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Unformatted text preview: The Nine Chapters .
on the Mathematical Art COMPANION AND COMMENTARY SHEN KANGSHEN Department of Mathematics
Zhejiang University, People’s Republic of China JOHN N. CROSSLEY School of Computer Science and Software Engineering
Monash University, Australia ANTHONY W.—C. LUN Department of Mathematics and Statistics Mow/i, (farewell: Australia OXFORD UNIVERSITY PRESS Science Press, Beijing FOREWORD BY WU WENTSUN The Nine Chapters on the Mathematical Art is the supreme classical Chinese
mathematical work. The book has not only remained the cornerstone of tra—
ditional Chinese mathematics in its development over the last 2000 years, but
has also exerted a profound influence on the development of mathematics in
other countries and regions. Traditional Chinese mathematics has its own dis
tinctive theoretical system and formulation. It is quite different, both in subject
matter and methodology, from the axiomatic system presented by Euclid. The
Nine Chapters on the M athematz'cal Art and Euclid’s Elements of Geometry pro—
vide a fascinating contrast between East and West. Unquestionably, these two
masterpieces have proVed to be the essential sources of modern mathematics.
Further, Liu Hui’s (3rd century AD) commentary is a remarkable achievement.
His comments, on the one hand, contain deep and innovative discoveries and,
on the other, present fundamental concepts in precise mathematical terminol
ogy. Using synthesis, analysis and even proof by contradiction, Liu Hui gives
strict proofs of the results that were merely stated in the Nine Chapters on the
Mathematical Art. Liu Hui works within an ancient geometrical tradition and 7graduallydevelopseraksystemi ofemathematics with a"distinctivemcharacter.thatﬂ , 7, aims at perfection. Liu’s discoveries have inspired all the later generations of Chinese mathemati
cians, even up to today’s modern mathematicians who still draw lessons from
his work. In terms of their contributions to mathematical science, Liu Hui and
Euclid should be mentioned in the same breath. It is a sad fact that contemporary generations cannot easily appreciate the
book because of the language barrier that affects both Chinese and non—Chinese,
and I believe it is therefore appropriate to have an English translation and anno—'
tation of the whole text, including the commentary by Liu Hui and the later one
by the Tang dynasty mathematician Li Chunfeng. I am honoured to introduce
the work of Professor Sheri Kangshen (Hangzhou University, People’s Republic of
China), Professor J .N. Crossley (Monash University, Australia) and Dr A.W.C.
Lun (Monash University, Australia) and congratulate them on this publication. Wu Wentsiin it 145;
Academia Sinica
Beijing PR .China
December 1994 FOREWORD BY HO PENG YOKE The influence of the Jiuzhang Suanshu (Nine Chapters on the Mathematical Art)
by Liu Hui in traditional China is comparable to that of Euclid’s writings in pre
modern Europe. This book also enjoys much attention among contemporary
historians of Chinese mathematics. However, although it formed the topic of a
Ph.D. dissertation of the University of Cambridge as early as in the 19505, it has
never been fully translated into English. Furthermore, all the studies made on
the Jiuzhang Suanshu so far are through the viewpoint of historians of Chinese
mathematics. This book provides both mathematicians and historians of mathematics a full
English translation of the text of the Jiuzhang Suanshu and the commentaries by
Liu Hui (3rd century AD) and Li Chunfeng (7th century AD) with annotations.
It is the result of international collaboration by three scholars in the Eastern
Hemisphere. Professor Shen Kangshen from the Hangzhou University in China
is an established historian of Chinese mathematics and has contributed many
learned articles in the subject. Although interested in intercultural development,
he lacks proﬁciency in English to make a balanced overview and to communicate 
with the English reader. John Crossley is a professor of mathematics at a leading
Australian University and, without being a scholar in Chinese, what he sees
is only mathematics. Anthony Lun, a lecturer in mathematics at the same
university (formerly a lecturer in Hong Kong), acted as the intermediary between
Shen and Crossley. This is not the ﬁrst time that Lun played such a role.
He and Crossley had previously translated the book Chinese mathematics: a
Concise History by Li Yan and Du Shiran. The joint effort of these three scholars
renders the Jiazhang Suanshu accessible to mathematicians and historians of mathematics alike. ‘ Ho Regrets"
Needham Research Institute
Cambridge, UK February 1995 r 1wwﬁ‘rmmgmwmwammﬁii'ﬁrg’ﬂaygw‘ WM W < A I , < 4 V ~ A ~ w r A l
may .: MESA 111:7 «gangsta; 3%? ammms sawfﬁﬁzmm" :rrr, ”zmcanr;_ PREFACE The Nine Chapters on the Mathematical Art has played a central role in Ori
ental mathematics somewhat similar to Euclid’s Elements of Geometry in the West. However, the Nine Chapters has always been more involved in methods for ﬁnding an algorithm to solve a problem, so that its influence has been both
pedagogical and practical. Instead of theorems, which Western readers are ac
customed to ﬁnd in works writtenin the Euclidean tradition, the Nine Chapters
provides algorithmic Rules.1 These Rules are justiﬁed in the text and anno—
tations of Liu Hui. Moreover, it is impossible to understand the development
of Chinese mathematics from its early beginnings right up to the present day
without substantial study of the Nine Chapters. The basic text consists of nine chapters, each of which contains problems,
their answers and terse descriptions of the method of solution.2 It was probably
written no later than 100 BC. However, according to Liu Hui ($1! tilt) the tradition of Jiushn ( 214$, Nine
Arithmetical Arts) became the Nine Chapters and the Jiashn dates from the time of Zhou Gong (11th century BC), the ﬁrst Prime Minister of the Zhou Dynasty. ' Themethods,ofisolutionﬁin the basicgtextiare extremely hard to decipher. This difﬁculty, however, is dramatically reduced when one’reads‘thé‘commnt‘ary’of’ * " ~ Liu Hui who lived in the third century AD. . Liu’s commentary is a revelation. Besides patiently explaining simple calcu
lations such as the Rule of Three or the Rule of Double False Position — we are
using the English names since we believe the reader will be familiar with them —
Liu also provides careful descriptions of much more intricate calculations. Most
Of these are given in the form of algorithms and this approach marks the clear—
est difference from the West. In the Nine Chapters and Liu’s commentary, as
Well as in most Chinese mathematics before Westerners entered China around
1600 AD, the goal is to achieve methods that will provide answers. Indeed, the
problems considered are ones that arise from everyday life and these are much
leSS contrived than those found in most Western mathematical textbooks even
today. When it is necessary, however, Liu will provide a proof. This contrasts 1A complete list of these Rules will be found at the end of the Index, p. 595. 2The only Western language translation before the present one was a German one by Vogel [?] in 1968. A French translation is being prepared by Chemla and Guo
Shuchung . viii PREFACE with the Euclidean tradition where proofs are paramount and applications are
not considered so important. Liu’s proofs are not written in an axiomatic style but they do conform to the
usual standards one expects in ordinary mathematics. There is one exception.
Liu was unable to complete the proof of the formula for the volume of a sphere.
(This was ﬁnally achieved by Zu Chongzhi (ﬁnial? Z.) and his son Zu Geng (in H43)
about 250 years later, see [32]). This “failure” in fact demonstrates Liu’s great
wisdom and modesty. He explicitly says he is waiting for someone better to
complete the proof. Again, in the development of the techniques one can see how one is being
led from simpler questions to more complicated ones through the sequence of
problems. In addition to Liu’s commentary there is also one by Li Chunfeng ($53?—
EL, 604—672 AD), Liang Shu ( Wang‘Zhenru (i s: it?) and others dating
from the seventh century AD. These Tang Dynasty scholars wrote commentaries
on the Ten Books of Mathematical Classics (Shibu suanjing, +%‘)K;iéi£§:). This
collection of books, which includes the Nine Chapters, formed the basic texts for
training successive generations of Chinese mathematicians. We have included
this commentary, in particular because it contains the only surviving record of I
Zu Geng’s calculation of the volume of the sphere. In the process of reading the book and Liu’s comments one learns a surprising
amount about life in China at the beginning of the Christian era. The staple  foods, millet and rice, of which the former was particularly important in north
ern China, give their names to a Chapter. The importance of barter as opposed
to cash transactions is also reflected in the text. The discussion of the construc
tion of earthworks for canals in irrigation and other engineering shows how the
government organized thousands of workers on large projects —' a phenomenon
that we still see in China today. Finally we have also included our own annotations to help the presentday
reader. Together with the explanation of'technicalities we have included notes ” mi W ’7 W" 7 "7' '7" "7' ﬂ“ 7' ' 7 “A” 7' "W" 7‘” ’ HOD» the ‘mathern'atica'ltreatment’of'simil'ar’prob’lems,’indeed'quite ’oft'en’the’v'ery“ same problems, in other countries from both East and West. We hope thereby
to help the reader to see how ancient Chinese mathematics completes the picture
of world mathematics. The more we have studied the Nine Chapters and Liu’s commentary, the more
fascinating the work has become for us. We hope that some of this enchantment
will be conveyed to you, gentle reader, through this book. Most of the basic Work was done by Professor Shen Kangshen in China (see
his Afterward, p. 560). In 1992 and 1994—5, he made extended visits, totalling
about half a year, to Monash University, Australia, to work with Professor John
N. Crossley and Dr Anthony W.—C. Lun supported by the Departments of Math
ematics and of Computer Science. The task was made much easier by the excel 1...J. __11...A.:.... .L‘1.1.. .. ISL—1.21....l‘l...L1....._L!_ :., 1.1.. T‘rr..... . 1:1. ,. a istance. Thanks to Andrei Sherstyuk for assistance with the Russian items
7nd to Li Ma who has read some of the manuscript and been most generous
ﬁnd helpful. Our colleague at Monash, Associate Professor John Stillwell, has
ead the whole manuscript and we have beneﬁted greatly from his encyclopaedi<
’nowledge of the history of mathematics. He has saved us from some errors
Those that remain are our sole responsibility and we crave our readers’ indul
nce. The initial typing was all done by AnneMarie Vandenberg for whose
i'recision, elegance and sheer hard work we are extremely grateful. 15W, th<
escendant of Donald Knuth’s wonderfully logicalsystem TEX, made subsequem
amendments as easy as possible. The Chinese characters were inserted by La IﬁDHr}_vlv1vw ' Chau Ping using the Chinese E‘x’IFXpackage CJK. We are grateful t<
ur editors Elizabeth Johnstone and Julia Tompson at Oxford University Pres:
‘ or their coninuing support, interest and constructive criticism and for handling the diagrams. We are pleased to acknowledge the'helpful co—operation betweer
'. Oxford University Press and Science Press, Beijing, and hope our combined worl
will contribute to better international understanding. '
Shen Kangshen John N. Crossley Anthony W.—C. Lun
WE? ' $15 ﬁiﬁ {aﬁﬁ '
Department of School of Computer Department of
i Mathematics, Science and Mathematics
3 College of Natural Software Engineering, and Statistics, a $91,399,533;  ,,, ., ,,._1VI,9,HES,13 Universitygm , quash University,
Zhejiang University, ' Clayton, Vic. 3168, WClayton,WVic.r731033,,m Hangzhou, Zhejiang, Australia. Australia. 310028, PR. China. 30 December 199: w ~. ~ v ~.
 \v i . ACKNOWLEDGMENT Figures 10.1, 10.2 (a)—(c), 10.3, 10.4, 10.5 and (ii)), 10.15 and 10.22 arr reproduced from Needham, J. Science and Civilisation in China, vol.3, Part I
(©Carnbridge'U'niversity Press 1959) with the kind permission of Cambridgi
University Press. 0
INTRODUCTION 0.1 The Nine Chapters on the Mathematical Art and Liu Hui’s Com—
mentaries 0.1.1 The content of the Nine Chapters The Nine Chapters on the Mathematical Art (Jiazhang Suanshu, hﬁﬁiﬁ;
hereafter, the Nine Chapters for brevity) has dominated the history of Chinese
mathematics. It served as a textbook not only in China but also in the neigh
bouring countries and regions until Western science was introduced into the Far
East around 1600 AD. In many ways the Nine Chapters can be considered the
Eastern counterpart of Euclid’s Elements. The book is anonymous like many
Chinese classics. The ancient tradition was for successive writers to modify the
original text over many generations and, even when a work is ascribed to an
author, the latest text may contain contributions from a number of later writers.
The 246 Problems in the Nine Chapters and their solutions, some of which aslashes}:ﬁghefgrethegin,(as) Dryeastyr<2217207139>@811 into ninesatsssriss making up the nine chaptersimThe book that has rbeenﬂhandedﬂdown to us wasmw H 7 recompiled by Zhang Cang (§%, 3—152 BC) and Geng Shouchang (like? 5,
1st century BC) of the Former Han Dynasty (206 BC—8 AD).
In 263 AD Liu Hui (else), the ﬁrst annotator, supplemented it using his p own approach based on novel theories and ideas. He also wrote the Sea Island Mathematical Manual (Haidao Suanjing, in ,% ﬁgs, hereafter the Sea Island),
which appears as a supplement to his commentary on Chapter 9 of the Nine
Chapters. Later scholars added further annotations, based on Liu’s work. Such
scholars were Zu Chongzhi (ini‘t‘z, 429—500 AD), his son Zu Geng (in an, 5th—
6th century AD), and Li Chunfeng ($53?— El” 604—672 AD). There are a number
of other independent books, such as Yang Hui’s (57%) A Detailed Analysis
Of the Nine Chapters (Xiangjie Jiuzhang Suanfa , 11$ ﬁfty 1261) and Li
Huang’s ($235; ?—1811) A Detailed Commentary on the Nine Chapters with
Diagrams (Jiuzhang Suanshu Xicao Tu Shoo, iLi’stﬁthém iii E that provide
exegeses of the Nine Chapters. In 1964, the Zhonghua Book Company reprinted
a collated version, with more than 460 notes, by Qian Baocong %‘ 15‘s, 1892—
1974), whose careful comparison of various editions of the Nine Chapters has
Provided us with the best version of the original text. In addition the notes have
allowed us to make very valuable material available to the non—Chinese reader.
In China, Li Yan (ii/ti, 1892—1963) and Qian Baocong pioneered the study
Of the history of Chinese mathematics from the 19205. Subsequently, interest
in the topics of the Nine Chapters and later traditional Chinese mathematics “earthworks 'and'pil'eS‘of'grain‘.’ "" 7"“ "m '7' ' A'"‘"’ 2 INTRODUCTION has grown throughout the World. To a large extent this has been due to Sir Joseph Needham (1900—1995) and his co—workers, whose ongoing work Science and Civilisation in China, see [N], treats the whole of Chinese science. Part 1
of Volume 3 (hereafter [N] refers to this Part 1 only) is devoted to mathematics
and contains an account of the Nine Chapters. Since the latter part of last
century there has been an increasing number of papers and books, ranging from
general to quite specialized works, that contribute to a proper appreciation and
full explanation of traditional Chinese mathematics. (A brief account of these
will be found in Section 0.8 of this Introduction, p. 41.) The Nine Chapters is the fundamental source of traditional Chinese math—
ematics. It is ﬁrmly based on practical needs and these determine the chapter
divisions. The contents of the Nine Chapters are as follows: ' Chapter 1, Rectangular Fields (Fangtian, 7‘5 3], 38 Problems). This Chapter
discusses land measurement, giving formulae for the areas of different shapes of
ﬁelds. Calculations with fractions are also studied in detail. Chapter 2, Millet and Rice (Sumi, gilt, 46 Problems), and Chapter 3, Dis
tribution by Proportion (Cuifen, is}, 20 Problems). The Problems in these
two Chapters are solved, for the most part, by the Jinyou Rule (sets) that is
known in the West as the Rule of Three. They contain a variety of interesting
problems from agriculture, manufacturing and commerce. A few of them involve
arithmetic or geometric progressions. Chapter 4, Short Width (Shaoguang, 9‘ E, 24 Problems), This curiously
titled Chapter gets its name from considering changing the dimensions of a ﬁeld
and maintaining its area. The text mainly concerns the addition of unit frac—
tions and the extraction of square and cube roots together with applications to
calculations concerning the circle and the sphere. This Chapter also introduces
some properties of irrational roots. ' Chapter 5, Construction Consultations (Shanggong, $25 1}], 28 Problems).
This Chapter gives formulae for the volumes of various shapes of buildings, Chapter 6, Fair Levies (Junsha, £3 at}, 28 Problems). This title comes from
considerations of taxes and corvée labour. The Problems in this Chapter are of
a higher order of complexity, providing a continuation and development of ideas
from Chapters 2 and 3. Chapter 7, Excess and Deﬁcit (Ying Buzu, ﬁZ; &, 20 Problems). In this
Chapter the Rule of Double False Position is introduced and applied to solve a
range of problems. Chapter 8, Rectangular Arrays (Fangcheng, 73%;, 18 Problems). The method.)
of solution of systems of linear equations given in this Chapter is equivalent to
Gaussian Elimination (nineteenth century) both in theory and in practice, but
was written about 2000 years earlier. This Chapter also gives the rules for ,.1_ 1.: l1 1 Chapter 9, Right—angled Triangles (Gougu, £7 )19'_, 24 Problems). This Chapter
gives the Gougu ( £7 Hi) Rule, which is the Chinese version of what Westerners call
Pythagoras’ Theorem. It also treats problems on similar triangles. Pythagorean
triples and quadratic equations are also introduced. 0.1.2 Lin Hui — The earliest notable Chinese mathematician. In the Book of Music and the Calendar (Lu Li Zhi, (if; in the History of
the Sui Dynasty (Sui Shu, FE a, 581—618 AD) we read: “In the fourth year3
of the Jingyuan (7%7‘5) Reign of Prince Chenliu (Fig? of the Wei [Dynasty],
Liu Hui commented on the Nine Chapters.” This brief remark constitutes the
only record we have of Liu’s personal history. Liu was the ﬁrst notable Chinese
mathematical scholar, yet his achievements are barely recorded in the oiﬁcial
histories. His comments on the Nine Chapters are numerous; they provide the
mathematical justiﬁcation for the rules and solutions of the problems in the basic
text. The picture of Liu Hui emerging from his comments is that of a scholar who
has a remarkable command of many techniques — not only the ones that are
hinted at in the original text of the Nine Chapters, but also of new techniques
that he introduces. On the other hand, some of his comments show him as a
modest man; for example, his comments on his attempt to ﬁnd the volume of
a sphere (Chapter 4, Problem 24). “I am afraid that it would be unreasonable ., tomakeconjectures neglectingthe [difference in], shapejbetwgeJIJhG ﬁphgygggé the joined umbrellas]. Let us leave the problem to whomever can tell the truth.”
(See p. 229.) Here we restrict ourselves to a few major items. ' 0.1.2.1 On quantities
Liu had a deep understanding of ﬁnding the greatest common divisor (dengshu,
iféﬁ) of natural numbers by mutual subtraction.5 He also tells us in detail
how to ﬁnd the least common multiple of several natural numbers. However,
the techniques are subtly different from Western ones that utilize the concept
of prime number, which does not exist in traditional Chinese mathematics. He
explains the operations on fractions by rules that correspond to the presentday
theory of fractions. ' ' He mastered the notions of rate (la, and of the Homogenization and 3The year 263 AD. 4Prince Chenliu was the title conferred on the last ruler, Emperor Yuandi (77:. th),
of the Wei Dynasty (220—265 AD) during the Three Kingdoms Period. It is therefore
possible that Liu Hui would have completed his Commentary on the Nine Chapters
prior to the forced abdication of the last Wei Emperor. However, it is possible that,
1n the traditional way, Liu could have retained his ofﬁcial position under the newly
founded Jin Dynasty. 5The method is equivalent to the Euclidean algorithm and is known as “mutual
subtraction” (hujian, 5715!?) in Chinese. ...
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