The Foundations of Celestial Reckoning: Three Ancient Chinese Astronomical Systems, 1st Edition (Hardback) book cover

The Foundations of Celestial Reckoning

Three Ancient Chinese Astronomical Systems, 1st Edition

By Christopher Cullen


434 pages

Look Inside
Purchasing Options:$ = USD
Hardback: 9781138101173
pub: 2016-11-21
SAVE ~$33.00
eBook (VitalSource) : 9781315657158
pub: 2016-12-01
from $28.98

FREE Standard Shipping!


The Foundations of Celestial Reckoning gives the reader direct access to the foundational documents of the tradition of calculation created by astronomers of the early Chinese empire between the late second century BCE and the third century CE. The paradigm they established was to shape East Asian thought and practice in the field of mathematical astronomy for centuries to come. It was in many ways radically different from better known traditions of astronomy in other parts of the ancient world. 

This book includes full English translations of the first three systems of mathematical astronomy adopted for use by imperial astronomical officials, together with introductory material explaining the origin and nature of each system, and a general introduction to the work as a whole. The translations, which are accompanied by the original Chinese text, give a consistent rendering of all technical terms, and include detailed explanatory notes. The text in which the second of the three systems is found also includes a unique collection of documents compiled around 178 CE by two experts in the field, one of whom was the author of the third system translated in this book. Using material transcribed from government archives of the two preceding centuries, these scholars carefully document and review controversies and large-scale official debates on astronomical matters up to their own time. Nothing equivalent in detail and clarity has survived from any other ancient culture. The availability of the totality of this material in English opens new perspectives to all historians of pre-modern astronomy.


Christopher Cullen's expert translations and commentaries provide long-needed access to three foundational texts of Chinese astronomy, and an opening to a much wider understanding of an ancient scientific tradition comparable in richness and sophistication to that of the Mesopotamians and Greeks.

- Alexander Jones, Institute for the Study of the Ancient World, New York University, USA

"This book stands out as one of the notable landmarks in the progress of researching on Chinese systems. I highly recommend this important and well-written book for historians of science and all who are interested in Chinese mathematical astronomy."

- Tang Quan, Xianyang Normal University, Journal for the History of Astronomy, August 2017

"Cullen has provided another excellent contribution toward making Chinese astronomical texts available to a wider audience and giving the history of Chinese mathematical astronomy its well-deserved place within the general history of non-Western and premodern astronomy."

- Benno van Dalen, Isis journal, March 2018

"This volume will be a fundamental for the study of the history of ancient astronomy, not only Chinese, but also Arabic and, because of the transmission of Arabic astronomy to Byzantium in the late period Byzantine."

- Touwaide, Byzantinische Zeitschrift issue 110 (= 2017/4).

Table of Contents

Chapter 1. General Introduction

A. The world of early imperial China

B. What was a li 曆, and what did it do?

C. How did a li work?

D. Dates

E. How was a li calculated?

F. How were the data of a li determined?

G. A note on translating computational procedures

H. Arrangement of the text

I. Works consulted


Chapter 2. The Triple Concordance astronomical system: San tong li 三統曆

I. Introduction

A. An astronomical system and its creator

B. Structure of the system

C. Organisation of the text

D. Text used

II. Text, translation and comments

A. Concordance constants

B. Sequence Constants

C. The Five Pacers

D. Concordance Workings

E. Sequence Workings

F. Year Workings

G. [The Twelve Stations]

H. The 28 Lodges

I. [The Nine Roads]

J. [Concordances and Rule Heads]

K. APPENDIX 1: Intercalations in theory and practice


Chapter 3. The Han Quarter Remainder system: Han si fen li 漢四分曆

I. Introduction

A. The system and its background

B. Authorship of the text

C. Structure of the system

D. Organisation of the text

E. Methods of calculation

F. Text used

II. Text, translation and comments

A. Methods for [astronomical] systems

B. Basic solar and lunar constants

C. [Lunar eclipse constants]

D. Procedures for sun and moon

E. Planetary constants

F. Planetary Procedures

G. Planetary Motions

H. [Months, qi and lodges]

I. [The solar table]

J. The system origin

K. Discussion


Chapter 4. The Uranic Manifestation astronomical system: Qian xiang li乾象曆

I. Introduction

A. Liu Hong and his work

B. The historical background of Liu Hong’s work

C. The structure of the system

D. Organisation of the text

E. The question of ‘months’ and the finding of true lunar motion

F. Text used

II. Text, translation and comments

A. [System Origin]

B. [Basic solar and lunar constants]

C. [Predicting times of lunar and solar events]

D. [Predicting positions of sun and moon]

E. [Lunar eclipses]

F. [Predicting hexagrams and phases in effect]

G. [Finding times of day of events]

H. Advance and retardation

I. [Retardation and acceleration of lunar motion]

J. [Lunar speed sequence]

K. [Lunar speed constants]

L. [Calculating true lunar motion]

M. [Lunar latitude]

N. [Constants for lunar latitude]

O. [Calculating lunar latitude]

P. [Epoch; repeated]

Q. Predictions for the Five Stars [sc. planets]

R. [Constants for the Five Planets]

S. Predictions for the Five Planets

T. [Planetary Phases]

U. [Planetary Motions]


Chapter 5. Han discussions of astronomical systems and their development: two texts

I. Introduction

II. Ban Gu 班固 on li c. 90 CE

A. [Astronomical systems from High Antiquity to Qin]

B. [The Grand Inception reform]

C. [Zhang Shouwang]

D. [Liu Xiang and Liu Xin]

III. The Documentary Collection of Cai Yong 蔡邕 and Liu Hong 劉洪, 178 CE

A. [Materials presented]

B. [Failure of the Grand Inception system in early Eastern Han; proposals for revision deferred.]

C. [Observed errors in lunar eclipse predictions in 62 - 69 CE lead to experiments in partial use of Quarter Remainder methods.]

D. [The fruitless discussions of 66 CE; increased divergence of prediction from observation; Zhangdi’s commission of 85 CE to Bian Xin and Li Fan.]

E. [The edict of March 18th 85 CE reviews the imperative need for revision, and commands use of the Quarter Remainder system.]

F. [Bian Xin and Deng Fan propose use of an initial long month; Jia Kui successfully controverts this.]

G. [The memorial of Jia Kui, 92 CE]

H. [103 CE: The edict to construct Jia Kui’s instrument; failure of officials to use it.]

I. [Jia Kui on the varying speed of lunar motion]

J. [Editorial note on two late Eastern Han attempts to reconstruct a ‘Nine Roads’ procedure for the moon]

K. [Seasonal changes in clepsydra graduations]

L. [Eastern Han disputes on system origin: introduction.]

M. [Lunar eclipses: Eastern Han discussions.]




About the Author

Christopher Cullen is Emeritus Director of the Needham Research Institute, Emeritus Fellow of Darwin College, and Emeritus Honorary Professor of the University of Cambridge. He is editor of the Needham Research Institute Series (Routledge).

About the Series

Scientific Writings from the Ancient and Medieval World

Scientific texts provide our main source for understanding the history of science in the ancient and medieval world. The aim of this series is to provide clear and accurate English translations of key scientific texts accompanied by up-to-date commentaries dealing with both textual and scientific aspects of the works and accessible contextual introductions setting the works within the broader history of ancient science. In doing so, the series makes these works accessible to scholars and students in a variety of disciplines including history of science, the sciences, and history (including Classics, Assyriology, East Asian Studies, Near Eastern Studies and Indology).

Texts will be included from all branches of early science including astronomy, mathematics, medicine, biology, and physics, and which are written in a range of ancient languages including Akkadian, Arabic, Chinese, Greek, Latin, and Sanskrit. Thus, the series is of use both in the teaching of history and the history of science and for researchers on ancient science. In addition, the series provides a venue for the publication of original research on early scientific texts, in particular through the commentaries.

Learn more…

Subject Categories

BISAC Subject Codes/Headings:
HISTORY / Ancient / General