to those who lived in China in 762. Once again, as in the case of the year 700 above, this new naming irregularity also induces surreptitiously the notion of an overall regularity, the last month of the year being always called ‘twelfth month’. To sum up the full enumeration of its months is the following: 1 (zhengyue), 2, 3, 4, 5, 4, 5 , 6, 7, 8, 9, 10, 11, 12. Part II Calculations CHAPTER 3 NUMBERS AND CALCULATION Modes of Representation of Numbers Convinced of the inherent limitation of predictive mathematics, the Chinese have not easily associated their astronomical canons with closed and immutable systems. In particular, they have exploited ever changing numerical patterns, not only with respect to their numerous astronomical canons but also from the more limited standpoint of the specific purposes of calculations concerning a given astronomical canon. They have thus not necessarily deemed identical number systems concerning specific aspects of the solar and lunar component of a given astronomical canon and, for instance, seasonal indicators and moon phases, have often been attributed their own modes of representation of numbers. Yet, as the observation of an extended corpus of specific examples1 clearly shows, these representations all follow a very general mathematical pattern, based on the idea of the division of the day j (or any other main unit) into b1 parts themselves subdivided into b2 parts and, when appropriate, into b3 , . . . , bi parts again. Hence sequences of submultiples j1 , j2 , . . . ji of j such that: j1 = j , b1 j2 = j1 , b2 ..., ji = ji−1 . bi (3.1) Let a0 , a1 , a2 , . . . be a sequence of integer coefficients. Then, for this system, the general expression of any duration, t, corresponds to: t = a0 j + a1 j1 + a2 j2 + . . . + ai ji . 1 Appendixes (3.2) F and G below. © Springer-Verlag Berlin Heidelberg 2016 J.-C. Martzloff, Astronomy and Calendars – The Other Chinese Mathematics, DOI 10.1007/978-3-662-49718-0_3 107 108 NUMBERS AND CALCULATIONS In practice, however, similar expressions, such as the following, are also attested: t = a0 j + a1 j j j + a2 + a3 +... b1 b1 × b2 b1 × b2 × b3 (3.3) or a1 a2 a3 + + +... (3.4) b1 b1 × b2 b1 × b2 × b3 Quite often too, some units ji are missing but, as a rule, the total number of such units rarely exceeds three. Moreover, when non-temporal units of interest in positional astronomy (mostly angular distances but also ordinary lengths in the case of gnomon shadows) are involved, the same irregularities also widely occur. In all cases, however, the numerical representations of the coefficients ai are always decimal, exactly like the quasi-totality of numbers attested in all sorts of Chinese texts, technical and non-technical alike. These representations are thus somewhat similar to those used in our notations for hour, minutes and seconds or angles. Beyond the Chinese case, numerical representations based on the same principle frequently occur in antique and medieval mathematics and astronomy. For instance, in the Hebrew calendar, the day is decomposed into 24 hours, divided in their turn into 1080 h.alakim and, once again, into 76 regaim.2 In the famous Liber Abaci, the same technique gives rise to a variety of complex fractional representations concerning monetary units, a large part of this famous treatise revolving around commercial arithmetic.3 Such examples could be easily multiplied. Despite the inherent variability of these representations, their general structure do not exclude regular patterns however: when the bi all have the same value, 10, 60, or 100, for instance, we obtain decimal, sexagesimal or centesimal number representations as particular cases of the above general decomposition 3.3. t = a0 + 2 U.C. Merzbach 1983, p. 24; U. Bouchet 1868, p. 232. N. Dershowitz and E.M. Reingold 1997, p. 87. 3 Very numerous examples of such representations are provided in L.E. Sigler 2003. More generally, J. Tropfke 1980, p. 113 and 114, offers a systematic inventory of such representations. MODES OF REPRESENTATION OF NUMBERS 109 Regular sexagesimal and centesimal divisions of a main unit are attested in medieval China, but mostly in the limited case of translations of foreign works. The sexagesimal division first occurs in an astronomical handbook, the Jiuzhi li ddd, (The ‘Nine Upholders’),4 adapted into Chinese from Indian sources at the beginning of the eighth century AD.5 Subsequently, the same technique also appears in the Huihui li ddd (Muslim Astronomical Canon), a practical handbook of Islamic origin, mainly composed of astronomical tables and translated into Chinese during the Ming dynasty.6 Eventually, the sexagesimal system replaced previous Chinese traditional modes of representation of numbers, in most astronomical and mathematical works of European origin adapted into Chinese by Jesuit missionaries, during the XVIIth and XVIIIth centuries.7 In these three cases, the diffusion of a technique of Greek origin, itself influenced by a much more ancient Babylonian technique, is at stake. In its turn, the centesimal division occurs in the table of the solar equation of the unofficial Futian li ddd, an atypical canon of unknown origin, compiled between 780 and 783, as already noted.8 No less importantly, centesimal systems of Indian origin also reached China during the Tang dynasty.9 A few centuries later, the centesimal division was also widely used in the two last traditional Chinese official canons, the Shoushi li ddd (1281–1384) and the Datong li ddd (1364– 1644). While this important feature of Chinese astronomical canons 4 This rendering of the Sanskrit term navagraha refers to real and imaginary astrological entities endowed with malevolent powers, the graha. They include the five classical planets, the sun, the moon and two other bizarre celestial entities associated with the nodes of the moon, in relation to eclipses. 5 The text of the Jiuzhi li is included in the j. 104 of the Kaiyuan zhanjing For further details about this important astrological source, see K. Yabuuchi 1963a/1988*; M. Yano, 1992 and 2004; J.-C. Martzloff 1997*/2006*, p. 100 and 207. See also footnote 54, p. 81 above. 6 See Chen Jiujin 1996; Ma Mingda and Chen Jing 1996; M. Yano 1999; Benno van Dalen 1999, 2000, 2002a and 2002b. 7 Most such tables rely on the sexagesimal numeration system. See, for instance, Pan Nai 1993. 8 See p. 22 above. See also S. Nakayama 1964, p. 62; Qu Anjing, Ji Zhigang and Wang Rongbin 1994, p. 291. 9 See J.-C. Martzloff, ibid., p. 97. 110 NUMBERS AND CALCULATIONS has not been too often taken into account by contemporary historians of Chinese astronomy, it has already been rightly underlined several centuries ago by Antoine Gaubil (1689–1759). In particular, the famous pioneer historian of Chinese astronomy provides a clear transcription of an astronomical table from the Shoushi li ddd, where the different centesimal orders of units are mutually differentiated by regrouping their decimal digits into pairs and by using the symbols ′ and ′′ for the Chinese centesimal units fen d and miao d (Fig. 3.1 below), which are equivalents of the grade and centigrade, introduced well after him during the French Revolution, in a more general mathematical context. Figure 3.1. Gaubil’s full awareness of the centesimal character of the representation of astronomical quantities in the Shoushi li (in this extract, the French opening sentence means ‘Supposedly known numbers in Guo Shoujing ddd’s methods’. Then, various astronomical constants, respectively corresponding to the sidereal year, the synodic month, its fourth part and its half are listed. In each case, the Chinese names of units are omitted and the equivalence 1 day = 100 ke d is assumed. See É. Souciet 1732, tome 3, p. 69 (from the copy preserved at the Bibliothèque nationale de France, V6362). Here, Gaubil’s Chinese source is Yuanshi, j. 54, ‘li 3’, p. 1192. More generally, on Gaubil’s works on Chinese astronomy see the notice on p. 404 below). Such regular modes of representations of numbers – or rather quantities – are not necessarily frequently documented in Chinese sources, however. For instance, decimal fractions occur very rarely in Chinese astronomical canons. The universally admitted idea of their absolute MODES OF REPRESENTATION OF NUMBERS 111 predominance in Chinese mathematics is thus inexact.10 Nonetheless, mixed representations, half-decimal, half-centesimal, are attested. In a table from a Song astronomical canon,11 for instance, meridian gnomon shadows are expressed in zhang d, chi d, cun d, fen d and xiaofen dd, each of these units being equal to ten times the preceding save the last one which is subdivided into one hundred parts (1 fen d = 100 xiaofen dd). See Fig. 3.2, p. 113 below. Yet, as the following representative examples show, most number representations are irregular. Example 3.1 The Dayan li ddd (729–761) numerical system. The Dayan li divides the day into 3040 parts determined by numerological considerations: as explained in this famous astronomical canon, 3040 is the result of sequences of arithmetical operations, introduced in the Yijing, and justified with reference to symbolic correlations between numbers, the Heavens, the Earth and divinatory rods: on the one hand, 1200 ÷ 4 = 300 ; 300 × 10 = 3000, 5 × 8 = 40 and 3000 + 40 = 3040; on the other hand, 2 × (1 + 2 + 3 + 4 + 5)(6 + 7 + 8 + 9 + 10) = 1200, where 1, 2, 3, 4, 5 and 6, 7, 8, 9 and 10 are the numbers of the Heavens and those of the Earth, respectively. In addition, the divisor ‘4’, appearing in the division 1200 ÷ 4, is said to come from a subdivision of a set of divinatory rods into four equal groups for divinatory purposes.12 Its solar year, Y , is thus attributed the unexpected value 1,110,343 = 3040 743 d 365 3040 and its further subdivisions –respectively determined by its so10 This widespread idea is the consequence of the limitation of research into Chinese mathematics of the ‘Nine Chapters’ variety – another name for logistics –, a restricted domain where this conclusion is unquestionably exact. It remains important to note, however, that logistics was deemed an elementary domain of knowledge in ancient China, the sort of mathematics really perceived as advanced being its predictive variety, abundantly developed in astronomical canons. Significantly, the Songshi (j. 68, ‘lüli 1’, p. 1493) explains that mathematics such as that of the ‘Nine Chapters’ belong to ‘elementary knowledge’ xiao xue dd. The great importance granted to predictive mathematical astronomy in Chinese histories and, at the same time, the scant attention given to other forms of mathematics in the same fundamental sources is an eloquent witness of Chinese priorities in this respect. 11 Songshi, j. 76, ‘ lüli 9 ’, p. 1765. 12 Xin Tangshu, j. 27A, ‘li 3a’, p. 588 and Chouren zhuan, j. 14, p. 163–164 (notice p. 391 below). 112 NUMBERS AND CALCULATIONS lar periods, phases of domination of Yijing13 hexagrams, seasonal indicators and the five phases14 – are expressed as indicated in the following table: Div. of the Sol. Year Original Formulations 1 Y 24 1, 110, 343 3040 × 24 2 Y 60 1, 110, 343 3040 × 60 3 Y 72 4 Y 120 15 d 664 d 7 15 yu 664 miao 7 Meanings 15 + 664 7 + 3040 3040 × 24 6 d 265 d 86 6 yu 265 miao 86 6+ 265 86 + 3040 3040 × 120 1, 110, 343 3040 × 72 5 d 221 d 31 5 yu 221 miao 31 5+ 221 31 + 3040 3040 × 60 1, 110, 343 3040 × 120 3 d 132 d 103 3 yu 132 miao 103 3+ 132 103 + 3040 3040 × 120 Table 3.1. Various subdivisions of the solar year in the Dayan li and their numerical representations, where the coefficients, 15, 664, 7, . . . , follow the usual Chinese decimal numeration system. In this example, the day is first divided into b1 = 3040 parts and b1 into b2,1 = 24, b2,2 = 120 and b2,3 = 60 parts, successively. Hence, a j unit of time of first order, j1 = 3040 , and three others of second order j1 j1 j1 j2,1 = 24 , j2,2 = 120 , and j2,3 = 60 . Moreover, j1 , is called yu d, a term meaning ‘remainder’, while the three other units, j2, i , i = 1, 2, 3, are given exactly the same name, miao d, ‘second’, although they are distinct.15 Therefore, what they stand for cannot be ascertained in the abstract. Most astronomical canons from all periods contain a tremendous variety of similar examples and such peculiarities are representative of a general trend. The great variability of these units suggests that they are not governed by any regular principle. Such is not the case, however, because an analysis of the underlying arithmetical structures shows that the numerical expressions listed in the above Table 3.1, for example, can be ob13 Famous divination classic (The Book of Changes). Tangshu, j. 28A, ‘li 4a’, p. 638–639. These various notions are introduced in the supplementary volume of the Grand dictionnaire Ricci de la langue chinoise (COLL. 2001b), p. 324–331. 15 Seconds also occurs everywhere in Greek, Arabo-Persian and Indian astronomical treatises but their value is determined once for all. 14 Xin MODES OF REPRESENTATION OF NUMBERS 113 Figure 3.2. The lengths li , i = 0, 1, 2 . . . of meridian shadows of a gnomon, tabulated day by day over a whole solar year, from the winter solstice (churi dd or initial day, i = 0) to the next winter solstice. The first row lists the first differences of the second. For instance, l3 = 1.28327, l4 = 1.28192 and l3 − l4 = 0.00135 (the main unit of length is the zhang d and its decimal submultiples are the chi d, the cun dand the fen d; the last unit, xiaofen dd (literally ‘small fraction’), is centesimal). The first difference, 0.00135, corresponds to 1 fen, xiaofen 35 and l3 = 1.28327 to 1 zhang 2 chi 8 cun 3 fen, xiaofen 27. The name of the centesimal unit, xiaofen, is placed in front of the quantity it concerns contrary to what happens with decimal units. Source: Songshi, j. 29, ‘lüli 9’ (For a reproduction of the original, see COL-astron, vol. 3, p. 1092). 114 NUMBERS AND CALCULATIONS tained quasi-mechanically from predetermined sequences of arithmetical operations. For instance, the above expressions can be obtained as follows: Y 24 = 15 + 15,943 3040×24 = 15 + ( 15,943 24 ) Y 60 = 6+ 15,943 3040×60 = 6+ ( 15,943×2 120 ) Y 72 = 5+ 15,943 3040×72 = 5+ ( 15,943 72 ) Y 120 15,943 = 3 + 3040×120 = 3+ 3040 3040 3040 ( 15,943 120 ) 3040 = 15 + 7 664+ 24 3040 664 = 15 + 3040 + = 6+ 86 265+ 120 3040 265 86 = 6 + 3040 + 3040×120 , = 5+ 221+ 31 72 3040 221 = 5 + 3040 + = 3+ 132+ 103 120 3040 132 103 = 3 + 3040 + 3040×120 . 7 3040×24 , 31 3040×60 , Therefore, the units j2,i so obtained do not correspond to any instrumental unit of time. Rather they constitute what might be called operational units, i.e. units whose values are determined by arithmetical operations. The fact that the above unit of the first order is called yu, a term meaning ‘remainder’, reinforces this interpretation because the term remainder designates the result of some arithmetical operation and not a metrological unit, even though various kinds of remainders should be differentiated: Chinese remainders are not necessarily restrictively associated with the remainder of a subtraction or a division for they also possibly refer to some other output of an arithmetical operation. For instance, even the quotient of a division is sometimes assimilated to a sort of remainder. For instance, the coefficients 664, 265, 221 and 132, appearing in the above sequences of arithmetical operations, are ‘remainders’ of the division of 15,943 (or the double of this dividend in the second case) by 24, 120, 72 and 120, respectively. Moreover, these operational units always precede the numbers they are associated with in all cases. By contrast, ordinary metrological units are postposed, when they exist. Example 3.2 Interval between consecutive mean lunar phases in the Jingchu li ddd (237–451). MODES OF REPRESENTATION OF NUMBERS 115 16 In such a case, the length of the lunar month is equal to 134,630 4559 days and the mean interval of time P between two consecutive lunar phases j is expressed by means of three units, the day j, j1 = 4559 and j2 = j21 ,17 as follows: Original Formulation dd 7 dd 1744 dd 1 dayu 7 xiaoyu 1744 xiaofen 1 Meaning 7+ 1744 1 + 4559 4559 × 2 Table 3.2. Interval between consecutive mean lunar phases in the Jingchu li. As before, a series of arithmetical operations also leads mechanically to such a decomposition: ( 134, 630 = 7+ 4559 × 4 3489 2 4559 ) 1744 × 2 + 1 1744 1 2 = 7+ = 7+ + . 4559 4559 4559 × 2 Here, the new time units called dayu dd, xiaoyu dd and xiaofen d d (‘great remainder’, ‘small remainder’ and ‘small part’, respectively) result from the following successive divisions: 7 is the quotient obtained when dividing 134,630 by 4 × 4559; likewise, 1744 is obtained by dividing 3489 by 2 while the final ‘1’ is the remainder of the same division. Obviously, the above remark on the operational nature of this kind of unit also apply to this example. Example 3.3 Interval between consecutive mean lunar phases in the Daye li ddd (597–618). The lunar month of this astronomical canon, contains 33,783 1144 days and, as indicated in the following table, the length P of any interval between 16 The precise origin of this unlikely denominator (why ‘4559’ rather than, say, 4560 or 5000?) is unknown but numerological manipulations, such as those introduced above in the case of the Dayan li, are certainly at stake. Anyway, it is not difficult to check that this unexpected fraction correctly represents an approximation of the length of the lunar month. 17 Jinshu , j. 18, ‘lüli 3’, p. 541. 116 NUMBERS AND CALCULATIONS its consecutive moon phases requires three units, j (the day), j1 = and j2 = j41 : j 1144 Original formulation Meaning (3) dd 7 dd 437 d 437 7+ + 4 dayu 7 xiaoyu 437 tai 1144 1144 Table 3.3. Interval between consecutive mean lunar phases in the Daye li. As in the two previous examples, the decomposition of the fraction follows from a sequence of mechanical calculations: 33,783 1144 ( 33, 783 = 7+ 1144 × 4 1751 4 1144 ) 3 3 4 = 7 + 437 + . 1144 1144 1144 × 4 437 + = 7+ Once again, this decomposition of P conforms to the Chinese original,18 modulo a small irregularity: a single Chinese character, tai d, which is not a unit, designates synthetically the fraction 3/4 exactly in the same way as the two characters shao d and ban d sometimes represent 1/4 and 1/2, respectively.19 Although such small irregularities occur from time to time, the mode of representation of numbers used in Chinese astronomical canons have tended to become more and more regular with time. Sometimes, however, as the next example shows, this increased regularity does not exactly correspond to what we would expect in this respect. Example 3.4 Fractions reduced to the same numerator. As Table 3.4 below shows, the fractions representing the solar year and the lunar month in four astronomical canons from the sixth century AD – the Zhengguang li ddd (Orthodox Brilliance Canon), the Xinghe li ddd (Ascendant Harmony Canon), the Tianbao li dd d (Celestial Preservation Canon) and the Tianhe li ddd (Celestial Harmony Canon) – are formally, but not arithmetically, more regular 18 Suishu, 19 The j. 17, ‘lüli 2’, p. 437. same passage of the Suishu explicitly mentions these possibilities. MODES OF REPRESENTATION OF NUMBERS 117 than those of older astronomical canons in the sense that their respective numerators are identical. Astronomical Canons Zhengguang li ddd Xinghe li ddd Tianbao li ddd Tianhe li ddd Solar Years Lunar Months 2,213,377 6060 6,158,017 16,860 8,641,687 23,660 8,568,631 23,460 d d d d 2,213,377 74,952 6,158,017 208,530 8,641,687 292,635 8,568,631 290,160 d d d d Table 3.4. Lunar and solar fractional constants with identical numerators and different denominators. Although the form of these fractions runs counter to the simplicity of calculations, its numerological appeal must have been great: a numerological regularity is still a regularity and at last, despite the non-linear character of the trend towards simplifications, it is a fact that, during the Tang dynasty (618–907), fractions associated with the fundamental solar and lunar periods have eventually been limited to those having the same and unique denominator. Later, mainly during the Song dynasty, this essential arithmetical simplification was never called into question and, at the same time, the coefficients bi defined above also tended to become less and less irregular. At the same time, subdivisions into 100 or 10,000 parts of certain units of interest in calendrical calculations have appeared more and more often.20 Finally, in the two last Chinese traditional astronomical canons (Shoushi li ddd (1281–1384) and Datong li ddd (1368– 1644)), the expression of time has essentially relied on a centesimal system based on the following equivalences: 1 day = 100 ke d, 1 ke = 100 fen d, 1 fen = 100 miao d. In practice, however, the numerical expressions really used are not as regular as these equivalences indicate because, in practice, some units 20 Examples of subdivision into 100 parts: Appendix G (table of lunar constants), items 23, 24, 26. Examples of subdivisions into 10,000 parts: items n◦ 34, 36, 39, 40, 41, 42, 44, 47. 118 NUMBERS AND CALCULATIONS are sometimes missing. For instance, the length of the lunar month21 is likewise equal to 29.530593d (decimal notation) in these two canons but both express this quantity in the two following different ways, where the day (ri d) and the ke (d) are missing each time: 1. dddddddddddddd22 2. ddddddddddddddd23 ( d = 2; d = 10; d = 9; d = 10,000; d = 5; d = 1000; d = 3; d = 100; d = 0; d = 5; 1 fen d = 10−4 days; ddd = 93; 1 miao d = 10−6 days.) With wholly regular notations, we should have had in both cases: ddddddddddddddd or 29 days (ri d) 53 ke (d) 05 fen (d) 93 miao (d). Moreover, the second expression also witnesses a very important innovation: a written zero in the form of a small circle. Various Zeroes, Non-Written and Written Far from being limited to the case of the Datong li ddd, the written zero-circle, O, is also fairly common in earlier Chinese mathematical treatises from the end of the Song and the beginning of the Yuan dynasties. In fact, its earliest known (and phletoric) occurrence is attested in the famous Shushu jiuzhang dddd (Computational Techniques in Nine Chapters),24 dated 1247. From then on, or slightly earlier, China has thus been aware of the existence of the universal zero, represented by a symbol similar to the nine other digits of their henceforth decimal and positional system of numeration. As far as we know, however, this novelty could not have been very ancient in China because, as stated in the preface of the same treatise, “in old books we find empty places, none of them uses a circle”.25 Therefore, despite the lack of precision of this testimony, the sudden presence of the zero-circle in China seems dd and shuoce dd, respectively. j. 35, ‘li 5’, p. 687. 23 Yuanshi, j. 54, ‘li 3’, p. 1191. 24 Notice, p. 398 below. See also U. Libbrecht 1973, p. 69. 25 See U. Libbrecht, ibid., p. 69. 21 shuoshi 22 Mingshi, VARIOUS ZEROES 119 like a sort of deus ex machina, a mathematical miracle born of nothing. That is why a more or less plausible reconstruction of events leading to the apparition of a written zero has been propounded. By relying on an interpretation of terse explanations, recorded in the famous Sunzi suanjing dddd (The Mathematical Canon of Master Sun),26 mainly, and concerning the way arithmetical operations with counting-rods were performed, the fully decimal and positional character of numeration systems used in China prior to 1247 has been tentatively asserted in the following form: prior to the Song dynasty (960– 1279), the Chinese constantly used a non-written an purely operational form of zero, consisting in a void space inserted between counting-rods when certain decimal units were missing, in order to ease the non-written practice of arithmetical operations. Still, an analysis of extant written notations of numbers attested in an authentic arithmetical manuscript from the Dunhuang caves, and clearly related to the same Sunzi suanjing, leads to a quite different appraisal of the question since void spaces can in no way be interpreted as referring to zeroes in such a case.27 Even with such a counterexample, and even without any zero symbol, some notion of zero is not necessarily wholly excluded in all circumstances. For instance, contrary to all expectations, a zero exists and is named as such in the writings of Dionysius Exiguus (Denys the Little), a famous monk from Scythia: “ The frequent complaint [. . . ] that [this monk] was unfamiliar with the concept of zero has no basis in the facts. The Easter table of Dyonisius begins with a new moon [. . . ]. In some tables, such a new moon was designated as the 30th day of the lunar cycle. Dyonisius, however, counts it as zero (nulla). The lack of a symbol for zero in Roman numerals and in the Greek alphabetical system of numbering does not mean that the ancients had no notion of the concept.”28 26 The Sunzi suanjing (fifth century AD very approximately) is one of the manuals included in the important Suanjing shishu dddd (The Ten Mathematical Manuals) collection, a collection giving access to the most essential Chinese mathematical sources handed down to us from the Han to the Tang dynasties. For a complete English translation of this work, see Lam Lay Yong and Ang Tian Se 1992*/2004*. 27 J.-C. Martzloff 1997*/2006*, p. 204–207. 28 A.A. Mosshammer 2009, p. 33. 120 NUMBERS AND CALCULATIONS Likewise, in some Chinese astronomical tables from the Tang dynasty, the initial day of the moon, for instance, is called chu d (initial [day]).29 In such cases, we are confronted with what might be called an ordinal zero.30 However, the knowledge of the existence of a symbol for zero, in one form or another, does not guarantee as such its acceptance. As the already mentioned Chinese translation of an Indian astronomical handbook shows, the Jiuzhi li31 ddd from the beginning of the Tang dynasty, some Chinese were certainly aware of the existence of a written zero represented in the form of a dot (bindu). We do know, however, that this innovation and the written arithmetic that goes with it, with its nine written decimal digits other than the zero-dot, was judged negatively by Chinese specialists: “Their calculation techniques [i.e. those of the Indians] depend on writing skills and do not rely on counting-rods. They are so involved and confused than they lead to the sought results only by chance. Therefore, they cannot become the basis of [a new] method”. ddddddd ddddd dddddddddd dddddd32 With the advantage of hindsight, this negative judgment seems somewhat unwise. However, when we think about the very slow and difficult acceptance of written calculation in Renaissance Europe, more than eight centuries after the Jiuzhi li, it seems hardly surprising. As the influential French historian Lucien Febvre (1878–1956) once pertinently remarked, written calculations that appear so convenient and easy to us seemed stupendously difficult for sixteenth century men and reserved for mathematical elites.33 In a passage devoted to the lack of adequate tools and of a scientific language during the Renaissance, he also explains that, in Europe, calculations still relied currently on counters until the end of the eighteenth century, long after the introduction of Hindu-Arabic numerals. On this specific point, this admirable work is 29 See, for instance, Xin Tangshu, j. 28A, ‘li 4a’, p. 648 (anomalistic lunar month); Ibid., j. 30A, ‘li 6a’, p. 646 (solar table); j. 30B, ‘li 6b’, p. 793 (non-exhaustive list). 30 This notion was first introduced by the linguist A. Cauty. See, for instance, A. Cauty, 2012, p. 33. 31 See p. 109 above. 32 Xin Tangshu, j. 28B, ‘li 4b’, p. 692. 33 Lucien Febvre 1947, p. 424. VARIOUS ZEROES 121 more enlightening than many histories of mathematics. Why should we not regard more positively the fact that written calculations were perceived in such a way in Tang China, where efficient non-written methods of calculation were currently available and operational? Still, the emergence of written calculations and of written zeroes refers to quite different modes of appraisal of number systems and calculations. When arithmetical operations are performed with calculating instruments, as was the case in China before 1247, the different orders of units of a given number are often bound to be physically distinct from each other. With counting-rods, for instance, the successive digits of numbers were generally allocated to specific columns, mutually differentiated by their relative positions. Likewise, in the case of the Roman abacus, the digits of a number took the form of small balls devoted straightaway to some grooves concerning tenths, hundredths, and so on. More generally, in all such cases, calculations often followed implicitly the principles of positional numeration without any need for the calculator to be fully aware of this fact since the values of the various digits of a number were often constrained by the physical structure of his calculating instruments. By contrast, when the results of calculations obtained by means of calculating instruments are written down, nothing necessarily ensures a positional differentiation between their different orders of units. Rather, the linguistic expressions of numbers, with or without further indications taken from the overall context, are generally liable to provide clues in this respect and such is still the case in modern languages when numbers are described in ordinary words. It is thus absolutely necessary to distinguish between the practice of arithmetical operations on the one hand and the writing of numbers on the other hand. In this latter respect, the study of Chinese astronomical tables reveals the existence of a written zero, taking the form of the Chinese character kong d (void).34 More precisely, the earliest known occurrence of this zero is attested in some astronomical tables of the Dayan li dd d (729–761)35 and, soon afterwards, in the already mentioned solar long time ago, in 1949, the historian of Chinese mathematics Yan Dunjie dd d (1917–1988) published an important but forgotten article where he accumulates terse but pertinent remarks on Chinese notations of numbers, including the kong zero. See Yan Dunjie 1947. 35 Xin Tangshu, j. 28A, ‘li 4a’, p. 657, ‘li 4b’, p. 667 and 677. 34 A 122 NUMBERS AND CALCULATIONS table of the Futian li ddd, in order to indicate the absence of centesimal seconds.36 Later, it regularly appears in practically all subsequent Chinese solar, lunar or planetary tables, from the Song and Yuan periods and, despite the fact that the character kong d is quite common,37 it apparently does not occur elsewhere, save, as will be explained more precisely soon, in late mathematical and astronomical texts from the end of the Song dynasty. The lasting limitation of the kong d zero to astronomical tables is puzzling but it perhaps betrays its origin38 because, in general, tables certainly have a very particular layout, quite distinct from those of other textual sources containing numbers, generally bound to display linear sequences of Chinese characters, irrespective of their numerical or nonnumerical character. In the case of tables, the immediate legibility of the large quantity of numbers they necessarily contain is fundamental for their users whereas, at the same time, the amount of space they devote to each individual number is always drastically limited. Therefore, such a requirement cannot be fulfilled in the same way as in ordinary texts, where it always remains possible to provide verbal explanations, if need be, and where the overall context is naturally much less limited. An analysis of astronomical tables shows that, when they contain a character kong d, then: 36 From the reproduction of the Futian li in COL-astron, vol. 1, p. 137. character kong d occurs in all sorts of domains associated with the notions of hollowness, emptiness or absence, not necessarily having something to do with numbers. In traditional Chinese medicine, for instance, it refers to inner cavities, in Chinese Buddhist sutras, it designates the void, or the vacuity, and is an equivalent of the Sanskrit sunya a polysemic term not only endowed with numerous metaphysical connotations but also a possible name of the Indian zero. Hence a potential connection with India in this respect. So far, however, not a single occurrence of kong d as an equivalent of sunya has been discovered anywhere in places where it could be expected to occur, such as Chinese mathematical works or even in various other sources such as, for example, the monumental Taishō shinshū daizōkyō ddddddd (modern revised Japanese edition of Chinese Buddhist sources edited during the Japanese Taishō dynastic era (1912–1926)). This huge collection is in no way limited to metaphysical questions: as the huge and very useful indices of this monumental work eloquently witness, it also relies on a great wealth of numerical notions and even mentions some Chinese astronomical canons. 38 See p. 128 below. 37 The VARIOUS ZEROES 123 1. a given cell of such a table contains no number while other cells, attributed to the same function, contain non-zero numbers. In this quite common case, the character kong d occurs alone in its cell (See Fig. 3.3, p. 124 below); 2. some coefficient of a given unit is missing and other units are left out due to lack of space. For example, expressions such as kong du wushi’er ddddd are often met with (literal meaning: ‘zero du [degree]39 52’)40 or qishiwu du kong dddd d (literally 75 du [degrees] zero).41 However, the character kong d appears almost always in initial or final position but almost never in intermediary positions; 42 3. no arithmetical operation is necessary while in contrast, numbers contained in parallel cells of the same table require an addition, a subtraction or some other operation. For example: ji kong d d (literally ‘the sum amounts to zero’ jia kong dd (‘add zero’ or ‘add nothing’) jian kong dd (‘subtract zero’ or ‘subtract nothing’) (Fig. 3.3). In the first case (a cell containing no number), kong d only means ‘nothing’ and cannot normally be regarded as a number. However, insofar as many cells of the same table have the same function as the cell in question and contain non-zero numbers, always dealt with in the same way, it appears to be a sort of proto-zero by virtue of the parallelism of the situation. In the second case, kong refers to a precise metrological unit, explicitly named.43 39 As a rule, the degrees in question are neither sexagesimal nor decimal. On the contrary, they generally depend on more complex systems of units such as those introduced above. Nevertheless, we make no attempt to explain the intended numerical systems in each case for we are only interested in the question of zero. 40 Songshi, j. 80, ‘lüli 13’, p. 1896 (table of Venus). 41 Xin Tangshu, j. 28A, ‘li 4a’, p. 649 (Moon table). 42 It seems that the only examples of intermediary zeroes are those recorded in the Songshi, j. 83, ‘lüli 16’, p. 1961 and 1963–1967 (solar inequality table). 43 For further remarks on the semantics of zero, see B. Rotman 1987’s interesting remarks. 124 NUMBERS AND CALCULATIONS Lastly, in the third case, kong is not identical with a void space since it occurs in arithmetical operations like an ordinary number: in expressions such as jia kong dd or jian kong dd it can be replaced by any number. Figure 3.3. Astronomical table, devoted to the motion of Venus (Taibai dd) with cells containing ‘jia kong dd’ (‘add nothing’), ‘jian kong dd’ (‘subtract nothing’) or isolated kong d zeroes (“nothing”). From Xin Tangshu, ‘li zhi’, j. 20A (for a reproduction of the original, see COL-astron, vol. 3, p. 719). Therefore, these two expressions are not formally distinguishable from expressions such as jia yi dd, jia er dd (add one) or jian sanshi qi dddd (subtract thirty seven) involving non-zero numbers. Once again, by virtue of the parallelism of these examples, kong is a sort of zero and recalls the well-known Indian examples where zero is dealt with as a full-fledged number, submitted to arithmetical operations, even though the complexity of the latter is much greater since it also THE ZERO-CIRCLE 125 involves the formulation of specific arithmetical rules.44 The Astronomical Zero and the Zero-Circle Quite unexpectedly, the kong d zero appears to be closely connected with the zero-circle (O), attested for the first time in Qin Jiushao’s Shushu jiuzhang dddd. Indeed, in the astronomical part of this famous mathematical treatise,45 a zero-circle, O, is explicitly called kong d and, as shown in one of its numerous computational diagrams (Fig. 3.4 (a), p. 127 below), the reader is asked to set down a kong d below a ‘celestial unit’, denoted |, that is, a unit in the system of rod-numerals (dddddd ddddd).46 Despite the fact that this by now familiar O is clearly called kong d, this example is probably not fully convincing because it relies on a very late reconstitution and reprint of the Shushu jiuzhang dddd and not on its original version (1247) or, at least, one of its early printed versions. Fortunately, however, another considerably more ancient extant Chinese mathematical text leads exactly to the same conclusion. The text in question is the Suanxue qimeng dddd (An Introduction to Computational Techniques) and, although its original, supposed to have been first released in 1299, is similarly no longer extant, like all Chinese mathematical books from the same period, one of its early Korean reprints, published with movable types during the fifteenth century, has nevertheless reached us.47 not only in the form of a plain text, but with fully explicit diagrams attached to three problems also of interest in 44 J. Tropfke 1980, p. 142 f. provides adequate and numerous references to the subject. The most ancient known example of an arithmetical use of a zero appears in a work of Brahmagupta, born in 598 (ibid.). 45 Shushu jiuzhang, j. 1, problem 1 (problem I-1). 46 The present analysis is restricted to what strictly concerns zero. Detailed explanations concerning the meaning of the so-called celestial unit and of the underlying mathematics are easily available. See, for instance, U. Libbrecht 1973, p. 388–391; Li Yan and Wang Shouyi 1992, p. 39–49 (modern Chinese translation of the problem in question with its solution). 47 A full reproduction is available in A. Kodama 1966. 126 NUMBERS AND CALCULATIONS the history of the zero-circle in China, in relation to the kong d zero.48 One of these49 states that “four, five and six feet (chi d) of red (hong d), green (qing d) and yellow (huang d) floss silk are respectively worth 300 coins (wen d) plus one foot of red, green and yellow floss silk, respectively” and asks the price of one foot of each piece of fabric. As usual, the solution depends on the square array method (fangcheng shu ddd) but a special diagram of the Suanxue qimeng dd dd, displaying a number of peculiarities unseen in earlier times, is inserted into the main text of the problem after having been introduced with the following words: dddddd, meaning “procedure: arrange the counting-rods according to the following diagram” (Fig. 3.4, next page). Here, written representations of rod-numerals are freely used and we note that, for instance, |||| and || respectively mean 4 and −1 (Fig. 3.4, p. 127, diagram (b), right column). More generally, the principles underlying this system of rod-numerals are the same as those explained in all books about ancient Chinese mathematics. In particular, the three ||| appearing on the last line of the same diagram mean ‘300’ and not ‘3’ (their correct values follow from what is indicated on their right sides: dd (three hundred). Beyond ordinary numerals, we also note in passing that the sizes of the three circular zeroes of this diagram are slightly different from each other but it is obviously impossible to attribute this irregularity to the underlying mathematics. Rather, the craftsman responsible for the woodcut necessary for the xylographic printing of this diagram could not perceive their identity, probably because their dimensions were slightly different in the handwritten manuscript he had been asked to carve. Similarly, the poor quality of the first diagram, (a), also explains the slightly different shapes of its two zeroes. More importantly for our present purpose, the specific meanings of the rod-numerals and of the zero-circle are specified on their right sides qimeng, j. 3, problems 2, 6 and 7, section fangcheng zhengfu ddd d (Rectangular arrays (linear systems) with positive and negative quantities). From A. Kodama 1966, op. cit., p. 154 and 155. 49 Ibid., problem 6, p. 155. 48 Suanxue THE ZERO-CIRCLE 127 and, in particular, the three circles of diagram (b) representing a zero are said to mean kong d. Figure 3.4. (a) Partial graphical representation of problem I-1 of the Shushu jiuzhang dddd (1247) where the zero-circle, O, is called kong (From Shushu jiuzhang, Yijiatang collection, 1842, j. 1, p. 5b). (b) Schema from the Suanxue qimeng dddd (1299) displaying the zero-circle (O) as well as the zero kong d and witnessing the link between both (original Chinese characters have been replaced by modern types). For a reproduction of the original, see A. Kodama 1966, op. cit., p. 154 and 155). (c) schematic translation of (b). Therefore, there is no doubt that the new zero foreign to Chinese writing, O, is closely associated with the former one, kong d, in some way. The kong d zero and the zero-circle are thus not independent from each other, at least from the end of the Song dynasty. Moreover, the former first occurs in astronomical tables from the Tang dynasty and is 128 NUMBERS AND CALCULATIONS thus possibly an ancestor of the later. Still, both are conceptually distinct since the zero-circle is used in the same way as any other decimal digit while kong d has always been previously used in a quite restricted manner. Therefore, kong d should be rather regarded as a kind of protozero, limited to astronomical tables for many centuries. The History of Zero Revisited According to the foregoing, we know that a late connection between the zero-circle and the kong d zero exists. But are both of them possibly related to earlier forms of zeroes? The zero-circle is perhaps a variant of the Indian dot or the result of its transformation into a small circle but its possible connection with the kong d zero cannot be excluded. If the latter assumption is correct, then it follows that the kong d zero, initially limited to astronomical tables, was later incorporated into more ordinary Chinese mathematics and was finally replaced by the new symbol O. In support of this possibility, we note that the famous author of the Shushu jiuzhang dddd, Qin Jiushao, is known to have mastered not only mathematics but also astronomical calculations. As he clearly explains: “In my youth, I was living in the capital, so that I was enabled to study in the Board [or Bureau] of Astronomy, subsequently I was instructed in mathematics by a recluse scholar.”50 Moreover, astronomical and mathematical concerns are both present in his Shushu jiuzhang dddd and this famous treatise equally relies on the two above forms of zeroes. We are thus led to examine the most recent results about the early history of the zero-circle. In astrological Greek papyri surviving from the Roman Empire and recovered in Egypt at Bahnasa (Oxyrhynchus in classical times), between 1896 and 1906, and now kept at the Ashmolean Museum (Oxford), an overlined zero (O, but with a longer stroke and numerous handwritten variants), represents a common form of zero.51 In this respect, the historian of astronomy A. Jones notes that: “It represents either an ‘empty’ place in a sexagesimal fraction or ‘no whole units’ (usually preceding a sexagesimal fraction). It was always 50 See 51 In U. Libbrecht 1973, p. 62. later periods, only the overlined zero was used. THE HISTORY ZERO REVISITED 129 closely tied to the sexagesimal notation and occurs only in astronomical contexts.”52 Not often mentioned in general histories of mathematics, this Greek zero can be seen as the counterpart of an earlier cuneiform separation symbol which was used in sexagesimal numerals (Fig. 3.5). But it is also the direct ancestor of the particular form of zero attested in Arabic and Byzantine astronomical manuscripts. For instance, it occurs in the Zı̄j al-Sanjarı̄ of G. Chionades (late thirteenth century).53 Likewise, but much later, this Greek tradition was kept alive even during the late Middle Ages (see, for instance, the recent critical edition of an important astronomical manuscript, from the fifteenth century, due to the Byzantine astronomer G.G. Plethon).54 Could it be, then, that this unquestionably very ancient and lasting overlined Greek zero was related to the Chinese kong astronomical zero? Figure 3.5. The cuneiform zero (third row) denoting missing sexagesimal units. From F.X. Kugler 1900, Table II, p. 34. Although we have no direct evidence of such a connection, it should be noted that elements of Greek horoscopic astrology are know to have reached China not later that the Tang dynasty. As first noted, a long time ago, by A. Wylie55 and, once again one century later by Joseph Needham,56 two Greek technical terms of interest in astronomy and astrology have been identified in Chinese astrological documents from the Tang, in the form of phonetical transliterations each time. The two terms in 52 A. Jones 1999, p. 61–62 (underlined by us). Library, Vat. Ar. 761, fol. 168 (I owe this reference to R. Mercier; see http://www.unicode.org/L2/L2004/04054r-greek-zero.pdf (accessed on 02/05/2015). On G. Chionades, see also J.G. Leichter 2004. 54 A. Tihon and R. Mercier 1998. 55 A. Wylie, 1897/1966*, p. 97. 56 J. Needham 1959, p. 176, note b. 53 Vatican 130 NUMBERS AND CALCULATIONS ' ρα (hour)57 and λ επτ óν (minute) first adapted in Sanquestion are ’ω skrit as hōra and liptā and then in Chinese as huoluo dd and liduo dd. The Sanskrit original has not been identified or is lost, but the Chinese corresponding text – a Tantric Buddhist text entitled Fantian huoluo jiuyao dddddd (The Hōra and Navagraha of Brahma) – is still extant.58 Moreover, the same A. Wylie also writes that a commentary of the Jiuzhi li ddd he had access to also uses such phonetic transliterations from the Greek language.59 However, the late Chinese zero, O, is not overlined. Still, if it is the case that the Song Chinese were aware of the Greek astronomical zero in one way or another, they could have attempted to adopt it later, but certainly not directly because an overlined circle meaning five times any odd power of ten, such as 50 or 5000, already exists,60 . Be that as it may, the kong d zero is not often met with in Chinese documents posterior to the Yuan dynasty. Rather, under the Ming dynasty, the Chinese zero takes the form of the character ling d, which has remained in current use in the Chinese language up to now.61 Far from being only a character of writing endowed with an utterly new meaning, the ling d, zero has so dominated Chinese numerical practices that the zero-circle has, at 57 This word also has many other meanings (a period of time, determined or not (season, year, a night, etc.)) but in the present context of horoscopic astrology, its appropriate meaning is either a spatial unit denoting half of a sign of the zodiac sign or a temporal unit equal to the 24th part of a day, that is, an hour. 58 M. Yano 1992, p. 2 gives more details on this subject: navagraha means approximately ‘nine demons’ and concerns medicine as well as astrology. In this latter domain, its meaning extends to planets. See also M. Yano 2003, p. 381 and 2004, p. 331 f. 59 A. Wylie, op. cit., p. 97. 60 See U. Libbrecht 1973 p. 69. Hence, perhaps, a slight simplification of the original overlined zero consisting in omitting its stroke while keeping the circle. Moreover, we also note in passing that, in Chinese mathematical texts from the end of the Song and the beginning of the Yuan dynasty, the symbol O is also used as a punctuation mark, signaling the end of a portion of text endowed with a strong semantic unity such as, for example, the statement of a problem or its solution. Numerous examples of this usage are documented in the Korean version of the Suanxue qimeng dddd consulted here. However, this O is quite different from the numerical zero and it is cannot be confused with it, if only because of its larger size: used as a punctuation mark, the zero-circle, O, occupies the same space as an ordinary Chinese character of writing whereas the numerical zero is a small circle whose diameter is approximately equal to one-thirdofthatofthenon-mathematical O. 61 On ling d, see J. Needham 1959, p. 16–17. NUMERICAL CONSTANTS 131 last, been systematically pronounced ling instead of kong. However, the pronunciation kong of the zero-circle has discretely but surely continued to perpetuate itself in Min dialects62 from Fujian province and Taiwan.63 Lastly, another form of zero, quan d, (literally ‘a circle’) also exists but it is extremely rare and documented much later, in tables of logarithms translated into Chinese from those of John Napier and Henry Briggs.64 Numerical Constants Chinese calendrical calculations depend on all sorts of numerical constants, astronomical and non-astronomical, which are systematically attributed multisyllabic names and specific numerical values each time. Most of them are regularly listed one by one at the beginning of the various sections of astronomical canons devoted to particular topics and, quite often, they also occur autonomously in astronomical tables. Insofar as elements of knowledge essential for us – such as their definitions, the values of their metrological units or the modes of representation of numbers they rely on – are mostly left implicit, it is often not easy to determine their nature and function. In fact, such difficulties are unavoidable because what is as stake is the operational orientation of the calculations, not the logical economy of the related procedures and still less their explanatory power. That is why, in particular, the quantity of numerical constants is never reduced to a minimum in Chinese astronomical canons. On the contrary, they are multiplied almost ad libitum for purposes of calculations. It thus often happens that several distinct constants refer, in fact, to a single underlying logical unit. For instance, Chinese authors often disso62 On the peculiarities of Chinese dialects, see S.R. Ramsey 1987/1989*, p. 107 f. Hanyu fangyan da cidian ddddddd (Great Dictionary of Chinese Dialects), collective work, Fudan University (Fudan Daxue dddd) and the Kyoto University of Foreign Languages (Kyoto, Gaikokugo Daigaku ddddddd), Beijing, Zhonghua Shuju dddd, 1999, vol. 3, p. 3695. Moreover, a relatively recent study of Chinese historical phonetics shows that the pronunciation of the character kong d towards 1300 was not very different from the present one: E.G. Pulleyblank, 1991b, p. 174, indicates that, for the Mongol period (Yuan dynasty) the reconstituted pronunciation of kong d is kh ung, with the same tone as in modern Chinese when it means ‘void’. 64 Chouren zhuan, j. 43, p. 561 (notice p. 391 below). 63 See 132 NUMBERS AND CALCULATIONS ciate the numerators and denominators of fractional constants since this dismemberment makes it easier to describe the successive steps of calculations in certain cases. Similarly, certain numerical constants listed apart are not independent, but only expressed with different units. In the opposite direction, several numerical constants that we would regard as independent entities are combined with each other in order to give rise to new ready-made entities prompted by the necessity of relying on computational shortcuts. The number of numerical constants being thus significantly larger than what the mere logic of the calculations procedures would imply,65 each particular topic often needs several tens of such constants and the whole of a given astronomical canon, hundreds. When all canons are taken into account, their number certainly reach at the very least one thousand. Since the end of the eighteenth century, Chinese historians of these technical matters have attempted to unravel this complexity. For instance, the philologist Li Rui dd (1768–1817) devoted considerable attention to an elucidation of the meanings of the various constants listed in some Han and Song astronomical canons.66 Much later, during the first half of the twentieth century, the question of the elucidation of the meaning of numerous constants recorded in all Chinese astronomical canons has been undertaken anew by the Chinese astronomer Zhu Wenxin67 ddd (1883–1939). Unfortunately, this pioneering work68 is less useful than it could have been because it contains numerous errors. A number of other more or less similar but often less extensive studies, such as Gao Pingzi ddd ’s analysis of Han astronomical canons69 have also been published. However, the most essential work in this do65 A critical translation of some such lists of numerical constants is provided, for example, in M. Teboul 1983, p. 1 f. (Santong li ddd case); Ang Tian Se 1979, p. 78 f. (Dayan li ddd case) and N. Sivin 2009, p. 389-550 (Shoushi li ddd case). 66 Li Rui’s works in this domain have been published posthumously (see p. 394 below). 67 On Zhu Wenxin, see Chen Meidong and Chen Kaige 2008 . 68 Zhu Wenxin 1934. 69 See Gao Pingzi 1987. NUMERICAL CONSTANTS 133 main is certainly Wang Yingwei ddd ’s very important study70 which was printed in 1998, after having been diffused confidentially for many years. In this essential work, designed as an aid to reading Chinese astronomical canons, the author provides not only a detailed analysis of a very large number of numerical constants used in thirty one major Chinese astronomical canons but also a minute explanation of the rationale at work in a large part of the individual procedures they are involved in. In addition, Liu Hongtao71 ddd’s study of pre-Tang sources72 and various isolated articles mentioned in our bibliography of secondary sources are also quite useful in this respect. Whatever the individual merits of these works made over several centuries, the philological works of wider groups of historians have also contributed in their own way to the advancement of these studies because, like all ancient texts, Chinese astronomical canons have been slowly but surely distorted during their process of transmission over long sequences of centuries. The critical edition of the Chinese official histories, released from 1975 to 1976, constitute a major advance in this respect: the Chinese scholars responsible of their edition have particularly taken care of the textual aspect of astronomical canons73 and provided an impressing critical apparatus in this respect. Yet, even so, not all textual problems have been settled. For example, the historian of astronomy Chen Meidong has shown that the edition of many astronomical tables contained in this edition often remains unsatisfactory.74 Whereas further progress of our understanding of all the details of these difficult texts will certainly still need a tremendous amount of 70 After scientific studies in Japan, Wang Yingwei ddd (1877–1964) became one of the first astronomers of the Chinese Republic, not long after 1916. After retirement, he devoted himself to the elucidation of Chinese astronomical canons and the first version of his work was presented to the Chinese astronomical association in 1962, when he was 85 years old. See Wang Yingwei 1998, p. 2. 71 Liu Hongtao ddd (1943–2001) (Nankai Daxue, department of history). 72 Liu Hongtao 2003. 73 We recall here that as far as astronomical canons are concerned, our main sources are the Chinese dynastic histories (see their complete list on p. 385 below). 74 Chen Meidong 1995, p. 298–304 and p. 318–321 proposes a critical study of these tables and very numerous corrections induced by the supposed regularity of their structure. 134 NUMBERS AND CALCULATIONS work, what we suggest to do here is more limited: instead of describing each numerical constant one by one, we have, on the contrary, attempted to determine their logical kernel by reducing their number to a minimum, from their division into primary and secondary constants, as follows: Definition 3.1 (Primary Constants) A primary constant is an autonomous logical, numerical or physical entity without which calendrical calculations cannot be performed completely. For example, the length of the solar year (tropical year) and synodic months are primary constants. More generally, primary constants introduced in the present work practically belong to one of the following categories: 1. constants concerning the way the epoch is related to a particular instant of the temporal chain, determined by a given year whose calendar is required; 2. constants giving the values of different kinds of solar years and lunar months, determined once for all in a given astronomical canon; 3. constants expressing the lengths of supra-annual cycles (or periods) used in a given astronomical canon; 4. constants giving an equivalence between a number of solar years and lunar months, such as ‘235 lunar months are equivalent to 19 solar years’; 5. numerical coefficients listed in solar or lunar tables; 6. lastly, an omnipresent constant is the number of different binomials the sexagenary cycle is made of. Definition 3.2 (Secondary Constants) A secondary constant is any numerical constant deducible from one or several primary constants in any mathematical way. THE EPOCH 135 For instance, this definition implies that if A and B are two primary constants, then all the following derived constants are secondary: 19A, 76A, A/24, A/72, A/120, A − 365, ⌊A − 360⌋, ⌊A⌋, numer(A/24) and denom(A/(A − 360)). The Epoch The epoch is the starting instant of all calendrical calculations. As such, it belongs to the deep structure of the calendar and in no way to its surface structure, where the origin of time is defined by means of dynastic eras. Chinese astronomical canons are divided into two kinds according to the way they define their epoch. In the first case, the epoch is conventionally located in a past instant, immensely distant from the present, at least equal to a few thousand years and at most three hundred million years. In the second case, the epoch is located on the contrary in an instant contemporary of an astronomical reform. During the history of China, astronomical canons based on the first kind of epoch dominate, both in number and longevity: the 48 astronomical canons officially promulgated between 104 BC and 1280 all belong to this category whereas only the two last ones, the Shoushi li d dd (1281–1384) and the Datong li ddd (1368–1644), rely on an epoch of the second kind. In the sequel, astronomical canons of the first kind will be said to be based on a Superior Epoch and the others on a Contemporary Epoch. Superior Epochs are defined by fictitious coincidences between the beginnings of the solar, lunar and sexagenary cycles (see definition 3.3 on next page). Contemporary Epochs are on the contrary based on astronomical observations of luni-solar phenomena having a quantifiable degree of precision. Apart from this fundamental difference, the simplicity of calendrical calculations is slightly different in both cases. With a Contemporary Epoch, calculations must necessarily rely on two distinct procedures, in accordance with the anteriority75 or posteriority of calendrical events, 75 Retrospective calendrical calculations have never ceased to be deemed important by the Chinese, notably in order to check the correctness of their new predictive techniques. 136 NUMBERS AND CALCULATIONS with respect to the epoch. By contrast, the Superior Epoch precedes by far all recorded historical events and consequently, all calendrical or astronomical events are necessarily posterior to the epoch. Regardless, the time between the epoch O and any given calendrical event E is represented by a quantity whose value is determined by the length t of the interval OE, the underlying units of time mostly used in calculations being generally the day and its multifarious fractional subdivisions. In the latter case, the mode of quantification of time is analogous to the Julian Day number system, another continuous count of days and fractions thereof from a fixed origin, which is now universally used by astronomers, historians of astronomy and chronologists.76 The Superior Epoch From 104 BC to 1280, Chinese official astronomical canons admit a ‘Superior Epoch’, shang yuan dd, defined as follows: Definition 3.3 (The Superior Epoch) The Superior Epoch belongs to the deep structure of the calendar and is characterized by the coincidence of the initial winter solstice and initial new moon (initial lunisolar conjunction), these two phenomena happening simultaneously at the instant of midnight of the first day of the sexagenary cycle, jiazi dd, ((1, 1) or #1). 76 The Julian Day number system was first introduced by the famous philologist and chronologist Joseph-Juste Scaliger (1540–1609) in calendrical year counts (for a full translation of the relevant passage in Scaliger’s works, see A. Grafton 1993, p. 249–253; see also L.E. Doggett 1992, p. 600 and C. Dumoulin and J.-P. Parisot 1987 , p. 53, for further historical details). However, this technique was adopted much later and became widespread only in 1849, when the astronomer John Herschel (1792–1871) began to use it in his works. We also note that the beginning of the Julian Day number system is 1st January −4712 at Greenwich mean noon (12h Universal Time), the year −4712 having been obtained from a combination of three supra-annual cycles composed of respectively 15, 28 and 19 years. The first, second and third cycle are, respectively: (a) a census for tax-gathering, held every 15 years in the Roman Empire; (b) the number of years in a solar cycle and involves the planetary week; (c) the number of solar years in the Metonic cycle. As in the Chinese case, the Julian Day epoch is located in an instant preceding all known historical events. However, the Chinese idea of a Superior Epoch is much more ancient and is already documented under the Han dynasty. Yet, the unceasing reforms of Chinese astronomical canons have resulted in the creation of very numerous different such epochs and related day-counts. THE SUPPORT YEAR 137 Whereas this definition is generally observed, some exceptions exist, however: the initial lunisolar conjunction does not always happen on the first day of the sexagenary cycle in all astronomical canons: in the Jiyuan li ddd (1106–1135), for instance, it coincides with a jimao d d day ((6, 4) or #16).77 Sometimes too, the initial lunisolar conjunction do not involve the winter solstice but another solar breath: the Yuanjia li ddd (445–509) starts from Rain Water, q5 , instead.78 Yet, such special cases can be dealt with quite easily once the general case has been mastered. Consequently, only the latter will be explained in detail in the sequel. Thus let us now suppose that a given astronomical canon admits a Superior Epoch. Then, its calendrical calculations cannot be performed without having also fixed the conventional beginnings of the solar and lunar years of the calendar. From 104 BC to 1281, these beginnings are essentially determined by the ‘Xia norm’, as already noted. The Lunar Year and its Support Year In astronomical canons based on the Xia norm, the Superior Epoch takes place at the instant of the initial winter solstice coinciding with the initial eleventh new moon and both belong to the end of the year immediately preceding the first theoretical year liable to be taken into account in calendrical calculations. Therefore, both can be denoted q1 (0) and n11 (0) and we have q1 (0) = n11 (0) = 0. In addition, this year zero is limited to its two last months, the eleventh and the twelfth, and the next month is the one beginning with the first new moon of the year 1, n1 (1). More generally, the first new moon of a given lunar year x and its previous nearest winter solstice always belong to the years x and x − 1, respectively. Let x be a given year. Then, Chinese sources respectively call q1 (x − 1) and n11 (x − 1) sui qian tianzheng dongzhi dddddd and dddddd (literally: ‘The winter solstice based on the celestial norm and preceding the calendrical year’ and ‘The initial mean79 new moon based on the celestial norm and preceding the calendrical year’). 77 See Bo Shuren 2003, p. 369. Songshu, j. 13, ‘lüli 3’, p. 274. 79 In general, the Chinese term jing d associated here with the idea of mean value means ‘fixed, regular, immutable’. In Han mathematics, however, it refers to a share-out into equal parts and, therefore, its connection with the notion of mean value is obvious 78 See 138 NUMBERS AND CALCULATIONS In view of a mathematical analysis of a situation which is as linguistically complex as it is mathematically simple, it seems better, however, to avoid such longwinded literal renderings and to use instead some straightforward mode of expression.80 We have thus coined the following very simple definition once and for all: Definition 3.4 (The Support Year) The support year of a given year x is the year x − 1. With this definition, the determination of the calendar of a given year x can be said to start from the calculation of the solar and lunar elements of its support year. In general the corresponding calculations only concern the eleventh and the twelfth months of the year x − 1 but the situation is sometimes slightly more complex since the possibility of existence of an intercalary month belonging to the year x − 1 cannot be excluded a priori. In such a case, three and not two months belonging to the year x − 1 are implied in the calculation of the calendar of the year x. The Emerging Year Definition 3.5 (The Emerging Year) The first calendrical year based on the calculations of a new astronomical canon is called its emerging year. This definition may seem superfluous because the year so defined looks identical to the year of official adoption of an astronomical canon. However, this is not always the case and the two notions have to be differentiated from one another. The Number of Years from the Superior Epoch Let x be a year of an astronomical canon having a Superior Epoch. Then, the fundamental time parameter of calendrical calculations is equal to (see Guo Shuchun 1990, p. 187, commentary on problem 18 of the Jiuzhang suanshu dddd (Computational Techniques in Nine Chapters). 80 Certain historians of the Chinese calendar sometimes use less literal renderings and rely instead on expressions such as the ‘winter solstice in the astronomical first month’ (see, for instance, N. Sivin 2009, p. 392 and 395). CHANGES OF ORIGIN 139 the number t(x) of solar years81 contained in the interval between the two winter solstices of the Superior Epoch and of the year x. Chinese sources call this number jinian dd (literally ‘accumulated number of [solar] years’). From a purely mathematical perspective, calendrical calculations can be done for any value of t(x). In practice, however, it remains necessary to connect t(x) to some dated historical event since the Superior Epoch is located outside historical time. To do so, each astronomical canon provides the integer number t0 of solar years contained in the interval between the two winters solstices, q1 (0) = 0 of the Superior Epoch, O, and q1 (x0 ) of a later year x0 , and where x0 is a number of a year determined once and for all (Fig. 3.6). Then, for any year x, we have: t(x) = t0 + (x − x0 ) solar years O q1 (0) = 0 t0 q1 (x0 ) t(x) (x − x0 ) (3.5) q1 (x) Figure 3.6. The connection between the winter solstices q1 (0) and q1 (x0 ) of the Superior Epoch (year 0) and of a later year, x0 (q1 (x) represents the winter solstice of the year x). The values of t0 and x0 attached to a given astronomical canon are most frequently provided in original texts through the usual dating system of dynastic eras; otherwise they can be reconstituted hypothetically with a variable degree of certainty.82 Changes of Origin A priori, the calculation of the calendar of a given lunar year x depends on t(x − 1), that is, on a generally very large integer number of solar years. Pre-Tang (618–907) Chinese calendars makers have thus often 81 As tables of the Chinese calendar show, a one-to-one correspondence between solar years and lunar years, ordinary or intercalary, exists. They can thus be numbered alike with the same variable x. 82 See Appendix E below. 140 NUMBERS AND CALCULATIONS attempted to simplify their calculations. The technique they have chosen has always been the same and consists in eliminating the greatest possible number of supra-annual periods contained in t(x − 1) solar years. The corresponding techniques imply rather abundant developments but the underlying mathematics is essentially trivial. Therefore, only a representative example of this kind of technique will be developed (see p. 243 f.). Number of Support Days In calendrical and astronomical calculations, the parameter t is first determined from the above expression 3.5 and the integer number of solar years so obtained is then converted into a number of days j. In all cases, j is obtained by multiplying t by the number of days Y contained in a mean solar year and the corresponding result is called zhongji d d ‘mean accumulation’ (zhong = mean, ji = accumulation): j(t) = Y t. (3.6) In most cases, Y is a primary astronomical constant but it sometimes happens that the length of the solar year is subjected to very small variations, only detectable when the century is taken as a new unit of time. In such cases, the analysis of the underlying procedures shows that the expression 3.6 should be replaced by: j(t) = Y (t)t (3.7) where Y (t) is a simple algebraic expression, giving the mean value of the solar year with respect to the interval between the two winter solstices of the epoch and a later year x.83 The epoch in question can then be a Superior Epoch (Tongtian li ddd case) or a Contemporary Epoch (Shoushi li ddd case). Example 3.5 The Shoushi li ddd procedures indicate that the initial length of the solar year for the year 1280 is equal to 365.2425 d and that its mean value, with respect to intervals such as [x, 1280] or [1280, x], where x designates any year, increases or diminishes by 0.0001 days 83 For a comparison with the modern notion of secular variations, see S. Nakayama 1982, p. 125 f. BINOMIAL REPRESENTATIONS 141 per century,84 depending on whether the year in question is anterior or posterior to 1280. According to the terse original formulation: “In the case of calculations towards the past, add ‘one’ per century; towards the future, subtract ‘one’.”85 ddddd ddddd dddddd dddddd Here, the first and the second yi d (one) should both be interpreted as referring to a unit of time (equal to 0.0001 days).86 Hence: Y (t) = 365.2425 ± 10−4 ⌊t/100⌋ d . (3.8) This symbolic representation of secular variations is of course extremely different from the procedural formulation of the original but, from a purely operational perspective, the results of the calculations are identical in both cases provided that the above Chinese sentence is understood as implying discontinuous variations by steps of one century. Another interpretation involving continuous variations is also possible, and would be readily obtained by replacing ⌊t/100⌋ by t/100. Yet, even without authentic examples of calculations from the Yuan dynasty at our disposal, this possibility seems much less likely because the above procedure explicitly stresses the notion of century. Binomial Representations The theoretical length of the solar year being not equal to an integer number of days, the interval between the Superior Epoch and the winter solstice of a given year x is generally not composed of an integer number of days. Consequently, Chinese calendar makers systematically 84 The underlying notion of century should certainly be related to the fact that the representation of numbers used in the Shoushi li ddd is based on the centesimal system, a system which is possibly the consequence of Chinese translations of Indian Buddhists texts currently using such a system in order to represent large numbers (J.-C. Martzloff 1997*/2006*, p. 97). This notion of century has of course nothing to do with the European periodization of historical phenomenons by periods of one hundred years used in Europe from the end of the sixteenth century See J. Leduc 1999, p. 97 f. 85 Yuanshi, j. 54, ‘li 3’, p. 1192. 86 Similar renderings are also possible, for instance yi d = ‘one counting-rod’, ‘one digit’ and so on. 142 NUMBERS AND CALCULATIONS dissociate this number into two components dealt with apart in calculations, namely an integer number of days, a, generally very large, and a fractional part, b, inferior to one day. Hence a pair < a; b > of numbers. Moreover, a is systematically reduced modulo 60. Hence new pairs < a mod 60; b >, where a mod 60 is a result between 0 and 59 rather than 1 and 60. It is thus often convenient to number the sixty sexagenary binomials from zero rather than from one so that 0 corresponds to (1, 1) or jiazi and so on. In its turn, b is expressed as a sum of fractions determined by divisions of the day into a series of always finer units, possibly different not only from one astronomical canon to the other but also according to the purpose of the calculations, as already noted on p. 107 above. In practice, given the bewildering variety of these representations, the form taken by b in such and such a case will be made explicit only when dealing with specific examples of calculations: the overall structure of the underlying procedures appears more clearly when the numerical representation of b in such and such a case is given once and for all and then left unspecified, as is the case in the original Chinese sources. Beyond the initial transformation of < a; b > into < a mod 60; b >, b is also quite often regarded as an integer rather than a fraction, exactly like our notations of decimal fractions leaving implicit the powers of ten they refer to. In the same spirit, and given the importance of division and reduction modulo 60 in Chinese calendrical calculations, it is also useful to analyze their procedures in terms of pairs of integers x and y giving rise to reduced binomials, bin(x, y), defined in the following way: bin(x, y) = < ⌊x/y⌋ mod 60; x mod y > . def (3.9) In other words, the left part of the binomial obtained from the two integers x and y is the integer quotient of x divided by y reduced modulo sixty, while the right part is the integer remainder of their division. In Chinese sources, ⌊x/y⌋ mod 60 and (x mod y) are called ‘the great remainder’, dayu dd, and ‘the small remainder’, xiaoyu dd, respectively, the first ‘remainder’ being the result of casting out sixty and the second, the ordinary remainder of a division of two integers. FRACTIONAL REPRESENTATIONS 143 These two ‘remainders’ are quite ancient since they already occur in the calendrical treatise of the Shiji.87 Later, they remained omnipresent in Chinese calendrical calculations from all periods. Hence their importance. Fractional Representations of Time Although the three notations < a; b >, < a mod 60; b > and bin(x, y) introduced above would allow us to deal with all numerical expressions appearing in Chinese calendrical calculations, it is often better not to stick rigidly to these fixed notations but to also rely on slightly simpler notations, particularly when large quantities of numbers endowed with a similar structure are listed in tables. We will thus frequently omit the right and left delimiters (‘<’ and ‘>’) and use instead the simpler notation a;b, the left term, a, not necessarily being reduced modulo 60. When the context will be sufficiently clear, denominators will also 664 7 be omitted. For instance: < 15; 3040 + 3040×24 > will be replaced by < 15; 664, 7 > or, more simply, 15;664, 7 and the same will be done in all similar cases provided that the context indicates sufficiently clearly that the main unit of time is the day and that the denominators to be restored are clearly identifiable. Still more radically, < a; 0 > will sometimes be replaced by a. In addition, in order to avoid any ambiguity, one or several zeroes will sometimes be inserted before certain digits even though nothing of 7 70 the sort exists in Chinese sources. For instance, 17 + 940 ; 9 + 940 and 700 59 + 940 will be replaced by 17;007, 9;070 and 59;700, respectively. Lastly, some additional examples will also allow us to introduce some similar numerical notations often met with. Example 3.6 Numerical representation of the winter solstice of the year Kaiyuan 15 according to the Dayan li ddd. From any concordance table of the Chinese calendar, the year Kaiyuan 15 corresponds to 72788 and, from the formula 4.11, p. 164 below, 87 See Shiji, j. 26, ‘lishu’, p. 1265-1287 (numerous occurrences). this example offers an example of proleptic calculation (the Dayan li ddd was promulgated in 729). 88 Incidentally, 144 NUMBERS AND CALCULATIONS applied to the case of the Dayan li, the winter solstice q1 of the year Kaiyuan 15 occurs 107,660,793,718,192 days after its Superior Epoch. Hence 3040 its representation: < 35, 414, 734, 775; 2192 3040 >. Moreover, 35, 414, 734, 775 mod 60 = 35. Hence the reduced binomial form < 35; 2192 3040 >. This kind of notation being somewhat cumbersome, we only write q1 =< 35; 2192 > or even, still more simply, q1 = 35;2192, as soon as the context will make obvious the implicit denominator, 3040. Written in one way or another, this result means that the winter solstice q1 happens on a day numbered 35 + 1 with respect to the sexagenary cycle, a day corresponding to (6, 12) or jihai dd. Moreover, q1 occurs 2,192 3,040 days after the instant of midnight of the day jihai in question. Lastly, the sum a + b also leads to another expression of time by 108,592 means of an improper fraction: a+b = 35+ 2192 3040 = 3040 represents the number of days and fractions thereof elapsed between the last beginning of the sexagenary cycle and q1 . Example 3.7 Representation of a mean solar period in the Jingchu li d dd. 402 11 This mean solar period is composed of 15 + 1843 + 1843×12 days. Hence the abridged notation 15;402,11. If need be, a few other notations will be introduced. For instance, in the case of lists of tabulated values, the notation 60/a/b will be taken to mean that the first number is the rank of a computed sexagesimal binomial (between 0 and 59) whereas the two other numbers are the numerators of fractions having a and a × b as denominators (see, notably, Table 9.1, p. 263 below). Likewise, days/a/b means ‘an integer number of days plus two fractions thereof, having a and a × b as denominators’. Mean and True Elements Definitions Chinese astronomical canons fall into two categories according to the mean or true nature of their solar and lunar elements. MEAN AND TRUE ELEMENTS 145 With respect to the Chinese calendrical calculations studied here, mean and true elements are purely temporal notions, only concerning the time of occurrence of lunar and solar phenomena (solar breaths, seasonal indicators, moon phases) and never astronomical positions. Mean elements are average quantities fixed once and for all. By contrast, true elements are variable quantities obtained from positive or negative small corrections affecting their mean values. The first case corresponds to an approximation, often sufficient for calendrical calculations, based on the idea of the complete linearity and uniformity of lunisolar phenomena. The second case reflects a much more elaborate astronomical conception, viewing these phenomena in terms of unceasing deviations from average values. From a historical perspective, the most ancient calendrical calculations are based on mean elements. All belong to the pre-Tang period and have thus been issued before 618. By contrast, one year after the beginning of the Tang dynasty, true elements were systematically taken into consideration, and this new technique was first implemented in the Wuyin li ddd. Historical Aspects Despite this simple bipartition of calendrical calculations into mean and true elements, the situation is slightly more complex because solar and lunar elements have not been dealt with in the same way throughout the course of history. From the Tang to the end of the Ming dynasty, the lunar component of the calendar has always been calculated with true elements and its solar counterpart only with mean elements. Consequently, the equinoxes of Chinese calendars issued during this long period always fall on a day where the lengths of day and night are never exactly equal (the difference between these calendrical equinoxes and their astronomical counterparts generally reaches two days). From the second year of the Qing dynasty (1644–1911), i.e. from 1645,89 the lunar and solar components of the calendar have both been calculated from true elements. Consequently, solar phenomena taken 89 The calendar of the year 1644 still relies on the Datong li, the astronomical canon in force during the Ming dynasty (1368–1644). 146 NUMBERS AND CALCULATIONS into account in Chinese calendars suddenly became more exact from an astronomical standpoint. As will be explained at the end of the present chapter, as soon as the necessary notions concerning the Chinese technique of intercalation were introduced, this modification provoked insidious but temporary perturbations of the calendrical structure. Notation and Terminology When it will be necessary to distinguish between mean and true elements, the former will be overlined. With this notation, the respective lengths of the lunar month of the Sifen li ddd and of the Daxiang li ddd, for example, should 28,422 d d be denoted m = 29 + 499 940 and m = 29 + 53,563 . But since these two astronomical canons do not oppose mean and true elements, m and m will not be distinguished from each other in such a case. Despite this simplicity, the technical terminology attested in Chinese sources in this respect is not reduced to binary oppositions, however. Without giving all the details, we note that the Chinese generally use the following terms in order to qualify mean elements: jing d, zhong d, chang d, heng d, and ping d. The first term, jing d, only qualifies mean new moons; it has already been briefly analyzed on p. 137 above. The second term, zhong d, has a wider scope and designates averages values, denoting some natural balance between sets of stronger or weaker values with respect to the fluctuations of the two complementary yin and yang principles.90 The two following terms, chang d and heng d, are also quite common but they mostly appear in astronomical tables devoted to the solar and lunar inequalities.91 kind of interpretation is particularly obvious in the case of the Dayan li d dd, an astronomical canon whose technical terminology is almost wholly borrowed from the Yijing dd, the ‘Bible’ of Chinese divination; Xin Tangshu, j. 28A, ‘li 4a’, p. 637 f.). 91 In general, chang d means ‘constant’ and is not necessarily used only in astronomical tables (see Xin Tangshu, j. 28A, ‘li 4a’, p. 639; heng d has exactly the same meaning but seems limited to astronomical tables (see, for instance, Yuanshi, j. 56, ‘li 5’, p. 1272 (solar inequality). 90 This FUNDAMENTAL ELEMENTS 147 The last term, ping d (non-technical meanings: level, even, average, ordinary, common, usual), seems less frequent.92 Quite differently, the monosyllabic term ding d designates true values in the original Chinese sources. Its literal meaning is ‘[which has been] determined’, i.e. ‘a quantity corrected in order to agree with observations’ by contrast with mean values which most often do not correspond to what is observed. Whereas modern Chinese historians of astronomy still call ding d true values, they prefer ping d when they refer to mean values. Hence the relatively modern expressions pingqi d d and pingshuo d d (mean solar breaths, mean new moons, respectively). Moreover, ping is still used in modern astronomical science in expressions such as ‘mean position’ ping weizhi ddd, ‘mean equator’ (ping ji) dd and the like. By contrast, ding d has been discarded by modern astronomers in favor of zhen d, the exact equivalent of ‘true’ in English. Fundamental Lunisolar Elements Let the calendar of a given year x be calculated. Then, the calculations begin with the prior determination of the following lunisolar elements: 1. all solar breaths, from q1 (x − 1) to a little beyond q1 (x); 2. all new moons, from n11 (x − 1) to n12 (x) or exceptionally one more month, when an intercalary twelfth month exists; 3. the lunisolar shift, or epact, e(x), defined and analyzed in detail on p. 149 below. Moreover, qi and ni give rise to solar and lunar months defined as follows: Definition 3.6 (Solar Months, Deep Structure) A solar month is any interval [qi , qi+2 [ i = 1, 3, . . . whose first and last elements are consecutive odd solar breaths. 92 The expression pingshuo dd (mean new moon) occurs, for example, in the Xin Tangshu, j. 25, ‘li 1’, p. 538. In modern Chinese, ping corresponds to the bisyllabic term pingjun dd (mean or average). 148 NUMBERS AND CALCULATIONS Definition 3.7 (Lunar Months, Deep Structure) A lunar month is any interval [ni , ni+1 [ composed of successive new moons (i = 1, 2, . . . or i = 11, 12 . . . according to their mode of indexation). With these two definitions, the last element of any solar or lunar interval also marks the beginning of the next solar or lunar month. Hence the choice of intervals closed on the left and open on the right. Next, the analogous definitions corresponding to the surface structure are readily obtained: Definition 3.8 (Solar Months, Surface Structure) A solar month is any interval of the form [⌊qi ⌋, ⌊qi+1 ⌋[. Definition 3.9 (Lunar Months, Surface Structure) A lunar month is any interval of the form [⌊ni ⌋, ⌊ni+1 ⌋[. The Last Solar Breath of a Lunar Year Given that Chinese solar and lunar years overlap, a few solar breaths belonging to the year x and posterior to the winter solstice q1 (x) exist but, as long as n1 (x + 1) has not been calculated, the last breath of the year x cannot be determined. The Numbering of New Moons The calculation of the lunar component of the year x generally starts from n11 (x − 1) rather than n1 (x) and this feature slightly complicates the numbering of new moons: they can only be numbered temporarily because, when an intercalary month exists, their final numbering must wait for the determination of its rank. Moreover, with such a starting point, the possibility of an intercalary month belonging to the year x − 1 cannot always be excluded a priori. During the early phase of calculations, it is therefore advisable to number new moons independently of the year they belong to. In this respect, the most straightforward technique consists in numbering them in natural order n1 , n2 , . . ., exactly in the same way as solar breaths, and to rely on the results of subsequent calendrical calculations in order to retain those belonging to the year x. THE LUNISOLAR SHIFT 149 The Lunisolar Shift Introduction The difference between the lengths of a solar year and twelve lunar months gives rise to the notion of epact but, in practice, this quantity is defined in various ways. In the Gregorian and Julian calendars, for example, the epact of a given year is an integer denoting the age of the moon on January 1st, by which the date of the Easter moon can be determined.93 Moreover, in such cases, its calculation depends on the usual Metonic cycle. In the Chinese case, on the contrary, the epact of a given year is not an integer number of days but an integer with respect to another unit of time and, although it sometimes also depends on some sort of Metonic cycle, it does not refer to a fixed calendrical date, determined from a day number and a lunar month. The Epact Definition 3.10 (The epact) In all Chinese calendars, the epact of a given year x, e(x), is equal to the value of the luni-solar shift with respect to the winter solstice q1 (x − 1), i.e., the length of the interval [n11 , q1 [, expressed in such a way as to obtain an integer related to the length of the lunar month. For instance, the fact that the epact of a certain year x of the Sifen li d dd is equal to ‘7’ means more precisely that the length of the interval 7 in question is equal to 19 lunar months, as explained on p. 159 f. below. Therefore, given that this canon is such that one lunar month = 27759 940 days, this epact amounts to 10.879 days. Moreover, the epact can be measured either in mean or true value but Chinese procedures only take mean values into account. Therefore, in the case of astronomical canons based on true elements, the epact so obtained is always approximate. What can be deduced from such an approximation is thus not always correct. When the true epact of the year x is really required, however, it should be noted that the calculations 93 G.V. Coyne, M.A. Hoskin and O. Pedersen 1983, 1983, p. 306; U. Bouchet 1868, p. 53 and 170. More generally, for an appraisal of the notion of epact in classical antiquity, see A.A. Mosshammer 2009, p. 75–80 . 150 NUMBERS AND CALCULATIONS only require the determination of n11 (x − 1) in true value because the true and mean value of the winter solstice are never differentiated from each other in all Chinese astronomical canons issued from 104 BC to AD 1644. The intercalary remainder (Runyu) Another important point of interest is the terminology associated with the Chinese notion of epact: Chinese astronomical canons call it runyu dd, a term whose literal meaning is ‘intercalary remainder’. From the above definitions 2.1, p. 76 and 3.10, p. 149, concerning the epact and the intercalary month, respectively, the reason for this appellation is not completely obvious but the precise connection between intercalary months and intercalary remainders will be explained in what follows. The Monthly Epact and the Intercalary Month Definition 3.11 (The Monthly Epact) Let us suppose that new moons ni and odd solar breaths, q2i−1 , i = 1, 2, . . ., are likewise enumerated in natural order from the Superior Epoch. Then, the monthly epact or, monthly lunisolar shift δi , relating to the odd solar breath q2i−1 , is equal def to the length l of the interval [ni , q2i−1 [: δi = l([ni , q2i−1 [). Figure 3.7 below shows the various elements of this definition: at the instant of the Superior Epoch, O, we have n1 = q1 = 0. Therefore δ1 = 0. One month later, δ2 is equal to the difference between the lengths of a solar month and a lunar month. O q1 = 0 n1 = 0 q3 n2 q2i−1 q5 n3 ni δi ni+1 Figure 3.7. The Monthly Lunisolar Shift δi+1 . More generally, the successive shifts δi are equal to their preceding value, δi−1 , plus the difference of length between the two kinds of months. Since a solar month is always longer than a lunar month, the result is of course a strictly positive quantity and the sequence of monthly lunisolar shifts, δi , is strictly increasing: THE LUNISOLAR SHIFT 0 = δ1 < δ2 < . . . < δi . 151 (3.10) As long as δi is smaller than a lunar month, the solar breath q2i−1 remains coupled with the lunar month [ni , ni+1 [ (Fig. 3.7 above) but when δi becomes greater than a lunar month, the coupling is broken. Consequently, the lunar month [ni , ni+1 [ necessarily becomes devoid of any odd solar breath, and its intercalary character seems unavoidable. The definition of the intercalary month, however, does not relate to the deep structure of the calendar but only to its surface structure (see 2.1, p. 76 above) and the two structures can in no way be confused. In fact, the lunar month [ni , ni+1 [ can possibly fulfill the definition of the intercalary month from the standpoint of the calendrical deep structure, while nothing of the sort remains true with respect to the corresponding surface structure: for instance, let us suppose that a lunar month [ni , ni+1 [ is such that: q2 j−1 < ni < ni+1 < q2 j+1 (3.11) and let us also suppose that q2 j−1 and ni both fall on the same day. Then, the surface lunar month whose first day contains ni also contains an odd solar breath. Therefore, it can in no way be regarded as intercalary. It is thus absolutely necessary to also take into account the integer parts of the solar breaths and new moons immediately preceding and following the two initial new moons ni and ni+1 in order to determine whether or not a given lunar month is intercalary. Hence the following criterion: Criterion 3.1 (Intercalary Month) The lunar month [⌊ni ⌋, ⌊ni+1 ⌋[ of the calendrical surface structure is regarded as intercalary when a solar month [q2 j−1 , q2 j+1 [ such that ⌊q2 j−1 ⌋ < ⌊ni ⌋ and ⌊ni+1 ⌋ ≤ ⌊q2 j+1 ⌋ exists. In other words, the lunar month [⌊ni ⌋, ⌊ni+1 ⌋[ is intercalary when it is included in the solar month [⌊q2 j−1 ⌋, ⌊q2 j+1 ⌋[. In such a case, the first odd solar breath falls on a day just preceding the first day of the intercalary month and the second odd solar breath happens after just after the end of its last day. 152 NUMBERS AND CALCULATIONS Equivalently, when a month is intercalary it contains a single even solar breath falling in its middle and conversely, when this situation happens, the corresponding month is necessarily intercalary. Consequences Given that the preceding double inequality allowing us to determine intercalary months is anything but obvious, the practical realization of well-defined systems of calendrical calculations leading to the periodical insertion of intercalary months in a consistent way is not warranted in advance. Consequently, it would certainly not be difficult to imagine calculation techniques making impossible the determination of such months. But it is also true that astronomical canons only based on mean elements would lead more easily to well-defined calendars in this respect because, in such a case, a lunar month is always smaller than a solar month while nothing of the sort is certain when true elements are taken into account: from an astronomical perspective, the length of a solar month corresponds to an increase of 360/12 = 30◦ in solar longitude. But the solar velocity is higher in winter with respect to the other seasons. Consequently, certain solar months are liable to become exceptionally shorter than certain lunar months. When this happens, the preceding criterion 3.1, p. 151, defining the intercalary month can become impossible to fulfill. Hence a perturbation of the lunisolar coupling, a solar month being included in a lunar month instead of the opposite situation, typical of the intercalation pattern. In such a case, the lunar and solar months [ni , ni+1 [ and [q2 j−1 , q2 j+1 [, respectively, are such that: ni < q2 j−1 < q2 j < q2 j+1 < ni+1 (3.12) and when the same inequality remains valid in the surface calendar, [ni , ni+1 [ contains three solar breaths, q2 j−1 , q2 j+1 and q2 j . The fundamental lunisolar coupling typical of the Chinese calendar is thus violated. The intercalary month cannot be properly determined and, as the example of the eight ‘pathological’ years of the Qing (1644–1911) dynasty prove, various other anomalies also manifest themselves. THE LUNISOLAR SHIFT 153 Pathological Calendars Although the following eight examples of calendars do not belong to our period of study, we have retained them because they illustrate particularly well the question of the perturbations induced by the concomitant usage of true solar and lunar months in the Chinese calendar. As clearly and correctly explained in a series of reports emanating from the Qing Bureau of Astronomy94 the phenomenon is a consequence of the Jesuit reform of Chinese astronomy having led to the replacement of a previous harmonious combination of mean solar elements and true lunar elements by true elements in both cases. The following table, beginning on next page, showing the lunisolar structure of a few lunar months belonging to sixteen different lunar years displays several anomalies, always concerning winter lunar months, as expected: first, eight of these have a lunar month containing two consecutive odd solar breaths and, consequently, three solar breaths (see the third line of each table). Second, two lunar months (denoted —), having no odd solar breath but separated from one another by at most six lunar months, exist each time. In the first case, the lunisolar coupling is violated because a lunar month can contain only two solar breaths and never three; in the second case, the situation is equally abnormal, but for a different reason: two months should be regarded as intercalary each time but this is impossible since consecutive intercalary months can never be that close to each other. Nevertheless, as the table shows, one of these two months has been regarded as intercalary each time but the decision to choose one of them rather than the other must have been dependent on nonmathematical considerations. However, as chronological tables of the Chinese calendar show, the calendars of the years in question are only temporarily anomalous. This pathology is not documented before the Qing reform of astronomy but when the logic behind certain calendrical calculations is taken to its limits, the same phenomenon could also have manifested itself, at least theoretically. With calculations based on secular variations of the tropical year, namely those of the Tongtian li ddd 95 (1199–1207) 94 Chen 95 See Zhanyun 1986. S. Nakayama 1982. 154 NUMBERS AND CALCULATIONS and Shoushi li ddd 96 (1281–1384), the mean value of the tropical year always diminishes slowly towards the future so that the length of the solar month is bound to become smaller than the length of the lunar month. In practice, however, Chinese astronomical canons have always been ‘kept alive’ for a limited number of years, well before the collapse of their mathematics, and of course, nothing of the sort ever happened. I 1661 1662 Month 5 6 7 7* 8 9 10 11 12 1 Breaths q13 q15 q17 — q19 q21 q23 q1 q3 — q5 5 6 7 8 9 10 11 12 1 q23 q1 q3 — q5 II Month Breaths 1680 q13 q15 q17 III Month Breaths — 1681 q21 1699 7 q17 7* — IV Month q19 8* 8 q19 9 q21 1700 10 q23 11 12 1 2 3 q1 q3 q5 q7 — q9 1775 10* 11 12 1 2 3 4 5 6 q23 — q1 q3 — q5 q7 q9 q11 q13 q15 Month 8 9 10 11 12 1 2 2* 3 4 Breaths q19 — q21 q23 q1 q3 q5 q7 — q9 q11 Breaths 10 1776 V 96 See 1813 Chapter 6, p. 197 below. 1814 THE LUNISOLAR SHIFT VI Month Breaths 1832 9 9* 155 1833 10 11 12 1 2 3 4 5 q23 q1 q3 q5 — q7 q9 q11 q13 q21 — Month 8 8* 9 10 11 12 1 2 3 4 Breath q19 — q21 q23 q1 q3 q5 q7 — q9 q11 10 10* 11 12 1 2 3 4 5 6 — q1 q3 — q5 q7 q9 q11 q13 q15 VII 1851 VIII Month Breath 1852 1870 q23 1871 Table 3.5. The eight irregular years of the Qing dynasty. CHAPTER 4 MEAN ELEMENTS Mean Elements in Practice The mean solar and lunar elements of all Chinese astronomical canons issued between 104 BC and AD 1644 are always obtained from fixed values of solar and lunar constants but the related calculations are not wholly identical in both cases because, on the one hand, Metonic canons also take avail of equivalences between given numbers of solar years and lunar months whereas, on the other hand, canons based on true elements only take the concerned mean values into account. The difference between the two approaches is thus purely operational and, in both cases, the most fundamental mean elements depend on the mean time between consecutive solar breaths or on mean new moons, regarded as components of the calendrical deep structure. Metonic constants Metonic constants are so called in connection with a classical equivalence between 19 solar years and 235 lunar months, composed of 228 ordinary months and 7 intercalary months, generally attributed to the Greek astronomer Meton of Athens (ca. 430 BC) or to one of his contemporaries.1 Only the most ancient Chinese astronomical canons rely on this classical equivalence whereas about ten other canons from the pre-Tang period use similar, but non-classical, equivalences between other numbers of solar years and lunar months. In the following, these equivalences 1 O. Neugebauer 1975, vol. 1, p. 354 and 541; J.P. Britton 1999, p. 239; R. Hannah 2005; S. Stern 2012, note 87, p. 50. The traditional attribution of the invention of the 19 year cycle to this Greek astronomer is tentative since the same idea is also present in Babylonian documents from approximately the same period. © Springer-Verlag Berlin Heidelberg 2016 J.-C. Martzloff, Astronomy and Calendars – The Other Chinese Mathematics, DOI 10.1007/978-3-662-49718-0_4 157 158 MEAN ELEMENTS will be referred to by means of pairs of integers α /β , respectively denoting an integer number of solar years and the corresponding number of intercalary months (rather than lunar months). With this notation, the list of known values is the following: 19/7 (classical); 391/144; 410/151; 429/158; 448/165; 505/186; 562/207; 600/221; 619/228; 657/242; 676/249 (non-classical).2 The two constants α and β sufficiently characterize these various Metonic equivalences, classical and non-classical. Quite naturally, however, Chinese sources also mention the corresponding number of lunar months, γ : α solar years = γ lunar months = (12α + β ) lunar months. (4.1) Moreover, like all other Chinese astronomical canons, Metonic canons also use improper fractions a/b and c/d in order to represent the mean lengths of their solar years and lunar months, respectively, the denominators b and d generally being different from one another. The five constants α , β , γ , a/b and c/d are of course not mutually independent. Therefore, it would be possible to deduce all related Chinese calendrical procedures from a more limited set of constants. Yet, despite these redundancies, it seems more appropriate to follow Chinese sources as closely as possible. Metonic Calculations Let us suppose that the constants α , β , γ , a/b and c/d belong to a given astronomical canon: Then, the determination of the calendar of any year x also relies on the value of the time parameter t(x − 1) = t, obtained as indicated on p. 139 above and representing the integer number of solar years elapsed between the Superior Epoch of the canon in question and the winter solstice of its support year. Let also m(x − 1), e(x − 1), j(x − 1), q1 (x − 1) and n11 (x − 1) be, respectively, the integer number of lunar months contained in the interval from the Superior Epoch to the winter solstice of the support year of the year x, the epact of the year x (definition on p. 149 above), the number 2 Further chronological, historical and terminological details are provided on p. 354 below. METONIC CONSTANTS 159 of whole days in m(x − 1) lunar months, the winter solstice of the year x − 1 and the new moon of the eleventh month of the same year. Then, these various quantities are obtained by means of the following formulae (A). Moreover, various related statements (B), formulated by taking avail of the above definitions of bin(x, y) and x mod y (p. 142 and p. 10, respectively) are also essential in various respects: (A) Formulae m(x − 1) = ⌊γ t/α ⌋ (4.2) e(x − 1) = γ t mod α (4.3) j(x − 1) = ⌊cm/d⌋ (4.4) q1 (x − 1) = bin(at, b) (4.5) n11 (x − 1) = bin(cm, d) (4.6) (B) Other Statements Determination of the full or hollow character of lunar months, year type (ordinary or intercalary) and, if need be, approximate rank of the intercalary month of the year x: Criterion 4.1 (Full and Hollow Months) Let c/d = (29 + c′ /d) d be the length of a lunar month and ni = < g; f > an arbitrary new moon. Then, if f ≥ d − c′ the month mi = [ni , ni+1 [ if full and hollow otherwise. Quasi-Criterion 4.1 If e(x − 1) ≥ α − β (intercalary limit) then the year x has an intercalary month. Approximate Result 4.1 If the quasi-criterion 4.1 is verified, then the integer ⌊12(α − e)/β ⌋ gives an approximate value of the number of lunar months contained in the interval between n11 (x − 1) and of the new moon marking the probable beginning of the intercalary month in question. The winter solstice q1 (x − 1) and the new moon n11 (x − 1) being determined from the above formulae 4.2 to 4.6, the values of all subsequent mean solar breaths and mean new moons required for the construction 160 MEAN ELEMENTS of the surface calendar of the year x are readily obtained from repeated additions of a/24b and c/d to their respective values. Then, a reduction modulo 60 of the integer parts of these results leads to the temporary list of the solar breaths and new moons of the year x. Of course, the final numbering of these new moons also depends on the previous determination of the rank of a possible intercalary month. When such a month exists, however, its exact rank is easily obtained by checking all possible cases of a double inequality (criterion 3.1, p. 151 above). By contrast, the above quasi-criterion 4.1, p. 159, is less useful in this respect, because it sometimes delivers negative results when the year x is intercalary. In fact, we sometimes come up against this difficulty because 4.1 has been obtained, as will be shown below, by restrictively checking the existence of an intercalary month between [q1 (x − 1) and q1 (x)[ and not in the year x itself. The two following examples clearly illustrate this point. Example 4.1 Is the year Tianjin 14 (515) intercalary? Answer: the astronomical canon used in 515 is the Daming li (510–589) and we have: t(x) = 51, 939 + (x − 462), α = 391, β = 144, γ = 4836. (4.7) (4.8) Moreover, from 4.3, p. 159 above, e(514) = 227 and α − β = 247. Therefore, the inequality e(514) ≥ 247 is false and, from the quasicriterion 4.1, the year 515 is not intercalary. However, all available tables of the Chinese calendar indicate, on the contrary, that its last month, 12*, is intercalary. Example 4.2 Is the year Zhide 1 (583) intercalary? Answer: The year 583 also depends on the Daming li and, we have e(582) = 244 < 247. Therefore, the same quasi-criterion concludes negatively again. Yet, tables of the Chinese calendar indicate, on the contrary, that its month 11* is intercalary. METONIC CONSTANTS 161 Likewise, in the case of the years 534 and 572, this quasi-criterion also delivers a false result (their month 12* is intercalary each time). Apart from these rare exceptions, however, all tables of the Chinese calendar show that all the years comprised between 510 and 589 (years of official validity of the Daming li) have intercalary months not falling outside the interval [q1 (x − 1), q1 (x)[ and, in their case, the quasicriterion 4.1 delivers a correct result. More generally, the same conclusion holds for all other Metonic astronomical canons, intercalary months 11* or 12* being extremely rare. Justifications Formula 4.2, p. 159 Let t be the number of solar years of the interval [O, q1 ], where O and q1 are respectively the winter solstice of the Superior Epoch and of the year (x − 1). Let also n11 and n12 be the two new moons of the year x − 1, immediately preceding and following q1 : n11 O (Superior Epoch) q1 n12 e t solar years Then, the interval [O, q1 [ can be decomposed into two contiguous intervals [O, n11 ] and [n11 , q1 ] and, from the fundamental coupling between lunar months and solar breaths, the winter solstice q1 always belong to the month [n11 , n12 [. Consequently: n11 ≤ q1 ≤ n12 . The length of the interval [n11 , q1 ] is thus necessarily smaller than a lunar month and the sought integer number of months is equal to the integer number of months contained in the interval [O, n11 ]. Moreover, from the Metonic equivalence 4.1, each solar year is composed of γ /α lunar months. The interval [O, q1 ] is thus composed of γ t/α lunar months. The integer part of this result, ⌊γ t/α ⌋, is thus equal to the integer number of lunar months, m(x − 1), contained in the interval [O, q1 ]. 162 MEAN ELEMENTS Formula 4.3, p. 159 The epact of the year x, e(x − 1) or e, is equal to the integer remainder of the division of γ t by α or, in our notation, γ t mod α . The length of the interval [n11 , q1 ] is thus equal to e/α lunar months. The original Chinese sources, however, only indicate ‘e’. Moreover, given that the length of the lunar month generally differs from one astronomical canon to the next, it must be noted that the fact that when two such values relating to two different astronomical canons happen to be equal, they do not necessarily represent the same number of days. Formula 4.4, p. 159 The justification is trivial: once the mean value c/d of the lunar month is given, the integer number of days contained in m(x − 1) lunar months is known. Formula 4.5, p. 159 The interval between the Superior Epoch O and the winter solstice q1 (x − 1) contains atb days because it is composed of t solar years having ab days each. Hence the following binomial expression, bin(at, b), of this winter solstice: q1 (x − 1) =< ⌊ at mod 60⌋; at mod b > . b (4.9) Formula 4.6, p. 159 The reasoning is the same as in the case of the winter solstice, at being replaced by cm and b by d: the number of days between the Superior Epoch O and n11 (x − 1) is equal to cm d . Hence the binomial expression, bin(cm, d), of this new moon: n11 (x − 1) =< ⌊ cm mod 60⌋; cm mod d > . d (4.10) Criterion 4.1, p. 159 Let us count the time elapsed since the instant of midnight marking the beginning of the day to which a new moon ni belongs and let us suppose that this new moon happens in a later instant of the same day, determined by the fraction df , smaller than one day. Then, if the length of a NON-METONIC CANONS 163 lunar month is equal to 29 + c′ /d, the instant of occurrence of the new ′ ′ ′ moon ni+1 is equal to df + 29 + cd = 29 + f +c d . Hence, if f + c ≥ d, ′ or f ≥ d − c , the sum in question is at least equal to 30. Consequently, from the point of view of the calendrical surface structure, the lunar month [ni , ni+1 [ is necessarily full. Quasi-Criterion 4.1, p. 159 With the lunar month as unit of time, the value of the epact e(x − 1) (or e) of the year x is equal to αe lunar months. But Metonic canons are such that α solar years = (12α + β ) lunar months. The yearly increase of the epact is thus equal to αβ lunar months. Now, supposing that the β β ≥ α− epact e(x − 1) ≥ α − β amounts to saying that e(x−1) α α = 1− α. Therefore, the epact becomes at least equal to one lunar month some time during the solar year determined by the interval [q1 (x − 1), q1 (x)]. Consequently, an intercalary month necessarily exists, somewhere in this interval, except that the definition of intercalary months is not taken into account here. Approximate Result 4.1, p. 159 As already noted, the increase of the epact e(x − 1) = e of the year x is equal to αβ lunar months per annum, with respect to a given Metonic canon determined by α and β . Its monthly increase is therefore equal to 12βα lunar months per lunar month. After k lunar months, its value is βk thus equal to 12 α lunar months. βk Consequently, when αe + 12 α = 1 lunar month for a certain integer k, the value of the epact becomes equal to one lunar month and the integer part of the value of k deduced from this result, that is ⌊ 12(αβ−e) ⌋, gives the sought approximate number of lunar months. Non-Metonic Canons The determination of the mean solar and lunar elements of a given year x in non-Metonic astronomical canons depends on two improper fractions a/b and c/b, respectively equal to the length of a solar year and of a lunar month and having the same denominator b. With the previous conventions and notations, the number of days elapsed between the Superior Epoch and the winter solstice q1 (x − 1) is 164 MEAN ELEMENTS equal to at/b, and the binomial representation of this winter solstice is identical to what it would have been in the case of a Metonic canon: q(x − 1) = bin(at, b). (4.11) In its turn, the epact e(x − 1) is now calculated from the solar and lunar constants a and c, instead of Metonic constants, and we readily obtain: e(x − 1) = at mod c. (4.12) Moreover, when converted into days, its value is: at mod c . (4.13) b Then, from this latter expression and by taking into account the fact that the epact e(x − 1) is equal to the age of the moon at the instant of the winter solstice q1 (x − 1), the number of days between the Superior Epoch and n11 (x − 1) is equal to: e(x − 1) = at − at mod c . b Hence the binomial representation of n11 (x − 1): j= n11 (x − 1) = bin(at − e, b). (4.14) (4.15) Lastly, supposing that the cumulated epact is at least equal to a lunar month, we have: a c c e(x − 1) + ( − 12 × ) ≥ . b b b (4.16) Hence: e(x − 1) ≥ 13c − a . b (4.17) The numerator of this fraction, 13c − a, is called runxian dd (intercalary limit), and is similar to the constant (α − β ) of Metonic astronomical canons; moreover the following pseudo-criterion analogous to the preceding one (4.1, p. 159 above) also exists: NON-METONIC CANONS 165 Quasi-Criterion 4.2 If e(x − 1) ≥ 13c − a then an intercalary month occurs somewhere in the year x. The fundamental elements necessary for the calculation of the calendar of the year x being so determined, its subsequent mean new moons and solar breaths are then obtained as before. However, the previous Metonic calculations are not excluded because, once given two fractions a/b and c/b (equal to the mean lengths of the solar year and lunar month), it is generally possible to deduce three Metonic constants α , β and γ (see 4.1, p. 158 above) from the following indeterminate equation: α a c = (12α + β ) . b b (4.18) Example 4.3 Metonic constants in the Xuanming li ddd. In this case, we have: a 3, 068, 055 d = , b 8400 c 248, 057 d lunar month = = . b 8400 solar years = With these values, the above fundamental Metonic relation 4.1, p. 158 becomes: 91, 371α − 248, 057β = 0. (4.19) Hence α = 248, 057k and β = 91, 371k and the smallest strictly positive solution of this equation is obtained with k = 1. Therefore, the Xuanming li ddd can be regarded as a Metonic astronomical canon. As such, this example may seem devoid of interest since the formula 4.11, p. 164 to 4.15, p. 164, provides the values of the fundamental solar and lunar elements of the Xuanming li ddd much more easily. It remains, however, that the Metonic constants α = 248, 057 and β = 91, 371 so obtained both belong to its list of constants in Chinese sources,3 as though all the calculations of the Xuanming li ddd were 3 Xin Tangshu, j. 30A, ‘li 6a’, p. 745. 166 MEAN ELEMENTS Metonic, while, in fact, only their restriction to its mean elements is so. However, the Xuanming li calculations never use them. They can thus be regarded as ‘fossil’ constants. Calculation Variants Although the preceding techniques uniformly apply to the calculation of mean elements in all astronomical canons, variants also exist. The already mentioned Sifen li ddd, and the Daye li ddd procedures offer an example of such a variant, based on the prior calculation of the epact. In both canons, the calculation of the new moon n11 (x − 1) takes the following unusual form,4 supposing that the epact e(x) has been previously obtained by calculating 235t mod 19 and 410t mod 151, in agreement with the above general technique 4.3 concerning Metonic canons of type 19/7 and 410/151, respectively. In the first case, for example, n11 (x − 1) is calculated from: n11 (x − 1) = 235 × 1461t − 1461e . 940 (4.20) This not particularly intuitive formula can be justified as follows: in the Sifen li ddd, the length of the solar year is equal to 1461/4 d and, in our usual notation, the number of days between the Superior Epoch and the winter solstice q1 (x − 1) is equal to: 1461t 235 × 1461t = . 4 940 (4.21) Moreover, when taking the day as main unit, the value of the epact e(x − 1) of the year x is equal to: e 27, 759 1461e × = . 19 940 940 (4.22) Hence the above variant 4.20, obtained by subtracting this last value from 4.21. 4 Hou Hanshu, zhi 3, ‘lüli 3’, p. 3062–3063, Suishu, zhi 12, ‘lüli zhong’, p. 436. CALCULATION VARIANTS 167 Quite differently, the Jiyuan li ddd (Era Epoch Canon) (1106– 1135), the Kaixi li ddd (Incipient Auspiciousness Canon) (1208– 1251) and a few other astronomical canons from the Song dynasty, calculate their binomial representation of the winter solstice q1 (x − 1) from the relatively complex expression:5 q1 (x − 1) = < ⌊ at mod 60b ⌋; (at mod 60b) mod b > . b (4.23) This new formula can be justified in the following way: Let t(x − 1) (or more simply t) be the integer number of years between the Superior Epoch and the winter solstice q1 (x − 1). Then: a q1 (x − 1) = t mod 60. b (4.24) Now, taking avail of the following property of distributivity:6 k(x mod y) = (kx) mod (ky), (4.25) where k is any strictly positive coefficient, the equation 4.24 can be transformed by distributing the coefficient b in the same way so that: q1 (x − 1) = at mod 60b . b (4.26) Hence 4.23. The initial reduction modulo 60 is thus replaced by a reduction modulo 60b but the final result is identical to the one which would have been obtained more simply by using the above formula 4.5. 5 Songshi , j. 79, ‘lüli 12’, p. 1848–1849 and ibid., j. 84, ‘lüli 17’, p. 2025. the generalized modulo and its distributivity property, see R.L. Graham, D.E. Knuth and O. Patashnik 1990, p. 82. 6 On CHAPTER 5 TRUE ELEMENTS (618–1280) Introduction Calendrical calculations based on true elements1 are naturally much more complex than those depending on mean elements and the Chinese case fully confirms this obvious point because its determination of moon phases requires two distinct lunar and solar corrections modifying their mean values in a rather complex way. Two main types of calculations with true elements are documented in Chinese sources, viz. those concerning astronomical canons adopted between 619 and 1280 and those of the two latest canons, the Shoushi li (1281–1364) and the Datong li (1385–1644). This chapter and the next one are respectively devoted to an operational presentation of the corresponding techniques. Let pi , ∆⊙ (pi ) and ∆$ (pi ) be a mean lunar phase and its lunar and solar corrections, respectively. Then, the calculation of these corrections involves two kinds of lunar months, namely the already well-known mean lunar (synodic) month and the mean anomalistic month.2 Two 1 True elements and their chronology have already been briefly introduced on p. 144 f. above. 2 In Chinese astronomical canons, the mean anomalistic month is a period determined by the return of the lunar motion to the same angular velocity, which is very uneven and varies from 10 to 14 degrees of longitude per day. It has 27.55 days, approximately (decimal notation) and it starts either when the moon reaches its highest or slowest velocity, as the case may be. In modern terms, the mean anomalistic month is equal to the mean interval of time between successive passages of the moon through its perigee, its closest point to earth. By contrast, Chinese astronomical canons never use any notion of distance between celestial bodies. The anomalistic month was first introduced in the Qianxiang li ddd (223–280) (Chen Meidong, 1995, p. 237) but it was then only used for questions of positional astronomy and not for calendrical calculations. Concerning its values in this latter case, see Appendix G, p. 365 below. © Springer-Verlag Berlin Heidelberg 2016 J.-C. Martzloff, Astronomy and Calendars – The Other Chinese Mathematics, DOI 10.1007/978-3-662-49718-0_5 169 170 TRUE ELEMENTS (618–1280) new time parameters are also required. The first is generally called ruqi dd and the second ruli dd (these two technical terms3 respectively mean ‘degree of advancement [of a lunar phase] into a solar breath’ and ‘degree of advancement [of a lunar phase] into an anomalistic month’).4 In addition, the solar breaths implied in the definition of the ruqi d d are sometimes taken in true value despite the fact that the calculation of the solar component of the calendar never relies on true values. Lastly, the general form of the resulting calculation is the following: pi = pi ± ∆⊙ (pi ) ± ∆$(pi ). (5.1) True Solar Breaths The determination of the 24 true solar breaths of a given solar year, [q1 (x − 1), q1 (x)[, necessitates the successive calculation of: 1. its mean solar breaths, q1 (x − 1), q2 (x − 1), . . . ; 2. the lengths li = l([qi , qi+1 [) , i = 1, 2, . . . , 24 of its true solar periods; 3. the values of the sought true solar breaths from trivial additions. In practice, let us suppose that the length of the solar year is equal to a d The mean length of a solar period, l, is thus equal to 24b . Next, let us also suppose that we have extracted 24 coefficients δi from a readymade Chinese table devoted to the solar inequality. Then, the li are obtained as follows: ad b . li = a δi + 24b b i = 1, 2, . . . , 24. (5.2) Some Peculiarities Leading to Simplifications It is generally not necessary to calculate the totality of the 24 li because the true values of the two solstices, q1 and q13 , are always identical to lunar anomalistic month is sometimes also called zhuan d. Consequently, ‘ruzhuan dd’ and ruli have the same meaning (the literal meaning of zhuan is ‘revolution’, a term as vague as li d (epoch or age, here)). 4 See their precise definitions on p. 172 and 179 below. 3 The TRUE SOLAR BREATHS 171 their mean values. Moreover, the existence of symmetries induced by omnipresent yin-yang considerations imply the equality of solar breaths equally distant from the summer solstice. Therefore, the calculation of only (24 − 2)/2 = 11 different values is necessary in order to determine all the li . Despite this simplification, the calculations are often slightly more involved than meets the eye, for they also implicitly involve various modes of representations of numbers by means of compound fractions, omnipresent in all Chinese astronomical canons. There is a rewarding counterpart, however, because these calculations exclude any approximation and deliver exact results, wholly identical with those which would have been originally obtained by Chinese calendars makers, independently of the peculiarities of their arithmetical operations. Moreover, the results remain valid for all solar years because the variations of length of solar periods are deemed identical from year to year. Example 5.1 Calculation of the 24 li (Dayan li) ddd. In the Dayan li the mean lengths of the solar year and of a solar period are respectively equal to: a 1, 110, 343 d = b 3040 and ( ) a 664 7 d = 15 + + 24b 3040 3040 × 24 and the list of the 24 δi , i = 1, 2, . . . , 24, is the following:5 [−2353, −1845, −1390, −976, −588, −214, +214, +588, +976, +1390, +1845, +2353, +2353, +1845, +1390, +976, +588, +214, −214, −588, −976, −1390, −1845, −2353] 5 See Jiu Tangshu , j. 34, ‘li 3’, p. 1237–1239 and Xin Tangshu, j. 28A, ‘li 4a’, p. 643–644. See also Zhang Peiyu 1982; Zhang Peiyu, Lu Yang and Liu Guixia 1986; Wang Yingwei 1998, p. 204–205. The signs that we have appended to the δi (next page) are such that ying d = ‘−′ and suo d = ‘+′ but, given the usual meaning of these two characters ying d (profit, increase, benefit, etc) and suo d (loss, decrease, slackening, etc.), this association may seem somewhat counterintuitive. However, when the solar motion is in a phase of excess (or of expansion), ying d, solar periods become shorter than their mean value. The corresponding δi must thus be subtracted from the mean value in question. By contrast, in the case of a suo d phase, the reverse is true. Similarly, when a phenomenon belongs to a phase of decrease, it sometimes happens that ying d designates an increase of something diminishing. See Qu Anjing, Ji Zhigang and Wang Rongbin 1994, p. 263 . 172 TRUE ELEMENTS (618–1280) Therefore: ( ) 664 7 δi d li = 15 + + + 3040 3040 × 24 3040 i = 1, 2, . . . , 24. (5.3) Hence the following table giving the values of the 24 li by means of compound fractions whose successive denominators are 3040 and 3040 × 24: i 1 or 24 2 or 23 3 or 22 4 or 21 li (days/3040/24) 14;1351,7 14;1859,7 14;2314,7 14;2728,7 i 5 or 20 6 or 19 7 or 18 8 or 17 li (days/3040/24) 15;0076,7 15;0450,7 15;0878,7 15;1252,7 i 9 or 16 10 or 15 11 or 14 12 or 13 li (days/3040/24) 15;1640,7 15;2054,7 15;2509,7 15;3017,7 Table 5.1. The lengths li of true solar periods (Dayan li dddd). As noted above, these li are repeated symmetrically in the sense that li = l25−i for 1 ≤ i ≤ 12. Then, the true values qi of all solar breaths are obtained as follows: q1 = q1 q2 = q1 + l1 q3 = q1 + l1 + l2 = q2 + l2 and so on. A Technical Term: The ruqi Definition 5.1 (The ruqi) Let a mean lunar phase p be given. Then ruqi(p) is equal to the length l of the time interval between p and the nearest preceding solar breath qi , taken either in mean or true value according to the astronomical canon under consideration (Fig. 5.1). qi p̄ ruqi( p̄) Solar Period Figure 5.1. The ruqi dd. qi+1 A TECHNICAL TERM: THE RUQI 173 From this definition, the value of the ruqi is thus necessarily such that: 0 ≤ ruqi(p) < l([qi , qi+1 [). (5.4) Moreover, if qi ≤ p ≤ qi+1 then ruqi(p) = l([qi , p[), with mean or true solar breaths, as the case may be. In astronomical canons from the Tang dynasty (618–907), the values of the ruqi depend on true solar breaths but, from the Chongxuan li d dd (893–938), only mean solar breaths enter into the picture. As the following examples show, the complexity of the calculations is significantly different in each case but the previous determination of the fundamental lunar and solar elements of the calendar of the concerned year always remains essential. A General Mode of Calculation of the ruqi Given an astronomical canon based on mean or true elements, let us find the successive values of the ruqi required for the calculation of the calendar of a year x. Given that the earliest event taken into account in the calculation of the calendar of a year x is the new moon n11 of its support year, x − 1, the first sought value, ruqi(n11 ), involves the last solar breath preceding n11 (x − 1). Hence an unexpected difficulty because, as a rule, calendrical calculations start from n11 (x − 1) and not from any previous calendrical event. From figures 5.2 and 5.3 below, it appears that the last solar breath preceding n11 (x − 1) is either q24 or q23 . However, it is not necessary to take these two quantities as such into account because the same figures directly provide the sought result in the following form, where e is the epact of the year x: { l24 − e e ≤ l24 , ruqi(n11 (x − 1)) = l24 + l23 − e e > l24 . (5.5) 174 TRUE ELEMENTS (618–1280) q23 q24 q1 n11 e ruqi(n11 ) l24 Figure 5.2. The calculation of ruqi(n11 ) (first case). q23 q24 n11 q1 e ruqi(n11 ) l23 + l24 Figure 5.3. The calculation of ruqi(n11 ) (second case). Therefore, ruqi(n11 ) can be obtained either from l24 , or l24 and l23 , regardless of q23 (x − 1) and q24 (x − 1), that is without taking into account the specific solar breaths preceding n11 (x − 1).6 Apart from this peculiarity, all the other values of the ruqi can be readily obtained from its definition as soon as the solar and lunar elements of the year x have been established and listed in sequential order. It must be noted, however, that the solar breath implied in the calculation of a given moon phase is not necessarily always the same from year to year. This direct mode of calculation is of course independent of the mean or true character of the solar breaths. In the case of mean solar breaths, however, the calculations can be done in a much more systematic way, as the example of the Jiyuan li ddd calculations clearly shows.7 The corresponding technique will be presented on p. 175 f. below. Another Mode of Calculation The preceding method is not really satisfactory because calendrical calculations normally imply reductions modulo 60 of moon phases and solar breaths which are thus only related to the last origin of the sexagenary cycle and not to a common origin. The following method, which Xuanming dd canon explicitly uses this mode of calculation. See Koryǒ sa/Gaoli shi, j. 50, ‘li 1’, p. 88 (notice p. 399 below). 7 See Songshi, j. 79, ‘lüli 12 ’, p. 1856, section qiu jing-shuo-xian-wang ruqi dd ddddd (Determination of the ruqi for new moons and other moon phases). 6 The A TECHNICAL TERM: THE RUQI 175 does not appear in Chinese sources, refers all calendrical events to a single common origin, the winter solstice q1 (x − 1), and renders them ipso facto mutually comparable. As before, let li , i = 1, 2, . . . , 24 be the variable lengths of the solar breaths. With the new origin of time, the abscissas of the successive solar breaths, q1 , q2 , . . . , q24 are thus equal to: 0, l1 , l2 , . . . , l1 + l2 + . . . + l23 . Let also msyn , e and n1 , n2 , . . . be the respective mean values of the lunar month, epact and new moons, temporarily numbered in natural order from the first one located after the new origin of time. Then, the abscissas of the mean successive new moons are: n1 = msyn − e n2 = 2msyn − e ... nk = kmsyn − e ... The abscissas of all other phases of the moon would of course be easily obtained in the same manner. Once these calculations are done, the successive ruqi(nk ), k = 2, 3, . . . are directly obtained from definition 5.1 above, i.e. by determining for each k an index j such that: q j ≤ kmsyn − e ≤ q j+1 . This procedure provides all sought values from ruqi(n12 ). For the four previous phases of the moon, an easy adaptation of this method is still necessary. The Calculation of the ruqi from Mean Solar Breaths The preceding techniques remain valid for mean solar breaths but a more appropriate mode of calculation, recorded in Chinese sources,8 is also available in their case. 8 See Songshi, ibid., j. 79, ‘lüli 12’ p. 1856. 176 TRUE ELEMENTS (618–1280) Let us number the relevant solar breaths q1 (x − 1), q2 (x − 1), . . . and moon phases from n11 (x − 1) in natural order, in both cases, so that n11 is temporarily denoted n1 . Then, the following single formula solves the question of the ruqi dd calculations once and for all: ruqi(pi ) = ( at ) e msyn a − + (i − 1) mod b b 4 24b i = 0, 1, . . . (5.6) Justification The left expression between parentheses is equal to the time between the Superior Epoch and any subsequent mean lunar phase. The calculation of specific solar breaths is thus not required in order to determine the length of the interval of time between the mean phase in question and the last preceding solar breath. A mere reduction of the left a interval of time modulo 24b is sufficient. The following example shows the details of the calculations in the case of a Song astronomical canon. Example 5.2 Calculation of the ruqi dd for the moon phases of the first month of the year Jiading 11 (1218). In this example, the relevant astronomical canon is the Kaixi li dd d and, as usual, a previous determination of its solar and lunar mean elements, together with the ordinary or intercalary character of its months is necessary.9 With our usual notations for the solar year and the lunar month, the following elements must first be taken into account: t(x) = 7, 848, 183 + (x − 1206) a 6, 172, 608 d = b 16, 900 ( ) a 3692 d = 15 + 24b 16, 900 c 499, 067 d = b 16, 900 ( ) c 6466 3 d = 7+ + 4b 16, 900 ×4 13c − a = 315, 263 9 Songshi, ibid. j. 84, ‘lüli 17’, p. 2023 f. (solar years) (5.7) (solar year) (5.8) (solar period) (5.9) (lunar month) (5.10) (lunar phase) (5.11) (intercalary limit) (5.12) A TECHNICAL TERM: THE RUQI 177 Since the support year of the year 1218 is the year 1217, 5.7 implies that t(1217) = 7, 848, 194 solar years and, from the above technique of calculation for the epact (4.12 p. 164), e(1218) = 241, 692. This result being inferior to the ‘intercalary limit’ (runxian dd), 315,263, the year 1218 is ordinary and is thus composed of 12 lunar months. Consequently, the correspondence between new moons and other phases of the moon of the years 1217 and 1218, denoted this time by p1 , p2 , . . . on the one hand and n11 , n12 , . . ., on the other hand, is wholly determined by the following correspondence, partially given here but easily extended. p1 n11 p2 p3 p4 p5 n12 p6 p7 p8 p9 n1 p10 p11 p12 p13 n2 Then, from 4.23, p. 167 above, the winter solstice of the year 1217 is obtained by calculating at mod 60b: 6, 172, 608 × 7, 848, 194) mod 60 × 16, 900 = 419, 952 and from: ⌊ 419, 952 ⌋ = 24 and 419, 952 mod 16, 900 = 14, 352. 16, 900 Hence q1 (1217) =< 24; 14 352 >. Then, the list of the other mean solar breaths is obtained as usual and, from the definition of the epact (3.10, p. 149): n11 (1217) = 24 + 14, 352 241, 692 9260 − = 10 + = 10;09260. 16, 900 16, 900 16, 900 In the same spirit, the values of all other mean lunar phases is readily obtained. From these preliminary results (Table 5.2, next page), the calculation of the ruqi can be tackled. In agreement with the preceding analysis, we propound hereafter three different methods. Method 1 Order the solar breaths qi and lunar phases pi in calendrical order and apply directly the definition of the ruqi. 178 TRUE ELEMENTS (618–1280) j qi /pi 1 q3 2 p9 3 q4 Values 55;04836 9;10294 10;08528 j qi /pi 4 p10 6 p11 5 q5 Values 16;16760,3 24;06327,2 25;12220 j qi /pi 7 p12 8 p13 9 q6 Values 31;12794,1 39;02361 40;15912 Table 5.2. Calculated lunar phases (first lunar month, year (Jiading 11 (1218)). ruqi(p 9 ) = p9 − q3 = 60 + 9;10294 − 55;04836 = 14;05458 ruqi(p10 ) = p10 − q4 = 16;16760 − 10;08528 = 6;08232,3 ruqi(p11 ) = p11 − q4 = 24;06327,2 − 10;08528 = 13;14699,2 ruqi(p12 ) = p12 − q5 = 31; 12794,1 − 25;08528 = 6;00574,1 Method 2 Direct application of formula 5.6 above. ( ruqi(pi ) = ) 6, 172, 608 241, 692 499, 067 × 7, 848, 194 − +( )(i − 1) 16, 900 16 900 16, 900 × 4 ( ) 6, 172, 608 mod i = 1, 2, . . . 16, 900 × 24 6,172,608 Method 3 Given that e ≤ l24 = 16,900×24 , calculate ruqi(n11 ) accordingly (see 5.5, p. 173 above) and use 5.6, p. 176, with mean solar breaths (the Kaixi li procedure only uses such breaths): ruqi(n11 ) = l24 − e = ( Hence ruqi(pi ) = 6, 172, 608 241, 692 15, 500 d − = 16, 900 × 24 16, 900 16, 900 ) 15, 500 479, 067 + (i − 1) 16, 900 16, 900 × 4 ( ) 6, 172, 608 mod 16 900 × 24 i = 1, 2, . . . ANOTHER TECHNICAL TERM: THE RULI 179 Another Technical Term: The ruli Definition 5.2 (The ruli) Let p be a mean lunar phase. Then, the corresponding ruli is equal to the length of the interval between p and the last beginning of the anomalistic month (Fig. 5.4). ai ai+1 p̄ ruli( p̄) anomalistic month Figure 5.4. The ruli. This definition makes sense because, if p is a given mean lunar phase, an anomalistic month [ai , ai+1 [ such that ai ≤ p < ai+1 always exists. In our notation, ruli(p = l([ai , p]) (Fig. 5.4). Hence ruli(p) < man . The ruli The anomalistic month being never taken otherwise than in mean value, the calculation of the ruli dd is similar to that of the ruqi dd evaluated with respect to mean solar breaths (see 5.6 above). Consequently, only a reduction of the time elapsed between the Superior Epoch and the concerned lunar phase, modulo the length of the anomalistic month, is sufficient: ruli(pi ) = ( at ) e msyn − + (i − 1) mod man b b 4 i = 1, 2, . . . (5.13) In cases where the required values of the ruli only concern new moons, a replacement of msyn /4 by msyn is of course necessary.10 Tables and Interpolation Techniques Once ruqi(pi ) and ruli(pi ) have been determined and when the calculation of the true phases of the moon depends on solar and lunar tables, interpolation techniques are required in order to evaluate ∆⊙ (pi ) and ∆$ (pi ). 10 This procedure is notably used in Songshi, j. 79, ‘lüli 12’, p. 1867. 180 TRUE ELEMENTS (618–1280) In general, these interpolation techniques are either linear or nonlinear and the latter belong essentially to the quadratic variety. We are not certain, however, that such techniques were systematically used for calendrical calculations because the Linde dd and the Xuanming dd canons state that, when such calculations are at stake, interpolation techniques can be restricted to their linear variety either when it is known in advance that no solar or lunar eclipse will occur or when fast calculations are required.11 While this remark is certainly important, our limited knowledge of ancient Chinese eclipse calculations impedes our understanding of the situation. The question of the influence of ancient Chinese mathematical prediction of eclipses on calendrical calculations must thus be left open.12 The allusion to fast calculations is no less puzzling: why such calculations were required? Perhaps in order to check the results of nonsimplified calculations? Or because the official calendar was sometimes calculated by using less time-consuming methods? It was presumably not always available sufficiently in advance in certain regions of the Chinese empire far from the capital and local officials in charge of the calendar would possibly have been required to perform its calculations by using simplified methods at the expense of obtaining calendars slightly different from their wholly regular variety? We do not know. Anyway, the following presentation will concentrate on the most important techniques of interpolation and since most are based on tables of the solar and lunar inequalities, these will be first analyzed. Before giving the details, it is important to note that the overall pattern followed in both cases is formally similar (Fig. 5.5). Solar Tables In China, the solar inequality is said to have been discovered towards 560 by Zhang Zixin d d d, an astronomer of the Northern Qi dynasty (ca. 570 AD). Then, this innovation was first implemented in the Huangji li ddd (Sovereign-Pole canon), an influential, but nonofficial, astronomical canon, where a pattern of quadratic interpolation 11 Jiu Tangshu, j. 33, ‘li 2’, p. 1185 and Koryǒ sa/Gaoli shi, j. 50, ‘li 1’, p. 94. course, these historical procedures should not be confused with the retrocalculation of ancient eclipses by means of modern astronomical theories. 12 Of TABLES AND INTERPOLATION TECHNIQUES 181 deviation O A B C D days Figure 5.5. The cosine-like appearance of the solar and lunar inequalities (Chinese canons with true elements). In the case of the sun, OD = a solar year (from a winter solstice O to the next D) and A, C and B are respectively equal to the instants of occurrence of the two theoretical equinoxes and of the summer solstice. In the case of the moon, OD = an anomalistic month, extending from one lunar perigee to the next and A, B, and C are the instants of time associated with the division of the anomalistic month into four equal intervals. In certain canons, however, the anomalistic month extends from a lunar apogee to the next but the lunar inequality still follows an inverted but similar pattern. For a minute description of specific instances of these inequalities, see also Qu Anjing, Ji Zhigang and Wang Rongbin 1994, p. 226–235 and p. 269 f. (The present schema is extremely simplified. In particular, the attested patterns do not necessarily display continuous variations, but the general trend is the same each time). very often taken up again in later official canons is defined for the first time, modulo some variations. Following the initial model of the Huangji li, the tables of the solar inequality are double-entry tables, generally providing four lists of twenty-four coefficients, associated one to one with the 24 solar periods, but not with smallers intervals, such as the day. Hence the necessity of an interpolation process in order to calculate the value of the solar inequality corresponding to any given instant located between two consecutive solar breaths. Without going into too much detail, the overall structure of these lists can be described as follows: 182 TRUE ELEMENTS (618–1280) First list names of the 24 solar periods, referred to from the name of their initial solar breath. As usual, these solar periods are not necessarily always listed in exactly the same order but the first period is always the one beginning with the winter solstice. Sometimes too, as in the case of the Dayan canon, their lengths are variable but most often, they are taken in mean value only. In the latter case, their lengths l are thus equal to one solar year divided by 24 but in practice, the value really used is slightly different. For Y 4946 instance, in the Kaixi li ddd, l = 3135 206 rather than 24 = 325 . The exact rationale behind such simplifications is unknown. Second list D = [D1 , D2 , . . . Di , . . . , D24 ]: values of the successive differences between the mean and true solar motion over each solar period. Third list d = [0, D1 , D1 + D2 , D1 + D2 + D3 , . . .]: cumulative sums of the elements of the preceding list. Remarkably, the first and thirteenth elements so obtained are always equal to zero, a fact reflecting the Chinese idea of the equality of the true and mean values of the solar inequality at the instant of the winter and summer solstices. Fourth list ∆ = [∆1 , ∆2 , . . . , ∆24 ]: results of the division of the elements of the third list by the mean apparent lunar daily sidereal motion, an astronomical constant equal to approximately k = 13.37 Chinese degrees (du d) per day (a du contains as many degrees as the number of days in a sidereal year). Fifth list δ : this list is derived from the fourth exactly in the same way as the third from the second. From this simplified description, it is obvious that the various coefficients of these lists are not independent from each other. Consequently, it would be easy to provide a more compact description of their tables by taking avail of obvious simplifications induced by such a characteristic. In practice, however, it remains quite desirable to describe the original procedures in terms of the above coefficients, even though some simplifications are not easily avoidable. TABLES AND INTERPOLATION TECHNIQUES qi 1 2 3 4 5 6 7 8 9 10 11 12 Di d 2353 d 1845 d 1390 d 976 d 588 d 214 d 214 d 588 d 976 d 1390 d 1845 d 2353 di dd d 2353 d 4198 d 5588 d 6564 d 7152 d 7366 d 7152 d 6564 d 5588 d 4198 d 2353 ∆i d 176 d 138 d 104 d 73 d 44 d 16 d 16 d 44 d 73 d 104 d 138 d 176 δi dd d 176 d 314 d 418 d 491 d 535 d 551 d 535 d 491 d 418 d 314 d 176 qi 13 14 15 16 17 18 19 20 21 22 23 24 Di d 2353 d 1845 d 1390 d 976 d 588 d 214 d 214 d 588 d 976 d 1390 d 1845 d 2353 di dd d 2353 d 4198 d 5588 d 6564 d 7152 d 7366 d 7152 d 6564 d5588 d 4198 d 2353 ∆i d 176 d 138 d 104 d 73 d 44 d 16 d 16 d 44 d 73 d 104 d 138 d 176 183 δi dd d 176 d 314 d 418 d 491 d 535 d 551 d 535 d 491 d 418 d 314 d 176 Table 5.3. The solar table of the Dayan Astronomical canon. For instance, as already noted in a particular case on p. 171 above and as explained again below, the identification between Chinese quantities seemingly having a sign and usual positive or negative quantities is generally anything but straightforward. Typically, the way the coefficients of the above Table 5.3 are represented offers a complex situation because, in its case, no fixed association between opposite characters, such as ying d and suo d, and positive or negative quantities exists. Following original sources to the letter in this respect would disclose many interesting aspects of the related procedures. But the calculations may also be performed more directly, by means of an adequate fixed attribution of a positive or a negative algebraic sign to all relevant Chinese quantities without modifying in the least the final results, and that is precisely what will be done in the following. Example: The Solar Table of the Dayan li and its ‘Signs’ The Dayan li solar table (Table 5.3 above) is composed of pairs of coefficients having the same value and preceded with characters having opposite meanings, namely ying/suo d/d (expansion/contraction), xian/hou d/ d (advance/retardation), yi/sun d/ d (profit/loss), and tiao/nü d/ d (waning/waxing). Contrary to what might be expected, however, these four pairs are not reducible to binary oppositions between signed quantities because an analysis of the related procedures 184 qi 1 2 3 4 5 6 7 8 9 10 11 12 TRUE ELEMENTS (618–1280) Di 2353 1845 1390 976 588 214 −214 −588 −976 −1390 −1845 −2353 di 0 2353 4198 5588 6564 7152 7366 7152 6564 5588 4198 2353 ∆i 176 138 104 73 44 16 −16 −44 −73 −104 −138 −176 δi 0 176 314 418 491 535 551 535 491 418 314 176 qi 13 14 15 16 17 18 19 20 21 22 23 24 Di −2353 −1845 −1390 −976 −588 −214 214 588 976 1390 1845 2353 di 0 −2353 −4198 −5588 −6564 −7152 −7366 −7152 −6564 −5588 −4198 −2353 ∆i −176 −138 −104 −73 −44 −16 16 44 73 104 138 176 δi 0 −176 −314 −418 −491 −535 −551 −535 −491 −418 −314 −176 Table 5.4. The solar table of the Dayan li ddd (modified version, adapted to a simplified, but equivalent, formulation of calendrical and astronomical calculations). shows that, when xian d = ‘+’ then ying d = ‘+’ and suo d = ‘−’ whereas, when hou d = ‘−’ then, on the contrary, ying d = ‘−’ and suo d = ‘+’13 . Therefore, the interdependence between these three characters implies that ying d cannot always remain associated in a fixed manner with ‘+’ and suo d with ‘−’, or with ‘−’ and ‘+’, respectively. These characters are thus not algebraic signs but, rather, contextdependent binary indicators, impossible to associate with fixed signs.14 Still, it turns out that it is always possible to modify the tables in order to obtain coefficients astronomically conforming to the solar inequality (Table 5.4 above), without contradicting the Chinese related procedures. In a different order of ideas, we also note that the second line of this table contains the following pairs of characters instead of specific numbers: xian duan dd (initial limit); nü chu dd (beginning of waxing); hou duan dd (final limit); and dd (beginning of waning). In fact, judging from the fact that the mean and true values of q1 and q13 13 See 14 See Qu Anjing 2008, p. 141. also the note 5, p. 171 above. TABLES AND INTERPOLATION TECHNIQUES 185 –the instants of the two solstices– are equal, these four verbal indications must be interpreted as meaning zero. Lunar Tables Whereas only one sort of solar table exists, lunar tables fall into two categories depending on whether they start from what is for us the perigee of the lunar motion or its apogee, as already noted at the beginning of the present chapter. In each case, they thus cover either the interval from one perigee to the next or from one apogee to the next. Despite this double possibility, all lunar tables are formally built on the same pattern, modulo numerous philological variations and other peculiarities, concerning notably the choice of units. We have chosen here to analyze the lunar table of the Linde li d dd, or rather one of its slightly modified versions, obtained from a replacement of its binary indicators by fixed algebraic signs, in the same spirit as the previous solar tables, but with a further omission of all ‘+’ signs and by following M. Uchida 1975’s interpretation (see Table 5.5, p. 193 below). The Linde dd lunar table extends from one lunar perigee to the next and is based on a division of the anomalistic month (equal to 27.55d ) into 28 intervals, one day long each. It contains the following lists of coefficients: First list coefficients Λi called licheng dd (lunar journey) representing the daily lunar true motion; Second list coefficients λi called zengjian lü ddd (additive-subtractive rates) and equal to (Λi −mmanan)×1340 , where 1340 is the common denominator of all fractions expressing various sorts of years and months in the Linde li dd and where man is the mean value of the anomalistic month, equal to 895.79.15 Third list cumulated sums of the preceding values. 15 895.79 = solar year + 1 (with solar year = 489428 d and lunar month = 39571 d ). lunar month 1340 1340 Moreover, 895/67 = 13.37 du d (a du is a Chinese degree, that is an angular unit such that 360 ordinary degrees= 365.25 du and ‘67’ is a primary constant of the Linde li). 186 TRUE ELEMENTS (618–1280) Quite differently from solar tables, certain entries of lunar tables contain two coefficients instead of a single one. Their role will be explained in the section below devoted to the calculation of the lunar correction. The Solar Correction Let the solar correction concerning a given mean lunar phase p be required.16 Then, an index i such that qi ≤ p < qi+1 always exists and the first thing to do is to calculate ruqi(p) = t (or ruqi(p) = t as the case may be) by using one of the above procedures. Then, noting that qi ≤ t < qi+1 , we know that the coefficients of the solar table to be used in order to determine the interpolated value of the solar correction depend on the solar period [qi , qi+1 [, and thus correspond to its i-th entry. When this correction is limited to a linear interpolation, the calculations are straightforward and no further explanations are necessary. By contrast, when the retained technique of interpolation is quadratic, two techniques of calculation should be distinguished. The first considers t as a continuous variable while the second consists in calculating the values of the solar correction corresponding to the integer part of t and to determine the contribution of its non-integer part from a linear interpolation (rule of three). Both methods provide not very different results but only the second one will be presented here since it essentially agrees with original procedures. Let us now suppose that the ruqi dd, t, related to the mean new moon n12 is such that q1 ≤ t < q2 . Let us also admit that the concerned astronomical canon is the Xuanming li ddd, this canon being chosen instead of any other because it is representative of the majority of quadratic interpolation techniques. In such a case, the procedure begins with the determination of the lengths l1 and l2 of the two first solar periods (Table 10.4, p. 284 below): 5 d l1 = 14; 4235, 5 = 14 + 4235 8400 + 8400×8 ≃ 14.504240 (decimal value), 5 d l2 = 14; 5235, 5 = 14 + 5235 8400 + 8400×8 ≃ 14.623289 (idem). Here, these lengths – as well as all other numbers of interest – have been here exceptionally converted into decimal approximations because 16 Depending on the astronomical canon used, the solar breaths q and q i i+1 are taken either in true or mean value. THE SOLAR CORRECTION 187 not all numbers encountered in this technique of interpolation can be exactly represented with fractions having 8400 and 8400 × 8 as denominators (original calculations necessarily suppose the existence of some, non-documented, technique of approximation). Next, two coefficients, ∆1 = 449 and ∆2 = 374, must be extracted from the solar table of the Xuanming li ddd, and the following quantities are successively named and evaluated step by step: 1. qi zhong lü ddd (mean solar ratio) = ∆1 ; l1 2. hou zhong lü ddd (posterior solar ratio) = 3. he cha dd (joint difference) = ∆1 ∆2 − ; l1 l2 4. zhong cha dd (mean difference) = 5. chu łü dd (initial ratio) = 6. mo lü dd (final ratio) = l1 ∆1 ∆2 ( − ); l1 + l2 l1 l2 ∆1 l1 ∆1 ∆2 + ( − ); l1 l1 + l2 l1 l2 ∆1 l1 ∆1 ∆2 − ( − ); l1 l1 + l2 l1 l2 7. ri cha dd (daily difference) = 8. ding lü dd17 = ∆2 ; l2 2 ∆1 ∆2 ( − ); l1 + l2 l1 l2 ∆1 l1 ∆1 ∆2 1 ∆1 ∆2 + ( − )± ( − ). l1 l1 + l2 l1 l2 l1 + l2 l1 l2 After these preliminaries, the Chinese procedure explains how these building blocks should be combined. In the present case, with the preceding set of data (values of l1 , l2 , ∆1 and ∆2 ), the calculations depend on three quantities a1 , b1 and c1 such that: a1 = 0, ∆1 l1 ∆1 ∆2 1 ∆1 ∆2 b1 = + ( − )− ( − ) = 33.4511, l1 l1 + l2 l1 l2 l1 + l2 l1 l2 17 ‘determined’, i.e. true ratio. 188 TRUE ELEMENTS (618–1280) c1 = − 2 ∆1 ∆2 ( − ) = −0.3695. l1 + l2 l1 l2 Next, the ‘diminution-augmentation rates’ sunyi lü ddd, i.e. the values of S1 (n) = b1 + n × c1 , are calculated for n = 0, 1, 2, . . . , 14,18 that is, for the successive days of the winter solstice period and new quantities called tiaonü shu ddd (‘waning-waxing numbers’), T1 (n), n = 1, 2, . . ., are defined as follows: T1 (0) = δ1 = 0 and, for n ≥ 1, T1 (n) = ∑n−1 i=0 S1 (i). Hence, at last, the following table: n 0 1 2 3 4 5 6 7 S1 (n) 33.4511 33.0816 32.7121 32.3426 31.9731 31.6036 31.2341 30.8646 T1 (n) δ1 = 0 33.4511 66.5327 99.2448 131.5874 163.5605 195.1641 226.3982 n 8 9 10 11 12 13 14 S1 (n) 30.4951 30.1256 29.7561 29.3866 29.0171 28.6476 28.2781 T1 (n) 257.2628 287.7579 317.8835 347.6396 377.0262 406.0433 434.6909 With S1 (n) = b1 + nc1 we have: 1 T1 (n) = a1 + nb1 + n(n − 1)c1 2 (in this particular case, a1 = 0 but, in other cases, the first term is not necessarily equal to zero). With this technique, we now have everything we need to calculate the solar correction corresponding to any value t of the ruqi. Suppose, for example, that t = 5.2653 (decimal notation). Then, the main part of the correction is obtained from the integer part of t, 5, by noting that T1 (5) = 163.5605. Lastly, a further small correction is obtained from the non-integer value of t by calculating 0.5605 × S1 (5). Hence 18 ‘14’ because l1 = 14.5042 days. THE SOLAR CORRECTION 189 ∆⊙ = 163.5605 + 17.7138 = 181.2743. Likewise, for any other instant t of the same solar period: ∆⊙ (t) = T1 (⌊t⌋) + (t − ⌊t⌋) × S1 (⌊t⌋). (5.14) More generally, an analysis of the Chinese procedures shows that each new case can be described in a similar way since only the two following expressions are involved each time: Si (n) = bi + nci , (5.15) 1 Ti (n) = ai + nbi + n(n − 1)ci . 2 (5.16) Consequently, the full description of the general case needs nothing more than the calculation of 72 (= 24 × 3) coefficients ai , bi and ci , i = 1, 2, . . . , 24, even though these are not referred to as such in Chinese sources. But they are wholly determined from arithmetical combinations of coefficients listed in our modified solar tables – in the sequel we will refer to this as Uchida’s method, from the name of the Japanese historian of the calendar who first introduced this way of viewing these calculations.19 Now, let us suppose that the values of the li have been calculated (see TAB 10.4, p. 284) and let: ∆ = [449, 374, 299, 224, 135, 45, −45, −135, −224, −299, −374, −449, − 449, −374, −299, −224, −135, −45, 45, 135, 224, 299, 374, 449], δ = [0, 449, 823, 1122, 1346, 1481, 1526, 1481, 1346, 1122, 823, 449, 0, − 449, −823, −1122, −1346, −1481, −1526, −1481, −1346, −1122, − 823, −449]. Let us also define two complementary series I and J of indices, the first composed of all the successive numbers from 1 to 24 save 6, 12, 18 and 24 and the second limited to these four indices. Then, the ai , bi and ci can be determined as follows: a i = δi 19 M. Uchida 1975, p. 511–521. i = 1, 2, . . . , 24, (5.17) 190 TRUE ELEMENTS (618–1280) ∆i li ∆i ∆i+1 1 ∆i ∆i+1 ( − )− ( − ) + li li + li+1 li li+1 li + li+1 li li+1 bi = ∆ l ∆ ∆ 1 ∆i−1 ∆i i + i−1 ( i−1 − i ) − ( − ) li li−1 + li li−1 li li−1 + li li−1 li −2 ∆i ∆i+1 ( − ) i in I, li + li+1 li li+1 ci = −2 ∆i−1 ∆i ( − ) i in J. li−1 + li li−1 li i in I, i in J, (5.18) (5.19) Lastly, the solar correction related to a value t of the ruqi such that qi ≤ t < qi+1 is still obtained from the analogous form of the expression 5.14, p. 189 above, involving both Ti and Si . Further Remarks On the Solar Correction The preceding technique also applies to most astronomical canons from the Song dynasty relying on a solar table but the calculation of Uchida’s coefficients is slightly simpler because, in such a case, only mean solar periods are used so that all li have the same value. Sometimes, however, some difficulties, more apparent than real, do occur because the coefficients in question are expressed by means of unexpected units. For instance, the numerators and the denominators of the fractions involved in the Dayan li ddd calculations are both multiplied by 12. By contrast, the more ancient Linde li ddd technique has its own features. On the one hand, its solar periods [qi , qi+1 [ are attributed two possible lengths as follows: { 7 15 + 12 18 ≤ i ≤ 24, li = (5.20) 8 14 + 12 otherwise. Yet, against all expectations, this peculiarity only concerns its solar table; for other purposes true solar periods of variable lengths are calculated in the same way as in other canons. On the other hand, the Linde li ddd obtains the coefficients ai as already explained but, otherwise, it distinguishes eight particular cases, THE LUNAR CORRECTION 191 determined by the set of indices I = [3, 6, 9, 12, 15, 18, 21, 24] – instead of the above one – and its coefficients bi and ci are accordingly calculated as follows: ∆i + ∆i+1 ∆i − ∆i+1 + 2li li bi = ∆ − ∆ i−1 i 2li i not in I, (5.21) i in I, −2 ∆i ∆i+1 ( − ) i not in I, l +l l li+1 ci = i i+1 i −2 ∆i−1 ∆i ( − ) i in I. li−1 + li li−1 li (5.22) No matter how the solar correction is calculated, the above techniques always distinguish two sets of indices. The reason for this distinction is not immediately obvious but it is a consequence of the fact that, sometimes, ∆i = −∆i+1 so that the previous formulae vanish. It thus becomes temporarily impossible to calculate the bi and ci . A new mode of calculation has thus been imagined. The Lunar Correction Exactly as in the case of the solar correction, the lunar correction can be obtained by using either a linear or a quadratic technique of interpolation but, in principle, calendrical calculations only depend on the former technique while the latter only concerns more complex astronomical calculations.20 As usual, the details of the calculations are almost never provided in Chinese sources and, in addition, the very peculiar structure of the lunar tables is such that these calculations are not precisely obvious. However, a late astronomical canon from the Northern and Southern Song dynasties, the Jiyuan li ddd,21 develops at length the whole linear procedure in such a manner that its instructions obviously remain 20 These two different techniques have both been studied by historians of Chinese astronomy, notably Qu Anjing 2008, p. 316 f., but much work still remain to be done in order to ascertain all the details of the latter, notably its exact piecewise formulation. 21 Songshi, j. 79, ‘lüli zhi’ 12, p. 1870. 192 TRUE ELEMENTS (618–1280) globally valid for all canons relying on such a correction, even though some peculiarities, like those of the Xuanming li ddd do exist.22 From this procedure23 it follows that the calculations do not depend on a table but on what might be called a quasi-table, that is a tabular structure designed in such a way that certain of its inner squares do not contain a single number, as expected, but on the contrary several numbers, intended to be used in particular ways, quite different from the treatment reserved for its other more ‘regular’ numbers. For instance, the Linde li ddd table, fully reproduced below (Table 5.5, p. 193), provides two unusual coefficients for the days 7, 14 and 21 and attribute a constant Λi to 25 of the 28 days into which the anomalistic lunar month is divided. The reason for these peculiarities will be explained later but let us first justify what happens with day no. 28: in its case, lunar tables extend beyond the duration of the anomalistic month, man . Therefore, only the fraction of a day smaller than man is used in the calculations while greater values are attributed to the first day of the table. Consequently, the constant Λ28 only concerns the value of the lunar inequality over the interval extending from the beginning of day no. 27 to the end of the anomalistic month (ca. 27.5546 days). The reason why two coefficients are required for days no. 7, 14 and 21 is different and not immediately obvious. It is due to the fact that the variations of the lunar inequality from values greater or smaller than man do not occur at the beginning of intervals of one day corresponding to entries of the lunar quasi-table but on the contrary somewhere between the beginnings and ends of such intervals. It is thus necessary to determine when the value of the lunar inequality is equal to zero in order to distinguish the two cases of a positive or negative lunar inequality. These variations being determined by a division of the anomalistic month into four equal intervals, the instants corresponding to a lunar an 3×man inequality equal to zero are respectively equal to: m4an , 2×m and 4 , 4 man days. for instance, Liu Jinyi and Zhao Chengqiu 1984, p. 67 (Linde li ddd case), Lin Jin-Chyuan 1998, p. 25 (Kaixi li ddd case). The Xuanming li ddd calculations do not use the whole anomalistic month but only its half, man /2. Consequently only days 7 and 14 need a special treatment (see p. 286 below). 23 Songshi, ibid., p. 1867–1870. 22 See, THE LUNAR CORRECTION Day (i) 1 2 3 4 5 6 Di 985 974 962 948 933 918 7 902 8 9 10 11 12 13 886 870 854 839 826 815 14 808 Λi −134 −117 −99 −78 −56 −33 { −9 0 { cumuls 0 −134 −251 −350 −428 −484 193 Λi 128 115 95 74 52 28 { 4 0 Day (i) 15 16 17 18 19 20 Di 810 819 832 846 861 877 cumuls 29 157 272 367 441 493 −517 21 893 14 38 62 85 104 121 −526 −512 −474 −412 −327 −223 22 23 24 25 26 27 909 925 941 955 968 979 −20 −44 −68 −89 −108 −125 525 505 461 393 304 196 102 29 −102 28 985 −71 71 521 Table 5.5. A modified version of the lunar quasi-table of the Linde li adapted to a simplified, but equivalent, formulation of calendrical and astronomical calculations. (The meanings of the coefficients are explained on p. 185 above.) In practice, however, the quasi-tables do not use such exact values but only approximations. More precisely, it appears that the length of the anomalistic month is replaced by 29 59 d . Consequently, the lunar inequality becomes equal to zero when the time elapsed from the beginning of the anomalistic month is respectively equal to 6 89 d , 13 79 d , 20 69 d and 27 59 d , that is at instants belonging to the days respectively numbered 7, 14, 21 and 28 in all lunar quasi-tables. Moreover, these four values are not referred to as such but related to the division of the day in the canon in question. For example, in the Linde li ddd the day is divided into 1340 parts and 89 becomes equal to 1340 × 89 ≃ 1191 1340 and so on. Day no. 7 is thus divided into two parts 1191 149 determined by the two fractions 1191 1340 and 1 − 1340 = 1340 . 194 TRUE ELEMENTS (618–1280) The cases of the other days which are a multiple of 7 are dealt with in a similar way. In addition, other small approximations, not always easily justified, are also introduced without warning. Anyway, Chinese sources take these calculations into account by inserting the numerators of the special numbers associated with the days 7, 14, 21 and 28 into the cells of their lunar quasi-tables. Hence pairs of numbers l71 and l72 , l141 and l142 , . . . Moreover, these lki are such that lk1 + lk2 = 1 day, and they are expressed with a unit of time u such that one day = b such units, b being the denominator used in the expression of the astronomical quantities of the concerned canon. For example, in the Linde canon, 1 day = 1340 u. In addition, the lki are associated one to one with coefficients αi1 and αi2 , one written below the other (left braces, quasi-table 5.5 on the preceding page) and playing the same role as the Λi but with respect to intervals smaller than one day (days a multiple of 7). Now, let us suppose that the mean new moon n is such that: ruli(n) =< x; y > and let b be the denominator of the fraction having y as numerator. Then, from the detailed procedure of the Jiyuan li,24 the lunar correction ∆$ should be calculated in the following way: (a) when i is not a multiple of 7: y ∆$ = λi + Λi × ; b (b) when i is a multiple of 7: αi1 × y λi ± li1 ∆$ = α (y − li1 ) λi + [αi1 − i2 ] li1 (5.23) y < li1 , y ≥ li1 ; (5.24) (c) when i = 14 ∆$ = α141 (y − l141 ) . α142 (5.25) The first formula is obviously a direct application of a process of linear interpolation. The justification of the others is less obvious, but they 24 Songshi, j. 79, p. 1870; Bo Shuren 2003, p. 420–421. CALCULATIONS WITHOUT TABLES 195 have also been devised in the same way, the only difference being that in some cases, the time has been reckoned from the end of an interval of one day rather than from its beginning. Calculations Without Tables In the three canons Yitian (Celestial Ritual) dd (1001–1023),25 Mingtian (Resplendent Heaven) dd (1065–1067)26 and Guantian (Contemplation of Heaven) dd (1094–1102)27 tabular techniques are sometimes provided for the calculation of the solar and lunar corrections but, remarkably, as shown by the Japanese historian of Chinese astronomy S. Nakayama,28 the solar correction ∆⊙ is also expressed by means of a simple parabolic function, documented for the first time in the nonofficial Futian dd canon (ca. 780–783). In his notation: ∆⊙ = l(182 − l) , 3300 (5.26) where l represents the mean solar longitude.29 The Futian canon was brought to Japan in 957 by a Buddhist monk and two Japanese horoscopes, at least, were calculated with its methods.30 25 Songshi, j. 68, ‘lüli 1’, p. 1491–1518. j. 74, ‘lüli 8’, p. 1709–1741. 27 Songshi, j. 77, ‘lüli 10 and 11’, p. 1797–1845. 28 S. Nakayama 1969, p. 62. 29 S. Nakayama 1987, p. 135. Further technical details on the Futian canon and the above three official astronomical canons are provided in Qu Anjing, Ji Zhigang and Wang Rongbin 1994, p. 289 f. and Qu Anjing 2008, p. 159–160. See also Wang Yingwei 1998, p. 617 f. 30 S. Nakayama 1969, p. 62. 26 Songshi, CHAPTER 6 LATER ASTRONOMICAL CANONS (1281–1644) The Supremacy of the Inception Granting Canon The two last Chinese astronomical canons – the Shoushi li d d d (Inception Granting Canon) (1281–1384) and the Datong li d d d (Great Unification Canon) (1385–1644) – are representative of Chinese traditional predictive astronomical techniques before the successful Jesuit reform of astronomy (1644). Contrary to what might be expected, however, only the Shoushi li ddd has been studied in depth by historians of Chinese astronomy because there is a consensus in favor of the identity of these two astronomical canons. Yet, as will be noted below, they differ in various respects. Regardless, the overwhelming prestige of the Shoushi li ddd and, correlatively, the oversight of the Datong li ddd is also the consequence of the very positive appraisal of eminent Chinese scholars from various periods and of the lasting impact of the Shoushi li on Korean and Japanese astronomies, even after 1644.1 More than one century and a half after the European reform of Chinese astronomy by Jesuit astronomers, the Shoushi li was still regarded in China as the summit of Chinese traditional astronomical techniques before the arrival of the Europeans in the Middle kingdom. After having explained that this famous astronomical canon fulfills the traditional Chinese criterion of superiority, namely the increased precision of its mathematical predictions with respect to all other Chinese astronomical canons, the distinguished evidential scholar Ruan Yuan dd (1764– 1849)2 highly praises its excellency, in the following terms: 1 See 2 B. p. 399 below. Elman 1984. © Springer-Verlag Berlin Heidelberg 2016 J.-C. Martzloff, Astronomy and Calendars – The Other Chinese Mathematics, DOI 10.1007/978-3-662-49718-0_6 197 198 LATER ASTRONOMICAL CANONS “From the Santong ddd astronomical canon (104 BC–84) onwards more that seventy [astronomical] schools have emerged, but none can be compared with the Shoushi school.” ddddddddddddddddddd3 In Korea, the Shoushi li was officially adopted in 1309 and in 1423; its Chinese text was reprinted and Korean new studies of the subject were issued.4 In Japan, the Shoushi li was reprinted in 1672 and studied by numerous scholars,5 including Takebe Katahiro dddd (1664–1739), a famous mathematician, attendant of the shōgun Tokugawa Ienobu (reign 1709–1712).6 More surprisingly, some of its mathematical procedures, including those concerning eclipse forecasts, were partially translated into French in 1732, on the occasion of the publication in France of the first history of Chinese astronomy ever published in a Western language.7 The Two Last Chinese Astronomical Canons A priori, the alleged identity of the two last traditional astronomical canons can be accounted for in various ways but in fact, the authoritative judgment of the Chinese historians responsible for the compilation of the section of the Mingshi (Ming History), devoted to the Datong li d dd, is probably decisive in this respect: “The predictive techniques of the Datong li derive from those of the Shoushi li; the Datong li has rejected its secular variations of the tropical year, nothing more.” ddddddddddddddddd8 To be sure, the secular variations of the length of the tropical year are so infinitesimal that they cannot easily induce significant differences 3 Chouren zhuan, j. 25, p. 305 (notice p. 391 below). 4 Lee Eun-Hee and Jing Bing 1998, p. 4–5; Lee Eun-Hee 1997. p. 399 below. 5 M. Sugimoto and D.L. Swain 1978, p. 252 f. 6 A. Horiuchi 2010, p. 116 f. 7 See p. 404 below. 8 Mingshi , j. 35, ‘li 5’, p. 685. For more details, see THE TWO LAST ASTRONOMICAL CANONS 199 between the results of calculations obtained from both canons. Still, we have not crosschecked systematically all their forecasts and the possibility of side-effects cannot wholly be ruled out a priori: when almost identical canons respectively assign the occurrence of a given calendrical or astronomical event slightly before or after the instant of midnight, a difference of one day in the final calendar or ephemeris is always possible. Apart from the question of the secular variations of the tropical year,9 the Shoushi li and the Datong li also differ formally from one another because they do not organize their subject matter in the same way: they do not use exactly the same terminology, their transcriptions of numbers are not wholly identical and the number of significant digits they take into account is not necessarily identical. Moreover, despite their purported logical equivalence, their astronomical tables are designed slightly differently, those of the latter generally relying on a division of fundamental astronomical periods into days rather than larger units. Lastly, contrary to the Shoushi li, the Datong li contains geometrical figures and even embryonic logical justifications.10 However, as far as we know, this latter feature has no obvious influence on its mathematical procedures. The existence of another disagreement between the techniques of calculations of true new moons according to the Shoushi li and the Datong li, evidenced in broad outline for the first time by contemporary historians of Chinese mathematical astronomy,11 is probably more significant. From preliminary investigations, however, it would seem that the operational differences between both modes of calculations are so limited that no example of Shoushi or Datong calculations leading to new moons having different dates12 has been discovered. But, of course, nothing of the sort can be ruled out a priori. Consequently, save mention to the contrary, the following analysis will be limited to the Shoushi li 9 Mingshi, idem. j. 32, ‘li 2’, p. 570, 571, 572 and 584 (figures) and j. 33, ‘li 3’, p. 585 f. (logical justifications). 11 Li Yong and Zhang Peiyu 1996 and, above all, Qu Anjing 2008, p. 326 (the first reference is very allusive. By contrast, despite its synthetic character, the second is somewhat more explicit). 12 Li Yong and Zhang Peiyu 1996, ibid. 10 Mingshi, 200 LATER ASTRONOMICAL CANONS calculations and regarded as also valid in the case of the Datong li. Nevertheless, when a better understanding of the minutiae of both canons will become available, a full comparison between both procedures will certainly have to be undertaken. Units of Time Despite various minor linguistic irregularities alluded to above,13 the fundamental time units of the Shoushi li are the day, ri d, and its centesimal divisions, the ke d, the fen d, and the miao d. Angular quantities also follow the same system, the corresponding units being the du d, whose value is equal to 365.2425/360 usual sexagesimal degrees, the fen d and the miao d. The Epoch Unlike astronomical canons having a Superior Epoch, the epoch of the Shoushi li is not defined by the coincidence of an initial winter solstice with an initial new moon, at the instant of midnight. On the contrary, this famous canon admits a Contemporary Epoch and therefore dissociates these three instants. Definition 6.1 (The Epoch) The epoch of the Shoushi li coincide with the instant of midnight of the first jiazi sexagenary day, #1, preceding both the mean new moon n11 (1280) and the winter solstice q1 (1280). This midnight, O, is thus such that O < n11 < q1 (Fig. 6.1 below). Moreover, [O, q1 ] is composed of days entirely belonging to a single sexagenary cycle. Therefore, O, n11 and q1 fall on days having different sexagenary numbers and, of course, O, q1 = q1 and n11 all belong to the year 1280. A quantitative analysis of the primary constants of the Shoushi li and of the way they are combined with other elements in its various procedures also shows that q1 (1280) is located 55.06 d after the epoch O and that the length of the interval [n11 , q1 ] is equal to 20.1850 d . 13 See p. 117 and 118. CONCORDANCES WITH JULIAN DATES O n̄11 (1280) q̄1 (1280) 201 n̄12 (1280) r q Figure 6.1. The epoch of the Shoushi li (O) and its shift constants, q and r, defined with respect to n̄11 (1280) and q̄1 (1280) = q1 (1280). Quite noticeably, the Shoushi li respectively calls these two quantities qiying14 dd (solar breath delay)15 and runying dd (intercalary delay).16 In the sequel, these two constants will be denoted q and r, respectively (Fig. 6.1) and similar ones will also be referred to as ‘shift constants’. Concordances with Julian Dates The Julian dates of O, q1 (1280) and n11 (1280) are obtained from any concordance table of the Chinese calendar. We have: • O: 20/10/1280, instant of midnight (sexagenary cycle: day #1, jiazi); • q1 (1280): 14/12/1280 (day #56, jiwei), 0.06 d after midnight, that is towards 1h 35 m ; • n11 (1280): 23/11/1280 (day #35 or wuxu), 0.875 d after midnight (because n11 (1280) = 55.0600 − 20.1850 = 34.8750), towards 21rmh . In addition, as already noted, no distinction exists between the true and mean values of the winter solstice q1 (1280), the two values being 14 Yuanshi, j. 54, ‘li 3’, p. 1192. d means ‘to respond to’, ‘to echo something’, as though the winter solstice was ‘echoing’ or ‘responding to the epoch’ with some delay, contrary to what happens in astronomical canons older than the Shoushi li. Hence the tentative rendering, ‘delay’. Other translations are possible. For instance, N. Sivin proposes qiying = ‘Ch’i [= qi] Interval Constant’. See N. Sivin 2009, p. 392. Moreover, similar constants, called cha d (difference, discrepancy, deviation), already exist in a more ancient astronomical canon based on a Superior Epoch, the Tongtian li ddd (1199–1207). See Chen Meidong 2003a, p. 534. 16 This ‘intercalary delay’ is equal to the age of the moon at the instant of the winter solstice q1 (1280) and can be regarded as an initial epact. 15 Ying 202 LATER ASTRONOMICAL CANONS equal. On the contrary, the true value of the initial new moon n11 (1280) is not equal to its mean value and the calculations below show that its value is equal to 35.2112. This initial true new moon thus happens on a jihai day, #36, 0.2112 d after its initial midnight, that is towards 5h 4m .17 The Reform of the Shift Constants A little more than ten years after the promulgation of the Shoushi li, in 1293, the values of all its shift constants save the first one, the qiying dd, were modified.18 In particular, the runying, initially equal to 20.1850 d , was replaced by a slightly higher value, 20.205 d . The year 1294 was thus the first concerned by this adjustment. These new values of some shift constants introduce an additional complication because, with the way the epoch of the Shoushi li is defined, two different modes of calculations must already be distinguished, depending on whether a year is prior or posterior to 1280. Therefore, the following three time-intervals must be taken into account: ] . . . , 1280], ]1281, 1293] and ]1293, . . .] The first interval concerns the proleptic usage of the Shoushi li, the second and the third its plain usage. Apart from the runying and the qiying, the Shoushi li also needs other shift constants. One of these, the zhuanying dd, is determined by the anomalistic mean lunar month. By definition, this is a sort of ‘anomalistic age of the moon’ whose value is equal to the length of the interval of time between the instant of the winter solstice (of either the year 1280 or 1293) and the nearest previous beginning of the anomalistic month. In the first case, the corresponding value given is equal to 13.1904 d and in the second, to 13.0205 d . 17 The values of q1 (1280) and n11 (1280) obtained from the Shoushi li calculations are not very different from those of the retro-calculated new moons and solstices listed in Zhang Peiyu’s tables (Zhang Peiyu 1990*/1997*): the difference is of the order of a quarter of an hour each time. More generally, Chen Meidong 2003a, p. 536, also provides further results on the precision of other quantities of interest. 18 The Mingshi, j. 34, ‘li 5’, p. 687, only mentions that [some] values of the shift constants were reformed ‘after 1281’ but the Lidai changshu jiyao, j. 9, p. 19b, and the Gujin tuibu zhushu kao, j. 2, p. 23b both indicate that this revision took place in 1293 (on these important works, see the notice on p. 378 below). MEAN ELEMENTS 203 Other shift constants are the jiaoying dd and the zhouying dd.19 Both are defined like the preceding ones as equal to the age of the moon at the instant of the winter solstice with respect to the beginning of their respective cycles. The first is important for eclipse forecasts and the second for positional astronomy. However, none of these are used in calendrical calculations. Mean Elements Let x be a given year, anterior or posterior to 1280. Then, the Shoushi li calculations start from the following determination of the number t(x) of solar years between the two winter solstices q1 (1280) and q1 (x):20 t(x) = |x − 1280|. (6.1) Next, taking into account the mean value, Y (t), of the tropical year over any interval of the form [x, 1280] or [1280, x] (see the expression 3.8, p. 141 above), the winter solstice q1 (x), the mean epact e(x) and the mean new moon n11 (x) are calculated as follows: { (55.06 − tY (t)) mod 60 q1 (x) = (tY (t) + 55.06) mod 60 (20.185 − tY (t)) mod msyn e(x) = (tY (t) + 20.185) mod msyn (tY (t) + 20.205) mod msyn x ≤ 1280, x ≥ 1281, x ≤ 1280, 1281 ≤ x < 1294, x ≥ 1294, n11 (x) = (q1 (x) − e(x)) mod 60. (6.2) (6.3) (6.4) Lastly, the successive mean solar breaths, seasonal indicators and mean phases of the moon, are obtained in the same way as in previous astronomical canons. 19 Zhang Peiyu 1994, p. 35. calendrical purposes, q1 (x) is a fundamental component of calculations concerning the year x + 1. 20 For 204 LATER ASTRONOMICAL CANONS Justifications For the calculation of the winter solstice q1 (x) = q1 (x), two cases must be distinguished according to the position of the year x with respect to the epoch O (definition 6.1, p. 200 above). First Case (year x ≥ 1280) Let q1 (1280) and q1 (x) be the two winter solstices of the years 1280 and x. Then, the time elapsed between the epoch O and q1 (x) is equal to the sum of the lengths of [O, q1 (1280)[ and [q1 (1280), q1 (x)[, that is (q + tY (t)) with q = 55.06d ) (Fig. 6.2). Hence the sought result. Once the winter solstice q1 (x) is obtained, the determination of e1 (x) and n11 (x) are straightforward. Second Case (year x < 1280) Let q′1 (x) be a winter solstice located before the epoch O. Then, the length of the interval [q′1 (x), q1 (x)[ is equal to (tY (t) − 55.06) (Fig. 6.2). However, this result should be taken negatively since the days of the sexagenary cycle are now enumerated backwards. q′1 (x) O n11 (1280) q1 (1280) q1 (x) r q Figure 6.2. The two possible positions of a winter solstice (q′1 (x) and q1 (x)) with respect to the Contemporary Epoch O. True Lunar Phases True New Moons Let n(t, i) i = 1, 2, . . . be the sequence of true new moons determined by t(x) (or t). Let also n(t, 1) = n11 (x), n(t, 2) = n12 (x) and so on. According to the Shoushi li procedures, the calculations depend either on tables or on direct calculations of polynomial values. In the sequel, only the latter technique will be used because it renders their overall pattern much more apparent. The complete list of what is required in this re- TRUE LUNAR PHASES 205 spect is first given and the explicit values of constants, polynomials and other functions are indicated immediately afterwards: • Four new primary numerical constants (B, C, D, k); • the length of the synodic month msyn = 29.530593 d ; • the length of the anomalistic month man = 27.5546 d ; • three polynomials of degree three, f , g and h; • another polynomial of degree two, ∆. • two parameters t⊙ (t, i) and t$ (t, i) (more simply, t⊙ and t$ ); • three piecewise functions, δ⊙ (t, i), δ$ (t, i) and ν (t, i), defined from f , g, h and ∆.21 B = 88.909225 d (6.5) C = 93.712025 d (6.6) D = 1.0962 (6.7) k = 0.082 (6.8) f (x) = 10−8 (5, 133, 200 − (31x + 24, 600)x)x (6.9) g(x) = 10−8 (4, 870, 600 − (27x + 22, 100)x)x (6.10) h(x) = 10−8 (11, 110, 000 − (325x + 28, 100)x)x (6.11) ∆(x) = 0.11081575 − 0.0005815x − 0.00000975x(x − 1) (6.12) t⊙ (t, i) = (−e(t) + msyn × i) mod Y (t) (6.13) 21 The present formulation is partly based on previous ideas of H. Hirose 1979, Li Yong, 1996 and Li Yong and Zhang Peiyu 1996. 206 LATER ASTRONOMICAL CANONS (−tY (t) + 13.1904 − e(t) + msyn × i) mod man t$ (t, i) = (tY (t) + 13.1904 − e(t) + msyn × i) mod man (tY (t) + 13.0205 − e(t) + msyn × i) mod man x ≤ 1280, 1281 ≤ x < 1294, x ≤ 1294, (6.14) f (t⊙ ) ( ) Y g − t ⊙ 2 δ⊙ (t, i) = ( ) Y −g t − ⊙ 2 − f (Y − t ) ⊙ δ$ (t, i) = 0 ≤ t⊙ < B, B ≤ t⊙ < Y , 2 (6.15) Y Y ≤ t⊙ < +C, 2 2 Y +C ≤ t⊙ < A, 2 ( ) t$ man 0 ≤ t$ < , −h k 4 man − t$ −h 2 man ≤ t$ < man , k 4 2 man h t$ − 2 k ( ) man − t$ h k (6.16) man 3man ≤ t$ < , 2 4 3man ≤ t$ < man , 4 TRUE LUNAR PHASES (t ) $ D + ∆ k D man | − t$ | D − ∆ 2 ν (t, i) = k D man | − t$ | 2 D + ∆ k 207 0 < t$ < 81k, man ≤ t$ < 86k, 4 86k ≤ t$ < 249k, (6.17) 249k ≤ t$ < 254k, 254k ≤ t$ < man . Lastly, the true new moons, n(t, i), are obtained by adding to or subtracting from n(t, i) a positive or negative corrective factor called jiajian cha ddd (additive or negative deviation) as follows: ( n(t, i) = n(t, i) + k ) δ⊙ (t⊙ (t, i)) + δ$ (t$ (t, i)) . ν (t$ (t, i)) (6.18) True Lunar Phases More generally, the same technique also applies to the case of true lunar phases by replacing everywhere msyn and n(t, i) by msyn /4 and p(t, i), i = 1, 2, . . ., respectively, the latter expression denoting the successive mean phases of the moon beginning from the mean new moon of the eleventh month of the year x. To sum up, the calculation of true lunar phases required for the calendar of the year (x + 1) relies on t(x), p(t, i), t⊙ , δ⊙ , t$ , δ$ , ν and p(t, i). Notes The Constants B, C, D and k B and C (p. 205 above) are two primary constants, equal to one of the two possible lengths of the four seasons: B = 88.909225 d is the com- 208 LATER ASTRONOMICAL CANONS mon length of winter and autumn with respect to the deep structure of the calendar and C = 93.712025 d is likewise the common length of spring and summer (Table 6.1, p. 211); D = 1.0962 is the mean lunar diurnal motion (p. 211); k = 0.082 (p. 205) is the result of the division of the anomalistic month, man = 27.5546 d22 , into 336 equal intervals called xian d. The Polynomials f and g f and g (p. 205) are two third degree polynomials used to define δ⊙ . Most remarkably, their representations23 follow the well-known parenthetical form of ‘Horner’s schema’ rather than the usual sums of weighted powers of the variable. In this way, no calculation of squares or cubes of numbers are necessary. Hence fewer additions, subtractions or multiplications than usual. Nonetheless, the symbolic notations we have used in this respect are utterly different from the purely verbal formulations of the original Chinese sources. In particular, they do not use anything even remotely similar to our 10−8 whose sole purpose is an operational representation of these calculations. The Polynomials h and ∆ h and ∆ (p. 205) are two polynomials designed in the same way as f and g but used as components of δ$ and ν (t, i). We have reconstituted them from the Datong procedures which are much more developed than those of the Shoushi li (among many other useful details, the relevant Datong table lists the first differences of h, thus making clear the relation between ∆ and h: ∆(x) = h(x + 1) − h(x)24 ). The Parameter t⊙ t⊙ (t, i) (p. 205) – called ru yingsuo li dddd ‘degree of advancement ru d [of a given lunar phase] into [any solar phase] of ‘expansion’ or ‘contraction’ of the solar motion’ – is the length of the time interval be22 Yuanshi, j. 54, ‘li 3’, p. 1213. ibid., p. 1198. 24 Mingshi, j. 34, ‘li 4’, p. 657–672. 23 Yuanshi, TRUE LUNAR PHASES 209 tween a given mean lunar phase and the last winter solstice.25 Therefore, t⊙ (t, i) is such that: 0 ≤ t⊙ (t, i) < Y (t). (6.19) Two phases are defined: the first extends from any initial winter solstice to the next summer solstice and corresponds to an interval of excess or ‘expansion’ (ying d) of the solar motion, with respect to its mean value. The second extends from the same summer solstice to the next winter solstice (Fig. 6.3) and reflects a deficit or ‘contraction’ (suo d), of the same motion, always with respect to its mean value. winter summer winter solstice solstice solstice q1 (x − 1) q13 (x − 1) q1 (x) Phase of Expansion (ying) Phase of Contraction (suo) Figure 6.3. The two phases of expansion and contraction of the solar motion. Therefore, n11 (x − 1) belongs to a phase of contraction because it is located between the summer solstice q13 (x − 1) and the winter solstice q1 (x − 1). By contrast, n12 (x − 1) and n1 (x) both belong to a phase of expansion because they occur between the winter solstice q1 (x − 1) and the summer solstice q13 (x). In their turn q1 (x − 1) and q13 (x) belong to both because, as already noted, their mean and true values are identical. Let us now suppose that the time between two calendrical events is calculated from the winter solstice of the year (x − 1), the first such solstice preceding n11 (x). Then, given that the time elapsed between the two consecutive winter solstices q1 (x − 1) and q1 (x) is equal to Y (definition of the solar year) and that the new moon n11 (x) occurs e days before q1 (x) (definition of the epact), the value of the parameter t⊙ relating to n11 (x − 1) is equal to: t⊙ = Y (t) − e. (6.20) 25 This notion is similar to the ruqi used in older astronomical canons based on true elements save that the calculation of the latter does not depend on a single solar breath fixed once and for all. 210 LATER ASTRONOMICAL CANONS The value of t⊙ relating to a new moon ni follows from the value of the number of days between q1 (x − 1) and n11 (x) and from a reordering of the indices of the concerned elements. Hence 6.13, p. 205, after a reduction modulo Y (t). Of course, the values of t⊙ with respect to the phases of the moon are dealt with similarly by replacing everywhere man by man /4. The Parameter t$ The technical appellation of t$ (t, i) (p. 206 above) is ruzhuan dd, a term whose literal meaning is ‘degree of advancement ru d [of any phase of the moon] into an anomalistic month’. By definition, its value is equal to the length of the time interval between a given phase of the moon and the last beginning of the anomalistic month. Hence: 0 ≤ t$ (t, i) < man . Formally, the expression 6.14 used in order to calculate t$ is analogous to the one concerning the epact e. The piecewise function δ⊙ δ⊙ (t, i) (p. 206 above) is based on the piecewise division of the mean solar year into four intervals of unequal lengths, corresponding to the astronomical seasons determined from the solstices and the equinoxes and not from their calendrical definition (p. 64). The solar year is thus first divided into two equal intervals, each being composed of (Y /2) d , as already explained above (p. 209). Then, these two intervals are divided in their turn into two smaller intervals of unequal lengths, as shown in Table 6.1 below. The Piecewise Function δ$ From the last expression, p. 206 above, δ$ (t, i) is an expression of the lunar inequality parallel to the one used in the case of the sun. Here, the lunar motion is determined by the division of the anomalistic month into four intervals (Table 6.2, p. 212 below), each composed of 336/4 = 84 xian (1 xian = 0.082 d ). Conversely 1 day = 12.1939712426 , a value rounded up to 12.2 xian in the Datong li.27 26 H. Hirose 1979, p. 29 . , j. 35, ‘li 5’, p. 688. 27 Mingshi TRUE LUNAR PHASES Phases initial expansion ying chu dd final expansion 2 ying mo dd initial contraction 3 suo chu dd final contraction 4 suo mo dd 1 Intervals Lengths Winter B = 88.909225 d Spring C = 93.712025 d 211 Summer C = 93.712025 d Autumn B = 88.909225 d Table 6.1. The lengths of the seasons according to the Shoushi li. Lastly, the value of the temporal variable to be taken into account in this system is t/k and not t. The Piecewise Function ν ν (t, i) (p. 207 above) is called chiji xianxia xingdu dddddd ‘degree of the [lunar] motion per xian’.28 The constant 1.0962, used in ν (t, i), corresponds to the mean diurnal tropical motion of the moon per xian, expressed in degrees du d (as before, 360◦ = 365.2425 du).29 Its value is not explicitly listed in the list of constants of the Shoushi li but the Datong li mentions it under the name of xian pingxing du dddd (mean movement by xian expressed in degrees du), in its section devoted to the lunar motion.30 True Lunar Phases The above mode of calculation of true lunar phases (equation (6.18), p. 207) and the procedure devoted to the same subject in Ptolemy’s Mathematical Syntaxis31 are formally similar because the latter can be formulated as follows: 28 Literally, xianxia means ‘under [each] xian’. that 1 day ∼ = 12.2 xian, as noted above, p. 210, this mean movement per xian should be equal to 13.36875 du/12.2 = 1.0957, a value slightly different from 1.0962. 30 Mingshi, j. 35, ‘li 5’, p. 698. 31 O. Pedersen 1974, formula 7.41, p. 223. 29 Given 212 LATER ASTRONOMICAL CANONS Phases Lengths chi chu dd (initial slowing down) chi mo dd 2 (final slowing down) ji chu dd 3 (initial speeding up) ji mo dd 4 (final speeding up) man 4 1 man 4 man 4 man 4 Table 6.2. The four intervals attributed to the variations of the lunar motion in the Shoushi li calculations. τn = tn + ∆tn = tn + λ⊙ (tn ) − λ$ (tn ) . λ˙$ − λ˙⊙ (6.21) Here, τn , tn , λ⊙ and λ$ , respectively designate the instants of occurrence of the mean and true lunisolar conjunctions, the mean longitudes of the sun and moon and their true angular velocities. Formally, these various quantities are similar to the n(t, i), n(t, i), δ⊙ (t, i), δ$ (t, i) of the above formula (6.18). Moreover, all available studies of the Shoushi li32 understand the denominator ν (t, i) as built from the first differences of h. Hence the analogy with the differential λ˙$ . Of course, Ptolemy relies on ecliptic longitudes while the Chinese formula is expressed in purely temporal terms. Moreover, the analogue of λ˙⊙ is missing. However, as shown recently by Qu Anjing, this latter quantity is taken into account in the Datong li.33 The origin of the Shoushi li formula is unknown, but the possibility of a Chinese adaptation of foreign astronomical knowledge resulting from ancient contacts between China, India as well as the Central Asian and Islamic worlds from the Tang dynasty (618–907) onwards cannot 32 Among those already mentioned above on p. 210, H. Hirose 1979, op. cit., is fully explicit in this respect. 33 Qu Anjing 2008, p. 324–326. HORARY SYSTEM 213 be rejected,34 even though the quantities involved in these calculations obviously belong to radically different epistemological contexts. In the case of Ptolemy, the formula has a geometrical and deductive origin while its Chinese counterpart is a purely arithmetical technique whose rationale is utterly unclear. The Horary System of the Datong li The Datong li divides the day into 12 double-hours, 24 equal singlehours, 100 ke d (marks)35 and 10,000 fen.36 Then, each fen is furthermore divided into 12 parts. Hence a new tiny time-unit, denoted u here but left unnamed in original sources, despite its important role in the procedure devoted to the conversion of the results of Datong li calculations into double-hours and ke.37 To sum up, 1 fen = 12 u and 1 day= 120, 000u, 1 double-hour = 10, 000u and 1 single-hour = 5, 000u. A further analysis of the Datong li procedures also shows that each single-hour is divided into four intervals, ke d, not all having the same length: the three first ke of a single-hour are 1, 200u long while the fourth ke is 1, 400u long, or one sixth longer. The day is thus divided into 4 × 24 = 96 ke of unequal lengths, 72 ke being 1, 200u long and the remaining 24 others 1, 400u long, or a little more or a little less than one quarter of a single-hour in each case. Unlike modern practice, however, divisions of time smaller than the ke do not show up in Chinese calendars.38 Given that theoretical calculations lead to results expressed in fen, they must be converted into hours and ke d in view of their insertion 34 On astronomical exchanges between China and India under the Tang, see K. Yabuuchi 1963a/1988*. During the same period, a fascinating glimpse into contacts between China, Central Asia and Persia is available, see Lai Swee Fo 2003. Lastly, contacts between China and the Islamic world have been evoked in an ancient, but always pertinent, article by K. Tasaka 1957. See also the following more recent studies: B. van Dalen 2002a and 2002b; Y. Isahaya 2009. 35 So called from the graduations on waterclocks. 36 Yuanshi, j. 54, ‘li 3’, p. 1197 and Mingshi, j. 35, ‘li 5’, p. 692. 37 Mingshi, ibid. 38 Astronomical canons from the Song dynasty frequently use similar divisions of the day, relying on a division of single-hours into equal and unequal ke. See Lin Jin-Chyuan 1998, p. 5. 214 LATER ASTRONOMICAL CANONS into the final calendar. In practice, with such a system, the number of fen of calendrical events is first multiplied by 12 in order to become related to the time-unit u and the result is then converted into doublehours, single-hours and ke, owing to successive divisions by 10,000; 5,000 and 1,200. Lastly, double-hours are associated with the twelve branches, as indicated in the following table, giving their modern equivalents. Then, the 12 double-hours are further divided into 12 initial (or first) chu d (3rd line of the table) and 12 second single-hours zheng, d (4th line of the table): The 12 Double Hours 1 zi 2 chou 3 yin 4 mao 5 chen 6 si 7 wu 8 wei 9 shen 10 you 11 xu 12 hai Chu d (First) Zheng d (Second) 23 24 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 Table 6.3. Double and single hours. Now, when an event happens between the beginning of a single-hour and its first ke, it is said to belong to its initial ke (chu ke dd). The order of enumeration of the successive ke of a single-hour is thus: initial ke, first ke, second ke and third ke. For instance, an event happening 25mn after the beginning of an hour is denoted yi ke dd, ‘first ke’, an expression meaning ‘first ke [already elapsed]’ or, more precisely, ‘at least one [full] ke but less than two’. Example 6.1 According to the Datong li calculations, the value of the Autumn Equinox of the year Yongle 15 (1417), q19 , is 21.971875 (see Table. 11.1, p. 306 below). Convert this value into the Datong horary system. Answer: this result means that this event falls on a day yiyou of the sexagenary cycle (#22 or (2, 10)), 0.971875 d after the instant of midnight marking its beginning. In order to convert this value into the above timesystem, we first note that this non-integer quantity means 9,718 fen 75 miao. Then, as explained above, we calculate 9718.75 × 12 = 116, 625. Next, we successively divide this result by 10,000; 5,000 and 1,200 so that 116, 625 = 11 × 10, 000 + 1 × 5000 + 1 × 1200 + 425. Consequently, the Autumn Equinox in question occurs 11 doublehours, 1 single-hour and a little more than 1 ke, but less than 2, after THE DURATIONS OF DAY AND NIGHT 215 midnight. The event in question thus belongs to the single hour,numbered 23. Hence the formulation initial zi hour’‘ (zi chu d d) (Table 6.3, above). Lastly, given that the value of an initial ke d is equal to 1200u and that 1 × 1200u + 425u > 1200u, the above result is transformed in its turn into yi ke dd (first ke) and the sought result is thus: zi chu yi ke dddd, or ‘initial zi hour and one [full] ke’. The Durations of Day and Night The Shoushi li and the Datong li divides the nycthemeron into the following seven characteristic instants:39 1. the initial midnight, O1 ; 2. the end of the night (or the beginning of twilight), C1 ; 3. the end of twilight coinciding with sunrise, L; 4. midday, M; 5. sunset, coinciding with the beginning of dawn, S; 6. the end of dawn (or the beginning of the night), C2 ; 7. the second midnight, O2 , marking the end of the day in question and the beginning of the next. As figure 6.4 shows, the instants O1 , C1 , L, on the one hand and S, C2 , O2 on the other hand, are respectively symmetrical with respect to the instant of midday, M. In addition, the quantity O1C1 = a – called chenfen dd ‘the length of the interval from midnight to the beginning of twilight’ – varies from day to day during a solar year. Therefore, given that the duration of twilight is taken equal to 250 fen in the Shoushi li and Datong li, a sunrise always happens 250 fen (= 0.025 d ) or approximately 36mn after the end of the night, in an instant L such that O1 L = a + 0.025 days.40 39 Yuanshi, j. 54, p. 1234–1235; N. Sivin 2009, p. 492 f. (translation); Mingshi, j. 34, ‘li 4’, p. 656. 40 See Yuanshi, j. 55, ‘li 4’, p. 1226–1234; Mingshi, j. 34, ‘li 4’, p. 640–656; K. Yabuuchi and S. Nakayama 2006, p. 31 f. 216 LATER ASTRONOMICAL CANONS midnight night tw. = twilight daylight tw. C1 L O1 midday daylight M tw. midnight S C2 O2 Figure 6.4. The division of the nycthemeron according to the Datong li. The Epoch of the Great Unification Canon From what has already been said about the equivalence between the Shoushi li and Datong li, it would seem that both canons have the same epoch but this is not the case: the epoch of the Datong li belongs to the year 1383 and not to the year 1280.41 Nevertheless, the fundamental components q1 , e and n11 of the support year of the first year of the Datong li recorded in the Mingshi (1383)42 are readily obtained from the Shoushi li procedures. The new epoch is thus neither the consequence of a reappraisal of earlier astronomical constants nor of a modification of former procedures but only the result of a purely mechanical change of origin. From the expression 6.1, p. 203 above, concerning the Shoushi li calculations, t(1383) = 103 solar years. Moreover, the values of q1 (1283), e(1383) and n11 (1383) are obtained from the formulas 6.2, 6.3 and 6.4, p. 203 above, in the following way: q1 (1383) = (103 × 365.2425 + 55.06) mod 60 e(1383) = (103 × 365.2425 + 20.205) mod 60 n11 (1383) = q1 (1383) − e(1383) = 55.0375, = 18.18702, = 36.85048. The first result shows that the initial winter solstice of the Datong li, q1 (1383), determined by the Shoushi li calculations, take place on a jiwei day ((6, 8) or #56), that is a day associated with a sexagenary binomial identical with the one of the initial winter solstice of the Shoushi li. In its turn, its fractional part, 0.0375, is equal to the value of the Datong shift constant, still called qiying.43 In addition, concordance tables show that the corresponding Julian date is 14/12/1283. 41 Mingshi, j. 35, ‘li 5’, p. 685. p. 689. 43 Ibid, p. 686. 42 Ibid., THE EPOCH 217 The second result shows that e(1383) = 0.18702 – the value of the other shift constant of the Datong li– is still called runying.44 The third result indicates that the initial new moon n11 (1383) takes place on a gengzi day ((7, 1) or #37) whose Julian date is 25/11/1283. Lastly, the midnight of the first jiazi day ((1, 1) or #1) preceding both q1 (1283) and n11 (1383) defining the new origin of the Datong li corresponds to the Julian date 20/10/1283. 44 Idem. CHAPTER 7 MO AND MIE DAYS Introduction The elements of the calendrical deep structure, called Mo dand Mie d are well-determined instants of the temporal continuum, comparable to points located on a straight line, easy to spot once their distance from the epoch or abscissa is known. From the viewpoint of the calendrical surface structure, however, they are neither instants of time nor geometrical points but only particular days containing a Mo or a Mie point. Hence their names: Mori dd (Mo days) and Mieri dd (Mie days) respectively, save in the case of the Datong li which calls them Yingri dd and Xuri dd, respectively.1 Taken literally, the terms Mori and Mieri do not seem very different from each other since their apparent meanings – ‘days of disappearance’ and ‘days of annihilation’ – are similar. But these direct translations do not necessarily convey the intended meanings since the nature of the auspicious and inauspicious daily activities associated with the corresponding days do not always fit well with such renderings.2 But, such 1 Mingshi, j. 35, ‘li 5’ (Datong li), p. 692 and 693. The identity between the names given to these particular days follows from the formal identity between their respective calculation procedures. 2 Some manuscript calendars from the Dunhuang collection – all issued between 829 and 993 – advise their readers not to go boating or even not to go near deep waters or rivers on Mo or Mie days. Consequently, these days clearly involve a belief in a risk of drowning (A. Arrault 2003, p. 105), in agreement with the ideas of disappearance and annihilation. Moreover, the two characters are written with the key of water (the two dots and the stroke on their left) in both cases. This correspondence should not be restricted to such a semantic field, however, because the Japanese historian of the calendar Y. Nishizawa has shown that Mo and Mie days are often inauspicious for all sorts of activities and not only those connected with water (Y. Nishizawa 2005–2006, vol. 3, p. 294-296). Moreover, a perusal of authentic calendars from the Ming dynasty shows © Springer-Verlag Berlin Heidelberg 2016 J.-C. Martzloff, Astronomy and Calendars – The Other Chinese Mathematics, DOI 10.1007/978-3-662-49718-0_7 219 220 MO AND MIE DAYS an approach is probably too simplistic and the question of the meaning of these two terms should either be left open or, at least, tackled differently.3 By contrast, the terms Yingri dd and Xuri dd – ‘full days’ and ‘empty days’ respectively – convey opposite meanings and, from a lapidary statement of the Mingshi dd 4 clarified below, what they express is clear:5 the first days are so called because, when a solar period contains a ‘full’ day, its number of days is greater than usual (16 days instead of 15 days). In a reverse direction, the second term indicates that, when a mean lunar month contains an ‘empty’ day it becomes hollow and its length is thus reduced to 29 days. Irrespective of their particular appellations, there is no doubt that the techniques of calculation of Mo and Mie days presented in Chinese official canons are formulated in such a way that a full analysis of the underlying mathematical structures remains possible in all cases, despite numerous obstacles suggesting the contrary (formal multiplicity of procedures, complex and unstable technical nomenclature, utter absence of logical justifications). Consequently, we approach the subject from a synthetic and global angle, without concern for the endless details of their luxuriant terminological maze and variant, but logically equivalent, formulations. In the following, we take for granted the results of all available specialized investigations concerning numerous astronomical canons and we have consequently coined general definitions and procedures for reckoning Mo and Mie days in all cases. Moreover, we have also taken advantage of these results in order to draw general conclusions about the structure of all surface calendars obtained from astronomical canons based on mean elements, notably from the standpoint of the types of successions of hollow and full months they allow. that, after having been attributed their new names, Mo and Mie days have been associated with a priori non-threatening activities such as visiting friends or construction works (see the official calendar for the year Yongle 15 (1417)). 3 See p. 234 below. 4 Mingshi, ibid. 5 Results 1 and 2, p. 232. DEFINITIONS 221 Definitions A study of the calculation procedures of all astronomical canons promulgated between 104 BC and AD 1644 shows that the Mo points have always been defined in the same way. By contrast, the definition of the Mie points has been modified from the Daye li (597–618) onwards.6 Consequently, three definitions are necessary for them, a single one for the Mo of any period and two others for what will be called the Mie of the first and second types in the sequel. Definition 7.1 (Mo Points) Let Y = a/b days be the length of the solar year in a given astronomical canon and suppose that Y − 360 = r. Let also O be the point corresponding to its Superior Epoch. Then, in geometrical terms and from the viewpoint of the calendrical deep structure, its Mo points are the set of points M0 [= O], M1 , M2 , . . . , Mi , . . ., located Y /r days apart from each other, from M0 towards the future. (Fig. 7.1). O M0 M1 k M2 k M3 k Mi Mi+1 k Mi+2 k Figure 7.1. The successive Mo points (M0 , M1 , . . .) of a given astronomical canon, the first, M0 , coinciding with its Superior Epoch O (k = Y /r). Definition 7.2 (Mie Points of the First Type) The Mie points of the first type are the Mo having a fractional part equal to zero or, in other words, a binomial representation equal to < mi ; 0 >, where mi is an integer. Definition 7.3 (Mie Points of the Second Type) Let m be the length (in days and fractions thereof) of the mean lunar month of a given astronomical canon. Then, in terms of its deep structure, the distance m between two consecutive Mie points of the second type is equal to 30−m days, the first such Mie coinciding with the Superior Epoch. 6 Wang Rongbin 1995, p. 255–256. 222 MO AND MIE DAYS Obviously, the geometrical representation of this new situation is the same as the preceding one concerning Mo points, but with different lengths between Mie points. In conformity with these definitions, the following analysis of what they imply will be limited to astronomical canons having a Superior Epoch. It should be noted, however, that similar conclusions also apply to the case of canons having a Contemporary Epoch, such as the Shoushi li and Datong li. Immediate Consequences of the Definitions From definition 7.1 above and the fact that the length of the solar year is approximately equal to 365.25 d , it follows that the theoretical interval between two consecutive Mo points is roughly equal to 69.57 d . Therefore, two consecutive Mo days occur every 69 or 70 days in the surface calendar. By contrast, the number of days between two Mie days of the first type in the surface calendar cannot be ascertained once and for all because they are liable to be significantly different according to the theoretical length of the solar year. For instance, this length is equal to 365 14 d in the case of the Sifen Y d apart from li. Two consecutive Mo points are thus located Y −360 = 487 7 each other. A Mie of the first type having a fractional part equal to zero, its abscissa must correspond to an integer multiple of 487 7 and since the smaller such multiple is 487, the corresponding Mie days of the surface calendar happen every 487 days, that is, every 1.3 years, the first one coinciding with the Superior Epoch. In the Yuanjia li, the solar year is 111,035 304 days long and the distance Y d , an irbetween two consecutive Mo points is equal to Y −360 = 22,207 319 reducible fraction. The smallest integer multiple of 22,207 319 is thus equal to 22, 207 d and the corresponding Mie days of the first kind occur every 60.8 years. These two examples clearly show that the number of days between two consecutive Mie days of the first type in the surface calendar can vary in considerable proportions. Unlike the Mie points of the first type, however, the Mie points of the second type have an autonomous definition which makes them entities CALCULATIONS TECHNIQUES 223 solely determined by the length of the lunar month and not by particular Mo points. This length being approximately equal to 29.53 d , the definition 7.3 above shows that the distance between two such Mie points is approximately equal to 62.83 d . Consequently, the distance between two consecutive Mie days of the second type is equal either to 62 or 63 days, in the surface calendar. Calculations Techniques (Mo and Mie Days) A priori, the three preceding definitions are sufficient to determine all Mo and Mie points of all astronomical canons having a Superior Epoch: their calculation only involves mere reduction modulo 60 of the values of sequences ⌊i × k⌋, i = 0, 1, 2 . . . with k = Yr or m λ (with λ = 30 − m), respectively, the lengths, Y , of the solar year and, m, of the lunar month being given in advance. Despite its simplicity, however, this method is obviously devoid of any practical interest because it leaves aside the question of the connection between the values so obtained and specific lunar years of interest to calendar makers. Consequently, the Chinese have devised several techniques in order to circumvent the difficulty. Although always elementary, some of these are particularly ingenious and efficient but their link with the above definitions is anything but obvious. By grouping together these methods by families determined by the calculation techniques recorded in the original Chinese sources, modulo some trivially equivalent variants not taken into account here, we distinguish hereafter four different such methods, 1, 2, 3 and 4. Method 1 (winter solstice method) Let x be a given lunar year. Calculate the binomial associated with the last Mo point preceding the winter solstice q1 (x − 1) of its support year (deep structure). First, suppose that the solar year of a given astronomical canon contains µ Mo (for example, in the above case of the Sifen li, µ = 21 4 ). Second, let us suppose that the interval I = [O, q1 (x−1)[ is composed of s solar years. Then, it contains s × µ Mo points and the fractional part of this result determines the length of the interval [M, q1 (x − 1)[ between the winter solstice q1 (x − 1) and the nearest preceding Mo point, M. M 224 MO AND MIE DAYS being determined in this way, a mere subtraction between the easily calculated value of q1 (x − 1) and the length of the interval in question determine the binomial relating to M and, in agreement with the definition 7.1 above, the binomials of the following Mo are obtained by reiterated additions of k = Yr . Lastly, the integer part of these results determine the Mo days of the surface calendar. In practice, owing to a change of coordinates, obtained from an elimination of the greatest possible number of supra-annual periods elapsed between the Superior Epoch and the winter solstice of the year (x − 1), the calculations are frequently performed with numbers much smaller than those determined by this method.7 Method 2 Mie days of the first kind (same technique as Method 1). By Definition, Mie of the first kind are particular Mo having a fractional part equal to zero. Once the corresponding Mo have been calculated from Method 1, their determination is trivial. Method 3 Mo Days (four steps method). This sophisticated method is essentially recorded in astronomical canons based on lunisolar constants all expressed with the same main denominator, denoted b in the sequel (for example, b = 3040 in the Dayan li). Four steps are necessary: First Step Determination of the binomial representations qi = < ai ; fi > of all solar breaths qi required for the calendar of the year x, with respect to a given astronomical canon (as usual ai designates an integer number of days, not necessarily reduced modulo 60, and fi is the numerator of a fraction having b as denominator). Second Step Test 1 (Existence of a Mo point in a Solar Period) Let [qi , qi+1 [ be a solar period beginning with the solar breath qi =< ai ; fi >. Then, as r soon as bfi > 1 − 24 , it contains a Mo point. 7 This 255. method is used, in particular, in the Sifen li. See Wang Rongbin 1995, p. 254– CALCULATIONS TECHNIQUES 225 This slightly modified version of the original test relies on a comparison between fractions but the Chinese procedures only take into account their numerators. In particular, they retain the numerator of the fraction r 1 − 24 and call it ‘Moxian’ dd, (literally, ‘the Mo limit’). The original form of this test thus implies a comparison between the numerator fi of the solar breath in question and this Moxian. The two procedures are of course equivalent, but the underlying logic is slightly more easily uncovered by using fractions. Third Step If the test 1 is positive, the integer number of days Ji elapsed between the instant of midnight of the day containing the solar breath qi and the sought Mo day is determined as follows: ⌊ ⌋ a − 360 fi Ji = . (7.1) a − 360b In this formula, a and b respectively represent the numerator and the denominator of the fraction Y = a/b, expressing the length of the solar year in a given astronomical canon. Fourth Step Determination of the sexagenary number of the sought Mo by adding the integer part of qi to Ji , modulo 60. Method 4 Mie Days of the Second Kind (four steps method). This method is similar to the preceding one; the only modifications concern its numerical parameters. Four steps are also necessary: First Step Determination of the mean new moons ni = < ai ; fi > required for the year x, with respect to a given astronomical canon. Second Step Test 2 (Existence of a Mie point of the second type in a lunar month) Let m be the mean length of the lunar month and suppose that the binomial representation of the mean new moon ni is equal to < ai ; fi >. Then, if bfi < 30 − m, the lunar month [ni , ni+1 [ contains a Mie point of the second type. 226 MO AND MIE DAYS As before, the relevant Chinese sources only rely on numerators and never on the underlying fractions. In the present case, only the numerator of the fraction (30 − m), called Miexian dd, that is, literally, ‘the Mie limit’, is taken into account. Third Step If the test 2 is positive, determination of the integer number of days Ji between the instant of midnight of the day containing the new moon ni belongs and the sought Mie point is as follows: ⌊ ⌋ 30 fi Ji = . (7.2) b(30 − m) Fourth Step Determination of the sexagenary number corresponding to the binomial representation of the sought Mie by adding to Ji the integer part of ni reduced modulo 60. Justifications Methods 1 and 2 These justifications are trivial. Method 3 Let us suppose that the solar period [Qi , Qi+1 [ from the deep structure of the calendar contains a Mo point (see test 1 above): i solar periods Q0 = 0 one solar period P1 Qi P2 Qi+1 one mo Let also P2 be the Mo point in question, the ‘2’ being chosen in reference to the preceding Mo, P1 , itself necessarily located before Qi , the interval between two consecutive Mo being greater than a solar period. Let us now suppose, without loss of generality, that Q0 = O. Then, we will try to determine the length of the interval Qi P2 from those of OQi and P1 Qi . Next, the result will be compared with the length of a solar period. CALCULATIONS TECHNIQUES 227 Y being equal to the length of the solar year, each solar period contains Y /24 days and since solar breaths are now enumerated from zero, the interval OQi contains i solar periods. Hence: Y d ) . (7.3) 24 Of course, OQi is generally not equal to an integer number of days. Consequently, it can be decomposed into an integer number of days, Ni , and a fractional part, equal to a fraction fi /b < 1 d whose denominator is equal to b: OQi = (i × OQi = (Ni + fi d ) . b (7.4) fi >. Moreb Yd over, OQi can also be evaluated by taking the length r between two consecutive Mo points as a new unit of time instead of the more usual day (see definition 7.1 above). With this new unit, the mo, the expression of OQi becomes:8 The binomial representation of Qi is thus equal to < Ni ; ( )mo (Y × i)/24 r Y − 360 Y OQi = = i× = i× = i × − 15i . (7.5) Y /r 24 24 24 Y days and i ×Y /24 = Ni + fi /b. Therefore r [( ) ] fi Y d OQi = Ni − 15i + × . (7.6) b r Now, one Mo has Consequently, OQi is decomposable into an integer number of Mo, Ni − 15i, plus a fraction of Mo, ( bfi )mo . Now, P1 being the last Mo preceding Qi , the length of OP1 is precisely equal to this integer number of mo units. Therefore, the remaining fraction measures the length of the interval P1 Qi : fi PQi = ( )mo . b 8 See Qu Anjing, Li Caiping and Han Qiheng 1998. (7.7) 228 MO AND MIE DAYS Based on this, the fraction bfi expresses at the same time the value of P1 Qi and the non-integer part of the solar breath Qi but the respective units of time are not the same in both cases: the first case represents a number of mo units and the second a number of days. Yet, given that a Mo contains Yr days, the value of P1 Qi can be converted into days: P1 Qi = ( fi Y d × ) . b r (7.8) Moreover, P2 belongs to the solar period [Qi , Qi+1 [. The number of days of the interval Qi P2 is thus strictly smaller than a solar period and we have: Qi P2 < Y 24 and Qi P2 = P1 P2 − P1 Qi = 1 mo − P1 Qi . (7.9) Hence: Y fi Y Y − × < . r b r 24 (7.10) fi r > 1− . b 24 (7.11) Lastly: Now, reasoning backwards, a justification of test 1, p. 224, is obtained. Then, the expression Ji relating to the number of days mentioned on p. 225 above is readily obtained with the help of the following figure: ji days P1 I Y r fi b Qi P2 days where: 1. the Mo point P1 precedes the solar breath Qi ; 2. the Mo point P2 is posterior to the same solar breath; Qi+1 CALCULATIONS TECHNIQUES 229 3. the instant I indicates the midnight of the day of occurrence of the solar breath Qi ; 4. IP2 = ji days and IQi = fi /b days. Then, with the same conventions as before, and by using the fact that P1 Qi = bfi × Yr once again, it follows that the length of the interval P1 P2 , expressed in days, is equal to: P1 P2 = Y fi Y fi = P1 Qi + IP2 − IQi = × + ji − . r b r b (7.12) Hence: fi Y ji = + b r ( fi 1− b ) . (7.13) Lastly, writing a/b and a/b − 360 instead of Y and r, respectively, this expression can be simplified as follows: ji = a − 360 fi . a − 360b (7.14) The sought number of days, Ji , is thus equal to the integer part of ji . Method 4 Let us suppose now that the lunar month [Li , Li+1 [ contains a Mie point of the second type (or, more simply, a Mie), say M2 :9 i lunar months L0 = 0 one lunar month M1 Li M2 Li+1 one mie The interval OLi being composed of i lunar months m, its number of days is generally not an integer. Therefore, exactly in the same way as before, we have: 9 The interval between two consecutive Mie being greater than a lunar month, the Mie point M1 immediately preceding M2 is located before the new moon Li . 230 MO AND MIE DAYS OLi = i × m = Ni + fi b with fi < 1 d. b (7.15) < Ni ; bfi > is thus the binomial representation of the new moon Li . m d The number of days between two consecutive Mie, 30−m , being taken as a new unit of time (see definition 7.3 above) – the mie – the interval OLi is composed of a number of mie units equal to: ( m OLi = (m × i)/ 30 − m ) ( fi = i(30 − m) = 30i − Ni − b )Mie . (7.16) As such, however, this result cannot be used directly as was the case in the previous justification of the method 3 because it contains a difference and not a sum of an integer plus a fraction. But it can be easily expressed as a sum without changing its value by a simultaneous addition and subtraction of one mie unit, its integer and fractional parts thus becoming distinguishable from each other: [ ( fi OLi = (30i − Ni − 1) + 1 − b )]Mie . (7.17) To sum up, the two components of OLi , (30i − Ni − 1) and (1 − bfi ), are respectively equal to an integer number of mie units and a fraction of such an unit. In geometrical terms, OLi is decomposed into an interval OM1 composed of an integer number of intervals one mie unit long each and another interval M1 Li smaller than one mie unit (see the preceding figure). Therefore, (1 − bfi ) represents the length of the interval M1 Li . But M1 M2 = 1 mie unit. Consequently: fi mie (7.18) b Moreover, M2 belongs to the lunar month [Li , Li+1 [. The number of days of the interval Li M2 is thus strictly smaller than a lunar month and we have: Li M2 = fi m × < m. b 30 − m (7.19) CALCULATIONS TECHNIQUES 231 Hence: fi < 30 − m. b (7.20) Now, reasoning backwards as before, test 2, p. 225, is obtained. Lastly, it remains to show how to obtain the value of Ji (expression 7.2 above). To this end, we shall use a method parallel to the one used above for the Mo and based on the following figure: ji days M1 I fi Li M2 Li+1 one mie Let I be the instant of midnight marking the beginning of the day containing the mean new moon Li and suppose also that the length of the two intervals IM2 = ji and ILi = bfi are expressed by taking the day as main unit. Then: M1 M2 = M1 Li + IM2 − ILi . (7.21) Now, when M1 M2 , M1 Li , IM2 and ILi are replaced by their respective values, this equality is changed into the following: ( ) m fi m fi = 1− + ji − . 30 − m b 30 − m b (7.22) Hence: ji = 30 fi d . b(30 − m) (7.23) Consequently Ji = ⌊ ji ⌋. Lastly, the replacement of m by its value, c/b, shows that: ⌊ 30 fi Ji = 30b − c ⌋ d . (7.24) 232 MO AND MIE DAYS Supplementary Results The inequalities of the above tests 1 and 2, intended to determine the presence of a Mo day in a solar period or of a Mie day of the second kind in a mean lunar month, allow us to obtain the following three supplementary results, all related to the surface structure of Chinese calendars calculated with mean elements: Result 1 (Number of days of a Solar Period) In any surface calendar determined from mean elements, a solar period is composed of 16 or 15 days depending on whether it contains a Mo or not. Result 2 (Number of Days of a mean Lunar Month) In any surface calendar determined from mean elements, a mean lunar month containing a Mie is hollow. Result 3 (Repartition of Full and Hollow Months) In any surface calendar determined from mean elements, no succession of two hollow months ever happens and every hollow month is necessarily followed by a full month. On the contrary, successions of at most two full months are possible. Justifications The following justifications are only intended to provide an overall idea of the corresponding reasonings. Result 1 Suppose that a solar period contains a Mo. Then, from test 1 above, the fractional part, bfi , of its initial solar breath, qi = < ai ; fi >, is such that fi r b > 1 − 24 , where r is the non-integer number of days which should be added to 360 days in order to obtain a full solar year (see the above definition 7.1). Now, from the point of view of the deep structure of the calendar, the r d length of a mean solar period is equal to (15 + 24 ) and qi+1 is obtained from qi by adding to it ai + bfi . Lastly, whereas the two fractions involved in this sum are each inferior to one day, their sum is on the contrary greater than one day because: SUPPLEMENTARY RESULTS ( fi r r r ) + > 1− + = 1 d. b 24 24 24 233 (7.25) Consequently, the number of days between the two solar breaths of the surface calendar corresponding to the qi and qi+1 of its deep structure is necessarily equal to one more day than their usual 15 days, that is 16 days. Result 2 From test 2 above, if a lunar month [ni , ni+1 [ contains a Mie of the second type, its new moon ni =< ai ; fi > is such that bfi < 30 − m. But the sum c′ of the fractional part of a lunar month, , and of bfi is necessarily smaller b than one day because: ( ) f i c′ c′ c′ + < 30 − 29 + + = 1. b b b b (7.26) Therefore, the surface lunar month corresponding to [ni , ni+1 [ has less than 30 days. Result 3 Two successive full months are possibly devoid of any Mie because their total number of days, 2 × 30 = 60, is smaller than the distance between two consecutive Mie, 62 or 63 days: this situation can happen when a first Mie is located not far from the beginning of the first month while the second falls just after the end of the second. In practice, numerous examples of such consecutive full months are easily located. Nevertheless, successions of more than two full months cannot exist in the surface calendar because, the distance between two consecutive Mie being equal to 62 or 63 days, the interval of 90 days composing such three months necessarily contain a Mie day. Hence, from the second result above, the month containing this Mie is necessarily hollow. The same conclusion also holds in the case of more than three months, of course, but when the calendar is based on true elements, this conclusion is no longer true. 234 MO AND MIE DAYS The Hypothetical Indian origin of Mo and Mie days Unlike other ancient elements of the Chinese calendar, such as solar breaths or seasonal indicators, already documented well before the beginning of our era, Mo and Mie days appear suddenly in Chinese calendars for the first time in those obtained from the Sifen canon, adopted in 85 BC under the Posterior Han dynasty.10 Moreover, they are never mentioned in any more ancient non-calendrical Chinese source. In two recent articles, the historian of Indian astronomy Y. Ōhashi has conjectured for the first time the possibility of an Indian origin of Mo and Mie days by noting that, on the one hand, ancient Indian calendars use a division of the solar year and of the lunar month into 360 and 30 equal parts, respectively11 and, on the other hand, that the Chinese Mie of the second type are formally akin to the Indian ks.aya-tithi.12 In the Artha-sastra, an Indian treatise going back to approximately 300 BC, well before the Later Han dynasty (85–263), the solar year is indeed divided into 360 equal parts or ‘artificial solar days’ called saura-divasa or saura-dina, all slightly longer than ordinary solar days. Similarly, other Indian sources divide the lunar month into 30 equal parts called tithi or ‘artificial lunar days’, all slightly shorter than an ordinary day.13 If a connection between Indian and Chinese notions is to be established in this respect, however, we must acknowledge that, as far as we can surmise from the scant extant documentation, it should have been quite indirect because Chinese astronomical canons never use artificial solar and lunar days as such but only natural days beginning and finishing at midnight and never at other moments during the period studied in the present work. Regardless, it remains possible to establish a surprising connection between Mo days, Mie days and the two artificial divisions of the solar year and of the lunar month into 360 and 30 equal parts. 10 Hou Hanshu, zhi 1, ‘lüli 3’, p. 3063. Ōhashi 2000 and 2001. 12 Y. Ōhashi 2000, p. 267. However, Indian days similar to the Chinese Mo have not been identified. 13 The tithi is a Sanskrit term equally known in Babylonian astronomy. See O. Neugebauer 1975, vol. 1, p. 358. 11 Y. THE INDIAN ORIGIN OF MO AND MIE DAYS 235 Given that we have defined above three different kinds of Mo and Mie days and not two, however, we must analyze the following three cases: Case 1 The Mo days – that is those containing a Mo point – are all ordinary and are always included in a saura-divasa, like those of the Artha-sastra mentioned by Y. Ōhashi. In other words, when such a day is present in the calendar, an artificial solar day beginning earlier and finishing later always exists (Fig. 7.2 below). Now, in order to understand the idea, we shall start from a simple example based on the Sifen li calculations. Example Ordinary and artificial days in the Sifen li. In the Sifen li, with the day taken as mean unit of time, we have: 1 solar year = 1461 4 and 1 artificial solar day = 1461 4×360 = 487 480 . Then, the successive abscissas of the theoretical first Mo points of the Sifen li, enumerated from the Superior Epoch, are equal to 487k 7 days with k = 0, 1, . . . (definition 7.1 above). Moreover, the corresponding Mo days are all included in artificial solar days whose beginnings and 487 480k 487 487 14 ends are respectively equal to: ⌊ 480k 7 ⌋ × 480 and ⌊ 7 ⌋ × 480 + 480 . Hence the following table, showing that the 487 first theoretical days of the Sifen li contain, on the one hand, six Mo days, which are all included in six artificial solar days and, on the other hand, a Mie day of the first type, [487, 488[, whose beginning coincides with the beginning of an artificial solar day: Likewise, other astronomical canons based on mean elements would lead to similar results. Hence, at least, a formal link between the Chinese notions of Mo and Mie and the above Indian divisions of the solar year and lunar month. 14 Starting from the Superior Epoch of the Sifen li, the first artificial solar day is the interval [0, 487 480 [ and, for each k, the integer number of artificial solar days contained in 487k 487 480k the interval [0, 487k 7 [ is obtained by dividing 7 by 480 , that is by calculating ⌊ 7 ⌋. 480k 487 The abscissas of the beginnings of these solar days are thus equal to ⌊ 7 ⌋ × 480 , k = 1, 2, . . .. 236 k 1 2 3 4 5 6 7 MO AND MIE DAYS Beginnings of Mo Artificial Solar Days Points 68 119 120 138 479 480 207 95 96 277 239 240 346 79 80 416 159 160 487 69 139 208 278 347 417 487 4 7 1 7 5 7 2 7 6 7 3 7 Mo Days Extremities of Artificial Solar Days [69, 70[ [139, 140[ [208, 209[ [278, 279[ [347, 348[ [417, 418[ [487, 488[ 1 70 160 1 140 80 1 209 240 1 279 96 1 348 480 1 418 20 7 488 480 Case 2 With the preceding division of the solar year into 360 parts when a Mo is a Mie of the first type then its beginning coincides with the solar day saura-divasa in question, but not its extremity (Fig. 7.3 on next page). Case 3 When a lunar month is divided into 30 tithi then, Mie days of the second type always contain a tithi. In other words, the instant of midnight of a Mie day precedes the beginning of a certain tithi and the corresponding artificial day ends before the instant of midnight of the Mie day in question (Fig. 7.4 below). THE INDIAN ORIGIN OF MO AND MIE DAYS 237 Mo day saura divasa Figure 7.2. The inclusion of Mo days (Mori) into artificial solar days saura-divasa. Mie day saura divasa Figure 7.3. The inclusion of Mie days (Mieri) of the first type into artificial solar days saura-divasa. Mie day tithi Figure 7.4. The inclusion of artificial lunar days (tithi) into Mie days (Mieri) of the second type. Part III Examples of Calculations CHAPTER 8 THE QUARTER-REMAINDER CANON Importance The Quarter-remainder canon – Sifen li1 – is the official astronomical canon adopted in 85 AD under the Later Han dynasty (25–220). It stands out due to its exceptional longevity (179 years) and its ability to outlast dynastic changes: after the division of China into three kingdoms, Wei (220–265), Shu (221–263) and Wu (222–280), the two first new dynasties still kept it in force until 236 and 263, respectively, rather than reforming their astronomical canons.2 Fundamental Parameters Number of years t(x)3 elapsed between the two winter solstices of the Superior Epoch and a given year x, primary and secondary solar and lunar constants:4 t(x) = 9366 + (x − 85) ( ) a 1461 1 d = = 365 + b 4 4 (solar years) (8.1) (solar year) (8.2) 1 Hou Hanshu , zhi 3, ‘lüli 3’, p. 3055–3100; Zhu Wenxin, 1934, p. 82–85; Chen Zungui 1984, p. 1433–1436, Yan Dunjie 1989a, LIFA, p. 302 f. 2 Chen Zungui, ibid., p. 1399, note 6. 3 The expression of t(x) follows from its general definition (p. 139 above) but its number 9366 of years is not mentioned in the Sifen li procedures. Instead, it has been obtained from the numerical data relating to ancient Chinese astronomical canons listed in a later astrological treatise of the Tang dynasty (618–907), the Kaiyuan zhanjing (notice p. 394 below). See Appendix E, note 6, p. 358. 4 Hou Hanshu, zhi 3, ‘lüli 3’, p. 3059. © Springer-Verlag Berlin Heidelberg 2016 J.-C. Martzloff, Astronomy and Calendars – The Other Chinese Mathematics, DOI 10.1007/978-3-662-49718-0_8 241 242 THE QUARTER-REMAINDER CANON ( ) a 487 7 d = = 15 + 24b 32 32 ( ) c 27, 759 499 d = = 29 + d 940 940 ( ) c 359 3 d = 7+ + 4d 940 940 × 4 (solar period) (8.3) (lunar month) (8.4) (lunar phases) (8.5) α = 19 β = 7 γ = 235 (Metonic constants). (8.6) The Calculation of the Calendar of the Year 119 Initial Calculations The general calculation techniques of Chinese Metonic canons, applied to the particular case of the Sifen li, allow us to readily obtain the solar and lunar sexagenary dates of any given year. However, no comparison with any extant official calendar is possible since no authentic such calendar from the later Han or other dynasties has reached us. We have thus no reason to deem more important a given year than any other. The example of the year Yuanchu 6 (119) retained here is as much or as less representative as any other, save perhaps the fact that, formerly, the Chinese historian of astronomy Gao Pingzi ddd has explained the most essential aspects of the Sifen li arithmetic and applied them to its case. The possibility to check the validity of our calculations was thus offered to us at an early stage of the present research.5 In order to calculate the calendar of the year 119, we have to start from its support year, the year 118. From the above formula 8.1, p. 241, the relevant number of solar years is equal to t(118) = 9399, and the previous general formulas 4.5, 4.2 and 4.6, p. 159, lead mechanically to the following preliminary results: q1 (118) = bin(1461 × 9399, 4) m = ⌊(235 × 9399)/19⌋ n11 (118) = bin(27759 × 116250, 940) 5 Gao Pingzi 1987, p. 118–121. = < 24; 3 >, = 116, 250 lunar months, = < 1; 410 > . THE YEAR 119 243 The first result means that the winter solstice of the year 118 happens on a day whose sexagenary binomial is equal to #25 ((5, 1), or wuzi), three quarters of a day after its instant of midnight, that is, for us, at 18h , the ‘3’ of the binomial < 24; 3 > meaning 34 d . The second result (number of lunar months) is an intermediary result. The third result means that the new moon of the eleventh month of d the year 118 occurs on a sexagenary day #2 ((2, 2) or yichou), 410 940 after its instant of midnight, or slightly after 10h 28m in the morning. Insofar as the number system of the solar breaths rely here on fractions whose denominator is 32 rather than 4, the first result above should be written q1 (118) =< 24; 24 > rather than < 24; 3 > since 3/4 = 24/32. More generally, the same trivial modification should be applied to similar results. Another Procedure Although the preceding calculations lead to results consistent with those of the original Sifen li procedures, their original version uses an initial number of solar years smaller that t(x). In fact, this different, but equivalent approach, takes avail of the fact that t(x) is composed of several supra-annual periods having a number of years greater than the 19 years of the Metonic cycle. One of these, the bu d, is composed of 76 years or four ordinary Metonic cycles (76 = 19 × 4). Like the Callipic period,6 it implies that the solar breaths and new moons of the Sifen li repeat themselves periodically every 76 years (see p. 254 below). Taking this property into account, the initial number of solar years t(x) can be replaced by t(x) modulo 76. Hence a smaller number of years, composed of at most 76 solar years. Always with the same example, the initial interval of 9399 years can thus be replaced by another smaller interval containing only 9399 mod 76 = 51 solar years. However, the new origin of time determined by its left extremity is not necessarily a day #1 ((1, 1) or jiazi). That is why the Sifen li lists the possible 6 Callipus (or Kallipos) is the name of the Greek astronomer who is supposed to have invented the cycle of 76 years. See D.R. Dicks 1970/1985*, p. 190 f.; G. Rocca-Serra 1980, p. 28 and 29, R. Hannah 2005, p. 56. 244 THE QUARTER-REMAINDER CANON sexagenary binomials of all initial days of a cycle of 76 years in a table analogous to the following:7 bu Bin. no. bu Bin. no. 1 #1 11 #31 2 #40 12 #10 3 #19 13 #49 4 #58 14 #28 5 #37 15 #7 6 #16 16 #46 7 #55 17 #25 8 #34 18 #4 9 #13 19 #43 10 #52 20 #22 Table 8.1. The ranks of the sexagenary binomials of the first days of the twenty different cycles of 76 years the Sifen li is composed of. The Sifen li does not explain the rationale of this table, but its derivation is straightforward because a bu contains 365.25 × 76 = 27759 days. The ranks of the sexagenary binomials of their first days are thus equal to: 27759(i − 1) mod 60, i = 1, 2, . . ., with results comprised between 0 and 59. Modulo a trivial modification, we thus get twenty different ranks comprised between 1 and 60 and this fact implies that the Sifen li calculations also give rise to another supra-annual period composed of 1520 solar years (20 × 76). Moreover, a still greater supra-annual period, also taking into account the sexagenary numbering of solar years is also used in the Sifen li. Hence a new cycle composed of 60 successive bu d, or 60 × 76 = 4560 solar years.8 The first day of a bu d is thus a jiazi day (1, 1) every 1520 solar years (a period called tong d), and two consecutive jiazi solar years occur every 4560 solar years (yuan d). To sum up, the Sifen li uses four supra-annual periods, or cycles: the ordinary Metonic cycle of 19 years (zhang) d, the bu d, the tong d and the yuan d, respectively composed of 20 and 60 bu.9 While these various supra-annual periods are deeply conditioned by various astrological, political and philosophical assumptions concerning the cyclical character of the cosmos,10 they also play a much more down to earth role in the Sifen calculations: exactly in the same way as in the case 7 Hou Hanshu zhi 2, ‘lüli 2’, p. 3061. Hou Hanshu , ibid., p. 3061–3062. See also Gao Pingzi 1987’s analysis of the subject, p. 112–113. 9 See J. Needham 1959, p. 406. 10 N. Sivin 1969. 8 See THE YEAR 119 245 of the bu, they are used to diminish significantly the value of the initial number of solar years owing to a succession of divisions of t(x) by their respective numbers of years. For example, in the case of the year 119, 9399 is successively divided by 4560, 1520 and 76: 9399 = 2 × 4560 + 0 × 1520 + 3 × 76 + 51. (8.7) This decomposition of the initial time interval of 9399 years can be represented geometrically by the following diagram, where the 9399 years separating the two winter solstices of the Superior Epoch, O and of the support year of the year 119, is divided into two intervals of 4560 years, 3 intervals [O1 O2 [, [O2 O3 [ and [O3 O4 [ of 76 years and, a remaining interval of 51 years [O4 S[: 1st yuan 2st yuan O (Superior Epoch) 3st yuan O1 O2 O3 O4 S 1st bu 2nd bu 3rd bu 51 years Here, the Superior Epoch, O, and the beginning of the first bu, O1 , are 2×4560 years distant from one another and the day corresponding to O1 indicates the beginning of the third yuan period. Its first day is thus numbered #1 and corresponds to (1, 1) or jiazi. In addition, as noted in Table 8.1, O2 , O3 and O4 (the beginnings of the second, third and fourth bu, respectively), are successively numbered #40, #19 and #58. At last, the replacement of t = 9399 by 51 years in the general formulas above (from 4.2 to 4.6, p. 159) imply that days are enumerated from an original day #58 instead of #1 and the results of the calculations have to be modified accordingly by adding 57 to them (57 = 58 − 1). But, apart from this minor adjustment, the general formulas 4.5, 4.2 and 4.6, p. 159 above, are still wholly usable without any further modification: q1 (118) = bin(1461 × 51, 4) m = ⌊(235 × 51)/19⌋ n11 (118) = bin(27759 × 630, 940) = < 27; 3 >, = 630 lunar months, = < 4; 410 > . 246 THE QUARTER-REMAINDER CANON As already noted, the fractional part of q1 (118) should still have 32 as denominator so that q1 (118) = < 27; 24 > and with the new origin of time, q1 (118) becomes equal to < (27 + 57) mod 60; 24 >=< 24; 24 >. Likewise, n11 (118) =< (4 + 57) mod 60; 940 >=< 1; 410 >. Other Solar and Lunar Elements q1 (118) and n11 (118) being obtained in one way or another, the successive solar breaths of the year 119 are readily obtained by adding as many times as necessary the constant length of a solar period and of a lunar month to these values, respectively. Therefore: qi = (24 + 24 7 ) + (15 + )(i − 1) i = 1, 2, . . . 32 32 (8.8) and ni = (1 + 410 499 ) + (29 + )(i − 1) 940 940 i = 1, 2, . . . (8.9) Similarly, the mean phases of the moon ni,k ; k = 1, 2, 3, 4, i = 1, 2 . . ., of each lunar month ni , are obtained by adding the successive multiples of the constant 7;359,3 to ni (mean time between two consecutive moon phases): ni,k = ni + (7 + 359 3 + )(i − 1) i = 1, 2, . . . 940 940 × 4 (8.10) The integer parts of these results should then be reduced modulo 60. Hence the following tables 8.2 (solar breaths) and 8.3 (lunar phases): i qi 1 24;24 2 39;31 3 55;06 i qi 4 10;13 5 25;20 6 40;27 i qi 7 56;02 8 11;09 9 26;16 i qi 10 41;23 11 56;30 12 12;05 Table 8.2. Initial values of solar breaths qi required for the calculation of the b calendar of the year 119 (0 ≤ a ≤ 59 and a; b = a + 32 ). THE YEAR 119 i 1 1 1 1 qi (60/940/4) 1;410,0 8;769,3 16;189,2 23;549,1 i 2 2 2 3 qi (60/940/4) 30;909,0 38;328,3 45;688,2 53;108,1 247 i 3 3 3 3 qi (60/940/4) 0;468,0 7;827,3 15;247,2 22;607,1 Table 8.3. Initial values of lunar phases required for the calculation of the calendar of the year 119 (each column concerns a particular month (1, 2 or 3) and the four moon phases of a given month are listed in its column). Next, it remains to renumber new moons in their calendrical order n11 , n12 , n1 , n2 . . . but this cannot be fully done without having determined the possible existence of an intercalary month, liable to induce a further perturbation of the sequence of their indices. This will be done below but, no less importantly, it must also be noted that the determination of moon phases other than new moons requires some further developments because of its dependence on the following criterion: Criterion 8.1 (Phases of the Moon Other than New Moons) If the fractional part of any moon phase other than a new moon is smaller than 260 940 , then the phase in question should be noted in the calendar one day earlier than its calculated occurrence.11 From Li Rui12 ’s interpretation, the fraction 260 940 – introduced without any justification in the Sifen li procedures – represents a rough approximation of the theoretical duration of the interval from midnight to sunrise. Therefore, the above criterion means that when a calculated moon phase other than a new moon is located between these two limits, 11 Hou Hanshu, zhi 3, ‘lüli 3’, p. 3063. Rui dd (1765–1814) is the author of the Han Sifen shu dddd (The Sifen li of the Han Dynasty), an important critical study of the Sifen li, (see COL-astron, vol. 2, p. 760). He is also a well-known scholar and member of the evidential research movement kaozhengxue ddd which resulted in deep changes in the admitted modes of textual explanation, typical of traditional Confucian scholarship (see B.A. Elman 1984). He has also contributed in an essential way to the elaboration of the Chouren zhuan (notice p. 391 below). On Li Rui, see A.W. Hummel 1943/1970*, p. 144; B.A. Elman 2005, p. 269 f.; Horng Wann-sheng 1993. 12 Li 248 THE QUARTER-REMAINDER CANON it should be regarded as having occurred one day earlier. More precisely, he explains that the table of the Sifen li, giving the durations of day and night in relation to the 24 solar periods of the Chinese calendar,13 mentions that the night is 55 ke long during the winter solstice period (1 ke = 1 day/100). Now, supposing that this duration is valid during the whole solar year, it follows that its half, 27.5 ke, represents the interval of time between the instant of midnight and sunrise. The day being divided into 940 parts in the Sifen li, 27.5 ke are equivalent to (940 × 27.5)/100 = 258.5 or 260 parts, approximately. Hence the fraction 260 940 appearing in the above criterion. The complete calculation of the year 119 needs the application of this criterion several times, for example in the case of the full moon of the 2 260 first month of the year 119, equal to 15;247,2, since 247 940 + 940×4 < 940 . The following Table 8.4 (next pages), contains all results concerning the calculations of the year 119 with, if need be, a shift of the relevant phases of the moon which have been asterisked accordingly. The Intercalary Month of the Year 119 From the expression 4.3, p. 159, the epact of the year 119 is equal to 235 × 9399 mod 19 = 15 and is greater than 12. The year 119 is thus intercalary and a systematic comparison between q1 , q2 , . . . and the new moons n1 , n2 , . . ., numbered in natural order each time, reveals that q13 < n8 < q14 < n9 < q15 because: 143 19 642 26 27 + 12 32 < 28 + 940 < 42 + 32 < 57 + 940 < 57 + 32 . Consequently, from criterion 3.1 on p. 151 (determination of intercalary months), the month [n8 , n9 [ contains no odd solar breath and is thus intercalary. Then, by replacing n1 by n11 , n2 by n12 , n3 by n1 and so on, in order to have months numbered in the order of the final calendar, [n8 , n9 [ corresponds to [n6 , n7 [ and the preceding month to [n5 , n6 [. Therefore, the intercalary month follows the 5th month and should be denoted 5∗ . The final numbering of the successive lunar months of the year 119 is thus 1, 2, 3, 4, 5, 5∗ , 6 . . . This last result and the preceding ones fully determine the lunisolar component of the calendar of the 13 Hou Hanshu, ibid., p. 3077–3079 . THE YEAR 119 249 year 119. Hence the following table containing the fundamental solar and lunar elements of the year 11914 and where the odd and even solar breaths have not been listed in the same column in order to highlight the lunisolar coupling: Months Year 118 (Months 11 and 12) and 119 (Months 1, 2, . . . , 12) Lunar Phases Solar Breaths Dates 11 NM FQ FM LQ 1;410,0 8;769,3 16;189,2* 23;549,1 12 NM FQ FM LQ 30;909,0 38;328,3 45;688,2 53;108,1* NM FQ FM LQ 0;468,0 7;827,3 15;247,2* 22;607,1 NM FQ FM LQ 30;027,0 37;386,3 44;746,2 52;166,1* NM FQ FM LQ 59;526,0 6;885,3 14;305,2 21;665,1 1 2 3 1/12/118 q1 24;24 30/12/119 q3 q2 39;31 q4 10;13 55;06 29/1/119 q5 25;20 28/2/119 q6 q7 40;27 56;02 29/3/119 q8 q9 11;09 26;16 14 The first solar breath of the lunar month n (118) is not included in this table 11 because the calculation of the calendar of the year 119 does not require it any more than all those of its support year. 250 THE QUARTER-REMAINDER CANON Months Year 118 (Months 11 and 12) and 119 (Months 1, 2, . . . , 12) Lunar Phases Solar Breaths Dates 4 NM FQ FM LQ 29;085,0 36;444,3 43;804,2 51;224,1* 5 NM FQ FM LQ 58;584,0 6;003,3* 13;363,2 20;723,1 NM FQ FM LQ 28;143,0 35;502,3 42;862,2 50;282,1 NM FQ FM LQ 5* 6 7 8 8 9 10 11 q10 q11 41;23 28/4/119 56;30 27/5/119 q12 q13 27;12 q14 42;19 57;642,0 5;061,3* 12;421,2 19;781,1 q15 57;26 NM FQ FM LQ 27;201,0 34;560,3 41;920,2 49;340,1 q17 NM FQ FM LQ 56;700,0 4;119,3* 11;479,2 18;839,1 q19 NM FQ FM LQ 26;259,0 33;618,3 41;038,2* 48;398,1 q21 NM FQ FM LQ 55;758,0 3;177,3* 10;537,2 17;897,1 q23 NM FQ 25;317,0 32;676,3 q1 12;05 26/6/119 25/7/119 q16 13;01 28;08 24/8/119 q18 43;15 58;22 22/9/119 q20 13;29 29;04 22/10/119 q22 44;11 59;18 20/11/119 q24 30;0 14;25 20/12/119 THE YEAR 119 Months 251 Year 118 (Months 11 and 12) and 119 (Months 1, 2, . . . , 12) Lunar Phases Solar Breaths Dates 11 FM LQ 40;096,2* 47;456,1 12 NM FQ FM LQ 54;816,0 2;235,3* 9;595,2 17;015,1* q2 q3 45;07 0;14 18/1/119 q4 15;21 Table 8.4. List of numerical results for the year 119 (end of the year 118 and year 119). With these results, the surface calendar of the year 119 is obtained by numbering the days of all lunar months from 1 to 29 or 30 according to their full or hollow character. Hence the following table, where the ranks of the sexagenary binomials (Bin.) are indicated in the third column: Months 1 full 2 hollow 3 full Year 119 Binomials Moon Ph. Day No. Bin. 1 8 11 15 23 26 #1 #8 #11 #15 #23 #26 (1 , 1) (8 , 8) (1 , 11) (5 , 3) (3 , 11) (6 , 2) NM FQ 1 8 11 15 22 27 #31 #38 #41 #45 #52 #57 (1 , 7) (8 , 2) (1 , 5) (5 , 9) (2 , 4) (7 , 9) NM FQ 1 8 13 16 23 28 #60 #7 #11 #15 #22 #27 (10 , 12) (7 , 7) (2 , 12) (5 , 3) (2 , 10) (7 , 3) NM FQ Solar Breaths Dates 29/1/119 q4 FM LQ q5 28/2/119 q6 FM LQ q7 29/3/119 q8 FM LQ q9 252 THE QUARTER-REMAINDER CANON Months Day No. Bin. Year 119 Binomials Moon Ph. 4 hollow 1 8 13 15 22 28 #30 #37 #42 #44 #51 #57 (10 , 6) (7 , 1) (2 , 6) (4 , 8) (1 , 3) (7 , 9) 1 8 15 16 23 30 #59 #6 #13 #14 #21 #28 (9 , 11) (6 , 6) (3 , 1) (4 , 2) (1 , 9) (8 , 4) NM FQ 1 8 15 #29 #36 #43 (9 , 5) (6 , 12) (3 , 7) NM FQ FM q14 6 full 23 1 #51 #58 (1 , 3) (8 , 10) LQ NM q15 6 full 8 16 17 23 #5 #13 #14 #20 (5 , 5) (3 , 1) (4 , 2) (10 , 8) FQ FM 1 2 8 15 17 23 #28 #29 #35 #42 #44 #50 (8 , 4) (9 , 5) (5 , 11) (2 , 6) (4 , 8) (10 , 2) NM 1 3 8 16 18 23 #57 #59 #4 #12 #14 #19 (7 , 9) (9 , 11) (4 , 4) (2 , 12) (4 , 2) (9 , 7) NM 1 4 #27 #30 5 full 5* hollow 7 hollow 7 8 full 9 hollow (7 , 3) (10 , 6) Solar Breaths NM FQ Dates 28/4/119 q10 FM LQ q11 27/5/119 q12 FM LQ q13 26/6/119 25/7/119 q16 LQ 24/8/119 q17 FQ FM q18 LQ 22/9/119 q19 FQ FM q20 LQ NM 22/10/119 q21 THE YEAR 119 Year 119 Binomials Moon Ph. Months Day No. Bin. 9 hollow 8 15 19 23 #34 #41 #45 #49 (4 , 10) (1 , 5) (5 , 9) (9 , 1) FQ FM 1 5 8 16 20 23 #56 #60 #3 #11 #15 #18 (6 , 8) (10 , 12) (3 , 3) (1 , 11) (5 , 3) (8 , 6) NM 1 6 8 15 21 23 #26 #31 #33 #40 # 46 # 48 (6 , 2) (1 , 7) (3 , 9) (10 , 4) (6 , 10) (8 , 12) NM 1 7 8 16 22 23 #55 #1 #2 #10 #16 #17 (5 , 7) (1 , 1) (2 , 2) (10 , 10) (6 , 4) (7 , 5) NM 10 full 11 hollow 12 full 253 Solar Breaths Dates q22 LQ 20/11/119 q23 FQ FM q24 LQ 20/12/119 q1 FQ FM q2 DQ 18/1/120 q3 FQ FM q4 LQ Table 8.5. The lunisolar elements (moon phases and solar breaths) of the intercalary year 119. A comparison between the dates of the above solar breaths and new moons15 with those of Chinese concordance tables16 confirms the correctness of their Julian dates. Quite differently, but no less usefully, the general criterion 4.1, p. 159, provides an easy way to check the hollow or full character of all lunar months: 15 No table listing moon phases has ever been published. for instance, Zhang Peiyu 1990*/1997*. 16 See, 254 THE QUARTER-REMAINDER CANON Criterion 8.2 (Full and Hollow Months) Depending on whether the numerator of the fractional part of a given new moon of the Sifen li is superior or equal to 441 (= 940 − 499) or not, the corresponding month is full or hollow. For example, n5 = 58; 584, 0 and since 584 > 441, the fifth lunar month of the year 119 is necessarily full. More generally, a repeated application of this criterion also shows that the year 119 is composed of an alternating sequence of full and hollow months, beginning with a full month. The General Structure of the Sifen li This section provides a full characterization of the monthly lunar structure of all calendars obtained from the Sifen li procedures and the results so obtained are also used in order to describe the structure of the Chinese calendrical chronology concerning the years of official validity of the Sifen li. To this end, we first ask ourselves whether the surface calendars obtained from the Sifen procedures reproduce themselves periodically or not and if so, what is the relevant number of years. First of all, we note that the Sifen li is a 19/7 Metonic canon. It would thus seem that its surface calendars are governed by a cycle of 19 solar years and that its sequences of full and hollow months and the ranks of its intercalary months are reproduced identically every 19 years. A careful observation of the structure of years during the years of validity of the Sifen li shows, however, that this is not the case. The situation is thus more complex than it at first appears but we will tackle the question from the easier, but equivalent, case of initial sequences of Sifen calendars, calculated from the winter solstice of the Superior Epoch. By definition, the initial new moon and winter solstice occur simultaneously at the midnight of the Superior Epoch and since 19 × 365.25 = 235 × (27, 759/940) = 6939.75 d , the last lunisolar conjunction of the first Metonic cycle takes place at what corresponds to 18h for us. Similarly, the last lunisolar conjunctions of the second and third Metonic cycle take place at 12h and 6h , since 2 × 6939.75 = 13879.5 d and 3 × 6939.75 = 20, 819.25 d , respectively. When the fourth cycle of GENERAL STRUCTURE 255 19 years has been exhausted, the initial conjunction of the next cycle occurs once again at midnight since 4 × 19 = 27, 759 d = 76 solar years. Therefore, the calculated new moons of the Sifen li follow a Callipic cycle of 4 × 19 = 76 solar years and, as can be readily checked, the same is true of its solar breaths. If the sexagenary dates of new moons and solar breaths are also taken into account, however, this conclusion does not hold because the number of days contained in 76 solar years, 27,759, is not divisible by 60. In particular, the initial day of the second cycle of 76 years cannot be a jiazi day (#1), as already noted. Still, it is obviously true that the sexagenary cycle only concerns the numbering of days and years but not the lunisolar skeleton of the calendar, represented by the hollow or full character of its lunar months, the ranks of its intercalary months, the relative position of the solar breaths with respect to new moons and their lunar dates. In other words, the Callipic period is the only really fundamental reference period of the Sifen li. Consequently, we need to calculate all the fundamental solar and lunar elements of the Sifen li only for the 76 years of any Callipic cycle. In particular, the sought results can be checked by choosing a set of years belonging to the period of validity of the Sifen li, (85–263). Among these, the years 144, 145, ..., 219 constitute a full Callipic period where the year 144 – calculated by taking into account the winter solstice of its support year (143) – corresponds to its first year because: t(143) = 9366 + (143 − 85) = 9424 = 124 × 76.17 Moreover, the 19 years of a Metonic cycle always contain 7 intercalary years. Consequently, the 76 years of the interval [144, 219] have 28 intercalary years. Let x be one of these years. Then the rank of its intercalary month necessitates the calculation of all its solar breaths and new moons determined by the usual procedures and the application of the criterion of intercalation (3.1, p. 151 above). Hence the following table giving the various ranks of all intercalary months of the 4 successive Metonic cycles of a Callipic cycle: 17 See p. 241 above. 256 THE QUARTER-REMAINDER CANON Rank of a Year 3 6 8 Cycle no. 1 Cycle no. 2 Cycle no. 3 Cycle no. 4 6* 7* 7* 6* 3* 3* 4* 3* 12* 11* 12* 12* 11 14 17 9* 8* 8* 8* 5* 5* 5* 4* 19 1* 10* 1* 9* 2* 10* 1* 10* Table 8.6. Possible ranks of intercalary years and corresponding intercalary months over a Callipic cycle of 76 years (Sifen li). These results show that any month of a lunar year is liable to be intercalary albeit with different frequencies: the months 2* and 11* appear only once in 76 years, 4*, 6*, 7* and 9* two times and 1*, 3*, 5*, 8* 10* and 12* three times, respectively. The same calculations would also provide the types of successions of lunar months, full or hollow, of a Callipic cycle but it is also possible to rely on a simpler technique depending on the one hand, on the criterion 8.2, p. 254 above, and on the other hand, on the fact that the calculated values of the new moons of the 76 years of a Callipic cycle are easily obtained. To this end, let us determine the 235×4 = 940 successive numerators of the fractional part of the new moons of this cycle by starting from the Superior Epoch and by calculating 27,759(i−1) , i = 1, 2, . . ., the resulting 940 fractions being reduced to the same denominator 940. Then, we readily obtain the following sequence where the initial ‘0’ corresponds to the fact that the fractional part of the initial new moon n11 (0) = < 0; 0 > is equal to 0 while the second value, 499, is equal to the fractional part of n12 (0): 0, 499, 58, 557, 116, 615, 174, 673, 232, 731, 290, 789, 348, 847, . . . A comparison between these values and 441 (criterion 8.2, p. 254) leads immediately to a transformation of this list into another one indicating the full (F) or hollow (H) character of each successive lunar month, H, F, H, F, H, F, H, F, H, F, H, F . . . GENERAL STRUCTURE 257 Hence the two tables 8.8 and 8.9 below, listing all the results obtained in this manner and where the ordinary and intercalary years have been dealt with apart. The first table (ordinary years) highlights the fact that among the 76 years of the Callipic cycle, only 13 different types of years exist (first column). For instance, the line of the table corresponding to the eighth type of year is composed of the following sequence of full and hollow months: FFHFHFHFHFHF. Moreover, these 13 years contain either 354 or 355 days, any other possibility being excluded. Those having 354 days are made of 6 full months and 6 hollow months and those of 355 days have 7 full months and 5 hollow months. The second table (intercalary years) gives the rank of each intercalary year with respect to the Callipic cycle (last column). One of these (Type 1) has 6 full months and 7 hollow months and is thus composed of 383 days. All the others have 384 days, obtained from various sequences of 7 full months and 6 hollow months. Globally, a given Callipic cycle of 76 years contains 27 different types of years, 13 ordinary and 14 intercalary. More precisely: 22 years are composed of an alternate succession of full and hollow months whereas 12 years begin with a full month and 10 others with a hollow month. These regular years are not distributed in any easily predictable regular way among the 76 years of the Callipic cycle, however: some years, like years no. 47 and 48 are consecutive while, in contrast, there is a gap of 3 and 4 years, respectively, between years no. 9 and 13 on the one hand, and no. 26 and 31 on the other. The remaining 76 − 22 = 54 years are characterized differently by the fact that they always possess two full consecutive months (lian da yue ddd), located absolutely anywhere. Nevertheless, no sequence of more than two full consecutive months exists, no more than sequences of two, three or more hollow months. But some sequences of two consecutive full months are unique while others appear several times in the same year in an apparently erratic way, impossible to predict without calculations. The sequence H F H F H F F H F H F H for example, where the two full months are the 6th and the 7th , represents years no. 7, 37, 258 THE QUARTER-REMAINDER CANON 58 and 62 of the Callipic cycle and not any other year; the sequence F H F H F H F H F H F F , where the two full months appear at the end of the year represents only years no. 34 and 59. In both cases, no obvious link between these sequences of full and hollow months and the ranks of the years they belong to can be detected. Conversely, the numbers of days the years of a Callipic cycle are composed of are more regular since only the following four possibilities occur, namely 354, 355, 383 or 384 days, the two last values concerning uniquely intercalary years, of course. Still, the number of years respectively having these numbers of days are strikingly different: on the one hand, 32 ordinary years of 354 days exist but only 16 years of 355 days. On the other hand, 27 intercalary years of 384 days occur but only a single such year has 383 days, namely the 66th year of the Callipic cycle which is composed of 6 full months and 7 hollow months (6 × 30 + 7 × 29 = 383), regularly alternated and beginning with a hollow month. Lastly, the numbers of days of each of the four Metonic cycles of a Callipic cycle are obtained as follows (Fig. 8.7): Cycle nº 1 : Cycle nº 2 : Cycle nº 3 : Cycle nº 4 : 354 × 9 + 355 × 3 354 × 8 + 355 × 4 354 × 8 + 355 × 4 354 × 7 + 355 × 5 + 383 × 1 +384 × 7 + 384 × 7 + 384 × 7 + 384 × 6 = = = = 6939 d 6940 d 6940 d 6940 d Table 8.7. The numbers of days of each Metonic cycle of the Sifen li. To sum up, the complex surface structure of calendars obtained from the Sifen li procedures contrasts sharply with those of other calendars having a regular monthly structure such as the Julian and Gregorian calendars, where the numbers of days of each month and their modes of succession from year to year are easy to memorize, even in the case of bissextile years. The calculation procedures of this famous astronomical canon from the Han dynasty are nonetheless wholly regular and able to generate a sequence of calendars endowed with a seemingly unpredictable arrangement of full and hollow months. In other words, the regularities induced by Sifen li calculations are globally obvious when considering GENERAL STRUCTURE 259 great numbers of years but its irregularities are at the same time striking from the local point of view of small numbers of years. In other words, the structure of the Sifen li presents a good example of the “Chinese conception of an infinite number of unpredictable irregularities within general regularities”.18 In the case of other Metonic canons, ordinary or generalized, it would be likewise possible to replace available chronological tables by simpler tables which would allow us to compare systematically Chinese chronology with the results of theoretical calculations and to display characteristics similar to those of the Sifen li. In practice, however, the limited temporal validity, which is a consequence of the numerous reforms of Chinese astronomical canons, cannot reveal the periodical aspect of Chinese calendars. On the contrary, this feature emphasizes their local irregularities. Unlike the Sifen li, most other Chinese astronomical canons became outdated before having revealed their regularities over sufficiently large numbers of years. Chinese chronology is thus necessarily dependent on voluminous chronological tables displaying endless irregularities. 18 J. Gernet 2005, p. 55. 260 THE QUARTER-REMAINDER CANON Year Type 1 2 Lunar Months, Full (F) or Hollow (H) Num. 3 4 5 6 7 8 9 10 11 12 of Days 1 H F H H F H F 354 2 3 4 5 6 H H H H H F F F F F F H F H F H F H F F H F H F H F H F F H F H F H F H F F H F H F H F H H H H H F F F F F F H H H H H 354 354 354 354 354 7 F H F H F H F H 354 8 9 10 11 12 13 F F F F F F F H F H F H F H F F H F H F H F H F F H F H F H F H F F H F H F H F H H F H F H F H H H H H F F F F F F F H H H H H H F F F F F F F 355 355 355 355 355 355 F H H F F H H F F Year Rank 1 , 5, 26, 31, 35 47, 51, 56, 72 2, 23 28, 32, 53 7, 37, 58, 62 12, 16, 67 21, 42 9, 13, 18, 39, 43 48, 64, 69, 73 10, 61 15, 40, 70 24, 45, 66 20, 50, 54, 75 4, 29 34, 59 Table 8.8. The repartition of lunar months according to their full or hollow character, with respect to the Callipic cycle of 76 years, in the case of ordinary years. Year Type 1 2 3 1 2 3 4 5 6 7 8 9 10 11 12 13 14 H H H H H H H F F F F F F F F F F F F F F H F H H H H H H F H H H H H F H F F F F F Lunar Months, full (F) or hollow (H) Num. 4 5 6 7 8 9 10 11 12 13 of Days F H F F F F F H F F H H H H H F F H H H H F H H F F F F F H H F F F F H F F F H H H H F F F H H H F H H H F F F F H H H F F F H F F F F H H H F F F F H H F H H H H F F F H H H H F F H F F F F F H H F F F F F H F H H H H H F F H H H H H F H F F F F F F H F F F F F F F H H H H H H 383 384 384 384 384 384 384 384 384 384 384 384 384 384 Year Rank 60 6, 27, 57 11, 36 41 71 25, 46, 76 30, 55 22, 68, 52 14, 44, 65 19, 74 49 3 8, 33, 63 17, 38 Table 8.9. The repartition of lunar months according to their full or hollow character, with respect to the Callipic cycle of 76 years, in the case of intercalary years. CHAPTER 9 THE LUMINOUS INCEPTION CANON Importance Like the Sifen li ddd, the Jingchu li ddd (Luminous Inception canon) concerns several dynasties. First, it was officially promulgated during the Three Kingdoms period, under the Wei dynasty, from 237 to 265; second it was adopted successively by three parallel dynasties: the Jin (one of the Six dynasties), from 265 to 420; the Liu Song, from 420 to 444 and lastly, the Northern Wei (or Toba Wei), from 398 to 451.1 Fundamental Parameters Number of years t(x) elapsed between the two winter solstices of the Superior Epoch and a given year x, primary and secondary solar and lunar constants: t(x) = 4045 + (x − 236) ( ) a 673, 150 455 d = = 365 + b 1843 1813 ( ) a 402 11 d = 15 + + 24b 1843 1843 × 12 ( ) c 134, 630 2419 d = = 29 + d 4559 4559 ( ) c 1744 1 d = 7+ + 4d 4559 4559 × 2 α = 19 β = 7 γ = 235 1 Chen (solar years) (9.1) (solar year) (9.2) (solar period) (9.3) (lunar month) (9.4) (lunar phases) (9.5) (Metonic constants). (9.6) Zungui 1984, note 4, p. 1400; Lin Jin-Chyuan 2008, p. 47. © Springer-Verlag Berlin Heidelberg 2016 J.-C. Martzloff, Astronomy and Calendars – The Other Chinese Mathematics, DOI 10.1007/978-3-662-49718-0_9 261 262 THE LUMINOUS INCEPTION CANON Apart from its Metonic cycle of 19 years, the Jingchu li also admits two other supra-annual periods, respectively composed of 1843 and 11,058 years (19 × 97 and 1843 × 6, respectively). They will not be taken into account in the following because the logical structure of the calculations is more apparent without them and allows us to obtain exactly the same results. Moreover, the results of the following calculations will be compared with the content of two extant calendar manuscripts for the years 450 and 451 (see p. 267 below). The Calendars of the Years 450 and 451 As usual, the determination of the time parameter attached to the year 450 starts from its support year, the year 449, so that t(449) = 4258 solar years. Then, the expressions 4.2 f. and the arithmetical instructions listed on p. 159 above lead to the following calculations: q1 = bin(673150 × 4258, 1843) = < 21; 0397 >, m = ⌊(235 × 4258/19⌋ = 52, 664 lunar months, e = ⌊(235 × 4258) mod 19⌋ = 14, n11 = bin(134630 × 52664, 4559) = < 59; 2079 > . The first result means that the calculated winter solstice of the year 397 449, q1 (449), falls on a day #22 ((2, 10) or yiyou), 1843 days after its instant of midnight; the second means that the interval between the Superior Epoch and the winter solstice of the year 449 contains 52,664 lunar months; the third indicates that the age of the moon at the instant of the same solstice (the epact) is equal to 14, or 14 19 lunar months, more explicitly. Lastly, the fourth result shows that n11 (449) takes places a day #60 ((10, 12) or guihai), 2079 4559 days after the instant of midnight, or shortly before noon. The solar breaths and the new moons posterior to q1 and n11 are then obtained by adding as many times as necessary the lengths of a solar period and of a lunar month to q1 and n11 , respectively, and by reducing the integer parts of the results modulo 60. Insofar as the Jingchu li only rely on mean elements, this simple mode of calculation remains valid even for the year 451, by calculating a sufficient number of qi and ni in the same way: THE YEARS 450 AND 451 263 ( ) 397 402 11 qi = 21 + + 15 + + (i − 1) i = 1, 2, . . . ; 1843 1843 1843 × 12 (9.7) 2079 134, 630 ni = 59 + +( )(i − 1) i = 1, 2, . . . (9.8) 4559 4559 i 1 2 3 4 5 6 7 8 9 10 qi (60/1843/12) 21;0397,00 36;0799,11 51;1202,10 6;1605,09 22;0165,08 37;0568,07 52;0971,06 7;1374,05 22;1777,04 38;0337,03 i 11 12 13 14 15 16 17 18 19 20 qi (60/1843/12) 53;0740,02 8;1143,01 23;1546,00 39;0105,11 54;0508,10 9;0911,09 24;1314,08 39;1717,07 55;0277,06 10;0680,05 i 21 22 23 24 1 2 3 4 qi (60/1843/12) 25;1083,04 40;1486,03 56;0046,02 11;0449,01 26;0852,00 41;1254,11 56;1657,10 12;0217,09 Table 9.1. The values of all solar breaths qi necessary for the calendar of the year 450 (end of the year 449 – complete year 450). The two tables 9.1 and 9.2 are limited to the year 450 and our notation of their results respect the formats 60/1843/12 and 60/4559, in order to indicate that they are composed of sexagenary binomials, numbered from 0 to 59, and fractions having 1843 and 1843 × 12 (Table 9.1) or 4559 (Table 9.2) as denominators, respectively.2 Next, the epact e(450) = 14, being superior to 19−7 = 12, the quasicriterion 4.1, p. 159 above, strongly suggests that the year 450 is intercalary. Then, the existence and the rank of its intercalary month follow 2 Y. Nishizawa, 2005–2006, vol. 1, p. 60, also provides similar results but with decimal approximations instead of exact results. 264 THE LUMINOUS INCEPTION CANON i 1 2 3 4 5 ni (60/4559) 59;2079 28;4498 58;2358 28;0218 57;2637 i 6 7 8 9 10 ni (60/4559) 27;0497 56;2916 26;0776 55;3195 25;1055 i 11 12 13 14 15 ni (60/4559) 54;3474 24;1334 53;3753 23;1613 52;4032 Table 9.2. Values, enumerated in natural order, of all new moons necessary for the construction of the calendar of the year 450 (end of the year 449 and year 450). from a systematic comparison between successive new moons and odd solar breaths until the double inequality of the criterion 3.1, p. 151, is fulfilled. We have: q2×8+1 = q17 = 24 + 1314 8 + ; 1843 1843 × 12 1055 ; 4559 3474 n11 = 54 + ; 4559 n10 = 25 + q2×9+1 = q19 = 55 + 277 6 + . 1843 1843 × 12 Therefore ⌊q17 ⌋ < ⌊n10 ⌋ and ⌊n11 ⌋ < ⌊q19 ⌋ and the month [n10 , n11 [ is intercalary. Lastly, the indices of the two new moons obtained in this way still have to be renumbered n11 , n12 , . . . instead of n1 , n2 , . . .. This being done, it appears that 7* is the corresponding intercalary month of the surface calendar. Less importantly but still usefully, the criterion 4.1, p. 159, allows us to readily determine the hollow or full character of all months of the year 450: if the numerator of the fractional part of a given lunar month is greater than 4559 − 2419 = 2140, then the month in question is full and hollow otherwise. For example, the month n1 = < 59; 2079 > is hollow because 2079 < 2140 and more generally, the double numbering THE YEARS 450 AND 451 265 (natural and calendrical order) and the type (full (F) or hollow (H) of all months from n11 (449) to n12 (450) correspond to: Year 449 Year 450 i 1 2 i* 11 12 Type H F 3 4 5 6 7 8 9 10 11 12 13 14 15 1 2 3 4 5 6 7 7* 8 9 10 11 12 F H F H F H F H F H F H F Table 9.3. Full and hollow months obtained from the calculation of the year 450. The number of days of the two last lunar months of the year 449 and of those of the year 450 being known, the sexagenary values of their new moons and solar breaths are easily ranked against each other. Moreover, most chronological tables of the Chinese calendar give, directly or indirectly, the Julian dates of the new moons in question. Hence the following table: Month 1 Full Day no. 1 #59 9 #7 25 #23 Binomial Solar Breath (9 , 11) (7 , 7) (3 , 11) 2 1 #29 Hollow 10 #38 25 #53 (9 , (8 , (3 , 3 Full (8 , 10) (8 , 8) (3 , 11) 1 #58 11 #8 26 #23 4 1 #28 Hollow 12 #39 27 #54 (8 , (9 , (4 , 5) 2) 5) 4) 3) 6) Date 29/1/450 q4 q5 28/2/450 q6 q7 29/3/450 q8 q9 28/4/450 q10 q11 266 THE LUMINOUS INCEPTION CANON Month 5 Full Day no. 1 #57 13 #9 28 #24 Binomial Solar Breath (7 , 9) (9 , 9) (4 , 12) 6 1 #27 Hollow 14 #40 29 #55 (7 , (10 , (5 , 7 Full (6 , 8) (10 , 10) (5 , 1) 1 #56 15 #10 30 #25 7* 1 #26 Hollow 15 #40 (6 , (10 , 3) 4) 7) 2) 4) 8 Full 1 #55 2 #56 17 #11 (5 , 7) (6 , 8) (1 , 11) 9 Hollow 1 #25 2 #26 17 #41 (5 , (6 , (1 , 10 Full 1 #54 4 #57 19 #12 (4 , 6) (7 , 9) (2 , 12) 11 Hollow 1 #24 4 #27 19 #42 (4 , 12) (7 , 3) (2 , 6) 12 Full 1 #53 5 #57 21 #13 (3 , (7 , (3 , 1) 2) 5) 5) 9) 1) Date 27/5/450 q12 q13 26/6/450 q14 q15 25/7/450 q16 q17 24/8/450 q18 22/9/450 q19 q20 22/10/450 q21 q22 20/11/450 q23 q24 20/12/450 q1 q2 18/1/451 q3 q4 Table 9.4. The lunisolar structure of the year 450. THE YEARS 450 AND 451 267 The Manuscript Calendars of the Years 450 and 451 In 1934, two draft manuscript calendars of the years AD 450 and 451 were privately purchased in Dunhuang. Since then, their originals never surfaced again but they were copied and several critical editions, not very different from each other, were published subsequently.3 All are easily available and, in particular, an almost complete but rough facsimile reproduction of the original is easily available.4 Of course, a direct access to the originals would also be important but, to my knowledge, if they are still extant, their present location is unknown. The first version of the following analysis was based on two research articles by Deng Wenkuan5 (analysis of surface calendars) and the above Jingchu li procedures; the present one also takes into account the innovative approach of the Taiwanese historian Lin Jin-Chyuan which both depends on a double analysis of the relevant surface and deep calendrical structures, contrary to most previous publications where the underlying mathematics is left aside.6 As often happens in the case of Dunhuang manuscripts, these two calendars were inscribed on the blank side of a sheet of paper, containing a text related to an early Chinese text from the 5th century BC on its other side, the Guoyu dd (The Discourses of the States). They bear the following titles: Taiping zhen jun shiyi [shi’er] nian li [ri] ddd dddd [ ddd ] dddd that is, literally, “Calendar of the 11th year [450] (respectively 12th year [451]) of the Era of the True Lord of the Great Peace”.7 These two years are the last ones of the interval of validity of the Jingchu li, an astronomical canon adopted by the Toba Wei dynasty from 398 to 451, as already noted. The manuscripts pro3 Lin Jin-Chyuan 2008 provides all known details in this respect. COL-astron, vol. 1, p. 275–276. This reproduction has been handed down to Deng Wenkuan by the Japanese scholar Ikeda On ddd. 5 Deng Wenkuan 1996, p. 101–110 and Deng Wenkuan 2002c. 6 Lin Jin-Chyuan 2008, op. cit. 7 The name of the reign-period (440–451) of the Emperor Taiwu (423–452). The expression zhenjun ‘True Lord’ is taken from Zhuangzi dd and represents a Taoist title of deference given to true men, immortals and divinities. In its turn, the ‘Great Peace’ refers to the last of the three historical ages described in the Gongyang zhuan d dd, commentary of the Chunqiu dd (Spring and Autumn Annals (or Annals of Lu)) by Gongyang Gao ddd, a work going back to the Warring States period (475–221 BC). 4 See 268 THE LUMINOUS INCEPTION CANON vide no explicit mention of their mode of calculation but, given that the totality of their content agrees with the Jingchu procedures, including eclipse forecasts, it is highly probable that they have not been obtained in any other way, if we exclude the possibility of a mere copy from some former original version of these calendars.8 In particular, as Lin Jin-Chyuan has shown,9 the calendar of the year 451 mentions two partial lunar eclipses whose dates wholly conform to those obtained from the Jingchu li procedures.10 Unfortunately, however, this last point does not prove completely that these manuscripts are authentic nor whether they were really composed in 450–451 or later, inasmuch as the Jingchu techniques have been made public in two dynastic histories, published after the fall of the Wei dynasty.11 In this respect, a scientific analysis of the paper they are composed of (physical properties and chemical analysis, nature of their fibers, notably) and a comparison with authentic paper documents from this period – for instance the paper of the fragmentary calendar discovered in the Turfan region and dated from 47812 – in order to obtain their datation and possible origin would be required. A Partial Translation of the Manuscript Calendar of the Year 450 Guidelines The following proposal of partial translation of the calendar for the year 450 retains the totality of its fundamental lunisolar component and all the days noted in its manuscript. We follow here Deng Wenkuan and Lin Jin-Chyuan’s critical editions of the text13 and we essentially leave aside the numerous difficulties arising from the establishment of the text, particularly the conjectural restitution of certain Chinese characters which 8 The mathematical analysis of Y. Nishizawa 2005–2006, op. cit., vol. 1, p. 46, Lin Jin-Chyuan, 2008, op cit., and what precedes confirms this point. 9 Idem. 10 See below, p. 274 The unusual mention of lunar eclipses in this calendar is highly puzzling because, as a rule, these phenomena are absolutely never taken into account in Chinese calendars. 11 The Jingchu li is dealt with both in Songshu, ‘lüli zhong’, p. 233 f. and Jinshu, ‘lüli xia’, p. 536 f. 12 See Chen Hao 2007. 13 Deng Wenkuan 1996, p. 101–110, Lin Jin-Chyuan 2008. THE YEARS 450 AND 451 269 are either demonstrably erroneous, partially erased or even illegible.14 In order to respect the concision of the original, we have avoided word for word translations and paraphrases, and we have kept the pinyin appellations of the trunks and branches without attempting to coin English equivalents. If need be, however, we have supplied, between square brackets, various indications not explicitly present in the original and we have also numbered its successive columns. Lastly, we have also inserted some footnotes and further explanations after the translation itself. Translation [1] dddddddd[d] [d] ddddddddd [dd] Calendar of the 11th year [450] of the Era of the True Lord of the Great Peace [Taiping zhenjun] ; Taisui [dd, Great Year (spirit)], associated with gengyin [#27]; Empress of the year [ dd i.e. Taiyin dd ] associated with zi [d]15 Great General Da jiangjun [ddd correlated with zi [d].16 [2] ddd ddddd ddddddd ddddd First month, full; day 1: renxu [#59]; [jianchu term:] shou [reception]; day 9: Beginning of Spring [q4 ], even breath of the first month; day 25: Rain Water [q5 ]. [3] ddd ddddd ddddddd ddddd ddd dd Second month: hollow; day 1: renchen [#29]; [jianchu term:] man [fullness]; day 10: Waking of Insects [q6 ], even breath of the second month; day 25: Spring Equinox [q7 ]; day 27: cult of soil god (she d). [4] ddd ddddd dddddddd ddddd Third month: full; day 1: xinyou [#58]; [jianchu term:] po (de14 The reader will nonetheless find everything he would need in this respect in Deng Wenkuan, 1996, ibid. and Y. Nishizawa 2005–2006, vol. 1, op. cit., p. 35 f., the more complete presently available study of the subject. 15 zi refers to, notably, to the north direction. The other branches are also associated with cardinal points determining all sorts of prescriptions. See B. Frank 1998. 16 This translation needs here three lines of text but the Chinese original uses only one column composed of only 19 characters. 270 THE LUMINOUS INCEPTION CANON struction); day 11: Pure Brightness [q8 ], even breath of the third month; day 26: Grain Rain [q9 ]. [5] ddd ddddd17 dddddddd ddddd Fourth month: hollow; day 1: xinmao [#28]; [jianchu term:] bi (closure), day 12: Beginning of Summer [q10 ], even breath of the fourth month; day 27: Grain Full [q11 ]. [6] ddd ddddd dddddddd ddddd Fifth month: full; day 1: gengshen [#57]; [jianchu term:] ping [Stability]; day 13: Bearded Grain [q12 ], even breath of the fifth month; day 28: Summer Solstice [q13 ]. [7] ddd ddddd dddddddd ddddd Sixth month: hollow; day 1: gengyin [# 27]; [jianchu term:] cheng (maturity); day 14: Slight Heat [q14 ], Even breath of the sixth month; day 29: Great Heat [q15 ]. [8] ddd ddddd dddddddd dddd Seventh month: full; day 1: jiwei [#56], [jianchu term:] jian (institution), day 15: Beginning of Autumn [q16 ], even breath of the seventh month; day 30: Limit of Heat [q17 ]. [9] ddd ddddd dddddddd Intercalary month [7*]: hollow; day 1: jichou [#26]; [jianchu term:] zhi (stable state); day 15: White Dew [q18 ], even breath of the eight month. [10] ddd dddddd dddd dddddddd Eight month: full; day 1: wuwu [#55]; [jianchu term:] shou (reception); day 1: cult of soil god (she); day 2: Autumn Equinox [q19 ]; day 17: Cold Dew [q20 ], even breath of the ninth month. [11] ddd ddddd dddd dddddddd Ninth month: hollow; day 1: wuzi [#25], [jianchu term:] man (fullness); day 2: Descent of Frost [q21 ]; day 17: Beginning of Winter [q22 ], even breath of the tenth month. 17 A jianchu dd term is expected here. Therefore, the erroneous character yong d of the original manuscript should be replaced by bi d. THE YEARS 450 AND 451 271 [12] ddd ddddd dddd ddddddddd Tenth month: full; day 1: dingsi [#54]; [jianchu term:] po (destruction); day 4: Slight Snow [q23 ]; day 19: Great Snow [q24 ], even breath of the eleventh month. [13] dddd ddddd18 dddd ddddddddd Eleventh month: hollow; day 1: dinghai [#24]; [jianchu term:] bi (Closure); day 4: Winter Solstice [q1 ], day 19: Slight Cold [q2 ], even breath of the twelfth month. [14] dddd ddddd dddd dddd ddddd ddd Twelfth month: full; day 1: bingchen [#53], [jianchu term:] ping (Balance); day 5: Great Cold [q3 ]; day 13: La festival; day 21: Beginning of Spring, even breath of the first month [q4 ]. Notes [1] The first column of the original contains two sorts of indications: first, the year of the calendar expressed in the chronological system of dynastic eras and second, indications concerning its calendrical annual spirits, shen d. Overall, calendrical spirits are occult powers, limited to the calendar and governing various calendrical functions, mainly prescriptions about daily activities, determined by correspondences with the denary, duodecimal and sexagenary cycles.19 Depending on their scope, they are referred to as nianshen dd, yueshen dd, or rishen dd, (yearly, monthly or daily spirits, respectively). The first such spirit, Taisui dd, is the master of all calendrical spirits. It governs the whole lunar year,20 and frequently determines when constructions and earthworks are to be undertaken favorably. Historically, this important calendrical spirit stems from the association of the planet Jupiter with the calendar derived from the sidereal revolution of the latter, approximately equal to twelve years and thus considered as 18 See footnote 17 above. A. Arrault 2003, p. 106–108. 20 Xieji bianfang shu, j. 3, p. 146 (notice p. 397 below); A. Forke 1907/1962*, vol. 2, p. 402 f.; L. Vandermeersch 1980, p. 345; M. Kalinowski 2003, p. 243. 19 See 272 THE LUMINOUS INCEPTION CANON a sort of Great Year.21 Moreover, the fact that 12 × 5 = 60 explains its connection with the sexagenary numbering of years.22 In practice, given that, as indicated in chronological tables, the year 124, for instance, is associated with the first binomial of the sexagenary cycle, jiazi ((1, 1) or #1), years having a rank of the form 124 + 60k, k = 1, 2, . . . are necessarily jiazi years. In the case of our manuscript, the year 450 is associated with the binomial (7, 3), or gengyin, because 450 − 124 = 5 × 60 + 26. Its sexagesimal rank is thus equal to #27 (26 + 1). Similar considerations related to the duodecimal cycle also apply in the case of the two other calendrical spirits mentioned above, Taiyin d d (The Great Yin)23 and Da jiangjun ddd (The Great General).24 In the following table, they are associated with particular elements of the duodecimal cycle, according to the branch of the year (first line), that is, according to the second element of its binomial:25 Year Branch Taiyin Da Jiangjun d d zi chou d d xu hai d d you you d d d d yin mao chen si d d d d zi chou yin mao d d d d zi zi zi mao d d d wu wei shen d d d chen si wu d d d mao mao wu d d you xu d d wei shen d d wu wu d hai d you d you Table 9.5. Correspondence between two calendrical spirits and the branch of a year. Since the year 450 is a gengyin year, dd [#27], its branch is yin d, (first line of the table). Therefore, Taiyin is associated with the branch zi d and the same table also shows that the Da jiangjun is associated with the same branch. 21 M. Kalinowski 2003, op. cit. note 62, p. 86 above. 23 Xieji bianfang shu, j. 3, p. 153 (notice p. 397). 24 Xieji bianfang shu, op. cit., p. 148. See also the substantial developments propounded in B. Frank 1998, (op. cit.) aiming at the study of Japan but often applicable to the Chinese case. 25 From Deng Wenkuan 2002a, p. 73 and 74. 22 See THE YEARS 450 AND 451 273 [2]–[14] From the above translation, we know that the calendar of the year 450 systematically contains the following elements, written in the same order from one month to the next: • the name of the month; • its type (full or hollow); • the sexagenary binomial of its first day; • the jianchu term associated with the same day; • the numbers of the two days containing the two solar breaths generally coupled with each month, save when an intercalary month exists. The nomenclature of the names of its months, its solar breaths and the sexagenary numbering of its days agree perfectly with the general principles of the lunisolar component of the Chinese calendar, described in Chapter 2 above (possible types of lunar months, lunisolar coupling definition of the intercalary month). Therefore, the intercalary month 7* is preceded and followed, as it should be, by two odd solar breaths, namely the Limit of Heat, q17 , and the Autumn Equinox, q19 , respectively located at the end of the preceding month and at the beginning of the following (days 30 and 2, respectively). Therefore, this particular month only contains a single solar breath of even order, White Dew, q18 , as it should. The jianchu pseudo-cycle with reduplications used in the present calendar also wholly agree with the rules explained on p. 94 f. above. In a different order of ideas, the dates of the cult of soil god, she, d, the Winter Sacrifice, la d, and the Beginning of the ploughing ceremony, Shi geng dd, (not included in our preceding table of the year 450, limited to lunisolar elements) call for a minimum of attention because they have not always been determined in the same way. During the Toba Wei dynasty, the cult of the soil god, she d, was a movable feast taking place on the fifth wu d day located either after q4 or q16 ,26 in other words, either after the Beginning of Spring or the 26 From Li Yongkuang and Wang Xi 1995, p. 186. 274 THE LUMINOUS INCEPTION CANON Beginning of Autumn. From this rule, ‘the wu day’ designates any day whose binomial has wu d as first term, or the fifth trunk. In this manuscript, q4 occurs on the Chinese date 9/I, a gengwu dd day27 (#7 or (7,7)). The first binomial posterior to (7, 7) having wu d as first term is thus (5, 3) or wuyin dd (#15), the second d d (#25 or (5,1)) and then #35, #45, up to dd #55. Therefore, given that our preceding table of the year 450 mentions that q7 occurs on day no. #53 of its second month, the feast in question necessarily happens two days later (Chinese date: 27/II). Similarly, the second she sacrifice occurs on 1/VIII and, as can be readily checked, these two dates, 27/II and 1/VIII, are the same as those of the manuscript. The Winter sacrifice, la d, of the year 450 should be placed the fourth day after the winter solstice whose branch is chen d, the fifth one.28 Consequently, its Chinese date is 13/XII. Lastly, the Beginning of the ploughing ceremony, Shi geng dd, occurs on day #12 (yihai, (2, 12)) of the first month (14/I). The Two Lunar Eclipses of the Year 451 The mathematical treatment of eclipses according to the Jingchu li29 procedures depends both on an eclipse cycle,30 composed of E = 790,110 4559 days, or 5.87 months, and the angular distances of the moon from a node 27 This wu d, pronounced wǔ (third tone) in modern Chinese, should not be mistaken with the wu d of the rule, also pronounced wu, but with a fourth tone (wù). Moreover, the first is a branch and the second a trunk. 28 A. Arrault 2003, p. 121. 29 Songshu, “lüli zhong”, j. 12, p. 241–242. 30 Eclipse cycles were maintained in China at least until the Jiyuan li ddd (1106– 1127), but, from the Jingchu li on, more sophisticated techniques, involving true elements, were also used in order to predict not only their day of occurrence but also their hour and other finer time subdivisions. Still, the results of these more precise calculations are not included in the present calendar and, consequently, they are omitted here (for the details of such calculations, see Che Yixiong 1984, p. 101–106; Wang Yingwei 1998, p. 25–45; Lin Jin-Chyuan 2008, p. 55–60. For the principles underlying eclipse calculations in Chinese history, see also Qu Anjing 2008, p. 394. On the history of eclipses in general, see, notably, J.M. Steele 2000, A.P. Cohen and R.R. Newton 1981– 1983, Zhuang Weifeng 2009. For a modern presentation of the subject, see also COLL 2005, p. 175 f., notably. THE YEARS 450 AND 451 275 at the moments of syzygies.31 As shown in Fig. 9.1 below, the origin of this new cycle is determined in an unusual way because it starts from a point O1 preceding the Superior Epoch, O2 . Then, the existence of a solar or lunar eclipse on the day of a new or full moon, as the case may be, is determined by reducing the length of the interval between O1 and the moon phase in question modulo E and by comparing the result so obtained, e, with fixed quantities called ‘eclipse limits’ shixian d d, not expressed in terms of degrees of longitude but by using a unit of 1 d time u such that u = 4559 . O1 O2 N M P n11 (x − 1) n12 (x − 1) n1 (x) Q R n2 (x) Figure 9.1. Schema for the calculation of eclipses according to the Jingchu li. In the case of the year 451, for example, let Q be a point representing the instant of the full moon of the first month of the year 451. Then, the length l of the interval O1 Q is first determined by noting that l = O1 O2 + O2 N + NM + MP + PQ (Fig. 9.1). Omitting the unit of time u, we have: 1. O1 O2 = 412, 919 (primary constant of the Jingchu li); 2. O2 N = ⌊ 4259×235 × 134, 630⌋ because, on the one hand, the sup19 port year of the year 451 is the year 450 so that t(450) = 4259 and on the other hand, the number of months to be taken into account is obtained from the formula 4.2, p. 159, applied to the case of a 19/7 Metonic equivalence, and with a month composed of 134, 630 units u; 3. NM + MP + PQ = 2 × 134, 630 + 134,630 = 336, 575 (two and a 2 half lunar months). Therefore: e = l mod E = 626, 644 and then, the repeated addition of one month, 134,630, reduced modulo E, leads to the successive values 31 This explanation is often reproduced. However, interpretations in terms of ancient Chinese conceptions concerning, for instance, the ‘nine roads of the moon’ and other such notions would perhaps shed another light on the subject. 276 THE LUMINOUS INCEPTION CANON of e. Writing these results with indices equal to final month numbers, we have: e(1) = 626, 644, e(2) = 761, 274 . . . e(8) = 778, 944 . . . Then, the Jingchu li procedure relies on the following criterion: if e(i) is smaller than 67,315 or greater than 722,79532 then a solar eclipse at new moon or a lunar eclipse at full moon is predicted. In the present case, e(2) and e(8) are both greater than 761, 274. Therefore, a lunar eclipse occurs at the full moons of the second and eight lunar months of the year 451.33 Once the calendar of the year 451 has been fully established, it follows that these eclipses occur on the following Chinese dates: 16/II and 16/VIII (Julian dates 2/4/451 and 26/9/451). The Jingchu li procedures do not stop at this stage but also tackle the question of the intervals of occurrence of these eclipses to within one Chinese double-hour by means of a technique based, inter alia, on true lunar elements. According to Lin Jin-Chyuan’s calculations based on the original Jingchu li procedure34 (omitted), the first happens between 7h and 9h AM and the second between 5h and 7h AM. From an astronomical perspective, the calculated dates of these two eclipses are correct but not their times: the Chinese astronomer Zhang Peiyu35 remarks that the first lasted from 11h 53 to 12h 53 and the second from 1h 10 to 4h 13. Moreover, only the second eclipse was visible in China and this conclusion is corroborated by a record preserved in a Chinese history.36 However, the date of the eclipse indicated in the calendar of the year 451 is 15/VIII, not 16/VIII, perhaps because phenomena occurring during the night were then attributed to the previous day. 32 67,315 and 722,795 correspond to an angular distance from a lunar node equal to = 15.55 [Chinese] degrees du d and 722795 4559 = 156.26 du, respectively (a du has as many degrees as the number of days in a sidereal year). 33 The Jingchu li procedure involves trivial calculations based on supra-annual cycles, equivalent to the present formulation. For more details, see Chen Meidong 1995 , p. 347 f.; Lin Jin-Chyuan 2008; LIFA, p. 394–399. 34 Lin Jin-Chyuan 2008, ibid., p. 55–60. 35 From Deng Wenkuan 2002c, p. 197–199. 36 Songshu, j. 13, ‘lüli 2’, p. 310. 67315 4559 CHAPTER 10 THE MANIFEST ENLIGHTENMENT CANON Importance The Xuanming li (Manifest Enlightenment canon) was not only officially adopted in China during 71 years, from 822 to 892 (Tang dynasty), but also in Japan, for 823 years, from 862 to 16841 and in Korea, during the IXth and Xth centuries.2 Unlike other Tang astronomical canons, its interest is exceptional because the content of an almanac for the year 877, issued during its period of official validity, which is also one of the most ancient printed document, fully agrees with its procedures (see p. 296 below). Fundamental Parameters Number of years t(x) between the two winter solstices of the Superior Epoch and a given year x, primary and secondary solar and lunar constants:3 t(x) = 7, 070, 138 + (x − 821) a 3, 068, 055 2055 d = = (365 + ) b 8400 8400 (solar years) (10.1) (solar year) (10.2) 1 M. Sugimoto and D.L. Swain 1978, p. 72–73 and p. 254; M. Uchida 1975, p. 511 f.; J.M. Steele 1998a, 1998b and 2000, p. 218–220. 2 The years of official validity of the Xuanming li in Korea are not precisely known. See Lee Eun-Hee, 1997, p. 339. 3 Xin Tangshu, j. 30A, ‘li 6a’, p. 745 and Koryǒ sa/Gaoli shi (Korean History), j. 50, p. 81 f. (notice p. 399 below). © Springer-Verlag Berlin Heidelberg 2016 J.-C. Martzloff, Astronomy and Calendars – The Other Chinese Mathematics, DOI 10.1007/978-3-662-49718-0_10 277 278 THE MANIFEST ENLIGHTENMENT CANON ( ) a 1835 5 d = 15 + + 24b 8400 8400 × 8 ( ) a 611 7 d = 5+ + 72b 8400 8400 × 8 ( ) c 248, 057 4457 d msyn = = = 29 + b 8400 8400 ( ) 4658 19 d man = 27 + + 8400 8400 × 100 (solar period) (10.3) (seasonal period) (10.4) (synodic month) (10.5) (anomalistic month) (10.6) α = 248, 057, β = 91, 371, γ = 3, 068, 055 (Metonic constants) (10.7) α − β = 156, 686 (intercalary limit). (10.8) The four first solar and lunar constants are similar to those of more ancient astronomical canons, based on mean elements. However, the anomalistic month concerns calculations with true elements. The Xuanming li calls it lizhou dd, and represents it in the following unusual form of a sum of two fractions: lizhou [=] ri 1 yu 4659 miao 19 (d dd 27 d 4659 d 19), where ri = day and where the values of the units yu and miao are obvious from 10.6 above.4 Moreover, as already noted on p. 165 above, the three Metonic constants α , β and γ and the intercalary limit are quite unexpected here because the Xuanming li is not Metonic. Nevertheless, when restricted to its mean elements, it becomes Metonic and these various constants can still be used in order to determine the approximate rank of its intercalary months, when they exist. 4 This lunar month was first integrated explicitly into the list of constants of Chinese astronomical canons in the Linde li (665–728) where it was already represented in the same way. The same peculiarity was maintained in later canons. See Appendix G. THE YEAR 877 279 The Calculation of the Calendar of the Year 877 Former Studies The almanac of the year 877 has given rise to a relatively important number of research reports5 which have mainly been carried out independently.6 Yet, their authors have not often provided the details of the calculations.7 Hence an emphasis on this essential aspect of the Xuanming li in what follows. The Mean Elements of the Year 877 As usual, the calculation of the mean elements of the year 877 is based on the previous determination of the winter solstice q1 (876) of the year 876 (the support year of the year 877); the mean epact e(876), defined as equal to the length of the interval [n11 , q1 [ and the mean new moon n11 (876). 5 M. Uchida 1975; Yan Dunjie 1989b; Zhang Peiyu, Wang Guifen et al. 1992; Y. Okada , K. Itō et al. 1993, vol. 4, p. 197–203 (notice p. 375 below); Deng Wenkuan 1996, p. 198–231; Y. Nishizawa 2005–2006, vol. 1, p. 299–430. 6 Yan Dunjie 1989b provides terse explanations concerning simplified Xuanming li calculations and a list of values of new moons for the year 877. In addition, he also explains the meaning of a large number of other hemerological elements, such as the days of the planetary week or the nine color palaces. Zhang Peiyu, Wang Guifen et al. 1992, propound a brief and synthetic account of the general principles of the Xuanming li calculations and a table of the totality of all the new moons belonging to its years of validity (these results, however, do not wholly conform to Chinese chronology and the reason for this discrepancy remains unexplained). Lastly, Y. Okada, K. Itō et al. 1993, vol. 4, op. cit.) offer a quasi-complete reconstitution of the Japanese year 877 from the Xuanming li procedures. The result of their calculations, however, differs slightly from the Chinese calendar for the year 877 deduced from all chronological tables of the Chinese official calendar: the 6th and 7th months of their Japanese calendar are respectively full and hollow whereas they are both hollow in its Chinese counterpart. From this, should we conclude that these calendars have not been obtained only from calculations? Or else, that Chinese and Japanese procedures are not wholly identical? Or, alternatively, have political factors or calendrical taboos triggered a modification of the full or hollow character of the lunar months in question? Or is Y. Okada, K. Itō et al., op. cit.,’s unpublished interpretation of the Xuanming calculations different from all those available? These various questions remain open, but beyond the isolated case of the year 877, a comparison between Chinese and Japanese calendrical tables clearly shows that this case is not unique. See Chen Meidong 1997. 7 M. Uchida 1975, op. cit., is exceptional in this respect. 280 THE MANIFEST ENLIGHTENMENT CANON From the above expression 10.1, p. 277, t(876) = 7, 070, 193 solar years and from the above formulas 4.2 to 4.6, p. 159, we have: q1 (876) = bin(3068055 × 7070193, 8400) = 37;1815, e(876) = (3068055 × 7070193) mod 248057 = 224, 529, n11 (876) = bin(3068055 × 7070193 − 248057, 8400) = 10;4086. Then, the mean solar breaths qi , the seasonal indicators hi and the mean new moons ni , numbered in natural order, are obtained as follows: qi = 37; 1815 + 15;1835, 5(i − 1) i = 1, 2 . . . (10.9) hi = 37; 1815 + 5;661,7(i − 1) i = 1, 2 . . . (10.10) ni = 10; 4086 + 29;4457(i − 1) i = 1, 2 . . . (10.11) Hence the three following tables 10.1, 10.2 and 10.3, obtained after a reduction modulo 60 of these results.8 In addition, since the value of the mean epact obtained previously, 224,529, is greater than the intercalary limit, 156,686, the year 877 seems9 intercalary. However, the rank of this hypothetical intercalary month cannot be determined wholly confidently only from means elements: true elements should be used instead (see ‘True New Moons’, p. 289 f.) The True Elements of the Year 877 The Steps of the Calculations The determination of the true elements of the year 877 requires the calculation of the solar and lunar corrections ∆⊙ and ∆$ of the 15 mean new moons ni , determined as explained above and whose particular values are listed below (Table 10.3, p. 283). In what follows, the solar correction ∆⊙ is obtained from the values of ruqi(ni ) and by using Uchida’s method (p. 189 above). In its turn, the calculation of the lunar correction ∆$ is determined by the values of ruli(ni ) and by the lunar Table 10.7, p. 288 below. 8 At the present stage, it is impossible to determine up to where the calculations should be extended. 9 We write ‘seems’ and not ‘is’ because this putative intercalary month possibly belongs to the end of the year 876. THE YEAR 877 i 1 2 3 4 5 6 7 8 9 10 11 12 13 qi 60/8400/8 37;1815,0 52;3650,5 7;5486,2 22;7321,7 38;0757,4 53;2593,1 8;4428,6 23;6264,3 38;8100,0 54;1535,5 9;3371,2 24;5206,7 39;7042,4 qi 60/8400/8 37;1815,0 51;6050,5 6;2886,2 21;0721,7 35;7957,4 50;7993,1 6;0828,6 21;3264,3 36;6900 52;3335,5 8;0771,2 23;7606,7 39;7042,4 i 14 15 16 17 18 19 20 21 22 23 24 1 2 281 qi 60/8400/8 55;0478,1 10;2313,6 25;4149,3 40;5985,0 55;7820,5 11;1256,2 26;3091,7 41;4927,4 56;6763,1 12;0198,6 27;2034,3 42;3870,0 57;5705,5 qi 60/8400/8 55;6478,1 11;4913,6 27;2349,3 42;7185 58;2420,5 13;4856,2 28;6091,7 43;6127,4 58;4963,1 13;2798,6 27;8034,3 42;3870 56;8105,5 Table 10.1. Mean and true solar breaths (qi and qi respectively) for the calculation of the year 877. The ruqi From definition 5.1, p. 172 above, the values of ruqi(ni ) are obtained from the lengths li of all solar periods included in the year 877, by using the coefficients δi [−60, −50, −40, −30, −18, −6, 6, 18, 30, 40, 50, 60, 60, 50, 40, 30, 18, 6, −6, −18, −30, −40, −50, −60] as follows: li = (15 + 1835 5 100 × δi + )+ 8400 8400 × 8 8400 i = 1, 2 . . . Moreover, since the equality li = l25−i holds for 1 ≤ i ≤ 12, these values are repeated symmetrically (Table10.4). In their turn, the true solar breaths qi , i = 1, 2, . . . , 24 (Table 10.1) are obtained as indicated on p. 170 above. Once their values are ascertained, the ruqi calculations can be tackled and, as noted on p. 172 f., several techniques are available. 282 THE MANIFEST ENLIGHTENMENT CANON i 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 hi (60/8400/8) 37;1815,0 42;2426,7 47;3038,6 52;3650,5 57;4262,4 2;4874,3 7;5486,2 12;6098,1 17;6710,0 22;7321,7 27;7933,6 33;0145,5 38;0757,4 43;1369,3 48;1981,2 53;2593,1 58;3205,0 3;3816,7 8;4428,6 13;5040,5 18;5652,4 23;6264,3 28;6876,2 33;7488,1 38;8100,0 44;0311,7 49;0923,6 54;1535,5 i 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 hi (60/8400/8) 59;2147,4 4;2759,3 9;3371,2 14;3983,1 19;4595,0 24;5206,7 29;5818,6 34;6430,5 39;7042,4 44;7654,3 49;8266,2 55;0478,1 0;1090,0 5;1701,7 10;2313,6 15;2925,5 20;3537,4 25;4149,3 30;4761,2 35;5373,1 40;5985,0 45;6596,7 50;7208,6 55;7820,5 1;0032,4 6;0644,3 11;1256,2 16;1868,1 i 57 58 59 60 61 62 63 64 65 66 67 68 69 70 71 72 1 2 3 4 5 6 7 8 9 10 hi (60/8400/8) 21;2480,0 26;3091,7 31;3703,6 36;4315,5 41;4927,4 46;5539,3 51;6151,2 56;6763,1 1;7375,0 6;7986,7 12;0198,6 17;0810,5 22;1422,4 27;2034,3 32;2646,2 37;3258,1 42;3870,0 47;4481,7 52;5093,6 57;5705,5 2;6317,4 7;6929,3 12;7541,2 17;8153,1 23;0365,0 28;0976,7 Table 10.2. Seasonal indicators for the calendar of the year 877 (end of the year 876 and complete year 877). THE YEAR 877 i 1 2 3 4 5 ni (60/8400) 10;4086 40;0143 9;4600 39;0657 8;5114 i 6 7 8 9 10 ni (60/8400) 38;1171 7;5628 37;1685 6;6142 36;2199 283 i 11 12 13 14 15 ni (60/8400) 5;6656 35;2713 4;7170 34;3227 3;7684 Table 10.3. Values, numbered in natural order, of all mean new moons required for the calendar of the year 877 (end of the year 876 and year 877). With the first technique, the initial new moon of the support year of the year 877 is set apart. Let n1 be this new moon (denoted n11 later). Then, the formula 5.5, p. 173 above, shows that two cases, determined by the value of the epact, are possible a priori. Now, since e = 224, 529/8400, or 26;6129 d , and l24 = 14; 4235, 5, this epact is greater than l24 . Therefore: ruqi(n1 ) = l23 + l24 − e = 14;5235,5d + 14;4235,5d − 26;6129d = 2;3342,2d . Then, the successive values of the other ruqi follow directly from their definition by mutually ordering mean new moons (Table 10.3) and true solar breaths (Table 10.1). We have: ruqi(n2 ) ruqi(n3 ) ruqi(n4 ) = = = n2 − q1 n3 − q3 n4 − q5 = = = 40;0143 − 37; 1815 9;4600 − 6;2886,2 39;0657 − 35;7957,4 = = = 2;6728 3;1713,6 3;1099,4 ruqi(n5 ) ruqi(n6 ) = = n5 − q7 n6 − q9 = = 8;5114 − 6;0828,6 38;1171 − 36;6900 = = 2;4285,2 1;2671 ruqi(n7 ) = n7 − q10 = 60 + 7;5628 − 52;3335,5 = 15;2292,3 284 THE MANIFEST ENLIGHTENMENT CANON ruqi(n8 ) ruqi(n9 ) ruqi(n10 ) = = = n8 − q12 n9 − q14 n10 − q16 = = = 37;1685 − 23;7606,7 60 + 6;6142 − 55;6478,1 36;2199 − 27;2349,3 = = = 13;2478,1 10;8063,7 8;8249, 5 ruqi(n11 ) ruqi(n12 ) = = n11 − q18 n12 − q20 = = 60 + 5;6656 − 58;2420,5 35;2713 − 28;6091,7 = = 7;4235,3 6;5021,7 ruqi(n13 ) ruqi(n14 ) ruqi(n15 ) = = = n13 − q22 n14 − q1 n15 − q2 = = = 60 + 4;7170 − 58;4963,1 34;3227 − 27;8034,3 60 + 3;7684 − 56;8105,5 = = = 6;2206,7 6;3592,5 6;7978,3 i 1 2 3 4 5 6 7 8 li (days/8400/8) 14;4235,5 14;5235,5 14;6235,5 14;7235,5 15;0035,5 15;1235,5 15;2435,5 15;3635,5 i 9 10 11 12 13 14 15 16 li (days/8400/8) 15;4835,5 15;5835,5 15;6835,5 15;7835,5 15;7835,5 15;6835,5 15;5835,5 15;4835,5 i 17 18 19 20 21 22 23 24 li (days/8400/8) 15;3635,5 15;2435,5 15;1235,5 15;0035,5 14;7235,5 14;6235,5 14;5235,5 14;4235,5 Table 10.4. The lengths li of the true solar periods of the Xuanming li. With the second technique (p. 174 above), we have q1 = 0, q2 = l1 and since 0 < 2;6728 < 14; 4235,5 the double inequality q1 < m − e < q2 is verified. Therefore, ruqi(n2 ) = m − e = 2;6728. Next, l1 + l2 = 29;1071,2 ; 2m − e = 32;2785, l1 + l2 + l3 = 43;7306,7. Consequently, l1 + l2 = 29;1071,2 < 2m − e < l1 + l2 + l3 < q4 . Lastly, ruqi(n3 ) = 2m − e − (l1 + l2 ) = 32;2785 − 29;1071,2 = 3;1713,6. The other results are obtained in the same way. THE YEAR 877 285 The Solar Correction As already explained on p. 186 above, the solar inequality ∆⊙ concerning a given mean new moon n both depends on the value of ruqi(n) = t;y and on the true solar breath qi precisely implied in the definition of the ruqi. With the method of M. Uchida, each solar breath is attributed an index i from 1 to 24 and this index is associated in its turn with three coefficients ai , bi and ci (Table 10.5) which are sufficient in order to do the calculations once t and y are known (see p. 189 above). i 1 2 3 4 5 6 7 8 9 10 11 12 ai 0 449 823 1122 1346 1481 1526 1481 1346 1122 823 449 bi 33.4511 28.0315 22.6998 17.8923 11.7966 5.7986 −0.2433 −6.1254 −12.2048 −16.9060 −21.5362 −26.0498 ci −0.3695 −0.3606 −0.3519 −0.4068 −0.3998 −0.3998 −0.3779 −0.3634 −0.2987 −0.2919 −0.2854 −0.2854 i 13 14 15 16 17 18 19 20 21 22 23 24 ai 0 −449 −823 −1122 −1346 −1481 −1526 −1481 −1346 −1122 −823 −449 bi −30.3119 −25.8126 −21.2454 −17.0296 −11.4744 −5.6429 0.1432 6.1488 12.6336 17.8043 23.0590 28.4618 ci 0.2854 0.2919 0.2987 0.3634 0.3779 0.3779 0.3998 0.4068 0.3519 0.3606 0.3695 0.3695 Table 10.5. Uchida’s 72 coefficients ai , bi and ci . For instance, the ruqi attached to the mean new moon n3 is equal to 3;1713,6. Therefore, t = 3 and this value determines the index i = 6 3, related to q3 . Moreover, y = 1713 8400 + 8400×8 , always with the same notations. Hence, as indicated in Table 10.5, the three coefficients of M. Uchida: a3 = 823, b3 = 22.6998 and c3 = −0.3519. Then, as explained on page 189 above, this data determines two polynomials of the first and second degree, S3 and T3 , and the sought correction is obtained as follows: 286 THE MANIFEST ENLIGHTENMENT CANON ∆⊙ (n3 ) = S3 (3)( 1713 6 + ) + T3 (3) ∼ = 894 or 8400 8400 × 8 894 . 8400 The other results (Table 10.8, p. 289 below) are obtained similarly. The ruli The general expression 5.13, p. 179 above, indicates that the successive values of ruli(ni ), concerning the 15 new moons required for the calculation of the calendar of the year 877 – the first two belonging to the year 876 and the others to the year 877 – can be obtained from: ( ) 3, 068, 055 224, 529 248, 057 × 7, 070, 193 − + (i − 1) 8400 8400 8400 ( ) 4658 19 mod 27 + + i = 1, 2, . . . 15. 8400 8400 × 100 In the particular case of the Xuanming li, however, these calculations are not sufficient because this celebrated canon divides the anomalistic month into two equal intervals (‘phase I’ and ‘phase II’) and evaluates the successive values of the ruli as follows: { calculated ruli calculated ruli ≤ man /2, final ruli = calculated ruli − man /2 calculated ruli > man /2. Moreover, when a subtraction is necessary, the result is always exa b 1 pressed in the form 8400 + 8400×100 + 8400×100×2 . The last digit of the numbers recorded below (Table 10.6, next page, column ‘phase II’) is thus equal to ‘1’ in all cases. The lunar correction The values of the ruli determined by each mean new moon being ascertained (Table 10.6, next page), the corresponding lunar corrections directly depend on the coefficients of a table (see Table 10.7, p. 288) or, more exactly, a quasi-table, as explained in the section beginning on p. 191 above). THE YEAR 877 i 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 287 ruli(ni ) (60/8400/100/2) Final Temporary Phase I Phase II 24;4071,69 10;5942,59,1 26;3870,50 12;5741,40,1 0;7411,12 0;7411,12 2;7209,93 2;7209,93 4;7008,74 4;7008,74 6;6807,55 6;6807,55 8;6606,36 8;6606,36 10;6405,17 10;6405,17 12;6203,98 12;6203,98 14;6002,79 0;7873,69,1 16;5801,60 2;7672,50,1 18;5600,41 4;7471,31,1 20;5399,22 6;7270,12,1 22;5198,03 8;7068,93,1 24;4996,84 10;6867,74,1 Table 10.6. Values of the ruli for the calculation of the calendar of the year 877. More precisely, the correction modifying the value of the mean new moon n such that ruli(n) = x;y is obtained by using the constant coefficients αi and λi of the line x + 1 of the table in question, appearing either in its column ‘phase I’ or ‘phase II’, as the case may be. For example, since ruli(n7 ) = 8;6606,36 belongs to ‘phase I’, the corresponding coefficients must be taken from the 9th line of the table because its first digit is ‘8’. In addition, α9 = 3136 and λ9 = −224, and, as explained on p. 194 above, the sought lunar correction is calculated as follows: 36 6606 + 100 ∼ ∆$ (n7 ) = 3136 − 224 × = 2960 8400 or 2960 . 8400 (10.12) 288 THE MANIFEST ENLIGHTENMENT CANON i 1 2 3 4 5 6 αi 0 830 1556 2162 2633 2970 { 7 3172 8 9 10 11 12 13 14 Phase I λi 3218 3136 2912 2546 2037 1394 646 830 726 606 471 337 202 53 7465 −7 935 −82 −224 −366 −509 −643 −748 −646 αi Phase II λi 0 −830 −1556 −2154 −2618 −2947 { −3142 −3188 −3106 −2881 −2515 −2014 −1386 −646 −830 −726 −598 −464 −329 −195 −53 6529 7 1871 82 225 366 501 628 740 646 Table 10.7. The coefficients required for the calculation of the lunar inequality ∆$ of the Xuanming li. But this modus operandi is not valid in all cases because, unfortunately, lunar quasi-tables are never completely regular. Some particular cases should thus also be taken into account. For example, the correction concerning the new moon n6 whose ruli is equal to 6;6807,55, is obtained from the three coefficients α7 = 3172, m1 = 53 and l1 = 7465 listed in line 7 of the same quasi-table (phase I) because, in the particular case of the index ‘7’ (see p. 194 above), the calculations take the following form: 6807 ∼ = 3220 or 3220 8400 . 7565 In all cases, results such as 3220 must be divided by 8400 in order to obtain a number of days. ∆$ (n6 ) = 3172 + 53 × THE YEAR 877 i 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 289 ni ∆⊙ ∆$ ni 10;4086 40;0143 9;4600 39;0657 8;5114 38;1171 7;5628 37;1685 6;6142 36;2199 5;6656 35;2713 4;7170 34;3227 3;7684 −0;0767 0;0092 0;0894 0;1381 0;1524 0;1329 0;0831 0;0079 −0;0715 −0;1261 −0;1514 −0;1432 −0;1004 −0;0259 0;0636 −0;2160 −0;0880 0;0732 0;2076 0;2914 0;3220 0;2960 0;2158 0;0842 −0;0777 −0;2102 −0;2911 −0;3141 −0;2917 −0;2105 10;1159 39;7755 9;6226 39;4114 9;1152 38;5720 8;1019 37;3922 6;6269 36;0161 5;3040 34;6770 4;3025 34;0051 3;6215 Table 10.8. The quantities intervening in the calculation of the true new moons of the year 877 and the values of the latter. True New Moons The values of ∆⊙ and ∆$ relating to the 15 mean new moons ni i = 1, . . . , 15 being ascertained, the corresponding true new moons are obtained by calculating: ni + ∆⊙ + ∆$ = ni . (10.13) Next, the true new moons so obtained (Table 10.8) are easily compared with the mean solar breaths (Table 10.1) in order to determine the rank of the intercalary month. We have: q7 = 8; 4428,6 < n5 = 9;0772 < n6 = 38;5720 < q9 = 38;8100. The inequalities being strict, the month [n5 , n6 [, belonging to the deep structure of the calendar, contains no odd solar breath. The corresponding month of its surface structure is thus intercalary (see 3.1, p. 151 290 THE MANIFEST ENLIGHTENMENT CANON above) and the final numbering of the lunar months follows from the replacement of the temporary numbering T of the lunar months by the final F calendrical numbering: T F 1 2 3 4 11 12 1 2 5 6 7 8 9 10 11 12 13 14 15 2* 3 4 5 6 7 8 9 10 11 12 Lastly, the mutual comparison between the ni and the ni (table 10.8, preceding page) shows that the taking into account of true new moons instead of mean ones, imply a modification of the dates of the beginnings of the lunar months in four cases: those first numbered 2, 5, 7 and 12, correspond to the final ones 12 (876), 2* (877), 4 (877) and 9 (877), respectively. Among these, 12 (876) and 9 (877), occur one day earlier than the corresponding mean new moons and the two others one day later. The Calendar of the Year 877 The preceding results allow us to draw up the following long Table 10.9, beginning on next page , recording the main lunar and solar elements of the calendar of the year 877 (new moons, solar breaths and seasonal indicators). In addition, the following non-astronomical elements have also been inserted at their proper dates – the particular rules they have been obtained from are explained below, p. 296 f.: (i) the initial day of the period of governance of the Earth, tuwang dd; (ii) the two first Sundays,10 mi d; (iii) the three canicular days, (initial, median and final) ( f u1 , f u2 and f u3 ), san fu dd. 10 From the fifth lunar month, the dates of a large number of Sundays are erroneous (Deng Wenkuan 1996, p. 206 f.). THE CALENDAR OF THE YEAR 877 Month 1 full 2 full 2 full 2* hollow 3 full Day 1 4 9 14 19 25 30 Bin. #10 (10,10) #13 (3, 1) #18 (8, 6) #23 (3,11) #28 (8, 4) #34 (4,10) #39 (9, 3) 1 5 10 15 #40 #44 #49 #54 (10, (4, (9, (4, 4) 8) 1) 6) 20 22 25 30 1 5 10 15 20 25 27 #59 #1 #4 #9 #10 #14 #19 #24 #29 #34 #36 (9,11) (1, 1) (4, 4) (9, 9) (10,10) (4, 2) (9, 7) (4,12) (9, 5) (4,10) (6,12) 1 7 12 17 22 27 #39 #45 #50 #55 #60 #5 (9, 3) (5, 9) (10, 2) (5, 7) (10,12) (5, 5) qi hi q5 h8 h9 h10 h11 h12 h13 q6 h14 h15 h16 q4 Various 291 Date 18/1/877 17/2/877 Sunday 3/3/877 Sunday 10/3/877 h17 q7 q8 h18 h19 18/3/877 19/3/877 h20 h21 h22 h23 h24 tuwang q9 q10 h25 h26 h27 h28 h29 h30 17/4/877 292 THE MANIFEST ENLIGHTENMENT CANON Month Day 4 hollow 1 2 7 12 17 22 27 #9 #10 #15 #20 #25 #30 #35 (9, 9) (10,10) (5, 3) (10, 8) (5, 1) (10, 6) (5,11) 1 3 8 13 19 24 29 30 #38 #40 #45 #50 #56 #1 #6 #7 (8, (10, (5, (10, (6, (1, (6, (7, 1 4 9 10 14 19 20 24 29 1 5 10 15 20 26 #8 #11 #16 #17 #21 #26 #27 #31 #36 #37 #41 #46 #51 #56 #2 5 full 6 hollow 7 hollow Bin. 2) 4) 9) 2) 8) 1) 6) 7) (8, 8) (1,11) (6, 4) (7, 5) (1, 9) (6, 2) (7, 3) (1, 7) (6,12) (7, 1) (1, 5) (6,10) (1, 3) (6, 8) (2, 2) qi hi q11 h31 h32 h33 h34 h35 h36 Various Date 17/5/877 q12 q13 q14 15/6/877 17/6/877 h37 h38 h39 h40 h41 h42 f u1 tuwang q15 15/7/877 h43 h44 f u2 q16 h45 h46 f u3 h47 h48 13/8/877 q17 q18 h49 h50 h51 h52 h53 THE CALENDAR OF THE YEAR 877 Month 8 full 9 hollow 10 full 10 full 11 hollow Day 1 2 7 12 17 22 27 1 2 4 7 12 17 22 27 1 3 9 #6 #7 #12 #17 #22 #27 #32 #36 #37 #39 #42 #47 #52 #57 #2 #5 #7 #13 Bin. (6, 6) (7, 7) (2,12) (7, 5) (2,10) (7, 3) (2, 8) (6,12) (7, 1) (9, 3) (2, 6) (7,11) (2, 4) (7, 9) (2, 2) (5, 5) (7, 7) (3, 1) 14 19 24 29 1 4 9 14 19 24 29 #18 #23 #28 #33 #35 #38 #43 #48 #53 #58 #3 (8, 6) (3,11) (8, 4) (3, 9) (5,11) (8, 2) (3, 7) (8,12) (3, 5) (8,10) (3, 3) qi q19 q20 hi Various h54 h55 h56 h57 h58 h59 293 Date 11/9/877 17/9/877 11/10/877 h60 tuwang q21 q22 h61 h62 h63 h64 h65 9/11/877 q23 q24 h66 h67 h68 h69 h70 h71 9/12/877 q1 q2 h72 h1 h2 h3 h4 h5 17/12/877 294 Month 12 full THE MANIFEST ENLIGHTENMENT CANON Day 1 5 7 10 15 21 26 30 Bin. #4 #8 #10 #13 #18 #24 #29 #33 qi (4, 4) (8, 8) (10,10) (3, 1) (8, 6) (4,12) (9, 5) (3, 9) hi Various Date 7/1/878 h6 tuwang q3 q4 h7 h8 h9 h10 5/2/878 Table 10.9. The year 877. The Period of Governance of the Earth By definition, the period of governance of the Earth (tuwang dd) is such that the Earth governs the four following intervals: [ j4 , q4 [, [ j10 , q10 [, [ j16 , q16 [, [ j22 , q22 [, where the ji are determined by the fact that the common length of these four intervals is equal to Y /20, with Y = one solar year. Although these intervals are wholly determined by the values of the qi , i = 4, 10, 16, 22 and this constant, the Chinese procedure is slightly different and depends on the values of the preceding qi , i = 3, 9, 15, 21 by taking avail of the following inequalities: j4 < q3 < q4 , j10 < q9 < q10 , j16 < q15 < q16 , j22 < q21 < q22 . THE CALENDAR OF THE YEAR 877 295 The common length of the intervals [ ji , qi [, i = 3, 9, 15, 21, is thus equal to: Y /20 −Y /24 = Y /120d . The origins of the intervals in question are thus obtained by subtracting this new quantity from q3 , q9 , q15 and q21 , respectively. However, the first tuwang so obtained do not belong to the year 877 because its first solar breath is q4 . Consequently, q3 and j4 both belong to the preceding year. The first subtraction to be taken into account is thus the second one and the last solar breath located at the end of the year 877 corresponds to q3 . We have: q9 −Y /120 = 38;8100,2 − 3;0367,1 = 35;7732,7 q15 −Y /120 = 10;2313,6 − 3;0367,1 = 7;1946,5 q21 −Y /120 = 41;4927,4 − 3;0367,1 = 38;4560,3 q3 −Y /120 = 12;7541,2 − 3;367,1 = 9;7174,1. The ranks of the sexagenary binomials of the tuwang days are thus #36, #8, #39 and #10, respectively, and with the help of the preceding table, their Chinese dates are: 27/II*, 2/VI, 4/IX and 7/XII, respectively. More generally, insofar as the definition of the tuwang has remained unchanged during our period of study, similar calculations are also valid in the case of other astronomical canons. The Sundays mi d of the Planetary Week Although the Xuanming li does not voice a single word about the days of the planetary week, the consultation of any concordance table of Chinese chronology shows that all days marked mi d in the calculated calendar of the year 877 occur on days chronologically associated with Sundays in the Julian calendar of the same year because they have the 296 THE MANIFEST ENLIGHTENMENT CANON same Julian day numbers.11 In the above Table 10.9, p. 294, only two Sundays have been recorded but it would be easy to restore them all from this remark. The Three Canicular Days san fu dd By definition, the canicular days are the 3rd , 4th and 5th days posterior to the Summer Solstice, q13 , having geng d as first element of their sexagenary binomial, the 7th element of the denary cycle. Therefore, given that the above calculations show that q13 occurs on the Chinese date 3/V, their sought Chinese dates are: 30/V, 10/VI and 20/VI (they respectively correspond to gengwu, #7, gengchen, #17, and gengyin #27). A Printed Almanac of the Year 877 General Presentation At the beginning of the twentieth century, an impressive source of more than 40, 000 genuine manuscripts and a few printed documents (paper rolls, codex-like booklets, isolated fragments of paper, etc.) from the Tang period (618–907), the Five dynasties (907-960) and the beginning of the Song dynasty, was discovered in cave no. 17 of the Mogao troglodyte Buddhist cave complex, some twelve miles south-east the town of Dunhuang, located along the so-called Silk Road and now in Gansu province.12 While these documents are mostly devoted to religious topics, divinatory techniques and economic life, they also include fifty calendars or almanacs13 having numerous dates at variance with official Chinese calendrical chronology: typically, their non-conforming dates fall ahead or lag behind one or two days. Moreover, all of them are manuscripts save three which have unquestionably been printed.14 That is the case, in 11 On Sundays, see also p. 90 f. above. P. Hopkirk 1981 (history of this remarkable discovery); S. Whitfield 1999 and 2004 (life along the Silk Road and iconography). 13 See Huang Yi-long 1992b; Deng Wenkuan 1996; A. Arrault and J.-C. Martzloff 2003; Y. Nishizawa, 2005–2006, vol. 1-3. 14 See A. Arrault 2003, p. 86. 12 See A PRINTED ALMANAC OF THE YEAR 877 297 particular, of the almanac S-P6 rº concerning the year 877 and presently held by the British Library.15 As a physical object, S-P6 rº is a rectangular sheet of paper printed on its recto and wound around a wooden stick from the end of the text; the almanac itself is a woodblock print, obtained from ink on paper, composed of two equal rectangular plates (height: 29 cm, width: 115.5 cm).16 Like most almanacs, its layout is particularly complex and its content exceptionally rich. Apart from its orderly succession of days and months, it also displays a wealth of drawings and diagrams concerning all sorts of mantic practices such as, for example, a talisman for the stabilization of residences with indications about auspicious orientations determined by the five Chinese patronymic groups; a table providing the palaces associated with the year of birth of men and women for years comprised between 784 and 877; drawings of the twelve cyclical animals and of the five demons of illnesses with the indication of the corresponding calendrical spirits; the five-drum method for retrieving lost property, and so on.17 However, S-P6 rº has unfortunately not been fully preserved and, in particular, its title and the year it refers to are missing. Nevertheless, an analysis of specific aspects of its cycles and enumerating systems shows beyond any doubt that it concerns the year 877.18 More precisely, the loss is not enormous for its 13 months are still displayed more or less fully in its unaltered part: the totality of their 29 or 30 days has been kept intact in eleven cases whereas its two first months are incomplete. The former only contain the days 17 to 30 and the latter the days 1 to 4 and 10 to 30. S-P6 rº also contains the 24 solar breaths, many seasonal indicators, the 3 canicular days, san fu dd; the days of the soil god cult, she d; the color palaces of the 3rd and 12th months, the nayin dd and 15 The initial ‘S’ in S-P6 rº recalls the name of its discoverer, the archaeologist and explorer of Central Asia, Sir Marc Aurel Stein (1862–1943). See P. Hopkirk 1981, ibid., p. 85 f. 16 S. Whitfield 2004, p. 302. 17 See A. Arrault 2003, ibid., p. 89; A. Fujieda, 1973, p. 395; S. Whitfield 1998, p. 14; Deng Wenkuan 2001; A. Arrault and J.-C. Martzloff 2003, p. 200–203. 18 Deng Wenkuan 1996, p. 198–231. 298 THE MANIFEST ENLIGHTENMENT CANON jianchu dd cycles associated with the successive days of the calendar. Yet, the first days of governance of the Earth, tuwang dd, are omitted. Whereas several reproductions of S-P6 rº have been published, their interest remains limited because their relatively poor quality renders them not always easily legible,19 even in the case of the more appealing color reproduction of the whole almanac published in a book about the Silk Road.20 Nevertheless, several specialized studies are available. First and foremost, Y. Nishizawa has published, in 2005–2006, a complete handwritten transcription of this almanac together with a series of unabridged reproductions of previous research articles on the subject as well as substantial critical notes concerning the totality of its content.21 In addition, a team of Chinese historians published, in 1993, two partial transcriptions of S-P6 rº limited to its lunisolar structure and a few other elements, notably those introduced in the present chapter.22 Lastly, in 1996, the Chinese historian of the calendar Deng Wenkuan has published a critical transcription and an overall study of S-P6 rº minutely explaining all sorts of difficult points.23 Some More Details A comparison between S-P6 rº and the above calculations reveals a full agreement, save in the case of some seasonal indicators. Apart from this mathematical aspect, the following figure – restricted to the first eighteen days of the ninth month of S-P6 rº and to a little less than the upper half of the corresponding printed text – is also intended to highlight, as far as possible, some prominent aspects of the layout and content of S-P6 rº. However, since the original is somewhat blurred, this portion of the almanac has been tentatively redrawn and the Chinese characters have been replaced by modern types for better legibility (Fig. 10.1 below). From this, it appears that its textual and graphical components are distributed among seven successive horizontal rows, of variable width, containing the following elements: 19 See the representative example of COL-astron, vol. 1, p. 359–361. Whitfield 2004, op. cit., p. 302–303. 21 Y. Nishizawa 2005–2006, op. cit., vol. 1, p. 299–430. 22 COL-astron, vol. 1, p. 363–377. 23 Deng Wenkuan 1996, op. cit. p. 198–231. 20 S. A PRINTED ALMANAC OF THE YEAR 877 299 1. First row: (a) name of the month and indication of its hollow character (jiuyue xiao ddd (ninth month, hollow)); (b) position of the Tiandao (the Celestial Way), a monthly calendrical spirit indicating the auspicious character of construction works located southwards (Tiandao nan xing d ddd (the Celestial Way makes headway southwards)); (c) mention of the sexagenary binomial gengxu, dd(7, 11), attached to the quinary enumeration of lunar months, by means of the expression yue jian dd (literally ‘the month is established upon’ [such and such sexagenary binomial]); (d) diagram of the color palace no. 9 (identified from the character zi d) of its central square; (e) three other less important monthly calendrical spirits, the Tiande dd, the Yuede dd and the Yuehe dd, deemed auspicious and having in common with the above Tiandao their relevance for construction works. Each time, the celestial trunk each of them is associated with is indicated. For example, the Tiande is associated with bing d;24 2. Second row: some Sundays, mi d, located above the numbers of the days they refer to and inscribed into small squares whose sides are slightly greater than the rectangular spaces attributed to each day; 3. Third row: the successive numbers of the days of the month in question, from 1 to 18; 4. Fourth row: their corresponding sexagesimal binomials; 24 This association is fixed once and for all for a given month. Most of these are provided in Y. Okada et al. 1993, vol. 4, p. 373 (notice p. 375 below) and have been obtained from hemerological treatises such as the Xieji bianfang shu (notice p. 397 below). 300 THE MANIFEST ENLIGHTENMENT CANON 5. Fifth row: the jianchu pseudo-cycle, with the character zhi d instead of zhi d25 (2 times); 6. Sixth row: three seasonal indicators and one solar breath: h60 , q21 , h62 , h63 (h61 is missing)26 mixed with hemerological elements and their interpretation in terms of auspicious activities (here, in particular, a marriage jiehun dd); 7. Seventh row: four cyclical animals: Monkey, Cock, Sheep, Horse (among a total of twelve). Another passage of S-P6 rº, not reproduced here, presents the various evils they are supposed to provoke.27 As noted by Deng Wenkuan,28 despite the quasi-correctness of everything contained in the present figure, the whole of S-P6 rº also contains a large number of various inaccuracies. Therefore, this almanac is probably not an official production.29 jianchu dd term is expected here. Therefore, the erroneous character zhi dof the original manuscript must necessarily be replaced by another character of this series, also spelt zhi: d. 26 More generally, numerous other seasonal indicators are missing in this manuscript and their dates are sometimes difficult to ascertain because the rectangular space where they are written overlaps three days of the calendar. 27 See A. Arrault and J.-C. Martzloff 2003, op. cit., p. 201. 28 Deng Wenkuan 1996, op. cit., p. 225–226. 29 On its probable author, see A. Arrault and J.-C. Martzloff 2003, ibid., p. 200. 25 A A PRINTED ALMANAC OF THE YEAR 877 301 Figure 10.1. The layout of a part of the ninth month of the printed almanac S-P6 rº from Dunhuang for the year 877. CHAPTER 11 THE GREAT UNIFICATION CANON Its importance The Great Unification canon (Datong li) is the astronomical canon officially adopted from 1384 to 1644, during the long-lasting Ming dynasty (1368–1644), hence its importance. Like previous canons, its imperfections began to show up well before its final rejection, but it was not seriously challenged despite an important but aborted project of reform1 and the parallel existence of the very important official Muslim canon, the Huihui li. Fundamental Parameters As explained above (chapter 6, p. 216), the Datong li calculations remain valid when performed with the epoch of the Shoushi li. Therefore, the number of solar years t(x) elapsed between the two winter solstices of 1280 and a given year x can always be obtained from the following formula: t(x) = |x − 1280| (solar years). (11.1) Likewise, its main primary and secondary lunisolar constants are identical with those of the Shoushi li but, as we know, secular variations are discarded: Y = 365.2425 d Y /24 = 15.2184375 d 1 See (solar year) (11.2) (solar period) (11.3) p. 28 above; W.J. Peterson 1986; Dai Nianzu 1986. © Springer-Verlag Berlin Heidelberg 2016 J.-C. Martzloff, Astronomy and Calendars – The Other Chinese Mathematics, DOI 10.1007/978-3-662-49718-0_11 303 304 THE GREAT UNIFICATION CANON Y /72 = 5.0728125 d (seasonal period) (11.4) msyn = 29.530593 d (synodic month) (11.5) (anomalistic month) (11.6) (intercalary limit). (11.7) man = 27.5546 d l = 18.655209 d Incidentally, the value of this intercalary limit fully agrees with the above quasi-criterion 4.2, p. 165: 13 × msyn −Y = 13 × 29.530593 − 365.2425 = 18.655209d . (11.8) The Calendar of the Year 1417 The Intercalary Character of the Year 1417 The calculations start from the support year of the year 1417. Therefore t(1416) = 1416 − 1280 = 136. Then, the mean epact of the year 1417 is obtained as follows:2 e = (136 × 365.2425 + 20.205) mod 29.530593 = 22.727574 d . This value being superior to the above intercalary limit, the quasicriterion 4.2, p. 165, suggests that the year 1417 is intercalary and later calculations confirm this result.3 The Datong li relying on true elements, its subsequent calculations are done in a two-steps process, respectively devoted to its mean and true elements. 2 Equation 6.3, p. 203 above. course, the taking into account of true elements can sometimes deliver an opposite result. For instance, in the case of the year 1365, we have t(1364) = 84 and the corresponding mean epact is equal to e = (365.2425 ∗ 84 + 20.205) mod 29.530593 = 18.288873. This result being smaller than the intercalary limit, 18.655209, the year 1365 should not be regarded as intercalary. Yet, all tables of the Chinese calendar indicate, on the contrary, that the month following the tenth month of the year 1365 is intercalary. In fact, the true value of this epact is equal to 18.7586 (see Zhang Peiyu 1994, p. 40) and is thus is slightly greater than the intercalary limit 18.655209. 3 Of THE YEAR 1417 305 The Mean Elements of the Year 1417 As usual, the determination of the relevant mean elements first depends on the winter solstice q1 (1416) = q1 (1416) and on the mean new moon n11 (1416), calculated as indicated on p. 203 above: q1 = q1 n11 = q1 − e = (136 × 365.2425 + 55.06) mod 60 = 48.04, (11.9) = 25.312426. (11.10) The subsequent mean solar breaths, seasonal indicators, mean new moons and mean moon phases, denoted here by p(x − 1, i), i = 1, 2 . . ., are then determined as follows: qi = 48.04 + 15.2184375(i − 1) 365.2425(i − 1) hi = 48.04 + 72 ni = 25.312426 + 29.530593(i − 1) 29.530593(i − 1) pi (136, i) = 25.312426 + 4 i = 1, 2 . . . (11.11) i = 1, 2 . . . (11.12) i = 1, 2 . . . (11.13) i = 1, 2 . . . (11.14) Then, these results are reduced modulo 60 (Tables 11.1, p. 306; 11.2, p. 307); 11.3, p. 310–311 below). The True Moons Phases of the Year 1417 The first mean moon phase of the year 1417 and the following ones are respectively equal to p(136, 1) = n11 (136) and p(136, i), i = 2, 3, ... Then, t⊙ (136, i), t$ (136, i), δ⊙ (t⊙ ), δ$ (t$ ), ν (t$ ) are determined sequentially. Lastly, the corrective factor, jiajian cha ddd, follows from the two formulas 6.13 and 6.18, p. 205 and p. 207 above. For instance, in the case of the true new moon n2 (1417) = p(136, 13) we have successively: p(136, 1) = 25.312426, p(136, 13) = 53.904205 = n2 (1417), 306 THE GREAT UNIFICATION CANON i qi Binomial 1 2 3 48.0400000 3.2584375 18.4768750 #49 #4 #19 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 1 2 3 4 33.6953125 48.9137500 4.1321875 19.3506250 34.5690625 49.7875000 5.0059375 20.2243750 35.4428125 50.6612500 5.8796875 21.0981250 36.3165625 51.5350000 6.7534375 21.9718750 37.1903125 52.4087500 7.6271875 22.8456250 38.0640625 53.2825000 8.5009375 23.7193750 38.9378125 #34 (4,10) #49 (9, 1) #5 (5, 5) #20 (10, 8) #35 (5,11) #50 (10, 2) #6 (6, 6) #21 (1, 9) #36 (6,12) #51 (1, 3) #6 (6, 6) #22 (2,10) #37 (7, 1) #52 (2, 4) #7 (7, 7) #22 (2,10) #38 (8, 2) #53 (3, 5) #8 (8, 8) #23 (3,11) #39 (9, 3) #54 (4, 6) #9 (9, 9) #24 (4,12) #39 (9, 3) (9, 1) (4, 4) (9, 7) Month Year 11 12 1416 1 2 3 4 5 5* 6 1417 7 8 9 10 11 12 Table 11.1. List of solar breaths (end of the year 1416 and complete year 1417). THE YEAR 1417 i 1 2 3 4 5 6 7 8 9 10 11 12 13 14 hi 48.040000 53.112813 58.185625 3.258438 8.331250 13.404063 18.476875 23.549688 28.622500 33.695313 38.768125 43.840938 48.913750 53.986563 307 Binomial Month #49 #54 #59 #4 #9 #14 #19 #24 #29 #34 #39 #44 #49 #54 (9, 1) (4, 6) (9, 11) (4, 4) (9, 9) (4, 2) (9, 7) (4, 12) (9, 5) (4, 10) (9, 3) (4, 8) (9, 1) (4, 6) Year 11 1416 12 1 1417 Table 11.2. Partial List of seasonal indicators (end of the year 1416–beginning of the year 1417). t⊙ (136, 13) = 65.864205, t$ (136, 13) = 15.811705, δ⊙ (t⊙ )(136, 13) = 2.225195, δ$ (t$ )(136, 13) = 2.533776, ν (t$ )(136, 13) = 1.005570, jiajian cha(136, 13) = 0.388074, n2 (1417) = 53.904205 + 0.388074 = 54.292279. Therefore, the true new moon n2 (1417) happens on a sexagenary day #55, one day later than the corresponding mean new moon (day #54). Likewise, similar calculations show that the true new moon n4 (1417) also happens one day later than the corresponding mean new moon, 308 THE GREAT UNIFICATION CANON n4 (1417). By contrast, the four following true new moons n11 (1416), n7 (1417), n9 (1417) and n11 (1417) all occur one day earlier than their corresponding mean new moons. Yet, n1 and n1 occur on the same day. Similar remarks can also be made for other moon phases. In their case, however, the above calculations are not sufficient because it is also necessary to take into account the variable duration of the night (see ‘Other Moon Phases’ below). Table 11.3, p. 311 below takes these peculiarities into account by adding an asterisk to true new moons (fourth column), when their values are different from those of their mean counterparts, and by indicating the nature of the shift concerning all moon phases in its last column. The Determination of the Intercalary Month Any concordance table of the Chinese calendar indicates that the month 5* of the year 1417 is intercalary and, from the viewpoint of our calculations, it is enough to check that the criterion 3.1, p. 151 above, also delivers the same result. Therefore, the two inequalities ⌊q13 ⌋ < ⌊n5 ∗⌋ and ⌊n6 ⌋ ≤ ⌊q15 ⌋ must hold. With our initial numbering of new moons and solar breaths,4 we have: ⌊q13 ⌋ < ⌊n8 ⌋ and ⌊n9 ⌋ ≤ ⌊q15 ⌋ and, since q13 = 50.661250, n8 = 52.322758, n9 = 21.629929, q15 = 21.098125. We obtain 50 < 52 and 21 ≤ 21. The month of the surface calendar – corresponding to [n8 , n9 [ (temporary notation) and to [n5 ∗, n6 [ (final notation) – is thus 5*, as expected. Other Moon Phases The Datong li also relies on the following criterion in order to determine the dates of moon phases other than new moons: Criterion 11.1 (Shifts of Moon Phases) When a moon phase other than a new moon happens before sunrise, it must fall on the previous day.5 The Datong li determines the duration of day and night from a table composed of 366 entries, one per day, in such a way that the solar year 4 In particular, These new moons respectively correspond to i = 29 and 33 (Table 11.3, below). 5 Mingshi, j. 35, ‘li 5’, p. 692. THE YEAR 1417 309 is divided into two symmetrical intervals having 183 days each and extending from one solstice to the next, the first one beginning from the winter solstice and the second from the summer solstice. Moreover, the initial days of each interval are called chu d (initial) and the subsequent days are numbered from 1 to 182, as though these initial days were both numbered zero (ordinal zero). With this mode of enumeration, all solar and lunar elements of interest for the calendar are attributed to one of these two intervals and ordered sequentially, within each of them, from their numerical values. For instance, the true full moon of the first month of 1417, p(136, 11) = 39.178198 falls on the day 51 of the interval [q1 (1416), q13 (1417)[ (Table 11.3 beginning on next page) and the table in question6 indicates that, on such a day, the end of night occurs 0.249918 d after the instant of midnight. Moreover, the Datong li7 hour system is such that the duration of twilight is equal to 0.025 d . Therefore, the sunrise of day 51 happens 0.249918 d + 0.025d = 0.274918 d after midnight. Lastly, p(316, 11) and the instant of sunrise of day 51 is such that 0.178198 < 0.274282 so that this full moon occurs before sunrise. From the above criterion this full moon should be written one day earlier in the calendar, on day 50 instead of 51 (sexagenary day #39 instead of #40). The dates of the other moon phases are determined in the same way but calculations are not always necessary. For instance, the lunar phase p(316, 27) = 36.927075 happens a long time after sunrise because its non-integer part is almost equal to one day. By contrast, the non-integer part of the lunar phase p(316, 35) = 36.083967 is almost equal to 0 and corresponds to an instant not far from midnight. In each case, the above criterion is easily applied. The following table lists all moon phases of the calendar for the year 1417, their final numbering, their true and mean values, day number, instant of sunrise and nature of their possible date-shifts, irrespective of their origins (consequence of true and mean elements having different integer parts) or application of the above criterion (±1 day or = (same date)). Lastly, the shifted new moons have been asterisked: 6 Mingshi 7 See , j. 34, ‘li 4’, p. 645. p. 213 above. 310 i 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 THE GREAT UNIFICATION CANON Lunar Phase NM (n11 ) FQ FM LQ NM (n12 ) FQ FM LQ NM (n1 ) FQ FM LQ NM (n2 ) FQ FM LQ NM (n3 ) FQ FM LQ NM (n4 ) FQ FM LQ NM (n5 ) FQ FM LQ NM (n5 *) FQ FM LQ NM (n6 ) p(136, i) 25.312426 32.695074 40.077723 47.460371 54.843019 2.225667 9.608316 16.990964 24.373612 31.756260 39.138909 46.521557 53.904205 1.286853 8.669502 16.052150 23.434798 30.817446 38.200095 45.582743 52.965391 0.348040 7.730688 15.113336 22.495984 29.878632 37.261281 44.643929 52.026577 59.409225 6.791874 14.174522 21.557170 p(136, i) 24.889964* 32.953640 40.290406 47.122217 54.674368 2.677023 9.759301 16.682319 24.507073 32.287444 39.178198 46.286496 54.292279* 1.782311 8.587453 15.934854 23.970229 31.181993 38.003324 45.612740 53.525532* 0.511675 7.449052 15.296425 22.972741 29.804074 36.927075* 44.917325 52.322758 59.097653 6.460226 14.465433 21.629929 Day Number 167 175 182 6 14 21 28 37 44 51 58 65 73 80 87 95 103 110 117 125 132 139 147 154 161 168 176 2 9 16 24 31 Sunrise 0.291328 0.292733 0.293166 0.291685 0.289839 0.287284 0.279181 0.274918 0.270414 0.260474 0.255843 0.251249 0.240820 0.236271 0.231763 0.222569 0.218713 0.214839 0.209788 0.208178 0.207107 0.207338 0.208441 0.210469 Shift −1 = −1 −1 = = = = = +1 −1 = +1 = = −1 = = −1 = +1 = = = = = −1 = = −1 = = = THE YEAR 1417 i 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 Lunar Phase FQ FM LQ NM (n7 ) FQ FM LQ NM (n8 ) FQ FM LQ NM (n9 ) FQ FM LQ NM (n10 ) FQ FM LQ NM (n11 ) FQ FM LQ NM (n12 ) FQ FM LQ p(136, i) 28.939818 36.322467 43.705115 51.087763 58.470411 5.853060 13.235708 20.618356 28.001004 35.383653 42.766301 50.148949 57.531597 4.914246 12.296894 19.679542 27.062190 34.444839 41.827487 49.210135 56.592783 3.975432 11.358080 18.740728 26.123376 33.506025 40.888673 311 p(136, i) Day Number 28.448303 36.083967 43.967996 50.936215* 57.896535 5.781781 13.436360 20.273327 27.451503 35.483463 42.873859 49.680021* 57.133700 5.158194 12.300940 19.178415 26.925680 34.785267 41.732222 48.790370* 56.775564 4.363164 11.198001 18.497155 26.559097 33.853184 40.679935 38 46 53 60 67 75 83 90 97 105 112 119 127 135 142 149 156 164 171 178 3 11 18 25 33 40 47 Sunrise 0.215915 0.219971 0.223949 0.232640 0.237810 0.243018 0.252153 0.257416 0.262060 0.271973 0.277024 0.281123 0.287852 0.290554 0.292152 0.293101 0.292252 0.290719 0.285061 0.281456 0.277392 Shift = −1 = −1 −1 = = = −1 = = −1 −1 = = = −1 = = −1 = +1 −1 = = = = Table 11.3. List of moon phases (end of the year 1416 and complete year 1417). The complete calculation of the year 1417 (omitted here) shows that the above calculations of all moon phases wholly conform with the content of the printed calendar of the year 1417 introduced below, p. 315. 312 THE GREAT UNIFICATION CANON Month First quarter 7 1 2 3 4 5 5* 6 7 8 9 10 11 12 8 × × × × × × × × × × Full Moon 9 15 16 17 × × × × × × × × × × × × × × × × Last quarter 22 23 24 × × × × × × × × × × × × × Table 11.4. The day-numbers of lunar phases other than new moons for the year 1417. More generally, this conclusion also applies to its other elements, notably its ‘empty days’8 (Xuri). Yet, the minute study of the first month of this calendar (Fig. 11.2, p. 323 and Table 11.5, p. 314) discloses the (fortuitous?) omission of the seasonal indicator h9 . The distribution of the dates of its moon phases can thus be studied confidently and, in particular, it appears that the day numbers of those other than new moons are always liable to take three different values: a first quarter, FQ, happens either on the 7, the 8 or the 9; a full moon, FM, on the 15, 16 or 17 and a last quarter, LQ, on the 22, 23 or 24. In addition, the modes of succession of these triplets of numbers from one month to the next appear wholly irregular and unpredictable without calculations (Table 11.4 above). More generally, the same conclusion applies to all Chinese astronomical canons based on true elements. 8 On the notion of ‘empty days’, see chapter 7, p. 219 above. THE YEAR 1417 313 Cycles and Pseudo-Cycles Once the luni-solar component of the year 1417 is known, it is easy to justify all dates associated with the fundamental cycles of the Chinese calendar from their definitions. In each case, they wholly conform with those of its extent printed calendar. The resulting schema, reproduced below (Table 11.5), is limited to the first month of 1417 but its extension is trivial. Justifications Nayin Column The association between the sexagenary binomials of the successive days of the first month of 1417 and the five phases of the nayin cycle readily follows from the procedure introduced on p. 96 above. Jianchu Column From the first two rules governing the elements of the reduplicated jianchu cycle (p. 94 above), the first term of the cycle, jian d, denoted 1 here, is associated with the first day posterior to the even solar breath q4 , whose sexagenary binomial is equal to #3, #15, #27, #39 or #51. In the present case, q4 , the Beginning of Spring, is numbered #34 and happens on the Chinese date 10/I (Table 11.5). Consequently, jian d must fall on the first day numbered #39 posterior to the day 10/I. Then, the backwards enumeration of the successive terms of the jianchu dd cycle from the day 10/I shows that the day containing q4 is associated with wei d, the eighth term of the jianchu cycle. Moreover, since q4 is an even solar breath, the second rule of the jianchu cycle requires its repetition on the preceding day. Then, the backwards enumeration becomes regular and it follows that the first day of the first month is associated with bi d, the twelfth term of the jianchu cycle. Lastly, the forward enumeration is also wholly regular. Xiu Column From any concordance table of the Chinese calendar, the Julian day number of the first day of the first lunar month of the year 1417 (Yongle 15) is equal to 2,238,635 and corresponds to Monday 18/1/1417 (Ju- 314 THE GREAT UNIFICATION CANON Days Binomials 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 #25 (5 , 1) #26 (6 , 2) #27 (7 , 3) #28 (8 , 4) #29 (9 , 5) #30 (10 , 6) #31 (1 , 7) #32 (2 , 8) #33 (3 , 9) #34 (4 , 10) #35 (5 , 11) #36 (6 , 12) #37 (7 , 1) #38 (8 , 2) #39 (9 , 3) #40 (10 , 4) #41 (1 , 5) #42 (2 , 6) #43 (3 , 7) #44 (4 , 8) #45 (5 , 9) #46 (6 , 10) #47 (7 , 11) #48 (8 , 12) #49 (9 , 1) #50 (10 , 2) #51 (1 , 3) #52 (2 , 4) #53 (3 , 5) #54 (4 , 6) Nayin Jianchu 28 Solar Seasonal Moon Phases Dates Cycle Cycle Xiu Breaths Indicators Mie/Xu Days (1417) 2 2 3 3 5 5 1 1 2 2 3 3 4 4 1 1 2 2 5 5 4 4 1 1 3 3 5 5 4 4 12 1 2 3 4 5 6 7 8 8 9 10 11 12 1 2 3 4 5 6 7 8 9 10 11 12 1 2 3 4 19 20 21 22 23 24 25 26 27 28 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 NM h9 FQ q4 h10 h11 FM h12 LQ Mie/Xu q5 h13 h14 18/1 19/1 20/1 21/1 22/1 23/1 24/1 25/1 26/1 27/1 28/1 29/1 30/1 31/1 1/2 2/2 3/2 4/2 5/2 6/2 7/2 8/2 9/2 10/2 11/2 12/2 13/2 14/2 15/2 16/2 Table 11.5. The first month of the year 1417. lian style),9 in complete agreement with the above table of the 28 Xiu (twenty-eight mansions) (p. 93), indicating that Bi d10 is one of the four different mansions associated with a Monday. 9 J. Meeus 1985, p. 23. to be confused with d, another mansion whose name is also spelt bi. 10 Not A CALENDAR FOR THE YEAR 1417 315 The Mie/Xu Day As justified in the next chapter (The Mie/Xu days of the Year Yongle 15 (1417), p. 330), and as already noted (Table 11.5, p. 314 above) a Mie/Xu day occurs on 24/I. The mean first month of 1417 is thus necessarily hollow and our previous calculations (Table 11.3, i = 9 and 13, p. 311 above) confirm this point. However, this is false in the surface calendar of 1417 because its calculations are based on true months: the first month of 1417 is full and contains a Mie/Xu day. In its case, the result 2, p. 232 above, is not valid. A Calendar for the Year 1417 Presentation The department of rare books of the Central National Library, Republic of China, Taipei,11 owns an authentic printed copy of an official calendar for the year 1417 (Yongle 15) and, to my knowledge, no more ancient copy of a wholly extant calendar from the Ming dynasty is extant.12 Remarkably, this is a sort of fine copy in the sense that it is legible without any difficulty (this is not so in many other cases).13 The calendar itself contains thirty pages altogether and the printed part of each is included in a rectangle whose dimensions are 13 × 23 cm (Fig. 11.2, p. 323). The whole has the appearance of a very thin Chinese traditional book, or more exactly of a booklet similar to those which have been currently printed from the Song dynasty. The title printed on the recto of its cover does not mention its sexagenary year (translation below, p. 318). Most conspicuously, however, it bears a rectangular stamp containing a warning, similar to those printed on modern banknotes, stipulating that only the appropriate state authority is allowed to reproduce the calendar and that counterfeiters incur severe punishments (full translation hereafter). By contrast, the verso of its cover contains no indication of any kind. Guoli zhongyang tushuguan shanben mulu ddddddddddd (Catalogue of Rare Books held at the Central National Library), Taipei, 1967, vol. 2, p. 500. 12 A tiny fragment of a calendar from Turfan, limited to a list of nine successive days, has recently been analyzed and dated from 1407, an earlier year of the Ming dynasty. See Deng Wenkuan 2002d. 13 I thank my colleague Alain Arrault for having made me aware of its existence. 11 See 316 THE GREAT UNIFICATION CANON As for the text of the booklet, it begins with four pages of preliminary data, including two pages devoted to the monthly structure of the calendar (full or hollow character of the totality of its thirteen months, sexagenary binomials of their new moons and even the precise indication of the calculated moments of occurrence of the solar breaths they are coupled with). Then, the two last preliminary pages are devoted to the ‘diagram of the directions of the annual spirits’ nianshen fangwei tu d dddd, a mantic diagram revealing auspicious or inauspicious daily activities for the year 1417 and the corresponding directional taboos.14 By contrast, its sequence of months has a perfectly regular and uniform layout in the sense that all months are presented in the same way: each of them always occupy two pages always subdivided in the same way and devoted to the same calendrical elements. The year 1417 being intercalary, it thus follows that its thirteen months are composed of twenty-six pages or, more precisely, 18 rectos and 18 versos. Monthly Structure Each month of the calendar uses at most four sizes of typefaces that we respectively call ‘huge’, ‘large’, ‘normal’ and ‘small’ (Fig. 11.2) and the first page of each month is divided into nine distinct zones, numbered 1, 2 . . . 9 by us, (diagram 11.1). The first four appear only on the right part of the first page of each month because they do not concern individual days but the whole month. By contrast, the five last zones are devoted to what concerns the successive days of the calendar, columnby-column. More precisely, each zone respectively contain the following data: zone 1. The name of the lunar month with the indication of its hollow or full character; zone 2. the sexagenary binomial of the month; zone 3. a list of various solar, astronomical or hemerological elements: a. the solar breaths of the month in question, with the indication of their day number and their exact beginnings, expressed by means of the Datong li horary system; 14 On this diagram, see Deng Wenkuan 2002b. A CALENDAR FOR THE YEAR 1417 317 5 6 2 1 7 8 3 9 4 Figure 11.1. The division of each month of the calendar of the year 1417 into nine zones. b. some details concerning its calendar spirits; c. the list of the main seasonal indicators of the month but without their dates; d. the instants of the sun’s entrance into the twelve stations of Jupiter, distributed along the Yellow Road, that is, along the ecliptic (richan huangdao ru shi’er ci shike dddddd dddd);15 zone 4. the diagram of the nine color palaces associated with the current lunar month; 15 These stations correspond to a division of the ecliptic into twelve zones whose limits are related to the twenty-eight mansions (xiu d) (see p. 92 above). They are first documented in the Tongtian li ddd (1199–1207) and in a few other canons from the Southern Song dynasty, in the Shoushi li and Datong li (see Lin Jin-Chyuan 1998, p. 38; N. Sivin 2009, p. 451–452; Yuanshi, j. 54, ‘li 3’, p. 1212–1213 and Mingshi, j. 35, ‘li 5’, p. 696–697, respectively). Insofar as the signs of the zodiac have been introduced in China well before the Song dynasty (see note 54, p. 81 above), the possibility of some connection with the Jupiter stations cannot be excluded, even though they belong to a well documented and very ancient Chinese tradition. 318 THE GREAT UNIFICATION CANON zone 5. isolated Chinese characters representing: a. the names of moon phases other than new moons; b. ‘full’ days Yingri dd and ‘empty’ days Xuri dd, (see p. 219 above); 1. cult of the soil god, she d; 2. the three days of scorching heat, chu fu dd, zhong fu d d, and mo fu dd (initial, median and final); zone 6. The successive numbers of the days of the current month, enumerated from 1 to 29 or 30, with ordinary Chinese numbers, their sexagenary binomials and the nayin cycle; zone 7. the jianchu pseudo-cycle; zone 8. the twenty-eight mansions associated by groups of four to the days of the planetary week (see p. 93 above); zone 9. a list of ‘elections’, that is daily activities to be followed or avoided according to their auspicious of inauspicious character, as well as various astronomical details, such as the occurrence of a solar breath, the length of day and night, the precise instant of sunrise. Translations The present section contains: a. a full translation of the cover of the calendar (title and stamp); b. a partial translation of the page of the calendar devoted to the thirteen first days of the first month of the year 1417 (Fig. 11.2). The Title of the Calendar The following title is printed on the left part of the cover and uses characters greater than those printed inside the stamp, on its right: “Datong Calendar for the fifteenth year of the Yongle era of the Great Ming.” dddddddddd A CALENDAR FOR THE YEAR 1417 319 The Cover Stamp “Calendar (liri dd) presented to the Emperor by the Bureau of Astronomy in view of its printing and diffusion in the whole Empire: By law, counterfeiters will be beheaded, those who will inform on such persons and facilitate their arrest will be awarded fifty ounces of silver (or taels) liang.16 Counterfeit calendars are firstly those devoid of the trustworthy seal of the Bureau of Astronomy and secondly private calendars”.17 ddddd18 ddddddddddddddddddd19 dd ddddddddddddddddddddddddd. The First Month of the Year 1417 zones 1 and 2. “First month: full, established upon20 renyin [#39].” zone 3. “Beginning of Spring lichun, even breath of the first month: 10th day, dingyou day [#34], second mark (ke) d, second hour shen d d [between 4h 29 and 4h 43 p.m., approximatively]. Since the Celestial Way21 moves southwards, traveling south is recommended and the mending of buildings located southwards is appropriate22 [...]. This month, the East Wind Dissipates the Cold (h10 ), Hibernating Creatures Begin to Stir (h11 ), Fish Ascend to the Ice (h12 ), Otter Sacrifices Fish (h13 ), Wild Geese Appears (h14 ), Plants Bud and Grow (h15 ). On the 28th day (yimao, #52), the sun enters Jüzi [Jupiter Station]. [. . . ]” zone 4. Color palace no. 8. zone 5. First quarter [9th day], full moon [15th day] (the remaining part of the first month contains the following elements: last quarter [23th day], (Xuri) [the 24]). 16 The usual translation of liang is tael – a word of Malay origin which made its way into European languages through the intermediary of Portuguese. One tael is approximately worth 36 grams of silver. 17 Such a warning was still present in the calendars printed not only during the long Ming dynasty but also much later, during the Qing dynasty. See R.J. Smith 1992, p. 7. 18 The slash indicates the end of a column of text in the original. 19 Illegible characters, reconstituted from R.J. Smith, ibid., p. 7. 20 This translation is literal and means that this month is associated with this binomial when lunar months are enumerated as explained on p. 86 above. 21 One of the very numerous monthly occult spirits associated with the calendar. 22 See Xieji bianfang shu, j. 5 , p. 198 (detailed reference and notice on p. 397 below). 320 THE GREAT UNIFICATION CANON zones 6, 7, 8, 9. “First day of the month: wuzi (#25); phase: Fire Huo d, [jianchu term:] bi d (Closure); mansion: Bi d (the Net). Suitable to perform one’s ablutions, to sew clothes, <the initial mark ke of the doublehour mao [from 5 a.m. to 7 a.m.] is fit for these activities>,23 to make transactions. Moving one’s house and acupuncture is forbidden.” “Second day jichou (#26), phase: Fire Huo d; [jianchu term:] jian d (Institution); mansion: Zi d (Beak). To perform sacrifices is in order. To go on a trip is forbidden.” “Third day: gengyin #27, phase: Wood Mu d; [jianchu term:] chu d; (removal); mansion: Shen (Triad). Suitable to call on government officials, meeting friends, to marry, [. . . ] < The double-hour chen [from 7 to 9 a.m.] is fit for these activities>, to make transactions, to heal oneself, to sweep one’s house, to bury the dead.” ‘Fourth day: xinmao (#28), phase: Wood Mu d; [jianchu term:] man d; (Fullness); mansion: Jing d; (the Well). Suitable for visiting government officials, meeting one’s friends, to marry, [. . . ] to mend one’s clothes <the double-hour mao [from 5 to 7 a.m.] is fit for these activities> to start commercial exchanges, to make transactions, to trade livestock. Moving one’s house is not appropriate.” “Fifth day: renchen #29; phase: Water Shui d; [jianchu term:] ping d; (balance); mansion: Gui d; (the Devils). Sunset: initial mark ke of the double-hour you [between 5 p.m. and 14mn later]”. “Sixth day: guisi (#30); phase: Water d; Shui; [jianchu term:] ding d, (Settlement); mansion: Liu d; (the Willow). [...] it is not suitable to travel.” “Seventh day: jiawu (#31); phase: Metal Jin d; [jianchu term:] zhi d, (stability), mansion: Xing d (Stars). Sunrise happens during the third mark of the double-hour mao [From 5 a.m to 5h 45mn at the latest]. [. . . ]’ ‘Eight day: [. . . ] Daylight: 44 ke; night: 56 ke. To make sacrifices is in order.’ 23 The text between < and > is a sort of footnote inserted into the main text and recognizable from the smaller size of its characters. A CALENDAR FOR THE YEAR 1417 321 Notes zones 1 and 2. The sexagenary binomial of the first lunar month, renyin, is fully determined by the numbering of the successive lunar months of the calendar according to a quinary cycle, as explained on p. 86 above. zone 3. Apart from the sun’s entrance into a Jupiter station not tackled here,24 the instant of the Beginning of Spring, q4 , given here agrees with our calculations, once converted into the Datong horary system (see p. 213 above). zone 4. The color palace reproduced here corresponds to the first lunar month of the year 1417, a year of the form 3k +1. The palace in question is thus the eight (see p. 89 above). zone 5. The dates of all moon phases wholly agree with our calculations. zones 6, 7, 8 and 9. Given that Chinese characters are written vertically, these four zones are divided into a number of columns equal to 29 or 30 according to the full or hollow character of the month. But most are not wholly filled up from top to bottom. The blank part of each could thus possibly have been used by owners of calendars in order to note various data of personal interest as if the calendar were an agenda. However that may be, the columns in question contains successively the following elements relating to the same day: its sexagenary binomial, its nayin and jianchu terms, the name of its mansion and consequently, albeit indirectly, the day of the planetary week associated with it. No less importantly, the day and night lengths attributed to certain days are noted in zone 9. For example, the calendar indicates that the durations of day and night on the eight day of the first month, 8/I, are respectively equal to 44 and 56 marks (ke d). Of course, this data follows from the rules of the Datong li. Indeed, according to the numbering conventions of the Datong li relating to the durations of day and night, the true new moon n1 happens on a day numbered ‘37’ of the solar year25 beginning with the winter solstice q1 (1416) (day 0). Therefore, the day 8/I corresponds to the entry ‘44’ of the same table. 24 The corresponding calculations are relatively complex and involve the determination of the position of the sun, inter alia. See LIFA, p. 653–659. 25 See Table 11.3, p. 309–311 above. 322 THE GREAT UNIFICATION CANON Then, given that the duration of night corresponding to this entry is equal to 0.25418126 and that one half night plus the duration of twilight equals 0.254181 + 0.025 = 0.279181 marks (ke), the length of the interval from midnight to sunrise is equal to 2 × 0.279181 = 0.558362 ∼ = 56 marks (ke), as noted in the calendar. Moreover, since the duration of the nychthemeron is equal to 100 marks, the corresponding day length is equal to 46 marks, as expected. 26 Mingshi , j. 34, ‘li 4’, p. 644. A CALENDAR FOR THE YEAR 1417 323 Figure 11.2. The thirteen first days of the first lunar month of the year 1417. Reproduced with the permission of the National Central Library, Taiwan, Republic of China (Guojia tushuguan ddddd), from microfilm no. 06283 of the calendar of the fifteenth year of the Yongle dynasty of the Ming dynasty, established by means of the Datong li (Da Ming Yongle shiwu nian Datong li dddddd dddd). CHAPTER 12 MO AND MIE DAYS Preliminary Remarks The procedures concerning the full determination of Mo d and Mie d days belonging to a given year x suppose the preliminary calculation of all its solar breaths and new moons. However, given that the present chapter only aims at highlighting what concerns more particularly these unusual days, the related calculations will be limited to what is absolutely necessary in this respect. Therefore, certain results, notably the lengthy calculations concerning the dates of the first new moon and of the last solar breath of the year x will be taken for granted without justification, save when they have already been determined in previous chapters or when articles on the subject are available. The Mo days of the year Jiading 11 (1218) In 1218 (Jiading 11) the Chinese official calendar was obtained from the procedures of the Kaixi li astronomical canon (1208–1251) where: t(x) = 7, 848, 183 + (x − 1206) ( ) a 6, 172, 608 d 4108 d = = 365 + b 16, 900 16, 900 ( ) a 3692 d = 15 + 24b 16, 900 (solar years), (12.1) (solar year), (12.2) (solar period). (12.3) Moreover, the related Moxian dd (Mo limit) is a secondary conr stant, obtained from ab by calculating r = ab − 360 = 88,608 16,900 , 1 − 24 = 13,208 16,900 and by retaining either this last fraction or its numerator, 13,208 (Test 1, p. 224 f. above). © Springer-Verlag Berlin Heidelberg 2016 J.-C. Martzloff, Astronomy and Calendars – The Other Chinese Mathematics, DOI 10.1007/978-3-662-49718-0_12 325 326 MO AND MIE DAYS With this data, the dates of the Mo days (mori dd) of the year 1218 are obtained as follows: 1. determination of the list of solar breaths required for the calculation of the year 1218 (q1 (1217), q2 (1217), . . .); 2. comparison between the numerator of the fractional part of each solar breath qi , i = 1, 2, . . . , 24 and the above Mo limit: if the former is greater than the latter, then the interval [qi , qi+1 [ contains a Mo point (see test 1, p. 224 above); 3. Lastly, if this test is positive, then the number of days between qi and the sought Mo day is obtained by calculating Ji (see p. 225 above). Now, according to the general rules of calendrical calculations (see p. 139 and 164 above), these mean solar breaths are obtained as follows: t(1217) = 7, 848, 183 + (1217 − 1206) = 7, 848, 194, q1 (1217) = bin(a × 1217, b) = < 24; 14, 352 >, 14, 352 3692 qi = 24 + + (i − 1)(15 + ), i = 1, 2 . . . 27. 16, 900 16, 900 In Table 12.1 below, the list of these qi has been extended up to the last solar breath of the year 1218, q3 (1218). Then, from Test 1, p. 224, those having a numerator greater than or equal to the Moxian d d, 13,208, have been asterisked (Table 12.1). Therefore, the interval [q1 (1217), q3 (1218)[ contains six Mo d days, namely q1 , q6 , q10 , q15 , q19 and q24 . However, the first of these, q1 , belongs to the year 1217. Therefore, the year 1218 has only five Mo d days. Next, let qi =< ai , fi >. Then the Mo d points in question are defi 6,172,608−360 fi termined by the successive values of Ji = ⌊ a−360 ⌋, a−360b ⌋ = ⌊ 88,608 i = 6, 10, 15, 19 and 24 (see p. 225 above). Hence Ji = 5, 13, 7, 15 and 9. Lastly, the sexagenary binomials of the sought Mo days are obtained from a simple addition, as indicated in Table 12.2, p. 327 below. THE MO DAYS OF THE YEAR JIADING 11 (1218) 327 Of course, these qi have been reduced modulo 60 but, given that the Mo d days are (i) necessarily posterior to the solar breaths from which they have been determined and (ii) always located between two consecutive solar breaths, it is not difficult to insert them into the final calendar. Nonetheless, the knowledge of the day of the month they belong to also supposes the calculation of the true new moons of the year 1218.1 q1 q2 q3 q4 q5 q6 q7 q8 q9 q10 q11 q12 q13 q14 24;14352* 40;01144 55;04836 10;08528 25;12220 40;15912* 56;02704 11;06396 26;10088 41;13780* 57;00572 12;04264 27;07956 42;11648 (5, 1) (1, 5) (6, 8) (1,11) (6, 2) (1, 5) (7, 9) (2,12) (7, 3) (2, 6) (8,10) (3, 1) (8, 4) (3, 7) q15 q16 q17 q18 q19 q20 q21 q22 q23 q24 q1 q2 q3 57;15340* 13;02132 28;05824 43;09516 58;13208* 14;00000 29;03692 44;07384 59;11076 14;14768* 30;01560 45;05252 0;08944 (8,10) (4, 2) (9, 5) (4, 8) (9,11) (5, 3) (10, 6) (5, 9) (10,12) (5, 3) (1, 7) (6,10) (1, 1) Table 12.1. The values of the solar breaths related to the calculation of the year 1218 (an asterisk signals those having a bearing on the determination of its Mo d days). q6 q10 q15 q19 q24 qi 40;15912 41;13780 57;15340 58;13208 14;14768 Ji 5 13 7 16 9 Calculations mod. 60 40 + 5 = 45 41 + 13 = 54 57 + 7 = 4 58 + 15 = 13 14 + 9 = 23 Mo Days #46 (6,10) #55 (5, 7) #5 (5, 5) #15 (4, 2) #24 (4,12) Table 12.2. The calculation of the Mo d days of the year 1218. 1 See Lin Jin-Chyuan 1998, p. 50 f. 328 MO AND MIE DAYS The Mie Days of the Second Kind of the Year Qianfu 4 (877) The astronomical canon in force in 877 is the Xuanming li (822–892) and its fundamental parameters have already been listed on p. 277–278 above. Moreover, its Mie d of the second kind depend on a ‘Mie limit’ (Miexian dd), whose value, 3943, is equal to numer(30 − m), where d (see Test 2, the length of the lunar month, m is such that m = bc = 248,057 8400 p. 225 above). From Method 4, p. 225 above, the calculations presuppose the determination of the mean new moons ni of the year 877 but their technique of calculation has already been explained earlier (p. 277 above). Moreover, they have been listed in a previous table (Table 10.3, p. 283 above). However, it is still necessary to renumber them in calendrical order. Once this is done, the Mie days (mieri dd) are detected from the mean new moons ni whose fractional parts are inferior or equal to the Miexian, 3943, that is n12 , n2 , n3 , n5 , n7 , n9 , n11 2 (Test 2, p. 225 above). Obviously, the first belongs to the year 876 and the six others to the year 30 fi 877. Then, the values of Ji = ⌊ 3943 ⌋, representing the numbers of days between each of these new moon and the sought Mie days (see p. 226 above), are readily obtained. For instance, n3 (877) =< 38; 1171 >. Hence f3 = 1171 and J3 = ⌊ 30×1171 3943 ⌋ = 8. Lastly, the sum of the integer part of n3 and J3 , 38 + 8, being equal to 46, the sexagenary binomial of the sought Mie is equal to #47 ((7, 11) or gengxu). The others results are obtained in the same way: n2 n3 n5 n7 n9 n11 ni 39.0657 38.1171 37.1685 36.2199 35.2713 34.3227 Ji 4 8 12 16 20 24 Calculations mod. 60 39 + 4 = 43 38 + 8 = 46 37 +12 = 49 36 +16 = 52 35 +20 = 55 34 +24 = 58 Mie Days #44 (4, 8) #47 (7,11) #50 (10, 2) #53 (3, 5) #56 (6, 8 ) #59 (9,11) Table 12.3. The calculation of the Mie days of the second kind (year 877). 2 This final numbering takes into account the fact that the month immediately following the second month of the year 877 is intercalary. THE MO DAYS OF 1417 329 Lastly, the determination of the full lunar component of the year 877 is obtained from the calculation techniques of the Xuanming li (Chapter 10 above). Hence, the dates of the sought Mie days: 5/II, 9/III, 13/V, 17/VII , 21/IX , 25/XI .3 The Mo days of the year Yongle 15 (1417) The year 1417 belongs to the interval of validity of the Datong li (1384– 1644), an astronomical canon devoid of Superior Epoch. Consequently, it would seem that Method 3, p. 224, used in order to calculate the Mo d days for astronomical canons having a such an epoch is not applicable. The analysis of the corresponding Datong techniques for the calculation of the Mo d days – called Ying d days, as already noted on p. 219 above – shows that the Method 3 in question remains valid, modulo a trivial modification of Ji (first defined on p. 225 above), consisting in a simultaneous division of its numerator and denominator by 24 × b. Therefore, given that the Datong solar year Y = ( ab )d is such that a = 3, 652, 425, b = 10, 000, the value of the Moxian, 1 − r/24, becomes equal to 0.78156254 and, lastly: Ji = ⌊ Y /24 − 15 fi 15.2184375 − 15 fi ⌋=⌊ ⌋. Y /24 − 15 0.2184375 Moreover, always with the same modification, fi becomes equal to the non-integer part of the corresponding qi . Now, the list of solar breaths for the year 1417 has already been calculated (Table 11.1, p. 306 above) and, in order to spot the solar periods [qi , qi+1 [ containing a Mo d point, those whose non-integer part is greater than or equal to the Moxian, 0.7815625 are retained (Test 2, p. 225 above). Hence Table 12.4 below, giving both the details of the calculations and the sexagenary binomials of the corresponding Mo d (or Ying d) days. 3 These dates wholly conform with those of the reconstituted Japanese calendar of the year 877, which is likewise based on the Xuanming li (see Y. Okada, K. Itō et al. 1993, vol. 4, p. 197–203). Unfortunately, however, the authentic Chinese almanac SP6 rº for the year 877 contains no Mie days. See A. Arrault and J.-C. Martzloff 2003, p. 200–203. 4Y /24 = 15.2184375 d , r = Y − 360 = 5.2425, 1 − r/24 = 0.7815625. 330 MO AND MIE DAYS Lastly, the Chinese dates of these Mo/Ying d /d days are readily determined from any calendrical table of the year 1417 or from direct calculations. We obtain the following list: 1/II, 12/IV, 23/V *, 4/VIII, 15/X , 26/ XII . Moreover, as readily checked, these dates wholly conform to those of the authentic official calendar of the year 1417. qi q5 q9 q14 q19 q23 q4 48.9137500 49.7875000 5.8796875 21.9718750 22.8456250 38.9378125 Ji 6 15 9 2 11 5 calculations mod. 60 48 + 6 = 54 49 + 15 = 4 5 + 9 = 14 21 + 2 = 23 22 + 11 = 33 38 + 5 = 43 Mo/Ying days #55 (5, 7) #5 (5, 5) #15 (5, 3) #24 (4, 12) #34 (4, 10) #44 (4, 8) Table 12.4. The calculation of the Mo/Ying days of the year 1417. The Mie days of the Year Yongle 15 (1417) The Mie d days of the Datong li, called Xu d days, are calculated in the same way as the Mie of the second kind of astronomical canons having a Superior Epoch. Consequently, their determination supposes the previous calculation of the mean new moons of the year 1417 and a comparison between their non-integer part and the Mie limit (Miexian). On the one hand, the new moons in question having already been obtained previously (see the more general Table 11.3, p. 311 above, concerning all moon phases and where new moons are denoted ‘NM’), we do not repeat their calculation here. On the other hand, the value of the Miexian is equal to 30 − m = 0.469407, where m = 29.530593d is the value of the mean lunar month of the Datong li. The Mie points having a non-integer value smaller than or equal to this constant are readily obtained. In the following table, they have been asterisked (Table 12.5, p. 331). Next, taking into account only those belonging to the year 1417, the 30 fi final calculations can be performed from Ji = ⌊ 30−29.530593 ⌋, an expression obtained from formula 7.2, p. 226 above, with non-integer values of fi , as previously (Table 12.6, p. 331). THE MIE DAYS OF 1417 331 Lastly, a comparison between the above sexagenary binomials and the dates of the lunar component of the calendar of the year 1417 shows that the list of the Chinese dates of its Mie/Xu days is the following: 24/I, 28/III , 2/ V*, 7/VII , 11/ IX and 15/XI . i 11 12 1 2 3 4 5 5* ni 25.312426* 54.843019 24.373612* 53.904205 23.434798* 52.965391 22.495984 52.026577* Bin. (6, 2) (5, 7) (5, 1) (4, 6) (4,12) (3, 5) (3,11) (3, 5) i 6 7 8 9 10 11 12 ni 21.557170 51.087763* 20.618356 50.148949* 19.679542 49.210135* 18.740728 Bin. (2,10) (2, 4) (1, 9) (1, 3) (10, 8) (10, 2) (9, 7) Table 12.5. The thirteen mean new moons (ni ) of the year 1417 and those containing a Mie day (*). n1 n3 n5∗ n7 n9 n11 ni 24.373612 23.434798 52.026577 51.087763 50.148949 49.210135 Ji 23 27 1 5 9 13 Calculations mod. 60 24 + 23 = 47 23 + 27 = 50 52 + 1 = 53 51 + 5 = 56 50 + 9 = 59 49 + 13 = 2 Mie/Xu Days #48 (8,12) #51 (1, 3) #54 (4, 6) #57 (7, 9) #60 (10,12) #3 (3, 3) Table 12.6. The calculation of the Mie/Xu days of the year 1417. AFTERTHOUGHTS The present research into the other Chinese mathematics can be extended 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 Astrologicum (late first century AD) and a Chinese translation – the Yusi jing ddd – 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 J.-C. Martzloff, Astronomy and Calendars – The Other Chinese Mathematics, DOI 10.1007/978-3-662-49718-0 333 334 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 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 extant because available tables of the Chinese calendar are in no way direct recordings of authentic dates but, on the contrary, the result of conjectural 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: 5 These 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.) 6 Research 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 (spurious, 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 nonofficial calendars, the deviations often reach one or two days. The Dunhuang 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 7 See R. Mercier 2002b. 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. 8 Initially, 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 ddd, 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 ddd 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 ddd 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 essen9 Huang Yilong 2001a. Yi-long 1992a. 11 Zhang Peiyu, Wang Guifen et al. 1992, p. 127. 10 Huang 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 allow 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 Taiwanese 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 suspected 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 char12 See also Huang Yi-long 1992b. Yi-long 1992a, p. 280. 13 Huang 338 AFTERTHOUGHTS acter of mathematics). In a quite different order of ideas too, problems such as the history the replacement of the kong d zero by the ling d 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. 1 This calculation technique corresponds to the notion of adjusted modulo. See N. Dershowitz and E.M. Reingold 1997, pp. 15,16, 19 and 20. © Springer-Verlag Berlin Heidelberg 2016 J.-C. Martzloff, Astronomy and Calendars – The Other Chinese Mathematics, DOI 10.1007/978-3-662-49718-0 341 342 APPENDICES #1 jiazi dd (1 , 1) #2 yichou dd (2 , 2) #3 bingyin dd (3 , 3) #4 dingmao dd (4 , 4) #5 wuchen dd (5 , 5) #6 jisi dd (6 , 6) #7 gengwu dd (7 , 7) #8 xinwei dd (8 , 8) #9 renshen dd (9 , 9) # 10 guiyou dd (10 , 10) # 11 jiaxu dd (1 , 11) # 12 yihai dd (2 , 12) # 13 bingzi dd (3 , 1) # 14 dingchou dd (4 , 2) # 15 wuyin dd (5 , 3) # 16 jimao dd (6 , 4) # 17 gengchen dd (7 , 5) # 18 xinsi dd (8 , 6) # 19 renwu dd (9 , 7) # 20 guiwei dd (10 , 8) # 21 jiashen dd (1 , 9) # 22 yiyou dd (2 , 10) # 23 bingxu dd (3 , 11) # 24 dinghai dd (4 , 12) # 25 wuzi dd (5 , 1) # 26 jichou dd (6 , 2) # 27 gengyin dd (7 , 3) # 28 xinmao dd (8 , 4) # 29 renchen dd (9 , 5) # 30 guisi dd (10 , 6) # 31 jiawu dd # 32 yiwei dd # 33 bingshen dd # 34 dingyou dd # 35 wuxu dd # 36 jihai dd # 37 gengzi dd # 38 xinchou dd # 39 renyin dd # 40 guimao dd # 41 jiachen dd # 42 yisi dd # 43 bingwu dd # 44 dingwei dd # 45 wushen dd # 46 jiyou dd # 47 gengxu dd # 48 xinhai dd # 49 renzi dd # 50 guichou dd # 51 jiayin dd # 52 yimao dd # 53 bingchen dd # 54 dingsi dd # 55 wuwu dd # 56 jiwei dd # 57 gengshen dd # 58 xinyou dd # 59 renxu dd # 60 guihai dd The sexagenary cycle. (1 , 7) (2 , 8) (3 , 9) (4 , 10) (5 , 11) (6 , 12) (7 , 1) (8 , 2) (9 , 3) (10 , 4) (1 , 5) (2 , 6) (3 , 7) (4 , 8) (5 , 9) (6 , 10) (7 , 11) (8 , 12) (9 , 1) (10 , 2) (1 , 3) (2 , 4) (3 , 5) (4 , 6) (5 , 7) (6 , 8) (7 , 9) (8 , 10) (9 , 11) (10 , 12) 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. © Springer-Verlag Berlin Heidelberg 2016 J.-C. Martzloff, Astronomy and Calendars – The Other Chinese Mathematics, DOI 10.1007/978-3-662-49718-0 343 344 APPENDICES The Twenty-Four Solar Breaths (104 BC – AD 1644) zhong d (Odd Breaths) jie d (Even Breaths) q1 q2 q3 q4 q5 q6 q7 q8 q9 q10 q11 q12 q13 q14 q15 q16 q17 q18 q19 q20 q21 q22 q23 q24 dd dd dd dd dd dd dd dd dd dd dd dd dd dd dd dd dd dd dd dd dd dd dd dd Dongzhi Xiaohan Dahan Lichun Yushui Jingzhe Chunfen Qingming Guyu Lixia Xiaoman mangzhong Xiazhi Xiaoshu Dashu Liqiu Chushu Bailu Qiufen Hanlu Shuangjiang Lidong Xiaoxue Daxue Winter Solstice Slight Cold Great Cold Beginning of Spring Rain Water Waking of Insects Spring Equinox Pure Brightness Grain Rain Beginning of Summer Grain Full Bearded Grain Summer Solstice Slight Heat Great Heat Beginning of Autumn Limit of Heat White Dew Autumn Equinox Cold Dew Descent of Frost Beginning of Winter Slight Snow Great Snow Approximate Dates (Limit Values) Dec. 25 Jan. 9 Jan. 24 Feb. 8 Feb. 23 Mar. 11 Mar. 26 Apr. 10 Apr. 26 May 10 May 26 10 June June 26 July 10 July 26 Aug. 10 Aug. 25 Sept. 9 Sept. 24 Oct. 10 Oct. 25 Nov. 9 Nov. 24 Dec. 10 – – – – – – – – – – – – – – – – – – – – – – – – Dec. 21 Jan. 5 Jan 20 Feb. 5 Feb. 20 Mar. 7 Mar. 22 Apr. 6 Apr. 22 May 7 May 22 6 June June 21 July 7 July 22 Aug. 6 Aug. 21 Sept. 6 Sept 21 Oct. 6 Oct. 21 Nov. 6 Nov. 22 Dec. 7 THE LUNISOLAR COUPLING 345 The Lunisolar Coupling Solar Breaths q24 q1 q2 q3 q4 q5 q6 q7 q8 q9 q10 q11 q12 q13 q14 q15 q16 q17 q18 q19 q20 q21 q22 q23 Appellations of the Solar Breaths in Relation to the Lunisolar Coupling dd shiyiyue jie dd shiyiyue zhong dd shi’eryue jie dd shi’eryue zhong dd zhengyue jie dd zhengyue zhong dd eryue jie dd eryue zhong dd sanyue jie dd sanyue zhong dd siyue jie dd siyue zhong dd wuyue jie dd wuyue zhong dd liuyue jie dd liuyue zhong dd qiyue jie dd qiyue zhong dd bayue jie dd bayue zhong dd jiuyue jie dd jiuyue zhong dd shiyue jie dd shiyue zhong dd dd dd dd dd dd dd dd ddd ddd ddd ddd ddd ddd ddd ddd ddd ddd ddd ddd ddd ddd ddd ddd ddd ddd ddd ddd even Breath of the 11th month odd Breath of the 11th month even Breath of the 12th month odd Breath of the 12th month even Breath of the 1st month odd Breath of the 1st month even Breath of the 2nd month odd Breath of the 2nd month even Breath of the 3rd month odd Breath of the 3rd month even Breath of the 4th month odd Breath of the 4th month even Breath of the 5th month odd Breath of the 5th month even Breath of the 6th month odd Breath of the 6th month even Breath of the 7th month odd Breath of the 7th month even Breath of the 8th month odd Breath of the 8th month even Breath of the 9th month odd Breath of the 9th month even Breath of the 10th month odd Breath of the 10th month Month Membership 10th or 11th 11th th 11 or 12th 12th th 12 or 1st 1st st 1 or 2nd 2nd nd 2 or 3rd 3rd rd 3 or 4th 4th 4th or 5th 5th th 5 or 6th 6th th 6 or 7th 7th th 7 or 8th 8th 8th or 9th 9th th 9 or 10th 10th APPENDIX C THE SEVENTY-TWO SEASONAL INDICATORS The 72 seasonal indicators (hou d) 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. qi q1 q2 q3 Initial Indicator chu hou dd The Seventy-two Seasonal Indicators Next Indicator Final Indicator ci hou dd mo hou dd ddd (h1 ) qiu yin jie Earth-worms Curl Up ddd (h2 ) mi jiao jie Elaphure Shed Antlers ddd (h3 ) shui quan dong Springs and Streams Stir ddd (h4 ) yan bei xiang Wild Geese Head Northwards ddd (h5 ) que shi chao Magpie Nests dddd (h6 ) yeji shi gou Pheasant Begin to Crow ddd (h7 ) ji shi ru Hens Begin to Brood dddd (h8 ) zhi niao li ji Birds of Prey Fierce and Quick dddd (h9 ) shui ze fu jian Rivers and Lakes Frozen Thick © Springer-Verlag Berlin Heidelberg 2016 J.-C. Martzloff, Astronomy and Calendars – The Other Chinese Mathematics, DOI 10.1007/978-3-662-49718-0 346 THE 72 SEASONAL INDICATORS qi Initial Indicator chu hou dd 347 The Seventy-two Seasonal Indicators Next Indicator Final Indicator ci hou dd mo hou dd q4 dddd (h10 ) dong feng jie dong East Wind Dissipates the Cold dddd (h11 ) zhi chong shi zhen Hibernating Creatures Begin to Stir ddd (h12 ) yu shang bing Fish Ascend to the Ice q5 ddd (h13 ) ta ji yu Otter Sacrifices Fish ddd (h14 ) dddd (h15 ) hong yan lai Wild Geese Appear caomu meng dong Plants Bud and Grow q6 ddd (h16 ) tao shi hua Peach Trees Begin to Blossom ddd (h17 ) cang geng ming Oriole Sings dddd (h18 ) ying hua wei jiu Hawks Transformed into Doves q7 ddd (h19 ) xuanniao zhi Black Bird (i.e. Swallow) Arrives dddd (h20 ) lei nai fasheng Thunder Sounds dd (h21 ) shi dian First Lightning q8 ddd (h22 ) tong shi hua Pawlownia Begins to Flower ddddd (h23 ) tianshu hua wei ru Moles Transformed into Button-quail ddd (h24 ) hong shi jian Rainbows begin to Appear q9 ddd (h25 ) ping shi sheng Duckweed Begins to Grow ddddd (h26 ) mingjiu fu qi yu Cooing-dove Preens ddddd (h27 ) dai sheng jiang yu sang Hoopoe Alights on Mulberry Trees q10 ddd (h28 ) lou guo ming Green Frog Croak ddd (h29 ) qiu yin chu Earthworms Emerge ddd (h30 ) wang gua sheng Royal Gourd Grows q11 ddd (h31 ) kucai xiu Sow-thistle in Seed ddd (h32 ) micao si Delicate Herbs Die ddd (h33 ) xiaoshu zhi Slight Heat Arrives 348 qi APPENDICES Initial Indicator chu hou dd The Seventy-two Seasonal Indicators Next Indicator Final Indicator ci hou dd mo hou dd ddd (h34 ) tanglang sheng Praying Mantis Born ddd (h35 ) ju shi ming Shrike Begins to Call dddd (h36 ) fanshe wu sheng Mockingbird Silent ddd (h37 ) lu jiao jie Deer Shed Antlers ddd (h38 ) tiao shi ming Cicadas Begin to Sing ddd (h39 ) banxia1 sheng Midsummer Plant Grows ddd (h40 ) wen feng zhi Warm Wind Arrives dddd (h41 ) xishuai ju bi Crickets Settle in the Walls dddd (h42 ) ying nai xue xi Young Hawks Learn to Fly q15 dddd (h43 ) fucao wei ying Decaying Grass Becomes Fireflies dddd (h44 ) tu run ru shu Ground Humid, Hot dddd (h45 ) da yu shi xing Heavy Rains Begin to Fall q16 ddd (h46 ) liang feng zhi Cool Wind Arrives ddd (h47 ) bai lu jiang White Dew Descends ddd (h48 ) hanchan ming Cold Cicada Chirps ddd (h49 ) ying ji niao Hawks Sacrifice Birds dddd (h50 ) tian di shi su Heaven and Earth Begin to be Severe ddd (h51 ) he nai deng Grain Presented ddd (h52 ) hong ying lai Wild Geese Arrive ddd (h53 ) xuan niao gui Black Bird (Swallow) Return dddd (h54 ) qun niao yang xiu All Birds Store up Provisions q12 q13 q14 q17 q18 Air 1 Medicinal Plant corresponding to the Pinellia ternata (or Arum ternatum) according to F. Fèvre and G. Métailié 2005, p. 24. THE 72 SEASONAL INDICATORS qi q19 q20 q21 q22 q23 q24 Initial Indicator chu hou dd 349 The Seventy-two Seasonal Indicators Next Indicator Final Indicator ci hou dd mo hou dd dddd (h55 ) lei nai shou sheng Thunder Restrains its Sound dddd (h56 ) zhi chong pei hu Hibernating Creatures Close their Burrows ddd (h57 ) shui shi he Water Begins to Dry Up dddd (h58 ) hong yan lai bin Wild Geese Come and Stay dddddd (h59 ) que ru dashui wei ge Small Birds Enter the Great Water and Become Bivalves dddd (or d) (h60 ) ju you huang hua Chrysanthemum Bears Yellow Blossom dddd (h61 ) chai nai ji shou Wild Dog Offer Prays dddd (h62 ) caomu huang luo Plants and Trees Turn Yellow and Drop Leaves dddd (h63 ) zhi chong xian fu Hibernating Insects All Burrow in ddd (h64 ) shui shi bing Water Begins to Turn to Ice ddd (h65 ) di shi dong Ground Begins Freeze dddddd (h66 ) yeji rushui wei shen Pheasants Enter Great Waters and Become Large Bivalves dddd (h67 ) hong cang bu jian dddddddd tianqi shangteng diqi xiajiang (h68 ) Sky Breath (qi d) Rises, Earth Breath Settles ddddd (h69 ) bi se ru cheng dong ddd (h71 ) hu shi jiao Tiger Begins to Mate ddd (h72 ) liting sheng liting2 Emerges Rainbow Hides and is Invisible dddd (h70 ) he niao bu ming Yellow Pheasant Silent 2 liting to All is Closed up: Winter Sets in 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 History 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 dd, are at variance with those of other ancient original sources.2 But is the Yuanshi dd faulty or on the contrary, are more ancient sources unreliable? In this respect, Wang Yuezhen’s Gujin tuibu zhushu kao ddddddd, first published much more than one century ago, in 1867, provides a well-argued refutation 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 1 See, 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. 2 See N. Sivin 2009, ibid., p. 43–52. 3 See Gujin tuibu zhushu kao, 1936, j. 2, p. 14a, 17b, 21a (notice p. 393 below). 4 N. Sivin 2009, ibid., p. 43–52. © Springer-Verlag Berlin Heidelberg 2016 J.-C. Martzloff, Astronomy and Calendars – The Other Chinese Mathematics, DOI 10.1007/978-3-662-49718-0 350 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 masterpiece, the Lidai changshu jiyao dddddd5 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 ddd ’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 5 Notice, p. 378 below. Sivin 2009, ibid., p. 42, rightly remarks that “many dates require further study”. 7 They are easily available nevertheless. See, for instance, Th.E. Deane 1989, Appendix D.; N. Sivin 2009, ibid., p. 43–56. 8 The 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. 9 It 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. 10 Yuanshi, j. 53, ‘li 2’, p. 1178–1188. 6 N. 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 1 2 Santong◦ Sifen◦ dd dd 3 4 Qianxiang◦ Jingchu* ◦ dd dd 5 6 Sanji jiazi yuan Yuanshi (or Xuanshi) 7 Yuanjia* ◦ dd 8 Daming* ◦ dd 9 Zhengguang* ◦ dd 10 Xinghe* ◦ dd 11 12 13 Tianbao◦ Tianhe◦ Daxiang◦ dd dd dd ddddd dd (or dd) Han Hou Han Shu Wei Wu Wei Jin Liu Song Northern Wei Later Qin Northern Liang Toba Wei Liu Song Qi Liu Song Liang Chen Toba Wei Oriental Wei Western Wei Northern Zhou Oriental Wei Northern Qi Northern Qi Northern Zhou Northern Zhou Sui Dates 104 BC–84 85–220 221–263 226–236 223–280 237–265 265–420 420–444 398–451 384–41711 412–439 452–522 445–479 479–50312 502–509 510–557 557–589 523–565 534–539 535–556 557–565 540–550 55013 551–577 566–578 579–581 581–583 11 Many authors write ’517’ here. For a justification of the year 417, see Chen Meidong 2003a, p. 254. 12 The Yuanjia li ddd was renamed Jianyuan li ddd under the Qi dynasty. 13 The Xinghe li has been in force only one year under the Northern Qi dynasty. OFFICIAL ASTRONOMICAL CANONS No. Canons Dynasties 14 15 16 17 18 19 20 21 22 23 24 Kaihuang ◦ Daye* ◦ Wuyin* ◦ Linde* ◦ Dayan* ◦ Zhide Wuji* ◦ Zhengyuan * Guanxiang Xuanming* ◦ Chongxuan* ◦ dd dd dd dd dd dd dd dd dd dd dd 25 Diaoyuan dd 26 Qintian* ◦ dd 27 28 29 Yingtian* ◦ Qianyuan *◦ Daming dd dd dd 30 31 Yitian ◦ Chongtian* ◦ dd dd 32 33 34 35 36 Mingtian* ◦ Fengyuan ◦ Guantian* ◦ Zhantian◦ Jiyuan* ◦ dd dd dd dd dd 37 38 Tongyuan ◦ Daming ◦ dd dd Sui Sui Tang Tang Tang Tang Tang Tang Tang Tang Tang Later Liang Later Tang Later Jin Later Jin Liao Later Zhou Song Northern Song Northern Song Liao Jin Northern Song Northern Song Northern Song Northern Song Later Zhou Northern Song Northern Song Southern Song Southern Song Jin 353 Dates 584–596 597–618 619–664 665–728 729–761 758–762 763–783 784–806 807–821 822–892 893–907 907–923 923–936 936–938 939–943 947–994 956–960 960–963 964–982 983–1000 995–1125 1123–1136 1001–1023 1024–1064 1068–1074 1065–1067 1075–1093 1094–1102 1103–1105 1106–1127 1133–113514 1136–1167 1137–1181 14 According to Chen Zungui 1984, p. 1404, the correct year span is 1106–1166. However, this author leaves unjustified the year 1166 whereas many chronologists propound the year 1127 instead. 354 No. APPENDICES Canons 39 40 41 Qiandao* ◦ Chunxi* ◦ Chongxiu Daming* ◦ 42 43 44 45 46 47 48 49 50 Huiyuan* ◦ Tongtian* ◦ Kaixi* ◦ Chunyou ◦ Huitian ◦ Chengtian *◦ Bentian Shoushi* ◦ Datong Dynasties dd dd dddd dd dd dd dd dd dd dd dd dd Southern Song Southern Song Jin Yuan Southern Song Southern Song Southern Song Southern Song Southern Song Southern Song Southern Song Yuan Ming Dates 1168–1176 1177–1190 1181–1234 1215–128015 1191–1198 1199–1207 1208–125116 1251–125217 1253–1270 1271–1276 1277–1279 1281–1384 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 k such that αβ = 19k+11 7k+4 . (In Chinese sources, α and β are respectively called zhangsui dd and zhangyue dd; sui d and yue d 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 d, 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. Nevertheless, we mention it here because a technique of derivation of new fractions from initial ones, consisting in mutual additions of their nu15 The year 1215 corresponds to the adoption of this astronomical canon by the Mongols before their conquest of China, in 1277. See Yabuuchi 1969a/1990*, p. 390. 16 These initial and final years are uncertain, various authors slightly vary in this respect (micro variations equal to ±1 year in each case). 17 From Chen Zungui 1984, p. 1406. However, COLL. 1980 only retains the sole year 1252. OFFICIAL ASTRONOMICAL CANONS N◦ 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 Canons Santong Sifen Qianxiang Jingchu Sanji Yuanshi Yuanjia Daming Zhengguang Xinghe Tianbao Tianhe Daxiang Kaihuang Daye dd dd dd dd dd dd dd dd dd dd dd dd dd dd dd 355 α /β k 19/7 19/7 19/7 19/7 ? 600/221 19/7 391/144 505/186 562/207 676/249 391/144 448/165 429/158 410/151 – – – – ? 31 – 20 26 29 35 20 23 22 21 Table D.2. List of Metonic constants. merators and denominators, is attested in Chinese sources.18 In the present case, fractions of the form 19k+11 7k+4 are formally obtained by start19 ing from the two initial fractions 7 and 11 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 in11 termediary between the two initial fractions, 19 7 and 4 : 18 See Chen Zungui 1984, ibid., note 3, p. 1447–1448; Chen Jiujin 1984; Liu Dun 1987; Li Jimin 1998. 19 This 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 For k positive integer 19 19k + 11 11 ≤ ≤ . 7 7k + 4 4 The Metonic constants α and β listed in the preceding table have perhaps been obtained in this way in order to obtain better Metonic approximations. However, the fraction 11 4 is apparently recorded in no extant Chinese source. Yet, similar Metonic fractions also occur in the nonofficial Kaiyuan taiyi li ddddd (the astronomical canon Taiyi20 from the Kaihuang reign-period (713–741).21 20 Literal meaning of this term: ‘The Great One’. See Ho Peng Yoke 2003, p. 36 f. Anjing 2005, p. 385, highlights the presence of Metonic fractions of the form 19k+11 235k+136 in this text. 21 Qu APPENDIX E TIME CONSTANTS The following table contains a partial list1 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 dd or suanshang dd ‘exhaustive counting’) or in ‘exclusive counting’ (suanwai dd ‘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. 1 For a more complete list, see Qu Anjing, Ji Zhigang and Wang Rongbin 1994, p. 154–155. 2 See p. 139 above. 3 See Gao Pingzi 1987, p. 112. 4 See, for example, E.G. Richards 1998, p. 81. © Springer-Verlag Berlin Heidelberg 2016 J.-C. Martzloff, Astronomy and Calendars – The Other Chinese Mathematics, DOI 10.1007/978-3-662-49718-0 357 358 Nº 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 APPENDICES Canon Santong Sifen Jingchu Daming Kaihuang Daye Wuyin Linde Dayan Wuji Xuanming Chongxuan Yingtian Qianyuan Chongtian Mingtian Fengyuan Guantian Zhantian Jiyuan Tongyuan Qiandao t0 143,1275 93666 4045 51,939 4,129,001 1,427,645 164,341 269,881 96,961,741 269,979 7,070,138 53,947,309 4,825,559 30,543,978 97,556,341 711,761 83,185,071 5,944,809 25,501,760 28,613,467 94,251,592 91,645,824 x0 −103 85 236 462 584 608 618 664 724 762 821 892 962 981 1024 1064 1074 1092 1103 1106 1135 1167 constant comes from Li Rui dd (1765–1814)’s Han Santong shu ddd d (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 dddd, 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. 5 This TIME CONSTANTS Nº Canon 23 24 25 26 Chunxi Huiyuan Kaixi Chunyou 359 t0 x0 52,421,973 25,494,768 7,848,183 120,267,647 1176 1191 1206 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 Moreover, the solar year of this canon is equal to 365 + 1539 = 562120 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: 7 See p. 164 above. 360 APPENDICES Canon Era x t(x) a b day Daming Kaihuang Daye Wuyin Linde Dayan Wuji Xuanming Chongxuan Yingtian Qianyuan Chongtian Mingtian Fengyuan Guantian Zhantian Daming 6 Kaihuang 12 Daye 3 Zhenguan 19 Zongzhang 3 Tianbao 9 Jianzhong 1 Dashun 1 Tiancheng 5 Kaibao 3 Zhidao 1 Huangyou 3 Zhiping 3 Yuanfeng 3 Yuanfu 2 Chongning 3 462 592 607 645 670 750 780 890 930 970 995 1051 1066 1080 1099 1104 51,939 4,129,009 1,427,644 164,368 269,882 96,961,767 26,999 7,070,207 53,947,347 4,825,567 30,543,992 97,556,368 711,763 83,185,077 5,944,816 25,501,761 14,423,804 37,605,463 15,573,963 3,456,675 489,428 1,110,343 489,428 3,068,055 4,930,801 3,653,175 1,073,820 3,867,940 14,244,500 8,656,273 4,393,880 10,256,040 39,491 102,960 42,640 9464 1340 3040 1340 8400 13,500 10,002 2940 10,590 39,000 23,700 12,030 28,080 26 47 7 26 37 36 #13 50 20 50 1 55 13 26 6 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. Likewise, 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 1 Santong Solar Year – 365 Solar Period –15 385 1539 1010 1539×3 1 4 2 Sifen 3 Qianxiang 7 4×8 145 589 4 Jingchu 455 1843 5 Sanji 605 2451 6 Yuanshi 1759 7200 7 Yuanjia 150 608 © Springer-Verlag Berlin Heidelberg 2016 J.-C. Martzloff, Astronomy and Calendars – The Other Chinese Mathematics, DOI 10.1007/978-3-662-49718-0 = 7 32 515 589×4 402 1843 11 + 1843×12 535 2451 1573 7200 5 + 2451×6 7 + 7200×24 132 608 22 + 608×24 361 362 APPENDICES Nº Canon Solar Year – 365 8 Daming 9 Zhengguang 9589 39,491 8626 39,491 5 + 39,491×6 1477 6060 1324 6060 1 + 6060×24 10 Xinghe 4117 16,860 3684 16,860 1 + 16,860×24 11 Tianbao 5787 23,660 5170 23,660 7 + 23,660×24 12 Tianhe 5761 23,460 5127 23,460 13 + 23,460×24 13 Daxiang 3167 12,992 25,063 102,960 14 Kaihuang 10,363 42,640 15 Daye 2838 12,992 22,494 102,960 5 + 12,992×8 7 + 102,960×24 9315 42,640 1 + 42,640×8 16 Wuyin 2315 9464 2068 9464 1 + 9464×8 17 Linde 328 1340 292 1340 5 + 1340×6 18 Dayan 743 3040 19 Zhide 20 Wuji 328 1340 21 Zhengyuan 268 1095 22 Guanxiang ? 2055 8400 3301 13,500 24 Chongxuan 22 Diaoyuan 26 Qintian 664 3040 7 + 3040×24 ? 23 Xuanming 1 Solar Period –15 ? 292 1340 239 1095 40 1 + 7200×100 7 + 1095×24 ? 1835 8400 2950 13,500 ? 1760 7200 5 + 1340×6 5 + 8400×8 1 + 13,500×24 ? 1573 7200 35 + 7200×100 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 Nº Canon 27 Yingtian 28 Qianyuan 29 Daming Solar Year – 365 2445 10,002 720 2940 363 Solar Period –15 2185 10,002 5 + 10,002×8 642 2940 ? 1 + 2940×2 ? 30 Yitian 2470 10,100 2207 10,100 3 + 10,100×36 31 Chongtian 2590 10,590 2314 10,590 6 + 10,590×36 32 Mingtian 9500 39,000 33 Fengyuan 5773 23,700 5178 23,700 1 + 23,700×24 34 Guantian 2930 12,030 2628 12,030 12 + 12,030×36 35 Zhantian 6840 28,080 36 Jiyuan 1776 7290 37 Tongyuan 1688 6930 38 Daming 1274 5230 8520 39,000 5 + 39,000×6 6135 28,080 1592 7290 1514 6930 3 + 7290×4 15 + 6930×180 1142 5230 2 + 5230×3 7308 30,000 6554 30,000 1 + 30,000×2 40 Chunxi 1374 5640 1232 5640 25 + 5640×100 41 Chongxiu Daming 1274 5230 1142 5230 60 + 5230×90 42 Huiyuan 9432 38,700 8455 38,700 1 + 38,700×2 43 Tongtian 2910 12,000 44 Kaixi 4108 16,900 39 Qiandao 2621 12,000 25 + 12,000×100 3692 16,900 45 Chunyou 857 3530 771 3530 1 + 3530×8 46 Huitian 2366 9740 2127 9740 3 + 9740×4 47 Chengtian 1801 7420 1620 7420 7 + 7420×8 364 APPENDICES Nº Canon Solar Year – 365 Solar Period –15 48 Bentian ? ? 49 Shoushi secular variations2 0.21843753 50 Datong 0.2425 2 See 0.2184375 p. 141 above. The Shoushi astronomical canon says nothing about the consequences of the variations of the tropical year on the lengths of solar periods. 3 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 official 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 81 — 499 940 — 3 Qianxiang 773 1457 — 4 Jingchu 2419 4559 — 5 Sanji 3217 6063 — 6 Yuanshi 47,251 89,052 — 7 Yuanjia 399 752 — 8 Daming 2090 3939 — 1 Santong 2 Sifen 1 These values have been checked with the help of Chen Meidong 1995, p. 237–244 and Wang Yingwei 1998. © Springer-Verlag Berlin Heidelberg 2016 J.-C. Martzloff, Astronomy and Calendars – The Other Chinese Mathematics, DOI 10.1007/978-3-662-49718-0 365 366 N◦ . Canon APPENDICES Syn. Month – 29 Anom. Month – 27 39,769 74,952 — 10 Xinghe 110,647 208,530 — 11 Tianbao 155,272 292,635 — 12 Tianhe 153,991 290,160 — 28,422 53,563 — 96,529 181,920 — 607 1144 — 6901 13,006 — 9 Zhengguang 13 Daxiang 14 Kaihuang 15 Daye 16 Wuyin 17 Linde 711 1340 743 1340 1 + 1340×12 18 Dayan 1613 1340 1685 1340 79 + 3040×80 19 Zhide ? 20 Wuji 711 1340 21 Zhengyuan 581 1095 22 Guanxiang ? 2 Chen Meidong 1995, p. 240. Peiyu,Wang Guifen et al. 1992, p. 122. 4 Wang Yingwei 1998, p. 526. 3 Zhang ? 7486 13,500 28 4 + 7200×100 5 2 + 1340×37 132 + 1095×10,000 7163 13,500 ? 3820 7200 607 1095 4658 8400 25 Diaoyuan 26 Qintian 743 1340 4457 8400 23 Xuanming 24 Chongxuan ? 19 3 + 8400×100 97 + 13,500×100 ? — LUNAR CONSTANTS N◦ . Canon Syn. Month – 29 27 Yingtian 28 Qianyuan 29 Daming 367 Anom. Month – 27 5307 5 10,002 5546 10,002 1560 2940 1620 2940 6210 6 + 10,002×10,000 6020 7 + 2940×10,000 ? ? 30 Yitian 5359 10,100 5601 10,100 165 + 10,100×10,000 31 Chongtian 5619 8 10590 5873 10,590 594 + 10,590×10,000 32 Mingtian 20,693 39,000 601,471,251 9 39,000×27,807 33 Fengyuan 12,575 23,700 ? 34 Guantian 6383 12,030 35 Zhantian 14,899 28,080 6672 12,030 + 389 12,030×10,000 ? 36 Jiyuan 3868 7290 4043 7290 990 + 7290×10,000 37 Tongyuan 3677 6930 3843 6930 2563 + 6930×10,000 38 Daming 39 Qiandao 40 Chunxi 41 Chongxiu Daming 42 Huiyuan 5 This ? 15,917 30,000 76 + 30,000×100 2992 5640 ? 16,637 30,000 7395 + 30,000×10,000 56 + 5640×100 3127 5640 9740 + 5640×10,000 2775 5230 2775 5230 6066 + 5230×10,000 20,534 38,700 21,461 38,700 7310 + 38,700×10,000 fraction is not irreducible and this is also the case for a few others. Yingwei 1998, ibid., p. 526. 7 According 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). 8 Wang Yingwei 1998, ibid., p. 576. 9 According 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 6 Wang 601, 47 1 , 251 instead. 368 N◦ . Canon APPENDICES Syn. Month – 29 43 Tongtian 6368 12,000 44 Kaixi 8967 16,900 Anom. Month – 27 6655 12,000 9372 16,900 5396 + 16,900×10,000 45 Chunyou 1873 3530 ? 46 Huitian 5168 9740 ? 47 Chengtian 3937 7420 4115 7420 1641 + 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 dd) 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. Apparently, 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 J.-C. Martzloff, Astronomy and Calendars – The Other Chinese Mathematics, DOI 10.1007/978-3-662-49718-0 371 372 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 modifications. 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 ddddddd (Five Thousand Years of Long Chinese Calendrical Chronology), Beijing, Qixiang Chubanshe ddddd. – 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, dd1926/1999*. Ershi shi shuorun biao dddd dd (Chronological Table of New Moons, Ordinary Lunar Months and Intercalary Months in the Twenty Dynastic Histories), Beijing, Zhonghua shuju dddd. – 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 ddd and FANG Xiaofen ddd 1987. Zhongguo shi liri he zhongxi liri duizhaobiao ddddddddddd dd (A Table of the Historical Chinese Calendar, With a Concordance Between its Dates and those of the Western Calendar), Shanghai, Shanghai Cishu Chubanshe ddddddd. – 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 Department. – 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 dddd1990. Tangdai de li dddd (The Astronomical Canons of the Tang Dynasty), Shanghai, Shanghai Guji Chubanshe ddddddd. (Chinese transl. from the Japanese of 374 BIBLIOGRAPHY Tōdai no koyomi, dddd Kyoto, Kyoto daigaku jimbun kagaku kenkyūjo ddddddddddd, 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 Calendar and a Concordance with the European Calendar. Zi-Ka-Wei, Printing Office of the Catholic Mission. Reedited by Le Cercle Sinologique 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 néoméniques chinoise et européenne, 2nd ed., Taichung, Kuangchi Press, (1st ed., Imprimerie de la Mission Catholique, Orphelinat de T’ou-sèwei, 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 dd d, 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); 10 The 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 ddd 2004. Liao, Song, Xia, Jin, Yuan wu chao rili ddddddddd (Chronological Tables of the Chinese Calendar for the Five Following Dynasties: Liao, Song, Xia, Jin and Yuan), Taipei, Zhongyang yanjiuyuan lishi yuyan yanjiusuo dddddd dddddd. – Critical reworking of Chen Yuan, 1926/1999* (already mentioned on p. 372 above). In order to ease its consultation and to render immediate 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 Yoshirō, dddd, ITŌ Kazuhiko, dddd ŌTANI Mitsuo dddd and FURUKAWA Kiichirō ddddd, 1993. Nihon rekijitsu sōran, gūchurekijitsu hen, ddddddddd ddd (A Survey of Japanese Calendars: Annotated Calendars), 20 vol. Tokyo, Honnotomo sha dddd. 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 ddd, (b) the Linde li/Gihōreki ddd and (c) the Xuanming li/Senmyōreki ddd 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 Japanese calendar (and thus the Chinese calendar also) is liable to contain, even its calendrical spirits shen d.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] ddd, 1960, Zhongguo nianli zongpu dddddd (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 dd). 12. WANG Huanchun ddd et al., 1991. Gong, Nong, Hui, Tai, Yi, Zang, Fo li he Rulüe ri duizhaobiao ddddddddddddd 11 On Japanese chronological systems, see R. Zöllner 2003. Y. Okada, K. Itō 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. 13 On calendrical spirits, see A. Arrault 2003, p. 106 f. 12 See TABLES OF THE CHINESE CALENDAR 377 dd (622–2050) (Concordance Tables between the Gregorian, Chinese, Muslim, Thai, Yi, Tibetan and Buddhist Calendars and Julian Days). Beijing, Kexue Chubanshe ddddd. – 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 ddd and LI Min dd, 1996. Zhonghua tongshi dali dian ddddddd (Great Chronology of Chinese History). 3 vol. Chengdu, Sichuan Minzu Chubanshe ddddddd. – Monumental compilation composed of approximately five thousand pages and giving all sorts of historical data beyond what is usually included in chronological tables, (various names of Chinese emperors, main historical events, etc.); – year span: 2674 BC–AD 2000. 14. *WANG Yuezhen ddd, 1866. Lidai changshu dddd (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 dddddddddddddd (Catalog of Ancient Texts and Rare Books of the Beijing Library, section devoted to technical works), Beijing, Shumu Wenxian Chubanshe d dddddd, 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 dd15 from Kuaiji sub-prefecture dd, Zhejiang province; he became juren dd (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 14 On Wang Yuezhen, see P. Hoang 1910/1968*, p. xi–xv (notice, p. 374 above); COL-astron, vol. 1, p. 717; Chouren zhuan sanbian ddddd 1898/1982*, j. 6, p. 823–826. 15 See 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 twentieth 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 ddd, 1867/1936*/1993*. Lidai changshu jiyao, fu gujin tuibu zhushu kao ddddddddddddd d (A Concise Handbook of the Long [Chinese] Chronology Calculated According to the Astronomical Canons of the Successive Dynasties (shu d), with an Annex Devoted to a Study of the Methods of Predictive Astronomical, Astrological and Hemerological Calculations (tuibu dd) Expounded in [Chinese] Astronomical Canons), Shanghai, Zhonghua shuju dddd, Sibu beiyao collection dd dd; 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 critically 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 d dd (1019–1086)’s Zizhi tongjian mulu dddddd (Chronol16 See 17 See the next notice. 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 Chinese 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 dd 18 Concerning this famous work, see Y. Hervouet 1978, p. 169 f. Original text: WYG, vol. 311, p. 321–787. 19 Sima 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 dddd. After publication, this work of Sima Guang became an unsurpassed reference. During the XVIIIth century evidential scholars such as Li Rui dd (1765–1814) and Qian Daxin ddd (1728–1804) tackled the subject anew. The first began to wrestle with various questions of chronology relating to Chinese 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 ddddddd (Research into the New Moons and Intercalations in the Song, Liao, Jin and Yuan Histories). 20 P. Hoang, 1910/1968*, p. xxvii (notice, p. 374 above). 380 BIBLIOGRAPHY dd, (b) 1936: edition made from the first one, without mention of corrections, Shanghai, Zhonghua shuju dddd, collection Sibu beiyao dddd, 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 d dd and ZHOU Yunqing ddd (ed.), 1957. Sibu zonglu, tianwen bian dddddddd (Catalog of the Four Bibliographical Departments: Astronomy, Shanghai, Shangwu Yinshuguan (Commercial Press), p. 562a–562b. – Another important notice on the same subject is BO Shuren dd d 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 ddd, 1992. Xinbian Zhongguo sanqian nian liri jiansuo biao dddddddddddd (New Table of Dates for Three Thousand Years of Chinese History), Renmin Jiaoyu Chubanshe d dddddd. – Concordance between the dates of the Chinese, Japanese and Muslim calendars; – year span: 1500 BC–2050. 17. XUE Zhongsan d d d and O UYANG Yi d d d, 1940/1957*. Liangqian nian Zhong-Xi li duizhaobiao d d d d d d d d d (Concordance Between the Chinese and Western Calendars for Two Thousand Years), Beijing, Sanlian shudian dddd. 21 i.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 ddddd, 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 ddd, 1987. Zhongguo xian Qin shi libiao d dddddd (Chronological Tables for Pre-Qin China). Jinan, Qi-Lu shushe dddd. – Retrospective astronomical calculations (dates of winter solstices, new moons, solar and lunar eclipses); – List of calendrical dates determined from the gu liu li ddd techniques, i.e. those of ‘the six ancient computus’;22 – year span: 1500 BC–105 BC. 19. *ZHANG Peiyu d d d, 1990*/1997*. Sanqian wubai nian liri tianxiang ddddddddd (Tables and Astronomical Phenomena for 3500 years), 1st ed.: Henan Jiaoyu Chubanshe ddddd dd; 2nd ed., Zhengzhou, Daxiang Chubanshe ddddd. – 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 dd; – 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 ddd (ed.), 1936/1985*. Jinshi zhong-xi shiri duizhaobiao ddddddddd (A Concordance Between Chi22 These ‘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 ddd but with different Superior Epochs each time. See Chen Meidong 2003a, p. 87–92. Moreover, it should be noted that, here, li d corresponds to ‘computus’ and not to ‘astronomical canon’ because their calculations only concern the luni-solar component of the calendar. 23 On 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 dddd. – Interesting historical introduction; – Concordance between the years of the Christian era, Chinese, Japanese 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, ddd. Zhuanxu rili biao ddddd (Tables of the Zhuanxu li), Beijing, Zhonghua shuju dddd, 2012. 2. – *Taichu rili biao ddd. Taichu liri biao ddddd (Tables of the Taichu li), Beijing, Zhonghua shuju dddd, 2013. 3. – *Hou Han Sifen rili biao ddddddd (Tables of the Sifen li during the Later Han Dynasty), Beijing, Zhonghua shuju dd dd, 2014. Computer Programs Computer programs delivering concordances between the Chinese, Julian, Gregorian and other calendars have been devised.24 Among these, 24 R. Mercier gives some references and an analysis of the specific difficulties programmers 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 dddd (Beijing) or from the former Bainaben ddd edition, Shangwu yinshuguan ddddd, (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: © Springer-Verlag Berlin Heidelberg 2016 J.-C. Martzloff, Astronomy and Calendars – The Other Chinese Mathematics, DOI 10.1007/978-3-662-49718-0 385 386 BIBLIOGRAPHY Shiji dd, (Records of the Historian), ca. 100 BC, Sima Qian dd d, (145 BC–86 BC?), j. 26, p. 1255–1287. Hanshu dd, (Former Han History), ca. 100, Ban Gu dd (32–92), j. 21A–21B, p. 955–1026. Hou Hanshu ddd (Later Han History), 450, Fan Ye dd (398–445) ‘zhi 1–3’, p. 2999–3100. Songshu dd (Song History or Book of Song), ca. 500, Shen Yue dd (441–513), j. 11-13, p. 203–325. Weishu dd (Wei History), 554, Wei Shou dd (506–572) et al., j. 107A , p. 2657–2731. Suishu dd (Sui History), 636, Wei Zheng dd (580–643) et al., j. 16– 18, p. 385–501. Jinshu dd (Jin History) ca. 646, Fang Xuanling ddd (579–648) et al., j. 16–18, p. 473–578. Jiu Tangshu ddd (Old Tang History), 945, Liu Xu dd, j. 32–34, p. 1151–1292. Xin Tangshu d d d (New Tang History), 1060, Ouyang Xiu d d d (1007–1072) and Song Qi dd (998–1061), j. 25–30B, p. 533– 804. Jiu Wudai shi dddd (Old History of the Five Dynasties), 974, Xue Juzheng ddd, j. 40, p. 1861–1880. Xin Wudai shi dddd (or Wudai shiji dddd), (New History of the Five Dynasties), 1072, Ouyang Xiu ddd (1007–1072), j. 58–59, p. 705–712. Songshi dd (Song History), 1345, Tuo Tuo dd [or Toktogha] et al., j. 68–84, p. 1491–2092. Liaoshi dd (Liao History), 1344, Tuo Tuo dd [or Toktogha] et al., j. 42–44, p. 517–683. Jinshi dd (Jin History), 1343, Tuo Tuo dd [or Toktogha] et al., j. 21–22, p. 441–532. Yuanshi dd (Yuan History), 1370, Song Lian dd (1310–1381) et al., j. 52–57, p. 1119–1344. Mingshi d d (Ming History), 1739, Zhang Tingyu d d d (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 Mongolia).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 ddd), essentially from the IXth and Xth centuries, were discovered on the eve of the XXth century at the famous Dunhuang site among thousands 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 dd, Inner Mongolia). It bears no mention of its year but the Chinese historian Deng Wenkuan has shown that it wholly conforms to Chinese official 25 Zhang Peiyu 1991; A. Arrault 2002. calendars have been studied, notably, by Zhang Peiyu 1991, ibid., Huang Yi-long, 1999, 2001a , 2001b, 2002. 27 See p. 296 above. 26 These 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 ddd procedures29 for the year 1211 (Jiading 4). No less importantly, the Kanazawa bunko dddd, (Yokohama) owns a fragmentary annotated calendar, demonstrably attributable to the year Jiading 11 (1218). Its content has been fully reconstituted from the Kaixi li ddd 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 ddd, an almost wholly lost astronomical canon. Nonetheless, owing to a critical analysis of the values of lunar and solar constants mentioned in the Songshi dd, Yuanshi dd and Wang Yinglin (1223– 1296) ddd’s Yuhai dd encyclopedia, the Taiwanese historian Lin Jin-Chyuan has managed to reconstruct its mean elements.33 Y. Nishizawa 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 Zhizheng 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 28 Deng Wenkuan 2002e. Chunxi (Pure Splendor) is both the name of an astronomical canon and of a reign-period (1174–1189). 29 Kaixi (Opening Auspiciousness) is both the name of an astronomical canon and of a reign-period (1205–1208). See Deng Wenkuan 2002f. 30 Lin Jin-Chyuan 1998. 31 Y. Nishizawa 2005–2006, vol. 3, p. 301–354. 32 Chen Zungui 1984 p. 1611 f. and, above all, COL-astron, vol. 1, p. 691–706 (full reproduction). 33 Lin Jin-Chyuan 1997, p. 1–27. 34 Y. Nishizawa 2005–2006, vol. 3, p. 367–414. 35 Zhang Peiyu, 1994 p. 30–58. 36 Zhang 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 d dd (ed.). Heicheng chutu wenshu (Hanwen wenshu juan) ddddd dddddddd (The documents discovered at Karakhoto (Chinese section)), Beijing, Kexue Chubanshe ddddd, 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 ddddddddddd (Catalog 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 Library also possess a similar number of calendars from the same period: see COLL., 1983, Beijing tushuguan guji shanben mulu, zi bu ddd ddddddddddd (Catalog of Ancient Texts and Rare Books from the Beijing Library, zi bu), Beijing, Shumu Wenxian Chubanshe ddddddd, 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 dddddddddddddd (Calendars from the Ming dynasty preserved at the National Library), Beijing, Beijing Tushuguan Chubanshe dddddddd, 2007. 37 Zhang 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 difficult 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 ddd (ed.), Zhongguo kexue jishu dianji tonghui, tianwen juan d d d d d d d d d dd d d d (General Collection of Chinese Scientific and Technical Works, Astronomy), 8 vol., Zhengzhou, Henan Jiaoyu Chubanshe dddddd d, 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 shoulderblades), 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 ddddddddd (Reproduction of the Siku quanshu Collection Preserved at the Wenyuange Library), 1500 vol., 1986 edited by Yun Lu dd (1695– 1767) et al., 1500 vol., Taipei, Shangwu yinshuguan ddddd, 1986; PRIMARY SOURCES 391 The Siku quanshu dddd 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 ddd (Biographies of chouren), 46 j., ca. 1810,40 RUAN Yuan d d 41 (1764–1849). Taipei, Shijie shuju d d d d, 1982. The Chouren zhuan is a compilation composed of a mosaic of quotations, 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 dd, 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 38 i.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. 39 See Guy, R. Kent 1987. 40 From Wang Ping 1974. 41 On Ruan Yuan, see Wang Ping 1974, op. cit., and Betty Peh-T’i Wei 2006. 42 This 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 d – the first of the compound chouren – is synonymous with chou d, 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 d, 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 longlasting 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 influential during the entire nineteenth century and several sequels to the Chouren zhuan were successively published, notably LUO Shilin ddd (1789?–1853)’s Xu Chouren zhuan dddd 1840/1982* (A Sequel to the Chouren zhuan), chapters (juan) numbered from 47 to 52, Taipei, Shijie shuju dddd. (This volume concerns Qing scholars and tends to view mathematics as more and more important than astronomy.) 5. Daxue yanyi bu ddddd (Complements to the ‘Great Learning’),44 160 j., 1487, QIU Jun dd (1420–1495).45 See WYG, vol. 713, p. 72–97. mostly, Liu’s Hui dd ’s proofs reproduced in his celebrated commentary of the Jiuzhang suanshu d d d d (Computational Techniques in Nine Chapters) (263 AD) (English translation in Guo Shuchun, J. Dauben and Xu Yibao 2013). 44 The Daxue yanyi dddd is a former work by a disciple of Zhu Xi (1130–1200). 45 See Wu Chi-hua and Ray Huang 1976. 43 See, 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 management, 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 ddddd (Research into Pitch-Pipes and Astronomical Canons, Ancient and Modern), 72 j.,48 1607, XING Yunlu ddd (ca. 1560 – ca. 1620), provincial judge (anchasi ddd) 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 ddddddd (Ancient and Modern Procedural Techniques (shu d) of Predictive (tuibu dd) 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 46 WYG, vol. 713. vol. 713, p. 96–97. 48 Chen Meidong 2003a, p. 618–620, provides an excellent presentation of the content of this treatise, chapter by chapter. 49 Gujin lüli kao ddddd, j. 36-59, in WYG, vol. 787, p. 413–653. 50 See F.R. Stephenson and M.A. Houlden 1986, p. 388. 51 Ibid., p. 551–560. 52 WYG, vol. 787, p. 561–566. 47 WYG, 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 dddd (Kaiyuan reign-period (713–741) Treatise on Astrology), 120 j., ca. 742, QUTAN Xida dddd. Quoted from the edition of the text published by Zhongguo Shudian ddd d, 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 nonofficial, 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 Superior Epoch and the values of some generalized Metonic constants (see Kaiyuan zhanjing, j. 103, p. 732–741). 9. Lishi yishu dddd (Posthumous works of Master Li), LI Rui54 d d (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 dddd (The Santong55 Calculation Procedures of the Hanshu), 3 j. See COL-astron, vol. 2, p. 708– 741. (b) Han Sifen shu dddd (The Sifen Calculation Procedures of the Han Dynasty), 3 j. See COL-astron, vol. 2, p. 741–778. 53 Yabuuchi 1963a/1988*, p. 6. Li Rui, see note 12, p. 247 above. 55 Santong means ‘Triple Concordance Astronomical Canon’, an allusion to its three supra-annual cycles, composed of 19, 76 and 1520 years, respectively. 54 On PRIMARY SOURCES 395 (c) Han Qianxiang shu dddd (The Qianxiang56 Calculations Procedures of the Han Dynasty),57 2 j. See COL-astron, vol. 2, p. 778–797. (d) Buxiu Song Fengyuan shu dddddd (The Fengyuan58 Calculation Procedures of the Song Dynasty, Revised and Reconstituted), 1 j. See COL-astron, vol. 2, p. 798–802. (e) Buxiu Song Zhantian shu dddddd (The Zhantian59 Calculation Procedures of the Song Dynasty, Revised and Reconstituted), 1 j. See COL-astron, vol. 2, p. 803–806. (f) Rifa shuoyu qiangruo kao ddddddd (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 dddd (An Explanation of the Linde61 Canon Calculation Procedures), 3 j., 1867, LI Shanlan ddd 62 (1811–1882). From the complete works of Li Shanlan – Zeguxizhai suanxue, dd dddd (The Mathematics of the ‘Studio Devoted to the Imitation of the Ancients’), 42 j., 1867, edited by Mo Youzhi ddd.63 56 Qianxiang = Supernatural Manifestation. Qianxiang li was elaborated during the Han dynasty but never officially used. See Hou Hanshu, zhi 2 “lüli 2”, p. 3043 (note 1). 58 Fengyuan = Oblatory Epoch (N. Sivin 2009, p. 50). 59 Zhantian = Augury of Heaven. 60 The 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. 61 Linde li = ‘Unicorn Virtue canon’ (an allusion to the rarity and precious character of this mythical animal). 62 On Li Shanlan, see Wang Yusheng 1990, Horng Wann-sheng 1991, J.-C. Martzloff 1997*/2006* p. 341–351. 63 For 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. 57 The 396 BIBLIOGRAPHY The Linde shu jie dddd offers a tentative interpretation of the elliptic text of the Xin Tangshu concerning the calculation of true elements according to the Linde li ddd64 procedures.65 11. Lüli rongtong dddd (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 ddddd mentioned below, the Lüli rongtong was composed in view of a reform of astronomy. 12. Mengqi bitan dddd, (Dream Pool Essays), 26 j., 1086, SHEN Gua dd (1031–1095). See HU Daojing ddd Mengqi bitan jiaozheng dddddd (Critical edition of the Mengqi bitan), Shanghai, Guji Chubanshe ddddd, 2 vol., 1987. The pen jotting litterature (biji) dd 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 dd (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 ddddd (Perpetual Astronomical Canon Dedicated to the Longevity of our Saint Emperor), 5 j., 1595, ZHU Zaiyu (1536–1611). WYG, vol. 786, p. 451–555. 64 Xin Tangshu, j. 26, ‘ li 2 ’, p. 559 f. 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. 66 This term designates here the moon and the planets. 67 For a minute study of this important source, see N. Sivin 1989. 65 An 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 dddddddd (Comprehensive Astrological Survey, Humbly Compiled and Limited to the Essentials), 80 j., 1653, HUANG Ding dd.72 Huang Ding was a Regional Commander zongbing73 dd of Zhejiang 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 ddd and even the Huihui li ddd.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 ddddd (A Comprehensive View on the Harmonies Between Cycles and the Distinction Between Allowed and Forbidden Directions), 36 j., ca. 1739, YUN Lu dd et al. See LI Ling dd (ed.), Zhongguo fangshu gaiguan, xuanze juan ddd ddddddd, (An Overview of Chinese Divinatory Techniques, 68 On 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. 69 In modern Chinese, tianwen dd means ‘astronomy’ but, in ancient texts, its literal meaning is ‘celestial drawings’ or ‘celestial signs’. It refers to judicial astrology. 70 The last term of this title, jiyao dd, means ‘limited to the essential’ and seems incompatible with the idea of comprehensiveness expressed by the term dacheng d d. 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. 71 The Chinese term guankui dd has an allusive value, impossible to understand without an awareness of its origin. It first occurs in Zhuangzi, j. 17, ‘Autumn Flood (Qiu shui dd)’ 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. 72 Biographical notice in Ding Fubao and Zhou Yunqing 1957, p. 52a and 52b. 73 Ch.O. Hucker 1985, item 7146. 74 Qu Anjing 1995. 75 J.-C. Martzloff 1998b. 398 BIBLIOGRAPHY ‘Elections’),76 2 vol., Beijing, Renmin Zhongguo Chubanshe dd ddddd, 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, ddd d) collection (WYG, vol. 811, p. 109–1022) devoted to divinatory techniques (shushu lei ddd). 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 d d d d (Computational Techniques in Nine Chapters), Yijiatang ddd 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, 76 This technical term designates the choice of auspicious and non-auspicious days. See J. Tester 1989, p. 88 f. 77 J.-C. Martzloff 1997*/2006*, p. 140; A. Bréard 2002. 78 See p. 125 above. PRIMARY SOURCES 399 notably concerning a famous astronomical canon, the Kaixi li dd d (1208–1251), and another one about the motion of Jupiter.79 17. Suanxue qimeng dddd (Introduction to Computational Science) [Zhu Shijie ddd (1299), in KODAMA Akihito dddd 1966. Jūgo seiki no Chosen kan dō-katusi-ji sūgaku sho ddddd dd dddddddd (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. Koryǒ sa/Gaoli shi ddd (1451), j. 50-52, Seoul, Yônse taehakkyo dddd, Tongbanhak-yôn’guso dddddd, 1955. Devoted to the official history of the Koryǒ 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/Sonmyong ryok81 ddd (j. 50, p. 81–110) and those of the Shoushi li /Susi ryok82 ddd (j. 51 and 52, p.112–183). The treatise of the Koryǒ 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 explanations, but not the details of its calculation procedures. For the same reason, its section devoted to the Shoushi li calculations should 79 See 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. 80 See K. Pratt and R. Rutt 1999, p. 245. 81 Koryǒ sa, j. 50, p. 81–111. 82 Ibid., j. 51 and 52, p. 112–182. 83 Xin 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 ddd 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 ddddd (The Shoushi li yi/Juji reki gi Evaluated),88 6 kan,89 ca. 1720–1730,90 TAKEBE Katahiro dddd. Quite legible undated manuscript, Tokyo Univ. (MS T30/95). 20. Juji reki jutsu kai ddddd, 4 kan, ca. 1720–1730, (The Calculation Procedures of the Shoushi li Explained). TAKEBE Katahiro dddd, Undated manuscript, Tokyo University (MS T30/99). 21. Juji reki sū kai dd d d d (The Numerical Constants of the91 Shoushi li), 2 kan, ca. 1720, TAKEBE Katahiro dddd. Undated manuscript, University of Tokyo (MS T30/102). 84 Yuanshi, j. 53 à 55 ‘li 2 to 4’, p. 1153–1264. Eun-Hee 1997 and Lee Eun-Hee and Jing Bing 1998. 86 See 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). 87 See K. Yabuuchi and S. Nakayama 2006, p. 2–3. 88 The 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). 89 kan d (chapter) corresponds to the juan d of the Chinese. 90 Datation from A. Horiuchi 2010, p. 227. 91 The literal meaning of the term sū d is ‘numbers’ and it refers more precisely to ‘numerical constants’ in this context. 85 Lee PRIMARY SOURCES 401 In these three manuscripts, the mathematician TAKEBE Katahiro d ddd comments in a particularly minute way on the techniques of calculation of the Shoushi li and those of the main Chinese official 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 shōgun 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 ddd (ed.), Haiwai zhenxi Zhongguo kexue jishu dianji jicheng dddddddddddddd (Rare and Precious Scientific and Technical Books Preserved Abroad), Hefei, Zhongguo Kexue Jishu Daxue Chubanshe ddddddddddd. 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 ddd (The Manifest Enlightenment Astronomical Canon), p. 1–32. A different edition of the same text has already been mentioned above (p. 399). (b) Shoushi li licheng ddddd (Handy Tables for the Inception Granting Astronomical Canon), p. 33–74. (c) Datong lifa tongggui dddddd (A Path to the Methods of the Great Unification Astronomical Canon), p. 75–210. (d) Shoushi li genian jiaoshi ddddddd (Eclipses Calculated for the Various Years of Validity of the Inception-Granting Astronomical Canon), p. 211–220. (e) Datong li zhu dddd (Hemerological Annotations for the Great Unification Astronomical Canon), p. 221–426. 92 See A. Horiuchi 2010, ibid., p. 118. 402 BIBLIOGRAPHY (f) Da Ming Datong lifa d d d d d d (The Methods of the Great Unification Astronomical Canon of the Great Ming Dynasty), p. 427–456. (g) Weidu Taiyang tongjing dddddd (General Canon of Solar Latitudes), p. 457–488. (h) Xuande shinian yue-wu-xing lingfan ddddddddd (The Assaults93 of the Moon and the Five Planets During the Tenth Year of the Xuande Era [1435]), p. 489–506. (i) Huihui lifa d d d d (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 dd (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 93 This term is a tentative rendering of the Chinese astrological expression lingfan d d, 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 d and fan d are also possible. See the footnotes 40 and 41, p. 16 above. 94 See 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 facsimile 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 dd (ed.). Chongzhen lishu, fu Xiyang xinfa lishu zengkan shi zhong ddddddddddddddd (Chongzhen reignperiod [1628–1644] Treatise on Astronomy with an Addition of Ten Other Works Taken from the Xiyang xinfa lishu). Shanghai, Shanghai Guji Chubanshe dddddddd. 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 ddd, Xu Guangqi ji dddd (Collected Works of Xu Guangqi), Shanghai Guji Chubanshe ddddddd, 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 Chinese 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 95 Among ancient but still precious studies on this subject, see, notably: BernardMaître 1945; R. Malek 1998. 96 C. 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, Astronomiques, 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 Chinese Books or Made in India and China by the Fathers of the Company 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 composition, 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 cherished endless speculations in the hope of determining the origins of 97 K. Hashimoto 1988. 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). 98 Long PRIMARY SOURCES 405 Chinese astronomy and they have often relied for that purpose on an often incredibly tenuous documentary basis. For example, the influential 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 extraordinary 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 dd: 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 extract 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] ddd, the author of the Gujin lüli kao ddddd, 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 99 See L. de Saussure’s biography in P. Pelliot 1925–26. (“[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”). 101 Souciet, ibid., p. 204–207. 100 L. de Saussure 1930, préface, p. v: 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 revolution – 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. 102 See 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 frequently 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 ddddddd (Chinese-English Dictionary of Astronomy ), Beijing, Kexue Chubanshe ddddd. – COLL. 1976. Dictionnaire Français de la langue chinoise, Paris, Institut Ricci, Centre d’Études chinoises. – COLL. 1980. Zhongguo da baikequanshu (tianwenxue) dddddddddd dd(The Great Encyclopedia: Astronomy), Beijing and Shanghai, Zhongguo Da Baikequanshu Chubanshe dddddddddd. – COLL. 1988. Zhongguo gudai tianxiang jilu zongji dddddddddd (General Catalog of Ancient Chinese Astronomical Observations), Jiangsu Kexue Jishu Chubanshe ddddddddd. – COLL. 1989a. Zhongguo gudai tianwen wenwu lunji dddddddddd (Collected Articles on the Chinese Cultural Heritage: the Astronomy of Ancient China), Beijing, Wenwu Chubanshe ddddd, 1989. – COLL. 1989b. Zhongguo tianwen shiliao huibian diyi juan d d d d d d dddddd (Materials for the History of Chinese Astronomy, First series), Beijing, Kexue Chubanshe ddddd. [To my knowledge, the following series have never been released]. – COLL. 1991–1994. Zhongguo xiandai kexuejia zhuanji ddddddddd (Biographies of Chinese Men of Science from the Modern Period), 6 vol., Beijing, Kexue Chubanshe. ddddd. – COLL. 1997. Hanyu da cidian ddddd (Great Chinese Dictionary) 3 vol., Shanghai, Hanyu Da Cidian Chubanshe dddddddd. – COLL. 1999. Hanyu fangyan da cidian ddddddd (Great Dictionary of Chinese Dialects), Collective work published by the Fudan University (Fudan daxue © Springer-Verlag Berlin Heidelberg 2016 J.-C. Martzloff, Astronomy and Calendars – The Other Chinese Mathematics, DOI 10.1007/978-3-662-49718-0 407 408 BIBLIOGRAPHY dddd) and the University of Foreign Languages (Kyoto) (Kyoto Gaikokugo Daigaku ddddddd), Beijing, Zhonghua Shuju dddd. – COLL. 2001a. Zhangjiashan Han mu zhujian : 247 hao mu, dddddddd ddddd (The Bamboo Slips Discovered in a Tomb at Zhangjiashan; The Tomb no. 247), Beijing, Wenwu Chubanshe ddddd. – 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 Taragaï (1394–1449) “Zidzh, Novye Guraganovy Astronomicheskie Tablitsy” (Ulug-Beg Muhammad Taragaï (1394–1449): Zı̄j, 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. 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Gudai tianwen lifa lunji dddddddd (Collected Articles on the Calendar and Ancient Chinese Astronomy), Guiyang, Guizhou Renmin Chubanshe ddddddd. – 2008/2009*. Gudai tianwen lifa jiangzuo dddddddd, (Lectures on Ancient Chinese Astronomy and Astronomical Canons), Guilin, Guangxi Shifan Daxue Chubanshe ddddddddd. ZHANG Yuzhe ddd (ed.), 1984. Tian wen: Zhongguo tianwen shi yanjiu, diyi ji dddddddddddddd (Questions of Astronomy: Research into the History of Chinese Astronomy, First Serie) Jiangsu Kexue Jishu Chubanshe ddd dddddd. ZHOU Yiping ddd and SHEN Chaying ddd, 1991. Suishi jishi cidian ddd ddd (Dictionary of the Chinese Calendar and its Festivals), Changsha, Hunan Chubanshe ddddd. ZHU Wenxin ddd, 1934. Lifa tongzhi dddd (A General Overview of Chinese Astronomical Canons), Shanghai, Shangwu Yinshuguan ddddd. ZHUANG Shen dd, 1960. ‘miri kao’ ddd (The Meaning of the mi Days [Sunday]), Guoli zhongyang yanjiuyuan lishi yuyan yanjiusuo jikan dddddddd dddddddd, [Bulletin of the Institute of History and Philology, Academia Sinica], Taipei, vol. 31, p. 271–301. ZHUANG Weifeng 2009. ddd (ed.) 2009. Zhongguo gudai tianxiang jilu de yanjiu yu yingyong dddddddddddddd (Research into Records of Ancient Celestial Phenomena and their Applications), collection Zhongguo tianwenxue shi daxi dddddddd (Great Encyclopedia of Chinese Astronomy), Beijing, Zhongguo Kexue Jishu Chubanshe, ddddddddd. ZÖLLNER Reinhard, 2003. Japanische Zeitrechnung, Ein Handbuch. München, Iudicium. GLOSSARY Chinese or Japanese Names Bo Shuren ddd Che Yixiong ddd Chen Hao dd Chen Jing dd Chen Jiujin ddd Chen Kaige ddd Chen Meidong ddd Chen Xiaozhong ddd Chen Yongzheng ddd Chen Yuan dd Chen Zhanyun ddd Chen Zungui ddd Dai Nianzu ddd Deng Wenkuan ddd Dong Yuyu ddd Du Yu dd Fujieda, Akira ddd Fukunaga, Mitsushi dddd Gao Pingzi ddd Gongyang Gao ddd Guo Moruo ddd Guo Shoujing ddd Guo Shuchun ddd Han Qiheng ddd Hashimoto, Keizo dddd Hirose, Hideo dddd Hong Jinfu ddd Horng Wannsheng ddd Huang Yi-long ddd Ikeda On ddd Itō, Kazuhiko dddd Ji Zhigang ddd Jiang Xiaoyuan ddd Jing Bing dd Kawahara, Hideki dddd Kodama, Akihito dddd Lai Swee Fo ddd Lee Eun-Hee ddd Li Caiping ddd Li Chongzhi ddd Li Heng dd Li Rui dd Li Yan dd Li Yong dd Li Yongkuang ddd Lin Jin-Chyuan ddd Liu Dun dd Liu Hongtao ddd Liu Jinyi ddd Liu Xin dd Lu Yang dd Luo Zhenyu ddd Ma Mingda ddd Nakayama, Shigeru ddd Nishizawa, Yūsō dddd Ōhashi, Yukio ddddd © Springer-Verlag Berlin Heidelberg 2016 J.-C. Martzloff, Astronomy and Calendars – The Other Chinese Mathematics, DOI 10.1007/978-3-662-49718-0 441 442 GLOSSARY Okada, Yoshirō dddd Onozawa, Seiichi ddddd Pan Nai dd Qian Baocong ddd Qin Jiushao ddd Qu Anjing ddd Ruan Yuan dd Shen Chaying ddd Sima Qian ddd Sun Xiaochun ddd Takebe, Katahiro dddd Uchida, Masao dddd Wang Baojuan ddd Wang Guifen ddd Wang Lixing ddd Wang Mang dd Wang Rongbin ddd Wang Shouyi ddd Wang Xi dd Wang Xiaohu ddd Wang Yingwei ddd Wang Yuezhen ddd Watanabe, Toshio dddd Wu Jiabi ddd Wu Zetian ddd Xi Zezong ddd Xin Deyong ddd Xu Guangqi ddd Xu Xiqi ddd Xu Yibao ddd Xu Zhentao ddd Yabuuchi, Kiyoshi ddd Yamada, Keiji dddd Yamanoi Yū ddd Yan Dunjie ddd Yano, Michio dddd Yao Dali ddd Ying Zong dd Yixing dd Zeng Xiongsheng ddd Zhang Peiyu ddd Zhang Shuli ddd Zhang Zixin ddd Zhao Chengqiu ddd Zhou Yiping ddd Zhu Wenxin ddd Zhu Xi dd Zhu Zaiyu ddd Zhuang Shen dd Zhuang Weifeng ddd Zhuangzi dd Zu Chongzhi ddd Various Items Amoy (Xiamen) dd bajie dd ce d Chongzhen lishu dddd cha d chunjie dd Chunqiu dd Chouren zhuan ddd chunjie dd da d Daming huidian dddd Daming li ddd dan d Datong li ddd GLOSSARY Dayan li ddd di zhong dd dizhi dd dongzhi dd du d Duanwu dd Duanyang dd du-liang-heng ddd Dunhuang dd fangcheng zhengfu dddd Fantian huoluo jiuyao dddd dd Futian li ddd Gongyang zhuan ddd gu liu li ddd gua d Guantian li ddd Gujin tuibu zhushu kao dddd ddd Guoyu dd Han Sifen shu dddd Hanshu dd Hanyu fangyan da cidian ddd dddd hao d Hou Hanshu ddd hou jiuyue ddd Huainan zi ddd Huangji li ddd Huihui li ddd Huitian li ddd jiaguwen ddd Jiaguwen heji ddddd jianchu dd jiaoying dd jie d 443 Jihe yuanben dddd Jingchu li ddd Jinshu dd jiu gong dd Jiu Tangshu ddd Jiuzhang suanshu dddd Jiuzhi li ddd Jiyuan li ddd juzhu li ddd Kaixi li ddd Kaiyuan zhanjing dddd kaogu dd kaozhengxue ddd Koryǒ sa/Gaoli shi ddd kong d layue dd li (astronomical canon) d li (principle of organization) d Liji dd lipu dd liri dd Lidai changshu jiyao ddddd d lifa dd Lifa xizhuan dddd Linde li ddd ling d lingfan li ddd Lixue dazhi dddd li zhou dd Lüli rongtong dddd mi d Mie d min li dd Mingshi dd Mingtian li ddd 444 GLOSSARY miri dd Mishu geju dddd Mishu jianzhi dddd Mo d mulu dd nayin dd nian d nianshen dd qi d Qingming dd Qintian jian ddd qiying dd qizheng chandu li ddddd quan d qubian dd rili dd rishen dd ruli dd runxian dd runying dd runyu dd runyue dd ruqi dd ruzhuan dd san zheng dd san fu dd Santong li ddd shangli dd shangxian dd shang yuan dd she d Shengshou wannian li dd d d d shi d shi ci dd Shi geng dd Shiji dd Shoushi li ddd shu (numbers, quantities) d shu (procedures) d Shujing dd shuo d Shuowen jiezi dddd Shushu jiuzhang dddd si li dd Sifen li ddd Sitian jian ddd Songshi dd Suanjing shishu dddd suanshu dd Suanxue qimeng dddd sui d Suishu dd Sunzi suanjing dddd Taishi dd Taishi jian ddd Taishi li ddd Taishi yuan ddd Taishō shinshū daizōkyō dddd ddd Tianbao li ddd tiangan dd tian guan dd Tianhe li ddd tianwen dd tianwen yue ddd tianxue dd tianyuan yi ddd tong d tongshu dd Tongtian li ddd tuibu dd GLOSSARY Tumubao ddd tuwang dd wang d wangli dd wangwang dd wannian dd Weishu dd Wu Beizhi ddd Wuyin li ddd Xi’an dd xia xian dd Xiamen (Amoy) dd xiao d xiao xue dd Xieji bianfang shu ddddd xin d Xu Guangqi ji dddd Xin Tangshu ddd Xinghe li ddd Xiyou ji ddd Xuanming li ddd Xuanye dd Xunzi, ‘zhenglun’ dd, dd Xuri dd Yao dian dd yaosenwen ddd Yijing dd ying d Yingri dd yin-yang dd Yitian li ddd yongjiu dd Yuandan dd Yuanjia li ddd yuannian dd Yuanshi dd Yuanxiao dd yue d yuejian ganzhi dddd yueshen dd Yufodan ddd Yulanpen ddd Yusi jing ddd Zhantian li ddd zheng d zhengyue dd Zhide li ddd Zhongyang dd zhouying dd zhuanying dd Zhu Xi quanshu dddd ziran dd Zhoubi suanjing dddd Zhouli dd zhu d 445 INDEX OF NAMES Akhmedov, A.A., 22 al-Sanjufı̄nı̄, 21 Ang Tian Se, 55, 119, 132, 334 Arrault, A., 34, 67, 75, 79, 82, 88, 91, 93, 98, 99, 219, 271, 274, 296, 329 Ascher, M., 333 Aubin, F., 21 Béhar, P., 333 Bäcker, J., 82, 83 Bazin, L., 82, 94 Biémont, É., 100 Billard, R. (1922–2000), 335 Blay, M., 45 Bo Shuren (1934–1997), 5, 6, 137, 194 Bodde, D. (1909–2003), 39, 63, 97, 99, 101 Bouchet, U., 29, 108, 149 Bredon, J., 97 Briggs, H. (1561–1639), 131 Brind’Amour, P. (1941–1995), 99 Britton, J.P., 157 Callataÿ, G. de, 333 Callipus (fl. 330 BC), 243 Cauty, A., 86, 120 Chambeau, G., 31, 61 Chang Chih-ch’eng, 41 Chavannes, É. (1865–1918), 91, 92 Che Yixiong, 274 Chen Hao, 268 Chen Jing, 109 Chen Jiujin, 5, 21, 30, 62, 109 Chen Kaige, 132 Chen Meidong (1942–2008), 5, 8, 28, 39, 55, 81, 93, 132, 133, 169, 201, 202, 276, 279 Chen Xiaozhong, 5, 54 Chen Yongzheng, 99 Chen Yuan (1880–1971), 78 Chen Zhanyun, 153 Chen Zungui (1901–1991), 56, 62, 66, 70, 80, 86, 241, 261 Chionades, G., 129 Chrisomalis, S., 333 Chu Pingyi, 41 Clavius, C. (1538–1612), 45– 46 Cook, A., 36 Copernicus, N. (1473–1543), 24, 45 Couvreur, S. (1835–1919), 58, 59, 67 Coyne, G.V., 29, 149 Cullen, C., 65, 66, 92 Dai Nianzu, 303 © Springer-Verlag Berlin Heidelberg 2016 J.-C. Martzloff, Astronomy and Calendars – The Other Chinese Mathematics, DOI 10.1007/978-3-662-49718-0 447 448 INDEX OF NAMES Dalen, Benno van, 21, 109 Danton, G.-J. (1759–1794), 49 De Ursis, Sabatino S.J. (1575– 1620), 93 Deane, T.E., 54, 59 Dehergne, J. (S.J.) (1903– 1990), 4 Deng Wenkuan, 14, 24, 25, 34, 75, 86, 95, 267, 268, 272, 276, 279, 290, 296–298, 300, 315, 316 Denys the Little, see Dionysius Exiguus Dershowitz, N., 79, 108 Dicks, D.R., 243 Dionysius Exiguus (ca. 475– 550), 119 Doggett, L.E., 29, 136 Dong Yuyu, 5 Dorotheus of Sidon (end of 1st cent. AD), 333 Du Shiran, 56 Du Yu (222–284), 46 Dumoulin, C., 136 Dux, G., 47 Forte, A., 91 Fujieda, A., 297 Fukunaga, M., 63 Fung Yu-lan (1895–1990), 63 Eade, J.-C., 333 Eberhard, W., 97 Elia, P. d’ S.J. (1890–1963), 94 Elman, B.A., 55, 197, 247 Engelfriet, P., 45 Escher, M.C. (1898–1972), 25 Halleux, R., 45 Han Qiheng, 227 Hannah, R., 157, 243 Harris, J., 78 Hashimoto, K., 24, 42 Havret, H., 31, 61 Herschel, J. (1792–1871), 136 Hirose, H., 6, 205, 210, 212 Ho Peng Yoke, 15, 16, 51, 54, 55 Hoang, P., 73, 76, 77, 100 Fairbank, J.K. (1907–1991), 14 Febvre, Lucien (1878–1956), 120 Forke, A., 271 Galileo, G. (1564–1642), 45 Gao Pingzi (1888–1970), 132, 242 Gassmann, R.H., 78 Gaubil, A. (1689–1759), 4, 110, 198 Gernet, J., 49, 97, 259, 337 Ginzel, F.K., 333 Goddu, A., 45 Golvers, N., 16, 54 Gongyang Gao (Warring States period (403–222 BC)), 267 Grafton, A., 136 Graham, R.L., 9, 167 Granet, M. (1884–1940)), 35 Guo Moruo (1892–1978), 80, 84 Guo Shoujing (1231–1316)), 55, 110 Guo Shuchun, 138 INDEX OF NAMES Hong Jinfu, 336 Hopkirk, P., 296 Horiuchi, A., 198 Horng Wann-sheng, 247 Hoskin, M.A., 29, 149 Hu, W.C., 97 Huang Chun-chieh, 32 Huang Yi-long, 15, 27, 33, 41, 296, 336, 337 Hummel, A.W., 247 Ikeda On, 267 Isahaya, Y., 22 Itō, K., 279, 329 Jardine, N., 39 Ji Zhigang, 7, 30, 109, 171, 181, 195 Jiang Xiaoyuan, 17 Jing Bing, 198 Jones, A., 128 Kalinowski, M., 81, 82, 92, 93, 96, 271 Kallipos, see Callipus Kawahara, H., 65 Kennedy, E.S. (1912–2009), 23 Kepler, J. (1571–1630), 24, 40, 41, 53 King, D.A., 22 Kistemaker, J., 93 Knobloch, J., 57 Knuth, D.E., 9, 167 Kodama, A., 125, 127 Krupp, E.C., 65 Kugler, F.X., 129 Kurath, H., 65 449 La Hire, Philippe de (1640– 1718), 39 Lænsberg, M., 29 Lai Swee Fo, 28, 54, 213 Lam Lay Yong, 119 Laurent, D., 65 Leduc, J., 141 Lee Eun-Hee, 198, 277 Lefort, J., 30 Lehoux, D., 50 Le Blanc, Ch., 17, 66 Li Caiping, 227 Li Chongzhi, 56, 78 Li Feng, 80 Li Rui (1768–1817), 132, 247 Li Yan (1917–1988), 125, 334 Li Yong, 7, 199, 205 Li Yongkuang, 97, 102, 273 Libbrecht, U., 57, 118, 125, 128, 130 Lin Jin-Chyuan, 7, 192, 213, 261, 267, 268, 317, 327 Liu Hongtao, 133 Liu Jinyi, 192 Liu Xin (ca. ?–AD 23), 65 Loewe, M., 25, 81, 94 Longomontanus, see Severin, Christian Lu Yang 2008, 5 Luo Zhenyu (1866–1940), 14 Ma Mingda, 21, 109 Macrobius (Vth century, philosopher and philologist), 100 Maeyama, Y., 335 Major, J.S., 17, 66 Mak, Bill M., 91, 333 450 INDEX OF NAMES Martzloff, J.-C., 24, 34, 45, 46, 57, 75, 91, 109, 119, 141, 296, 297, 300, 329 Maspero, H. (1883–1945), 30 Mathieu, R., 17, 66 Meeus, J., 314 Mercier, R., 39, 129, 335 Merzbach, U.C., 108 Meton of Athens (ca. 430 BC), 157 Mithra (solar divinity), 91 Mitrophanow, I., 97 Monier, R., 47 Mosshammer, A