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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 . . . . . . . . . . . .
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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
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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 . . . .
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118
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157
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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 . . . . . . . . . . . . . . . .
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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 . . . . . . . . . . . . . . . . . . . . .
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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 . . .
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219
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232
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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 . . . . . . . . . . .
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241
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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
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261
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10 The Manifest Enlightenment Canon
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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 . . . . . . . . .
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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
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325
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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 . . . . . . . . . . . . . . .
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385
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387
387
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399
400
xvi
CONTENTS
Rare Sources . . . . . . . . . . . . . . .
The Jesuit Reform of Chinese Astronomy
Antoine Gaubil . . . . . . . . . . . . . .
Philippe de La Hire . . . . . . . . . . . .
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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
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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.
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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.
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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
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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.
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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.
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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
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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.
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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
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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’,
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ARRAULT Alain, 2002. ‘Les premiers calendriers chinois du IIe siècle avant notre ère
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société dans la Chine médiévale, Paris, Bibliothèque nationale de France, 2003,
p. 85–123.
– 2004. ‘Jianlun Zhongguo gudai liri zhong de nian ba xiu zhu li’, dddddd
ddddddddd (On the Twenty-Eight Mansions in the Ancient Chinese
Calendar), Dunhuang Tulufan yanjiu ddddddd, Beijing, Zhonghua Shuju
dddd, vol. 7, p. 410–421.
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Costantino MORETTI, La fabrique du lisible, La mise en page des manuscrits de la
Chine ancienne et médiévale, Paris, Collège de France, Institut des Hautes Études
Chinoises, p. 99–111.
ARRAULT Alain et MARTZLOFF Jean-Claude, 2003. ‘Notices [sur les calendriers
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ASCHER Marcia, 1991. Ethnomathematics, A Multicultural View of Mathematics,
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BÄCKER Jörg, 2007. ‘Sur l’origine des signes cycliques chinois, quelques implications
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BALACZ E. (ed.), 1961. ‘L’histoire comme guide de la pratique bureaucratique (les
monographies, les encyclopédies, les recueils de statuts)’, in W. G. BEASLEY and
E. G. PULLEYBLANK, Historians of China and Japan, London, School of Oriental and African Studies, 1961, p. 78–94.
BAZIN Louis (ed.), 1991. Les systèmes chronologiques dans le monde turc ancien,
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BECK M. (ed.), 1990. The Treatises of Later Han: their author, sources, contents and
place in Chinese historiography, Leyden, E. J. Brill.
BEER A., HO Ping-yu, LU Gwei-djen, NEEDHAM Joseph et al., 1961. ‘An 8thCentury Meridian Line: I-Hsing’s Chain of Gnomons and the Prehistory of the
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BÉHAR Pierre, 1996. Les langues occultes de la Renaissance, Paris, Desjonquières.
BERNARD-MAÎTRE Henri, 1945. ‘Les adaptations chinoises d’ouvrages européens :
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BIÉMONT Émile, 2000. Rythmes du temps, Astronomie et calendriers, Paris,
Bruxelles, De Boeck.
BLAY M., HALLEUX R. et al. (ed.), 1998. La science classique (XVI e –XVIII e siècles), dictionnaire critique, Paris, Flammarion.
BO Shuren ddd, 1983. ‘Shitan Santong li he Taichu li de butong dian’ dddd
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– 1996. Zhongguo tianwenxue shi dddddd (A History of Chinese Astronomy),
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BODDE Derk, 1975. Festivals in Classical China, New Year and Other Annual Observances during the Han Dynasty, 206 BC – AD 220, Princeton, Princeton University
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BOUCHET Ulysse, 1868. Hémérologie ou traité pratique complet des calendriers
julien, grégorien, israélite et musulman. Ouvrage approuvé par l’Académie des
Sciences de l’Institut Impérial de France, Paris, E. Dentu.
BRÉARD, A. ‘Problems of Pursuit: Recreational Mathematics or Astronomy?’, in
Y. DOLD-SAMPLONIUS, J. W. DAUBEN , M. FOLKERTS and B. van DALEN,
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2002, Stuttgart, Franz Steiner Verlag, p. 57–86 [This article contains a large number
of specific examples of problems of pursuit].
BREDON Juliet and MITROPHANOW Igor, 1927. The Moon Year, A Record of Chinese Customs and Festivals, Shanghai, Kelly and Walsh.
BRIND’AMOUR Pierre, 1983. Le calendrier romain, recherches chronologiques,
Ottawa, Éditions de l’Université d’Ottawa, Collection d’Études Anciennes de l’Université d’Ottawa, no. 2.
BRITTON John P., 1999. ‘Lunar Anomaly in Babylonian Astronomy’ in N. M. SWERDLOW (ed.), Ancient Astronomy and Celestial Divination, Cambridge (Mass.), The
MIT Press, 1999, p. 187–254.
BRUIN Frans and Margaret, 1977. ‘The limit of Accuracy of Aperture Gnomons’, in
Y. MAEYAMA and W. G. SALZER (ed.), Prismata, Naturwissenschafts-geschichtliche Studien für Willy Hartner, Franz-Steiner, 1977, p. 21–42.
CALLATAŸ Godefroid de, 1996a. Annus Platonicus, A Study of World Cycles in Greek,
Latin and Arabic Sources, Louvain, Université Catholique de Louvain, Institut
Orientaliste, Louvain-La-Neuve.
– 1996b. Ikhwân al-S.afâ, Les révolutions et les cycles (Épitres des Frères de la Pureté,
xxxvi), Traduction de l’arabe, introduction, note et lexique. Beyrouth, Al-Bouraq
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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.A., 119, 149
Nakayama, S. (1928–2014), 5,
39, 43, 50, 109, 140, 153,
195, 215
Nan Wang, 15
Napier, J. (1550–1617), 131
Nas.ı̄r al-Din T.ūsı̄ (1201–1274),
22
Needham, J. (1900–1995), 14,
53, 66, 91, 92, 129, 130,
244
Neugebauer, O. (1889–1990),
157, 234
Nishizawa, Y., 263, 269, 279,
296, 298
Nivison, D.S., 93
North, J.D., 13
Ōhashi, Y., 234
Oirats (Mongolian related people), 41
Okada, Y., 279, 299, 329
Onozawa, S., 63
Ovid (43BC–AD17 or 18), 67
Pan Nai, 24, 93, 109
Panaino, A., 64
Parisot, J.-P., 136
Patashnik, O., 9, 167
Pedersen, O., 29, 149, 211
Pelliot, P. (1878–1945), 74, 91,
92
Peterson, J., 303
Peurbach, G. (1423–1461), 21
Plethon, G.G. (ca. 1360–1452),
129
Plofker, 333
Prager Branner, D., 80
Ptolemy, C. (ca. 85–165), 21,
24, 50, 211, 212
Pulleyblank, E.G., 80, 131
Qian Baocong, 334
Qin Jiushao (ca. 1202–1261),
125, 128
Qu Anjing, 6, 7, 30, 109, 171,
181, 191, 195, 199, 212,
227, 334
Qubilai (emperor, founder of
the Yuan dyn. in 1277), 19,
55
Ramsey, S.R., 131
Ramus (Pierre de la Ramée)
(1515–1572), 39
Reingold, E.M., 79, 108
Ricci, Matteo S.J. (1552–1610),
46, 112
Richards, E.G., 29
Rocca-Serra, G., 243
INDEX OF NAMES
Romano, A., 45, 54
Rotman, B., 123
Ruan Yuan (1764–1849), 197
Saint Ronan, 65
Samsó, J., 22
Scaliger, J.-J. (1540–1609), 136
Schafer, E.H., 28, 52, 54
Schuh, D., 333
Segonds, A., 39
Severin, Christian, 18
Shakespeare, W., 64
Shaughnessy, E.L., 80
Shen Shaying, 97
Sigler, L.E., 108
Sima Qian (135–86 BC), 52
Simon, G., 53
Sivin, N., 5, 8, 17, 21, 23, 43,
47, 48, 50, 55, 59, 132,
138, 201, 215, 244, 317
Smith, R.J., 16, 27, 33, 94, 319
Souciet, É. (1671–1744), 4, 110
Steele, J.M., 274, 277
Stein, Sir Marc Aurel (1862–
1943), 297
Stern, S., 157
Sugiki, T., 333
Sugimoto, M., 198, 277
Sun Xiaochun, 5, 93
Sung, Z.D., 57
Swain, D.L., 198, 277
Takebe, Katahiro (1664–1739),
198
Tasaka, K., 19, 213
Taub, L.C., 50
Teboul, M., 132
451
Teiser, S.F., 98
Tester, J., 58
Tihon, A., 129
Tokugawa Ienobu (shōgun)
(government: 1709–1712),
198
Toomer, G.J., 50
Troesch, A., 30
Trombert, E., 97
Tropfke, J., 108, 125
Twitchett, D.C., 14
Tycho Brahe (1546–1601), 18,
24
Uchida, M., 6, 10, 189, 190,
277, 279, 280, 285
Ulugh-Beg (1394–1449), 22
Ursis, see De Ursis
Vandermeersch, L., 53, 62, 63,
271
Vernus, M., 29
Vogel, H.U., 36, 47, 48
Wang Baojuan, 54
Wang Guifen, 279, 336
Wang Lixing, 30
Wang Mang (emperor (reign:
9–25 AD)), 56, 100
Wang Rongbin, 7, 30, 109, 171,
181, 195, 221, 224, 334
Wang Shouyi, 125
Wang Xi, 97, 102, 273
Wang Xiaohu, 5, 16, 42
Wang Yingwei (1877–1964),
56, 133, 171, 195, 274
Wang Yuezhen (1812–1881),
100
452
INDEX OF NAMES
Watanabe, T., 91
Welch Bjaaland, P., 97
Weschler, H.J., 55
Whitfield, S., 91, 296, 297
Wilkinson, E.P., 78, 97
Wu Yiyi, 51
Wu Zetian (empress (reign:
684–704)), 101
Wylie, A. (1815–1887), 91, 92,
129, 130
Xin Deyong, 56
Xu Guangqi (1562–1633), 18,
46
Xu Zhentao, 5
Yabuuchi, K. (1906–2000), 5,
109, 213, 215
Yamada Keiji, 55
Yamanoi, Y., 63
Yan Dunjie (1917–1988), 56,
121, 241, 279, 334
Yang Lien-sheng, 92
Yano, M., 21, 109
Yao Dali, 21
Ying Zong (Chinese emperor,
reign: 1435–1449), 41
Yixing (683–727), 55
Zeng Xiongsheng, 5
Zhang Peiyu, 5, 7, 8, 11, 100,
171, 199, 202, 205, 253,
276, 279, 304, 336
Zhang Shuli, 54
Zhang Zixin, 180
Zhao Chengqiu, 192
Zhou Yiping, 97
Zhu Wenxin (1883–1939), 132,
241
Zhu Xi, 50, 51
Zhu Zaiyu (1536–1611), 28, 37
Zhuang Shen, 91
Zhuang Weifeng, 5, 274
Zhuangzi (late 4th century BC),
267
Zu Chongzhi, 56
Zürcher, E. (1928–2008), 32
INDEX OF SUBJECTS
60/a/b, 144
< a; b >, 142
a;b, c, 143
< a mod 60; b >, 142
bin(x, y), 142
days/a/b, 144
⌊x⌋, 10
x mod y, 10
α , β , γ (Metonic constants),
158
administration, 33
age of the moon, 9, 29, 149,
164, 201–203, 262
Almagest, 24, 50
Almanach de Liège, 29
almanacs, 29
Amoy (Xiamen) (place name),
92
animals; bones of, 62; cyclical,
81, 297; lost, 92; self moving things, 46
archaeoastronomy, 65
arithmetic, see also logistics,
35, 213, 242, 262; meaning of shu (numbers or
calculations), 36; commercial, 108; Dunhuang
manuscript, 119; rules of,
53; written, 120
arithmetical mean, 25
arithmetical operations, 36, 53,
111, 114, 119, 121, 123,
124, 189
armillary sphere, 53
Artha-sastra, 234
Ashmolean Museum (Oxford),
128
astrology, 13, 16, 28, 33, 58,
334; also called tianxue
(the science of the heavens), 28; and military affairs, 28, 55; elections
(choices), 58; Greek, 129;
horoscopic, 129, 130;
Greek papyri, 128; inceptions, 58; Indian influence,
81; judicial, 4, 54
Astronomica Danica (C. Severin), 18
astronomical canon(s), 5, 6, 13,
17, 19, 21–25, 28, 31, 32,
34–37, 42, 52, 53, 55, 63,
65, 86, 107, 109–112, 114–
118, 120–122, 130–132,
220, 221, 224, 235, 241,
242, 259, 261, 267, 274,
277, 286, 295, 300, 317,
325, 328–330; atypical,
109; centesimal number
system, 109; key ideas, 37–
© Springer-Verlag Berlin Heidelberg 2016
J.-C. Martzloff, Astronomy and Calendars – The Other
Chinese Mathematics, DOI 10.1007/978-3-662-49718-0
453
454
INDEX OF SUBJECTS
41; later, 197–217, 303,
308, 323, 329, 330; mean
elements, 157–167, 220,
278; mentioned in the Chinese Tripitaka, 122; no artificial days, 234; predictive mathematics, 111; reform(s), 42–52, 259; their
name(s), 55–59; true elements, 169–195, 312
astronomical system, 17
astronomical tables, 8, 109,
110, 117, 120–122, 127,
128, 131, 146, 199, 204;
day and night durations,
308; European, 18, 109; lunar, 8, 123, 185, 191; modern, 202; Philippe de la
Hire, 40; political factors,
41; Rudolphine, 40; shadows, 111; solar, 8, 41, 109,
123, 180–185; Venus, 123,
124; zero, 122
astronomy, 13, 108; nonChinese; and theology, 53;
Arabo-Persian; sexagesimal number system, 112;
Chinese; Bureau of, 4, 52–
55, 128, 153; history, 5, 49,
73, 110; ignorance of traditional Chinese astronomy
in 18th China, 4; Islamic
(transl.), 109; its limitation,
41; its mathematical character, 4, 5; Jesuit reform
of, 4, 18, 23, 63, 197; posi-
tional, 11, 55, 108; predictive techniques, 49, 111;
table(s), 8, 109–111, 117,
122, 123, 192–194; Tribunal, 4; Chinese Muslim;
Bureau, 19; Greek, 129;
sexagesimal number system, 112; Indian; sexagesimal number system, 112;
non-Chinese; Chinese perception of, 19; positional,
334; tables, 127; technical
and contemporary, 3; without hypotheses, 38
Bahnasa (place name (modern Egypt), Oxyrhynchus
in antiquity), 128
bajie (the Eight Nodes (solar
breaths),64, 75
Beatty sequences, 30
Beijing, 74, 93, 131; Library,
16; capital of China, 41;
latitude, 41
Bibliothèque nationale de
France, 74
binary indicators, 184, 185
bindu (a dot (Skr.)), 120
binomials, 9
branche(s), 73, 76, 80–84, 86,
95, 269, 272, 274
Breton ethnology, 65
British Isles, 64
Buddhist sutras, 122
Bureau of Astronomy, 52
calculating instruments, 121
INDEX OF SUBJECTS
calendar(s) Chinese;
nonofficial, 74; contemporary;
Hong Kong, 25; Taiwan,
25; counterfeit, 319; Dunhuang, 25, 34, 82, 219,
267, 296, 301; at variance with official calendars, 79; P3247 vº, 74; Pelliot collection, 74; planetary week, 92; errors of
dates corrected by Huang
Yi-long, 336; lunisolar, 9;
manuscripts, 34; Ming dynasty, 5, 34; paradoxical
object(s), 24–27; popular,
15; printed, 34; private,
319; Qing dynasty, 33;
Shang-Yin dynasty; sexagenary, 62; Song dynasty,
5; sources for its history,
33–35
calendar(s) non-Chinese; Breton, 65; Celtic, 64; Gregorian, 63, 73, 97, 149, 258;
simplified version, 77; Hebrew; division of the day,
108; Julian, 73, 91, 101,
136, 149, 216, 217, 253,
258, 265, 295, 296, 313;
Roman pre-Julian; years of
confusion, 99; Zoroastrian
gāhānbār, 64
calendrical norms, 76
calendrical structure, 29–35;
deep, 9, 29, 62, 63, 70, 73,
135, 136, 147, 151, 157,
455
208, 219, 221, 223, 226,
232, 233, 289; uniform and
linear, 32; units of time, 31;
double history, 32; historical sources, 29; public, 29;
secret, 29; surface, 9, 29,
49, 61–63, 66, 70, 71, 76,
135, 148, 151, 152, 160,
163, 219, 220, 222–224,
232, 233, 251, 254, 258,
264, 289, 308; discrete, 30;
history, 33; lunar month,
151, 233; monthly, 73
Callipic period (or cycle), 243,
244, 255–258, 260
canon (= καν ών (Greek)), 23
Canon of Yao (Yao dian), see
Shujing
Carmen Astrologicum, 333
ce (divinatory rods), 53
celestial anomalies, 51
celestial motions; multifactorial
character, 51
celestial officers (tian guan), 52
century (in history), 141
cha (difference (shift constants)), 201
Chinese Thought (M. Granet),
35
Chongzhen lishu (Chongzhen
reign-period (1628–1644)
Treatise on Astronomy), 23
Chouren zhuan (Biographies
of Hereditary Specialists),
111, 131, 198, 247
Christians; Nestorian stele, 91
456
INDEX OF SUBJECTS
Chunjie (Spring Festival), 97
Chunqiu (Spring and Autumn
Annals), 267
clepsydra, 30
COL-astron (abbrev.), 16, 20,
113, 122, 124, 247, 267,
298
COLL. (abbrev.), 25, 112
color palaces (the nine) (jiu
gong), 87–90, 279, 297,
317, 319, 321
computistics, 12–14, 23, 29
computists, 29
computus, 17, 23
concrete mathematics, 9
correlative thinking, 36
cosmology, 24
counting-rods, 53, 119–121,
126
criterion, see also quasicriterion, 49; full and hollow months, 159, 162, 253,
264; intercalary month,
151, 248, 255, 264, 308;
of superiority, 197; shifts
of moon phases, 247, 248,
308
cult of soil god (she), 269, 270,
273, 297, 318
cultural manifolds, 5
cycle(s), 69, 79, 119, 297,
298; 28 mansions, 93; astronomical, 53; Callipic,
243, 244, 255–258, 260;
denary, 80, 296; duodecimal, 73, 81, 82, 272;
jianchu, 94–95, 298, 300,
321; Metonic, 29, 136,
149, 157, 243, 244, 254,
255, 258, 262; simple or
generalized, 9; nayin, 96,
297, 313, 321; number of
elements, 79; of the seasons, 39; pseudo, 94, 273,
298, 300; quasi-Buddhist,
50; sexagenary, 24, 26,
30, 32, 83, 86, 88, 134–
137, 144, 174, 200, 201,
204, 214, 255, 272; divinatory inscriptions, 84; enumeration of years, 89; origin, 83; simple, 79; simultaneous, 79; solar, 94,
136; supra-annual, 79, 134,
136; 5 years, 86; 12 years
(animals), 82; 60 years,
89; 76 years, 86; 180
years, 89; 1520 years, 244;
1539 years, 58; 4560 years,
244; Triple concordance
(san tong), 58; with reduplications, 79, 96, 273
da (full lunar month), 70
Daming huidian (Collected
Statutes of the Ming Dynasty), 28
Daming li (astron. canon), 42,
43, 56, 160
dan (one bushel), 48
date of birth, 54
Datong li (astron. canon), 18,
58, 63, 109, 117, 118, 145,
INDEX OF SUBJECTS
169, 197, 199, 208, 210–
216, 219, 222, 303, 304,
308, 316–318, 321, 323,
329, 330
Dayan li (astron. canon), 32,
37, 42, 43, 46, 57, 111,
112, 115, 121, 132, 143,
146, 171, 182–184, 190,
224
Daye li (astron. canon), 55, 115,
116, 166, 221
De Revolutionibus (Copernicus), 24
deep waters or rivers, 219
definition(s), 131; astronomical systems, 17; Contemporary Epoch, 135; cycle(s) with reduplications,
79; day, 62; emerging
year, 138; epact, 149, 177;
Epoch of the Shoushi li,
200; in a military encyclopedia, 55; intercalary
month(s), 76, 99; lunar
month(s), 148, 151; Mie
points (first type), 221; Mie
points (second type), 221;
Mo points, 221; monthly
epact, 150; primary constants, 9, 134; ruli, 179;
ruqi, 172; ruzhuan, 210;
secondary constants, 9,
134; simple cycles, 79; simultaneous cycles, 79; solar month(s), 147, 148; Superior Epoch, 136; support
457
year, 138; technical terms,
35; zhuanying, 202
di zhong (the middle of the
Earth), 41
divination, 33, 52, 57, 58, 81,
146
divinatory rods, 53, 111
DKW (abbrev.), 46, 64, 74, 80
Dog star, see Sirius
dominical letter, 29
dongzhi, see winter solstice
double-hour(s), 30, 79, 81, 213,
214, 320
du (degrees), 123
du-liang-heng (units of length,
capacity and weight), 36
Dunhuang manuscripts, 74,
296; almanacs S-P6 rº and
S612 rº, 82; arithmetic,
119; calendars, 25, 34;
deep waters, 219; deviant
dates, 335; for the year
877 (S-P6 rº), 296; for the
years 450 and 451, 267;
planetary week, 92; festivals; socio-economic aspects, 97; non-valid names
of dynastic eras, 79
dynastic era(s), 31, 55, 135,
139, 271; concordance tables, 78; naming system
first attested in 140 BC, 78;
Taishō (1912–1926), 122
East Asian Science, Technology
and Medicine, 5
East-Syrian Church, 91
458
INDEX OF SUBJECTS
Easter; date, 13; moon, 149; table of Dionysius Exiguus,
119
ecclesiastical moon, 13
eclipses, 109; A. Gaubil’s
French transl. of Chin.
proc., 198; and Chin. cal.
calculations, 180; cycle(s),
274; forecasts, 3, 11, 32,
274, 334; limits, 275; lunar; mentioned in the cal.
of the year 451, 16; on May
15, 1631, 18; on May 4,
1632, 18; partial, 268; unusual mention of, 268; observations, 42; solar and lunar; not recorded in Chinese calendars, 16
elections, 318
elementary knowledge (xiao
xue), 111
emerging year, 138
empirical verifications, 38
English (modern and middle),
64
epact, 9, 29, 147, 149, 150, 158,
163, 164, 166, 175, 177,
209, 210, 262, 263, 283;
approximate, 149; definition, 149; initial, 201;
mean, 203, 279, 280, 304;
monthly, 150; true, 149,
304
ephemeris, 16
epoch, 31, 51, 66, 134–136,
140; Contemporary, 135;
Datong li, 216, 329;
Shoushi li, 140, 200–202,
204, 303; and astronomical observations, 135; procedural complexity, 135;
deep structure, 135; Julian
day, 136; multiple, 31; Superior (shang yuan), 11,
135–141, 144, 150, 158,
161–164, 166, 167, 200,
329, 330, 333; Jingchu li,
261, 262; Sifen li, 241, 245,
254, 256; Tongtian li, 140;
Xuanming li, 277; definition, 136; does not belong
to the surface structure,
135; its fictitious character,
135; located outside historical time, 139; procedural
simplicity, 136
Euclidis Elementorum Libri XV
(Clavius), 45
‘ev-šambat (‘Sunday’ (Pehlvi)),
91
ezich (Lat., from zı̄j (Ar.)), 23
faithful simplifications, 10
fangcheng zhengfu (arrays with
positive and negative quantities), 126
Fantian huoluo jiuyao (Tantric
Buddhist text), 130
festivals (Chinese);
Duanyang, 98; Duanwu (Double Fifth), 98; Qingming (Tomb sweeping),
98; Winter Sacrifice, 101;
INDEX OF SUBJECTS
Yulanpen (Ghost), 98;
Yuanxiao (Lantern), 98;
Yuandan (New Year or
Spring Festival), 97; Yufodan (Bathing Buddha),
98; Zhongyang (Double
Ninth), 98
festivals non-Chinese; Celtic
New Year, 65; Halloween,
65
fictitious celestial entities, 15
five phases, 112
fossil constants, 166
fraction(s), 10, 162, 164, 185,
190, 192–194, 216, 221–
228, 230, 232, 233, 243,
246–248, 254, 256, 263,
264, 326, 328; centesimal,
113; compound, 171, 172;
decomposition of, 116; exact, 171; improper, 158,
163; irreducible, 222; lunar month, 115; monetary
units, 108; non-decimal,
37; reduced to a common numerator, 117; represented with a single Chinese character, 116; sexagesimal, 128; solar year,
116; solar and lunar periods, 117; sum of two, 278
Fudan University, 131
Futian li (astron. canon), 22,
109, 122, 195
geocentrism, 41
geomancy, 85
459
geometry, 24, 45
gnomon; as an instrument, 30,
51, 53; shadows, 39, 108,
111, 113
golden number, 29
Gongyang zhuan (a commentary), 267
graduations (waterclocks), 213
graha (upholders (astrological
‘planets’)), 109
Grand dictionnaire Ricci de la
langue chinoise, 112
Greece, 47
Gregorian calendar (or dates),
63, 73, 77, 149, 258
gua (hexagrams), 53
Guantian li (astron. canon), 195
Gujin tuibu zhushu kao
(Research into astron.
canons (Wang Yuzhen),
202
Guoyu (Early Chin. text, 5th
century BC), 267
h.alakim (Heb., time unit), 108
Han Sifen shu (The ‘Quarter
Remainder’ Procedures (Li
Rui)), 247
Hanshu (Han History), 14, 35–
38, 65, 66, 336
Hanyu fangyan da cidian
(Great Dictionary of Chinese Dialects), 131
hao (hair), 47
heavenly stems (tiangan), 80,
82
Hellenistic Egypt, 47
460
INDEX OF SUBJECTS
hemerology, 33
hexagram(s), 53, 57, 112
history; short-term, 33
History of Chinese Philosophy
(Fung Yu-lan), 63
Horner’s schema, 208
horoscope, 54
Hou Hanshu (Later Han History), 49, 76, 86, 166, 234,
241, 244, 248
hou jiuyue (intercalary month
after the ninth month), 70
Huainan zi (The Masters of
Huainan), 16, 66
Huangji li (astron. canon), 180,
181
Huihui li (astron. canon), 18,
21, 109, 303
Huitian li (astron. canon), 49
infinite space, 53
intercalary; delay (runying),
201; limit (runxian), 159,
163, 164, 177, 278, 280,
304; month(s) (runyue), 9,
69, 70, 75, 76, 86, 87,
90, 99–103, 138, 148, 150–
153, 157–161, 163, 165,
176, 247, 248, 254–256,
263, 264, 270, 273, 278,
280, 289, 308; remainder
(runyu), 150; year, 69, 70,
139, 159, 160, 248, 253,
255–258, 260, 263, 304,
308
inverted tree (arbor inversa), 83
irregularities within
regularities, 259
general
Jesuit(s), 4, 16, 23, 45, 46, 49,
63, 81, 93, 109, 153, 197
jiaguwen (oracle bones), 62
jianchu (pseudo-cycle), 94–95,
269–273, 313, 318
jiaoying (draconitic shift constant), 203
jie (node(s)), 75
Jihe yuanben (Chin. transl. of
Euclid’s Elements), 46
Jingchu li (astron. canon), 56,
114, 115, 144, 261–274
Jinshu (Jin History), 46, 47, 55,
75, 115, 268
Jiu Tangshu (Old Tang History), 171, 180
Jiuzhang suanshu (Computational Techniques in Nine
Chapters), 138
Jiuzhi li (astron. canon), 109,
120, 130
Jiyuan li (astron. canon), 42,
43, 137, 167, 174, 191,
194, 274
Julian calendar (or dates), 73,
149, 258
Jupiter; entrance of the sun
into its stations, 317, 321;
planet, 15; sidereal revolution, 63, 86, 271; signs of
the zodiac, 317; the Year
Star, 51
juzhu li (annotated calendar(s)),
15
INDEX OF SUBJECTS
Kaixi li (astron. canon), 49,
167, 176, 178, 182, 192,
325
Kaiyuan zhanjing (astrological
book), 81, 109, 241
kaogu (investigation of past
events), 42
kaozhengxue (evidential studies), 247
Koryǒ sa/Gaoli shi (Korean
History), 174, 180, 277
ks.aya-tithi (Skr.), 234
Kyoto University, 131
La pensée chinoise (M. Granet),
35
laws of Kepler, 40, 41
layue (month in which the Winter Sacrifice was held), 101
li (principle of organization), 50
li (astronomical canon), 13, 19,
24
Li Heng (emperor (reign: 756–
762)), 101
Liber Abaci, 108
Lidai changshu jiyao (cal. table), 202, 336
lifa (predictive methods of astronomical canons), 17, 50
LIFA (abbrev.), 5, 66, 241, 276,
321
Lifa xizhuan (History of Western Astronomy), 24
Liji (The Ritual), 67
limit; final, 184; initial, 184;
intercalary, 159, 163, 164,
176, 177, 278, 280, 304;
461
Mie, 226, 328, 330; Mo,
225, 325, 326; of Heat,
270, 273
limits; eclipses, 275
Linde li (astron. canon), 180,
185, 190–193, 278, 336
Linear congruences, 57
Lingfan li (‘calendars’ of ‘coercitions’ (a sort of astrological ephemeris)), 16
lipu (calendrical calculations),
14
liri (calendar), 14
Lixue dazhi (The Almagest), 24
lizhou (anomalistic month), 278
logistics, 36, 111
Lüli rongtong (astron. canon),
37
lunisolar coupling, 9, 73–76,
99, 152, 161, 249, 273
Manifest Enlightenment canon,
see Xuanming li
manicheism, 91
Maple (from Maplesoft), 10
Mars (planet), 15
martyrolog, 13
mathematical astronomy, 13
Mathematical Syntaxis, see Almagest
mathematics; as the best approximation to theology,
45; their limitation, 107
mean elements, 144–146, 153,
157–167, 169–171, 176,
182, 201, 262, 277–280,
304, 305, 309; anomalis-
462
INDEX OF SUBJECTS
tic month, 169, 179, 185;
apparent lunar daily sidereal motion, 182; epact,
203, 279, 280, 304; lunar; diurnal motion, 208,
211; longitude, 212; lunar month(s), 71, 175,
330; new moon(s), 157,
159, 175, 186, 194, 200,
203, 207, 212, 279, 280,
283, 285–287, 289, 290,
305, 307, 308, 328, 330;
phase(s) of the moon, 115,
169, 172, 176, 177, 179,
186, 203, 207, 209, 246;
seasonal indicators, 203;
solar; period(s), 157, 190;
breath(s), 159, 170, 173–
175, 177, 178, 203, 280,
281, 289, 305, 326; longitude, 195, 212; period(s),
171; year(s), 158, 171,
210; synodic month, 169;
terminology, 147; tropical
year, 203; value(s), 305
Mercury (planet), 15
Messager Boiteux, 29
metamorphosis, 67
method of deviations (R. Billard), 335
Metonic; calculations, (158,
)163; canon(s), 157, 242,
254, 259; constants, (157,
)158, 242, 261, 278; cycle(s), 9, 29, 136, 149, 243,
244, 254, 255, 258, 262
mi (Sunday), 91, 92, 290, 295,
299
micro-uncertainties (dates), 336
Midsummer Night’s Dream (A)
(Shakespeare), 64
Mie (points and days), 219–
236, 325–332
Mie limit (Miexian), 226, 328,
330
Mingshi (Ming History), 18, 52,
118, 198, 202, 210–216,
219, 220, 308, 309, 317,
322
Mingtian li (astron. canon), 195
Ministry of Rites, 54
minli (popular calendar), 15
miri (Sunday), see mi
Mishu geju (Board of the
Gallery of Secret Writings), 52, 54
Mishu jianzhi (Records of Secret Writings), 19
Mo (points and days), 219–236,
325–332
Mo limit (Moxian), 225, 325,
326
Mogao caves (near Dunhuang),
296
month(s); anomalistic, 169,
170, 179, 181, 185, 192,
193, 202, 205, 208, 210,
278, 286, 304; draconitic,
32; intercalary, 9, 14, 70,
99–101, 138, 148, 150–
152, 157–161, 163, 165,
176, 247, 248, 254–256,
INDEX OF SUBJECTS
263, 264, 273, 278, 280,
289, 304, 308, 335; 11*
or 12* are rare, 161; definition, 151; rank, 159;
skipped, 86; irregular, 101;
lunar, see also synodic, 30,
47, 61, 64, 69, 70, 73,
74, 79, 83, 87, 88, 90, 91,
96, 99–102, 134, 137, 138,
147, 148, 150–152, 157–
159, 161–164, 175, 177,
242, 246, 248, 249, 254,
256–258, 260–262, 264,
265, 269–271, 273, 290,
297–299, 301, 312, 313,
316–319, 321, 327; astronomical, 73, 138; calendrical spirits, 81; eleventh,
87, 90, 100, 207, 243, 271;
enumeration, 86, 87, 90;
first, 69, 75, 90, 96; full or
hollow, 9, 26, 70, 101, 159,
251, 253, 254, 256–258,
264, 273, 279, 321; length,
26, 115, 116, 118, 146,
154, 163, 328; mean, 163,
330; missing, 101; names,
31, 69, 99, 101, 273, 299;
non-intercalary, 69; number of, 69, 99–101, 158,
159, 161, 163, 243; numbered backwards, 89; numbering, 290; ordinary, 69,
70, 86, 90, 157; percentage
of full and hollow months,
71; rank, 31, 77; sequences
463
of, 26, 257; smaller than a
solar month, 152; synodic,
110, 134, 169, 176, 205,
278, 304; true, 153; Winter Sacrifice, 101; solar, 61,
147, 148, 150–152; length,
154; true, 153
moon, see also age, Easter, ecclesiastical, epact, mean elements, phases, true elements
mulu (planetary ephemeris), 15
mythology, 33
Nanjing; capital of China, 41;
latitude, 41; Observatory,
336
Nanjing University, 7
Nanking, see Nanjing
National Cheng-Kung University, 7
natural regularities, 13
navagraha (the Nine Demons
(astrological
entities)),
109, 130
nayin (cycle), 96, 318
negative numbers (or quantities), see positive or negative (numbers or quantities)
Nestorians, 90
nian (year), 62
nianshen (yearly spirits), 271
Nine Chapters, 111, 118, 138
nine roads of the moon, 275
Nishizawa, Y., 219
non-Metonic canons, 163
Northwest University (Xi’an), 7
464
INDEX OF SUBJECTS
novelty, 49
number systems; centesimal,
108–110, 113, 117; Futian li, 122; Shoushi li,
141, 200; of Indian origin, 109; decimal, 30, 37,
108, 110–113, 118, 119,
128, 169, 186, 188; digits,
110; approximations, 263;
decimal submultiples, 113;
positional, 119; degrees;
non-decimal, 123; nonsexagesimal, 123; mixed,
111; non-decimal, 37; sexagesimal, 108, 109; their
variability, 107
numerical constant(s), see also
Metonic constants, 6, 50,
158, 192, 211, 278, 287,
330; astronomical, 12, 27,
51, 182, 216; fossil, 9,
166; interval, 201; lunar,
157, 164, 224, 261, 277,
278, 303; primary, 9, 134,
185, 200, 205, 207, 241,
275; secondary, 9, 134,
135, 241, 325; shift, 51,
201–203, 216, 217; solar,
157, 164, 224, 261, 277,
278, 303
numerology, 36, 37, 111, 117
nychtemeron, 62
occult sciences, 53
overlined circle, 130
Oxyrhynchus (place name (ancient Egypt)), 128
paper, 33, 34, 267, 268, 296,
297; rare commodity, 33;
scientific analysis, 268
Pehlvi language, 91
Peking, see Beijing
People’s Republic of China, 77
Persian language, 91
phase(s) of the moon, 107
phonograms, 80
piecewise functions, 9, 205,
210, 211
pitch-pipes, 36, 37
planetary ephemeris, 15, 16; for
the year 1531, 16
planetary week, see week, 92
pocket computer Sharp PC
1475, 10
politics, 41–42, 50, 54–56, 96
Portuguese language, 93
positive or negative; integer,
89; lunar inequality, 192;
quantities, 126, 183, 338;
small corrections, 145
precision; Chin. astron. parameters and constants, 3, 12,
202; fictitious, 31; increased, 31, 197; lack of,
118; limited to one day,
43; of our historical knowledge, 33; predictive power,
13, 27, 42, 49; quantifiable
(Epochs), 135; surface versus deep structures, 62
prediction, 42, 44; and retrodiction, 42; eclipses, 180, 274,
334; fate of individuals, 54;
INDEX OF SUBJECTS
not explanation, 17; precision, 49, 197; wrong, 46
printing, 33
programming techniques, 10
proleptic calculations, 143, 202
pronunciation of kong (zero),
131
qi (breath or energy), 50, 63
Qintian jian (Directorate of the
Celestial Veneration), 52,
54
qiying (solar breaths delay); delay, 201
qizheng chandu li (planetary
ephemeris), 15
quasi-criterion, 9, 159–161,
163, 263, 304; intercalary
limit, 159
quasi-table(s), 9, 192–194, 286,
288
qubian (clever modification),
44
reductio ad absurdum, 44
reforms of Chinese astronomical canons, 25
regaim (Heb., time unit), 108
religion, 33
Republic of China, 78
Resplendent Heaven canon, see
Daming li
retrodictive calculations, 42
ri (day), 62
rili (calendar), 14
rishen (daily spirits), 271
rod-numerals, 125, 126
465
Roman abacus, 121
Roman Empire, 128
Rudolphine tables (Kepler), 40
ruli (parameter), 170, 179, 194,
280, 286, 287
runying (intercalary delay), 201
ruqi (parameter), 170, 172–179,
186, 188, 190, 209, 280,
281, 284, 287
ruzhuan (parameter), 170
sacrifice(s); of dogs, 99; Winter
(la), 273, 274
Saint Ronan (tomb of), 65
san fu (canicular days), 99, 290,
296, 297
san zheng (three norms (beginning of Chinese lunar
years)), 76
Sanjufı̄nı̄ zı̄j (astron. handbook),
21
Sanskrit language, 81, 98, 109,
122, 130, 234
Santong li (astron. canon), 58,
132, 198
Sasanian world, 91
Saturn (planet), 15
Saturnalia (by Macrobius), 100
saura-dina (Skr.), 234
saura-divasa (Skr.), 234
Science and Civilisation in
China, see also Needham,
J., 14
seasonal indicators, 107, 112
secular variations of the tropical year, 51, 140, 141, 153,
198
466
INDEX OF SUBJECTS
seven days week, see week
Shang norm, 76
shangli (calendar for princes),
15
shangxian (first quarter of the
moon), 70
Shengshou wannian li (astron.
canon), 28
shi (seasons or propitious moments), 58
shi ci (erratic planetary behavior), 51
Shi geng (Beginning of ploughing ceremony), 273, 274
shift constants, 201, 202, 203,
216, 217
Shiji (Records of the Historian),
52, 143, 336
Shoushi li (astron. canon), 5,
42–46, 49, 50, 57, 58, 109,
110, 117, 132, 135, 140,
154, 169, 197–204, 211,
212, 215, 216, 222, 303,
317
shu (numbers, quantities), 36
Shuihudi (place name (Hubei
province)), 94
Shujing (Canon of History), 15,
57, 58
shuo (first day of the lunar
month), 69
Shuowen jiezi (dictionary), 74
Shushu jiuzhang (math. book),
118, 125, 127, 128
si li (Enthronements of the four
seasons), 64, 68, 98
Sifen li (astron. canon), 56,
241–261
signs (+ and −), 184, 185
Silk Road, 296, 298
Sirius (heliacal rising of), 99
Sitian jian (Directorate of the
Celestial Administration),
52
six ancient computus (gu liu li),
23
SIXIANG (abbrev.), 31, 38, 48,
50
slaves, 47
solar breath(s) (qi), 9, 11, 64–
67, 73–75, 137, 145, 147,
148, 151, 152, 161, 170,
172–177, 179, 182, 186,
209, 243, 246, 249, 253,
255, 263, 265, 273, 281,
285, 290, 295, 297, 300,
305, 306, 308, 316, 318,
326, 327, 329; and intercalary months, 76; consecutive, 66, 157, 181, 327;
delay, 201; even, 75, 76,
152, 269–271, 273, 313,
319; internal, 74; mean,
147, 165, 170, 173–175,
177, 178, 203, 280, 289,
305, 326; number system,
243; numbering, 176; odd,
74–77, 150, 151, 248, 264,
273, 289; odd and consecutive, 148, 153, 262; temporary list, 160; true, 170,
172, 173, 281, 283, 285
INDEX OF SUBJECTS
solar period(s), 26, 36, 63, 68,
112, 117, 170, 171, 181,
182, 186, 189, 190, 241,
246, 248, 261, 262, 277,
281, 329; mean, 144, 190;
true, 190, 284
solar terms, see solar breath(s)
Songshi (Song History), 37, 47,
49, 53, 111, 113, 123, 167,
174, 176, 179, 191, 194,
195
Songshu (Book of Song), 137,
268, 274, 276
Sovereign-Pole canon, see
Huangji li
spontaneous generation, 67
structure, see calendrical structure
Suanjing shishu (math. book),
119
suanshu (logistics), 36
Suanxue qimeng (math. book),
125, 126, 130
sui (solar year), 62
Suishu (Sui History), 54, 89,
116, 166
Sunday, 29
sunya (void, (Skr.)), 122
Sunzi suanjing (math. book),
119
support days, 140
support year, 138, 223, 242
Tables astronomiques (Philippe
de La Hire), 40
tables of logarithms, 131
467
Taiping zhen jun shiyi [shi’er]
nian li [ri] (calendars
(years 450 and 451)), 267
Taishi jian (Office of the Grand
Historiographer), 52
Taishi li (astron. canon)), 56
Taishi yuan, see Taishi jian
Taishō shinshū daizōkyō (Revised Tripitaka)), 122
technical terminology, 8
ten days week, see week
terrestrial branches (dizhi), 80–
82
terrestrial branches, (dizhi), 73
tianguan (celestial officials), 52
Tianbao li (astron. canon, 116
Tianhe li (astron. canon), 116
tianwen (judicial astrology), 54
tianwen yue (astronomical
months), 73
tianyuan yi (celestial unit), 125
Tibet (Mongol viceroy of), 21
time; discrete, 30; instruments
of measure, 32; linear, 32;
local, 30; origin, 31; scholarly, 32; units of, 32
tithi (artificial lunar day (Skr.)),
234, 236
tong (cycle), 58
tongshu (almanacs), 15
Tongtian li (astron. canon), 42,
43, 50, 57, 140, 153, 201,
317
translation(s), 28, 47, 57,
58; Jinshu (Eng.), 55;
Sunzi suanjing (Eng.), 119;
468
INDEX OF SUBJECTS
Shoushi li (Eng., Jap.), 5;
Arabo-Persian zı̄j (Chin.),
21; cal. of the year 1417
(1st month), 318–322;
Chinese primary sources
(Eng.), 7; Clavius’s commentary to Euclid’s Elements (Chin.), 45; from
the Chinese, 7, 56, 58;
absence of consensus, 8;
from the Sanskrit, 81; in
medieval China, 109; in
the present work, 61; independent of philology,
63; lists of numerical constants, 132; ms. of the
year 450, 268–271; names
of Chinese astronomical
canons, 55–59; of Buddhist
texts (Chin.), 141; of the
terms Mo and Mie, 219;
passage from Scaliger’s
works, 136; problem from
the Shushu jiuzhang, 125
Tribunal of mathematics, 4
troglodyte Buddhist cave complex, see Mogao caves
true elements, 63, 144–146,
149, 152, 153, 157, 169–
195, 201, 209, 280, 304,
309, 312; angular velocities, 212; daily lunar motion, 185; epact, 149, 304;
new moon(s), 199, 202,
204, 211, 289, 290, 307,
308, 321, 327; phase(s) of
the moon, 179, 207, 308;
solar; breath(s), 170, 172–
174, 281, 285; month(s),
153; period(s), 284; terminology, 147
trunk(s), 80, 81, 83–85, 99, 269,
274, 299
tuibu (prediction and retrodiction of celestial phenomena), 42
Tumubao (place name), 41
Turkish world, 82
tuwang (period of governance
of the Earth), 68, 290–295,
298
twenty-eight mansions (the),
46, 92–94, 314, 317, 318
Tycho Brahe, 24
ullambana (hanging down
(Skr.)), 98
ullampana (salvation (Skr.)), 98
unpredictability, 44, 49, 73,
258, 259, 312
Uyghur; language, 94; roll
manuscript, 94
Venus (planet), 15
wall paintings; Dunhuang; zodiac, 81
wang (full moon), 70
wangli (royal calendar), 15
wangwang (disparition), 98
wannian (perpetual), 49
waterclocks, 213
INDEX OF SUBJECTS
week; of ten days, 92; planetary, 32, 33, 79, 90–93,
136, 279, 295, 318, 321
Weishu (Wei History), 75
Western Jin dynasty, 46
winter solstice(s), 39, 41, 43,
63–65, 73, 75, 100, 113,
136–141, 143, 144, 147,
150, 158, 161, 162, 164,
166, 167, 175, 177, 182,
188, 200–204, 209, 216,
241, 243, 245, 248, 254,
255, 261, 262, 271, 274,
277, 279, 303, 305, 321
written calculation, 120
Wu Beizhi (Treatise on Armament Technology), 55
Wuyin li (astron. canon), 65,
145
WYG (abbrev.), 28, 37
Xi’an (place name), 6, 91
Xia norm, 76, 87
xia xian (last quarter of the
moon), 70
xiao (hollow lunar month), 70
xiao xue (elementary knowledge), 111
Xieji bianfang shu (Treatise on
Hemerology), 91, 94, 95,
271, 272, 299, 319
xin (novelty), 48
Xin Tangshu (New Tang History), 46, 49, 53, 111, 120,
121, 124, 277
Xinghe li (astron. canon), 116
469
Xiyou ji (Journey to the West),
15
Xu Guangqi ji (Collected Works
of Xu Guangqi), 18
Xuanming li (astron. canon), 42,
43, 57, 165, 174, 180, 186,
187, 192, 277–300, 328,
329, 336
Xuanye (infinite space), 53
Xunzi, ‘zhenglun’, 57
Xuri (other appellation of Mie
days), 219, 312, 314, 318,
319, 330, 331
xylographic printing, 126
yakšambah (Sunday (Persian)),
91
yaosenwen (Sunday (Nestorian
inscription)), 91
year(s); bissextile (Julian or
Gregorian calendars), 73;
emerging, 138; intercalary,
69, 159, 160, 253, 255–
258; lunar, 30, 31, 61, 69–
73, 78, 83, 87, 90, 137,
256, 271; beginning, 76–
78; enumeration, 88, 94;
first month, 69; number of
days, 69, 71; number of
full and hollow months, 71;
ordinary, 69, 70, 159, 257,
258; of birth, 297; of confusion, 99; sidereal, 110,
182; solar, 9, 36, 39, 56,
61–64, 66–69, 73, 86, 111–
113, 134, 136, 137, 157,
158, 161, 163, 166, 170,
470
INDEX OF SUBJECTS
177, 181, 182, 203, 209,
210, 215, 216, 241–245,
248, 254, 255, 261, 277,
280, 303, 321, 325; beginning, 63; length, 116,
134, 162, 171; mean, 210;
number of, 161; quadripartition, 64, 65; sexagenary
numbering, 244; support,
9, 138, 158, 177, 245, 249,
255, 262, 279, 283, 304;
tropical, 51, 134, 198, 203
Yijing (Book of Changes), 57,
111, 112, 146
yin-yang, 39
Yingri (other appellation of Mo
days), 219, 318, 329, 330
Yitian li (astron. canon), 195
yongjiu (perpetual), 50
Yuanjia li (astron. canon), 55,
137, 222
yuannian (initial year of a dynastic era), 78
Yuanshi (Yuan History), 48, 50,
110, 118, 141, 146, 201,
208, 213, 215, 317
yue (lunar month), 69
yuejian ganzhi (sexagenary
enumeration of lunar
months), 86
yueshen (monthly spirits), 271
Yusi jing, 333
zero, 118–131, 182, 188, 221,
222, 224, 227, 338; astronomical, 125–128; Babylonian, 129; bindu (a dot
(Skr.)), 120; circle, 118,
125, 127, 128, 130, 131;
overlined, 128, 129; dot,
120; Greek, 129; history
of, 128; Indian, 120, 122,
124, 128; kong, 121, 122,
338; ling, 130, 338; lunar
inequality, 192, 193; missing units, 143; nulla (Lat.),
119; numbering of binomials, 142; ordinal, 120, 309;
quan, 131; separation symbol, 129; sunya, 122; universal, 118; void space(s),
119
Zhantian li (astron. canon), 57
zheng (norm), 69
zhengyue (first month of the lunar year), 69
Zhide li (Perfect Virtue canon),
57
Zhou norm, 76
Zhoubi suanjing (Zhou Dynasty
Canon of the Gnomon), 66
Zhouli (The Rites of Zhou), 46
zhouying (sidereal shift constant), 203
zhu (unit of weight), 36, 48
Zhu Xi quanshu (Complete
Works of Zhu Xi), 50
Zhuangzi (book), 267
zhuanying (anomalistic age of
the moon), 202
zich (Lat., from zı̄j (Ar.)), 23
zı̄j (astronomical handbook
(Ar.)), 21–23
INDEX OF SUBJECTS
Zı̄j
al-Sanjarı̄ (astronomical
handbook), 129
Zijinshan Observatory (Nanjing), 5
471
ziran (that-which-is-of-itselfwhat-it-is), 47
zodiac, 81; introduction in
China, 81; signs of the,
130, 317