The Nine Chapters on the History of Chinese Mathematics by Rik Brandenburg & Keimpe Nevenzeel (2007).pdf

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The Nine Chapters on the

History of Chinese Mathematics

Rik Brandenburg

∗

Keimpe Nevenzeel

†

15 July 2007

Abstract

This article explores Chinese mathematics from the ﬁrst archeological

evidence of numbers on oracle bones (14

century BC) to the time Chi-

nese mathematics became a part universal mathematics (halfway the 19

century AD).

First a concise overview of Chinese history and in philosophy is given.

The ethical oriented Confucianism was the dominant philosophy and con-

sequently little attention was given to the natural world, hindering the

development of natural sciences and mathematics. Due to historical and

philosophical reasons, Chinese mathematics took quite a diﬀerent path

than its Western counterpart: Chinese mathematics was focused on alge-

bra and practical applications instead of geometry and theoretical reason-

ing.

The

Nine Chapters on the Mathematical Art

(ca. 1

century AD) is

by far the most inﬂuential work: it would shape Chinese mathematics

for centuries to come. Between the 3

and the 11

century AD, Bud-

dhist and Indian ideas got a ﬁrm grip on China, yet curiously Chinese

mathematics is barely inﬂuenced. In the ‘Chinese Renaissance’ and the

subsequent Mongol occupation between the 12

and 14

century Chinese

mathematics will reach its zenith.

In the 15

and 16

centuries mathematical development waned and

important achievements were forgotten. Only after the arrival of Eu-

ropean missionary-scientists at the end of the 16

and during the 17

century mathematics made progress again. The Opium Wars of the 19

century mark the end of the classical China and the indigenous Chinese

mathematics would be assimilated by universal mathematics.

Two conclusions are reached: (i) war seems to be good for mathe-

matical progress and (ii) from the 15

century on Chinese mathematics

passed away in particular due ot the lack of a general algebraic structure,

which prevented structural insight.

Keywords: Chinese mathematics, history of mathematics, Nine Chapters

on the Mathematical Art.

Introduction

China is one of the oldest civilizations, comparable only to Egypt and Baby-

lonia, which were well versed in mathematics (for a discourse on Babylonian

∗

IWI,

†

IWI

University of Groningen

- Kapteyn Institute, University of Groningen

and Egyptian mathematics, see

Klassieke algebra door de tijden heen

by Van

der Elsken, N. and Van der Pol, E.; for an extensive treatment of Egypt, see

Wiskunde voor het leven op papyrus beschreven

by Jaarsma, K. and Van Oost,

K.). But unlike these nations, China developed upon her own until modern

times.

Since it is impossible to Rule an empire as large as or larger than Rome without

arithmetic and given the millennia that Chinese culture developed relatively

unimpeded, her mathematics achievements are intriguing.

In this article the authors aim to share their discoveries about those achieve-

ments from the very start of mathematical development until its assimilation

by Western mathematics.

To do so, we have divided the history of Chinese mathematics into six periods,

treating them from chapter 3 to chapter 8:

•

Chapter 3, Prelude (< ca. -200): As will be explained later, not much is

known about the mathematics before the 2

century BC in China. This

chapter will focus on the things we

know.

•

Chapter 4, Foundations (ca. -200, ca. 600): Central theme of this chapter

will be the most inﬂuential book in Chinese mathematical history. Also

some wonderful achievements during the Chinese Dark Ages will be dis-

cussed. This book together with the mathematical developments during

the Dark Ages would form the foundations of Chinese mathematics.

•

Chapter 5, Buddhism (ca. 600 - ca. 1200): After the Dark Ages a new dy-

nasty took control of China, who reformed the educational system. Mean-

while Buddhistic and Indian inﬂuences were at their best.

•

Chapter 6, Zenith (ca. 1200 - ca. 1400): A new war was unleashed, after

which China was conquered by the Mongols. In this period of war and

bloodshed, ironically, Chinese mathematics ﬂourished more than ever; its

achievements would be the zenith of Chinese mathematics, unparalleled

in Chinese history both before and after. When compared to the West,

the Chinese where centuries ahead.

•

Chapter 7, Decline (ca. 1400 - ca. 1850): China freed herself from her

Mongolian invaders, but her mathematics would drop

points. She

would, however, slowly recover with the aid of Europeans.

•

Chapter 8, Westernization (ca. 1850 - ca. 1920): The end of Chinese

mathematics. At gunpoint, Chinese mathematics was assimilated by its

Western counterpart.

But before we immerse ourselves in the development of China’s mathematics,

we should keep in mind the following quote:

Our initial view of the science as a gradual accumulation of iso-

lated discoveries, connected only by their common end is in the last

analysis teleological. It can point

as a step on the way to

but

it cannot lead to an understanding of how

evolved out of the in-

ner necessity, historical experience, and social consequences of

...

did not appear at a certain time because of some ﬁnal cause

ordained its time had come, but rather because it was coherent with

other ideas, attitudes, and prejudices of its time.

In other words:

Science has shaped human history. The advent of agriculture, the mastery of

metallurgy, the development of a script, all this proto-scientiﬁc developments

changed the course of events. Especially after the Scientiﬁc Revolution at the

end of the European Renainssance, scientiﬁc advancements have played a crucial

role in determining the balance of power; it is above all the high level of science

and technology that gives the Western world her current dominant position in

the world.

But the converse is true as well: the development of science – just as any

other human enterprise – is inextricably linked with the social context in which

those who develop the science live. The value a society attaches to science and

the manner in which scientists are regarded is of profound inﬂuence on scientiﬁc

development.

Therefore the ﬁrst two chapters will discuss the political history (chapter 1)

and the Chinese philosophy (chapter 2), since these may explain why Chinese

mathematics developed the way she did.

Our summary and conclusions can be found in chapter 9.

The History of China

Contrary to Western history and philosophy, we expect that most of the readers

will only have rudimentary knowledge of their Chinese counterparts. Therefore,

our exploration of the history of Chinese mathematics will start with an overview

of the history and philosophy of China itself.

1.1

Linear vs. Cyclic historiography: historical patterns

When history is viewed from the western side of Eurasia, the natural

pattern seems to be for civilizations and empires to rise and wane.

By 2000 BC Egypt had eclipsed Sumeria; by 1000 BC Egypt was on

the decline but Babylonia was an impressive power. In time, how-

ever, it would be surpassed by the Persians, then the Greeks, then the

Romans.

(...)

Underlying this view of history is an unspoken analogy

between civilizations and human lives or perhaps the lives of compet-

itive individual warrior-heroes. Civilizations have an early, creative,

aggressive stage followed by the strong, mature age, but over time

lose vigour and become less ﬂexible until eventually they are defeated

in battle or succumb of old age.

...

When history is viewed from the eastern edge of Eurasia, a very dif-

ferent pattern emerges as natural and normal. There is no sense that

younger civilizations supplant older ones, but that civilization pro-

gresses through a series of yin-yang-like reversals of direction from

excessive disorder to excessive order and back again.

(Ebrey, pg.33)

Few historians today accept the cyclic view of history, but nevertheless the

history which will unfold itself in the coming paragraphs shows some striking

patterns. A large part of Chinese history can indeed be described as a succes-

sion of dynasties. In general, a capable military leader unites a divided China

under a strongly centralized civil government, with a scholar-oﬃcial bureau-

cracy. Often, China itself is enlarged, because the momentum of the of internal

uniﬁcation victories leads the armies beyond the Chinese borders. “Their suc-

cessors, however, would not all be supermen able to prevent power struggles at

court, keep the cost of defense and local administration low, and preserve or

enhance sources of revenue, all the while inspiring loyalty through their bearing

and virtue. In this view of history, men of ability and integrity – both emperors

and their counselors – could arrest decline or even temporarily reverse it, but

inevitably the dynasty would weaken and eventually fall.” (Ebrey, pg. 135.)

Consequently, barbarian tribes who are eager to invade the rich China cannot

be hold back any longer. These tribes conquer China and found own dynasty.

But because (i) their lack of experience in governing a large state; (ii) their

much smaller population size and (iii) their attempts to accommodate the Chi-

nese, Chinese scholars are appointed to organize this new state. With the state

based on Confucian values and the immersion of the foreign elite in the Chinese

culture, the barbarian tribes are siniﬁed within a few generations, eﬀectively

making the alien dynasty a Chinese one. This dynasty would be strong at ﬁrst,

though slowly decline and eventually there would be civil war, till a capable

military leader again unites China, etc.

So ironically, with the occupation of China the conquerers would not end Chi-

nese civilization, but their own. As a result, Chinese cultural tradition, with

its ﬁrst traces in 8 000 BC, has survived almost ten millennia, although China

has been occupied by foreign peoples many times. In the coming paragraphs we

will outline the ten millennia of Chinese civilization, starting by the invention

of the family and ending in the 19

century, when the last dynasty falls.

1.2

In the beginning...

not some Divine Creator, but brilliant human beings where responsible for the

steps that transformed the Chinese from primitive people to a civilization, or

so the Chinese creation legends say (Ebrey, pg. 10). The Ox-farmer Fu Xi

invented the family and the domestication of animals, Shen Nong, (whose name

means literally Divine Farmer) invented agriculture and commerce and the Yel-

low Emperor expelled barbarian tribes from the Yellow River plane, giving his

people a place to live. Furthermore, the Yellow Emperor ordered 7 of his sub-

jects to respectively observe the sun, observe the moon, observe the stars, ﬁx

the musical scales, construct a sexagimal system

, create arithmetic and to use

the previous 6 to develop a calendar (Mikami,pg. 2).

The view that the cosmos came into being on its own, without a Creator to

set everything in motion, would become a fundamental diﬀerence between the

Chinese and Western philosophical schools, having major repercussions on the

development of Chinese and Western science, as we will see in chapter 2.

But despite a fundamental diﬀerent cosmological view, it is safe to say that

the Chinese are among the most ancient civilizations, comparable only to the

Babylonians and Egyptians (Mikami, pg. 1).

Archeological excavations reveal that around 8000 BC a transition from a hunter-

gatherer to an agricultural society occurred (Gernet, pg. 39), around two thou-

sand years later the agricultural civilization was well entrenched. Stone, bone

and (later) ceramic tools were used. The ﬁrst bronze alloys are dated at 1700

– 1600 BC (Ebrey, pg. 25) and gradually the Chinese mastered the techniques

to work with bronze to an unparalleled degree of sophistication (Gernet, pg. 40).

The ﬁrst proof of a written language stems from ‘oracle bones’ used for reli-

gious purposes. Most inscriptions are inquiries from the royalty to spirits, as to

inform when was a proper time to start a war, the worshipping of ancestors and

divinities, agricultural campaigns, etcetera. The writing was rather complex:

on the about 100 000 oracle bones found so far, with in total 5 000 diﬀerent

characters, of which 1 500 have been deciphered. On these bones also the ﬁrst

numbers are found, as will be described in further detail in chapter 3. (Gernet,

pg. 47 and Schirokauer, pg. 9.)

numerical system with 60 numbers in its base.

there are Chinese creation myths which stage a Creator who sets everything in

motion. However, these myths are younger than the legends regarding the Yellow Emperor

described here (Schirokauer, pg. 23).

Actually,

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